6.4 6.5 6.5 2.1 2.1 0.0

constrain the reduced models to obtain both the FS length and volume at the same time. As discussed before (Section 5.1), the reduced models hold relatively smaller ice thickness and hence underestimate the ice volume, even if they yield the same glacier length. We could constrain the models so that they simulate the evolution of ice volume properly; in such cases the associated glacier lengths, however, would have been overestimated. This is not recommended, becasue ice volumes are unknown for most real-world glaciers.

Based on the reactions of ice volume and glacier length (Figs. 5c–5d), we note the following: (1) initially both the ice volume and glacier length decrease in each model case, even in the positive mass balance scenario, as a consequence of the generally decreasing trend of the past mass balance; (2) the times after which the glaciers start gaining ice volume and length (for the positive Δ*m*(*t*) scenario) are roughly equivalent to the corresponding values of *tv* and *tl*, respectively; (3) in the latter half of simulations, all models follow the similar trend for both the volume and length evolutions; (4) interestingly, the PS2 and SIA2 models slightly overestimate the glacier lengths, when they forecast lower ice volumes. The projected ice volumes and lengths after *t* = 100 years are summarized in Table 1. Due to its quick response to climate change and lack of resistances associated with high-order dynamics, the SIA2 model predicts reduced ice volumes, respectively by 6.4% and 14.0% on an average, than the PS2 and FS models. The reduced models, however, predict longer glaciers by 2.1%; this is probably because the FS model needs a longer time (*tl* = 42 years, compared to flowline models, i.e. *tl* ≈ 21 years) to fully adjust the glacier length.

#### **6. Summary and outlook**

Based on the distinct physical mechanisms of land-based glacier dynamics, we present a simplified classification of models, namely the SD, PS and FS models. SD models are simplest and are based on the shallow-ice approximation, where local driving stress is solely balanced by the basal drag. The FS models, which solve a complete set of Stokes equations, are the most comprehensive. In addition to the basal drag, they also capture the effects of longitudinal stress gradients and lateral drag. The PS models, the intermediate complexity models, only deal with the basal drag and the longitudinal stress gradients. The main advantage of having these classes of models is that the scope of each model is clear. Since the dominant physical mechanisms in a given glaciological condition can easily be identified, one can choose the optimal model accordingly to yield realistic simulations. In the interior of the large ice sheets where the shallow-ice theory holds, for example, the computationally expedient SD models are optimal. For proper simulations of a narrow and steep valley glacier, however, FS models are essential, although they are numerically intensive.

We consider three different models, one from each model family, and compare them numerically at several stages of valley glacier modelling. Results are summarized in Table 1. We find that: (1) the high-order resistances are crucial to control the dynamics of gravity-driven ice flow, (2) absence of such resistances makes the reduced models yield larger velocities in diagnostic simulations and reduced ice thickness in prognostic simulations, (3) simpler flowline models, without accounting for the effects of varying glacier width, are not sufficient to yield realistic estimates of response times; they predict response times that are 50% too rapid to 3D models, and (4) constraining a model for the particular glacier application is not straightforward, as it is difficult to properly simulate the ice volume, glacier length and surface velocities altogether.

16 Will-be-set-by-IN-TECH

constrain the reduced models to obtain both the FS length and volume at the same time. As discussed before (Section 5.1), the reduced models hold relatively smaller ice thickness and hence underestimate the ice volume, even if they yield the same glacier length. We could constrain the models so that they simulate the evolution of ice volume properly; in such cases the associated glacier lengths, however, would have been overestimated. This is not

Based on the reactions of ice volume and glacier length (Figs. 5c–5d), we note the following: (1) initially both the ice volume and glacier length decrease in each model case, even in the positive mass balance scenario, as a consequence of the generally decreasing trend of the past mass balance; (2) the times after which the glaciers start gaining ice volume and length (for the positive Δ*m*(*t*) scenario) are roughly equivalent to the corresponding values of *tv* and *tl*, respectively; (3) in the latter half of simulations, all models follow the similar trend for both the volume and length evolutions; (4) interestingly, the PS2 and SIA2 models slightly overestimate the glacier lengths, when they forecast lower ice volumes. The projected ice volumes and lengths after *t* = 100 years are summarized in Table 1. Due to its quick response to climate change and lack of resistances associated with high-order dynamics, the SIA2 model predicts reduced ice volumes, respectively by 6.4% and 14.0% on an average, than the PS2 and FS models. The reduced models, however, predict longer glaciers by 2.1%; this is probably because the FS model needs a longer time (*tl* = 42 years, compared to flowline models, i.e.

Based on the distinct physical mechanisms of land-based glacier dynamics, we present a simplified classification of models, namely the SD, PS and FS models. SD models are simplest and are based on the shallow-ice approximation, where local driving stress is solely balanced by the basal drag. The FS models, which solve a complete set of Stokes equations, are the most comprehensive. In addition to the basal drag, they also capture the effects of longitudinal stress gradients and lateral drag. The PS models, the intermediate complexity models, only deal with the basal drag and the longitudinal stress gradients. The main advantage of having these classes of models is that the scope of each model is clear. Since the dominant physical mechanisms in a given glaciological condition can easily be identified, one can choose the optimal model accordingly to yield realistic simulations. In the interior of the large ice sheets where the shallow-ice theory holds, for example, the computationally expedient SD models are optimal. For proper simulations of a narrow and steep valley glacier, however, FS models

We consider three different models, one from each model family, and compare them numerically at several stages of valley glacier modelling. Results are summarized in Table 1. We find that: (1) the high-order resistances are crucial to control the dynamics of gravity-driven ice flow, (2) absence of such resistances makes the reduced models yield larger velocities in diagnostic simulations and reduced ice thickness in prognostic simulations, (3) simpler flowline models, without accounting for the effects of varying glacier width, are not sufficient to yield realistic estimates of response times; they predict response times that are 50% too rapid to 3D models, and (4) constraining a model for the particular glacier application is not straightforward, as it is difficult to properly simulate the ice volume, glacier length and

recommended, becasue ice volumes are unknown for most real-world glaciers.

*tl* ≈ 21 years) to fully adjust the glacier length.

are essential, although they are numerically intensive.

**6. Summary and outlook**

surface velocities altogether.


Table 1. Numerical comparison of several models, one from each model family. Errors between the models, e.g. *ePS*.*FS*, are given in percentage. Response timescales are also calculated for flowline models (\$). The future projection results represent for the central flowline; values are listed for both the positive (\*) and negative (#) mass balance (Fig. 5a)

There are more than 200 k small glaciers and ice caps on Earth; it is not feasible to use the numerically intensive FS model to simulate every valley glacier. Therefore, we mostly encounter simple flowline models, e.g. the SIA2 and PS2 models, being used in the valley glacier applications. Without adding the numerical complexity, the dynamical reach of such models can be extended through the introduction of parameterized correction factors. The effects of longitudinal stress gradients can be accounted for by embedding *L*-factors (Adhikari & Marshall, 2011); the lateral drag associated with the valley walls (Nye, 1965) and stick/slip basal interface (Adhikari & Marshall, in preperation) can also be captured via analogous correction factors. This offers a pragmatic middle ground for simulating glacier response to climate change.

Undoubtedly, the biggest challenge in glacier modelling is the lack of sufficient field data. Geometric and climatic data (e.g. basal and surface topographies, glacier length records, and mass balance fields), as well as observations of ice velocities, are not available in most cases. They are essential to justify the cost of using complex, 3D, FS models. Furthermore, the lack of proper theories and associated data to describe basal processes, e.g. basal sliding, is also a subject of concern that we have not discussed in the text; in many cases, poor characterization of basal flow is the limiting factor in modelling glacier dynamics.

#### **7. Acknowledgements**

We acknowledge support from the Western Canadian Cryospheric Network (WC2N), funded by the Canadian Foundation for Climate and Atmospheric Sciences (CFCAS), and the Natural Sciences and Engineering Research Council (NSERC) of Canada.

*∂ ∂x η ∂u ∂z* + *∂ ∂x η ∂w ∂x* <sup>+</sup> <sup>2</sup> *<sup>∂</sup> ∂z η ∂w ∂z* + *∂p ∂z*

2*�*˙ 2 *<sup>e</sup>* <sup>=</sup> *<sup>∂</sup>*2*<sup>u</sup> <sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

**A3: The SD model**

given by Equation (2) with

where the nonlinear effective viscosity, *η*, is given by Equation (2) with

*∂*2*w <sup>∂</sup>z*<sup>2</sup> <sup>+</sup>

(A9) form the governing equations, where *η* is given by Equation (2) with

*∂ ∂z η ∂v ∂z* + *∂p*

where the nonlinear effective viscosity, *η*, is given by Equation (2) with

2*�*˙ 2 *<sup>e</sup>* <sup>=</sup> <sup>1</sup> 2

> *∂ ∂z η ∂u ∂z* + *∂p*

> *∂ ∂z η ∂v ∂z* + *∂p*

2*�*˙ 2 *<sup>e</sup>* <sup>=</sup> <sup>1</sup> 2 *∂*2*v <sup>∂</sup>z*<sup>2</sup> <sup>+</sup>

(A1), the **SIA3 model** has the following governing equations,

2*�*˙ 2 *<sup>e</sup>* <sup>=</sup> *<sup>∂</sup>*2*<sup>u</sup> <sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

*∂ ∂z η ∂u ∂z* + *∂ ∂z η ∂w ∂x* + *∂p*

*∂ ∂x η ∂u ∂z* + *∂ ∂x η ∂w ∂x* + *∂p ∂z*

1 2 *∂*2*v <sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

The **PS2 model** is a 2D Stokes model, strictly following the plane-strain approximations (after neglecting the non-zero *τyy*, which is required to maintain *�*˙*yy* = 0). Equations (A2), (A7) and

Modelling Dynamics of Valley Glaciers 133

This family of models does not account for the effects of longitudinal stress gradients, as well as those of lateral drag. Consequently, the vertical shear stresses are the only non-zero stress components. In addition, the reduced definition of *�*˙ approximated in the PS models is also applied here. Along with Equation (A1), the governing equations for the **SD3 model** are,

> *∂w ∂x* + *∂u ∂z* <sup>2</sup>

Equations (A2), (A12) and (A14) are the governing equations for the **SD2 model**, where *η* is

 *∂w ∂x* + *∂u ∂z* <sup>2</sup>

Further approximations are needed to obtain the standard SIA models; (1) the horizontal gradients in stresses are negligible, i.e. *τij*,*<sup>x</sup>* = 0, and (2) the definition of *�*˙ is further reduced by excluding the horizontal gradients of velocities, i.e. *ui*,*<sup>x</sup>* = 0. Hence, in addition to Equation

1 2  *∂w ∂x* + *∂u ∂z* <sup>2</sup>

*∂*2*w <sup>∂</sup>z*<sup>2</sup> <sup>+</sup> *∂*2*v <sup>∂</sup>z*<sup>2</sup> <sup>+</sup>  *∂w ∂x* + *∂u ∂z* <sup>2</sup> 

+ *ρg* = 0, (A9)

. (A11)

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> 0, (A12)

+ *ρg* = 0, (A14)

. (A15)

. (A16)

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> 0, (A17)

*<sup>∂</sup><sup>y</sup>* <sup>=</sup> 0, (A18)

*<sup>∂</sup><sup>y</sup>* <sup>=</sup> 0, (A13)

. (A10)

#### **8. Appendix**

#### **Appendix A: Diagnostic equations**

We present the governing equations, with field variables *u* and *p*, for each model introduced in Section 3.1. The continuity equation (Eq. 5) holds in all cases; it takes the following respective form for 3D and 2D (flowline) models,

$$
\frac{\partial u\_x}{\partial x} + \frac{\partial u\_y}{\partial y} + \frac{\partial u\_z}{\partial z} = 0,\tag{A1}
$$

$$
\frac{
\partial u\_x
}{
\partial \mathbf{x}
} + \frac{
\partial u\_z
}{
\partial z
} = 0.
\tag{A2}
$$

Hereafter, the velocity components, *ux uy uz <sup>T</sup>*, are simply denoted by {*uvw*}*T*.

The differences between the models arise from the approximations made in the momentum balance equation itself (Eq. 6) and in the definition of strain-rate tensor (Eq. 4). Assuming that *gz* (hereafter denoted by *g*) is the only non-zero component of gravity vector, we obtain the governing equations associated with the momentum balance via Equations (7) and (1–4). The algebraic process is straightforward; we simply quote the results for each model.

#### **A1: The FS model**

This full-system 3D model has no such approximations at all. Along with Equation (A1), followings are the governing equations for the FS model,

$$2\frac{\partial}{\partial \mathbf{x}} \left( \eta \frac{\partial u}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial y} \left( \eta \frac{\partial u}{\partial y} \right) + \frac{\partial}{\partial z} \left( \eta \frac{\partial u}{\partial z} \right) + \frac{\partial}{\partial y} \left( \eta \frac{\partial v}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial z} \left( \eta \frac{\partial w}{\partial \mathbf{x}} \right) + \frac{\partial p}{\partial \mathbf{x}} = 0,\tag{A3}$$

$$2\frac{\partial}{\partial \mathbf{x}} \left(\eta \frac{\partial u}{\partial y}\right) + \frac{\partial}{\partial \mathbf{x}} \left(\eta \frac{\partial v}{\partial \mathbf{x}}\right) + 2\frac{\partial}{\partial y} \left(\eta \frac{\partial v}{\partial y}\right) + \frac{\partial}{\partial z} \left(\eta \frac{\partial v}{\partial z}\right) + \frac{\partial}{\partial z} \left(\eta \frac{\partial w}{\partial y}\right) + \frac{\partial p}{\partial y} = 0,\tag{A4}$$

$$\frac{\partial}{\partial x}\left(\eta\frac{\partial u}{\partial z}\right) + \frac{\partial}{\partial y}\left(\eta\frac{\partial v}{\partial z}\right) + \frac{\partial}{\partial x}\left(\eta\frac{\partial w}{\partial x}\right) + \frac{\partial}{\partial y}\left(\eta\frac{\partial w}{\partial y}\right) + 2\frac{\partial}{\partial z}\left(\eta\frac{\partial w}{\partial z}\right) + \frac{\partial p}{\partial z} + \rho g = 0,\tag{A5}$$

where the nonlinear effective viscosity, *η*, is given by Equation (2) with

$$2\dot{\varepsilon}\_{\varepsilon}^{2} = \frac{\partial^{2}u}{\partial x^{2}} + \frac{\partial^{2}v}{\partial y^{2}} + \frac{\partial^{2}w}{\partial z^{2}} + \frac{1}{2}\left[\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right)^{2} + \left(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}\right)^{2} + \left(\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}\right)^{2}\right].\tag{A6}$$

#### **A2: The PS model**

In this family of models, we make two approximations to ensure that the effects of lateral drag are completely absent. We neglect (1) the lateral variation of stress deviators, i.e. *τij*,*<sup>y</sup>* = 0, and (2) reduce the definition of *�*˙ by excluding the lateral variation of ice velocities, i.e. *ui*,*<sup>y</sup>* = 0. Along with Equation (A1), the **PS3 model** has the following governing equations,

$$2\frac{\partial}{\partial \mathbf{x}} \left( \eta \frac{\partial u}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial z} \left( \eta \frac{\partial u}{\partial z} \right) + \frac{\partial}{\partial z} \left( \eta \frac{\partial w}{\partial \mathbf{x}} \right) + \frac{\partial p}{\partial \mathbf{x}} = 0,\tag{A7}$$

$$
\frac{
\partial
}{
\partial \mathbf{x}
}
\left(
\eta \frac{
\partial v
}{
\partial \mathbf{x}
}
\right) + \frac{
\partial
}{
\partial \mathbf{z}
}
\left(
\eta \frac{
\partial v
}{
\partial \mathbf{z}
}
\right) + \frac{
\partial p
}{
\partial y
} = \mathbf{0},
\tag{A8}
$$

18 Will-be-set-by-IN-TECH

We present the governing equations, with field variables *u* and *p*, for each model introduced in Section 3.1. The continuity equation (Eq. 5) holds in all cases; it takes the following respective

*<sup>∂</sup><sup>z</sup>* <sup>=</sup> 0, (A1)

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> 0, (A3)

*<sup>∂</sup><sup>y</sup>* <sup>=</sup> 0, (A4)

+ *ρg* = 0, (A5)

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> 0, (A7)

*<sup>∂</sup><sup>y</sup>* <sup>=</sup> 0, (A8)

. (A6)

*<sup>∂</sup><sup>z</sup>* <sup>=</sup> 0. (A2)

*<sup>T</sup>*, are simply denoted by {*uvw*}*T*.

<sup>2</sup> + *∂w ∂x* + *∂u ∂z* <sup>2</sup> 

*∂ux ∂x* + *∂uy ∂y* + *∂uz*

> *∂ux ∂x* + *∂uz*

algebraic process is straightforward; we simply quote the results for each model.

followings are the governing equations for the FS model,

where the nonlinear effective viscosity, *η*, is given by Equation (2) with

*∂<sup>u</sup> ∂y* + *∂v ∂x* <sup>2</sup> + *∂v ∂z* + *∂w ∂y*

1 2 *ux uy uz*

The differences between the models arise from the approximations made in the momentum balance equation itself (Eq. 6) and in the definition of strain-rate tensor (Eq. 4). Assuming that *gz* (hereafter denoted by *g*) is the only non-zero component of gravity vector, we obtain the governing equations associated with the momentum balance via Equations (7) and (1–4). The

This full-system 3D model has no such approximations at all. Along with Equation (A1),

In this family of models, we make two approximations to ensure that the effects of lateral drag are completely absent. We neglect (1) the lateral variation of stress deviators, i.e. *τij*,*<sup>y</sup>* = 0, and (2) reduce the definition of *�*˙ by excluding the lateral variation of ice velocities, i.e. *ui*,*<sup>y</sup>* = 0.

Along with Equation (A1), the **PS3 model** has the following governing equations,

**8. Appendix**

**A1: The FS model**

2 *∂ ∂x η ∂u ∂x* + *∂ ∂y η ∂u ∂y* + *∂ ∂z η ∂u ∂z* + *∂ ∂y η ∂v ∂x* + *∂ ∂z η ∂w ∂x* + *∂p*

*∂ ∂x η ∂u ∂y* + *∂ ∂x η ∂v ∂x* <sup>+</sup> <sup>2</sup> *<sup>∂</sup> ∂y η ∂v ∂y* + *∂ ∂z η ∂v ∂z* + *∂ ∂z η ∂w ∂y* + *∂p*

2*�*˙ 2 *<sup>e</sup>* <sup>=</sup> *<sup>∂</sup>*2*<sup>u</sup> <sup>∂</sup>x*<sup>2</sup> <sup>+</sup>

**A2: The PS model**

*∂ ∂x η ∂u ∂z* + *∂ ∂y η ∂v ∂z* + *∂ ∂x η ∂w ∂x* + *∂ ∂y η ∂w ∂y* <sup>+</sup> <sup>2</sup> *<sup>∂</sup> ∂z η ∂w ∂z* + *∂p ∂z*

**Appendix A: Diagnostic equations**

form for 3D and 2D (flowline) models,

Hereafter, the velocity components,

*∂*2*v <sup>∂</sup>y*<sup>2</sup> <sup>+</sup>

> 2 *∂ ∂x η ∂u ∂x* + *∂ ∂z η ∂u ∂z* + *∂ ∂z η ∂w ∂x* + *∂p*

> > *∂ ∂x η ∂v ∂x* + *∂ ∂z η ∂v ∂z* + *∂p*

*∂*2*w <sup>∂</sup>z*<sup>2</sup> <sup>+</sup>

$$\frac{\partial}{\partial \mathbf{x}} \left( \eta \frac{\partial u}{\partial z} \right) + \frac{\partial}{\partial \mathbf{x}} \left( \eta \frac{\partial w}{\partial \mathbf{x}} \right) + 2 \frac{\partial}{\partial z} \left( \eta \frac{\partial w}{\partial z} \right) + \frac{\partial p}{\partial z} + \rho \mathbf{g} = \mathbf{0},\tag{A9}$$

where the nonlinear effective viscosity, *η*, is given by Equation (2) with

$$2\dot{\varepsilon}\_{\varepsilon}^{2} = \frac{\partial^{2}u}{\partial x^{2}} + \frac{\partial^{2}w}{\partial z^{2}} + \frac{1}{2} \left[ \frac{\partial^{2}v}{\partial x^{2}} + \frac{\partial^{2}v}{\partial z^{2}} + \left( \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z} \right)^{2} \right]. \tag{A10}$$

The **PS2 model** is a 2D Stokes model, strictly following the plane-strain approximations (after neglecting the non-zero *τyy*, which is required to maintain *�*˙*yy* = 0). Equations (A2), (A7) and (A9) form the governing equations, where *η* is given by Equation (2) with

$$2\dot{\varepsilon}\_{\varepsilon}^{2} = \frac{\partial^{2}u}{\partial x^{2}} + \frac{\partial^{2}w}{\partial z^{2}} + \frac{1}{2} \left(\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}\right)^{2}. \tag{A11}$$

#### **A3: The SD model**

This family of models does not account for the effects of longitudinal stress gradients, as well as those of lateral drag. Consequently, the vertical shear stresses are the only non-zero stress components. In addition, the reduced definition of *�*˙ approximated in the PS models is also applied here. Along with Equation (A1), the governing equations for the **SD3 model** are,

$$\frac{\partial}{\partial z}\left(\eta \frac{\partial u}{\partial z}\right) + \frac{\partial}{\partial z}\left(\eta \frac{\partial w}{\partial x}\right) + \frac{\partial p}{\partial x} = 0,\tag{A12}$$

$$\frac{\partial}{\partial z}\left(\eta \frac{\partial v}{\partial z}\right) + \frac{\partial p}{\partial y} = 0,\tag{A13}$$

$$
\frac{
\partial
}{
\partial \mathbf{x}
}
\left(
\eta \frac{
\partial \mu
}{
\partial z
}
\right) + \frac{
\partial
}{
\partial \mathbf{x}
}
\left(
\eta \frac{
\partial w
}{
\partial \mathbf{x}
}
\right) + \frac{
\partial p
}{
\partial z
} + \rho \mathbf{g} = \mathbf{0},
\tag{A14}
$$

where the nonlinear effective viscosity, *η*, is given by Equation (2) with

$$2\dot{\varepsilon}\_{\varepsilon}^{2} = \frac{1}{2} \left[ \frac{\partial^{2}v}{\partial z^{2}} + \left( \frac{\partial w}{\partial \mathbf{x}} + \frac{\partial u}{\partial z} \right)^{2} \right]. \tag{A15}$$

Equations (A2), (A12) and (A14) are the governing equations for the **SD2 model**, where *η* is given by Equation (2) with

$$2\dot{\varepsilon}\_{\varepsilon}^{2} = \frac{1}{2} \left( \frac{\partial w}{\partial \mathbf{x}} + \frac{\partial u}{\partial z} \right)^{2} . \tag{A16}$$

Further approximations are needed to obtain the standard SIA models; (1) the horizontal gradients in stresses are negligible, i.e. *τij*,*<sup>x</sup>* = 0, and (2) the definition of *�*˙ is further reduced by excluding the horizontal gradients of velocities, i.e. *ui*,*<sup>x</sup>* = 0. Hence, in addition to Equation (A1), the **SIA3 model** has the following governing equations,

$$\frac{\partial}{\partial z}\left(\eta \frac{\partial u}{\partial z}\right) + \frac{\partial p}{\partial x} = 0,\tag{A17}$$

$$
\frac{
\partial
}{
\partial z
}
\left(
\eta \frac{
\partial v
}{
\partial z
}
\right) + \frac{
\partial p
}{
\partial y
} = 0,
\tag{A18}
$$

 Ω*<sup>e</sup> η* 2 *∂ψ<sup>i</sup> ∂x*

> 2*η ∂u*˜ *∂x* + *p*˜

 Ω*<sup>e</sup> η* 2 *∂ψ<sup>i</sup> ∂x*

 Ω*<sup>e</sup> <sup>η</sup> ∂ψ<sup>i</sup> ∂x*

> Ω*<sup>e</sup> η ∂ψ<sup>i</sup> ∂x*

where *σy*

 Ω*<sup>e</sup> ψi ∂* [*ψ*]

*∂* [*ψ*] *∂x*

> Γ*e ψi* <sup>2</sup>*<sup>η</sup> ∂u*˜ *∂x* + *p*˜

*nx* + *η*

<sup>+</sup> *∂ψ<sup>i</sup> ∂y*

*<sup>∂</sup><sup>y</sup>* {*u*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>+</sup>

<sup>+</sup> *∂ψ<sup>i</sup> ∂y*

= *σxynx* + *σyyny* + *σyznz*

integrals in Equations (B5–B7) become zeros.

*<sup>∂</sup><sup>x</sup>* {*u*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>+</sup>

*∂* [*ψ*] *∂x*

> + Ω*<sup>e</sup> <sup>η</sup> ∂ψ<sup>i</sup> ∂z*

> + Ω*<sup>e</sup> <sup>η</sup> ∂ψ<sup>i</sup> ∂z*

*∂* [*ψ*] *∂x*

*∂* [*ψ*]

*∂u*˜ *∂y* + *∂v*˜ *∂x* 

> *∂* [*ψ*] *∂y*

> > *∂* [*ψ*]

 Ω*<sup>e</sup> η ∂ψ<sup>i</sup> ∂x*

*∂* [*ψ*]

 Ω*<sup>e</sup> <sup>η</sup> ∂ψ<sup>i</sup> ∂x*

> + 2 *∂ψ<sup>i</sup> ∂z*

> > and *σz*

The incompressibility criterion (Eq. 12) can simply be expanded as,

 Ω*<sup>e</sup> ψi ∂* [*ψ*]

*∂* [*ψ*] *∂y*

<sup>+</sup> *∂ψ<sup>i</sup> ∂y*

*∂* [*ψ*] *∂y*

> + Ω*<sup>e</sup> <sup>η</sup> ∂ψ<sup>i</sup> ∂z*

<sup>+</sup> *∂ψ<sup>i</sup> ∂z*

By substituting approximations of field variables (Eqs. 10–11) into Equation (B2), we obtain

Modelling Dynamics of Valley Glaciers 135

{*u*} d*x*d*y*d*z* +

*<sup>∂</sup><sup>x</sup>* {*w*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>+</sup>

 *∂u*˜ *∂y* + *∂v*˜ *∂x* 

Here, the RHS surface integral forms a Neumann or natural boundary condition. With Equations (4), (1), and (7), the terms inside large parentheses take the following forms,

> *∂u*˜ *∂z* + *∂w*˜ *∂x*

The RHS term above indicates the horizontal *x* component of the Cauchy stress tensor on the

{*u*} d*x*d*y*d*z* +

 Ω*<sup>e</sup> ∂ψ<sup>i</sup>*

+ 2 *∂ψ<sup>i</sup> ∂y*

> Ω*<sup>e</sup> ∂ψi*

*<sup>∂</sup><sup>z</sup>* {*u*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>+</sup>

{*w*} d*x*d*y*d*z* +

= Ω*<sup>e</sup>*

= *σxznx* + *σyzny* + *σzznz*

 Ω*<sup>e</sup> ψi ∂* [*ψ*]

 Ω*<sup>e</sup> <sup>η</sup> ∂ψ<sup>i</sup> ∂y*

 Ω*<sup>e</sup> ∂ψi*

*ny* + *η*

 Ω*<sup>e</sup> <sup>η</sup> ∂ψ<sup>i</sup> ∂y*

*∂* [*ψ*] *∂y*

> Ω*<sup>e</sup> <sup>η</sup> ∂ψ<sup>i</sup> ∂y*

*<sup>∂</sup><sup>x</sup>* [*ψ*] {*p*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>=</sup>

<sup>+</sup> *∂ψ<sup>i</sup> ∂z*

*<sup>∂</sup><sup>y</sup>* [*ψ*] {*p*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>=</sup>

 Ω*<sup>e</sup> ∂ψ<sup>i</sup>*

*ρgψ<sup>i</sup>* d*x*d*y*d*z* +

[*K*] {*φ*} = {*F*} , (B9)

*∂* [*ψ*]

 *∂u*˜ *∂z* + *∂w*˜ *∂x nz* d*S*.

*<sup>∂</sup><sup>x</sup>* {*v*} <sup>d</sup>*x*d*y*d*<sup>z</sup>*

(B3)

(B5)

(B6)

(B7)

*<sup>∂</sup><sup>x</sup>* [*ψ*] {*p*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>=</sup>

*nz* = *σxxnx* + *σxyny* + *σxznz*. (B4)

*<sup>∂</sup><sup>x</sup>* {*v*} <sup>d</sup>*x*d*y*d*<sup>z</sup>*

*ψiσ<sup>x</sup>* d*S*.

{*v*} d*x*d*y*d*z*

*ψiσ<sup>y</sup>* d*S*,

*ψiσ<sup>z</sup>* d*S*,

are *y* and *z* components

*<sup>∂</sup><sup>z</sup>* {*w*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>=</sup> 0. (B8)

 Γ*e*

 Γ*e*

*<sup>∂</sup><sup>z</sup>* {*v*} <sup>d</sup>*x*d*y*d*<sup>z</sup>*

*<sup>∂</sup><sup>z</sup>* [*ψ*] {*p*} <sup>d</sup>*x*d*y*d*<sup>z</sup>*

 Γ*e*

*∂* [*ψ*]

*∂* [*ψ*] *∂z*

*∂* [*ψ*]

*∂* [*ψ*]

*nx* + *η*

*ny* + *η*

*∂* [*ψ*] *∂z*

*<sup>∂</sup><sup>x</sup>* {*w*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>+</sup>

*<sup>∂</sup><sup>y</sup>* {*w*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>+</sup>

*∂* [*ψ*]

*∂* [*ψ*] *∂z*

of *σ* on the boundary surface, respectively. For the stress free boundary condition, the surface

*<sup>∂</sup><sup>y</sup>* {*v*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>+</sup>

Now, we write the linear equations (Eqs. B5–B8) in the following elemental matrix form,

Similar equations can be written, respectively for horizontal *y* and vertical *z* directions,

*∂* [*ψ*] *∂x*

boundary surface. Denoting this by *σx*, we rewrite Equation (B3) as,

<sup>+</sup> *∂ψ<sup>i</sup> ∂z*

*∂* [*ψ*] *∂z*

$$\frac{\partial p}{\partial z} + \rho g = 0,\tag{A19}$$

where the nonlinear effective viscosity, *η*, is given by Equation (2) with

$$2\dot{\varepsilon}\_{\varepsilon}^{2} = \frac{1}{2} \left( \frac{\partial^{2}v}{\partial z^{2}} + \frac{\partial^{2}u}{\partial z^{2}} \right) . \tag{A20}$$

The set of Equations (A2), (A17) and (A19) form the governing equations for the **SIA2 model**, where *η* is given by Equation (2) with

$$2\dot{\varepsilon}\_{\varepsilon}^{2} = \frac{1}{2} \frac{\partial^{2} u}{\partial z^{2}}. \tag{A21}$$

With a simple algebraic manipulation of the governing equations for SIA2 model, one can obtain the analytical solution for *u*(*x*, *z*) in a laminar flow (Eq. 8). The corresponding solution for *w*(*x*, *z*) follows directly from the incompressibility criterion (Eq. A2).

#### **Appendix B: FE formulation of diagnostic equations**

Here, we construct a linear system of Equations (12) and (13) for the FS model; those for the reduced models can be obtained in a similar manner. The LHSs of governing equations (Eqs. A3–A5 and A1), after imposing the approximations of field variables, represent the system residuals, i.e. the terms inside the large parentheses in Equations (12–13). So, expanding Equation (13) in horizontal *x* axis, for example, gives

$$\int\_{\Omega\_{\ell}} \psi\_{i} \left[ 2 \frac{\partial \left( \eta \frac{\partial \mathbf{d}}{\partial \mathbf{x}} \right)}{\partial \mathbf{x}} + \frac{\partial \left( \eta \frac{\partial \mathbf{d}}{\partial \mathbf{y}} \right)}{\partial \mathbf{y}} + \frac{\partial \left( \eta \frac{\partial \mathbf{d}}{\partial \mathbf{z}} \right)}{\partial \mathbf{z}} + \frac{\partial \left( \eta \frac{\partial \mathbf{d}}{\partial \mathbf{x}} \right)}{\partial \mathbf{y}} + \frac{\partial \left( \eta \frac{\partial \mathbf{d}}{\partial \mathbf{x}} \right)}{\partial \mathbf{z}} + \frac{\partial \tilde{\mathbf{p}}}{\partial \mathbf{x}} \right] \mathbf{d} \Omega\_{\ell} = \mathbf{0}. \tag{\text{B1}}$$

Similar equations can be obtained for the second horizontal and vertical directions, and also for the incompressibility criterion (Eq. 12).

Applying Green-Gauss theorem to integrate Equation (B1) by parts (Rao, 2005), we obtain,

− � Ω*<sup>e</sup>* 2 *∂ψi ∂x* � *η ∂u*˜ *∂x* � d*x*d*y*d*z* + � Γ*e* 2*ψ<sup>i</sup>* � *η ∂u*˜ *∂x* � *nx* d*S* − � Ω*<sup>e</sup> ∂ψi ∂y* � *η ∂u*˜ *∂y* � d*x*d*y*d*z* + + � Γ*e ψi* � *η ∂u*˜ *∂y* � *ny* d*S* − � Ω*<sup>e</sup> ∂ψ<sup>i</sup> ∂z* � *η ∂u*˜ *∂z* � d*x*d*y*d*z* + � Γ*e ψi* � *η ∂u*˜ *∂z* � *nz* d*S* + − � Ω*<sup>e</sup> ∂ψi ∂y* � *η ∂v*˜ *∂x* � d*x*d*y*d*z* + � Γ*e ψi* � *η ∂v*˜ *∂x* � *ny* d*S* − � Ω*<sup>e</sup> ∂ψi ∂z* � *η ∂w*˜ *∂x* � d*x*d*y*d*z* + + � Γ*e ψi* � *η ∂w*˜ *∂x* � *nz* d*S* − � Ω*<sup>e</sup> ∂ψ<sup>i</sup> <sup>∂</sup><sup>x</sup> <sup>p</sup>*˜ <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>+</sup> � Γ*e ψipn*˜ *<sup>x</sup>* d*S* = 0, (B2)

where d*<sup>S</sup>* <sup>∈</sup> <sup>Γ</sup>*<sup>e</sup>* is a boundary surface, and � *nx ny nz* �*<sup>T</sup>* are the *x*, *y* and *z* components of the unit normal at the boundary surface.

20 Will-be-set-by-IN-TECH

� *∂*2*v <sup>∂</sup>z*<sup>2</sup> <sup>+</sup>

The set of Equations (A2), (A17) and (A19) form the governing equations for the **SIA2 model**,

With a simple algebraic manipulation of the governing equations for SIA2 model, one can obtain the analytical solution for *u*(*x*, *z*) in a laminar flow (Eq. 8). The corresponding solution

Here, we construct a linear system of Equations (12) and (13) for the FS model; those for the reduced models can be obtained in a similar manner. The LHSs of governing equations (Eqs. A3–A5 and A1), after imposing the approximations of field variables, represent the system residuals, i.e. the terms inside the large parentheses in Equations (12–13). So, expanding

> + *∂* � *η <sup>∂</sup>v*˜ *∂x* �

Similar equations can be obtained for the second horizontal and vertical directions, and also

Applying Green-Gauss theorem to integrate Equation (B1) by parts (Rao, 2005), we obtain,

� *η ∂u*˜ *∂z* �

*nz* d*S* −

� Ω*<sup>e</sup> ∂ψ<sup>i</sup>*

*nx ny nz*

*∂y*

*nx* d*S* −

*ny* d*S* −

+ *∂* � *η <sup>∂</sup>w*˜ *∂x* �

> � Ω*<sup>e</sup> ∂ψi ∂y*

d*x*d*y*d*z* +

� Ω*<sup>e</sup> ∂ψi ∂z*

*<sup>∂</sup><sup>x</sup> <sup>p</sup>*˜ <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>+</sup>

*∂z*

� *η ∂u*˜ *∂y* �

� *η ∂w*˜ *∂x* �

> � Γ*e*

� Γ*e ψi* � *η ∂u*˜ *∂z* �

+ *∂p*˜ *∂x* ⎤

⎦ dΩ*<sup>e</sup>* = 0. (B1)

d*x*d*y*d*z* +

*nz* d*S* +

(B2)

d*x*d*y*d*z* +

*ψipn*˜ *<sup>x</sup>* d*S* = 0,

�*<sup>T</sup>* are the *x*, *y* and *z* components of the

*∂*2*u ∂z*<sup>2</sup> �

+ *ρg* = 0, (A19)

. (A20)

*<sup>∂</sup>z*<sup>2</sup> . (A21)

*∂p ∂z*

2*�*˙ 2 *<sup>e</sup>* <sup>=</sup> <sup>1</sup> 2 *∂*2*u*

where the nonlinear effective viscosity, *η*, is given by Equation (2) with

2*�*˙ 2 *<sup>e</sup>* <sup>=</sup> <sup>1</sup> 2

for *w*(*x*, *z*) follows directly from the incompressibility criterion (Eq. A2).

**Appendix B: FE formulation of diagnostic equations**

Equation (13) in horizontal *x* axis, for example, gives

*∂y*

d*x*d*y*d*z* +

*ny* d*S* −

d*x*d*y*d*z* +

+ � Γ*e ψi* � *η ∂w*˜ *∂x* �

where d*<sup>S</sup>* <sup>∈</sup> <sup>Γ</sup>*<sup>e</sup>* is a boundary surface, and �

unit normal at the boundary surface.

+ *∂* � *η <sup>∂</sup>u*˜ *∂z* �

> � Γ*e* 2*ψ<sup>i</sup>* � *η ∂u*˜ *∂x* �

� Ω*<sup>e</sup> ∂ψ<sup>i</sup> ∂z*

� Γ*e ψi* � *η ∂v*˜ *∂x* �

*∂z*

+ *∂* � *η <sup>∂</sup>u*˜ *∂y* �

for the incompressibility criterion (Eq. 12).

*∂x*

� *η ∂u*˜ *∂x* �

> � *η ∂v*˜ *∂x* �

+ � Γ*e ψi* � *η ∂u*˜ *∂y* �

� Ω*<sup>e</sup> ψi* ⎡ ⎣2 *∂* � *η <sup>∂</sup>u*˜ *∂x* �

− � Ω*<sup>e</sup>* 2 *∂ψi ∂x*

> − � Ω*<sup>e</sup> ∂ψi ∂y*

where *η* is given by Equation (2) with

By substituting approximations of field variables (Eqs. 10–11) into Equation (B2), we obtain

$$\begin{split} \int\_{\Omega\_{\varepsilon}} \eta \left( 2 \frac{\partial \psi\_{i}}{\partial x} \frac{\partial \left[ \psi \right]}{\partial x} + \frac{\partial \psi\_{i}}{\partial y} \frac{\partial \left[ \psi \right]}{\partial y} + \frac{\partial \psi\_{i}}{\partial z} \frac{\partial \left[ \psi \right]}{\partial z} \right) \{ u \} \, dxdydz + \int\_{\Omega\_{\varepsilon}} \eta \frac{\partial \psi\_{i}}{\partial y} \frac{\partial \left[ \psi \right]}{\partial x} \{ v \} \, dxdydz \\ + \int\_{\Omega\_{\varepsilon}} \eta \frac{\partial \psi\_{i}}{\partial z} \frac{\partial \left[ \psi \right]}{\partial x} \{ w \} \, dxdydz + \int\_{\Omega\_{\varepsilon}} \eta \frac{\partial \psi\_{i}}{\partial x} \{ \psi \} \{ p \} \, dxdydz = \tag{B3} \\ \int\_{\Gamma\_{\varepsilon}} \psi\_{i} \left[ \left( 2 \eta \frac{\partial \tilde{u}}{\partial x} + \tilde{p} \right) n\_{x} + \eta \left( \frac{\partial \tilde{u}}{\partial y} + \frac{\partial \tilde{v}}{\partial x} \right) n\_{y} + \eta \left( \frac{\partial \tilde{u}}{\partial z} + \frac{\partial \tilde{w}}{\partial x} \right) n\_{z} \right] \, d\mathcal{S}. \end{split}$$

Here, the RHS surface integral forms a Neumann or natural boundary condition. With Equations (4), (1), and (7), the terms inside large parentheses take the following forms,

$$\left(2\eta\frac{\partial\widetilde{u}}{\partial\mathbf{x}} + \widetilde{\boldsymbol{\rho}}\right)n\_{\mathbf{x}} + \eta\left(\frac{\partial\widetilde{u}}{\partial\mathbf{y}} + \frac{\partial\widetilde{v}}{\partial\mathbf{x}}\right)n\_{\mathbf{y}} + \eta\left(\frac{\partial\widetilde{u}}{\partial\mathbf{z}} + \frac{\partial\widetilde{w}}{\partial\mathbf{x}}\right)n\_{\mathbf{z}} = \sigma\_{\mathbf{xx}}n\_{\mathbf{x}} + \sigma\_{\mathbf{xy}}n\_{\mathbf{y}} + \sigma\_{\mathbf{xz}}n\_{\mathbf{z}}.\tag{\mathbb{B}4}$$

The RHS term above indicates the horizontal *x* component of the Cauchy stress tensor on the boundary surface. Denoting this by *σx*, we rewrite Equation (B3) as,

$$\begin{split} \int\_{\Omega\_{\varepsilon}} \eta \left( 2 \frac{\partial \psi\_{i}}{\partial \mathbf{x}} \frac{\partial \left[ \psi \right]}{\partial \mathbf{x}} + \frac{\partial \psi\_{i}}{\partial y} \frac{\partial \left[ \psi \right]}{\partial y} + \frac{\partial \psi\_{i}}{\partial z} \frac{\partial \left[ \psi \right]}{\partial z} \right) \{ u \} \, \mathrm{d} \mathrm{x} \mathrm{d}y \mathrm{d}z + \int\_{\Omega\_{\varepsilon}} \eta \frac{\partial \psi\_{i}}{\partial y} \frac{\partial \left[ \psi \right]}{\partial \mathbf{x}} \, \mathrm{d} \mathrm{x} \mathrm{d}y \mathrm{d}z \\ + \int\_{\Omega\_{\varepsilon}} \eta \frac{\partial \psi\_{i}}{\partial z} \frac{\partial \left[ \psi \right]}{\partial \mathbf{x}} \, \mathrm{d} \mathrm{x} \mathrm{d}y \mathrm{d}z + \int\_{\Omega\_{\varepsilon}} \frac{\partial \psi\_{i}}{\partial \mathbf{x}} \, \left[ \psi \right] \, \mathrm{d} \mathrm{x} \mathrm{d}y \mathrm{d}z = \int\_{\Gamma\_{\varepsilon}} \psi\_{i} \sigma\_{\mathrm{x}} \, \mathrm{d} \mathrm{S}. \end{split} \tag{\mathbb{B}5}$$

Similar equations can be written, respectively for horizontal *y* and vertical *z* directions,

 Ω*<sup>e</sup> <sup>η</sup> ∂ψ<sup>i</sup> ∂x ∂* [*ψ*] *<sup>∂</sup><sup>y</sup>* {*u*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>+</sup> Ω*<sup>e</sup> η ∂ψ<sup>i</sup> ∂x ∂* [*ψ*] *∂x* + 2 *∂ψ<sup>i</sup> ∂y ∂* [*ψ*] *∂y* <sup>+</sup> *∂ψ<sup>i</sup> ∂z ∂* [*ψ*] *∂z* {*v*} d*x*d*y*d*z* + Ω*<sup>e</sup> <sup>η</sup> ∂ψ<sup>i</sup> ∂z ∂* [*ψ*] *<sup>∂</sup><sup>y</sup>* {*w*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>+</sup> Ω*<sup>e</sup> ∂ψi <sup>∂</sup><sup>y</sup>* [*ψ*] {*p*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>=</sup> Γ*e ψiσ<sup>y</sup>* d*S*, (B6) Ω*<sup>e</sup> <sup>η</sup> ∂ψ<sup>i</sup> ∂x ∂* [*ψ*] *<sup>∂</sup><sup>z</sup>* {*u*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* <sup>+</sup> Ω*<sup>e</sup> <sup>η</sup> ∂ψ<sup>i</sup> ∂y ∂* [*ψ*] *<sup>∂</sup><sup>z</sup>* {*v*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* Ω*<sup>e</sup> η ∂ψ<sup>i</sup> ∂x ∂* [*ψ*] *∂x* <sup>+</sup> *∂ψ<sup>i</sup> ∂y ∂* [*ψ*] *∂y* + 2 *∂ψ<sup>i</sup> ∂z ∂* [*ψ*] *∂z* {*w*} d*x*d*y*d*z* + Ω*<sup>e</sup> ∂ψ<sup>i</sup> <sup>∂</sup><sup>z</sup>* [*ψ*] {*p*} <sup>d</sup>*x*d*y*d*<sup>z</sup>* = Ω*<sup>e</sup> ρgψ<sup>i</sup>* d*x*d*y*d*z* + Γ*e ψiσ<sup>z</sup>* d*S*, (B7)

where *σy* = *σxynx* + *σyyny* + *σyznz* and *σz* = *σxznx* + *σyzny* + *σzznz* are *y* and *z* components of *σ* on the boundary surface, respectively. For the stress free boundary condition, the surface integrals in Equations (B5–B7) become zeros.

The incompressibility criterion (Eq. 12) can simply be expanded as,

$$
\int\_{\Omega\_t} \psi\_i \frac{\partial \left[\psi\right]}{\partial x} \left\{ u \right\} dxdydz + \int\_{\Omega\_t} \psi\_i \frac{\partial \left[\psi\right]}{\partial y} \left\{ v \right\} dxdydz + \int\_{\Omega\_t} \psi\_i \frac{\partial \left[\psi\right]}{\partial z} \left\{ w \right\} dxdydz = 0. \tag{B8}
$$

Now, we write the linear equations (Eqs. B5–B8) in the following elemental matrix form,

$$\left[K\right]\left\{\phi\right\} = \left\{F\right\},\tag{B9}$$

*Rαβ <sup>i</sup>* <sup>=</sup> *<sup>∂</sup> ∂χβ*

obtained from Equation (A1),

 Γ*e ψ<sup>i</sup>* [*ψ*]

where

 [*M*] + 

vector, such that *<sup>δ</sup>*<sup>3</sup> = *hk*

 *<sup>η</sup> ∂ψ<sup>i</sup> ∂χα*

*Rα*<sup>0</sup> *<sup>i</sup>* <sup>=</sup> *∂ψ<sup>i</sup> ∂χα*

> *∂ψ<sup>i</sup> ∂x*

= *δ*<sup>1</sup> [*Rσ*]

*∂s*˜ *∂x* + *v ∂s*˜

 *Fstb*

**Appendix D: FE formulation and stabilization of prognostic equations**

With approximation of field variable (Eq. 17), Equation (D1) becomes,

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> [*K*] + 

*ψ<sup>i</sup>* (*w* + *m*) d*S*

[*Rs*] =

*ψiψ<sup>j</sup>* d*S*

 , *Mstb*

 , *Fstb*

Here, *δ*<sup>3</sup> is the stability parameter that is a function of element size, *hk*, and the norm of velocity

Matrices and vectors with superscript *stb* are once again the stabilization contributions. The elemental equation (Eq. D3) can be assembled into the global matrix, as for Equation (B9).

 *<sup>u</sup> ∂ψ<sup>i</sup> ∂x* + *v ∂ψi ∂y* 

 Γ*e*

where [*F*] is the force vector, given in the RHS of Equation (B10).

Equation (18), in a three-dimensional space, can be expanded as,

[*Ru*] =

RHS force vector take the following respective forms, *Kstb*

> Γ*e ψs i ∂s*˜ *<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>u</sup>*

*∂* {*s*} *∂t*

[*K*] =

{*F*} =

d*S* + Γ*e ψi u ∂* [*ψ*] *∂x*

which can be expressed in the following elemental matrix form,

*<sup>M</sup>stb <sup>∂</sup>* {*φ*}

[*M*] =

 Γ*e ψi u ∂ψ<sup>j</sup> ∂x* + *v ∂ψ<sup>j</sup> ∂y* d*S* , *Kstb*

 Γ*e* Modelling Dynamics of Valley Glaciers 137

Similarly, the corresponding residual matrix, [*Ru*], associated with the continuity equation is

*∂ψi ∂y*

Referring to Equations (14–15), the stabilization contributions to the elemental matrix and the

= *δ*<sup>1</sup> [*F*]

+ *v ∂* [*ψ*] *∂y*

*∂ψi <sup>∂</sup><sup>z</sup>* [0]

*<sup>T</sup>* [*Rσ*] + *<sup>δ</sup>*<sup>2</sup> [*Ru*]

*<sup>∂</sup><sup>y</sup>* <sup>−</sup> (*<sup>w</sup>* <sup>+</sup> *<sup>m</sup>*)

*<sup>K</sup>stb* {*φ*} <sup>=</sup>

<sup>2</sup>�*u*� . The mass matrix, [*M*], deals with the evolution of glacier surface.

{*s*} d*S* =

 {*F*} + 

 Γ*e*

; *α*, *β* = 1, 2, 3, (C4)

. (C6)

*<sup>T</sup>* [*Ru*] , (C7)

dΓ*<sup>e</sup>* = 0. (D1)

= �*ψi*, *δ*3*Rs*�, (D4)

= �(*w* + *m*), *δ*3*Rs*�, (D6)

, (D7)

= �*Rs*, *δ*3*Rs*�, (D5)

*ψ<sup>i</sup>* (*w* + *m*) d*S*, (D2)

*<sup>F</sup>stb* , (D3)

*<sup>T</sup>* [*Rσ*] , (C8)

; *α* = 1, 2, 3, (C5)

where [*K*] is the coefficient matrix, {*φ*} is comprised of unknown field variables, and {*F*} is the RHS force vector. They are given by,

$$
\begin{bmatrix}
\begin{bmatrix}
\mathbf{K}^{11}\_{\ast} \\
\mathbf{K}^{21}\_{\ast}
\end{bmatrix}
&
\begin{bmatrix}
\mathbf{K}^{12}\_{\ast} \\
\mathbf{K}^{22}\_{\ast}
\end{bmatrix}
&
\begin{bmatrix}
\mathbf{K}^{13}\_{\ast}
\end{bmatrix}
&
\begin{bmatrix}
\mathbf{K}^{10}
\end{bmatrix}
\\
\begin{bmatrix}
\mathbf{K}^{31}\_{\ast}
\end{bmatrix}
&
\begin{bmatrix}
\mathbf{K}^{32}\_{\ast}
\end{bmatrix}
&
\begin{bmatrix}
\mathbf{K}^{23}\_{\ast}
\end{bmatrix}
&
\begin{Bmatrix}
\mathbf{0}
\end{Bmatrix}
\end{bmatrix}
\begin{Bmatrix}
\begin{Bmatrix}
\mu
\end{Bmatrix}
\\
\begin{bmatrix}
\mathbf{K}^{33}\_{\ast}
\end{Bmatrix}
&
\begin{bmatrix}
\mathbf{K}^{33}\_{\ast}
\end{bmatrix}
\begin{Bmatrix}
\mathbf{K}^{30}
\end{Bmatrix}
\end{bmatrix}
=
\begin{Bmatrix}
\begin{Bmatrix}
\mathbf{F}^{1} \\
\mathbf{F}^{2}
\end{Bmatrix} \\
\begin{bmatrix}
\mathbf{K}^{10}
\end{bmatrix}
&
\begin{bmatrix}
\mathbf{K}^{20}
\end{bmatrix}
\begin{Bmatrix}
\mathbf{K}^{20}
\end{Bmatrix}
\begin{Bmatrix}
\mathbf{K}^{30}
\end{Bmatrix}
\end{Bmatrix}
\tag{\mathbf{B}10}
$$

Components of the coefficient matrix and the force vector are as follow,

$$\left[K\_{\ast}^{aa}\right] = \left[K^{aa}\right] + \left[K^{11}\right] + \left[K^{22}\right] + \left[K^{33}\right];\ \mathfrak{a} = 1,2,3,\tag{B11}$$

$$K\_{ij}^{a\beta} = \int\_{\Omega\_{\epsilon}} \eta \frac{\partial \psi\_i}{\partial \chi\_{\beta}} \frac{\partial \psi\_j}{\partial \chi\_a} \, \mathrm{d}x \mathrm{d}y \mathrm{d}z; \; a, \beta = 1, 2, 3,\tag{B12}$$

$$K\_{ij}^{\rm a0} = \int\_{\Omega\_{\varepsilon}} \eta \frac{\partial \psi\_i}{\partial \chi\_a} \psi\_j \,\mathrm{d}x \mathrm{d}y \mathrm{d}z;\ \mathrm{a} = 1,2,3,\tag{\text{B13}}$$

$$F^{\mathfrak{a}} = \int\_{\Gamma\_{\mathfrak{e}}} \psi\_i \sigma\_{\mathfrak{a}} \, \mathrm{d}S; \, \mathfrak{a} = 1,2,\tag{B14}$$

$$F^3 = \int\_{\Omega\_\varepsilon} \rho g \psi\_i \,\mathrm{d}x \mathrm{d}y \,\mathrm{d}z + \int\_{\Gamma\_\varepsilon} \psi\_i \sigma\_z \,\mathrm{d}S\_\prime \tag{\text{B15}}$$

where *χ* denotes the Cartesian coordinates with subscripts (1, 2, 3) for (*x*, *y*, *z*). Similarly, (*σ*1, *σ*2) in Equation (B14) denote (*σx*, *σy*), respectively.

Now, we assemble the elemental equation (Eq. B9) to get the global equation,

$$\left[\mathbf{K}\right]\left\{\Phi\right\} = \left\{\mathbf{F}\right\},\tag{\text{B16}}$$

where [**K**] = ∑*<sup>E</sup> <sup>e</sup>*=<sup>1</sup> [*K*], {**Φ**} <sup>=</sup> <sup>∑</sup>*<sup>E</sup> <sup>e</sup>*=<sup>1</sup> {*φ*}, {**F**} <sup>=</sup> <sup>∑</sup>*<sup>E</sup> <sup>e</sup>*=<sup>1</sup> {*F*}, and *E* is the total number of elements *e* within the global domain Ω.

#### **Appendix C: Diagnostic system stabilization**

We use superscript of *stb* to refer to the stabilization contributions to the elemental matrices and vectors. With stabilization terms, Equation (B9) takes the following form

$$\left\{ \left[ K \right] + \left[ K^{stb} \right] \right\} \left\{ \phi \right\} = \left\{ \left\{ F \right\} + \left\{ F^{stb} \right\} \right\}.\tag{C1}$$

For the FS model, the residual matrix, [*Rσ*], associated with the momentum balance equation (see Eq. 14) can be obtained from Equations (A3–A5),

$$
\begin{bmatrix}
\boldsymbol{R}\_{\sigma}
\end{bmatrix} = \begin{bmatrix}
\begin{bmatrix}
\boldsymbol{R}\_{\ast}^{11}
\end{bmatrix} & \begin{bmatrix}
\boldsymbol{R}^{12}
\end{bmatrix} & \begin{bmatrix}
\boldsymbol{R}^{13}
\end{bmatrix} \begin{bmatrix}
\boldsymbol{R}^{10}
\end{bmatrix} \\
\begin{bmatrix}
\boldsymbol{R}^{21}
\end{bmatrix} & \begin{bmatrix}
\boldsymbol{R}\_{\ast}^{22}
\end{bmatrix} & \begin{bmatrix}
\boldsymbol{R}^{23}
\end{bmatrix} \begin{bmatrix}
\boldsymbol{R}^{20}
\end{bmatrix} \\
\begin{bmatrix}
\boldsymbol{R}^{31}
\end{bmatrix} & \begin{bmatrix}
\boldsymbol{R}^{32}
\end{bmatrix} \begin{bmatrix}
\boldsymbol{R}\_{\ast}^{33}
\end{bmatrix} & \begin{bmatrix}
\boldsymbol{R}^{30}
\end{bmatrix} \\
\begin{bmatrix}
\boldsymbol{0}
\end{bmatrix} & \begin{bmatrix}
\boldsymbol{0}
\end{bmatrix} & \begin{bmatrix}
\boldsymbol{0}
\end{bmatrix}
\end{bmatrix} \tag{C2}
$$

where

$$\left[R\_{\ast}^{aa}\right] = \left[R^{aa}\right] + \left[R^{11}\right] + \left[R^{22}\right] + \left[R^{33}\right];\ \alpha = 1,2,3,\tag{C3}$$

22 Will-be-set-by-IN-TECH

where [*K*] is the coefficient matrix, {*φ*} is comprised of unknown field variables, and {*F*} is

⎤ ⎥ ⎥ ⎥ ⎦

⎧ ⎪⎪⎨

{*u*} {*v*} {*w*} {*p*} ⎫ ⎪⎪⎬ ⎧ ⎪⎪⎨ � *F*1 � ⎫ ⎪⎪⎬

⎪⎪⎭

d*x*d*y*d*z*; *α*, *β* = 1, 2, 3, (B12)

*ψ<sup>j</sup>* d*x*d*y*d*z*; *α* = 1, 2, 3, (B13)

*ψiσα* d*S*; *α* = 1, 2, (B14)

[**K**] {**Φ**} = {**F**} , (B16)

; *α* = 1, 2, 3, (B11)

*ψiσ<sup>z</sup>* d*S*, (B15)

*<sup>e</sup>*=<sup>1</sup> {*F*}, and *E* is the total number of

. (C1)

<sup>⎦</sup> , (C2)

; *α* = 1, 2, 3, (C3)

, (B10)

� *F*2 �

� *F*3 � {0}

⎪⎪⎩

⎪⎪⎭ =

⎪⎪⎩

� �*K*30�

� �*K*12� �*K*13� �*K*10�

� �*K*23� �*K*20�

*<sup>K</sup>*30�*<sup>T</sup>* [0]

*∂ψj ∂χα*

*ρgψ<sup>i</sup>* d*x*d*y*d*z* +

*<sup>e</sup>*=<sup>1</sup> {*φ*}, {**F**} <sup>=</sup> <sup>∑</sup>*<sup>E</sup>*

We use superscript of *stb* to refer to the stabilization contributions to the elemental matrices

� {*F*} + � *Fstb*��

� �*R*12� �*R*13� �*R*10�

∗

� �*R*23� �*R*20�

� �*R*30�

⎤ ⎥ ⎥

{*φ*} =

*R*21� �*R*<sup>22</sup> ∗

For the FS model, the residual matrix, [*Rσ*], associated with the momentum balance equation

*R*31� �*R*32� �*R*<sup>33</sup>

[0] [0] [0] [0]

where *χ* denotes the Cartesian coordinates with subscripts (1, 2, 3) for (*x*, *y*, *z*). Similarly,

� Γ*e*

∗

the RHS force vector. They are given by, ⎡ ⎢ ⎢ ⎢ ⎣

� *K*<sup>11</sup> ∗

�

�

� *<sup>K</sup>*10�*<sup>T</sup>* �

[*Kαα*

*K*21� �*K*<sup>22</sup> ∗

*K*31� �*K*32� �*K*<sup>33</sup>

<sup>∗</sup> ] <sup>=</sup> [*Kαα*] <sup>+</sup>

*Kα*<sup>0</sup> *ij* = � Ω*<sup>e</sup> <sup>η</sup> ∂ψ<sup>i</sup> ∂χα*

*F*<sup>3</sup> = � Ω*<sup>e</sup>*

*Kαβ ij* = � Ω*<sup>e</sup> <sup>η</sup> ∂ψ<sup>i</sup> ∂χβ*

(*σ*1, *σ*2) in Equation (B14) denote (*σx*, *σy*), respectively.

*<sup>e</sup>*=<sup>1</sup> [*K*], {**Φ**} <sup>=</sup> <sup>∑</sup>*<sup>E</sup>*

� [*K*] + � *Kstb*��

(see Eq. 14) can be obtained from Equations (A3–A5),

[*Rαα*

[*Rσ*] =

<sup>∗</sup> ] <sup>=</sup> [*Rαα*] <sup>+</sup>

⎡ ⎢ ⎢ ⎣ � *R*<sup>11</sup> ∗

�

�

� *R*11� + � *R*22� + � *R*33�

elements *e* within the global domain Ω.

**Appendix C: Diagnostic system stabilization**

where [**K**] = ∑*<sup>E</sup>*

where

*<sup>K</sup>*20�*<sup>T</sup>* �

Components of the coefficient matrix and the force vector are as follow,

� *K*11� + � *K*22� + � *K*33�

*F<sup>α</sup>* = � Γ*e*

Now, we assemble the elemental equation (Eq. B9) to get the global equation,

and vectors. With stabilization terms, Equation (B9) takes the following form

$$R\_i^{a\beta} = \frac{\partial}{\partial \chi\_{\beta}} \left( \eta \frac{\partial \psi\_i}{\partial \chi\_a} \right); \text{ a.} \ \beta = 1, 2, 3,\tag{C4}$$

$$R\_i^{a0} = \frac{\partial \psi\_i}{\partial \chi\_a}; \; a = 1,2,3,\tag{C5}$$

Similarly, the corresponding residual matrix, [*Ru*], associated with the continuity equation is obtained from Equation (A1),

$$[R\_u] = \begin{bmatrix} \frac{\partial \psi\_i}{\partial x} & \frac{\partial \psi\_i}{\partial y} & \frac{\partial \psi\_i}{\partial z} & [0] \end{bmatrix}. \tag{C6}$$

Referring to Equations (14–15), the stabilization contributions to the elemental matrix and the RHS force vector take the following respective forms,

$$\left[\boldsymbol{K}^{stb}\right] = \delta\_1 \left[\boldsymbol{R}\_\sigma\right]^T \left[\boldsymbol{R}\_\sigma\right] + \delta\_2 \left[\boldsymbol{R}\_u\right]^T \left[\boldsymbol{R}\_u\right] \tag{C7}$$

$$\left[F^{stb}\right] = \delta\_1 \left[F\right]^T \left[\mathcal{R}\_\sigma\right] \,\tag{C8}$$

where [*F*] is the force vector, given in the RHS of Equation (B10).

#### **Appendix D: FE formulation and stabilization of prognostic equations**

Equation (18), in a three-dimensional space, can be expanded as,

$$\int\_{\Gamma\_{\varepsilon}} \psi\_{i}^{s} \left[ \frac{\partial \mathfrak{s}}{\partial t} + u \frac{\partial \mathfrak{s}}{\partial x} + v \frac{\partial \mathfrak{s}}{\partial y} - (w + m) \right] d\Gamma\_{\varepsilon} = 0. \tag{D1}$$

With approximation of field variable (Eq. 17), Equation (D1) becomes,

$$\int\_{\Gamma\_{\varepsilon}} \psi\_{i} \left[ \psi \right] \frac{\partial \left\{ s \right\}}{\partial t} \, \mathrm{d}S + \int\_{\Gamma\_{\varepsilon}} \psi\_{i} \left( u \frac{\partial \left[ \psi \right]}{\partial x} + v \frac{\partial \left[ \psi \right]}{\partial y} \right) \left\{ s \right\} \, \mathrm{d}S = \int\_{\Gamma\_{\varepsilon}} \psi\_{i} \left( w + m \right) \, \mathrm{d}S,\tag{\text{D2}}$$

which can be expressed in the following elemental matrix form,

$$
\left\{ \left[ M \right] + \left[ M^{stb} \right] \right\} \frac{\partial \left\{ \Phi \right\}}{\partial t} + \left[ \left[ K \right] + \left[ K^{stb} \right] \right] \left\{ \Phi \right\} = \left\{ \left\{ F \right\} + \left\{ F^{stb} \right\} \right\}, \tag{D3}
$$

where

$$\begin{bmatrix} \mathbf{M} \end{bmatrix} = \left[ \int\_{\Gamma\_{\varepsilon}} \psi\_{i} \psi\_{j} \, \mathrm{d}S \right] \,, \left[ \mathrm{M}^{\mathrm{stb}} \right] = \left< \psi\_{i} , \delta\_{3} \mathbf{R}\_{\mathrm{s}} \right> , \tag{\text{D4}}$$

$$\begin{aligned} [K] = \left[ \int\_{\Gamma\_{\varepsilon}} \psi\_{i} \left( u \frac{\partial \psi\_{j}}{\partial x} + v \frac{\partial \psi\_{j}}{\partial y} \right) \mathrm{d}S \right], \; \left[ \mathrm{K}^{\mathrm{stb}} \right] = \left< \mathrm{R}\_{\mathrm{s}}, \delta\_{3} \mathrm{R}\_{\mathrm{s}} \right> \, \end{aligned} \tag{\text{D5}}$$

$$\{F\} = \left\{ \int\_{\Gamma\_{\varepsilon}} \psi\_i \,(w+m) \,\mathrm{d}S \right\} \,, \left\{ F^{stb} \right\} = \left\langle (w+m) \,, \delta\_3 \mathrm{R}\_s \right\rangle \,. \tag{\text{D6}}$$

$$\mathbb{E}\left[\mathcal{R}\_{\mathbb{S}}\right] = \left[u\frac{\partial\psi\_{i}}{\partial x} + v\frac{\partial\psi\_{i}}{\partial y}\right],\tag{D7}$$

Here, *δ*<sup>3</sup> is the stability parameter that is a function of element size, *hk*, and the norm of velocity vector, such that *<sup>δ</sup>*<sup>3</sup> = *hk* <sup>2</sup>�*u*� . The mass matrix, [*M*], deals with the evolution of glacier surface. Matrices and vectors with superscript *stb* are once again the stabilization contributions. The elemental equation (Eq. D3) can be assembled into the global matrix, as for Equation (B9).

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**7** 

*Algéria* 

**Numerical Modelling of Dynamic Nitrogen** 

**at Atmospheric Pressure in a Negative** 

*Laboratory of Theoretical Physics, University Abou Bakr Belkaid, Tlemcen* 

Plasmas are generated by supplying energy to a neutral gas causing the formation of charge carriers. Electrons and ions are produced in the gas phase when electrons or photons with sufficient energy collide with the neutral atoms and molecules in the feed gas (electron impact ionization or photoionization). The most widely used method for plasma generation utilizes the electrical breakdown of a neutral gas in the presence of an external electric field. Charge carriers accelerated in the electric field couple their energy into the plasma via collisions with other particles. Electrons retain most of their energy in elastic collisions with atoms and molecules because of their small mass and transfer their energy primarily in inelastic collisions. Discharges are classified as DC discharges, AC discharges, or pulsed discharges on the basis of the temporal behaviour of the sustaining electric field. The spatial and temporal characteristics of plasma depend to a large degree on the particular

Today, plasmas are increasingly used in industry [13]. There are two types of plasma, the so-called thermal plasmas and cold plasmas said. The corona is a process that could lead to the creation of the latter. The use of techniques involving the corona tends to grow in importance. Indeed, they are out and already widely used in the areas of destruction of pollutants and waste gas, the surface treatment (cleaning and surface erosion, deposition of films, modifying the surface chemistry). They are also used in other applications such as ozone generation and elimination of static electricity. Also, to minimize development costs,

In this work, we study the thermodynamics of the neutral gas subjected to energy injection as the result of electric discharge in the considered medium. This approach to the problem allows considering the discharge only on its energetic aspect. The discharge plays the role of an injection in the gas. To define the profile of this energy injection, we propose a mathematical function that represents the spatial dependence of the discharge density. The spatio-temporal evolution of the neutral gas particles is studied on the basis of hydrodynamic set of equations, i.e. equations of transport for mass, momentum and energy [4]. The hydrodynamic set of equations is solved by the F.C.T method (Flux

**1. Introduction** 

Corrected Transport).

application for which the plasma will be used [1].

recent research attempting to model the phenomena involved

**DC Corona Discharge** 

A.K. Ferouani, B. Liani and M. Lemerini


### **Numerical Modelling of Dynamic Nitrogen at Atmospheric Pressure in a Negative DC Corona Discharge**

A.K. Ferouani, B. Liani and M. Lemerini *Laboratory of Theoretical Physics, University Abou Bakr Belkaid, Tlemcen Algéria* 

#### **1. Introduction**

28 Will-be-set-by-IN-TECH

142 Numerical Modelling

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*Ann. Glaciol.* 29: 184–190.

T. (2003). Tropical glacier and ice core evidence of climate change on annual to

Plasmas are generated by supplying energy to a neutral gas causing the formation of charge carriers. Electrons and ions are produced in the gas phase when electrons or photons with sufficient energy collide with the neutral atoms and molecules in the feed gas (electron impact ionization or photoionization). The most widely used method for plasma generation utilizes the electrical breakdown of a neutral gas in the presence of an external electric field. Charge carriers accelerated in the electric field couple their energy into the plasma via collisions with other particles. Electrons retain most of their energy in elastic collisions with atoms and molecules because of their small mass and transfer their energy primarily in inelastic collisions. Discharges are classified as DC discharges, AC discharges, or pulsed discharges on the basis of the temporal behaviour of the sustaining electric field. The spatial and temporal characteristics of plasma depend to a large degree on the particular application for which the plasma will be used [1].

Today, plasmas are increasingly used in industry [13]. There are two types of plasma, the so-called thermal plasmas and cold plasmas said. The corona is a process that could lead to the creation of the latter. The use of techniques involving the corona tends to grow in importance. Indeed, they are out and already widely used in the areas of destruction of pollutants and waste gas, the surface treatment (cleaning and surface erosion, deposition of films, modifying the surface chemistry). They are also used in other applications such as ozone generation and elimination of static electricity. Also, to minimize development costs, recent research attempting to model the phenomena involved

In this work, we study the thermodynamics of the neutral gas subjected to energy injection as the result of electric discharge in the considered medium. This approach to the problem allows considering the discharge only on its energetic aspect. The discharge plays the role of an injection in the gas. To define the profile of this energy injection, we propose a mathematical function that represents the spatial dependence of the discharge density. The spatio-temporal evolution of the neutral gas particles is studied on the basis of hydrodynamic set of equations, i.e. equations of transport for mass, momentum and energy [4]. The hydrodynamic set of equations is solved by the F.C.T method (Flux Corrected Transport).

Numerical Modelling of Dynamic Nitrogen at

high curvature can be positive or negative [8-9].

thermodynamic equilibrium [10].

to be accelerated [11].

molecules occur [8].

(Fig. 2).

Atmospheric Pressure in a Negative DC Corona Discharge 145

Landfills crown are guy characterized by asymmetry of the electrodes, at least one of the two electrodes with high curvature. Reduces the electric field produced in the electrode gap, when applying a high voltage is strongly inhomogeneous. The name of corona discharge is the luminous halo-shaped crown that appears around the electrode with high curvature at the initiation of the discharge. Deferent types of geometry are used in the experiments: tipup, wire up, wire-wire and wire-cylinder. The high voltage applied to the electrode with

One of the main difficulties encountered with landfill type crown is the transition to the arc. This phenomenon is characterized by a strong rise in the current flowing in the discharge and a significant increase in the gas temperature. The plasma is then generated close to

In a point-to-plane configuration at atmospheric pressure, with the sharp electrode being supplied with a negative discharge DC [8], the corona discharge inception is principally due to the acceleration of background electrons (resulting from cosmic radiation) in the high electric field created by the small curvature radius of the point. The resulting space charge field, added to the 'geometrical' initial one, allows the electrons situated a little farther away

The corona discharge is initiated when the electric field near the wire is sufficient to ionize the gaseous species. The minimum electric field is a function of the wire radius, the surface roughness of the wire, Nitrogene temperature, and pressure. The free electrons produced in the initial ionization process are accelerated away from the wire in the imposed electric field. More frequent inelastic collisions of electrons and neutral gas

Numerous models of corona discharge have been proposed. In [5] a wire-to-cylinder corona discharge is modelled by means of electronic injectors with azimuth symmetry, assimilating the coaxial discharge to a succession of elementary point-to-cylinder electrical discharges

Fig. 2. Corona discharge in wire cylinder electrode geometry.

Nitrogene

Negative high voltage

Corona discharge

#### **2. Description of corona discharge**

Under the action of an electric field, the gas molecules undergo electron collisions, according to the complex mechanisms associated with shock [5]. The reactivity of the gas depends mainly on the shape of the energy delivered to the electrode system and generating the corona called "reactor ". Geometries are often very divergent and energy sources can be of multiple origins [6].

A corona discharge occurs when a current, power is created between two electrodes brought to a high potential and separated by an inert gas, usually air ionization plasma is created and the electric charges propagate through ions with neutral gas molecules. When the electric field at a point of a gas is sufficiently large, the gas ionizes around this point and becomes conductive. In particular, if one has been charged peaks, the electric field will be greater than elsewhere, this is usually as a corona discharge will occur, the phenomenon will tend to stabilize itself as the region becomes ionized conductive tip will apparently tend to disappear. The charged particles dissipate while under the influence of the electric force and neutralize an object in contact with opposite charge.

Corona discharges therefore generally occur between an electrode of small radius of curvature (for example: fault of the conductor forming a point) as the electric field surrounding area is large enough to allow the formation of a plasma. Corona discharge can be positive or negative depending on the polarity of the electrode with a small radius of curvature. If positive, it is called positive corona, otherwise negative crown [7]. Because of the difference in mass between electrons (negative) and ions (positive), the physics of these two types of corona is radically different.

Fig. 1. domain study.

Under the action of an electric field, the gas molecules undergo electron collisions, according to the complex mechanisms associated with shock [5]. The reactivity of the gas depends mainly on the shape of the energy delivered to the electrode system and generating the corona called "reactor ". Geometries are often very divergent and energy sources can be of

A corona discharge occurs when a current, power is created between two electrodes brought to a high potential and separated by an inert gas, usually air ionization plasma is created and the electric charges propagate through ions with neutral gas molecules. When the electric field at a point of a gas is sufficiently large, the gas ionizes around this point and becomes conductive. In particular, if one has been charged peaks, the electric field will be greater than elsewhere, this is usually as a corona discharge will occur, the phenomenon will tend to stabilize itself as the region becomes ionized conductive tip will apparently tend to disappear. The charged particles dissipate while under the influence of the electric force

Corona discharges therefore generally occur between an electrode of small radius of curvature (for example: fault of the conductor forming a point) as the electric field surrounding area is large enough to allow the formation of a plasma. Corona discharge can be positive or negative depending on the polarity of the electrode with a small radius of curvature. If positive, it is called positive corona, otherwise negative crown [7]. Because of the difference in mass between electrons (negative) and ions (positive), the physics of these

**2. Description of corona discharge** 

and neutralize an object in contact with opposite charge.

two types of corona is radically different.

Fig. 1. domain study.

multiple origins [6].

Landfills crown are guy characterized by asymmetry of the electrodes, at least one of the two electrodes with high curvature. Reduces the electric field produced in the electrode gap, when applying a high voltage is strongly inhomogeneous. The name of corona discharge is the luminous halo-shaped crown that appears around the electrode with high curvature at the initiation of the discharge. Deferent types of geometry are used in the experiments: tipup, wire up, wire-wire and wire-cylinder. The high voltage applied to the electrode with high curvature can be positive or negative [8-9].

One of the main difficulties encountered with landfill type crown is the transition to the arc. This phenomenon is characterized by a strong rise in the current flowing in the discharge and a significant increase in the gas temperature. The plasma is then generated close to thermodynamic equilibrium [10].

In a point-to-plane configuration at atmospheric pressure, with the sharp electrode being supplied with a negative discharge DC [8], the corona discharge inception is principally due to the acceleration of background electrons (resulting from cosmic radiation) in the high electric field created by the small curvature radius of the point. The resulting space charge field, added to the 'geometrical' initial one, allows the electrons situated a little farther away to be accelerated [11].

The corona discharge is initiated when the electric field near the wire is sufficient to ionize the gaseous species. The minimum electric field is a function of the wire radius, the surface roughness of the wire, Nitrogene temperature, and pressure. The free electrons produced in the initial ionization process are accelerated away from the wire in the imposed electric field. More frequent inelastic collisions of electrons and neutral gas molecules occur [8].

Numerous models of corona discharge have been proposed. In [5] a wire-to-cylinder corona discharge is modelled by means of electronic injectors with azimuth symmetry, assimilating the coaxial discharge to a succession of elementary point-to-cylinder electrical discharges (Fig. 2).

Fig. 2. Corona discharge in wire cylinder electrode geometry.

Numerical Modelling of Dynamic Nitrogen at

provides for this by the collision term (∂f/∂t)coll.

calculations one obtains the continuity equation:

where u is the average (fluid) velocity:

**3.2 The conservation equations** 

in velocity space.

density does not change the outfield. It is written as follows [17]:

Atmospheric Pressure in a Negative DC Corona Discharge 147

Boltzmann equation is useful to describe the evolution of a gas of charged particles in an external electromagnetic field, and it obviously implies that the particles are small enough

*df f F f vf f dt t m t* . .

Here F is the force acting on the particles, and (∂f/∂t)coll is the time rate of change of f due to collisions. Considering, for example, free electrons, this collision term must account for elastic and inelastic electron–neutral collisions, and, at relatively high degrees of ionization, for electron–electron and electron–ion collisions. The symbol stands, as usual, for the gradient in configuration space (x,y,z) while the symbol ∂/∂v or v stands for the gradient

Here, ∂f/∂t is the rate of change due to the explicit dependence on time. The next three terms are just v.f while the last three terms, taking into account Newton's third law m(dv/dt) = F are recognized as (F/m).(∂f/∂v). The total derivative df/dt can be interpreted as the rate of change as seen in a frame moving with the particles in the six dimensional (r, v) space. The Boltzmann equation simply says that df/dt is zero unless there are collisions. Collisions have the effect of removing a particle from one element of velocity space and replacing it in another, or even creating a new particle in the case of ionization. One

The conservation equations of density number, momentum and energy for a single species may be obtained by the method of moments. In this method, f (r, v, t) is multiplied with a function g(v) of the velocity and integrated over the entire velocity space. For the case that g(v) = 1, we obtain the continuity equation, if g(v) = m v we obtain the momentum conservation equation, and if g(v)= mv2/2, we obtain the energy conservation equation. The fluid equations are simply moments of the Boltzmann equation. The lowest moment is

*v*

*f Ff dv v <sup>f</sup> d v <sup>f</sup> dv dv*

where dv stands for a three-dimensional volume element in velocity space. By transforming the third term on the left-hand side by Green's theorem and after straightforward

> *<sup>n</sup> nu S <sup>t</sup>* ( )

*ur t v f r v t dv nr t*

<sup>1</sup> <sup>3</sup> (,) (, ,) (,) 

33 3 3

*t mt*

 

*coll*

(6)

(5)

(7)

obtained just by integrating this equation over velocity space [17-19]:

*v*

*coll*

(4)

During the inception and development of the plasma in a point to plane gas discharge [10], a spatio-temporel evolution of the temperature of the neutral gas occurs as a result of plasma-neutral molecules energy interaction [11]. The temperature gradient causes a phenomenon of diffusion and convection as a result of the accompanied strong heterogeneity in the neutral gas density and pressure [12]. The fundamental role of neutral heating in the inception of gas breakdown has been shown by theoretical studies [13-14], as well as by experimental studies [15]. The behaviour of a point to plane discharge has been optically and electrically analysed for a centimeter gaps in Nitrogene at atmospheric pressure [16].

#### **3. Introduction to kinetic theory**

There are many phenomena in ionized gases for which we need to consider the velocity distribution function of the particles, or at least of some particles such as free electrons, and to use a treatment called kinetic theory. In fluid theory, the velocity distribution of each species is assumed to be Maxwellian everywhere and is therefore uniquely specified by the species temperature T. Because inelastic collisions, especially between electrons and neutral particles, play a major role in low-temperature plasmas, significant deviations from thermal equilibrium are usually present in such media, which justifies the need for using the kinetic theory. By definition, the velocity distribution function f(r, v, t) of a given species represents the number of particles of that species per unit volume of the six dimensional phase space at position (r, v) and time t. This means that the number of particles per unit volume in configuration space with velocity components between vx and vx +dvx, vy and vy +dvy , and vz and vz +dvz at time t is:

$$f(\mathbf{x}, y, z, v\_x, v\_y, v\_z).dv\_xdv\_ydv\_z \tag{1}$$

When we consider velocity distributions, we therefore have seven independent scalar variables *(*r, v*,* t*)*. The density in configuration space n = n(r, t), which is a function of only four scalar variables, is obtained by integration of f(r, v, t) over velocity space, that is

$$m(r, t) = \int\_{-\infty}^{\infty} dv\_x \int\_{-\infty}^{v\_x} dv\_y \int\_{-\infty}^{v\_y} f(r, v, t) \, dv\_z \tag{2}$$

The distribution function of simple speed allows us to calculate, for each position r and time t, the average value of certain physical properties, resulting in the so-called macroscopic or hydrodynamic quantities. Let A (r, v, t) a molecular property of any kind. The most general definition of its average value, denoted by A (r, t), is given by the expression:

$$A(\vec{r},t) = \frac{1}{n(\vec{r},t)} \int\_{-\infty}^{\infty} A(\vec{r}, \vec{v}, t) \int (\vec{r}, \vec{v}, t) d\vec{v} \,\tag{3}$$

#### **3.1 The Boltzmann equation**

The Boltzmann equation is derived rigorously from the Liouville theorem,. However, we can get this equation quickly, but in a formal way, by first assuming the absence of collisions between particles and, in a second time, taking into account the effect of collisions. The

During the inception and development of the plasma in a point to plane gas discharge [10], a spatio-temporel evolution of the temperature of the neutral gas occurs as a result of plasma-neutral molecules energy interaction [11]. The temperature gradient causes a phenomenon of diffusion and convection as a result of the accompanied strong heterogeneity in the neutral gas density and pressure [12]. The fundamental role of neutral heating in the inception of gas breakdown has been shown by theoretical studies [13-14], as well as by experimental studies [15]. The behaviour of a point to plane discharge has been optically and electrically analysed for a centimeter gaps in Nitrogene at atmospheric

There are many phenomena in ionized gases for which we need to consider the velocity distribution function of the particles, or at least of some particles such as free electrons, and to use a treatment called kinetic theory. In fluid theory, the velocity distribution of each species is assumed to be Maxwellian everywhere and is therefore uniquely specified by the species temperature T. Because inelastic collisions, especially between electrons and neutral particles, play a major role in low-temperature plasmas, significant deviations from thermal equilibrium are usually present in such media, which justifies the need for using the kinetic theory. By definition, the velocity distribution function f(r, v, t) of a given species represents the number of particles of that species per unit volume of the six dimensional phase space at position (r, v) and time t. This means that the number of particles per unit volume in configuration space with velocity components between vx and vx +dvx, vy and vy +dvy , and

When we consider velocity distributions, we therefore have seven independent scalar variables *(*r, v*,* t*)*. The density in configuration space n = n(r, t), which is a function of only

*n r t dv dv f r v t dv <sup>x</sup> y z* (,) (, ,)

The distribution function of simple speed allows us to calculate, for each position r and time t, the average value of certain physical properties, resulting in the so-called macroscopic or hydrodynamic quantities. Let A (r, v, t) a molecular property of any kind. The most general

*Ar t Ar vt f r v t dv*

<sup>1</sup> <sup>3</sup> (,) (, ,) (, ,) (,)

The Boltzmann equation is derived rigorously from the Liouville theorem,. However, we can get this equation quickly, but in a formal way, by first assuming the absence of collisions between particles and, in a second time, taking into account the effect of collisions. The

four scalar variables, is obtained by integration of f(r, v, t) over velocity space, that is

definition of its average value, denoted by A (r, t), is given by the expression:

*nr t*

*<sup>x</sup> yz x y z f*( , , , , , ). *x y z v v v dv dv dv* (1)

(2)

(3)

pressure [16].

**3. Introduction to kinetic theory** 

vz and vz +dvz at time t is:

**3.1 The Boltzmann equation** 

Boltzmann equation is useful to describe the evolution of a gas of charged particles in an external electromagnetic field, and it obviously implies that the particles are small enough density does not change the outfield. It is written as follows [17]:

$$\frac{d\vec{f}}{dt} = \frac{\partial \vec{f}}{\partial t} + \vec{v} \cdot \vec{\nabla} \, f + \frac{\vec{F}}{m} \cdot \vec{\nabla}\_v \, f = \left(\frac{\partial \vec{f}}{\partial t}\right)\_{coll} \tag{4}$$

Here F is the force acting on the particles, and (∂f/∂t)coll is the time rate of change of f due to collisions. Considering, for example, free electrons, this collision term must account for elastic and inelastic electron–neutral collisions, and, at relatively high degrees of ionization, for electron–electron and electron–ion collisions. The symbol stands, as usual, for the gradient in configuration space (x,y,z) while the symbol ∂/∂v or v stands for the gradient in velocity space.

Here, ∂f/∂t is the rate of change due to the explicit dependence on time. The next three terms are just v.f while the last three terms, taking into account Newton's third law m(dv/dt) = F are recognized as (F/m).(∂f/∂v). The total derivative df/dt can be interpreted as the rate of change as seen in a frame moving with the particles in the six dimensional (r, v) space. The Boltzmann equation simply says that df/dt is zero unless there are collisions. Collisions have the effect of removing a particle from one element of velocity space and replacing it in another, or even creating a new particle in the case of ionization. One provides for this by the collision term (∂f/∂t)coll.

#### **3.2 The conservation equations**

The conservation equations of density number, momentum and energy for a single species may be obtained by the method of moments. In this method, f (r, v, t) is multiplied with a function g(v) of the velocity and integrated over the entire velocity space. For the case that g(v) = 1, we obtain the continuity equation, if g(v) = m v we obtain the momentum conservation equation, and if g(v)= mv2/2, we obtain the energy conservation equation. The fluid equations are simply moments of the Boltzmann equation. The lowest moment is obtained just by integrating this equation over velocity space [17-19]:

$$\int\_{-\infty}^{\overline{\circ}} \frac{\partial \overline{f}}{\partial t} \, d^3 \overline{v} + \int\_{-\infty}^{\overline{\circ}} \overline{v} \, \overline{\nabla} \, f \, \, d^3 \overline{v} + \int\_{-\overline{\circ}}^{\overline{\circ}} \frac{\overline{F}}{m} \, \overline{\nabla}\_v \, f \, \, d^3 \overline{v} = \int\_{-\overline{\circ}}^{\overline{\circ}} \left( \frac{\partial f}{\partial t} \right)\_{\text{coll}} \, \mathbf{d}^3 \overline{v} \tag{5}$$

where dv stands for a three-dimensional volume element in velocity space. By transforming the third term on the left-hand side by Green's theorem and after straightforward calculations one obtains the continuity equation:

$$\frac{\partial \mathbf{u}}{\partial t} + \overline{\nabla} (\nu \mathbf{u}) = \mathbf{S} \tag{6}$$

where u is the average (fluid) velocity:

$$u(\vec{r},t) = \frac{1}{n(\vec{r},t)} \int\_{-\infty}^{\eta} \vec{v} \cdot f(\vec{r}, \vec{v}, t) \, d\vec{v} \,\tag{7}$$

Numerical Modelling of Dynamic Nitrogen at

**4. Mathematical model** 

masse (continuity equation).

For momentum (equation of motion):

and energy, for a perfect gas,

where M is the masse of a gas molecule and

the coefficient of thermal conductivity.

The equation of state is:

velocity.

Atmospheric Pressure in a Negative DC Corona Discharge 149

The discharge column is considered to be cylindrically symmetric and longitudinally uniform. It is quasi-neutral, weakly-ionized, and collision-dominated. The column is characterized by a current density distribution j(r,t), witch is a function of radius r as well as time t, and a longitudinal voltage gradient E, which is assumed independent of both position in the column and time. The current density j goes to zero at a fixed radius RC. The ionized region, which is initially diffuse, is contained in an infinite background of perfect

The rate at which thermal energy is added to the gas per unit volume is given by j(r,t) E(r,t). That is, all the input power is assumed to be transferred from the electron to the background gas. As the temperature increases, the gas expands and its density decreases near the axis. Where the gas density decreases the electrical conductivity and current density increase,

The gas dynamics are described by the conservation equations for a viscous compressible fluid and the equation of state for a perfect gas. The equations are written in cylindrical coordinates, written rotational symmetry and axial uniformity, with gas flow in the radial direction only, and with zero body forces. The fluid equations are, for the conservation of

> *tr r* <sup>1</sup> ( ) <sup>0</sup>

*v rv r T Tp p <sup>N</sup> <sup>N</sup> MN C v C <sup>v</sup>*

*<sup>T</sup> v v vv r r rr jE rr r r r r r* 2 2 1 4 ( ) <sup>3</sup> 

where p is the gas pressure, *Cv* is the specific heat at constant volume, T temperature, is

*t rM t M t* (1 ) (1 )

where N is particle density, r is function of radius as will as time t and vr is the radial

*r r r r r r <sup>r</sup> v v <sup>p</sup> v vr v v MN v t r r r r rr r r r*

(12)

(11)

31 2 2 ( ) <sup>2</sup>

is the coefficient of viscosity.

*p Nk TB* (14)

(13)

gas of particle density (at t=0) uniform at N0 and T0, respectively.

thus enhancing the subsequent rate of heating and expansion [13].

*<sup>N</sup> Nv rr*

S represents the net creation rate of particles per unit volume as a result of collisions (for example, in the case of electrons, this term takes into account new electrons created by ionization and electron losses due to recombination with ions or attachment).

The next moment of the Boltzmann equation is obtained by multiplying it by mv and integrating over dv. We have:

$$m\int\_{-\alpha}^{\alpha} \stackrel{\circ}{v} \frac{\partial f}{\partial t} \, d^3 v + m \int\_{-\alpha}^{\alpha} (v \stackrel{\circ}{\nabla}) f \, d^3 v + m \int\_{-\alpha}^{\alpha} \stackrel{\circ}{v} (\stackrel{\circ}{\nabla} . \overline{\nabla}\_v) f \, d^3 v = \int\_{-\alpha}^{\alpha} \left(\frac{\partial f}{\partial t}\right)\_{coll} d^3 v \tag{8}$$

After calculation we obtain [4]:

$$
\frac{
\partial \mathbf{u} \mathbf{u} \mathbf{u} \mathbf{u}}{
\partial t} + \overline{\nabla} (\mathbf{u} \mathbf{u} \mathbf{\hat{u}}) + \overline{\nabla} \Pi + \overline{\nabla} \mathbf{p} - \mathbf{n} \overline{\mathbf{F}} = \int\_{-\infty}^{\infty} \mathbf{m} \overline{\mathbf{v}} \left( \frac{\partial \mathbf{f}}{\partial t} \right)\_{\text{coll}} \mathbf{d}^3 \mathbf{v} \tag{9}
$$

where denotes the viscosity, p denotes the pressure, and F the specific external forces exerted on the species. The first term of (9) represents the accumulation of the specific momentum, which is generally nonzero in a transient system. The second term denotes the momentum transport caused by the flow. The third term represents the viscous forces. The fourth term is the pressure gradient. Formany flowing systems, including the plasmas treated in this work, this is the driving force that causes the various plasma species to flow. The fifth term represents the external forces, thus the combined action of the electric force, the Lorentz force and gravity. Tight-hand side term represents the momentum gained and lost trough collisions with other species. This may include the transfer of momentum from other species, or the creation of species with nonzero momentum.

We have deduced the form of the equation of conservation of energy as a function of thermal energy, using the Fourier law for thermal conductivity and the ideal gas law [1,4]:

$$\frac{3}{2}\frac{\partial n k\_{\mathbf{B}} T}{\partial t} + \frac{3}{2}\vec{\nabla}(n k\_{\mathbf{B}} T \vec{u}) + \vec{p} \vec{\nabla} \vec{u} + (\vec{\nabla} \vec{u}) \Pi - \vec{\nabla}(\lambda \vec{\nabla} T) = \int\_{-\alpha}^{+\alpha} E^{T} \left(\frac{\partial f}{\partial t}\right)\_{\text{coll}} d^{3}v \tag{10}$$

with *k*B Boltzmann's Constant, *λ* the thermal conductivity and *E*T the thermal energy. By assuming the existence of a temperature T for the species, we implicitly assume Maxwell-Boltzmann equilibrium. However, (10) can readily be rewritten in terms of average particle energies if deviations from Maxwell-Boltzmann equilibrium are relevant.

The first term on the left-hand side of (10) denotes the accumulation of thermal energy, and generally is nonzero in the transient systems treated in this work. The second term represents the convective transport of energy by means of the systematic velocity of the species. The third term represents the expansion work. The fourth term is the production of thermal energy by viscous dissipation, which is in fact the transfer of directed kinetic energy to random thermal energy in the species. The fifth term represents the diffusive heat transport (thermal conduction).

The term on the right side represents the transfer of thermal energy by collision

#### **4. Mathematical model**

148 Numerical Modelling

S represents the net creation rate of particles per unit volume as a result of collisions (for example, in the case of electrons, this term takes into account new electrons created by

The next moment of the Boltzmann equation is obtained by multiplying it by mv and

*f f m v dv m v v <sup>f</sup> dv m vF <sup>f</sup> dv dv t t*

> *nmu <sup>f</sup> nuu p nF mv d v t t* <sup>3</sup> .( ) . .

where denotes the viscosity, p denotes the pressure, and F the specific external forces exerted on the species. The first term of (9) represents the accumulation of the specific momentum, which is generally nonzero in a transient system. The second term denotes the momentum transport caused by the flow. The third term represents the viscous forces. The fourth term is the pressure gradient. Formany flowing systems, including the plasmas treated in this work, this is the driving force that causes the various plasma species to flow. The fifth term represents the external forces, thus the combined action of the electric force, the Lorentz force and gravity. Tight-hand side term represents the momentum gained and lost trough collisions with other species. This may include the transfer of momentum from

We have deduced the form of the equation of conservation of energy as a function of thermal energy, using the Fourier law for thermal conductivity and the ideal gas law [1,4]:

> *nk T <sup>f</sup> nk Tu <sup>p</sup> u u T E dv t t* 3 3 <sup>3</sup> ( ) . ( .) ( ) 2 2

with *k*B Boltzmann's Constant, *λ* the thermal conductivity and *E*T the thermal energy. By assuming the existence of a temperature T for the species, we implicitly assume Maxwell-Boltzmann equilibrium. However, (10) can readily be rewritten in terms of average particle

The first term on the left-hand side of (10) denotes the accumulation of thermal energy, and generally is nonzero in the transient systems treated in this work. The second term represents the convective transport of energy by means of the systematic velocity of the species. The third term represents the expansion work. The fourth term is the production of thermal energy by viscous dissipation, which is in fact the transfer of directed kinetic energy to random thermal energy in the species. The fifth term represents the diffusive heat

**<sup>B</sup>**

(10)

other species, or the creation of species with nonzero momentum.

**B**

energies if deviations from Maxwell-Boltzmann equilibrium are relevant.

The term on the right side represents the transfer of thermal energy by collision

3 3 33 ( ) (. )

*v*

(9)

(8)

*coll*

*coll*

*T*

*coll*

ionization and electron losses due to recombination with ions or attachment).

 

integrating over dv. We have:

After calculation we obtain [4]:

transport (thermal conduction).

The discharge column is considered to be cylindrically symmetric and longitudinally uniform. It is quasi-neutral, weakly-ionized, and collision-dominated. The column is characterized by a current density distribution j(r,t), witch is a function of radius r as well as time t, and a longitudinal voltage gradient E, which is assumed independent of both position in the column and time. The current density j goes to zero at a fixed radius RC. The ionized region, which is initially diffuse, is contained in an infinite background of perfect gas of particle density (at t=0) uniform at N0 and T0, respectively.

The rate at which thermal energy is added to the gas per unit volume is given by j(r,t) E(r,t). That is, all the input power is assumed to be transferred from the electron to the background gas. As the temperature increases, the gas expands and its density decreases near the axis. Where the gas density decreases the electrical conductivity and current density increase, thus enhancing the subsequent rate of heating and expansion [13].

The gas dynamics are described by the conservation equations for a viscous compressible fluid and the equation of state for a perfect gas. The equations are written in cylindrical coordinates, written rotational symmetry and axial uniformity, with gas flow in the radial direction only, and with zero body forces. The fluid equations are, for the conservation of masse (continuity equation).

$$\frac{\partial \mathbf{N}}{\partial \mathbf{t}} + \frac{1}{r} \frac{\partial (\mathbf{N} \mathbf{v}\_r r)}{\partial r} = \mathbf{0} \tag{11}$$

where N is particle density, r is function of radius as will as time t and vr is the radial velocity.

For momentum (equation of motion):

$$\mathbf{MN}\left[\frac{\partial \mathbf{v}\_r}{\partial t} + \mathbf{v}\_r \frac{\partial \mathbf{v}\_r}{\partial r}\right] = -\frac{\partial p}{\partial r} + \frac{\partial}{\partial r}\left[\mu \left(2\frac{\partial \mathbf{v}\_r}{\partial r} - \frac{3}{2}\frac{1}{r}\frac{\partial \mathbf{v}\_r r}{\partial r}\right)\right] + \frac{2\mu}{r} (\frac{\partial \mathbf{v}\_r}{\partial r} - \frac{\mathbf{v}\_r}{r})\tag{12}$$

where M is the masse of a gas molecule and is the coefficient of viscosity.

and energy, for a perfect gas,

$$\begin{aligned} \text{MN} \left[ \mathbf{C}\_v \frac{\partial \mathbf{T}}{\partial t} + \mathbf{v}\_r \mathbf{C}\_v \frac{\partial \mathbf{T}}{\partial r} + \frac{p}{M} \frac{\partial (\mathbf{1}/\mathbf{N})}{\partial t} + \frac{p}{M} \mathbf{v}\_r \frac{\partial (\mathbf{1}/\mathbf{N})}{\partial t} \right] \\ = \mathbf{jE} + \frac{1}{r} \frac{\partial}{\partial r} (\lambda \frac{\partial \mathbf{T}}{\partial r}) + \frac{4\mu}{\Im} \left[ \left( \frac{\mathbf{v}\_r}{r} \right)^2 + \left( \frac{\partial \mathbf{v}\_r}{\partial r} \right)^2 - \frac{\mathbf{v}\_r}{r} \frac{\partial \mathbf{v}\_r}{\partial r} \right] \end{aligned} \tag{13}$$

where p is the gas pressure, *Cv* is the specific heat at constant volume, T temperature, is the coefficient of thermal conductivity.

The equation of state is:

$$p = \mathbf{N}k\_B T \tag{14}$$

Numerical Modelling of Dynamic Nitrogen at

form:

equation.

steps:

each node of the cell.

ambient temperature.

**6. Results and discussion** 

constant injection of energy.

Atmospheric Pressure in a Negative DC Corona Discharge 151

equations of the charged or the neutral particles defined previously obey the same generic

*t rr z*

the method of corrections of flow developed by Boris and Book [20].


*NTv ttt r rr*

(0,0, ) (0,0, ) (0,0, ) 0

almost constant near the edge and varies slowly near the cathode.

maximum temperature in the direction of the anode

step is necessary to find the accuracy of the transport step

*r z r rztv rztv rzt Srzt*

 

(19)

<sup>1</sup> (,,) (,,) (,,) (,,)

where r, z are space variables, t is temporal variable, φ(r,z,t) is the transported size (density, momentum or energy) and S(r,z,t) indicates the source term of the corresponding transport

The transport equations which are narrowly coupled are discretized by the method of volumes finished and are corrected by the method of the finished volume and corrected by

The transport equations were discretized on the mesh nodes using numerical schemes to avoid the problems of digital broadcasting, which is especially important. To simplify the presentation of the method we consider a time step Δt constant, we divide the twodimensional space into cells infinitely small. The application of this method involves three



The study domain is defined by figure 2. The limit velocity of the molecules on the surface is assumed equal to zero. As it is necessary to take into account the local heating effects, the temperature of the surface is assumed equal to the averaged temperature of the surrounding gas, and the temperature of the electrode body is assumed invariable and equal to the

In Figs. 3-6, the spatio-temporal evolution of temperature, density, pressure and speed of neutrals are shown, respectively, for the case of a negative point discharge, cold wall and

In Figure 3, we observe a growing neutral heating in the function of time. This transfer of heat is important for the discharge was near the center. Indeed 10 mm (from the point), the temperature passes from the value 350 K at t = 1 μs to 650 K at t6 = 50μs, whereas it remains

On the other hand the temperature increases rapidly with time, there is also a shift of the

and *T K v U kV <sup>r</sup>* 293 , 0, 12 (20)

#### where kB is Boltzmann constant

Energy transfer by radiation has been neglected. Eliminating the time derivative of N from the energy equation by using the continuity equation, replacing the pressure with NkBT, and rearranging terms, we can rewrite the conservation equations:

$$\frac{\partial \mathbf{N}}{\partial t} = -\frac{1}{r} \frac{\partial (\mathbf{N} \, \mathbf{v}\_r \, \mathbf{r})}{\partial r} \tag{15}$$

$$\frac{\partial \mathbf{v}\_r}{\partial t} = -\mathbf{v}\_r \frac{\partial \mathbf{v}\_r}{\partial r} - \frac{k}{\text{MN}} \frac{\partial (\text{NT})}{\partial r} + \frac{1}{\text{MN}} \frac{\partial}{\partial r} \left| \mu \left( 2 \frac{\partial \mathbf{v}\_r}{\partial r} - \frac{2}{3} \frac{1}{r} \frac{\partial \mathbf{v}\_r r}{\partial r} \right) \right| + \frac{2\mu}{\text{MN}r} (\frac{\partial \mathbf{v}\_r}{\partial r} - \frac{\mathbf{v}\_r}{r}) \tag{16}$$

and

$$\frac{\partial \mathbf{T}}{\partial t} = -\boldsymbol{\sigma}\_r \frac{\partial \mathbf{T}}{\partial t} - \frac{kT}{\mathbf{M} \mathbf{C}\_v \mathbf{N}} \frac{1}{r} \frac{\partial (\mathbf{N} \boldsymbol{\sigma}\_r \mathbf{r})}{\partial r} + \frac{k \mathbf{T} \boldsymbol{\sigma}\_r}{\mathbf{M} \mathbf{C}\_v \mathbf{N}} \frac{\partial \mathbf{N}}{\partial r}$$

$$+ \frac{1}{M \mathbf{N} \mathbf{C}\_v} \left\{ j \mathbf{E} + \frac{1}{r} \frac{\partial}{\partial r} (\lambda \frac{\partial \mathbf{T}}{\partial r}) + \frac{4\mu}{3} \left[ \left( \frac{\mathbf{\boldsymbol{\sigma}}\_r}{r} \right)^2 + \left( \frac{\partial \mathbf{\boldsymbol{\sigma}}\_r}{\partial r} \right)^2 - \frac{\mathbf{\boldsymbol{\sigma}}\_r}{r} \frac{\partial \mathbf{\boldsymbol{\sigma}}\_r}{\partial r} \right] \right\} \tag{17}$$

If viscosity and thermal conduction are neglected, the speed of sound in the gas vs can be written:

$$
\sigma\_s = \left(\frac{\gamma \, kT}{\mathcal{M}}\right)^{\mathcal{V}/2} \tag{18}
$$

Where is the ratio *Cp*/*Cv* of the specific heat at constant pressure to that at constant volume [13].

#### **5. Numerical analysis**

The discharge studied in our work requires that the method used to solve the equations of transport is efficient and has the ability to follow the strong density gradients while keeping a reasonable computation time. To this end, we opted for the scheme of Flux Corrected Transport Low phase error has already been used successfully in several areas such as solving the Boltzmann equation in weakly ionized gases. The diagram FCT (Flux Corrected Transport) is certainly one of the best choices to make while it is quite complex. Among its advantages are: the absence of spurious oscillations, numerical diffusion minimum; It can also calculate the evolution of profiles with very sharp spatial variations [20].

Our work of the simulation of the discharge in space is two-dimensional with cylindrical symmetry. The hydrodynamic set of equations is solved by the F.C.T method (Flux Corrected Transport) using the procedure of time splitting for the two space variables. An FCT algorithm consists conceptually of two major stages, a transport or convective stage (Stage I) followed by an antidiffusive or corrective stage (Stage II) [20-22]. All transport

Energy transfer by radiation has been neglected. Eliminating the time derivative of N from the energy equation by using the continuity equation, replacing the pressure with NkBT, and

> *N Nv rr trr*

*r r r r r r <sup>r</sup> v v k NT v vr v v*

*r r <sup>r</sup>*

*T T kT Nv r kTv N*

*t t MC N r r MC N r* <sup>1</sup> ( ) 

*<sup>T</sup> v v vv jE MNC r r r r r r r* 2 2 11 4 ( ) <sup>3</sup> 

If viscosity and thermal conduction are neglected, the speed of sound in the gas vs can be

*kT*

*M* 1 2 

Where is the ratio *Cp*/*Cv* of the specific heat at constant pressure to that at constant volume

The discharge studied in our work requires that the method used to solve the equations of transport is efficient and has the ability to follow the strong density gradients while keeping a reasonable computation time. To this end, we opted for the scheme of Flux Corrected Transport Low phase error has already been used successfully in several areas such as solving the Boltzmann equation in weakly ionized gases. The diagram FCT (Flux Corrected Transport) is certainly one of the best choices to make while it is quite complex. Among its advantages are: the absence of spurious oscillations, numerical diffusion minimum; It can also calculate the evolution of profiles with very sharp spatial variations

Our work of the simulation of the discharge in space is two-dimensional with cylindrical symmetry. The hydrodynamic set of equations is solved by the F.C.T method (Flux Corrected Transport) using the procedure of time splitting for the two space variables. An FCT algorithm consists conceptually of two major stages, a transport or convective stage (Stage I) followed by an antidiffusive or corrective stage (Stage II) [20-22]. All transport

*s*

*v*

(16)

*v v*

() 1 21 2 2 ( ) <sup>3</sup>

*r r rr*

(18)

*t r MN r MN r r r r MNr r r*

<sup>1</sup> ( ) (15)

(17)

rearranging terms, we can rewrite the conservation equations:

*v*

*v*

where kB is Boltzmann constant

*v*

and

written:

[13].

[20].

**5. Numerical analysis** 

equations of the charged or the neutral particles defined previously obey the same generic form:

$$\frac{\partial}{\partial t}\varrho(\mathbf{r},\mathbf{z},\mathbf{t}) + \frac{1}{r}\frac{\partial \operatorname{r}\varrho(\mathbf{r},\mathbf{z},\mathbf{t})\mathbf{v}\_r}{\partial \mathbf{r}} + \frac{\partial \operatorname{\boldsymbol{\phi}}(\mathbf{r},\mathbf{z},\mathbf{t})\mathbf{v}\_z}{\partial \mathbf{z}} = \mathbf{S}(\mathbf{r},\mathbf{z},\mathbf{t})\tag{19}$$

where r, z are space variables, t is temporal variable, φ(r,z,t) is the transported size (density, momentum or energy) and S(r,z,t) indicates the source term of the corresponding transport equation.

The transport equations which are narrowly coupled are discretized by the method of volumes finished and are corrected by the method of the finished volume and corrected by the method of corrections of flow developed by Boris and Book [20].

The transport equations were discretized on the mesh nodes using numerical schemes to avoid the problems of digital broadcasting, which is especially important. To simplify the presentation of the method we consider a time step Δt constant, we divide the twodimensional space into cells infinitely small. The application of this method involves three steps:


The study domain is defined by figure 2. The limit velocity of the molecules on the surface is assumed equal to zero. As it is necessary to take into account the local heating effects, the temperature of the surface is assumed equal to the averaged temperature of the surrounding gas, and the temperature of the electrode body is assumed invariable and equal to the ambient temperature.

$$\frac{\partial \mathbf{N}}{\partial \mathbf{r}}(0,0,\mathbf{t}) = \frac{\partial \mathbf{T}}{\partial \mathbf{r}}(0,0,\mathbf{t}) = \frac{\partial \mathbf{v}}{\partial \mathbf{r}}(0,0,\mathbf{t}) = 0 \quad \text{and} \quad \mathbf{T} = 293 \,\mathbf{K}, \quad \mathbf{v}\_r = 0, \quad \mathbf{U} = 12 \text{ kV} \tag{20}$$

#### **6. Results and discussion**

In Figs. 3-6, the spatio-temporal evolution of temperature, density, pressure and speed of neutrals are shown, respectively, for the case of a negative point discharge, cold wall and constant injection of energy.

In Figure 3, we observe a growing neutral heating in the function of time. This transfer of heat is important for the discharge was near the center. Indeed 10 mm (from the point), the temperature passes from the value 350 K at t = 1 μs to 650 K at t6 = 50μs, whereas it remains almost constant near the edge and varies slowly near the cathode.

On the other hand the temperature increases rapidly with time, there is also a shift of the maximum temperature in the direction of the anode

Numerical Modelling of Dynamic Nitrogen at

8.0x1018 1.0x1019 1.2x1019 1.4x1019 1.6x1019 1.8x1019 2.0x1019 2.2x1019 2.4x1019 2.6x1019 2.8x1019

wall, constant injection of energy).

400

cold wall, constant injection of energy).

500

600

700

800

P (Torr)

900

1000

1100

N (m-3 )

Atmospheric Pressure in a Negative DC Corona Discharge 153

t 1

t 2 t 3

> t 4

0 5 10 15 20

axial distance (mm)

0 5 10 15 20

radial distance (mm)

Fig. 5. The radial evolution of neutral pressure for several laps (negative point discharge,

Fig. 4. The axial evolution of neutral density for several laps (negative point discharge, cold

t

<sup>4</sup> t

3

t 2

t 5 t 6

t 1 =0.00 t 2 =1.00 µs

t 3 =5.00 µs

t 4 =10.0 µs

t 5 =40.0 µs

t 6 =50.0 µs

t 1 =0.00 t 2 =1.00 µs

t 3 =5.00 µs

t 4 =10.0 µs

t 5 =40.0 µs

t 6 =50.0 µs

t 1

t

<sup>5</sup> t

6

Fig. 3. The axial evolution of neutral temperature for several laps (negative point discharge, cold wall, constant injection of energy).

Figure 4 shows the evolution of the density of neutral in the function of time and space. We notice on all the curves a neutral depopulation in Inter electrode space. This decline results from the thermal footprint caused by the passage of the streamer discharge, it is more important at 10 mm (from the point), where there is a rate of 40% at t6 = 50 µs , while there is a decline up to 5 % in t2 =1μs. In the middle of the discharge are the ionization phenomena responsible for the decrease in the density of neutral or figures 5 and 6, which represents the evolution of the pressure and the module of the neutral speed, we notice, because of the inertia of molecules of gas, a phase shift between the maximum module speed and total maximum pressure. This gap is especially well marked on the axis, and at the beginning of the discharge. For other parts of the field, this phase shift is less accentuated because the disturbance created by the discharge is less important for intensity. As the time elapses, the evolution of pressure and speed module becomes constant.

From the moment 20 μs, we see a trend toward stationarity for all sizes (temperature, density, pressure and speed), because the heating in a comprehensive manner (contribution of all terms), decreases in intensity over time and the dissipation of energy becomes important. The result of all these processes, that all occurs as if a heating effect (known as heat wave) begins at the tip to spread towards the plan.

t 6

> t 5

0 5 10 15 20

axial distance (mm)

Fig. 3. The axial evolution of neutral temperature for several laps (negative point discharge,

Figure 4 shows the evolution of the density of neutral in the function of time and space. We notice on all the curves a neutral depopulation in Inter electrode space. This decline results from the thermal footprint caused by the passage of the streamer discharge, it is more important at 10 mm (from the point), where there is a rate of 40% at t6 = 50 µs , while there is a decline up to 5 % in t2 =1μs. In the middle of the discharge are the ionization phenomena responsible for the decrease in the density of neutral or figures 5 and 6, which represents the evolution of the pressure and the module of the neutral speed, we notice, because of the inertia of molecules of gas, a phase shift between the maximum module speed and total maximum pressure. This gap is especially well marked on the axis, and at the beginning of the discharge. For other parts of the field, this phase shift is less accentuated because the disturbance created by the discharge is less important for intensity. As the time elapses, the evolution of pressure and speed module becomes

From the moment 20 μs, we see a trend toward stationarity for all sizes (temperature, density, pressure and speed), because the heating in a comprehensive manner (contribution of all terms), decreases in intensity over time and the dissipation of energy becomes important. The result of all these processes, that all occurs as if a heating effect (known as

t 1

t 3

t 4

> t 2

t 1 =0.00 t 2 =1.00 µs

t 3 =5.00 µs

t 4 =10.0 µs

t 5 =40.0 µs

t 6 =50.0 µs

250

cold wall, constant injection of energy).

heat wave) begins at the tip to spread towards the plan.

300

350

400

450

T (K)

constant.

500

550

600

650

700

Fig. 4. The axial evolution of neutral density for several laps (negative point discharge, cold wall, constant injection of energy).

Fig. 5. The radial evolution of neutral pressure for several laps (negative point discharge, cold wall, constant injection of energy).

Numerical Modelling of Dynamic Nitrogen at

**8. References** 

6463

1999),123-130

Atmospheric Pressure in a Negative DC Corona Discharge 155


Due to its qualities of stability, accuracy and speed compared to digital technologies that preceded it, we can say that using the FCT method has opened new perspectives for

[1] Michel Moisan and Jacques Pelletier, P*hysique des plasmas collisionnels: Applications aux* 

[2] S.I. Medjahdi, A.K. Ferouani , M. Lemerini and S. Belhour, *International Review of* 

[3] Alexandre Labergue 1992, *Etude de décharges électriques dans l'air pour le développement* 

[6] J. Zhang, K. Adamiak and G.S.P Castle, *Journal of Electrostatics*, 65, (September 2007), PP.

[7] J Dupuy and A Gibert *J. Phys. D: Appl. Phys*., 15, ( April 1982),PP. 655-664. ISSN 1361-

[8] A.K.Ferouani, M. Lemerini and S. Belhour, *Plasma Science and Technology*, 12, (April

[9] Junhong Chen and Jane H. Davidson, Model of the Negative DC Corona Plasma:

[10] M. Lemerini,B. Bouhafs,B. Benyoucef and A Belaidi, *Rev. Energ. Ren*. 2, (September

[11] J.C Mateo-Velez, P. Degond, F. Rogier, A. Seraudie and F. Thivet, *J. Phys. D: Appl. Phys*.,

[12] K. Yanallah,S. Hadj-Ziane, A. Belasri and Y. Meslem , *Journal of Molecular Structure*. 777,

[15] O. Ducasse, O. Eichwald, N. Merbahi , D. Dubois, and M. Yousfi , *J. Appl. Phys*,101,

[16] A. Luque, U. Ebert, and W. Hundsdorfer, physical review letters,( December 2007),101,

[17] C.M. Ferreira and J Loureiro, Electron kinetics in atomic and molecular plasmas, *Plasma* 

[18] J.L. Delcroix , A. Bers Physique des plasmas, volume 1, EDP Sciences, 1994. ISBN,

[14] R. Morrow , *J. Phys. D: Appl. Phys*., 30, (June1997), PP. 3099-3114, ISSN 1361-6463

Comparison to the Positive DC Corona Plasma *Plasma Chemistry and Plasma* 

[4] B. Held, Physique des plasmas froids, Paris: Masson, 1994, ISBN 978-2-225-84580-2 [5] C. Soria, F. Pontiga and A. Castellanos , *Plasma Sources Sci Technol*., 13, (November 2004),

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modeling non-equilibrium discharges in general and in particular corona.

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maximum total pressure module, due to the inertia of the gas molecules.


Fig. 6. The radial evolution of total speed module of neutral for several laps (negative peak discharge, cold wall constant injection of energy).

#### **7. Conclusion**

In this paper, we have presented numerical calculations for neutral thermal effects produced by the negative dc corona discharge DC at atmospheric pressure, is conducted.

The objective of this study is to develop an efficient numerical model for solving transport equations. This allowed us to study the evolution of temperature and density of neutral particles as a function of axial distance, in the case of a negative corona discharge at atmospheric pressure for a better understanding of the evolution and heat transfer in situations of large variations in density and electric field. We completed this approach by a numerical parametric study on the behavior of radial profiles of pressure and velocity neutral particles.

The results obtained reveal the existence of the phenomena of interaction between charged particles and neutral particles, which are causing instability reaction electric shocks. These instabilities can come from two sources:


The results show that the stabilization of the neutral gas is mainly on the function of the energy injection distribution, and depopulation is more important than the plane advance. So, as soon as a current goes through the neutral gas, obviously a Joule heating effect increases the temperature locally. These results also show that:


Due to its qualities of stability, accuracy and speed compared to digital technologies that preceded it, we can say that using the FCT method has opened new perspectives for modeling non-equilibrium discharges in general and in particular corona.

#### **8. References**

154 Numerical Modelling

t

t 3 t 1 =1.00 µs

t 2 =5.00 µs

t 3 =10.0 µs

t 4 =40.0 µs

t 5 =50.0 µs

t 5

<sup>4</sup> <sup>t</sup>

2

t 1

0 5 10 15 20

Fig. 6. The radial evolution of total speed module of neutral for several laps (negative peak

In this paper, we have presented numerical calculations for neutral thermal effects produced

The objective of this study is to develop an efficient numerical model for solving transport equations. This allowed us to study the evolution of temperature and density of neutral particles as a function of axial distance, in the case of a negative corona discharge at atmospheric pressure for a better understanding of the evolution and heat transfer in situations of large variations in density and electric field. We completed this approach by a numerical parametric study on the behavior of radial profiles of pressure and velocity

The results obtained reveal the existence of the phenomena of interaction between charged particles and neutral particles, which are causing instability reaction electric shocks. These



The results show that the stabilization of the neutral gas is mainly on the function of the energy injection distribution, and depopulation is more important than the plane advance. So, as soon as a current goes through the neutral gas, obviously a Joule heating effect

by the negative dc corona discharge DC at atmospheric pressure, is conducted.

radial distance (mm)

0

discharge, cold wall constant injection of energy).

5

10

15

V (m/s)

**7. Conclusion** 

neutral particles.

field.

instabilities can come from two sources:

variations in temperature and density of the neutrals.

increases the temperature locally. These results also show that:

20

25

30


**Part 2** 

**Maxwell's Equations** 


## **Part 2**

**Maxwell's Equations** 

156 Numerical Modelling

[19] A.K.Ferouani, M.Lemerini, F.Boudahri and S.Belhour, *Conference Proceedings American* 

[20] J.P Boris and D. L. Book , journal of computational physics, 11,( November 1972), PP.

[21] T. S .Sergey and j. S. Shang, *Journal of Computational Physics* 199, (September 2004),PP.

[22] D. Kuzmin R. Löhner S. Turek, Flux-Corrected Transport : Principles, Algorithms, and Applications Springer-Verlag Berlin Heidelberg 2005, ISBN-10: 3540237305

*Institute of Physics* (September 2008),1047, 232-235. ISSN 0094-243X

38-69 ISSN 0021-9991

437–464 ISSN:0021-9991

**8** 

*UK* 

**Numerical Modelling and Design** 

Eddy current testing involves exciting a coil with a fixed frequency or pulse and bringing it into close proximity with a conductive material. The electrical impedance of the coil changes due to the influence of electrical 'eddy currents' in the material. Using an eddy current technique, the sizing of surface and sub-surface defects, measurements of thickness of metallic plates and of conductive and non-conductive coatings on metal substrates, assessment of corrosion, ductility, heat treatment and measurements of electrical conductivity and magnetic permeability are all possible and quantifiable. The eddy current method has become one of the most successful non-destructive techniques for testing

The data acquired from eddy current sensors however is affected by a large number of variables, which include sample conductivity; permeability; geometry and temperature as well as sensor lift off. The multivariable properties of sample coatings add an even greater level of complexity. Many of these problems have been overcome in the laboratory using precision wound air-cored coils, multiple excitation frequencies and theoretical inversion models. High levels of agreement between theoretical models and measurement however are only possible with accurately constructed coils, which are difficult to manufacture in practice. Coils are also prone to poor sensitivity, poor resolution, and a poor dynamic range as well as self-resonance at high frequencies, which make them unsuitable for online process control. Many of the problems associated with air-cored coils however can be overcome

Inversion models often make use of simplifying assumptions, which include symmetrically wound coils, constant current distributions in coil regions and ideal test materials. Ideal coils simply do not exist outside the laboratory; ideal test materials do not exist outside the laboratory either. An example of non-ideal test materials is hot-dip galvanising, where molten zinc reacts with steel to form distinct eutectic alloy layers (Langhill, 1999). Another example is case hardening in steel. Steel also has a magnetic permeability that is frequency

Other than non-ideal coils and test materials, a practical limit exists to the information that can be extracted through eddy current testing (Norton & Bowler, 1993). Eddy currents can

**1. Introduction** 

conductive coatings on conductive substrates.

when the coils use ferrite cores or cup cores.

dependent and subject to localised variation (Bowler, 2006).

**of an Eddy Current Sensor** 

Philip May1 and Erping Zhou2

*1Elcometer Ltd. and 2University of Bolton* 

## **Numerical Modelling and Design of an Eddy Current Sensor**

Philip May1 and Erping Zhou2 *1Elcometer Ltd. and 2University of Bolton UK* 

#### **1. Introduction**

Eddy current testing involves exciting a coil with a fixed frequency or pulse and bringing it into close proximity with a conductive material. The electrical impedance of the coil changes due to the influence of electrical 'eddy currents' in the material. Using an eddy current technique, the sizing of surface and sub-surface defects, measurements of thickness of metallic plates and of conductive and non-conductive coatings on metal substrates, assessment of corrosion, ductility, heat treatment and measurements of electrical conductivity and magnetic permeability are all possible and quantifiable. The eddy current method has become one of the most successful non-destructive techniques for testing conductive coatings on conductive substrates.

The data acquired from eddy current sensors however is affected by a large number of variables, which include sample conductivity; permeability; geometry and temperature as well as sensor lift off. The multivariable properties of sample coatings add an even greater level of complexity. Many of these problems have been overcome in the laboratory using precision wound air-cored coils, multiple excitation frequencies and theoretical inversion models. High levels of agreement between theoretical models and measurement however are only possible with accurately constructed coils, which are difficult to manufacture in practice. Coils are also prone to poor sensitivity, poor resolution, and a poor dynamic range as well as self-resonance at high frequencies, which make them unsuitable for online process control. Many of the problems associated with air-cored coils however can be overcome when the coils use ferrite cores or cup cores.

Inversion models often make use of simplifying assumptions, which include symmetrically wound coils, constant current distributions in coil regions and ideal test materials. Ideal coils simply do not exist outside the laboratory; ideal test materials do not exist outside the laboratory either. An example of non-ideal test materials is hot-dip galvanising, where molten zinc reacts with steel to form distinct eutectic alloy layers (Langhill, 1999). Another example is case hardening in steel. Steel also has a magnetic permeability that is frequency dependent and subject to localised variation (Bowler, 2006).

Other than non-ideal coils and test materials, a practical limit exists to the information that can be extracted through eddy current testing (Norton & Bowler, 1993). Eddy currents can

Numerical Modelling and Design of an Eddy Current Sensor 161

Fig. 1. Ferrite-Cored Eddy Current Sensor.

Cylindrical ferrite core Source coil *Nc* turns

generate no magnetic flux.

2). See figure 2.

density *Jce*, then

Certain assumptions are made about the sensor, which are listed below:

homogeneous; core conductivity is assumed to be negligible.

 The self-resonant frequency of each coil is greater than the maximum operating frequency of the sensor as a measuring system; corrections for coil self-capacitance or

Pick-up coils are matched and act into loads of infinite impedance. Pick-up coils

The sensor core is soft magnetic ferrite. The core is assumed to be linear, isotropic and

In order to begin an analysis of the sensor of Fig. 1, the ferrite core and pick-up coils were removed and the source coil replaced with a delta function coil. The free space region bounding the sensor was also divided into two regions, one above the plane of the delta function coil (region 1) and one below and extending to the surface of the medium (region

Using Maxwell's equations and the homogeneous wave equation, the PDE defining the

*<sup>S</sup> <sup>t</sup> A* 

 represents medium permeability and *Jt* is the total electric current density. If *Jt* is comprised of an impressed current density *Js* and an effective electric conduction current

> ( ) <sup>2</sup> *<sup>S</sup> <sup>s</sup> ce A*

*J* <sup>2</sup> (1)

Pick-up coil *Nc* turns

Medium

Free space µ0 <sup>0</sup>

*J J* (2)

source vector potential field *AS* in any of the regions of figure 2 is of the form:

coil-ferrite capacitance is not considered necessary (Harrison et al, 1996). The source coil is considered to be a region of constant current density.

only really sense the presence of layer boundaries owing to the integrating character of eddy current signals. Glorieux and co-workers give an example of this, observing that sharp material profile features appear smooth under reconstruction (Glorieux et al, 1999). Another limitation, which affects coatings on steel, is the permeability-conductivity ratio and coating conductivity-thickness product (Becker et al, 1988). One of these quantities must be known prior to inspection.

This chapter focuses on the development and testing of a new highly accurate and highly sensitive ferrite-cored sensor and a novel magnetic moment model of the sensor, which requires only the discretisation of the sensor core-air boundary interface. The chapter starts by developing a set of partial differential equations (PDE) to model the vector potential fields present in the regions bounding the sensor. Sensor regions were considered to be source-less with imaginary surface currents imposed at region interfaces. Green's functions were determined for all bounded regions. Basis functions were then used to represent the sensor cores surface current distribution, which were then formed into a set of *2N* linearly independent equations by applying the relevant boundary conditions. A matrix method was finally developed to solve these equations using a moment method.

The matrix method was further developed in this chapter in order to calculate sensor coil impedance and induced voltage. An efficient material profile function *m*() for modelling the interaction between the sensor and test material was also developed and verified. A novel form of parameterisation was adopted for *m*(). The accuracy and convergence of the vector potentials generated by the source coil and core-air boundary surface currents was reviewed and a new free-space Green's function introduced.

#### **2. Sensor theoretical model**

This section introduces a new ferrite-cored eddy current sensor and develops integral equations to characterise the source vector potential and core vector potential fields. Closed form solutions of the core equations are applied to the core-air boundary interface, generating 2*N* linearly independent equations with 2*N* unknown coefficients. The unknown coefficients are evaluated using the method of weighted residuals.

#### **2.1 Basic sensor design**

When a ferrite core is used in an eddy current sensor, the coil inductance, sensitivity and resolution increase significantly (Blitz, 1991, Moulder et al, 1992). A ferrite core is therefore incorporated into the sensor design used for this chapter, which is shown in figure 1. Coaxial to the ferrite core below are three coils, a central source coil, which carries a current *I* amps and two sense or pick-up coils. The sensor is a reflection sensor (or transformer style sensor) with pick-up coils in a differential configuration. Each coil is assumed to have *n*c coil turns per unit area with a total of *Nc* turns. The sensor is located in free space and positioned above and orthogonal to a medium, which is comprised of *M* planar layers. Each layer is considered to be linear, isotropic and homogeneous, where the *ith* layer has conductivity <sup>i</sup> and permeability i.

only really sense the presence of layer boundaries owing to the integrating character of eddy current signals. Glorieux and co-workers give an example of this, observing that sharp material profile features appear smooth under reconstruction (Glorieux et al, 1999). Another limitation, which affects coatings on steel, is the permeability-conductivity ratio and coating conductivity-thickness product (Becker et al, 1988). One of these quantities must be known

This chapter focuses on the development and testing of a new highly accurate and highly sensitive ferrite-cored sensor and a novel magnetic moment model of the sensor, which requires only the discretisation of the sensor core-air boundary interface. The chapter starts by developing a set of partial differential equations (PDE) to model the vector potential fields present in the regions bounding the sensor. Sensor regions were considered to be source-less with imaginary surface currents imposed at region interfaces. Green's functions were determined for all bounded regions. Basis functions were then used to represent the sensor cores surface current distribution, which were then formed into a set of *2N* linearly independent equations by applying the relevant boundary conditions. A matrix method was finally developed to solve these equations using a

The matrix method was further developed in this chapter in order to calculate sensor coil

the interaction between the sensor and test material was also developed and verified. A

vector potentials generated by the source coil and core-air boundary surface currents was

This section introduces a new ferrite-cored eddy current sensor and develops integral equations to characterise the source vector potential and core vector potential fields. Closed form solutions of the core equations are applied to the core-air boundary interface, generating 2*N* linearly independent equations with 2*N* unknown coefficients. The unknown

When a ferrite core is used in an eddy current sensor, the coil inductance, sensitivity and resolution increase significantly (Blitz, 1991, Moulder et al, 1992). A ferrite core is therefore incorporated into the sensor design used for this chapter, which is shown in figure 1. Coaxial to the ferrite core below are three coils, a central source coil, which carries a current *I* amps and two sense or pick-up coils. The sensor is a reflection sensor (or transformer style sensor) with pick-up coils in a differential configuration. Each coil is assumed to have *n*c coil turns per unit area with a total of *Nc* turns. The sensor is located in free space and positioned above and orthogonal to a medium, which is comprised of *M* planar layers. Each layer is considered to be linear, isotropic and homogeneous, where the *ith* layer has conductivity <sup>i</sup>

). The accuracy and convergence of the

) for modelling

impedance and induced voltage. An efficient material profile function *m*(

novel form of parameterisation was adopted for *m*(

**2. Sensor theoretical model** 

**2.1 Basic sensor design** 

and permeability i.

reviewed and a new free-space Green's function introduced.

coefficients are evaluated using the method of weighted residuals.

prior to inspection.

moment method.

Fig. 1. Ferrite-Cored Eddy Current Sensor.

Certain assumptions are made about the sensor, which are listed below:


In order to begin an analysis of the sensor of Fig. 1, the ferrite core and pick-up coils were removed and the source coil replaced with a delta function coil. The free space region bounding the sensor was also divided into two regions, one above the plane of the delta function coil (region 1) and one below and extending to the surface of the medium (region 2). See figure 2.

Using Maxwell's equations and the homogeneous wave equation, the PDE defining the source vector potential field *AS* in any of the regions of figure 2 is of the form:

$$
\nabla^2 A\_{\mathcal{S}} = \mu J\_t \tag{1}
$$

 represents medium permeability and *Jt* is the total electric current density. If *Jt* is comprised of an impressed current density *Js* and an effective electric conduction current density *Jce*, then

$$
\nabla^2 \mathcal{A}\_{\mathcal{S}} = \mu (\mathcal{J}\_s + \mathcal{J}\_{\mathcal{C}e}) \tag{2}
$$

*Bn* (

) and *Cn* (

sensor core-air interface.

function source:

**2.2 The influence of the sensor core** 

Numerical Modelling and Design of an Eddy Current Sensor 163

*A*

*r A*

1968). If *Gs*(*r*, *z*; *r0*, *z0*) is the Green's function for equation (9), then:

 

( , ; ', ') [ (

order and first kind; is defined for each region as follows:

surface equivalence theorem, where coefficients *Bn*(

0

 *j* 

*S*

 

*r A*

for the generalised *nth* media layer of figure 2:

( )

*s s s s*

gives the PDE for the delta function coil:

*r r A*

2

 

*A z r rA r*

/ 0 (1/ ) ( )/

*<sup>j</sup> <sup>A</sup> <sup>r</sup> <sup>r</sup> <sup>z</sup> <sup>z</sup> <sup>z</sup>*

) are media dependent functions and *J*1 (α*r*) is a Bessel function of the first

*A* (8)

. (9)

*<sup>s</sup>* . (10)

1

 ) 

<sup>2</sup> <sup>2</sup> . (13)

) and *Cn*(

) are determined by

) ] (

(11)

(12)

*<sup>S</sup> <sup>S</sup>*

( ) ( ) <sup>0</sup> <sup>1</sup>

 

*s*

0 0 2 2 2

Equation (9) is widely recognised as the PDE first used by Dodd and Deeds (Dodd & Deeds,

( , ) ( , ; , ) <sup>0</sup> <sup>0</sup> *A r z G r z r z <sup>s</sup>*

<sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> *<sup>A</sup>* (*r*,*z*) *<sup>G</sup>* (*r*,*z*;*<sup>r</sup>* ,*<sup>z</sup>* )*<sup>J</sup>* (*<sup>r</sup>* ,*<sup>z</sup>* )*dr dz <sup>s</sup>*

The Green's function for equation (11) first proposed by Cheng and co-workers (Cheng et al, 1971) and forming the basis of nearly all subsequent eddy current research, is given below

) (

*n n n*

The Cheng method imposes surface currents on media layer boundaries according to the

enforcing boundary conditions at each layer interface. The sensor core can be treated in exactly the same way. Figure 3 shows surface current *J* impressed on closed surface *S* at the

Figure 3 shows the sensor core partitioned into two separate regions, an external region (vector potential *AE*) and an internal source-less region (vector potential *AI*). Considering the internal region first, let a surface current *JI* be impressed on the closed surface *S*. Let *JI* be azimuthal, and at some arbitrary point *ρ* on *S*, let the limiting value of *JI* be a delta

*s s*

 *<sup>G</sup> <sup>r</sup> <sup>z</sup> <sup>r</sup> <sup>z</sup> <sup>B</sup> <sup>e</sup> <sup>C</sup> <sup>e</sup> <sup>J</sup> <sup>r</sup> <sup>d</sup> <sup>z</sup> <sup>n</sup> <sup>z</sup> <sup>n</sup> <sup>n</sup> <sup>s</sup> <sup>n</sup> <sup>n</sup>*

If the coil has a rectangular cross section and a source distribution *Js* = *Jsφ(r0, z0)aφ*, then:

*r z*

/ / /

*r r*

/ /

*r φ z*

*a a a*

Fig. 2. Delta Function Source Coil Located above a Layered Medium.

If a delta function coil *<sup>0</sup> <sup>φ</sup> I* )()( *a* <sup>0</sup> <sup>0</sup> *zzrr* is located at (*r*0, *z*0), where

$$\delta(r - r\_0)\delta(z - z\_0) = \begin{cases} \infty, & \text{if } & (r = r\_0), \ (z = z\_0) \\\\ 0 & \text{otherwise} \end{cases} \tag{3}$$

and current density *Jce AS j* , then:

$$
\nabla^2 \mathbf{A}\_{\mathcal{S}} - j\alpha\mu\sigma \mathbf{A}\_{\mathcal{S}} + \mu \mathbf{I}\_{\emptyset} = 0 \,. \tag{4}
$$

Using the vector identity *B B* )( *B* <sup>2</sup> gives

$$
\nabla(\nabla \cdot \mathcal{A}\_{\mathsf{S}}) - \nabla \times \nabla \times \mathcal{A}\_{\mathsf{S}} - j\alpha\mu\sigma \mathcal{A}\_{\mathsf{S}} + \mu \mathcal{I}\_{\mathsf{0}} = 0 \tag{5}
$$

Since the coil excitation is azimuthal and since both the media and sensor core have axial symmetry, then vector potential *AS* will also be azimuthal, hence let the source field be

$$\mathcal{A}\_{\mathcal{S}} = A\_{\mathcal{S}\emptyset}(r, z) \mathfrak{a}\_{\emptyset} \,. \tag{6}$$

Substituting the Coulomb gauge 0 *AS* into PDE (5) gives

$$-\nabla \times \nabla \times \mathbf{A\_S} - j\alpha\mu\sigma \mathbf{A\_S} + \mu \mathbf{I\_\theta} = 0 \tag{7}$$

and using:

$$\nabla \times \nabla \times \mathcal{A}\_{\mathcal{S}} = \begin{vmatrix} \mathbf{a}\_r/r & \mathbf{a}\_\phi & \mathbf{a}\_z/r \\\\ \mathbf{\hat{o}}/\partial r & \mathbf{\hat{o}}/\partial \phi & \mathbf{\hat{o}}/\partial \mathbf{\hat{z}} \\\\ -\mathbf{\hat{o}}A\_{S\phi}/\partial \mathbf{\hat{z}} & 0 & (1/r)\mathbf{\hat{o}}(rA\_{S\phi})/\partial r \end{vmatrix} \tag{8}$$

gives the PDE for the delta function coil:

162 Numerical Modelling

*ar* 

 *a*

Fig. 2. Delta Function Source Coil Located above a Layered Medium.

**I**

 

0

 

*zzrr* is located at (*r*0, *z*0), where

i, <sup>i</sup>

*zzrrif*

Region 1 1, <sup>1</sup>

Region 2 2, <sup>2</sup>

Medium layers

0 0

*IA 0S* . (4)

. (6)

*IA 0S* 0 (7)

*IA* (5)

, )(),(

<sup>0</sup> , (3)

*otherwise*

Since the coil excitation is azimuthal and since both the media and sensor core have axial symmetry, then vector potential *AS* will also be azimuthal, hence let the source field be

> *AS a<sup>φ</sup> zrA* ),( *<sup>S</sup>*

> >

If a delta function coil *<sup>0</sup> <sup>φ</sup> I* )()( *a* <sup>0</sup> <sup>0</sup>

 *z0* 

*r0 az* 

*AS j* , then:

<sup>0</sup> <sup>2</sup>*AS <sup>j</sup>*

Using the vector identity *B B* )( *B* <sup>2</sup> gives

Substituting the Coulomb gauge 0 *AS* into PDE (5) gives

*AS j*

)( 0 *<sup>S</sup> <sup>S</sup> 0S A A j*

*zzrr*

0 

and current density *Jce*

and using:

)()(

$$\frac{\left\|\hat{\mathcal{C}}^2 A\_{s\rho} + \frac{1}{r} \frac{\partial A\_{s\rho}}{\partial r} - \frac{A\_{s\rho}}{r^2} + \frac{\partial A\_{s\rho}}{\partial z^2} - j\alpha\mu\sigma A\_{s\rho} + \mu\delta(r - r\_0)\delta(z - z\_0) = 0 \,\text{.}\tag{9}$$

Equation (9) is widely recognised as the PDE first used by Dodd and Deeds (Dodd & Deeds, 1968). If *Gs*(*r*, *z*; *r0*, *z0*) is the Green's function for equation (9), then:

$$A\_{s\rho}(r, z) = \mu G\_s(r, z; r\_0, z\_0) \, \Big. \tag{10}$$

If the coil has a rectangular cross section and a source distribution *Js* = *Jsφ(r0, z0)aφ*, then:

$$A\_{s\rho}(r,z) = \mu \iiint G\_s(r,z;\eta\_0, z\_0) J\_{s\rho}(r\_0, z\_0) dr\_0 dz\_0 \tag{11}$$

The Green's function for equation (11) first proposed by Cheng and co-workers (Cheng et al, 1971) and forming the basis of nearly all subsequent eddy current research, is given below for the generalised *nth* media layer of figure 2:

$$G\_s^{(n)}(r, z; r', z') = \bigcap\_{0 \le s \le r}^{\infty} [B\_n(a)e^{-\alpha\_n z} + C\_n(a)e^{\alpha\_n z}] J\_1(ar) da \tag{12}$$

*Bn* () and *Cn* () are media dependent functions and *J*1 (α*r*) is a Bessel function of the first order and first kind; is defined for each region as follows:

$$
\alpha\_n^2 = \alpha^2 + j\alpha\mu\_n\sigma\_n \,. \tag{13}
$$

#### **2.2 The influence of the sensor core**

The Cheng method imposes surface currents on media layer boundaries according to the surface equivalence theorem, where coefficients *Bn*() and *Cn*() are determined by enforcing boundary conditions at each layer interface. The sensor core can be treated in exactly the same way. Figure 3 shows surface current *J* impressed on closed surface *S* at the sensor core-air interface.

Figure 3 shows the sensor core partitioned into two separate regions, an external region (vector potential *AE*) and an internal source-less region (vector potential *AI*). Considering the internal region first, let a surface current *JI* be impressed on the closed surface *S*. Let *JI* be azimuthal, and at some arbitrary point *ρ* on *S*, let the limiting value of *JI* be a delta function source:

Numerical Modelling and Design of an Eddy Current Sensor 165

where *ai* represents a basis function and *ui* the basis function coefficient. Substituting

*AR r z ui GR r z r z ai r z ds*

Basis functions were now chosen to accurately represent the anticipated unknown function *JR<sup>φ</sup>*. A piecewise constant sub-domain function was chosen to do this, which is of the form

( , ; ', ') ( ', ') '

*r r r r r*

1 ' ( ' , ' ), ' '

1 ' ( ' , ' ), ' '

*otherwise*

' / 4 *<sup>r</sup> <sup>i</sup>*1 *<sup>i</sup>*<sup>1</sup> *r r* and ' ' / 4 *<sup>z</sup> <sup>i</sup>*1 *<sup>i</sup>*<sup>1</sup> *z z* .

Sub-domains were divided into *N* sub-intervals and evenly distributed along the sensor core-air interface. Observation points (*r, z*) were located at the centre of sub-domains (see

When combined the source and scattered field gives the total vector potential *AEφ*. For the *nth*

( ', ') <sup>1</sup> <sup>1</sup> <sup>1</sup> <sup>1</sup>

*a r z <sup>i</sup> <sup>r</sup> <sup>i</sup> <sup>r</sup> <sup>i</sup> <sup>i</sup>*

*z z z z z*

1 1 1 1

*z*i + *<sup>z</sup>*

*z*i - <sup>z</sup>

Sensor core-air interface Curve *C* (shown in red)

*i z i z i i*

.

Observation points (*r*, *z*)

(20)

(21)

*i C*

1

*N*

( , )

 

0

'

 

figure 4) for greatest computational accuracy (Balanis, 1989).

Fig. 4. Discretization of Current *JR* on the Sensor Core Interface.

**2.3 The total field and internal core field** 

media layer outside the sensor core:

Observation point

Sensor axis *r* = 0

(*ri*, *zi*)

equation (19) into (18) gives the following:

*i*

shown below:

Fig. 3. Surface Current Distribution *J* on Core Surface *S*.

Since *JI* is azimuthal it follows that vector potential *AI* is likewise azimuthal, which leads To the following PDE for the component *AIφ*:

$$\frac{\left\|\widehat{\boldsymbol{\sigma}}^{2}A\_{I\varphi}\right\|}{\left\|\widehat{\boldsymbol{r}}r\right\|^{2}} + \frac{1}{r}\frac{\left\|A\_{I\varphi}\right\|}{\left\|\widehat{r}r\right\|} - \frac{A\_{I\varphi}}{r^{2}} + \frac{\left\|A\_{I\varphi}\right\|}{\left\|\boldsymbol{z}\right\|^{2}} + \mu\_{I}\delta(r - r^{\overset{\cdot}{\cdot}})\delta(z - z^{\overset{\cdot}{\cdot}}) = 0\tag{15}$$

with the solution for *JI* = *JIφ*(*r', z'*)*a<sup>φ</sup>*

$$\mathcal{A}\_{l\phi}(r,z) = \mu\_l \int\_{\mathbb{C}} \mathcal{G}\_l(r,z;r',z')|\_{l\phi}(r',z')ds'.\tag{16}$$

Integration is along a contour *C* on closed surface *S*. *GI* (*r, z; r, z*) and *Gs* (*r, z; r0, z0*) are clearly identical and differ only in media dependent functions *Bn* () and *Cn* (). A similar set of equations does not directly follow for the external core field *AE* due to the presence of source current *Js*. The field in this region must be regarded as the vector sum of the source field *As* and a source-less scattered field *AR* (Yildir et al, 1992):

$$A\_E = A\_R + A\_s \tag{17}$$

Concentrating on the source-less scattered field *AR* and impressing a scattering current *JR* on *S,* leads to the following:

$$A\_{R\wp}(r,z) = \mu \int\_C G\_R(r,z;r',z') J\_{R\wp}(r',z') ds' \tag{18}$$

*GR* (*r*, *z*, *r, z*) and *GS* (*r*, *z*; *r0*, *z0*) are the same function as both are determined for the same *M*  + 2 media layers. A solution to equation (18) proceeds by expanding surface current *JR<sup>φ</sup>* as follows (Balanis, 1989):

$$J\_{R\phi}(r',z') = \sum\_{l=1}^{N} \mu\_l a\_l(r',z') \tag{19}$$

164 Numerical Modelling

( ) ( ) ( ) <sup>3</sup> ' ' ' *JI*

. (14)

Surface current Distribution, *J* Am-1

(15)

) and *Gs* (*r, z; r0, z0*) are

. (18)

). A similar

(19)

 *JI* 

Since *JI* is azimuthal it follows that vector potential *AI* is likewise azimuthal, which leads

*A r z G r z r z J r z ds*

set of equations does not directly follow for the external core field *AE* due to the presence of source current *Js*. The field in this region must be regarded as the vector sum of the source

Concentrating on the source-less scattered field *AR* and impressing a scattering current *JR* on

*<sup>R</sup> <sup>R</sup> <sup>R</sup> A* (*r*,*z*) *G* (*r*,*z*;*r*',*z*')*J* (*r*',*z*')*ds*'

+ 2 media layers. A solution to equation (18) proceeds by expanding surface current *JR<sup>φ</sup>* as

 *<sup>N</sup> <sup>i</sup> <sup>R</sup> <sup>i</sup> <sup>i</sup> <sup>J</sup> <sup>r</sup> <sup>z</sup> <sup>u</sup> <sup>a</sup> <sup>r</sup> <sup>z</sup>* <sup>1</sup> ( ', ') ( ', ')

 *<sup>r</sup> <sup>r</sup> <sup>z</sup> <sup>z</sup> z A*

*C*

*r A*

 *C*

( ) ( ) <sup>0</sup> <sup>1</sup> ' '

Core internal region Core external region

*AI AE* 

*I*

 

*, z*) and *GS* (*r*, *z*; *r0*, *z0*) are the same function as both are determined for the same *M* 

 

. (16)

*, z*

*AE AR As* (17)

) and *Cn* (

*d*

2 2 2

*I I I I*

 

*r A*

( , ) ( , ; ', ') ( ', ') ' *I II I*

Integration is along a contour *C* on closed surface *S*. *GI* (*r, z; r*

field *As* and a source-less scattered field *AR* (Yildir et al, 1992):

clearly identical and differ only in media dependent functions *Bn* (

Fig. 3. Surface Current Distribution *J* on Core Surface *S*.

To the following PDE for the component *AIφ*:

Closed surface, *S*

2

*A*

with the solution for *JI* = *JIφ*(*r', z'*)*a<sup>φ</sup>*

*S,* leads to the following:

*GR* (*r*, *z*, *r*

follows (Balanis, 1989):

*r r*

 where *ai* represents a basis function and *ui* the basis function coefficient. Substituting equation (19) into (18) gives the following:

$$A\_{R\phi}(r,z) = \mu \sum\_{l=1}^{N} \mu\_l \int\_C G\_R(r,z;r',z')a\_l(r',z')ds'\tag{20}$$

Basis functions were now chosen to accurately represent the anticipated unknown function *JR<sup>φ</sup>*. A piecewise constant sub-domain function was chosen to do this, which is of the form shown below:

$$a\_{l}(r',z') = \begin{cases} 1 & z' \in (z'\_{l-1} + \Delta\_{z}, z'\_{l+1} - \Delta\_{z}), \quad z'\_{l+1} > z'\_{l-1} \\\\ 1 & r' \in (r'\_{l-1} + \Delta\_{r}, r'\_{l+1} - \Delta\_{r}), \quad r'\_{l+1} \neq r'\_{l-1} \\\\ 0 & \text{otherwise} \end{cases} \tag{21}$$
 
$$\Delta\_{r} = \left| r'\_{l+1} - r'\_{l-1} \right| / 4 \text{ and } \Delta\_{z} = \left| z'\_{l+1} - z'\_{l-1} \right| / 4 \text{ .}$$

Sub-domains were divided into *N* sub-intervals and evenly distributed along the sensor core-air interface. Observation points (*r, z*) were located at the centre of sub-domains (see figure 4) for greatest computational accuracy (Balanis, 1989).

Fig. 4. Discretization of Current *JR* on the Sensor Core Interface.

#### **2.3 The total field and internal core field**

When combined the source and scattered field gives the total vector potential *AEφ*. For the *nth* media layer outside the sensor core:

Numerical Modelling and Design of an Eddy Current Sensor 167

If *AI* is the vector potential inside the core and *AE* outside, then the boundary conditions for

*<sup>E</sup>* = 1 for free space gives:

*<sup>E</sup>* ) *AE <sup>n</sup>* (26)

*AI AE* (27)

*<sup>I</sup>* ) *AI AR*) *n* (28)

1,..., . (32)

. (33)

(30)

(34)

*<sup>I</sup>* ) *AI n* (1/

Evaluating equation (28) gives the following for the core upper and lower flat faces:

*AS AI AR* . (29)

*A r z z A r z A r z z <sup>s</sup> <sup>I</sup> <sup>I</sup> <sup>R</sup>* ( , ) ( ( , ) ( , ))

(1/*r* / *r*)*A* (*r*,*z*) (1 *r* / *r*)(*A* (*r*,*z*) *A* (*r*,*z*)) *<sup>s</sup>*

The two sensor core boundary equations define the relationship between unknown basis function coefficients *ui* and *vi*. The method of weighted residuals was used to solve these equations, which proceeds by grouping *ui* and *vi* together into a single 2*N*1 column matrix

> *<sup>v</sup> <sup>p</sup> <sup>N</sup> <sup>N</sup> u p N*

*p*:

( ( ; ') ( ') ' ( ; ') ( ') ') 1,...,

*ρ ρ ρ ρ ρ ρ a n*

( )

such that:

*G b ds G a ds p N N*

( ; ') ( ') ' ( ; ') ( ') ' 1,...,2

*G b ds G a ds p N*

*<sup>p</sup>* 1,...2

> 

(31)

*<sup>I</sup> <sup>R</sup>*

*I*

**2.5 Evaluation of expansion coefficients using the method of weighted residuals** 

 

*<sup>p</sup> <sup>n</sup> <sup>I</sup> <sup>p</sup> <sup>R</sup>*

*q q*

*ρ ρ ρ ρ ρ ρ*

*q q φ*

*<sup>p</sup> <sup>n</sup> <sup>I</sup> <sup>I</sup> <sup>p</sup> <sup>R</sup>*

*p p*

> ...

*k*

` *k k <sup>N</sup> <sup>N</sup>* <sup>1</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> *ΨK*

*c c*

*c c*

1 ( )

*AS n* ((1/

the core-air interface can be shown to be:

Substituting *AE AR AS* and assuming

and for the core's central cylindrical face:

*K* with the following elements:

 

 

*p*

Given *K* above, define a new 12*N* row matrix

The following can be seen to apply for element

and

and

(1/ 

$$\begin{aligned} A\_{E\phi}(r,z) &= \mu\_n \sum\_{l=1}^N \mu\_l \int\_C G\_R^{(n)}(r,z;r',z') a\_l(r',z') ds' \\ &+ \mu\_n \iint G\_s^{(n)}(r,z;r\_0,z\_0) J\_{s\phi}(r\_0,z\_0) dr\_0 dz\_0 \end{aligned} \tag{22}$$

where *Js<sup>φ</sup>* is the source current distribution:

$$J\_{s\varphi}(\eta\_0, z\_0) = N\_c \left/ (l\_2 - l\_1)(\nu\_2 - \eta\_1) \right. \tag{23}$$

(*l*2, *l*1) and (*r*2, *r*1) are the length and radial dimensions of the source coil, which is assumed to be rectangular in cross section. Since *AI<sup>φ</sup>* is solved in exactly the same way as scattered field *ARφ* expand *JIφ* into a similar *N* term series, letting *bi* represent the series basis function and *vi* the expansion coefficient. Given this *bi* is defined as follows:

$$b\_l(r', z') = \begin{cases} 1 & z' \in (z'\_{l-1} + \Delta\_z, z'\_{l+1} - \Delta\_z), \quad z'\_{l+1} > z'\_{l-1} \\\\ 1 & r' \in (r'\_{l-1} + \Delta\_r, r'\_{l+1} - \Delta\_r), \quad r'\_{l+1} \neq r'\_{l-1} \\\\ 0 & otherwise \end{cases} \tag{24}$$

Substitution of equation (24) into equation (16) gives:

$$A\_{I\phi}(r,z) = \mu\_I \sum\_{l=1}^{N} v\_l \int\_C G\_I(r,z;r',z')b\_l(r',z')ds'\tag{25}$$

#### **2.4 Sensor core boundary conditions**

Unknown expansion coefficients *ui* and *vi* are determined by applying the sensor core boundary conditions. Since the core is rod shaped, two surfaces exist where boundary conditions must be met. These surfaces are shown below in figure 5 with appropriate unit normal vectors *n*:

Fig. 5. Core Boundary Unit Normal Vectors *n.*

If *AI* is the vector potential inside the core and *AE* outside, then the boundary conditions for the core-air interface can be shown to be:

$$(\mathbf{l} \mid \boldsymbol{\mu}\_I) \nabla \times \mathbf{A}\_I \times \mathbf{n} = (\mathbf{l} \mid \boldsymbol{\mu}\_E) \nabla \times \mathbf{A}\_E \times \mathbf{n} \tag{26}$$

and

(22)

. (24)

(25)

166 Numerical Modelling

*<sup>i</sup> <sup>n</sup> <sup>E</sup> <sup>n</sup> <sup>i</sup> <sup>R</sup> <sup>A</sup> <sup>r</sup> <sup>z</sup> <sup>u</sup> <sup>G</sup> <sup>r</sup> <sup>z</sup> <sup>r</sup> <sup>z</sup> <sup>a</sup> <sup>r</sup> <sup>z</sup> ds*

 <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> ( ) *<sup>G</sup>* (*r*,*z*;*<sup>r</sup>* ,*<sup>z</sup>* )*<sup>J</sup>* (*<sup>r</sup>* ,*<sup>z</sup>* )*dr dz <sup>s</sup> <sup>n</sup>*

( , ) /( )( ) <sup>0</sup> <sup>0</sup> <sup>2</sup> <sup>1</sup> <sup>2</sup> <sup>1</sup> *J r z IN l l r r <sup>s</sup>*

(*l*2, *l*1) and (*r*2, *r*1) are the length and radial dimensions of the source coil, which is assumed to be rectangular in cross section. Since *AI<sup>φ</sup>* is solved in exactly the same way as scattered field *ARφ* expand *JIφ* into a similar *N* term series, letting *bi* represent the series basis function and *vi*

*r r r r r*

( , ; ', ') ( ', ') '

1 ' ( ' , ' ), ' '

*otherwise*

*<sup>I</sup> <sup>I</sup> <sup>i</sup> <sup>I</sup> <sup>i</sup> A r z v G r z r z b r z ds*

Unknown expansion coefficients *ui* and *vi* are determined by applying the sensor core boundary conditions. Since the core is rod shaped, two surfaces exist where boundary conditions must be met. These surfaces are shown below in figure 5 with appropriate unit

*n*

*AI*  = *I*

( ', ') <sup>1</sup> <sup>1</sup> <sup>1</sup> <sup>1</sup>

*b r z <sup>i</sup> <sup>r</sup> <sup>i</sup> <sup>r</sup> <sup>i</sup> <sup>i</sup>*

*i C*

1

*N*

1 ' ( ' , ' ), ' '

*z z z z z*

1 1 1 1

*i z i z i i*

( , ; ', ') ( ', ') '

*<sup>c</sup>* (23)

.

*n*

*AE*  = *E*

*i C*

1 ( ) ( , ) 

*N*

where *Js<sup>φ</sup>* is the source current distribution:

*i*

**2.4 Sensor core boundary conditions** 

Fig. 5. Core Boundary Unit Normal Vectors *n.*

*s* 

normal vectors *n*:

*n s*

the expansion coefficient. Given this *bi* is defined as follows:

 

0

 

( , )

Substitution of equation (24) into equation (16) gives:

$$A\_I = A\_E \tag{27}$$

Substituting *AE AR AS* and assuming *<sup>E</sup>* = 1 for free space gives:

$$\nabla \times \mathbf{A}\_{\mathbf{S}} \times \mathfrak{n} = ((1/\mu\_I)\nabla \times \mathbf{A}\_I - \nabla \times \mathbf{A}\_{\mathbf{R}}) \times \mathfrak{n} \tag{28}$$

and

$$A\_{\mathcal{S}} = A\_I - A\_{\mathcal{R}}.\tag{29}$$

Evaluating equation (28) gives the following for the core upper and lower flat faces:

$$
\left\langle \hat{\alpha} A\_{s\wp}(r, z) \right\rangle \hat{\otimes} z = \left\langle \hat{\alpha} \left( A\_{I\wp}(r, z) \right) \right\rangle \mu\_I - A\_{R\wp}(r, z) \rangle \left\langle \hat{\otimes} z \right\rangle \tag{30}
$$

and for the core's central cylindrical face:

$$(\mathbf{l}/r + \hat{\boldsymbol{\sigma}}/\hat{\boldsymbol{\sigma}}r)A\_{\mathbf{s}\boldsymbol{\varphi}}(\mathbf{r},z) = (\mathbf{l}/r + \hat{\boldsymbol{\sigma}}/\hat{\boldsymbol{\sigma}}r)(A\_{I\boldsymbol{\varphi}}(\mathbf{r},z)/\mu\_I - A\_{R\boldsymbol{\varphi}}(\mathbf{r},z))\tag{31}$$

#### **2.5 Evaluation of expansion coefficients using the method of weighted residuals**

The two sensor core boundary equations define the relationship between unknown basis function coefficients *ui* and *vi*. The method of weighted residuals was used to solve these equations, which proceeds by grouping *ui* and *vi* together into a single 2*N*1 column matrix *K* with the following elements:

$$k\_p = \begin{cases} \mu\_p & p = 1, \dots, N \\ \nu\_p & p = N + 1, \dots 2N \end{cases} \tag{32}$$

Given *K* above, define a new 12*N* row matrix such that:

$$\mathbf{^\frown}\mathbf{\bar{PK}} = \left(k\_{\mathbf{l}}\boldsymbol{\nu}\_{\mathbf{l}} + \ldots + k\_{\mathbf{2N}}\boldsymbol{\nu}\_{\mathbf{2N}}\right). \tag{33}$$

The following can be seen to apply for element *p*:

$$\nu\_{p} = \begin{cases} \nabla \times (\mu\_{I}^{-1} \int\_{c} G\_{I}(\rho\_{\mathbf{q}} \cdot \rho^{\star}) b\_{p}(\rho^{\star}) ds^{\star} - \int\_{c} G\_{R}^{(n)}(\rho\_{\mathbf{q}} \cdot \rho^{\star}) a\_{p}(\rho^{\star}) ds^{\star}) a\_{\mathbf{q}} \times \mathbf{n} & p = 1, \dots, N \\\\ \int\_{c} G\_{I}(\rho\_{\mathbf{q}} \cdot \rho^{\star}) b\_{p}(\rho^{\star}) ds^{\star} - \int\_{c} G\_{R}^{(n)}(\rho\_{\mathbf{q}} \cdot \rho^{\star}) a\_{p}(\rho^{\star}) ds^{\star} & p = N + 1, \dots, 2N \end{cases} \tag{34}$$

Numerical Modelling and Design of an Eddy Current Sensor 169

1 2

*N N N N N N N N N N*

1 1 1 2 1 1

*r z r z*

1 1 1 2 1 1

*r z r z r z r z*

*N N N N N*

( , ) .. .. .. .. ( , ) .. .. .. .. .. .. .. .. ( , ) .. .. .. .. ( , ) ( , ) .. .. .. .. ( , ) .. .. .. .. .. .. .. .. ( , ) .. .. .. .. ( , )

*r z r z*

1 2 2 2 2 2

In previous section, a set of partial differential equations were developed to model the vector potential fields present in the regions bounding a ferrite-cored eddy current sensor; sensor regions were considered to be source-less with imaginary surface currents imposed at region interfaces. Green's functions were determined for all bounded regions. A novel set of Basis functions were introduced to reproduce the surface currents present on the sensor core-air interface, which were then formed into a system of 2*N* linearly independent equations. A matrix method was finally developed to solve these equations using the

The matrix method was further developed in this section in order to calculate sensor coil

the interaction between the sensor and test material was also developed and verified. A

Section 2.1 introduced the medium as being comprised of M planar layers. Each layer had medium properties that were considered to be isotropic, homogeneous and linear. Any change between the electrical and magnetic properties of the layers was a step change. This approach enabled Cheng and co-workers to successfully model non-homogeneous materials using piecewise constant approximations (Cheng et al, 1971). Such non-linear material profiles might be produced, as an example, by coating a substrate, by case hardening, heat treatment, ion bombardment, or by chemical processing. A recently developed alternative method used hyperbolic tangential profiles to represent near surface changes in conductivity (Uzal et al, 1993). The method proposed by Cheng was adopted here due to it being more flexible. Applying boundary equations (26) and (27) to Green's function (12) for

potential fields generated is reviewed and a new free-space Green's function introduced.

[(1 ) (1 ) ] <sup>2</sup>

<sup>1</sup> 1, 1, <sup>1</sup> *<sup>n</sup> <sup>z</sup> <sup>n</sup> <sup>n</sup> <sup>n</sup> <sup>z</sup> <sup>n</sup> <sup>n</sup> <sup>z</sup> Bn <sup>e</sup> <sup>n</sup> <sup>e</sup> <sup>n</sup> <sup>n</sup> <sup>B</sup> <sup>e</sup> <sup>n</sup> nC*

 impedance and induced voltage. An efficient material profile function *m*(

 

 

 

novel form of parameterisation is adopted for *m*(

the *nth* media layer below the coil gives:

1

**3.1 The material profile function** *m***() for stratified layers** 

*N N N*

( , ) .. .. .. ( , ) ( , ) .. .. .. ( , )

**3. Numerical model implementation** 

*f r z*

method of weighted residuals.

2 2 2

*N N N N*

1 1 1 1

*f r z*

1 1 1

*f r z f r z*

  

 

 

). The accuracy and convergence of vector

(41)

) for modelling

*v*

2

.. .. .. *u*

1

*N*

*N*

*N N*

.. .. ..

1

(40)

*v u*

where q is a field point at (*rq*, *zq*), is a source point at (*r*, *z*) in sub-domain *p* and *n* is a unit normal vector at field point (*rq*, *zq*) on the core boundary surface. Since 2*N* unknowns in *K* require the formation of 2*N* linearly independent equations, define a 2*N*1 column matrix *F* for the source field at *N* points (*rq* ,*zq* ); *q* 1,..., *N* on *C*, with elements:

$$f\_q = \begin{cases} \nabla \times \left( \iint G\_s^{(n)}(r\_q, z\_q; r\_0, z\_0) J\_{s\phi}(r\_0, z\_0) d\eta\_0 dz\_0 \right) \mathbf{a}\_\Phi \times \mathbf{n} & q = \mathbf{l}, \dots, N \\\\ \iint G\_s^{(n)}(r\_q, z\_q; r\_0, z\_0) J\_{s\phi}(r\_0, z\_0) d\eta\_0 dz\_0 \end{cases} \tag{35}$$

In order to form 2*N* linearly independent equations, introduce a further 12*N* row matrix *W* and take the inner product *W Ψ <sup>T</sup>* , . Integration is along the entire length of the core-air interface (curve *C*) to minimise any residual error, giving:

$$\left\lfloor \int\_{c} W^{T} \Psi ds \right\rfloor \cdot \mathbf{K} = \int\_{c} W^{T} F ds \tag{36}$$

Weight vector *W* is selected according to one of the following methods (Sadiku, 1992):


Point collocation was selected because it provided acceptable accuracy for computational effort (Balanis, 1989). Point collocation uses the following weight vector *W*:

$$W = \left[\delta(c - c\_1), \dots, \delta(c - c\_{2N})\right] \tag{37}$$

Collocation points on *C* are chosen to coincide with basis function observation points, where the following applies for weight element *wq*:

$$\mathcal{S}(\mathbf{c} - \mathbf{c}\_q) = \begin{cases} \infty & \text{if} \quad (\mathbf{c} = \mathbf{c}\_q) \\ 0 & \text{otherwise} \end{cases} \tag{38}$$

Recognising that *<sup>i</sup> i x <sup>x</sup> <sup>i</sup>* (*x x* )*dx* 1 and inserting this into row *q* of equation (36), gives:

$$\begin{aligned} \prescript{C\_q^+}{}{\delta}\delta(\mathbf{c} - \mathbf{c}\_q)\mathbf{d}s &= \nu\_1 \mu\_1 \int \delta(\mathbf{c} - \mathbf{c}\_q)d\mathbf{s} + \cdots + \nu\_{2N} \nu\_{2N} \int \delta(\mathbf{c} - \mathbf{c}\_q)d\mathbf{s} \ . \end{aligned} \tag{39}$$

Evaluating equation (39) for all *N* collocation points leads to the matrix equation for *ui* and *vi*:

unit normal vector at field point (*rq*, *zq*) on the core boundary surface. Since 2*N* unknowns in *K* require the formation of 2*N* linearly independent equations, define a 2*N*1 column matrix

0 0 0 0 0 0

0 0 0 0 0 0

In order to form 2*N* linearly independent equations, introduce a further 12*N* row matrix *W* and take the inner product *W Ψ <sup>T</sup>* , . Integration is along the entire length of the core-air

*W Ψds K W Fds <sup>T</sup> <sup>T</sup>*

Point collocation was selected because it provided acceptable accuracy for computational

( ),..., ( ) *W*

Collocation points on *C* are chosen to coincide with basis function observation points, where

 *otherwise if <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>c</sup> <sup>q</sup> <sup>q</sup>* 0

*q*

Evaluating equation (39) for all *N* collocation points leads to the matrix equation for *ui* and *vi*:

( ) ( ) ( )

( )

(*x x* )*dx* 1 and inserting this into row *q* of equation (36), gives:

2 2

( ) (38)

*q*

*C*

*q*

. (39)

*C N N q*

 

*c c*

Weight vector *W* is selected according to one of the following methods (Sadiku, 1992):

*G r z r z J (r z )dr dz q N N*

( , ; , ) , ) 1,...,2

.

*G r z r z J (r z )dr dz q N*

*a<sup>φ</sup> n*

( ( , ; , ) , ) 1,...,

is a source point at (*r*, *z*) in sub-domain *p* and *n* is a

*c c*2*<sup>N</sup>* , (37)

. (35)

(36)

*F* for the source field at *N* points (*rq* ,*zq* ); *q* 1,..., *N* on *C*, with elements:

*<sup>q</sup> <sup>q</sup> <sup>s</sup> <sup>n</sup>*

 

effort (Balanis, 1989). Point collocation uses the following weight vector *W*:

*c c*<sup>1</sup>

*f c c ds u c c ds v c c ds*

1 1 

*C*

*C*

*q*

*<sup>q</sup> <sup>q</sup>*

*q*

*<sup>q</sup> <sup>q</sup> <sup>s</sup> <sup>n</sup>*

where 

q is a field point at (*rq*, *zq*),

 

the following applies for weight element *wq*:

*x*

*i*

*q*

*C*

*q*

*C*

*<sup>i</sup>*

*<sup>x</sup> <sup>i</sup>* 

*s*

( )

interface (curve *C*) to minimise any residual error, giving:

*s*

( )

 

*f*

 Point collocation. Sub-domain collocation.

 Least square. Galerkin.

Recognising that

*q*


#### **3. Numerical model implementation**

In previous section, a set of partial differential equations were developed to model the vector potential fields present in the regions bounding a ferrite-cored eddy current sensor; sensor regions were considered to be source-less with imaginary surface currents imposed at region interfaces. Green's functions were determined for all bounded regions. A novel set of Basis functions were introduced to reproduce the surface currents present on the sensor core-air interface, which were then formed into a system of 2*N* linearly independent equations. A matrix method was finally developed to solve these equations using the method of weighted residuals.

The matrix method was further developed in this section in order to calculate sensor coil impedance and induced voltage. An efficient material profile function *m*() for modelling the interaction between the sensor and test material was also developed and verified. A novel form of parameterisation is adopted for *m*(). The accuracy and convergence of vector potential fields generated is reviewed and a new free-space Green's function introduced.

#### **3.1 The material profile function** *m***() for stratified layers**

Section 2.1 introduced the medium as being comprised of M planar layers. Each layer had medium properties that were considered to be isotropic, homogeneous and linear. Any change between the electrical and magnetic properties of the layers was a step change. This approach enabled Cheng and co-workers to successfully model non-homogeneous materials using piecewise constant approximations (Cheng et al, 1971). Such non-linear material profiles might be produced, as an example, by coating a substrate, by case hardening, heat treatment, ion bombardment, or by chemical processing. A recently developed alternative method used hyperbolic tangential profiles to represent near surface changes in conductivity (Uzal et al, 1993). The method proposed by Cheng was adopted here due to it being more flexible. Applying boundary equations (26) and (27) to Green's function (12) for the *nth* media layer below the coil gives:

$$B\_{n-1} = \frac{1}{2} e^{\alpha\_{-1}z\_n} \left[ (\mathbf{l} + \beta\_{n-1,n}) e^{-\alpha\_n z\_n} B\_n + (\mathbf{l} - \beta\_{n-1,n}) e^{\alpha\_n z\_n} C\_n \right] \tag{41}$$

where

Region 1

Region 2

where

If *V* (*n*, *M* ) is a 2x2 matrix, then define:

(1)

(2)

Numerical Modelling and Design of an Eddy Current Sensor 171

*V* (*n*,*M* ) *T2,3 T3,4* ...*TM 2,M <sup>1</sup> TM 1,M* (54)

Evaluating the above gives the Green's functions for the regions bounding the coil:

*<sup>v</sup> <sup>M</sup> <sup>G</sup> <sup>r</sup> <sup>z</sup> <sup>r</sup> <sup>z</sup> <sup>r</sup> <sup>J</sup> <sup>r</sup> <sup>J</sup> <sup>r</sup> <sup>z</sup> <sup>z</sup> <sup>z</sup> <sup>S</sup>* <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup>

(2, ) ( , , , ) ( ) ( )

 

A new coil region (region 12) can also be defined for coils with finite length and radial dimensions by adding vector potential *A*(1) and *A*(2) and applying relevant boundary

*<sup>v</sup> <sup>M</sup> <sup>G</sup> <sup>r</sup> <sup>z</sup> <sup>r</sup> <sup>z</sup> <sup>r</sup> <sup>J</sup> <sup>r</sup> <sup>J</sup> <sup>r</sup> <sup>z</sup> <sup>z</sup> <sup>z</sup> <sup>S</sup>* <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup>

(2, ) ( , , , ) ( ) ( )

  12

12

(2, ) (2, )

22 12 *v M*

Material profile function *m* is dependent only on the media properties and is independent of coil geometry and coil lift-off; *m* not only applies to the source field *AS*, but also to the scattered field *AR*, which makes this function a universal profile function. If the material under test is comprised of two layers for simplicity, a conductive coating of thickness *Zc* (region 3) deposited on a magnetic substrate (region 4), then the material profile function *m*

3 3 4 4 3 3 4 4

 

 

> <sup>4</sup>

 

2

3 3

 

3 3 4 4 3 3 4 4 ( )( / ) ( )( / ) ( )( / ) ( )( / ) ( )

 

(2, )

(2, )

0 0 0 1 0 1

conditions on coil length *Z* (Dodd & Deeds, 1968).

can be shown to be (Dodd & Deeds, 1968):

 

<sup>3</sup> *j* 

 

From the above, the following relationship is evident:

0 0 0 1 0 1

<sup>0</sup> <sup>22</sup>

<sup>0</sup> <sup>22</sup>

 *CM* 0

*AM* . (53)

*<sup>v</sup> <sup>M</sup> <sup>m</sup>* (57)

 

 

(56)

*<sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>d</sup> <sup>v</sup> <sup>M</sup>*

(55)

*<sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>d</sup> <sup>v</sup> <sup>M</sup>*

3 3

2

*e*

 

 

*<sup>e</sup> <sup>m</sup>* (58)

 and <sup>4</sup> <sup>4</sup> 2

 *j* .

2

*c c Z Z*

and

$$C\_{n-1} = \frac{1}{2} e^{\alpha\_{n-1}z\_n} [ (\mathbf{l} - \beta\_{n-1,n}) e^{-\alpha\_n z\_n} B\_n + (\mathbf{l} + \beta\_{n-1,n}) e^{\alpha\_n z\_n} C\_n ] \tag{42}$$

where:

$$
\beta\_{n-1,n} = \beta\_n / \beta\_{n-1} \tag{43}
$$

and

$$
\beta\_n = \alpha\_n / \mu\_n \,. \tag{44}
$$

Taking the coil region (regions 1 and 2) to have the properties 0, 0 and 0, and giving consideration to the presence of a source give:

$$-B\_2 e^{-\alpha\_0 z\_0} + C\_2 e^{\alpha\_0 z\_0} = -B\_1 e^{-\alpha\_0 z\_0} + C\_1 e^{\alpha\_0 z\_0} + \frac{\alpha}{\beta\_0} r\_0 J\_1(\alpha r\_0) \,. \tag{45}$$

and

$$-B\_2 e^{-\alpha\_0 z\_0} + C\_2 e^{\alpha\_0 z\_0} = -B\_1 e^{-\alpha\_0 z\_0} + C\_1 e^{\alpha\_0 z\_0} \tag{46}$$

Cheng and co workers showed that unknown coefficients *Bn* and *Cn* were dramatically simplified using a matrix method (Cheng et al, 1971). This method defined the following matrices, a 2x1 coefficient column matrix:

$$\mathcal{A}\_n = \begin{bmatrix} B\_n \\ C\_n \end{bmatrix} \tag{47}$$

and a 2x2 transformation matrix *Tn-1,n*, with the elements:

$$((T\_{n-I,n})\_{11} = \frac{1}{2}(1+\beta\_{n-1,n})e^{(a\_{n-l}-a\_n)z\_n} \tag{48}$$

$$(T\_{n-I,n})\_{12} = \frac{1}{2}(1 - \beta\_{n-1,n})e^{(\alpha\_{n-1} + \alpha\_n)z\_n} \tag{49}$$

$$((T\_{n-I,n})\_{21} = \frac{1}{2}(1 - \beta\_{n-1,n})e^{-(a\_{n-1} + a\_n)z\_n} \tag{50}$$

$$(T\_{n-I,n})\_{22} = \frac{1}{2}(1+\beta\_{n-1,n})e^{-(a\_{n-1}-a\_n)z\_n} \tag{51}$$

Successive multiplication of *Tn-1,n* gives the following:

$$A\_2 = \begin{bmatrix} B\_2 \\ C\_2 \end{bmatrix} = T\_{2,3} \cdot T\_{3,4} \dots T\_{M-2,M-1} \cdot T\_{M-1,M} \cdot A\_M \tag{52}$$

where

170 Numerical Modelling

<sup>1</sup> 1, 1, <sup>1</sup> *<sup>n</sup> <sup>z</sup> <sup>n</sup> <sup>n</sup> <sup>n</sup> <sup>z</sup> <sup>n</sup> <sup>n</sup> <sup>z</sup> Cn <sup>e</sup> <sup>n</sup> <sup>n</sup> <sup>e</sup> <sup>n</sup> <sup>n</sup> <sup>B</sup> <sup>e</sup> <sup>n</sup> nC*

 *n* 

Taking the coil region (regions 1 and 2) to have the properties 0, 0 and 0, and giving

<sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>1</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> *<sup>B</sup> <sup>e</sup> <sup>C</sup> <sup>e</sup> <sup>B</sup> <sup>e</sup> <sup>C</sup> <sup>e</sup> <sup>r</sup> <sup>J</sup> <sup>r</sup> <sup>z</sup> <sup>z</sup> <sup>z</sup> <sup>z</sup>*

<sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>0</sup> <sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>1</sup> *<sup>z</sup> <sup>z</sup> <sup>z</sup> <sup>z</sup> <sup>B</sup> <sup>e</sup> <sup>C</sup> <sup>e</sup> <sup>B</sup> <sup>e</sup> <sup>C</sup> <sup>e</sup>*

Cheng and co workers showed that unknown coefficients *Bn* and *Cn* were dramatically simplified using a matrix method (Cheng et al, 1971). This method defined the following

> 

> > *<sup>n</sup> <sup>n</sup> nz <sup>n</sup> <sup>n</sup> <sup>e</sup>* ( )

*<sup>n</sup> <sup>n</sup> nz <sup>n</sup> <sup>n</sup> <sup>e</sup>* ( )

*<sup>n</sup> <sup>n</sup> nz <sup>n</sup> <sup>n</sup> <sup>e</sup>* ( )

*<sup>n</sup> <sup>n</sup> nz <sup>n</sup> <sup>n</sup> <sup>e</sup>* ( )

 *Tn 1,n* (51)

*<sup>B</sup>* (52)

*Tn 1,n* (50)

*Tn 1,n* (48)

*Tn 1,n* (49)

*n <sup>n</sup> <sup>n</sup> <sup>C</sup> B*

<sup>11</sup> 1, <sup>1</sup> (1 ) <sup>2</sup>

<sup>12</sup> 1, <sup>1</sup> (1 ) <sup>2</sup>

<sup>21</sup> 1, <sup>1</sup> (1 ) <sup>2</sup>

<sup>22</sup> 1, <sup>1</sup> (1 ) <sup>2</sup>

...

*<sup>n</sup>*1 (43)

*<sup>n</sup>* . (44)

( ) <sup>0</sup> <sup>1</sup> <sup>0</sup>

(46)

0

*A* (47)

. (45)

(42)

[(1 ) (1 ) ] <sup>2</sup>

*<sup>n</sup>*1,*<sup>n</sup>*

> *<sup>n</sup> n*

and a 2x2 transformation matrix *Tn-1,n*, with the elements:

<sup>1</sup> ( )

<sup>1</sup> ( )

<sup>1</sup> ( )

<sup>1</sup> ( )

 *A2 T2,3 T3,4 TM 2,M <sup>1</sup> TM 1,M AM* 

2 2 *C*

Successive multiplication of *Tn-1,n* gives the following:

1

consideration to the presence of a source give:

matrices, a 2x1 coefficient column matrix:

 and

where:

and

and

$$\mathcal{A}\_{\mathcal{M}} = \begin{bmatrix} 0 \\ C\_{\mathcal{M}} \end{bmatrix}. \tag{53}$$

If *V* (*n*, *M* ) is a 2x2 matrix, then define:

$$V(n,M) = T\_{2,3} \cdot T\_{3,4} \dots T\_{M-2,M-1} \cdot T\_{M-1,M} \tag{54}$$

Evaluating the above gives the Green's functions for the regions bounding the coil: Region 1

$$G\_S^{(1)}(r, z, \eta\_0, z\_0) = \int\_0^{\eta\_0} a\eta\_0 J\_1(ar\_0) J\_1(ar) \left(\frac{\upsilon\_{12}(2, M)}{\upsilon\_{22}(2, M)} \cdot e^{-a\_0 z\_0} + e^{a\_0 z\_0}\right) \cdot e^{-a\_0 z} da \tag{55}$$

Region 2

$$G\_S^{(2)}(r, z, v\_0, z\_0) = \bigcap\_{0}^{\alpha} ar\_0 J\_1(ar\_0) J\_1(ar) \left(\frac{v\_{12}(2, M)}{v\_{22}(2, M)} \cdot e^{-a\_0 z} + e^{a\_0 z}\right) \cdot e^{-a\_0 z\_0} da \tag{56}$$

A new coil region (region 12) can also be defined for coils with finite length and radial dimensions by adding vector potential *A*(1) and *A*(2) and applying relevant boundary conditions on coil length *Z* (Dodd & Deeds, 1968).

From the above, the following relationship is evident:

$$m = \frac{\nu\_{12}(2, M)}{\nu\_{22}(2, M)}\tag{57}$$

Material profile function *m* is dependent only on the media properties and is independent of coil geometry and coil lift-off; *m* not only applies to the source field *AS*, but also to the scattered field *AR*, which makes this function a universal profile function. If the material under test is comprised of two layers for simplicity, a conductive coating of thickness *Zc* (region 3) deposited on a magnetic substrate (region 4), then the material profile function *m* can be shown to be (Dodd & Deeds, 1968):

$$m(\alpha) = \frac{(\alpha + \alpha\_3)(\alpha\_3 - \alpha\_4 \mid \mu\_4) + (\alpha - \alpha\_3)(\alpha\_3 + \alpha\_4 \mid \mu\_4) \cdot e^{2 \cdot Z\_\varepsilon \cdot \alpha\_3}}{(\alpha - \alpha\_3)(\alpha\_3 - \alpha\_4 \mid \mu\_4) + (\alpha + \alpha\_3)(\alpha\_3 + \alpha\_4 \mid \mu\_4) \cdot e^{2 \cdot Z\_\varepsilon \cdot \alpha\_3}}\tag{58}$$

where

$$\alpha\_3 = \sqrt{\alpha^2 + j o \mu \mu\_3 \sigma\_3} \quad \text{and} \quad \alpha\_4 = \sqrt{\alpha^2 + j o \mu \mu\_4 \sigma\_4} \quad .$$

Numerical Modelling and Design of an Eddy Current Sensor 173

**0.5**

Fig. 6a. Real Component of Material Profile Function *m*().

**0 .7**

**0 .6**

**0 .5**

**0 .4**

**0 .3**

**0 .2**

**0 .1**

**0**

**0**

**0.5**

Re*m* Re *m* Re *m* Re *m*

I m *<sup>m</sup>* ( )) I m *<sup>m</sup>* ( )) I m *<sup>m</sup>* ( )) I m *<sup>m</sup>* ( )) **1**

**<sup>0</sup> 2000 4000 6000 <sup>8000</sup> <sup>1</sup> <sup>104</sup> <sup>1</sup>**

**<sup>0</sup> <sup>2000</sup> 4 000 <sup>6000</sup> <sup>8000</sup> <sup>1</sup> 1 0<sup>4</sup> 0 .8**

3 0 0 m icro n C u co a tin g o n Ste e l @ 3 0 kH z 3 0 0 m icro n C u co a tin g o n Ste e l @ 1 0 0H z

U n co ated S te el @ 30 kH z U n co ated S te el @ 100Hz

Fig. 6b. Imaginary Component of Material Profile Function *m*().

300 micron Cu coating on Steel @ 30kHz 300 micron Cu coating on Steel @ 100Hz

Uncoated Steel @ 30kHz Uncoated Steel @ 100Hz

#### **3.2 Implementation of the material profile function**

The material profile function *m* is actually a function of many variables. Most of these variables however can be regarded as constant for a given test, making the material profile function a function of spatial frequency α only. For a large number of medium layers, at least 40 to accurately represent continuously varying profiles (Uzal et al, 1993), the evaluation of *V*12 and *V*22 begins to become computationally prohibitive. Not only does the amount of matrix algebra required to calculate *m*() dramatically increase, but this calculation must be repeated for every element of matrix equation (40). A more efficient approach replaces *m*() in its matrix form with a spline curve. The oscillatory nature of high degree polynomial approximations, such as least squares regression, discounts their use. In order to assess the suitability of cubic spline interpolation it is necessary to determine the general form and amount of variation expected for *m*(). Given this, consider a two layer medium defined by equation (58), where angular frequency ranges from = 2100 rads/sec to = 23104 rads/sec and coating thickness *Zc* (region 3) ranges from *Zc* = 0 m to 300 m. A worse case of copper plating on steel is assumed. The following two graphs show real and imaginary components of *m*() for these conditions.

Examination of figure 6a and 6b clearly shows that *m*() has considerable variation below = 104, but that above this it is relatively smooth. Assuming that *m*() is defined on the interval {*a*, *b*} and that a clamped boundary is used, let cubic polynomial *S*j occur on subinterval [j, j+1]. Given this it can be shown that the maximum error occurs when: (Burden & Faires, 1989):

$$\max\_{a \le a \le b} \left| m(a) - S(a) \right| \le \frac{\\$ \cdot M\_a}{\\$ \\$4} \quad \begin{array}{l} \max \\ 0 \le j \le n - 1 \end{array} (a\_{j+1} - a\_j)^4 \tag{59}$$

where

$$M\_{\alpha} \ge \max\_{a \le \alpha \le b} \left| m^4(\alpha) \right| \text{ and } a = a\_0 \langle \alpha\_1 \rangle ... \langle \alpha\_n = b \dots$$

From above, interpolation error can be linked with the maximum subinterval step size max[j, j+1] and the maximum 4th derivative of *m*(). Since the maximum derivative error is always below = 2103 and since *m*() is almost linear above = 104, it seems reasonable to reduce subinterval step size for low and increase it above = 104. This adjustment enables the interpolating cubic polynomials *Sj* to more accurately reproduce data in regions of maximum variation, whilst minimising the total number of subinterval domains. Empirical study showed that the optimum choice for j is:

$$\alpha\_j = 0.035 \cdot ((j+1)^3 + j^3) \tag{60}$$

where

$$j \in \{0, 1, \dots, 111\}.$$

The material profile function *m* is actually a function of many variables. Most of these variables however can be regarded as constant for a given test, making the material profile function a function of spatial frequency α only. For a large number of medium layers, at least 40 to accurately represent continuously varying profiles (Uzal et al, 1993), the evaluation of *V*12 and *V*22 begins to become computationally prohibitive. Not only does the amount of matrix algebra required to calculate *m*() dramatically increase, but this calculation must be repeated for every element of matrix equation (40). A more efficient approach replaces *m*() in its matrix form with a spline curve. The oscillatory nature of high degree polynomial approximations, such as least squares regression, discounts their use. In order to assess the suitability of cubic spline interpolation it is necessary to determine the general form and amount of variation expected for *m*(

Given this, consider a two layer medium defined by equation (58), where angular frequency ranges from = 2100 rads/sec to = 23104 rads/sec and coating thickness *Zc* (region 3) ranges from *Zc* = 0 m to 300 m. A worse case of copper plating on steel is assumed. The following two graphs show real and imaginary components of

Examination of figure 6a and 6b clearly shows that *m*() has considerable variation below = 104, but that above this it is relatively smooth. Assuming that *m*() is defined on the interval {*a*, *b*} and that a clamped boundary is used, let cubic polynomial *S*j occur on subinterval [j, j+1]. Given this it can be shown that the maximum error occurs when:

384

From above, interpolation error can be linked with the maximum subinterval step size max[j, j+1] and the maximum 4th derivative of *m*(). Since the maximum derivative error is always below = 2103 and since *m*() is almost linear above = 104, it seems reasonable to reduce subinterval step size for low and increase it above = 104. This adjustment enables the interpolating cubic polynomials *Sj* to more accurately reproduce data in regions of maximum variation, whilst minimising the total number of subinterval domains. Empirical

( )

and *a b*

0.035 (( 1) ) <sup>3</sup> <sup>3</sup> *j j*

*j* {0, 1, … , 111}.

<sup>5</sup> ( ) ( )

max <sup>4</sup>

*<sup>m</sup> <sup>a</sup> <sup>b</sup>*

*<sup>M</sup> <sup>m</sup> <sup>S</sup> <sup>a</sup> <sup>b</sup>* 

).

4

<sup>1</sup> ( ) <sup>0</sup> <sup>1</sup>

(59)

01...*<sup>n</sup>* .

*<sup>j</sup> <sup>j</sup> j n*

*<sup>j</sup>* (60)

max

**3.2 Implementation of the material profile function** 

*m*() for these conditions.

(Burden & Faires, 1989):

where

where

max

*M*

study showed that the optimum choice for j is:

Fig. 6a. Real Component of Material Profile Function *m*().

Fig. 6b. Imaginary Component of Material Profile Function *m*().

Numerical Modelling and Design of an Eddy Current Sensor 175

 

*a*

<sup>1</sup> ( , ) ( ) { ( ) ( )} <sup>1</sup>

*<sup>B</sup> <sup>n</sup> <sup>I</sup> <sup>I</sup> <sup>r</sup> <sup>r</sup> <sup>J</sup> <sup>r</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>m</sup> <sup>e</sup> <sup>d</sup> <sup>l</sup> <sup>l</sup> <sup>z</sup> <sup>z</sup> <sup>c</sup> ( )*

<sup>12</sup> ( , ) ( ){2 ( ) ( )} <sup>1</sup>

<sup>1</sup> ( , ) ( ) { ( ) ( )} <sup>1</sup>

*<sup>B</sup> <sup>n</sup> <sup>I</sup> <sup>I</sup> <sup>r</sup> <sup>r</sup> <sup>J</sup> <sup>r</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>m</sup> <sup>e</sup> <sup>d</sup> <sup>l</sup> <sup>l</sup> <sup>z</sup> <sup>z</sup> <sup>c</sup> ( )*

<sup>2</sup> ( , ) ( )( )( ( ) ) <sup>1</sup>

Boundary equations (64) – (68) have a rate of convergence no worse than -2. Comparison with that of the source coil inductance indicates that the rate of convergence of source coil

*<sup>B</sup> <sup>n</sup> <sup>I</sup> <sup>I</sup> <sup>r</sup> <sup>r</sup> <sup>J</sup> <sup>r</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>m</sup> <sup>d</sup> <sup>z</sup> <sup>l</sup> <sup>l</sup> <sup>l</sup> <sup>l</sup> <sup>c</sup> ( )*

 

*<sup>B</sup> <sup>n</sup> <sup>I</sup> <sup>I</sup> <sup>r</sup> <sup>r</sup> <sup>J</sup> <sup>r</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>m</sup> <sup>d</sup> <sup>z</sup> <sup>l</sup> <sup>z</sup> <sup>l</sup> <sup>z</sup> <sup>l</sup> <sup>l</sup> <sup>c</sup> ( )*

<sup>2</sup> ( , ) ( )( )( ( ) ) <sup>1</sup>

1 2

( ) ( )

 

*<sup>B</sup> <sup>n</sup> <sup>I</sup> <sup>I</sup> <sup>r</sup> <sup>r</sup> <sup>J</sup> <sup>r</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>m</sup> <sup>d</sup> <sup>z</sup> <sup>l</sup> <sup>l</sup> <sup>l</sup> <sup>l</sup> <sup>c</sup> ( )*

*L n I r r l l e e e e m d <sup>l</sup> <sup>l</sup> <sup>l</sup> <sup>l</sup> <sup>l</sup> <sup>l</sup>*

The function shown above has a very rapid rate of convergence due to being raised to the 5*th* power. Application of the sensor core boundary equation (28) leads to five core equations

In this instance (62) is said to be convergent if *n* > 1. Given this, it seems reasonable to assume that a large positive value for *n* is required for a high rate of convergence. An example of this is given for the self inductance *L* of an air-cored coil (Dodd & Deeds, 1968):

<sup>2</sup> ( , ) {2 ( ) <sup>2</sup> <sup>2</sup> ( <sup>2</sup> ) } <sup>1</sup>

2 1 2 1 2 1

2 1 2 1

*Sz* (65)

2 1 1 2

2 1 2 1

*Sr* (68)

*Sr* (67)

1 2

*Sz* (66)

*Sz* (64)

*C* (63)

*n*

. Given this equation (61) is convergent. Stephenson

( ) 2 2 ( )

   

 *<sup>g</sup>*( ) (62)

If *n* > 1 then 0 

> 

 

0

2 1

<sup>1</sup>*<sup>n</sup>* for

generalises this further by redefining equation (61) as (Stephenson, G. 1974):

where g() is some arbitrary function that is bounded and non-zero.

2 1

for the source coil magnetic flux density *BS* , which are given below:

2 1 0 2

2 1 0 2

2 1 1 2

2 1 1 2

2

2 1 5

 

2 1

> 

2 1

field vectors is relatively poor.

2 1

0

 

2 1

0

 

0

0

 

0

2 1 0 2

#### **3.3 Material profile function testing and evaluation**

A typical spline curve is given in figure 7 for a two layer material: substrate (region 4: 4 = 100, 4 = 10 MS/m) and coating (region 3: 3 = 1, 3 = 58 MS/m, coating thickness *Zc*= 300 m) for an excitation frequency = 230103 rads/sec.

Fig. 7. Spline Approximation *S*() of Cheng Formula Re(*m*()).

The major benefit of this approach is that the cubic polynomial coefficients for all *Sj* need only be calculated once, making the method very rapid.

#### **3.4 The convergence of the source coil fields**

The vector potentials required for evaluating the regions bounding the source coil are derived from equations (11) and (12), which are improper integrals. Since an explicit antiderivative does not exist for these equations, numerical quadrature was be used. The convergence of these integrals can be studied by considering the form of their integrand for large , which can be represented in the following way:

$$\int\_{a}^{\xi} \frac{d\alpha}{\alpha^{n}} = \frac{\xi^{1-n} - a^{1-n}}{1 - n} \tag{61}$$

4 =

(61)

3 = 58 MS/m, coating thickness *Zc*= 300

A typical spline curve is given in figure 7 for a two layer material: substrate (region 4:

3 = 1, 

**<sup>0</sup> <sup>5000</sup> <sup>1</sup> <sup>104</sup> 1.5 10<sup>4</sup> <sup>2</sup> <sup>104</sup> 1.2**

**3.3 Material profile function testing and evaluation** 

m) for an excitation frequency = 230103 rads/sec.

4 = 10 MS/m) and coating (region 3:

**1**

Cubic spline

Fig. 7. Spline Approximation *S*() of Cheng Formula Re(*m*()).

only be calculated once, making the method very rapid.

large , which can be represented in the following way:

**3.4 The convergence of the source coil fields** 

Cheng et al material profile formula

The major benefit of this approach is that the cubic polynomial coefficients for all *Sj* need

The vector potentials required for evaluating the regions bounding the source coil are derived from equations (11) and (12), which are improper integrals. Since an explicit antiderivative does not exist for these equations, numerical quadrature was be used. The convergence of these integrals can be studied by considering the form of their integrand for

*d a <sup>n</sup> <sup>n</sup>*

1 1 

*<sup>n</sup>* 

<sup>1</sup>

*a*

*n*

**0.8**

**0.6**

**0.4**

**0.2**

**0**

**0.2**

100, 

> *S*( ) Re(*m*())

If *n* > 1 then 0 <sup>1</sup>*<sup>n</sup>* for . Given this equation (61) is convergent. Stephenson generalises this further by redefining equation (61) as (Stephenson, G. 1974):

$$\int\_{a}^{\zeta} \frac{\mathbf{g}(a)}{\alpha^{n}} \tag{62}$$

where g() is some arbitrary function that is bounded and non-zero.

In this instance (62) is said to be convergent if *n* > 1. Given this, it seems reasonable to assume that a large positive value for *n* is required for a high rate of convergence. An example of this is given for the self inductance *L* of an air-cored coil (Dodd & Deeds, 1968):

$$L = \mathfrak{m}\_C^2 \mu \left\{ \frac{1}{\alpha^5} I(r\_2, r\_1)^2 (2a(l\_2 - l\_1) + 2e^{-\alpha(l\_2 - l\_1)} - 2 + (e^{-2\alpha l\_2} + e^{-2\alpha l\_1} - 2e^{-\alpha(l\_2 - l\_1)})m) da \right. \tag{63}$$

The function shown above has a very rapid rate of convergence due to being raised to the 5*th* power. Application of the sensor core boundary equation (28) leads to five core equations for the source coil magnetic flux density *BS* , which are given below:

$$B\_{S\sharp}^{(1)} = \frac{1}{2} n\_c I \mu \left| \frac{1}{a^2} I(\eta\_2, \eta\_1) J\_0(ar) e^{-a\varepsilon} \langle e^{al\_2} - e^{al\_1} - (e^{-al\_2} - e^{-al\_1}) m(a) \rangle da \right. \tag{64}$$

$$B\_{Sz}^{(2)} = \frac{1}{2} n\_c I \mu \left\{ \frac{1}{\alpha^2} I(r\_2, \eta) J\_0(\alpha r) (e^{-\alpha l\_1} - e^{-\alpha l\_2}) (e^{\alpha z} - m(\alpha) e^{-\alpha z}) da \right. \tag{65}$$

$$B\_{\Sigma}^{(12)} = \frac{1}{2} n\_c I \mu \left\| \frac{1}{\alpha^2} I(r\_2, r\_1) J\_0(\alpha r) \left( 2 - e^{\alpha(z - l\_2)} - e^{-\alpha(z - l\_1)} + e^{-\alpha} (e^{-d\_1} - e^{-d\_2}) \mathfrak{m}(\alpha) \right) d\alpha \right\|\tag{66}$$

$$B\_{\rm Sb}^{(1)} = -\frac{1}{2} n\_c I \mu \int\_0^{\alpha} \frac{1}{\alpha^2} I(r\_2, \eta\_1) J\_1(\alpha r) e^{-\alpha \underline{x}} \langle e^{\alpha l\_2} - e^{\alpha l\_1} - (e^{-\alpha l\_2} - e^{-\alpha l\_1}) m(\alpha) \rangle d\alpha \tag{67}$$

$$B\_{\rm Sr}^{(2)} = -\frac{1}{2} n\_c I \mu \int\_0^{\alpha} \frac{1}{\alpha^2} I(r\_2, \eta) J\_1(\alpha r) (e^{-\alpha l\_1} - e^{-\alpha l\_2}) (e^{\alpha x} - m(\alpha) e^{-\alpha x}) d\alpha \tag{68}$$

Boundary equations (64) – (68) have a rate of convergence no worse than -2. Comparison with that of the source coil inductance indicates that the rate of convergence of source coil field vectors is relatively poor.

Numerical Modelling and Design of an Eddy Current Sensor 177

2 1

 

1 2

  (73)

*B u J r J r e e e d <sup>a</sup> <sup>a</sup> <sup>z</sup> <sup>l</sup> <sup>l</sup>*

<sup>1</sup> <sup>0</sup> ( ) ( ) ( ) ( ) <sup>2</sup>

*u J r J r e e e m d <sup>a</sup> <sup>a</sup> <sup>z</sup> <sup>l</sup> <sup>l</sup>*

Assuming that all other field equations can be treated in the same way, a comparison of the integrands of equation (73) is shown in a normalised form in figure 8, with *m*() = 1.0.

<sup>10</sup><sup>4</sup> 1.5

It is clear from Figure 8 that separating the field equations into two terms, and considering only the material dependent term *hm*() gives a function with a very rapid rate of convergence. An equation replacing the free-space or material independent term *hfs*() now needs to be determined. Let the delta function coil representing the material independent Green's function *G0* (*r*, *z*; *r*, *z*) be formed from discrete current elements *IdS*. See figure 9.

<sup>10</sup><sup>4</sup> <sup>2</sup>

<sup>10</sup><sup>4</sup> 2.5

104 <sup>3</sup>

104

<sup>1</sup> <sup>0</sup> ( ) ( ) { } <sup>2</sup>

 

*Rz i a*

1 

 

1 

0

0 5000 1

Fig. 8. Free-Space *hfs*() and Material Dependent *hm*() Convergence.

0

0.2

0.4

0.6

0

*<sup>h</sup> fs* ( ) <sup>d</sup>

0

 <sup>d</sup>

 *<sup>h</sup> <sup>m</sup>* ( ) 0.8

1

1.2

0

*i a*

#### **3.5 Convergence of the basis function fields**

The field generated by an *ith* basis function located on the cylindrical face of the sensor core is given by:

$$A\_{R\rho}^{\ast}(r,z) = \mu\iota\_i \int\_{z\_i-\Delta z}^{z\_i+\Delta z} G\_R^{(n)}(r,z;r',z')dz'\tag{69}$$

where the total scattered field *N i <sup>R</sup> <sup>R</sup> A r z A r z* 1 \* ( , ) ( , ) and *<sup>z</sup> zi*<sup>1</sup> *zi*<sup>1</sup> / 4 .

If the axial coordinates of the basis function are *l z z <sup>a</sup>*<sup>2</sup> *<sup>i</sup>* and *l z z <sup>a</sup>*<sup>1</sup> *<sup>i</sup>* , with radial coordinate *ra*, equation (69) becomes:

$$A\_{R\rho}^{\ast} = \frac{r\_a}{2} \mu \mu\_l \left[ \frac{1}{\alpha} J\_1(\alpha r\_a) J\_1(\alpha r) \langle 2 - e^{\alpha(z - l\_{a2})} - e^{-\alpha(z - l\_{a1})} + e^{-\alpha z} (e^{-d\_{a1}} - e^{-d\_{a2}}) m(\alpha) \right] d\alpha \tag{70}$$

Which assumes that field point (*r*, *z*) is bounded, with *a*<sup>2</sup> *<sup>a</sup>*<sup>1</sup> *l z l* .

The components of flux density *<sup>r</sup> <sup>z</sup> \* BR <sup>a</sup> <sup>a</sup> \* Rz \* BRr <sup>B</sup>* for equation (70) are:

$$B\_{Rr}^{\*} = \frac{r\_a}{2} \mu \mu\_l \begin{bmatrix} \int\_1^{\alpha} (\sigma r\_a) J\_1(\sigma r) \left( -e^{\alpha(z - l\_{a2})} - e^{-\alpha(z - l\_{a1})} - e^{-\alpha z} (e^{-d\_{a1}} - e^{-d\_{a2}}) \eta(\alpha) \right) d\alpha \end{bmatrix} \tag{71}$$

$$B\_{Rz}^{\*} = \frac{r\_a}{2} \mu \mu \int\_0^{\infty} J\_1(\alpha r\_a) J\_0(\alpha r) (2 - e^{\alpha(z - l\_{a2})} - e^{-\alpha(z - l\_{a1})} + e^{-\alpha z} (e^{-d\_{a1}} - e^{-d\_{a2}}) m(\alpha)) da \tag{72}$$

It is clear that a basis function vector potential has a rate of convergence that is poor, the convergence of it's flux density vector *BR* is even worse. A significant benefit in terms of computational efficiency and accuracy is gained if a method can be found to improve the

convergence of these field equations. Note that the fields above and below the basis function, as well as the basis functions on the end faces of the core, have been omitted for brevity.

#### **3.6 The modified free space green's function** *G0* **(***r***,** *z***;** *r***,** *z***)**

It is evident that field equations have a poor rate of convergence. Considering only the basis function fields, separate equation (72) into two parts, which are given on the following:

The field generated by an *ith* basis function located on the cylindrical face of the sensor core

( , ) ( , ; ', ') ' \* ( ) 

*u G r z r z dz*

(69)

 

 

 

and *<sup>z</sup> zi*<sup>1</sup> *zi*<sup>1</sup> / 4 .

*z z*

*i*

 *N*

*i <sup>R</sup> <sup>R</sup> A r z A r z* 1 \* ( , ) ( , )

( ) ( ) <sup>1</sup> <sup>1</sup> ( ) ( ){ ( ) ( )} <sup>2</sup>

( ) ( ) <sup>1</sup> <sup>0</sup> ( ) ( ){2 ( ) ( )} <sup>2</sup>

*<sup>u</sup> <sup>J</sup> <sup>r</sup> <sup>J</sup> <sup>r</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>m</sup> <sup>d</sup> <sup>r</sup> <sup>B</sup> <sup>a</sup> <sup>a</sup> <sup>a</sup> <sup>a</sup> <sup>z</sup> <sup>l</sup> <sup>z</sup> <sup>l</sup> <sup>z</sup> <sup>l</sup> <sup>l</sup>*

It is clear that a basis function vector potential has a rate of convergence that is poor, the convergence of it's flux density vector *BR* is even worse. A significant benefit in terms of computational efficiency and accuracy is gained if a method can be found to improve the

convergence of these field equations. Note that the fields above and below the basis function, as well as the basis functions on the end faces of the core, have been omitted for

It is evident that field equations have a poor rate of convergence. Considering only the basis function fields, separate equation (72) into two parts, which are given on the

*<sup>u</sup> <sup>J</sup> <sup>r</sup> <sup>J</sup> <sup>r</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>m</sup> <sup>d</sup> <sup>r</sup> <sup>B</sup> <sup>a</sup> <sup>a</sup> <sup>a</sup> <sup>a</sup> <sup>z</sup> <sup>l</sup> <sup>z</sup> <sup>l</sup> <sup>z</sup> <sup>l</sup> <sup>l</sup>*

*A r z*

Which assumes that field point (*r*, *z*) is bounded, with *a*<sup>2</sup> *<sup>a</sup>*<sup>1</sup> *l z l* .

**3.6 The modified free space green's function** *G0* **(***r***,** *z***;** *r***,** *z***)** 

*<sup>n</sup> <sup>R</sup> <sup>i</sup> <sup>R</sup>*

If the axial coordinates of the basis function are *l z z <sup>a</sup>*<sup>2</sup> *<sup>i</sup>* and *l z z <sup>a</sup>*<sup>1</sup> *<sup>i</sup>* , with radial

( ) ( ) <sup>1</sup> <sup>1</sup> ( ) ( ){2 ( ) ( )} <sup>1</sup>

*<sup>u</sup> <sup>J</sup> <sup>r</sup> <sup>J</sup> <sup>r</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>m</sup> <sup>d</sup> <sup>r</sup> <sup>A</sup> <sup>a</sup> <sup>a</sup> <sup>a</sup> <sup>a</sup> <sup>z</sup> <sup>l</sup> <sup>z</sup> <sup>l</sup> <sup>z</sup> <sup>l</sup> <sup>l</sup>*

*<sup>i</sup> <sup>a</sup> <sup>a</sup> <sup>R</sup>* (70)

2 1 1 2

*Rz \* BRr <sup>B</sup>* for equation (70) are:

> 

2 1 1 2

2 1 1 2

*<sup>i</sup> <sup>a</sup> <sup>a</sup> Rz* (72)

*<sup>i</sup> <sup>a</sup> <sup>a</sup> Rr* (71)

*z z*

*i*

**3.5 Convergence of the basis function fields** 

where the total scattered field

 

The components of flux density *<sup>r</sup> <sup>z</sup> \* BR <sup>a</sup> <sup>a</sup> \**

 

0

 

brevity.

following:

0

0

coordinate *ra*, equation (69) becomes:

2

is given by:

$$\begin{aligned} B\_{Rz}^{\*} &= \frac{1}{2} \mu \iota\_i \int\_0^\alpha J\_1(ar\_a) J\_0(ar) e^{-\alpha \varpi} \{e^{ad\_{a2}} - e^{ad\_{a1}}\} da + \\\\ \frac{1}{2} \mu \iota\_i \int\_0^\alpha J\_1(ar\_a) J\_0(ar) e^{-\alpha \varpi} (e^{-al\_{a1}} - e^{-al\_{a2}}) m(a) da \end{aligned} \tag{73}$$

Assuming that all other field equations can be treated in the same way, a comparison of the integrands of equation (73) is shown in a normalised form in figure 8, with *m*() = 1.0.

Fig. 8. Free-Space *hfs*() and Material Dependent *hm*() Convergence.

It is clear from Figure 8 that separating the field equations into two terms, and considering only the material dependent term *hm*() gives a function with a very rapid rate of convergence. An equation replacing the free-space or material independent term *hfs*() now needs to be determined. Let the delta function coil representing the material independent Green's function *G0* (*r*, *z*; *r*, *z*) be formed from discrete current elements *IdS*. See figure 9.

Numerical Modelling and Design of an Eddy Current Sensor 179

where *Nc* is the number of turns on the source coil. Coil impedance *Z* can be found by

Evaluation of the sensor model proceeded by defining the dimensional and physical properties of the sensor of section 2, which are shown below in table 1. Note that no

**Sensor Core Sensor Source Coil** 

Source Coil Turns *Nc*: 294

*r* = 95.6 and conductivity

=

FEM 0 μm lift-off FEM 250 μm lift-off FEM 500 μm lift-off NUM 0μm lift-off NUM 250 μm lift-of NUM 500 μm lift-of

Sensor Lift-off: 0.50 mm Source Coil *r1*: 1.45 mm Core Radius: 0.99 mm Source Coil *r2*: 3.175 mm Core Length: 6 mm Source Coil *l1*: 3.005 mm Core Permeability 1000 Source Coil *l2*: 3.845 mm

Matrix equation (40) was solved for coefficients *ui* and *vi* using Mathcad, version 11.0a and source coil impedance determined from equation (76). It was found empirically that 80

Source coil self inductance *L* and resistance *R* was calculated for differing sensor lift off over

8.4x106 S/m. As a comparison, the sensor model of table 1 was also simulated using the commercial FEM solver MagNet, version 6.25. The results of this, displayed in the form of a

collocation points spread evenly along the core-air interface *C* provided good results.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 R/ωLo

steel with the following properties: relative permeability

normalised impedance plane diagram, are shown in figure 10.

Fig. 10. Normalised Impedance Plane Diagram of Sensor.

information for the two pickup coils is given as this is the subject of future work.

simply dividing *V* by source current *I*.

Table 1. Sensor Properties.

0.875

0.8875

0.9125

0.925

0.9375

L/Lo

0.95

0.9625

0.975

0.9875

1

0.9

Fig. 9. Current Element *IdS* forming a Delta Function Coil.

The magnetic vector potential generated by *IdS* is given as:

$$\mathcal{A}\_{\mathbf{R}}^{\*} = \mu \mathbb{I} \oint\_{\Gamma} \frac{d\mathbf{S}}{R} \,, \tag{74}$$

where is media permeability and *I* is current.

If ))sin()(cos(' *<sup>x</sup> <sup>y</sup> S r a a* , *aaS' zx zr* and *R SS'* , then the free space Green's function *G0* (*r*, *z*; *r*, *z*) is of the following form:

$$G\_0(r, z; r', z') = \frac{I}{4\pi} \int\_0^\pi \frac{\mu \cdot r' \cdot \cos(\theta) d\theta}{\sqrt{\left(r - r' \cdot \cos(\theta)\right)^2 + \left(z - z'\right)^2 + \left(r' \cdot \sin(\theta)\right)^2}}\tag{75}$$

Integration of *G0* (*r, z; r*, *z*) over *r* or *z* gives the relevant equation for the basis functions.

Equation (75) satisfies all the requirements of the Green's function (Sadiku, 1992). Highly accurate calculations of field quantities were found to be possible using this equation.

#### **4. Testing and evaluation**

Nearly all eddy current investigations are conducted in the sensor coil region by determining coil impedance. Given this, if the source coil is densely and uniformly wound with a rectangular cross section, having the radial and axial dimensions (*r2, r1*) and (*l2, l1*), the induced voltage *V* across the coil will be equal to:

$$V = \frac{j\alpha 2\pi N\_c}{(l\_2 - l\_1)(r\_2 - r\_1)} \int\_{l\_1}^{l\_2} \int\_{\eta\_1}^{r\_2} r(A\_{S\rho}^{(12)}(r, z) + A\_{r\rho}(r, z)) dr dz \tag{76}$$

where *Nc* is the number of turns on the source coil. Coil impedance *Z* can be found by simply dividing *V* by source current *I*.

Evaluation of the sensor model proceeded by defining the dimensional and physical properties of the sensor of section 2, which are shown below in table 1. Note that no information for the two pickup coils is given as this is the subject of future work.


Table 1. Sensor Properties.

178 Numerical Modelling

Field point *S*

*<sup>z</sup>*Delta Function

*ay*

 *<sup>R</sup> <sup>d</sup> <sup>I</sup> <sup>S</sup> A\* R* 

<sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>0</sup> ))sin('()'())cos('(

Equation (75) satisfies all the requirements of the Green's function (Sadiku, 1992). Highly accurate calculations of field quantities were found to be possible using this equation.

Nearly all eddy current investigations are conducted in the sensor coil region by determining coil impedance. Given this, if the source coil is densely and uniformly wound with a rectangular cross section, having the radial and axial dimensions (*r2, r1*) and (*l2, l1*),

1

2 )12(

*l l*

2 1 )),(),(( ))((

*<sup>r</sup> <sup>S</sup> <sup>r</sup> <sup>c</sup> drdzzrAzrAr*

, (76)

*r*

*a* , *aaS' zx zr* and *R SS'* , then the free space Green's

*<sup>I</sup> dr zrzrG* (75)

)cos('

, (74)

Coil

Fig. 9. Current Element *IdS* forming a Delta Function Coil.

θ

*I* 

*r*

*R* 

*dS*

Γ

*r* 

*az*

The magnetic vector potential generated by *IdS* is given as:

is media permeability and *I* is current.

the induced voltage *V* across the coil will be equal to:

<sup>2</sup>

1212

*rrll Nj <sup>V</sup>*

0

<sup>4</sup> )',';,( *rr rzz*

Integration of *G0* (*r, z; r*, *z*) over *r* or *z* gives the relevant equation for the basis functions.

function *G0* (*r*, *z*; *r*, *z*) is of the following form:

where 

If ))sin()(cos(' *<sup>x</sup> <sup>y</sup> S r a*

*ax*

**4. Testing and evaluation** 

Matrix equation (40) was solved for coefficients *ui* and *vi* using Mathcad, version 11.0a and source coil impedance determined from equation (76). It was found empirically that 80 collocation points spread evenly along the core-air interface *C* provided good results.

Source coil self inductance *L* and resistance *R* was calculated for differing sensor lift off over steel with the following properties: relative permeability *r* = 95.6 and conductivity = 8.4x106 S/m. As a comparison, the sensor model of table 1 was also simulated using the commercial FEM solver MagNet, version 6.25. The results of this, displayed in the form of a normalised impedance plane diagram, are shown in figure 10.

Fig. 10. Normalised Impedance Plane Diagram of Sensor.

Numerical Modelling and Design of an Eddy Current Sensor 181

Future work entails implementing a Galerkin method of weighted residuals to replace the current collocation method and conducting detailed tests on non-linear material profiles.

Balanis, C. A. (1989). *Advanced Engineering Electromagnetics,* John Wiley & Sons, ISBN 0-471-

Becker, R.; Dobmann, G.; & Rodner, C. (1988) *Quantitative eddy current variants for* 

Blitz, J. (1991). *Electrical and Magnetic Methods of Non-destructive Testing*, 1st edition, Adam

Bowler, N. (2006). *Frequency dependence of relative permeability in steel.* In: Review of Progress

Burden, R. L. Faires, J. D. (1989). *Numerical Analysis,* 4th Edition, Boston: PWS-KENT

Cheng, C. C.; Dodd, C. V. & Deeds, W. E. (1971). *General Analysis of Probe Coils Near* 

Dodd, C. V. & Deeds, W. E. (1968) 'Analytical Solutions to Eddy-Current Probe-Coil

Glorieux C.; Moulder, J.; Basart, J. & Thoen, J. (1999). *The Determination of Electrical* 

Harrison, D. J.; Jones, L. D. & Burke, S. K. (1996). *Benchmark problems for defect size and shape* 

Langhill, T. J. (1999). *Painting Over Hot-Dip Galvanised Steel,* Materials Performance, pp. 44-

Moulder, J.; Uzal, E. & Rose, J. H. (1992). *Thickness and Conductivity of Metallic Layers from* 

Norton, S. J. & Bowler, J. R. (1993). *Theory of Eddy Current Inversion,* Journal Of Applied

Sadiku, M. N. O. (1992). *Numerical Techniques in Electromagnetics,* Florida: CRC Press, ISBN:

Stephenson, G. (1973). Mathematical Methods for Science Students' 2nd Edition, Published

Uzal, E.; Moulder, J. C.; Mitra, S. & Rose, J. H. (1993). *Impedance of Coils over Layered Metals* 

Journal of Applied Physics, Vol. 74, No. 3, pp. 2076-2089

*with Continuously Variable Conductivity and Permeability: Theory and Experiment,*

Problems' Journal of Applied Physics, Vol. 39, No. 6. pp 2829-2838

Chimenti, D. E. pp.1269-1276, CP820, AIP, ISBN 0-7354-0312-0

*micromagnetic microstructure multiparameter analysis 3MA.* In: *Review of Progress in Quantitative Non-Destructive Evaluation 7B,* edited by Thompson, D.O. & Chimenti,

in Quantitative Non-Destructive Evaluation, Vol. 25, edited by Thompson, D.O. &

*Stratified Conductors*, International Journal of Non-destructive Testing, Vol. 3, pp.

*Conductivity Using Neural Network Inversion of Multi-frequency Eddy Current Probe* 

*determination in eddy-current non-destructive evaluation*, Journal of Non-Destructive

*Eddy Current Measurements*, Review of Scientific Instruments, Vol. 63, No. 6, pp

**6. References** 

50316-9, New York

109-130

Hilger, ISBN 0-7503-0148-1

D. E. pp.1703 – 1707, Plenum Press, New York

Publishing Company, ISBN: 0-534-98059-7

*Data,* Journal of Physics D. Vol. 32, pp. 616-622

Evaluation, Vol. 15, No. 1, pp. 21-34

Physics, Vol. 73, No. 2, pp. 501-512

49, December 1999

3455-3465.

0-8493-4232-5

London Longman

Changes to the value of core permeability r were also simulated for the sensor held in free space, positioned above a solid copper substrate (*f* = 30 kHz) and finally above 100 m of copper plating on steel (*f* = 10 kHz). The results of this simulation are displayed in figure 11.

Fig. 11. Source Coil Inductance as a Function of Core Permeability.

#### **5. Conclusion**

The benefits of the magnetic moment method developed in this work are:


Future work entails implementing a Galerkin method of weighted residuals to replace the current collocation method and conducting detailed tests on non-linear material profiles.

#### **6. References**

180 Numerical Modelling

0 250 500 750 1000 1250 1500 1750 2000 Probe Tip Permeability μi

Fig. 11. Source Coil Inductance as a Function of Core Permeability.

with no penalty in terms of computation time or accuracy.

varied without the need to recalculate basis function fields.

computed once for any given simulation session. For a given lift-off and material profile function *m(*

The benefits of the magnetic moment method developed in this work are:

Only points on the closed surface bounding the sensor core are discretised.

 The use of a spline function to replace the material dependent matrices *V(M,1)12*/*V(M,1)22* of Cheng, Dodd and Deeds (Equation 7), allows for a potentially infinite number of stratified layers to be used to represent non-linear material profiles,

 Separating the basis functions into free-space static and substrate dependent dynamic terms, allows for more efficient computation of magnetic fields on the equivalent boundary surface. Static field components (free space components) only need to be

*)*, probe tip permeability i can be

free space, positioned above a solid copper substrate (*f* = 30 kHz) and finally above 100 m of copper plating on steel (*f* = 10 kHz). The results of this simulation are displayed in

r were also simulated for the sensor held in

numerical model FEM model

numerical model (copper plating @ 10kHz) FEM model (copper plating @ 10kHz) numerical model (solid copper @ 30kHz) FEM model (solid copper @ 30kHz)

Changes to the value of core permeability

figure 11.

**5. Conclusion** 

Probe Inductance (μH)


**0**

**9**

<sup>1</sup>*Belarus* <sup>2</sup>*Germany*

**Numerical Study of Diffusion of Interacting**

Magnetic fluids are stable colloidal suspensions of ferromagnetic nano-particles (of size 10-20 nm) in a nonmagnetic liquid carrier. An initially uniform particle distribution in the carrier becomes spatially inhomogeneous in nonuniform magnetic fields. Motion of particles in magnetic fluids under the action of magnetic fields is of particular interest for contemporary

The most theoretical models for the diffusion process in magnetic fluids assume no interaction between particles (Bashtovoi et al., 2007; 2008; Lavrova et al., 2010; Polevikov & Tobiska, 2008; 2011), which is valid for dilute fluids only. This assumption allows to construct an explicit dependence between equilibrium particle concentration and the magnetic field distribution, simplifying significantly the modelling. In case of concentrated magnetic fluids another theoretical model should be considered. Recently, a dynamic mass transfer equation for describing diffusion of interacting ferromagnetic particles in magnetic fluids was derived

"...In the case of high particle concentrations, the magnetic and diffusion problems are strictly interrelated, and the concentration profile depends markedly on steric, magnetodipole, and hydrodynamic interparticle interactions, whose counting is a

The present study is devoted to the classical problem of ferrohydrostatics on stability (known as the normal field instability or the Rosensweig instability) of a horizontal semi-infinite layer of a magnetic fluid under the influence of gravity and a uniform magnetic field normal to the plane free surface of the layer (Rosensweig, 1998). A periodic peak-shaped structure is formed on the fluid surface when the applied magnetic field exceeds a critical value. This

A number of papers are devoted to numerical investigations of the Rosensweig instability. The numerical results concern equilibrium states of the ferrofluid layer in (Aristidopoulou et al., 1996; Bashtovoi et al., 2002; Boudouvis et al., 1987; Gollwitzer et al., 2007; 2009; Lange et al., 2007; Lavrova et al., 2003; 2008; 2010) and analyze dynamical properties of the ferrofluid

phenomenon was observed first experimentally (Cowley & Rosensweig, 1967).

mathematical and numerical modelling in ferrohydrodynamics.

(Pshenichnikov et al., 2011). In this paper it is mentioned that

problem of great concern..."

**1. Introduction**

**Particles in a Magnetic Fluid Layer**

Olga Lavrova1, Viktor Polevikov1 and Lutz Tobiska2

<sup>1</sup>*Belarusian State University* <sup>2</sup>*Otto-von-Guericke University*

Yildir, Y. B.; Klimpke, B. W. & Zheng, D. (1992). *A computer program for 2D/RS eddy current problem based on boundary element method,* Available from: http://www.integratedsoft.com/papers/techdocs/tech\_2ox.pdf

### **Numerical Study of Diffusion of Interacting Particles in a Magnetic Fluid Layer**

Olga Lavrova1, Viktor Polevikov1 and Lutz Tobiska2 <sup>1</sup>*Belarusian State University* <sup>2</sup>*Otto-von-Guericke University* <sup>1</sup>*Belarus* <sup>2</sup>*Germany*

#### **1. Introduction**

182 Numerical Modelling

Yildir, Y. B.; Klimpke, B. W. & Zheng, D. (1992). *A computer program for 2D/RS eddy current* 

*problem based on boundary element method,* Available from: http://www.integratedsoft.com/papers/techdocs/tech\_2ox.pdf

> Magnetic fluids are stable colloidal suspensions of ferromagnetic nano-particles (of size 10-20 nm) in a nonmagnetic liquid carrier. An initially uniform particle distribution in the carrier becomes spatially inhomogeneous in nonuniform magnetic fields. Motion of particles in magnetic fluids under the action of magnetic fields is of particular interest for contemporary mathematical and numerical modelling in ferrohydrodynamics.

> The most theoretical models for the diffusion process in magnetic fluids assume no interaction between particles (Bashtovoi et al., 2007; 2008; Lavrova et al., 2010; Polevikov & Tobiska, 2008; 2011), which is valid for dilute fluids only. This assumption allows to construct an explicit dependence between equilibrium particle concentration and the magnetic field distribution, simplifying significantly the modelling. In case of concentrated magnetic fluids another theoretical model should be considered. Recently, a dynamic mass transfer equation for describing diffusion of interacting ferromagnetic particles in magnetic fluids was derived (Pshenichnikov et al., 2011). In this paper it is mentioned that

"...In the case of high particle concentrations, the magnetic and diffusion problems are strictly interrelated, and the concentration profile depends markedly on steric, magnetodipole, and hydrodynamic interparticle interactions, whose counting is a problem of great concern..."

The present study is devoted to the classical problem of ferrohydrostatics on stability (known as the normal field instability or the Rosensweig instability) of a horizontal semi-infinite layer of a magnetic fluid under the influence of gravity and a uniform magnetic field normal to the plane free surface of the layer (Rosensweig, 1998). A periodic peak-shaped structure is formed on the fluid surface when the applied magnetic field exceeds a critical value. This phenomenon was observed first experimentally (Cowley & Rosensweig, 1967).

A number of papers are devoted to numerical investigations of the Rosensweig instability. The numerical results concern equilibrium states of the ferrofluid layer in (Aristidopoulou et al., 1996; Bashtovoi et al., 2002; Boudouvis et al., 1987; Gollwitzer et al., 2007; 2009; Lange et al., 2007; Lavrova et al., 2003; 2008; 2010) and analyze dynamical properties of the ferrofluid

whereas the fluid corresponds to a region *z* < 0. The system is regarded under the action of gravity *g* = (0, 0, −*g*) and a uniform magnetic field *H*<sup>0</sup> = (0, 0, *H*0) normal to the plane free surface of the layer. We consider a single peak in the surface pattern with a cell Ω*cell* and a free surface Γ. The problem will be formulated in a cylinder Ω*cell* × (−∞, +∞), see Fig. 1 for

Numerical Study of Diffusion of Interacting Particles in a Magnetic Fluid Layer 185

✻

*z*

✲

*y*

*g H*❄ <sup>0</sup> ✻

Ω*cell*

*x*

Fig. 1. The computational domain.

 ✠

The mathematical model for a non-uniform equilibrium distribution of ferromagnetic particles in a magnetic fluid with a free surface leads to a coupled problem formulation consisting of three subproblems. The first subproblem describes the magnetic field structure inside the fluid and in the surrounding air by the Maxwell's equations. The second subproblem concerns the diffusion of particles in the bulk of the fluid as a steady-state concentration problem. Finally, the third subproblem is given by the generalized

see e.g. (Rosensweig, 1998). Here **H** denotes the magnetic field, **B** = *μ*0(**H** + **M**) the magnetic induction, *<sup>μ</sup>*<sup>0</sup> <sup>=</sup> <sup>4</sup>*<sup>π</sup>* <sup>×</sup> <sup>10</sup>−7H/m the permeability of vacuum. The magnetization vector **<sup>M</sup>** is parallel to the field vector and follows a magnetization law *M* = *M*(*H*, *C*) dependent on the field intensity *H* and a particle concentration *C*. An equilibrium magnetization of magnetic fluids with account for interparticle interaction is derived in (Ivanov & Kuznetsova, 2001) in

∇ × **H** = **0**, ∇ · **B** = 0 in Ω*cell* × (−∞, +∞), (1)

*Ms* 3 *C C*0 *<sup>L</sup>*(*γH*), *<sup>γ</sup>* <sup>=</sup> <sup>3</sup>*χ<sup>L</sup>*

*Ms*

. (2)

Young-Laplace equation for equilibrium free-surface shapes of the fluid-air interface.

*L*(*γHe*), *He* = *H* +

The Maxwell's equations inside the magnetic fluid and in the air are

the framework of the modified model of the effective field

*C C*0

*M*(*H*, *C*) = *Ms*

the case of a hexagonal cell.

behavior in (Knieling et al., 2007; Matthies & Tobiska, 2005). A non-uniform equilibrium distribution of particles in the ferrofluid layer is computed in (Lavrova et al., 2010) for the first time.

In order to reach the equilibrium between concentration and the magnetic field, quite a long time is needed. The concentration remains almost constant for much shorter time scales. That is why, the validity of the results, mentioned in the previous paragraph, will not be abolished by the preset contribution. The aging process of the Rosensweig instability was experimentally studied over the long time (up to 50 days) in (Sudo et al., 2006). The effect of evaporation and non-evaporation of magnetic fluid on the pattern aging were examined. It was found that the interfacial spike pattern of magnetic fluids changes with time dramatically. Namely, the cell pattern gradually bifurcates at the constant magnetic field, and the number of spikes increases with time.

The particles are in Brownian motion inside the ferrofluid layer, when no magnetic field is applied, and the particle concentration is constant over the fluid volume. This is correct under an assumption that the gravity force has a negligible influence to the diffusion of particle. When the applied field is switched on but the field intensity is too weak to perturb the plane surface, then the magnetic field inside the layer remains constant and the particle concentration is constant, as a consequence. The situation changes when the applied field intensity is strong enough to perturb the free surface. A nonuniformity of the magnetic field inside the fluid causes a redistribution of the particles. This is due to interactions between field and particles and interparticle interactions. An interaction between particles and the magnetic field of the fluid was taken into account for the modelling of the Rosensweig instability in our previous research in (Lavrova et al., 2010). This interaction plays a dominant role in dilute magnetic fluids. The main objective of the contribution is the extension to the case of interacting particles, which should be taken into account for concentrated magnetic fluids.

Mathematical model of the coupled problem consists of the magnetostatic subproblem, the concentration subproblem and the free-surface subproblem. The concentration subproblem is based on a recently developed mass transfer equation for describing diffusion of interacting ferromagnetic particles in magnetic fluids (Pshenichnikov et al., 2011). Three subproblems are strictly interrelated to each other and should be solved simultaneously to resolve equilibrium states of the system. An iterative decoupling strategy is applied for solving the coupled system of equations. The finite-element method is used for discretization of the magnetostatic subproblem in a fixed domain. The Newton method is applied to find an element-wise distribution of the concentration at the finite-element mesh. The finite-difference approach is used for the free-surface subproblem. Numerical results of three models (model 1 nonuniform particle distribution without particle interaction, model 2 - nonuniform particle distribution with particle interaction, model 3 - uniform particle distribution) are compared. The effect of particle interaction shows a considerable influence on behavior of the ferrofluid layer in a uniform applied magnetic field.

#### **2. Mathematical model**

We consider a semi-infinite ferrofluid layer with a horizontal plane free surface bounded from above by a nonmagnetic gas (air). The unperturbed free surface is defined by equation *z* = 0, whereas the fluid corresponds to a region *z* < 0. The system is regarded under the action of gravity *g* = (0, 0, −*g*) and a uniform magnetic field *H*<sup>0</sup> = (0, 0, *H*0) normal to the plane free surface of the layer. We consider a single peak in the surface pattern with a cell Ω*cell* and a free surface Γ. The problem will be formulated in a cylinder Ω*cell* × (−∞, +∞), see Fig. 1 for the case of a hexagonal cell.

Fig. 1. The computational domain.

2 Will-be-set-by-IN-TECH

behavior in (Knieling et al., 2007; Matthies & Tobiska, 2005). A non-uniform equilibrium distribution of particles in the ferrofluid layer is computed in (Lavrova et al., 2010) for the

In order to reach the equilibrium between concentration and the magnetic field, quite a long time is needed. The concentration remains almost constant for much shorter time scales. That is why, the validity of the results, mentioned in the previous paragraph, will not be abolished by the preset contribution. The aging process of the Rosensweig instability was experimentally studied over the long time (up to 50 days) in (Sudo et al., 2006). The effect of evaporation and non-evaporation of magnetic fluid on the pattern aging were examined. It was found that the interfacial spike pattern of magnetic fluids changes with time dramatically. Namely, the cell pattern gradually bifurcates at the constant magnetic field, and the number

The particles are in Brownian motion inside the ferrofluid layer, when no magnetic field is applied, and the particle concentration is constant over the fluid volume. This is correct under an assumption that the gravity force has a negligible influence to the diffusion of particle. When the applied field is switched on but the field intensity is too weak to perturb the plane surface, then the magnetic field inside the layer remains constant and the particle concentration is constant, as a consequence. The situation changes when the applied field intensity is strong enough to perturb the free surface. A nonuniformity of the magnetic field inside the fluid causes a redistribution of the particles. This is due to interactions between field and particles and interparticle interactions. An interaction between particles and the magnetic field of the fluid was taken into account for the modelling of the Rosensweig instability in our previous research in (Lavrova et al., 2010). This interaction plays a dominant role in dilute magnetic fluids. The main objective of the contribution is the extension to the case of interacting particles, which should be taken into account for concentrated magnetic fluids. Mathematical model of the coupled problem consists of the magnetostatic subproblem, the concentration subproblem and the free-surface subproblem. The concentration subproblem is based on a recently developed mass transfer equation for describing diffusion of interacting ferromagnetic particles in magnetic fluids (Pshenichnikov et al., 2011). Three subproblems are strictly interrelated to each other and should be solved simultaneously to resolve equilibrium states of the system. An iterative decoupling strategy is applied for solving the coupled system of equations. The finite-element method is used for discretization of the magnetostatic subproblem in a fixed domain. The Newton method is applied to find an element-wise distribution of the concentration at the finite-element mesh. The finite-difference approach is used for the free-surface subproblem. Numerical results of three models (model 1 nonuniform particle distribution without particle interaction, model 2 - nonuniform particle distribution with particle interaction, model 3 - uniform particle distribution) are compared. The effect of particle interaction shows a considerable influence on behavior of the ferrofluid

We consider a semi-infinite ferrofluid layer with a horizontal plane free surface bounded from above by a nonmagnetic gas (air). The unperturbed free surface is defined by equation *z* = 0,

first time.

of spikes increases with time.

layer in a uniform applied magnetic field.

**2. Mathematical model**

The mathematical model for a non-uniform equilibrium distribution of ferromagnetic particles in a magnetic fluid with a free surface leads to a coupled problem formulation consisting of three subproblems. The first subproblem describes the magnetic field structure inside the fluid and in the surrounding air by the Maxwell's equations. The second subproblem concerns the diffusion of particles in the bulk of the fluid as a steady-state concentration problem. Finally, the third subproblem is given by the generalized Young-Laplace equation for equilibrium free-surface shapes of the fluid-air interface.

The Maxwell's equations inside the magnetic fluid and in the air are

$$
\nabla \times \mathbf{H} = \mathbf{0}, \quad \nabla \cdot \mathbf{B} = 0 \quad \text{in } \Omega\_{\text{cell}} \times (-\infty, +\infty), \tag{1}
$$

see e.g. (Rosensweig, 1998). Here **H** denotes the magnetic field, **B** = *μ*0(**H** + **M**) the magnetic induction, *<sup>μ</sup>*<sup>0</sup> <sup>=</sup> <sup>4</sup>*<sup>π</sup>* <sup>×</sup> <sup>10</sup>−7H/m the permeability of vacuum. The magnetization vector **<sup>M</sup>** is parallel to the field vector and follows a magnetization law *M* = *M*(*H*, *C*) dependent on the field intensity *H* and a particle concentration *C*. An equilibrium magnetization of magnetic fluids with account for interparticle interaction is derived in (Ivanov & Kuznetsova, 2001) in the framework of the modified model of the effective field

$$M(H, \mathbb{C}) = M\_{\mathbb{S}} \frac{\mathbb{C}}{\mathbb{C}\_0} L(\gamma H\_{\varepsilon}), \quad H\_{\varepsilon} = H + \frac{M\_{\mathbb{S}}}{\mathbb{S}} \frac{\mathbb{C}}{\mathbb{C}\_0} L(\gamma H), \quad \gamma = \frac{\mathfrak{Z}\chi\_L}{M\_{\mathbb{S}}}.\tag{2}$$

our model and get

(7) to fix the constant *cc*.

*(Polevikov & Tobiska, 2008)*

*dependence (8).*

*<sup>G</sup>*(1, *<sup>C</sup>*) = 1.38667(<sup>1</sup> <sup>+</sup> 1.28972*<sup>C</sup>* <sup>+</sup> 0.72543*C*2)

Numerical Study of Diffusion of Interacting Particles in a Magnetic Fluid Layer 187

The static distribution of particles in the cavity is obtained by equating the full particle flux to

To fix the constant *cc* the condition of conservation for the concentration over the fluid domain

*<sup>R</sup>*(*C*) = 7.838(1.517 <sup>+</sup> *<sup>C</sup>*)(4.844 <sup>+</sup> *<sup>C</sup>*)(2.258 <sup>−</sup> 2.862*<sup>C</sup>* <sup>+</sup> *<sup>C</sup>*2)(0.509 <sup>−</sup> 0.222*<sup>C</sup>* <sup>+</sup> *<sup>C</sup>*2)(0.688 <sup>+</sup> 1.282*<sup>C</sup>* <sup>+</sup> *<sup>C</sup>*2)

The second subproblem of the mathematical model is presented by equation (8) and condition

*Such a representation is similar in form to the solution for the concentration in dilute approximation*

*<sup>γ</sup><sup>H</sup>* , *const* <sup>=</sup> *<sup>C</sup>*0|Ω*<sup>f</sup>* <sup>|</sup>/

*Thus, for concentrated fluids the explicit dependence of C on H (9) is replaced by the implicit*

The third subproblem of the model defines a shape of the interface Γ between the magnetic fluid and the air. Equilibrium shapes of a free magnetic-fluid surface are described by the generalized Young-Laplace equation, which presents the force balance at the fluid-air interface

> 2 *<sup>M</sup> Hn H*

zero. According to (Pshenichnikov et al., 2011), it gives

3 − *C*

ln *<sup>C</sup>* <sup>+</sup> *<sup>R</sup>*(*C*) = ln sinh (*γHe*)

where *R*(*C*) is a rational function of the concentration

**Remark 2.1.** *Equation (8) can be reformulated as*

*C* = *const*

*CeR*(*C*) = *const*

sinh (*γH*)

*<sup>σ</sup>*<sup>K</sup> <sup>+</sup> *<sup>p</sup>*<sup>0</sup> <sup>=</sup> *<sup>p</sup>* <sup>+</sup> *<sup>μ</sup>*<sup>0</sup>

(<sup>1</sup> <sup>−</sup> *<sup>C</sup>*)<sup>3</sup> <sup>−</sup> *<sup>∂</sup>*(*C*2*G*(*λ*, *<sup>C</sup>*))

 Ω*<sup>f</sup>*

*γHe*

sinh (*γHe*) *γHe*

will be used. Let us substitute the function *G*(1, *C*) to equation (6), then

ln *C* +

(<sup>1</sup> <sup>+</sup> 0.308*C*)(<sup>1</sup> <sup>+</sup> 0.83333*C*) .

*<sup>∂</sup><sup>C</sup>* <sup>=</sup> ln sinh (*γHe*)

<sup>+</sup> *cc*, *He* <sup>=</sup> *<sup>H</sup>* <sup>+</sup> *<sup>χ</sup><sup>L</sup>*

, *He* <sup>=</sup> *<sup>H</sup>* <sup>+</sup> *<sup>χ</sup><sup>L</sup>*

*γHe*

*Cd*Ω = *C*0|Ω*<sup>f</sup>* | (7)

*γ C C*0

(<sup>1</sup> <sup>−</sup> *<sup>C</sup>*)3(1.2 <sup>+</sup> *<sup>C</sup>*)2(3.247 <sup>+</sup> *<sup>C</sup>*)<sup>2</sup> .

*γ C C*0

Ω*<sup>f</sup>*

<sup>2</sup>

*L*(*γH*).

*<sup>γ</sup><sup>H</sup> <sup>d</sup>*Ω. (9)

on Γ. (10)

sinh (*γH*)

+ *cc*. (6)

*L*(*γH*), (8)

Here *Ms* is the saturation magnetization, *C* the volumetric concentration and *C*<sup>0</sup> = <sup>1</sup> |Ω*<sup>f</sup>* | Ω*<sup>f</sup> Cdx*

for the fluid domain Ω*<sup>f</sup>* with the volume |Ω*<sup>f</sup>* |. *L*(*ξ*) = coth (*ξ*) − 1/*ξ* is the Langevin function, *γ* the Langevin parameter, *χ<sup>L</sup>* the initial susceptibility of the Langevin magnetization, *He* the effective field. The magnetization of air equals to zero. The magnetic field satisfies continuity conditions at the interface Γ between ferrofluid and air, see (Rosensweig, 1998)

$$[\mathbf{H} \cdot \boldsymbol{\tau}\_k] = 0, k = 1, 2 \qquad [\mathbf{B} \cdot \mathbf{n}] = 0 \quad \text{on } \Gamma. \tag{3}$$

Here [·] denotes a jump over the interface, *τ<sup>k</sup>* and **n** are tangential and normal vectors to the interface. A symmetry condition is specified at the side of a cylinder domain

$$\mathbf{H} \cdot \mathbf{n} = 0 \quad \text{on } \partial \Omega\_{cell} \times (-\infty, +\infty). \tag{4}$$

The uniform applied field *H*<sup>0</sup> is perturbed only in a neighborhood of the interface Γ. That is why we introduce asymptotic boundaries *z* = ±*δ*, far enough from the interface, and specify there a uniform magnetic field

$$\mathbf{H} = \mathbf{H}\_0 \quad \text{for } z = \delta, \quad \mathbf{H} = \mathbf{H}\_0^1 \quad \text{for } z = -\delta,\tag{5}$$

where *δ* > 0. The intensity of the applied field *H*<sup>0</sup> presents a control parameter of the model, whereas the field *H*<sup>1</sup> <sup>0</sup> = (0, 0, *<sup>H</sup>*<sup>1</sup> <sup>0</sup> ) will be computed from the second transmission condition (3), satisfied at the flat interface *z* = 0. The Maxwell's equations (1) together with conditions (3)-(5) present the first subproblem of the mathematical model.

The second subproblem describes a magnetophoresis process, i.e. the diffusion of ferromagnetic particles in the magnetic fluid under the action of a nonuniform magnetic field. A dynamic mass transfer equation for describing diffusion of interacting ferromagnetic particles in magnetic fluids was derived in (Pshenichnikov et al., 2011)

$$\frac{\partial \mathbb{C}}{\partial t} = -\nabla \cdot \left[ D\_0 K(\mathbb{C}) \left\{ - \left( 1 + \frac{2\mathbb{C}(4 - \mathbb{C})}{(1 - \mathbb{C})^4} - \mathbb{C} \frac{\partial^2 (\mathbb{C}^2 G(\lambda, \mathbb{C}))}{\partial \mathbb{C}^2} \right) \nabla \mathbb{C} + \mathbb{C} L(\gamma H\_\ell) \nabla(\gamma H\_\ell) \right\} \right]$$

in Ω*<sup>f</sup>* , *t* > 0. The equation is presented with an assumption that the gravity force has a negligible influence to the diffusion of particles. The constant *D*<sup>0</sup> denotes Einstein's value of the diffusion coefficient for dilute solutions,

$$K(\mathbf{C}) = (1 - 6.55\mathbf{C})$$

is the relative mobility of particles in the magnetic fluid. A function

$$G(\lambda, \mathbb{C}) = \frac{4}{3} \lambda^2 \frac{(1 + 0.04 \lambda^2)}{(1 + 0.308 \lambda^2 \mathbb{C})} \frac{(1 + 1.28972 \mathbb{C} + 0.72543 \mathbb{C}^2)}{(1 + 0.83333 \lambda \mathbb{C})}$$

specifies the contribution of a magnetodipole interaction to the free energy of the dipolar hard sphere (Pshenichnikov et al., 2011). Here *λ* is the dipolar coupling constant or the aggregation parameter, estimating the intensity of the magnetodipole interaction in comparison with thermal energy. The modified effective field model, which is used to describe the equilibrium magnetization (2), is applicable for *λ* ≤ 2 (Pshenichnikov & Lebedev, 2004). We take *λ* = 1 in our model and get

4 Will-be-set-by-IN-TECH

for the fluid domain Ω*<sup>f</sup>* with the volume |Ω*<sup>f</sup>* |. *L*(*ξ*) = coth (*ξ*) − 1/*ξ* is the Langevin function, *γ* the Langevin parameter, *χ<sup>L</sup>* the initial susceptibility of the Langevin magnetization, *He* the effective field. The magnetization of air equals to zero. The magnetic field satisfies continuity

Here [·] denotes a jump over the interface, *τ<sup>k</sup>* and **n** are tangential and normal vectors to the

The uniform applied field *H*<sup>0</sup> is perturbed only in a neighborhood of the interface Γ. That is why we introduce asymptotic boundaries *z* = ±*δ*, far enough from the interface, and specify

where *δ* > 0. The intensity of the applied field *H*<sup>0</sup> presents a control parameter of the model,

(3), satisfied at the flat interface *z* = 0. The Maxwell's equations (1) together with conditions

The second subproblem describes a magnetophoresis process, i.e. the diffusion of ferromagnetic particles in the magnetic fluid under the action of a nonuniform magnetic field. A dynamic mass transfer equation for describing diffusion of interacting ferromagnetic

(<sup>1</sup> <sup>−</sup> *<sup>C</sup>*)<sup>4</sup> <sup>−</sup> *<sup>C</sup> <sup>∂</sup>*2(*C*2*G*(*λ*, *<sup>C</sup>*))

in Ω*<sup>f</sup>* , *t* > 0. The equation is presented with an assumption that the gravity force has a negligible influence to the diffusion of particles. The constant *D*<sup>0</sup> denotes Einstein's value

*K*(*C*)=(1 − 6.55*C*)

specifies the contribution of a magnetodipole interaction to the free energy of the dipolar hard sphere (Pshenichnikov et al., 2011). Here *λ* is the dipolar coupling constant or the aggregation parameter, estimating the intensity of the magnetodipole interaction in comparison with thermal energy. The modified effective field model, which is used to describe the equilibrium magnetization (2), is applicable for *λ* ≤ 2 (Pshenichnikov & Lebedev, 2004). We take *λ* = 1 in

*∂C*<sup>2</sup>

[**H** · *τk*] = 0, *k* = 1, 2 [**B** · **n**] = 0 on Γ. (3)

**H** · **n** = 0 on *∂*Ω*cell* × (−∞, +∞). (4)

<sup>0</sup> ) will be computed from the second transmission condition

(1 + 1.28972*C* + 0.72543*C*2) (1 + 0.83333*λC*)

<sup>0</sup> for *z* = −*δ*, (5)

∇*C* + *CL*(*γHe*)∇(*γHe*)


Here *Ms* is the saturation magnetization, *C* the volumetric concentration and *C*<sup>0</sup> = <sup>1</sup>

conditions at the interface Γ between ferrofluid and air, see (Rosensweig, 1998)

interface. A symmetry condition is specified at the side of a cylinder domain

*H* = *H*<sup>0</sup> for *z* = *δ*, *H* = *H*<sup>1</sup>

there a uniform magnetic field

<sup>0</sup> = (0, 0, *<sup>H</sup>*<sup>1</sup>

(3)-(5) present the first subproblem of the mathematical model.

particles in magnetic fluids was derived in (Pshenichnikov et al., 2011)

is the relative mobility of particles in the magnetic fluid. A function

*<sup>λ</sup>*<sup>2</sup> (<sup>1</sup> <sup>+</sup> 0.04*λ*2) (1 + 0.308*λ*2*C*)

2*C*(4 − *C*)

whereas the field *H*<sup>1</sup>

*∂C*

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> −∇ ·

*D*0*K*(*C*)

 − 1 +

of the diffusion coefficient for dilute solutions,

*<sup>G</sup>*(*λ*,*C*) = <sup>4</sup>

3

$$G(1,\mathsf{C}) = \frac{1.38667(1 + 1.28972\mathsf{C} + 0.72543\mathsf{C}^2)}{(1 + 0.308\mathsf{C})(1 + 0.83333\mathsf{C})}.$$

The static distribution of particles in the cavity is obtained by equating the full particle flux to zero. According to (Pshenichnikov et al., 2011), it gives

$$\ln \mathcal{C} + \frac{3 - \mathcal{C}}{(1 - \mathcal{C})^3} - \frac{\partial (\mathcal{C}^2 G(\lambda, \mathcal{C}))}{\partial \mathcal{C}} = \ln \left( \frac{\sinh \left( \gamma H\_\ell \right)}{\gamma H\_\ell} \right) + c\_\mathcal{C}. \tag{6}$$

To fix the constant *cc* the condition of conservation for the concentration over the fluid domain

$$\int\_{\Omega\_f} \mathbb{C}d\Omega = \mathbb{C}\_0|\Omega\_f| \tag{7}$$

will be used. Let us substitute the function *G*(1, *C*) to equation (6), then

$$\ln \mathbb{C} + R(\mathbb{C}) = \ln \left( \frac{\sinh \left( \gamma H\_{\ell} \right)}{\gamma H\_{\ell}} \right) + c\_{\text{c} \prime} \quad H\_{\ell} = H + \frac{\chi\_L}{\gamma} \frac{\mathbb{C}}{\mathbb{C}\_0} L(\gamma H) , \tag{8}$$

where *R*(*C*) is a rational function of the concentration

$$R(\mathbb{C}) = \frac{7.838(1.517 + \mathbb{C})(4.844 + \mathbb{C})(2.258 - 2.862\mathbb{C} + \mathbb{C}^2)(0.509 - 0.222\mathbb{C} + \mathbb{C}^2)(0.688 + 1.282\mathbb{C} + \mathbb{C}^2)}{(1 - \mathbb{C})^3(1.2 + \mathbb{C})^2(3.247 + \mathbb{C})^2}.$$

The second subproblem of the mathematical model is presented by equation (8) and condition (7) to fix the constant *cc*.

**Remark 2.1.** *Equation (8) can be reformulated as*

$$\mathcal{C}e^{R(\mathbb{C})} = const \frac{\sinh\left(\gamma H\_{\mathfrak{E}}\right)}{\gamma H\_{\mathfrak{E}}}, \quad H\_{\mathfrak{E}} = H + \frac{\chi\_L}{\gamma} \frac{\mathcal{C}}{\mathbb{C}\_0} L(\gamma H).$$

*Such a representation is similar in form to the solution for the concentration in dilute approximation (Polevikov & Tobiska, 2008)*

$$\mathcal{C} = \text{const} \frac{\sinh\left(\gamma H\right)}{\gamma H}, \quad \text{const} = \mathcal{C}\_0 |\Omega\_f| / \int \frac{\sinh\left(\gamma H\right)}{\gamma H} d\Omega. \tag{9}$$

*Thus, for concentrated fluids the explicit dependence of C on H (9) is replaced by the implicit dependence (8).*

The third subproblem of the model defines a shape of the interface Γ between the magnetic fluid and the air. Equilibrium shapes of a free magnetic-fluid surface are described by the generalized Young-Laplace equation, which presents the force balance at the fluid-air interface

$$
\sigma \mathcal{K} + p\_0 = p + \frac{\mu\_0}{2} \left( M \frac{H\_{\text{fl}}}{H} \right)^2 \quad \text{on } \Gamma. \tag{10}
$$

*a*

Fig. 2. Top view of the hexagonal surface pattern with wavelength *λhex*.

⎧ ⎨ ⎩

[*φ*] = 0, �

(0, *a*) × (−*δ*, *δ*) and Γ*<sup>N</sup>* = *∂*Ω*ax* \ Γ*D*. We have from condition (5) that

*undisturbed interface z* = 0*. The potential is given in this case as*

� *<sup>μ</sup> ∂φ ∂z* � 1 + <sup>3</sup>*χ<sup>L</sup> γ C C*0 *L* �

> *<sup>μ</sup> ∂φ ∂n* �

> > *∂φ*

The boundary Γ*<sup>D</sup>* consists of the top and bottom boundaries of the rectangular domain Ω*ax* =

*<sup>φ</sup>D*(*r*, *<sup>z</sup>*) = � *<sup>H</sup>*0*<sup>z</sup>* for *<sup>z</sup>* <sup>=</sup> *<sup>δ</sup>*, *H*<sup>1</sup>

*φ* = *H*0*z for z* > 0 *and φ* = *H*<sup>1</sup>

<sup>=</sup> <sup>0</sup> <sup>⇒</sup> *<sup>μ</sup>*(*C*0, *<sup>H</sup>*<sup>1</sup>

<sup>0</sup> *z* for *z* = −*δ*.

<sup>0</sup> )*H*<sup>1</sup>

<sup>0</sup> *is defined from the second transmission condition (14), satisfied for the*

<sup>0</sup> = *H*<sup>0</sup> ⇒

<sup>0</sup> *z for z* < 0.

Here Ω*ax* := Ω<sup>1</sup> ∪ Ω<sup>2</sup> is a meridional cross-section of the 3D-domain Ω, where Ω<sup>1</sup> corresponds to the fluid and Ω<sup>2</sup> to the air. Differential operators are though in cylindrical coordinates with an assumption of an axial symmetry for the potential. The magnetostatic problem is closed by

Numerical Study of Diffusion of Interacting Particles in a Magnetic Fluid Layer 189

due to different magnetization in the fluid and the air

*μ*(*r*, *z*, *C*, |∇*φ*|) =

a set of conditions

**Remark 3.1.** *The constant H*<sup>1</sup>

*We get*

*λhex*

−∇· (*μ*(*r*, *z*, *C*, |∇*φ*|)∇*φ*) = 0, in Ω*ax*, (13)

*<sup>C</sup>*<sup>0</sup> *<sup>L</sup>*(*γ*|∇*φ*|)

� |∇*φ*<sup>|</sup> in <sup>Ω</sup>1,

= 0 for (*r*, *z*) ∈ Γ, (14)

*φ* = *φ<sup>D</sup>* for (*r*, *z*) ∈ Γ*D*, (15)

*<sup>∂</sup><sup>n</sup>* <sup>=</sup> 0 for (*r*, *<sup>z</sup>*) <sup>∈</sup> <sup>Γ</sup>*N*. (16)

*<sup>γ</sup>*|∇*φ*|+*χ<sup>L</sup> <sup>C</sup>*

1 in Ω2.

Here *σ* is the surface tension coefficient, K the sum of principal curvatures, *p*<sup>0</sup> the pressure in the air and *p* is the fluid pressure. The equation of hydrostatics for magnetic fluids is

$$\nabla p = -\rho g e\_z + \mu\_0 M \nabla H - \nabla \left[ -\mu\_0 \int\_0^H \mathcal{C} \left( \frac{\partial M}{\partial \mathcal{C}} \right)\_H dH + \mu\_0 \int\_0^H M dH \right],$$

see e.g. in (Rosensweig, 1998), where *ρ* is the fluid density and *ez* = (0, 0, 1). This equation allows us to express in explicit form the fluid pressure as

$$p = -\rho gz + \mu\_0 \int\_0^H \mathbb{C}\left(\frac{\partial M}{\partial \mathbb{C}}\right)\_H dH + c\_{f'} \tag{11}$$

where *c <sup>f</sup>* is an integration constant. The equation (10) for the fluid pressure (11) is

$$
\sigma \mathcal{K} = -\rho gz + f(\mathbb{C}, H) + \mathfrak{c}\_f \quad \text{on } \Gamma,\tag{12}
$$

.

where

$$f(\mathbb{C}, H) = \mu\_0 \int\_0^H \mathbb{C}\left(\frac{\partial M}{\partial \mathbb{C}}\right)\_H dH + \frac{\mu\_0}{2} \left(M \frac{H\_{\mathbb{H}}}{H}\right)^2$$

The constant *c <sup>f</sup>* will be determined by integrating equation (12) over the surface Γ, see Section 3.3 for details.

A solution of the magnetostatic subproblem (1)-(5) depends on the particle concentration and on the shape of the fluid-air interface. The concentration distribution is a solution of the nonlinear equation (8) and depends on the magnetic field configuration inside the ferrofluid. The fluid-air interface satisfies equation (12), which depends on the magnetic field and the concentration at the free surface of the ferrofluid. Three subproblems are strictly interrelated to each other and should be solved simultaneously to resolve equilibrium states of the system.

#### **3. Numerical methods and tools**

Experiments show that the surface pattern of the Rosensweig instability is presented by a hexagonal or square array of spikes, see e.g. (Gollwitzer et al., 2006). For sake of simplicity, we assume that the cell has a circular shape of radius *a* inscribed to a hexagonal cell of the pattern, see Fig. 2. It allows us to study axisymmetric solutions of the presented model in a two-dimensional geometry of cylindrical coordinates. Axisymmetric solutions on a circular cell were compared with three-dimensional ones on a hexagonal cell in (Lavrova et al., 2003). It was found that the profile shapes of both models nearly completely coincides. The different geometry of the cell results in a 10 % smaller amplitude of an axisymmetric peak in comparison with a three-dimensional solution.

#### **3.1 Magnetostatic subproblem**

Let us consider the magnetostatic subproblem (1)-(5) in a domain with the fixed interface and assume that the spacial distribution of the particle concentration is given. We express the magnetic field in terms of a scalar magnetic potential *φ* as **H** = ∇*φ* and reformulate the magnetostatic subproblem (1)-(5). The first Maxwell's equation (1) is exactly satisfied, whereas the second one takes form of an elliptic partial differential equation with jumping coefficients,

Fig. 2. Top view of the hexagonal surface pattern with wavelength *λhex*.

due to different magnetization in the fluid and the air

$$-\nabla \cdot \left(\mu(r, z, \mathbb{C}\_{\prime}|\nabla \phi|) \nabla \phi\right) = 0, \quad \text{in } \Omega\_{\text{d}\text{x}\prime} \tag{13}$$

$$\mu(r, z, \mathbb{C}\_{\prime}|\nabla \phi|) = \begin{cases} 1 + \frac{3\chi\_{\text{L}}}{\gamma} \frac{\mathbb{C}}{\mathbb{C}\_{0}} \frac{\mathbb{L}\left(\gamma|\nabla \phi| + \chi\_{\text{L}} \bigoplus\_{0} \mathbb{L}\left(\gamma|\nabla \phi|\right)\right)}{|\nabla \phi|} & \text{in } \Omega\_{1\prime} \\ 1 & \text{in } \Omega\_{2}. \end{cases}$$

Here Ω*ax* := Ω<sup>1</sup> ∪ Ω<sup>2</sup> is a meridional cross-section of the 3D-domain Ω, where Ω<sup>1</sup> corresponds to the fluid and Ω<sup>2</sup> to the air. Differential operators are though in cylindrical coordinates with an assumption of an axial symmetry for the potential. The magnetostatic problem is closed by a set of conditions

$$\left[\phi\right] = 0, \quad \left[\mu \frac{\partial \phi}{\partial n}\right] = 0 \qquad \text{for } (r, z) \in \Gamma,\tag{14}$$

$$
\phi = \phi\_D \qquad \text{for } (r, z) \in \Gamma\_{D\prime} \tag{15}
$$

$$\frac{\partial \phi}{\partial n} = 0 \qquad \text{for } (r, z) \in \Gamma\_N. \tag{16}$$

The boundary Γ*<sup>D</sup>* consists of the top and bottom boundaries of the rectangular domain Ω*ax* = (0, *a*) × (−*δ*, *δ*) and Γ*<sup>N</sup>* = *∂*Ω*ax* \ Γ*D*. We have from condition (5) that

$$\phi\_D(r,z) = \begin{cases} H\_0 z & \text{for } z = \delta\_\prime \\ H\_0^1 z & \text{for } z = -\delta\_\prime \end{cases}$$

**Remark 3.1.** *The constant H*<sup>1</sup> <sup>0</sup> *is defined from the second transmission condition (14), satisfied for the undisturbed interface z* = 0*. The potential is given in this case as*

$$
\phi = H\_0 z \quad \text{for } z > 0 \quad \text{and} \quad \phi = H\_0^1 z \quad \text{for } z < 0.
$$

*We get*

6 Will-be-set-by-IN-TECH

Here *σ* is the surface tension coefficient, K the sum of principal curvatures, *p*<sup>0</sup> the pressure in

 *H* 0 *C ∂M ∂C H*

see e.g. in (Rosensweig, 1998), where *ρ* is the fluid density and *ez* = (0, 0, 1). This equation

The constant *c <sup>f</sup>* will be determined by integrating equation (12) over the surface Γ, see

A solution of the magnetostatic subproblem (1)-(5) depends on the particle concentration and on the shape of the fluid-air interface. The concentration distribution is a solution of the nonlinear equation (8) and depends on the magnetic field configuration inside the ferrofluid. The fluid-air interface satisfies equation (12), which depends on the magnetic field and the concentration at the free surface of the ferrofluid. Three subproblems are strictly interrelated to each other and should be solved simultaneously to resolve equilibrium states of the system.

Experiments show that the surface pattern of the Rosensweig instability is presented by a hexagonal or square array of spikes, see e.g. (Gollwitzer et al., 2006). For sake of simplicity, we assume that the cell has a circular shape of radius *a* inscribed to a hexagonal cell of the pattern, see Fig. 2. It allows us to study axisymmetric solutions of the presented model in a two-dimensional geometry of cylindrical coordinates. Axisymmetric solutions on a circular cell were compared with three-dimensional ones on a hexagonal cell in (Lavrova et al., 2003). It was found that the profile shapes of both models nearly completely coincides. The different geometry of the cell results in a 10 % smaller amplitude of an axisymmetric peak

Let us consider the magnetostatic subproblem (1)-(5) in a domain with the fixed interface and assume that the spacial distribution of the particle concentration is given. We express the magnetic field in terms of a scalar magnetic potential *φ* as **H** = ∇*φ* and reformulate the magnetostatic subproblem (1)-(5). The first Maxwell's equation (1) is exactly satisfied, whereas the second one takes form of an elliptic partial differential equation with jumping coefficients,

 *H* 0 *C ∂M ∂C H*

where *c <sup>f</sup>* is an integration constant. The equation (10) for the fluid pressure (11) is

 *H* 0 *C ∂M ∂C H* *dH* + *μ*<sup>0</sup>

*σ*K = −*ρgz* + *f*(*C*, *H*) + *c <sup>f</sup>* on Γ, (12)

<sup>2</sup> .

*dH* <sup>+</sup> *<sup>μ</sup>*<sup>0</sup> 2 *<sup>M</sup> Hn H*

 *H* 0

*MdH* ,

*dH* + *c <sup>f</sup>* , (11)

the air and *p* is the fluid pressure. The equation of hydrostatics for magnetic fluids is

 −*μ*<sup>0</sup>

∇*p* = −*ρgez* + *μ*0*M*∇*H* − ∇

where

Section 3.3 for details.

**3. Numerical methods and tools**

**3.1 Magnetostatic subproblem**

in comparison with a three-dimensional solution.

allows us to express in explicit form the fluid pressure as

*f*(*C*, *H*) = *μ*<sup>0</sup>

*p* = −*ρgz* + *μ*<sup>0</sup>

$$\left[\mu \frac{\partial \phi}{\partial z}\right] = 0 \quad \Rightarrow \quad \mu (\mathbf{C}\_0 \, H\_0^1) H\_0^1 = H\_0 \quad \Rightarrow$$

Here *Np* denotes the number of unknowns and *φ<sup>i</sup>* = *φh*(*ξi*) is the nodal value of the potential at the grid point *ξi*. The iterative process needs 5-10 iterations to converge independent of the

Numerical Study of Diffusion of Interacting Particles in a Magnetic Fluid Layer 191

−1

**3.2 Concentration subproblem**

the constant *cc* satisfying (20)

Let us consider algebraic equation (8)

Fig. 3. Schematic representation of the finite element mesh. Thick solid line denotes the fluid-air interface. Bilinear interpolation is used for the mesh construction: inner node (filled

circular marker) is defined by eight boundary nodes (empty circular markers).

<sup>Φ</sup>(*C*, *<sup>H</sup>*) :<sup>=</sup> ln *<sup>C</sup>* <sup>+</sup> *<sup>R</sup>*(*C*) <sup>−</sup> ln sinh (*γ<sup>H</sup>* <sup>+</sup> *<sup>χ</sup><sup>L</sup>*

0

1

1

*C <sup>C</sup>*<sup>0</sup> *L*(*γH*)) = *cc*. (20)

*C <sup>C</sup>*<sup>0</sup> *L*(*γH*)

*Fi*(*Ci*, *cc* ) := Φ(*Ci*, *Hi*) − *cc* = 0, *i* = 1, . . . , *M* (21)

*γH* + *χ<sup>L</sup>*

and assume that a spacial configuration of the magnetic field *H* is given. The magnetic field is computed from a solution of the magnetostatic problem (13)-(16) as *Hh* = |∇*φh*|. The function *Hh* is a piecewise-constant approximation of the field *H* over the finite-element mesh and is given by the values *H*1,... *HM*, where *Hi* corresponds to the cell *Ti* and *M* is a number of cells in the fluid domain Ω1. For the given field values *H*1,... *HM* we have to find *C*1,... *CM* and

mesh size.

$$H\_0^1 + \frac{3\chi\_L}{\gamma} L(\gamma H\_0^1 + \chi\_L L(\gamma H\_0^1)) = H\_0. \tag{17}$$

*The nonlinear equation (17) is solved by the Newton method. Starting from the value H*0*, the method converges for 3-5 iterations for the relative error* 10−10*.*

Due to the problem reformulation in cylindrical coordinates and the assumption of axial symmetry, a corresponding variational problem is formulated in weighted Sobolev spaces:

Find *φ* ∈ *V*(Ω) such that

$$\int\_{\Omega} \mu(r, z, \mathbb{C}, |\nabla \phi|) \nabla \phi \cdot \nabla v r dr dz = 0 \quad \text{for any } v \in V\_D(\Omega), \tag{18}$$

and *φ* − *φ*<sup>0</sup> ∈ *VD*(Ω) for any *φ*<sup>0</sup> ∈ *V*(Ω) such that *φ*0|Γ*<sup>D</sup>* = *φD*.

$$V(\Omega) = \{v|\int\_{\Omega} v^2 r dr dz < \infty, \int\_{\Omega} |\nabla v|^2 r dr dz < \infty\}, \quad V\_D(\Omega) = \{v|v \in V(\Omega), v|\_{\Gamma\_D} = 0\}.$$

A structured triangular mesh is used for discretization. A schematic representation of the mesh is shown in Fig. 3. An algorithm of bilinear interpolation is applied for the mesh construction. This algorithm defines every interior grid point (filled circular marker in Fig. 3) by interpolation of eight boundary points (empty circular markers)

$$\tilde{\xi}(\mathbf{s},t) = (1-\mathbf{s})\tilde{\xi}\_{\mathcal{W}} + \mathbf{s}\tilde{\xi}\_{\mathcal{E}} + (1-t)(\tilde{\xi}\_{\mathcal{S}} - (1-\mathbf{s})\tilde{\xi}\_{\mathcal{SW}} - \mathbf{s}\tilde{\xi}\_{\mathcal{E}}) + t(\tilde{\xi}\_{\mathcal{N}} - (1-\mathbf{s})\tilde{\xi}\_{\mathcal{NW}} - \mathbf{s}\tilde{\xi}\_{\mathcal{ME}})\_{\mathcal{H}}$$

where parameters *s*, *t* ∈ (0, 1) and indices represent the quarter directions of boundary points (N - north, S - south, W - west, E - east), relative to the inner point *x*. To produce an interface-fitted mesh, the algorithm is applied in Ω<sup>1</sup> and Ω<sup>2</sup> separately, based on the pointwise representation of the interface, as a solution of the free-surface subproblem, and a uniform distribution of grid points at the boundary sides of Ω<sup>1</sup> and Ω2, see Fig. 3. A finite element mesh is reconstructed every time, when the interface is changed. By construction all meshes have the same topology. It allows to define an initial approximation of the potential at the new mesh as a solution of the magnetostatic problem at the old mesh without any interpolation.

The variational problem (18) is discretized by continuous piecewise linear functions on triangles for the given concentration *C*. Nonlinearities in the discrete equations are treated by a fixed-point iteration

$$\int\_{\Omega\_{\rm int}} \mu(r, z, \mathbb{C} \, | \, \nabla \phi\_h^k |) \nabla \phi\_h^{k+1} \cdot \nabla v\_h r dr dz = 0. \tag{19}$$

One Gauss-Seidel step is applied to the resulting system of linear equations in each iteration. The iterative process (19) continues until the relative a-posteriori error will be smaller than *�p* (generally 10−5)

$$\frac{\max\_{1 \le i \le N\_p} \left| \phi\_i^{k+1} - \phi\_i^k \right|}{\frac{1}{N\_p} \sum\_{i=1}^{N\_p} \left| \phi\_i^{k+1} \right|} < \varepsilon\_p.$$

Here *Np* denotes the number of unknowns and *φ<sup>i</sup>* = *φh*(*ξi*) is the nodal value of the potential at the grid point *ξi*. The iterative process needs 5-10 iterations to converge independent of the mesh size.

Fig. 3. Schematic representation of the finite element mesh. Thick solid line denotes the fluid-air interface. Bilinear interpolation is used for the mesh construction: inner node (filled circular marker) is defined by eight boundary nodes (empty circular markers).

#### **3.2 Concentration subproblem**

8 Will-be-set-by-IN-TECH

*The nonlinear equation (17) is solved by the Newton method. Starting from the value H*0*, the method*

Due to the problem reformulation in cylindrical coordinates and the assumption of axial symmetry, a corresponding variational problem is formulated in weighted Sobolev spaces:

A structured triangular mesh is used for discretization. A schematic representation of the mesh is shown in Fig. 3. An algorithm of bilinear interpolation is applied for the mesh construction. This algorithm defines every interior grid point (filled circular marker in Fig.

*ξ*(*s*, *t*)=(1 − *s*)*ξ<sup>W</sup>* + *sξ<sup>E</sup>* + (1 − *t*)(*ξ<sup>S</sup>* − (1 − *s*)*ξSW* − *sξSE*) + *t*(*ξ<sup>N</sup>* − (1 − *s*)*ξNW* − *sξNE*),

where parameters *s*, *t* ∈ (0, 1) and indices represent the quarter directions of boundary points (N - north, S - south, W - west, E - east), relative to the inner point *x*. To produce an interface-fitted mesh, the algorithm is applied in Ω<sup>1</sup> and Ω<sup>2</sup> separately, based on the pointwise representation of the interface, as a solution of the free-surface subproblem, and a uniform distribution of grid points at the boundary sides of Ω<sup>1</sup> and Ω2, see Fig. 3. A finite element mesh is reconstructed every time, when the interface is changed. By construction all meshes have the same topology. It allows to define an initial approximation of the potential at the new mesh as a solution of the magnetostatic problem at the old mesh without any interpolation. The variational problem (18) is discretized by continuous piecewise linear functions on triangles for the given concentration *C*. Nonlinearities in the discrete equations are treated

*h*|)∇*φk*+<sup>1</sup>

One Gauss-Seidel step is applied to the resulting system of linear equations in each iteration. The iterative process (19) continues until the relative a-posteriori error will be smaller than *�p*

> *<sup>i</sup>*=<sup>1</sup> <sup>|</sup>*φk*+<sup>1</sup> *i* |

< *�p*.

<sup>0</sup> <sup>+</sup> *<sup>χ</sup>LL*(*γH*<sup>1</sup>

*μ*(*r*, *z*, *C*, |∇*φ*|)∇*φ* · ∇*vrdrdz* = 0 for any *v* ∈ *VD*(Ω), (18)

<sup>2</sup>*rdrdz* <sup>&</sup>lt; <sup>∞</sup>}, *VD*(Ω) = {*v*|*<sup>v</sup>* <sup>∈</sup> *<sup>V</sup>*(Ω), *<sup>v</sup>*|Γ*<sup>D</sup>* <sup>=</sup> <sup>0</sup>}.

*<sup>h</sup>* · ∇*vhrdrdz* = 0. (19)

<sup>0</sup> )) = *H*0. (17)

*L*(*γH*<sup>1</sup>

*H*1 <sup>0</sup> + 3*χ<sup>L</sup> γ*

and *φ* − *φ*<sup>0</sup> ∈ *VD*(Ω) for any *φ*<sup>0</sup> ∈ *V*(Ω) such that *φ*0|Γ*<sup>D</sup>* = *φD*.

 Ω |∇*v*|

3) by interpolation of eight boundary points (empty circular markers)

*converges for 3-5 iterations for the relative error* 10−10*.*

*v*2*rdrdz* < ∞,

 Ω*ax*

*<sup>μ</sup>*(*r*, *<sup>z</sup>*, *<sup>C</sup>*, |∇*φ<sup>k</sup>*

max 1≤*i*≤*Np*

> 1 *Np* <sup>∑</sup>*Np*

 *<sup>φ</sup>k*+<sup>1</sup> *<sup>i</sup>* <sup>−</sup> *<sup>φ</sup><sup>k</sup> i* 

Find *φ* ∈ *V*(Ω) such that

*V*(Ω) = {*v*|

by a fixed-point iteration

(generally 10−5)

 Ω

 Ω

Let us consider algebraic equation (8)

$$\Phi(\mathbb{C}, H) := \ln \mathbb{C} + \mathbb{R}(\mathbb{C}) - \ln \left( \frac{\sinh \left( \gamma H + \chi\_L \frac{\mathbb{C}}{\mathbb{C}\_0} L(\gamma H) \right)}{\gamma H + \chi\_L \frac{\mathbb{C}}{\mathbb{C}\_0} L(\gamma H)} \right) = \mathbb{c}\_{\mathbb{C}}.\tag{20}$$

and assume that a spacial configuration of the magnetic field *H* is given. The magnetic field is computed from a solution of the magnetostatic problem (13)-(16) as *Hh* = |∇*φh*|. The function *Hh* is a piecewise-constant approximation of the field *H* over the finite-element mesh and is given by the values *H*1,... *HM*, where *Hi* corresponds to the cell *Ti* and *M* is a number of cells in the fluid domain Ω1. For the given field values *H*1,... *HM* we have to find *C*1,... *CM* and the constant *cc* satisfying (20)

$$F\_i(\mathbb{C}\_{i\prime}\mathbb{c}\_{\mathcal{C}}) := \Phi(\mathbb{C}\_{i\prime}H\_i) - \mathbb{c}\_{\mathcal{C}} = 0, \quad i = 1, \ldots, M \tag{21}$$

**3.3 Free-surface subproblem**

Equation (25) takes a new form

*<sup>f</sup>*(*C*, *<sup>H</sup>*) = *<sup>λ</sup>*Si <sup>2</sup>*<sup>γ</sup>*

and the effective field *He* = *H* + *<sup>χ</sup><sup>L</sup>*

*fI*(*C*, *H*) := *C*

 *H* 0

*<sup>f</sup>*(*C*, *<sup>H</sup>*) <sup>≈</sup> *<sup>λ</sup>*Si <sup>2</sup>*<sup>γ</sup>*

= *C* 

= *C* 

*fI*(*C*, *H*)*dH* ≈ *h*

3*χ<sup>L</sup> h* 1

3*χ<sup>L</sup>*

*∂*

 *H* 0 *C ∂*

> *γ C*

*<sup>∂</sup><sup>C</sup>* [*CL*(*γHe*)]

*<sup>L</sup>*(*γHe*) + *<sup>C</sup>*

*<sup>L</sup>*(*γHe*) + *<sup>C</sup>*

 1

The final form of the function *f*(*C*, *H*), used in computations, is

*<sup>a</sup>* <sup>=</sup> <sup>−</sup>*aρgz* <sup>+</sup> *<sup>μ</sup>*<sup>0</sup>

*σ* K

where

Let us write equation (12) for space variables dimensionless over *a*

*dH* <sup>+</sup> *<sup>μ</sup>*<sup>0</sup> 2 

We assume that the magnetic field *H* and the concentration *C* are given at the interface Γ. The magnetic field is determined from the fluid side as *H* = |∇*φ*1| and *Hn* = ∇*φ*<sup>1</sup> · **n** for the normal vector **n**, external to the fluid domain Ω1. We introduce dimensionless parameters

Numerical Study of Diffusion of Interacting Particles in a Magnetic Fluid Layer 193

*<sup>ρ</sup>g*/*σ*, Si = *<sup>μ</sup>*0*M*<sup>2</sup>

*<sup>∂</sup><sup>C</sup>* [*CL*(*γHe*)]

equations (25) and (26) and will be fixed later. The integrand in (27) is further transformed

*χLL*(*γH*)*L*�

*χLL*(*γH*)

We approximate the integral term in (27) by a composite trapezoidal rule on a uniform grid

Here *h* = *H*/*n*, (*n* + 1) is the number of the grid points, *Hi* = *ih* and *Ci* denotes the

*n*−1 ∑ *i*=1

To find concentration *Ci*, corresponding to the integration points *Hi*, we apply the Newton method to equation (20) for the given field *H* = *Hi* and the given *cc*. The value of constant *cc* is defined in the process of solving the concentration subproblem. For details of computations

*fI*(*Ci*, *Hi*)

<sup>2</sup> (*fI*(*C*0, 0) + *fI*(*Cn*, *<sup>H</sup>*)) <sup>+</sup>

 *H*

*C*0

*C*0

concentration, corresponding to the field *Hi*. *fI*(*C*0, 0) = 0, because *L*(0) = 0 and *L*�

<sup>2</sup> *fI*(*Cn*, *<sup>H</sup>*) +

*M*(*H*, *C*)

√*ρgσ*).

<sup>K</sup> <sup>=</sup> <sup>−</sup>*λ*2*<sup>z</sup>* <sup>+</sup> *<sup>f</sup>*(*C*, *<sup>H</sup>*) + *<sup>c</sup> <sup>f</sup>* on <sup>Γ</sup>, (26)

*dH* <sup>+</sup> *<sup>λ</sup>*Si

(*γHe*)<sup>2</sup> <sup>−</sup> <sup>1</sup>

*n*−1 ∑ *i*=1

<sup>+</sup> *<sup>λ</sup>*Si

*<sup>C</sup>*<sup>0</sup> *L*(*γH*). The unknown constant *c <sup>f</sup>* is different in

*CL*(*γHe*)

(sinh (*γHe*))<sup>2</sup>

*fI*(*Ci*, *Hi*)

*CL*(*γHe*)

*Hn H*

.

*Hn H*

<sup>2</sup>

. (28)

(0) = 1/3.

. (29)

<sup>2</sup>

(27)

*<sup>s</sup>* /(2

 *H*

> (*γHe*)

1

*Hn H*

<sup>2</sup>

+ *c <sup>f</sup>* on Γ. (25)

 *H* 0 *C ∂M ∂C H*

*λ* = *a*

.

and the integral condition (7)

$$F\_{M+1}(\mathbb{C}\_1, \dots, \mathbb{C}\_M) := \sum\_{i=1}^M \omega\_i \mathbb{C}\_i - \mathbb{C}\_0 \left(\sum\_{i=1}^M \omega\_i\right) = 0, \quad \omega\_i = 2\pi \int\_{T\_i} r dr dz. \tag{22}$$

Here *ω<sup>i</sup>* is the area of a circular element, obtained by rotating of the cell *Ti*. A system of *M* nonlinear equations (21) and one linear equation (22)

$$\mathbf{F}(\mathbf{x}) = \mathbf{0},$$

where **F** = (*F*1,..., *FM*+1)*T*, **x** = (*C*1,..., *CM*, *cc* )*T*, will be solved by the Newton method **<sup>x</sup>***k*+<sup>1</sup> <sup>=</sup> **<sup>x</sup>***<sup>k</sup>* <sup>−</sup> (**F**� (**x***<sup>k</sup>* ))−1**F**(**x***k*) (23)

Here **x***<sup>k</sup>* = (*C<sup>k</sup>* <sup>1</sup>,..., *<sup>C</sup><sup>k</sup> <sup>M</sup>*, *<sup>c</sup><sup>k</sup> <sup>c</sup>* )*<sup>T</sup>* and

$$\mathbf{F}'(\mathbf{x}) = \begin{pmatrix} \frac{\partial \Phi}{\partial \mathbf{C}}(\mathbf{C}\_1, H\_1) & 0 & 0 & \cdots & 0 & -1 \\ & 0 & \frac{\partial \Phi}{\partial \mathbf{C}}(\mathbf{C}\_2, H\_2) & 0 & \cdots & 0 & -1 \\ & & & & \cdots & \\ & & & & & \\ 0 & 0 & 0 & \cdots & \frac{\partial \Phi}{\partial \mathbf{C}}(\mathbf{C}\_M, H\_M) & -1 \\ & \omega\_1 & \omega\_2 & \omega\_3 \cdots & \omega\_M & 0 \\ \end{pmatrix}.$$

Eliminating *Ck*+<sup>1</sup> <sup>1</sup> ,..., *<sup>C</sup>k*+<sup>1</sup> *<sup>M</sup>* from the system (23) we get

$$c\_c^{k+1} = \sum\_{i=1}^M \frac{\Phi(\mathbf{C}\_{i\prime}^k, H\_i)}{\frac{\partial \Phi}{\partial \mathbf{C}\_i}(\mathbf{C}\_{i\prime}^k, H\_i)} / \sum\_{i=1}^M \frac{\partial \Phi}{\partial \mathbf{C}}(\mathbf{C}\_{i\prime}^k, H\_i).$$

Finally we compute *Ck*+<sup>1</sup> *<sup>i</sup>* from the *<sup>i</sup>*-th equation of the system (23) for the given *<sup>c</sup>k*+<sup>1</sup> *<sup>c</sup>*

$$\mathbb{C}\_{i}^{k+1} = \mathbb{C}\_{i}^{k} - \frac{\Phi(\mathbb{C}\_{i}^{k}, H\_{i}) - \mathfrak{c}\_{\mathbb{C}}^{k+1}}{\frac{\partial \Phi}{\partial \mathbb{C}}(\mathbb{C}\_{i}^{k}, H\_{i})}, \quad i = 1, \dots, M.$$

The computations for *Ck*+<sup>1</sup> <sup>1</sup> ,..., *<sup>C</sup>k*+<sup>1</sup> *<sup>M</sup>* can be realized in parallel.

The Newton method requires 3 − 5 iterations to converge for the relative a-posteriori error <sup>|</sup>**x***k*+<sup>1</sup> <sup>−</sup> **<sup>x</sup>***k*<sup>|</sup> to be smaller than *�<sup>c</sup>* (generally 10−9). We set *<sup>C</sup>*<sup>0</sup> *<sup>i</sup>* = *C*<sup>0</sup> at the initial Newton step. An initial value *c*<sup>0</sup> *<sup>c</sup>* is defined from the condition that far from the interface the concentration and the magnetic field intensity are known and equal *C*<sup>0</sup> and *H*<sup>1</sup> <sup>0</sup> , respectively. From equation (20) we get

$$c\_c^0 = \ln \mathbb{C}\_0 + R(\mathbb{C}\_0) - \ln \left( \frac{\sinh \left( \gamma H\_0^1 + \chi\_L L(\gamma H\_0^1) \right)}{\gamma H\_0^1 + \chi\_L L(\gamma H\_0^1)} \right). \tag{24}$$

#### **3.3 Free-surface subproblem**

Let us write equation (12) for space variables dimensionless over *a*

$$
\sigma \frac{\mathcal{K}}{a} = -a\rho gz + \mu\_0 \int\_0^H \mathbb{C}\left(\frac{\partial M}{\partial \mathbb{C}}\right)\_H dH + \frac{\mu\_0}{2} \left(M(H, \mathbb{C}) \frac{H\_{\mathbb{R}}}{H}\right)^2 + c\_f \quad \text{on } \Gamma. \tag{25}
$$

We assume that the magnetic field *H* and the concentration *C* are given at the interface Γ. The magnetic field is determined from the fluid side as *H* = |∇*φ*1| and *Hn* = ∇*φ*<sup>1</sup> · **n** for the normal vector **n**, external to the fluid domain Ω1. We introduce dimensionless parameters

$$
\lambda = a\sqrt{\rho \mathbf{g}/\sigma}, \quad \text{Si} = \mu\_0 M\_\text{s}^2/(2\sqrt{\rho \mathbf{g}\sigma}).
$$

Equation (25) takes a new form

$$\mathcal{K} = -\lambda^2 z + f(\mathbb{C}, H) + \mathfrak{c}\_f \quad \text{on } \Gamma,\tag{26}$$

where

10 Will-be-set-by-IN-TECH

Here *ω<sup>i</sup>* is the area of a circular element, obtained by rotating of the cell *Ti*. A system of *M*

**F**(**x**) = **0**,

*<sup>∂</sup><sup>C</sup>* (*C*1, *H*1) 0 0 ··· 0 −1

0 00 ··· *<sup>∂</sup>*<sup>Φ</sup>

*<sup>∂</sup><sup>C</sup>* (*C*2, *H*2) 0 ··· 0 −1

···

*∂*Φ *<sup>∂</sup><sup>C</sup>* (*C<sup>k</sup>*

*<sup>i</sup>* , *Hi*) , *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>M</sup>*.

*<sup>c</sup>* is defined from the condition that far from the interface the concentration

<sup>0</sup> <sup>+</sup> *<sup>χ</sup>LL*(*γH*<sup>1</sup>

<sup>0</sup> <sup>+</sup> *<sup>χ</sup>LL*(*γH*<sup>1</sup>

<sup>0</sup> ))

�

0 )

sinh (*γH*<sup>1</sup>

*γH*<sup>1</sup>

*<sup>i</sup>* from the *<sup>i</sup>*-th equation of the system (23) for the given *<sup>c</sup>k*+<sup>1</sup> *<sup>c</sup>*

*<sup>i</sup>* , *Hi*) <sup>−</sup> *<sup>c</sup>k*+<sup>1</sup> *<sup>c</sup>*

*<sup>M</sup>* can be realized in parallel.

The Newton method requires 3 − 5 iterations to converge for the relative a-posteriori error

�

*ω*<sup>1</sup> *ω*<sup>2</sup> *ω*<sup>3</sup> ··· *ω<sup>M</sup>* 0

where **F** = (*F*1,..., *FM*+1)*T*, **x** = (*C*1,..., *CM*, *cc* )*T*, will be solved by the Newton method

**<sup>x</sup>***k*+<sup>1</sup> <sup>=</sup> **<sup>x</sup>***<sup>k</sup>* <sup>−</sup> (**F**�

0 *<sup>∂</sup>*<sup>Φ</sup>

*<sup>M</sup>* from the system (23) we get

*∂*Φ *∂Ci* (*C<sup>k</sup> <sup>i</sup>* , *Hi*) / *M* ∑ *i*=1

*<sup>i</sup>* <sup>−</sup> <sup>Φ</sup>(*C<sup>k</sup>*

Φ(*C<sup>k</sup> <sup>i</sup>* , *Hi*)

> *∂*Φ *<sup>∂</sup><sup>C</sup>* (*C<sup>k</sup>*

*M* ∑ *i*=1 � *M* ∑ *i*=1 *ωi* �

= 0, *ω<sup>i</sup>* = 2*π*

� *Ti*

(**x***k*))−1**F**(**x***k*) (23)

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ .

*<sup>i</sup>* = *C*<sup>0</sup> at the initial Newton step.

<sup>0</sup> , respectively. From equation

. (24)

*<sup>∂</sup><sup>C</sup>* (*CM*, *HM*) −1

*<sup>i</sup>* , *Hi*).

*rdrdz*. (22)

and the integral condition (7)

Here **x***<sup>k</sup>* = (*C<sup>k</sup>*

Eliminating *Ck*+<sup>1</sup>

Finally we compute *Ck*+<sup>1</sup>

The computations for *Ck*+<sup>1</sup>

An initial value *c*<sup>0</sup>

(20) we get

<sup>1</sup>,..., *<sup>C</sup><sup>k</sup>*

**F**� (**x**) =

<sup>1</sup> ,..., *<sup>C</sup>k*+<sup>1</sup>

*<sup>M</sup>*, *<sup>c</sup><sup>k</sup>*

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

*<sup>c</sup>* )*<sup>T</sup>* and

*∂*Φ

*<sup>c</sup>k*+<sup>1</sup> *<sup>c</sup>* =

<sup>1</sup> ,..., *<sup>C</sup>k*+<sup>1</sup>

<sup>|</sup>**x***k*+<sup>1</sup> <sup>−</sup> **<sup>x</sup>***k*<sup>|</sup> to be smaller than *�<sup>c</sup>* (generally 10−9). We set *<sup>C</sup>*<sup>0</sup>

and the magnetic field intensity are known and equal *C*<sup>0</sup> and *H*<sup>1</sup>

*<sup>c</sup>* = ln *C*<sup>0</sup> + *R*(*C*0) − ln

*Ck*+<sup>1</sup> *<sup>i</sup>* <sup>=</sup> *<sup>C</sup><sup>k</sup>*

*c*0

*FM*+1(*C*1,..., *CM*) :=

nonlinear equations (21) and one linear equation (22)

*M* ∑ *i*=1

*ωiCi* − *C*<sup>0</sup>

$$f(\mathbb{C}, H) = \lambda \text{Si} \frac{2\gamma}{3\chi\_L} \int\_0^H \mathbb{C}\left(\frac{\partial}{\partial \mathbb{C}} \left[\mathbb{C}L(\gamma H\_\ell)\right]\right)\_H dH + \lambda \text{Si}\left(\mathbb{C}L(\gamma H\_\ell)\frac{H\_\mathbb{II}}{H}\right)^2\tag{27}$$

and the effective field *He* = *H* + *<sup>χ</sup><sup>L</sup> γ C <sup>C</sup>*<sup>0</sup> *L*(*γH*). The unknown constant *c <sup>f</sup>* is different in equations (25) and (26) and will be fixed later. The integrand in (27) is further transformed

$$\begin{split} f\_{I}(\mathbb{C},H) &:= \mathbb{C}\left(\frac{\partial}{\partial \mathbb{C}}\left[\mathbb{C}L(\gamma H\_{\varepsilon})\right]\right)\_{H} \\ &= \mathbb{C}\left(L(\gamma H\_{\varepsilon}) + \frac{\mathbb{C}}{\mathbb{C}\_{0}}\chi\_{L}L(\gamma H)L'(\gamma H\_{\varepsilon})\right) \\ &= \mathbb{C}\left(L(\gamma H\_{\varepsilon}) + \frac{\mathbb{C}}{\mathbb{C}\_{0}}\chi\_{L}L(\gamma H)\left[\frac{1}{(\gamma H\_{\varepsilon})^{2}} - \frac{1}{(\sinh\left(\gamma H\_{\varepsilon}\right))^{2}}\right]\right). \end{split}$$

We approximate the integral term in (27) by a composite trapezoidal rule on a uniform grid

$$\int\_{0}^{H} f\_{\rm I}(\mathbb{C}, H) dH \approx h \left( \frac{1}{2} \left( f\_{\rm I}(\mathbb{C}\_{0}, 0) + f\_{\rm I}(\mathbb{C}\_{0}, H) \right) + \sum\_{i=1}^{n-1} f\_{\rm I}(\mathbb{C}\_{i}, H\_{i}) \right). \tag{28}$$

Here *h* = *H*/*n*, (*n* + 1) is the number of the grid points, *Hi* = *ih* and *Ci* denotes the concentration, corresponding to the field *Hi*. *fI*(*C*0, 0) = 0, because *L*(0) = 0 and *L*� (0) = 1/3. The final form of the function *f*(*C*, *H*), used in computations, is

$$f(\mathbb{C}, H) \approx \lambda \text{Si} \frac{2\gamma}{3\chi\_L} \text{h} \left( \frac{1}{2} f\_{\text{I}}(\mathbb{C}\_{\text{tr}}, H) + \sum\_{i=1}^{n-1} f\_{\text{I}}(\mathbb{C}\_i, H\_i) \right) + \lambda \text{Si} \left( \mathbb{C} L(\gamma H\_\ell) \frac{H\_\text{il}}{H} \right)^2. \tag{29}$$

To find concentration *Ci*, corresponding to the integration points *Hi*, we apply the Newton method to equation (20) for the given field *H* = *Hi* and the given *cc*. The value of constant *cc* is defined in the process of solving the concentration subproblem. For details of computations

approach to equations (30) with boundary conditions (32)

*r*¯ *k*+1 *<sup>N</sup>* − *r*¯

2 *Fk* <sup>0</sup> , *z*¯ *k*+1 *<sup>N</sup>* =

�<sup>2</sup> *z*¯ *k <sup>i</sup>* <sup>−</sup> <sup>1</sup> *r*¯*k N*

*k*+1 *N*−1 *<sup>h</sup>* <sup>=</sup> 1;

Numerical Study of Diffusion of Interacting Particles in a Magnetic Fluid Layer 195

� 1 *r*¯*k N*

*f*(*H<sup>k</sup>*

step size *h* = 1/*N*. The difference quotients correspond to the central derivatives (*r*¯◦

*<sup>i</sup>* ) + �

the second derivatives (*r*¯*ss*, *z*¯*ss*). Nonlinearities of equations (30) are resolved by iterations, resulting in a three-diagonal system for the unknown grid functions at the (*k* + 1)-th iteration. A relaxation technique with a parameter *τ<sup>f</sup>* is applied to improve numerical stability. We took

The model under study consists of the magnetostatic subproblem, the concentration subproblem and the free-surface subproblem. The magnetostatic subproblem is described by a nonlinear elliptic partial differential equation with jumping coefficients (13) for the magnetostatic potential, augmented by transmission and Dirichlet-Neumann boundary conditions (14)-(16). The concentration subproblem is presented by nonlinear algebraic equation (20) for the concentration, augmented by integral condition (7). The free-surface subproblem is described by a system of two nonlinear ordinary differential equations (30) for the parametric representation of the free surface, augmented by integral and boundary

We apply an iterative decoupling strategy for solving the coupled system of equations. It consists of three steps at every iteration. The first step is a numerical solution of the magnetostatic problem in a fixed domain and for a given distribution of the concentration. The finite element method is applied for the discretization, see Section 3.1 for details. The second step is a numerical solution of system of nonlinear equations (21)-(22) for the element-wise concentration at the finite-element mesh for the given magnetic field distribution, as a solution at the first step. The Newton method is applied for a solution of the system, see Section 3.2 for details. The third step is a numerical solution of the free-surface subproblem for the given magnetic field from the first step and the given concentration from the second step of the iterative procedure. The finite-difference method is applied for the discretization, see Section 3.3 for details. A relaxation technique is applied to the free surface representation at

*<sup>i</sup>* = 0, *i* = 1, ..., *N* − 1

� �*r*¯*<sup>k</sup>*

*<sup>i</sup>*−<sup>1</sup> <sup>+</sup> *<sup>r</sup>*¯*<sup>k</sup> i* 2

�<sup>2</sup> ⎤ ⎦;

> *s* , *z*¯◦ *s* ) and

*<sup>i</sup>* = 0, *i* = 1, ..., *N* − 1

⎡ ⎣ � *z*¯ *k <sup>i</sup>* − *z*¯ *k i*−1

�<sup>3</sup> *<sup>N</sup>* ∑ *i*=1

*<sup>i</sup>*=<sup>0</sup> are grid-functions uniformly distributed over the free surface with a

��� *r*¯ *k i* �2 − � *r*¯ *k i*−1 �2 � *f*(*H<sup>k</sup> i* ) � .

�<sup>2</sup> *<sup>N</sup>* ∑ *i*=1

> 1 *r*¯*k N*

1 *τf* � *r*¯ *k*+1 *ss*,*<sup>i</sup>* − *r*¯ *k ss*,*i* � + *r*¯ *k ss*,*<sup>i</sup>* + *z*¯ *k* ◦ *s*,*i Fk*

*r*¯ *k*+1 <sup>0</sup> = 0,

*k*+1 0 *<sup>h</sup>* <sup>=</sup> *<sup>h</sup>*

1 *τf* � *z*¯ *<sup>k</sup>*+<sup>1</sup> *ss*,*<sup>i</sup>* <sup>−</sup> *<sup>z</sup>*¯ *k ss*,*i* � + *z*¯ *k ss*,*<sup>i</sup>* − *r*¯ *k* ◦ *s*,*i Fk*

*z*¯ *k*+1 <sup>1</sup> − *z*¯

*z*¯ *k* ◦ *s*,*i r*¯ *k i*

*τ<sup>f</sup>* = 0.01 in computations.

**3.4 Decoupling strategy**

conditions (31)-(32).

+ *λ*<sup>2</sup> � 1 *r*¯*k N*

*<sup>i</sup>*=<sup>0</sup> and {*z*¯*i*}*<sup>N</sup>*

*Fk <sup>i</sup>* = −

Here {*r*¯*i*}*<sup>N</sup>*

see Section 3.2. Test computations of the integral approximation (28) for *n* = 10 and *n* = 20 show changes in the fifth significant digit. We use *n* = 20 for our computations.

The free boundary Γ is represented by an arc-length parametrization

$$\Gamma = \{(r, z) \mid r = r(s), z = z(s), s = [0, \ell]\}\_{\prime}$$

where the parameter *s* is measured from the top of the peak (*s* = 0) to the peak foot (*s* = �). Following the approach in (Polevikov, 2004), we reformulate equation (26) as a system of second-order ordinary differential equations

$$
\begin{aligned}
\tilde{\tau}^{\prime\prime} &= -\tilde{\tau}^{\prime}\mathbf{F}, \quad \tilde{\boldsymbol{z}}^{\prime\prime} = \tilde{\tau}^{\prime}\boldsymbol{F} \qquad 0 < \tilde{\mathbf{s}} < 1; \\
\boldsymbol{F} &= -\frac{\tilde{\mathbf{z}}^{\prime}}{\tilde{\mathbf{r}}} + \lambda^2 \ell^2 \bar{\boldsymbol{z}} - \ell f(\boldsymbol{H}) + \mathbf{c}\_f. \end{aligned} \tag{30}
$$

Here *r*¯(*s*¯) = *r*(*s*)/�, *z*¯(*s*¯) = *z*(*s*)/� and *s*¯ = *s*/� are scaled versions of the space variables, introduced to have a formulation at the fixed domain [0, 1] instead of the changing and a-priori unknown domain [0, �]. The boundary condition *r*(�) = 1, transformed to *r*¯(1) = 1/¯ � in new variables, allows us to express the unknown parameter � of the problem as

$$\ell = \frac{1}{\bar{r}(1)}.$$

The constant *c <sup>f</sup>* is determined by integrating equation (30) over *s*¯

$$\int\_0^1 \left(\overline{r}\overline{z}'\right)' d\overline{s} = \int\_0^1 \left(\lambda^2 \ell^2 \overline{z} - \ell f(H) + c\_f\right) \overline{r}\overline{r}' d\overline{s}.$$

The left-hand side equals zero, because *r*¯(0) = 0 and *z*¯� (1) = 0. The right-hand side gives that

$$c\_f = 2\ell^3 \int\_0^1 f(H) \overline{r} \overline{r}' d\overline{s}\_\prime$$

using the volume conservation condition

$$\int\_{0}^{1} \overline{z}\overline{r}\overline{r}'d\overline{s} = 0. \tag{31}$$

Equations (30) are augmented by boundary conditions

$$\begin{array}{l} \overline{\tau}(0) = 0, \\ \overline{\tau}'(1) = 1, \\ \overline{z}'(0) = 0, \\ \overline{z}(1) = \ell^2 \int\_0^1 \overline{r}^2 \overline{z}' d\overline{s}. \end{array} \tag{32}$$

The nonlocal boundary condition is due to the integration by parts of the volume conservation condition (31).

An iterative finite-difference scheme of the second order approximation for the parametric Young-Laplace equations was developed in (Polevikov, 2004). We apply the developed approach to equations (30) with boundary conditions (32)

$$\begin{aligned} \frac{1}{\tau\_f} \left( \overline{r}\_{ss,i}^{k+1} - \overline{r}\_{ss,i}^k \right) + \overline{r}\_{ss,i}^k + \overline{z}\_{\stackrel{\circ}{s},i}^k F\_i^k &= 0, \quad i = 1, \ldots, N - 1, \\\ \overline{r}\_0^{k+1} &= 0, \quad \frac{\overline{r}\_N^{k+1} - \overline{r}\_{N-1}^{k+1}}{h} = 1; \end{aligned}$$

$$\begin{split} & \frac{1}{\tau\_{f}} \left( \tilde{z}\_{\mathrm{ss},i}^{k+1} - \bar{z}\_{\mathrm{ss},i}^{k} \right) + \bar{z}\_{\mathrm{ss},i}^{k} - \bar{r}\_{\mathrm{s},i}^{k} F\_{i}^{k} = 0, \quad i = 1, \ldots, N - 1 \\ & \frac{\bar{z}\_{1}^{k+1} - \bar{z}\_{0}^{k+1}}{h} = \frac{h}{2} F\_{0'}^{k}, \bar{z}\_{N}^{k+1} = \left( \frac{1}{\bar{r}\_{N}^{k}} \right)^{2} \sum\_{i=1}^{N} \left[ \left( \bar{z}\_{i}^{k} - \bar{z}\_{i-1}^{k} \right) \left( \frac{\bar{r}\_{i-1}^{k} + \bar{r}\_{i}^{k}}{2} \right)^{2} \right]; \\\\ & F\_{i}^{k} = -\frac{\bar{z}\_{\mathrm{s},i}^{k}}{\bar{r}\_{i}^{k}} + \lambda^{2} \left( \frac{1}{\bar{r}\_{N}^{k}} \right)^{2} \bar{z}\_{i}^{k} - \frac{1}{\bar{r}\_{N}^{k}} f(H\_{i}^{k}) + \left( \frac{1}{\bar{r}\_{N}^{k}} \right)^{3} \sum\_{i=1}^{N} \left[ \left( \left( \bar{r}\_{i}^{k} \right)^{2} - \left( \bar{r}\_{i-1}^{k} \right)^{2} \right) f(H\_{i}^{k}) \right]. \end{split}$$

Here {*r*¯*i*}*<sup>N</sup> <sup>i</sup>*=<sup>0</sup> and {*z*¯*i*}*<sup>N</sup> <sup>i</sup>*=<sup>0</sup> are grid-functions uniformly distributed over the free surface with a step size *h* = 1/*N*. The difference quotients correspond to the central derivatives (*r*¯◦ *s* , *z*¯◦ *s* ) and the second derivatives (*r*¯*ss*, *z*¯*ss*). Nonlinearities of equations (30) are resolved by iterations, resulting in a three-diagonal system for the unknown grid functions at the (*k* + 1)-th iteration. A relaxation technique with a parameter *τ<sup>f</sup>* is applied to improve numerical stability. We took *τ<sup>f</sup>* = 0.01 in computations.

#### **3.4 Decoupling strategy**

12 Will-be-set-by-IN-TECH

see Section 3.2. Test computations of the integral approximation (28) for *n* = 10 and *n* = 20

Γ = {(*r*, *z*) | *r* = *r*(*s*), *z* = *z*(*s*),*s* = [0, �]},

where the parameter *s* is measured from the top of the peak (*s* = 0) to the peak foot (*s* = �). Following the approach in (Polevikov, 2004), we reformulate equation (26) as a system of

> �� = *r*¯ �

Here *r*¯(*s*¯) = *r*(*s*)/�, *z*¯(*s*¯) = *z*(*s*)/� and *s*¯ = *s*/� are scaled versions of the space variables, introduced to have a formulation at the fixed domain [0, 1] instead of the changing and a-priori unknown domain [0, �]. The boundary condition *r*(�) = 1, transformed to *r*¯(1) = 1/¯

> � <sup>=</sup> <sup>1</sup> *r*¯(1) .

> > 1 0

*<sup>λ</sup>*2�2*z*¯ <sup>−</sup> � *<sup>f</sup>*(*H*) + *<sup>c</sup> <sup>f</sup>*

*f*(*H*)*r*¯*r*¯ � *ds*¯,

<sup>0</sup> *<sup>r</sup>*¯2*z*¯� *ds*¯.

The nonlocal boundary condition is due to the integration by parts of the volume conservation

An iterative finite-difference scheme of the second order approximation for the parametric Young-Laplace equations was developed in (Polevikov, 2004). We apply the developed

*F* 0 < *s*¯ < 1;

*<sup>r</sup>*¯ <sup>+</sup> *<sup>λ</sup>*2�2*z*¯ <sup>−</sup> � *<sup>f</sup>*(*H*) + *<sup>c</sup> <sup>f</sup>* . (30)

 *r*¯*r*¯ � *ds*¯.

(1) = 0. The right-hand side gives that

*ds*¯ = 0. (31)

� in new

(32)

show changes in the fifth significant digit. We use *n* = 20 for our computations.

The free boundary Γ is represented by an arc-length parametrization

second-order ordinary differential equations

*r*¯ �� = −*z*¯ � *F*, *z*¯

*<sup>F</sup>* <sup>=</sup> <sup>−</sup> *<sup>z</sup>*¯ �

The constant *c <sup>f</sup>* is determined by integrating equation (30) over *s*¯

 1 0 *r*¯*z*¯ � � *ds*¯ =

using the volume conservation condition

condition (31).

The left-hand side equals zero, because *r*¯(0) = 0 and *z*¯�

Equations (30) are augmented by boundary conditions

variables, allows us to express the unknown parameter � of the problem as

 1 0 

*c <sup>f</sup>* = 2�<sup>3</sup>

 1 0 *z*¯*r*¯*r*¯ �

*r*¯(0) = 0, *r*¯�

*z*¯�

(1) = 1,

(0) = 0, *z*¯(1) = �<sup>2</sup> <sup>1</sup> The model under study consists of the magnetostatic subproblem, the concentration subproblem and the free-surface subproblem. The magnetostatic subproblem is described by a nonlinear elliptic partial differential equation with jumping coefficients (13) for the magnetostatic potential, augmented by transmission and Dirichlet-Neumann boundary conditions (14)-(16). The concentration subproblem is presented by nonlinear algebraic equation (20) for the concentration, augmented by integral condition (7). The free-surface subproblem is described by a system of two nonlinear ordinary differential equations (30) for the parametric representation of the free surface, augmented by integral and boundary conditions (31)-(32).

We apply an iterative decoupling strategy for solving the coupled system of equations. It consists of three steps at every iteration. The first step is a numerical solution of the magnetostatic problem in a fixed domain and for a given distribution of the concentration. The finite element method is applied for the discretization, see Section 3.1 for details. The second step is a numerical solution of system of nonlinear equations (21)-(22) for the element-wise concentration at the finite-element mesh for the given magnetic field distribution, as a solution at the first step. The Newton method is applied for a solution of the system, see Section 3.2 for details. The third step is a numerical solution of the free-surface subproblem for the given magnetic field from the first step and the given concentration from the second step of the iterative procedure. The finite-difference method is applied for the discretization, see Section 3.3 for details. A relaxation technique is applied to the free surface representation at

.

*z* = 0

whereas

in Fig. 4-Fig. 6.

model 3 in what follows.

of the instability occurs at *H*∗

instability occurs in a weaker field *H*∗

value of the pattern wavelength

radius *a* of the circular cell we have

The intensity *Hc* corresponds to the fluid domain. The critical field intensity at the air domain *H*<sup>∗</sup> is found from the transmission condition [**B** · **n**] = 0, satisfied at the unperturbed interface

Numerical Study of Diffusion of Interacting Particles in a Magnetic Fluid Layer 197

We get *H*<sup>∗</sup> ≈ 9.11 kA/m for the considered ferrofluid. The stability theory predicts a critical

We assume that the hexagonal pattern wavelength *λhex*, see Fig. 3, equals *λc*. Then for the

√

*ρg*/*σ* = 2*π*/

Two meshes have been used for computations to analyze an influence of the discretization refinement to the computational predictions. Table 1 shows the critical field, the maximum value of the particle concentration over the fluid domain and *z*-coordinate of the equilibrium free surface at the peak axis and the peak foot for the applied field *H*<sup>0</sup> = 9.2 kA/m at different meshes. The found difference in values allows us to conclude that computations at the mesh with 160 × 1600 nodes are accurate enough. This mesh has been used to get results

> 80 × 800 8.71 1.111383 1.186572 −0.258553 160 × 1600 8.65 1.115006 1.222114 -0.261704 % Difference 0.7 0.3 3 1.2

Results of two models, which account for a nonuniform particle distribution inside the ferrofluid layer will be compared with the results of the model with a uniform particle distribution. The first model assumes no interaction between particles and it was numerically studied in (Lavrova et al., 2010). The second model accounts for interaction between particles and is the subject of this contribution. The model with a uniform particle distribution is called

Computations for the first and the third models in (Lavrova et al., 2010) show that the onset

with the result of the linear stability analysis *H*<sup>∗</sup> ≈ 9.11 kA/m, which assumes a uniform particle distribution. It means, that the concentration effect does not influence to the onset of the instability in the frame of the model without particle interactions. Computations for the second model, which takes into account particle interactions, show that the onset of the

case influences to the critical field. A possible reason for this effect is that the interparticle interaction can intensify considerably the fluid mangetization and a small initial surface

<sup>1</sup> = *H*<sup>∗</sup>

<sup>2</sup> = 8.65 ± 0.01 kA/m. A concentration effect in this

<sup>3</sup> . This value nearly coincides

<sup>1</sup> = 9.12 ± 0.01 kA/m and *H*<sup>∗</sup>

Table 1. Values of some control parameters and their difference at different meshes.

*ρg*/*σ*.

3 = 2*π*/

*Hc* = *H*∗.

*ρg*/*σ*/

√ 3.

<sup>2</sup> , kA/m max (*C*)/*C*<sup>0</sup> *z*(0)/*a z*()/*a*

√ 3,

*M*(*Hc*, *C*0) *Hc*

*λ<sup>c</sup>* = 2*π*/

3 = *λc*/

 1 +

*a* = *λhex*/

Mesh *H*∗

√

*λ* = *a*

We assume that the parameter *λ* is fixed for any applied field intensity.

every iteration

$$r\_i^{n+1} = r\_i^n + \tau (r\_i^{n'} - r\_i^n), \quad z\_i^{n+1} = z\_i^n + \tau (z\_i^{n'} - z\_i^n), \quad i = \overline{0, M}.$$

It allows to suppress a rapid change of free surface shapes during iterations. We take initially *τ* = 0.1 and decrease this value to *τ* = 0.01 in the case of strong shape changes.

An initial surface configuration at the first iteration of the presented iterative algorithm is defined as a small perturbation of the flat surface with an amplitude of around 1 % of the cell radius. An initial concentration equals *C*0. The iterations are stopped when the change in the surface shape is smaller than a prescribed threshold *�* ( generally 10−7)

$$\max\_{0 \le i \le M} \left( \left| r\_i^{n+1} - r\_i^n \right|, \left| z\_i^{n+1} - z\_i^n \right| \right) < \epsilon.$$

The iterative process is controlled by the threshold *�*, whereas three subproblems are controlled by own thresholds *�p*, *�<sup>c</sup>* and *� <sup>f</sup>* .

All algorithms, discussed in Section 3, and the coupling of three subproblems were implemented in Fortran with the help of the software tools, earlier developed for the Rosensweig instability computations. Numerical results of the previous computations are published in (Bashtovoi et al., 2002; Lavrova et al., 2008; 2010).

#### **4. Results of computations**

Numerical calculations were performed for the magnetic fluid EMG 901 (Ferrotec) with the following characteristic properties: the initial susceptibility *χ* = 2.2, the density *ρ* = 1406 kg/m3, the surface tension coefficient *σ* = 0.025 kg/s2, the saturation magnetization *Ms* = 48 kA/m, the volumetric concentration, corresponding to the uniform particle distribution, *C*<sup>0</sup> = 0.1. The initial susceptibility of the Langevin magnetization *χ<sup>L</sup>* is related to the initial susceptibility of the effective-field magnetization (2) as

$$\chi = \chi\_L (1 + \chi\_L/3)\text{.}$$

see e.g. (Pshenichnikov et al., 1996). For the considered magnetic fluid we have that *χ* = 2.2 and *χ<sup>L</sup>* ≈ 1.47489. The control parameter of the model is the applied field intensity *H*0.

Computations are performed at the computational domain Ω*ax* = (0, *a*) × (−*δ*, *δ*) with *δ* = 5*a*. Computations with different *δ* have shown that the error caused by replacing the unbounded domain by a bounded one is less than 1%.

A linear stability analysis for the Rosensweig instability was carried out under the assumption of a uniform particles distribution in a layer of infinite thickness (Rosensweig, 1998). The stability theory predicts a critical value of the magnetic field intensity *Hc*, corresponding to the onset of instability, as a solution of the nonlinear equation

$$M(H\_{\mathfrak{c}})^2 = \frac{2\sqrt{\rho \mathfrak{g} \sigma}}{\mu\_0} \left( 1 + \left( 1 + \frac{M(H\_{\mathfrak{c}})}{H\_{\mathfrak{c}}} \right)^{-1/2} \left( 1 + \frac{\partial M}{\partial H}(H\_{\mathfrak{c}}) \right)^{-1/2} \right).$$

The intensity *Hc* corresponds to the fluid domain. The critical field intensity at the air domain *H*<sup>∗</sup> is found from the transmission condition [**B** · **n**] = 0, satisfied at the unperturbed interface *z* = 0

$$\left(1 + \frac{M(H\_{\mathcal{C}}, \mathbb{C}\_0)}{H\_{\mathcal{C}}}\right) H\_{\mathcal{C}} = H^\*.$$

We get *H*<sup>∗</sup> ≈ 9.11 kA/m for the considered ferrofluid. The stability theory predicts a critical value of the pattern wavelength

$$
\lambda\_{\mathfrak{c}} = 2\pi / \sqrt{\rho \mathfrak{g} / \sigma}.
$$

We assume that the hexagonal pattern wavelength *λhex*, see Fig. 3, equals *λc*. Then for the radius *a* of the circular cell we have

$$a = \lambda\_{\text{flex}} / \sqrt{3} = \lambda\_{\text{c}} / \sqrt{3} = 2\pi / \sqrt{\rho \lg \sigma} / \sqrt{3} \text{s}$$

whereas

14 Will-be-set-by-IN-TECH

*<sup>i</sup>* <sup>=</sup> *<sup>z</sup><sup>n</sup>*

It allows to suppress a rapid change of free surface shapes during iterations. We take initially

An initial surface configuration at the first iteration of the presented iterative algorithm is defined as a small perturbation of the flat surface with an amplitude of around 1 % of the cell radius. An initial concentration equals *C*0. The iterations are stopped when the change in the

The iterative process is controlled by the threshold *�*, whereas three subproblems are

All algorithms, discussed in Section 3, and the coupling of three subproblems were implemented in Fortran with the help of the software tools, earlier developed for the Rosensweig instability computations. Numerical results of the previous computations are

Numerical calculations were performed for the magnetic fluid EMG 901 (Ferrotec) with the following characteristic properties: the initial susceptibility *χ* = 2.2, the density *ρ* = 1406 kg/m3, the surface tension coefficient *σ* = 0.025 kg/s2, the saturation magnetization *Ms* = 48 kA/m, the volumetric concentration, corresponding to the uniform particle distribution, *C*<sup>0</sup> = 0.1. The initial susceptibility of the Langevin magnetization *χ<sup>L</sup>* is related to the initial

*χ* = *χL*(1 + *χL*/3),

see e.g. (Pshenichnikov et al., 1996). For the considered magnetic fluid we have that *χ* = 2.2 and *χ<sup>L</sup>* ≈ 1.47489. The control parameter of the model is the applied field intensity *H*0.

Computations are performed at the computational domain Ω*ax* = (0, *a*) × (−*δ*, *δ*) with *δ* = 5*a*. Computations with different *δ* have shown that the error caused by replacing the unbounded

A linear stability analysis for the Rosensweig instability was carried out under the assumption of a uniform particles distribution in a layer of infinite thickness (Rosensweig, 1998). The stability theory predicts a critical value of the magnetic field intensity *Hc*, corresponding to

> *M*(*Hc*) *Hc*

−1/2

1 + *∂M <sup>∂</sup><sup>H</sup>* (*Hc*)

−1/2

.

*<sup>i</sup>* <sup>+</sup> *<sup>τ</sup>*(*zn*�

*<sup>i</sup>* <sup>−</sup> *<sup>z</sup><sup>n</sup>*

*<sup>i</sup>* ), *i* = 0, *M*.

every iteration

*rn*+<sup>1</sup> *<sup>i</sup>* <sup>=</sup> *<sup>r</sup><sup>n</sup>*

controlled by own thresholds *�p*, *�<sup>c</sup>* and *� <sup>f</sup>* .

**4. Results of computations**

*<sup>i</sup>* <sup>+</sup> *<sup>τ</sup>*(*rn*�

*<sup>i</sup>* <sup>−</sup> *<sup>r</sup><sup>n</sup>*

surface shape is smaller than a prescribed threshold *�* ( generally 10−7)

 *rn*+<sup>1</sup> *<sup>i</sup>* <sup>−</sup> *<sup>r</sup><sup>n</sup> i* , *zn*+<sup>1</sup> *<sup>i</sup>* <sup>−</sup> *<sup>z</sup><sup>n</sup> i* < *�*.

max 0≤*i*≤*M*

published in (Bashtovoi et al., 2002; Lavrova et al., 2008; 2010).

susceptibility of the effective-field magnetization (2) as

the onset of instability, as a solution of the nonlinear equation

 1 + 1 +

√*ρgσ μ*0

domain by a bounded one is less than 1%.

*<sup>M</sup>*(*Hc*)<sup>2</sup> <sup>=</sup> <sup>2</sup>

*<sup>i</sup>* ), *<sup>z</sup>n*+<sup>1</sup>

*τ* = 0.1 and decrease this value to *τ* = 0.01 in the case of strong shape changes.

$$
\lambda = a\sqrt{\rho g/\sigma} = 2\pi/\sqrt{3}.
$$

We assume that the parameter *λ* is fixed for any applied field intensity.

Two meshes have been used for computations to analyze an influence of the discretization refinement to the computational predictions. Table 1 shows the critical field, the maximum value of the particle concentration over the fluid domain and *z*-coordinate of the equilibrium free surface at the peak axis and the peak foot for the applied field *H*<sup>0</sup> = 9.2 kA/m at different meshes. The found difference in values allows us to conclude that computations at the mesh with 160 × 1600 nodes are accurate enough. This mesh has been used to get results in Fig. 4-Fig. 6.


Table 1. Values of some control parameters and their difference at different meshes.

Results of two models, which account for a nonuniform particle distribution inside the ferrofluid layer will be compared with the results of the model with a uniform particle distribution. The first model assumes no interaction between particles and it was numerically studied in (Lavrova et al., 2010). The second model accounts for interaction between particles and is the subject of this contribution. The model with a uniform particle distribution is called model 3 in what follows.

Computations for the first and the third models in (Lavrova et al., 2010) show that the onset of the instability occurs at *H*∗ <sup>1</sup> = 9.12 ± 0.01 kA/m and *H*<sup>∗</sup> <sup>1</sup> = *H*<sup>∗</sup> <sup>3</sup> . This value nearly coincides with the result of the linear stability analysis *H*<sup>∗</sup> ≈ 9.11 kA/m, which assumes a uniform particle distribution. It means, that the concentration effect does not influence to the onset of the instability in the frame of the model without particle interactions. Computations for the second model, which takes into account particle interactions, show that the onset of the instability occurs in a weaker field *H*∗ <sup>2</sup> = 8.65 ± 0.01 kA/m. A concentration effect in this case influences to the critical field. A possible reason for this effect is that the interparticle interaction can intensify considerably the fluid mangetization and a small initial surface

but a more elongated shape. Fig. 5 shows that the concentration equals the volumetric value *C*<sup>0</sup> for *z*/*a* < −1 and the particle diffusion mechanism is present only near the free surface.

Numerical Study of Diffusion of Interacting Particles in a Magnetic Fluid Layer 199

Fig. 5. Equilibrium distribution of the particle concentration over the peak axis at the applied field *H*<sup>0</sup> = 9.2 kA/m. 1 - nonuniform particle distribution without particle interaction, 2 nonuniform particle distribution with particle interaction, 3 - uniform particle distribution.

The distribution of a *z*-component of the magnetic field vector inside of the ferrofluid and in the air is presented in Fig. 6 for three models. The magnetic field is uniform inside of

Fig. 6. Equilibrium distribution of the field intensity over the peak axis at the applied field *H*<sup>0</sup> = 9.2 kA/m. 1 - nonuniform particle distribution without particle interaction, 2 nonuniform particle distribution with particle interaction, 3 - uniform particle distribution.

perturbation is developed to a surface pattern in a weaker field if to compare with the models without particle interactions.

Fig. 4 shows equilibrium free-surface shapes for three models. A more elongated peak region is formed for solutions with a nonuniform particle distribution. Namely, the peak is 20 % higher for the model without particle interactions and 60 % higher for the model with particle interactions in comparison with the uniform case. A 20 % difference of the first model is due to the concentration effect, which intensifies a spacial nonuniformity of the fluid magnetization in the peak region. A 60 % difference of the second model is influenced also by the fact that the onset of the instability for the second model occurs at the weaker field *H*∗ <sup>2</sup> < *H*<sup>∗</sup> <sup>3</sup> . A further increase in the field strength results in the increasing peak amplitude, which can lead to a sizable difference in the peak height if we compare results of the second and the third model at the overcritical field *H*<sup>0</sup> = 9.2 kA/m. Fig. 4 contains isolines of the equilibrium concentration for the second model. The main inhomogeneity of the particle distribution occurs at the peak region. The concentration takes the greatest value at the top of the peak and the smallest value at the peak foot.

Fig. 4. Overcritical equilibrium free-surface shapes at the applied field *H*<sup>0</sup> = 9.2 kA/m. 1 - nonuniform particle distribution without particle interaction, 2 - nonuniform particle distribution with particle interaction, 3 - uniform particle distribution. Isolines of the concentration corresponds to *C*/*C*<sup>0</sup> = {0.995, 0.999, 1.001, 1.02, 1.05, 1.08, 1.1, 1.11}.

Fig. 5 shows the equilibrium distribution of the particle concentration over the peak axis for three models. The concentration increases monotonically in *z*-direction, moving along the peak axis, for the models with the nonuniform particle distribution and the concentration is constant for the third model. The concentration takes a value at the peak top which is about 25 % greater than in the fluid bulk for the first model and about 11 % greater for the second model. Taking into account the particle interaction, we get a smaller concentration at the peak 16 Will-be-set-by-IN-TECH

perturbation is developed to a surface pattern in a weaker field if to compare with the models

Fig. 4 shows equilibrium free-surface shapes for three models. A more elongated peak region is formed for solutions with a nonuniform particle distribution. Namely, the peak is 20 % higher for the model without particle interactions and 60 % higher for the model with particle interactions in comparison with the uniform case. A 20 % difference of the first model is due to the concentration effect, which intensifies a spacial nonuniformity of the fluid magnetization in the peak region. A 60 % difference of the second model is influenced also by the fact that the

increase in the field strength results in the increasing peak amplitude, which can lead to a sizable difference in the peak height if we compare results of the second and the third model at the overcritical field *H*<sup>0</sup> = 9.2 kA/m. Fig. 4 contains isolines of the equilibrium concentration for the second model. The main inhomogeneity of the particle distribution occurs at the peak region. The concentration takes the greatest value at the top of the peak and the smallest value

Fig. 4. Overcritical equilibrium free-surface shapes at the applied field *H*<sup>0</sup> = 9.2 kA/m. 1 - nonuniform particle distribution without particle interaction, 2 - nonuniform particle distribution with particle interaction, 3 - uniform particle distribution. Isolines of the concentration corresponds to *C*/*C*<sup>0</sup> = {0.995, 0.999, 1.001, 1.02, 1.05, 1.08, 1.1, 1.11}.

Fig. 5 shows the equilibrium distribution of the particle concentration over the peak axis for three models. The concentration increases monotonically in *z*-direction, moving along the peak axis, for the models with the nonuniform particle distribution and the concentration is constant for the third model. The concentration takes a value at the peak top which is about 25 % greater than in the fluid bulk for the first model and about 11 % greater for the second model. Taking into account the particle interaction, we get a smaller concentration at the peak

<sup>2</sup> < *H*<sup>∗</sup>

<sup>3</sup> . A further

onset of the instability for the second model occurs at the weaker field *H*∗

without particle interactions.

at the peak foot.

but a more elongated shape. Fig. 5 shows that the concentration equals the volumetric value *C*<sup>0</sup> for *z*/*a* < −1 and the particle diffusion mechanism is present only near the free surface.

Fig. 5. Equilibrium distribution of the particle concentration over the peak axis at the applied field *H*<sup>0</sup> = 9.2 kA/m. 1 - nonuniform particle distribution without particle interaction, 2 nonuniform particle distribution with particle interaction, 3 - uniform particle distribution.

The distribution of a *z*-component of the magnetic field vector inside of the ferrofluid and in the air is presented in Fig. 6 for three models. The magnetic field is uniform inside of

Fig. 6. Equilibrium distribution of the field intensity over the peak axis at the applied field *H*<sup>0</sup> = 9.2 kA/m. 1 - nonuniform particle distribution without particle interaction, 2 nonuniform particle distribution with particle interaction, 3 - uniform particle distribution.

**7. References**

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Aristidopoulou, A.A.; Papaioannou, A.T. & Boudouvis, A.G. (1996). Computational analysis

Numerical Study of Diffusion of Interacting Particles in a Magnetic Fluid Layer 201

Bashtovoi, V.G.; Lavrova, O.A.; Polevikov, V.K. & Tobiska, L. (2002). Computer modeling of

Bashtovoi, V.G.; Polevikov, V.K.; Suprun, A.E.; Stroots, A.V. & Beresnev, S.A. (2007). Influence

Bashtovoi, V.G.; Polevikov, V.K.; Suprun, A.E.; Stroots, A.V. & Beresnev, S.A. (2008). The effect

Boudouvis, A.G.; Puchala, J.L.; Scriven, L.E. & Rosensweig, R.E. (1987). Normal field

Cowley, M & Rosensweig, R. (1967). The interfacial instability of a ferromagnetic fluid. *Journal*

Gollwitzer, C.; Richter, R. & Rehberg, I. (2006). Via hexagons to squares in ferrofluids:

Gollwitzer, C.; Spyropoulos, A.N.; Papathanasiou, A.G.; Boudouvis, A.G. & Richter, R. (2009).

Ivanov, A.O. & Kuznetsova, O.B. (2001). Magnetic properties of dense ferrofluids: An influence of interparticle correlations. *Physical Review E*, Vol. 64, 041405. Knieling, H.; Richter, R.; Rehberg, I.; Matthies, G. & Lange, A. (2007). Growth of surface undulations at the Rosensweig instability. *Physical Review E*, Vol. 76, No. 6, 066301. Lange, A.; Richter, R. & Tobiska, L. (2007). Linear and nonlinear approach to the Rosensweig

Lavrova, O.; Matthies, G.; Mitkova, T.; Polevikov, V. & Tobiska, L. (2003). Finite

Lavrova, O.; Matthies, G. & Tobiska, L. (2008). Numerical study of soliton-like

Matthies, G. & Tobiska, L. (2005). Numerical simulation of normal-flied instability in the static and dynamic case. *Journal of Magnetism and Magnetic Materials*, Vol. 289, 346–349.

element methods for coupled problems in ferrohydrodynamics, In: *Lecture Notes in Computational Science and Engineering*, Bänsch, E. (Ed.), Vol. 35, 160-183,

surface configurations on a magnetic fluid layer in the Rosensweig instability. *Communications in Nonlinear Science and Numerical Simulation*, Vol. 13, 1302–1310. Lavrova, O.; Polevikov, V. & Tobiska, L. (2010). Numerical Study of the Rosensweig Instability

in a Magnetic Fluid Subject to Diffusion of Magnetic Particles. *Mathematical Modelling*

magnetic field. *Journal of Physics: Condensed Matter*, Vol. 18, S2643–2656. Gollwitzer, C.; Matthies, G.; Richter, R.; Rehberg, I. & Tobiska, L. (2007). The surface

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computations. *New Journal of Physics*, Vol. 11, 053016.

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fluid. *Magnetohydrodynamics*, Vol. 44, No. 2, 121–126.

*of Fluid Mechanics*, Vol. 30, No. 4, 671–688.

of free surface magnetohydrostatic equilibrium: force versus energy formulation.

the instability of a horizontal magnetic-fluid layer in a uniform magnetic field. *Journal*

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instability and patterns in pools of ferrofluid. *Journal of Magnetism and Magnetic*

experiments on hysteretic surface transformations under variation of the normal

topography of a magnetic fluid – a quantitative comparison between experiment and

The normal field instability under side-wall effects: comparison of experiments and

the ferrofluid far from the interface, *Hz* = *H*<sup>1</sup> <sup>0</sup> , and the particle diffusion mechanism is absent there. The field distribution for three models coincides for *z*/*a* < −1. The intensity of the field increases monotonically in *z*-direction, moving along the peak axis inside the ferrofluid, up to the value close to the applied field intensity *H*<sup>0</sup> at the interface. Crossing the interface, the field intensity jumps to the value 3.17*H*<sup>0</sup> for the first model, 2.79*H*<sup>0</sup> for the second model and 2.7*H*<sup>0</sup> for the third model. The intensity of the field decreases monotonically in *z*-direction, moving along the peak axis inside the air, up to the value of the applied field intensity *H*<sup>0</sup> far from the interface. All considered models show that the field non-uniformity occurs at the region −*a* < *z* < 2*a* and the field is nearly uniform outside of this region. It means that a restriction of the unbounded domain by a bounded one with −5*a* < *z* < 5*a* will introduce an insignificant error in computations.

#### **5. Conclusions**

The effect of particle interaction in ferrofluid has been taken into account in numerical simulations of the Rosensweig instability for the first time. The mathematical model is based on a recently developed mass transfer equation for describing diffusion of interacting ferromagnetic particles in ferrofluids in (Pshenichnikov et al., 2011). The mathematical model for the Rosensweig instability in homogeneous ferrofluids is augmented by the concentration subproblem in the form of the nonlinear algebraic equation of the particle concentration and the magnetic field intensity inside the ferrofluid. We suggest an approach for numerical solution of the concentration subproblem and for the coupling of the concentration subproblem to the magnetic field and free surface computations.

Based on comparison between three models (model 1 - nonuniform particle distribution without particle interaction, model 2 - nonuniform particle distribution with particle interaction, model 3 - uniform particle distribution), it has been shown that the effect of particle interaction cannot be neglected or replaced by simpler models which try to capture the diffusion of particles. It was found that the onset of the instability occurs in a weaker field and the peak region is more elongated, if we take into account the particle interactions.

The effect of particle interaction on the pattern wavelength can not be numerically analyzed in the frame of the presented mathematical model. The pattern wavelength is fixed by the critical value obtained in the frame of the linear stability analysis. The experimental study of the aging variation of the Rosensweig instability in (Sudo et al., 2006) shows, however, that the pattern wavelength changes with time dramatically. Namely, the cell pattern gradually bifurcates at the constant magnetic field, and the number of spikes increases with time. That is why we plan for the future to consider a soliton-like surface configurations in the Rosensweig instability (Richter & Barashenkov, 2005). The pattern wavelength is a variable of this problem and the effect of particle interaction to the wavelength selection can be numerically analyzed.

#### **6. Acknowledgements**

The authors would like to thank Prof. A. Ivanov for stimulating discussions and Prof. A. Pshenichnikov for the helpful comments on the formulation of the mathematical model.

#### **7. References**

18 Will-be-set-by-IN-TECH

there. The field distribution for three models coincides for *z*/*a* < −1. The intensity of the field increases monotonically in *z*-direction, moving along the peak axis inside the ferrofluid, up to the value close to the applied field intensity *H*<sup>0</sup> at the interface. Crossing the interface, the field intensity jumps to the value 3.17*H*<sup>0</sup> for the first model, 2.79*H*<sup>0</sup> for the second model and 2.7*H*<sup>0</sup> for the third model. The intensity of the field decreases monotonically in *z*-direction, moving along the peak axis inside the air, up to the value of the applied field intensity *H*<sup>0</sup> far from the interface. All considered models show that the field non-uniformity occurs at the region −*a* < *z* < 2*a* and the field is nearly uniform outside of this region. It means that a restriction of the unbounded domain by a bounded one with −5*a* < *z* < 5*a* will introduce an

The effect of particle interaction in ferrofluid has been taken into account in numerical simulations of the Rosensweig instability for the first time. The mathematical model is based on a recently developed mass transfer equation for describing diffusion of interacting ferromagnetic particles in ferrofluids in (Pshenichnikov et al., 2011). The mathematical model for the Rosensweig instability in homogeneous ferrofluids is augmented by the concentration subproblem in the form of the nonlinear algebraic equation of the particle concentration and the magnetic field intensity inside the ferrofluid. We suggest an approach for numerical solution of the concentration subproblem and for the coupling of the concentration

Based on comparison between three models (model 1 - nonuniform particle distribution without particle interaction, model 2 - nonuniform particle distribution with particle interaction, model 3 - uniform particle distribution), it has been shown that the effect of particle interaction cannot be neglected or replaced by simpler models which try to capture the diffusion of particles. It was found that the onset of the instability occurs in a weaker field and the peak region is more elongated, if we take into account the particle interactions.

The effect of particle interaction on the pattern wavelength can not be numerically analyzed in the frame of the presented mathematical model. The pattern wavelength is fixed by the critical value obtained in the frame of the linear stability analysis. The experimental study of the aging variation of the Rosensweig instability in (Sudo et al., 2006) shows, however, that the pattern wavelength changes with time dramatically. Namely, the cell pattern gradually bifurcates at the constant magnetic field, and the number of spikes increases with time. That is why we plan for the future to consider a soliton-like surface configurations in the Rosensweig instability (Richter & Barashenkov, 2005). The pattern wavelength is a variable of this problem and the effect of particle interaction to the wavelength selection can be numerically analyzed.

The authors would like to thank Prof. A. Ivanov for stimulating discussions and Prof. A. Pshenichnikov for the helpful comments on the formulation of the mathematical model.

subproblem to the magnetic field and free surface computations.

<sup>0</sup> , and the particle diffusion mechanism is absent

the ferrofluid far from the interface, *Hz* = *H*<sup>1</sup>

insignificant error in computations.

**5. Conclusions**

**6. Acknowledgements**


**10** 

*Algeria* 

**Finite Element Method Applied to the** 

M'hemed Rachek and Tarik Merzouki *University Mouloud Mammeri of Tizi-Ouzou* 

**Modelling and Analysis of Induction Motors** 

During the past decades, the development of solution methods and the growth of computer capacities have made its possible to solve more and more involved magnetic field problems. Thus, numerical techniques essentially based on the Finite Elements Method (FEM) have been used and has gradually become a standard in electrical machine modelling-design, analysis and optimisation. Electrical machines are electromagnetic devices with combined constrains such as complex geometries and several physical phenomena's. To model them, we must solve the magnetic field non-linear Partial Differential Equation (PDE) derived from the Maxwell's equations combined to the materials properties, and their coupling with phenomena that exist in electromagnetic structures, such as electric circuits, and mechanical

Induction Motor (IM) is an electromagnetic-mechanical actuator where strongly interacts several phenomena such as magnetic field, electrical circuits, mechanical motion. The aim of this chapter is to present an implementation of the finite element method for the modelling of rotating electrical machines, especially the squirrel cage three-phase induction motors. The generalized model consists firstly on strong coupling between the partial differential equation of the magnetic field diffusion and the electric circuits equations obtained from Kirchhoff laws. The model integrates as well realistic geometries, and the non-linear properties of the magnetic materials, as voltage supply of the stator windings. Secondly, the mechanical equation including the rotor movement effects is coupled to the electromagnetic phenomenon through the magnetic force responsible of

The governing magnetic field time-dependent equation derived from Maxwell formalism is expressed in term of Magnetic Vector Potential (MVP) with only z-direction component for the cases of two dimensional (x,y) cartesian coordinates. The induction motors stator windings are usually in star or delta connection, then the source term of the magnetic field is explicitly an applied line voltage or implicitly the magnetizing current. The squirrel rotor cage is formed by massive conductive bars short-circuited at their ends through massive and conductive end-rings. Mathematically, the squirrel rotor cage appear as a polyphases circuits modelled by the same way that the stator windings but with affecting a zero voltage

**1. Introduction** 

the rotor motion.

motional equations. (Arkkio, 1987; Benali, 1997).

for each adjacent bars with theirs end-rings portion.


### **Finite Element Method Applied to the Modelling and Analysis of Induction Motors**

M'hemed Rachek and Tarik Merzouki *University Mouloud Mammeri of Tizi-Ouzou Algeria* 

#### **1. Introduction**

20 Will-be-set-by-IN-TECH

202 Numerical Modelling

Polevikov, V. (2004). Methods for numerical modeling of two-dimensional capillary surfaces.

Polevikov, V. & Tobiska, L. (2008). On the solution of the steady-state diffusion problem for

Polevikov, V. & Tobiska, L. (2011). Influence of diffusion of magnetic particles on stability of a

Pshenichnikov, A.F.; Mekhonoshin, V.V. & Lebedev, A.V. (1996). Magneto-granulometric

Pshenichnikov, A.F. & Lebedev, A.V. (2004). Low-temperature susceptibility of concentrated magnetic fluids. *Journal of Chemical Physics*, Vol. 121, No. 11, 5455–5467. Pshenichnikov, A.F.; Elfimova, E.A. & Ivanov, A.O. (2011). Magnetophoresis, sedimentation,

Richter, R. & Barashenkov, I. (2005). Two-dimensional solitons on the surface of magnetic

Sudo, S.; Yano, T. & Nakagawa, A. (2006). The aging variation of interfacial instability of magnetic fluids. *Journal of Advanced Science*, Vol. 18, No. 1-2, 119–122.

ferromagnetic particles in a magnetic fluid. *Mathematical Modelling and Analysis*, Vol.

static magnetic fluid seal under the action of external pressure drop. *Communications*

analysis of concentrated ferrocolloids. *Journal of Magnetism and Magnetic Materials*,

and diffusion of particles in concentrated magnetic fluids. *Journal of Chemical Physics*,

*Computational Methods in Applied Mathematics*, Vol. 4, No. 1, 66–93.

*in Nonlinear Science and Numerical Simulation*, Vol. 16, 4021–4027.

fluids. *Physical Review Letters*, Vol. 94, 184503–184506.

Rosensweig, R.E. (1998). *Ferrohydrodynamics*, Dover Pubns.

13, No. 2, 233–240.

Vol. 161, 94–102.

Vol. 134, 184508.

During the past decades, the development of solution methods and the growth of computer capacities have made its possible to solve more and more involved magnetic field problems. Thus, numerical techniques essentially based on the Finite Elements Method (FEM) have been used and has gradually become a standard in electrical machine modelling-design, analysis and optimisation. Electrical machines are electromagnetic devices with combined constrains such as complex geometries and several physical phenomena's. To model them, we must solve the magnetic field non-linear Partial Differential Equation (PDE) derived from the Maxwell's equations combined to the materials properties, and their coupling with phenomena that exist in electromagnetic structures, such as electric circuits, and mechanical motional equations. (Arkkio, 1987; Benali, 1997).

Induction Motor (IM) is an electromagnetic-mechanical actuator where strongly interacts several phenomena such as magnetic field, electrical circuits, mechanical motion. The aim of this chapter is to present an implementation of the finite element method for the modelling of rotating electrical machines, especially the squirrel cage three-phase induction motors. The generalized model consists firstly on strong coupling between the partial differential equation of the magnetic field diffusion and the electric circuits equations obtained from Kirchhoff laws. The model integrates as well realistic geometries, and the non-linear properties of the magnetic materials, as voltage supply of the stator windings. Secondly, the mechanical equation including the rotor movement effects is coupled to the electromagnetic phenomenon through the magnetic force responsible of the rotor motion.

The governing magnetic field time-dependent equation derived from Maxwell formalism is expressed in term of Magnetic Vector Potential (MVP) with only z-direction component for the cases of two dimensional (x,y) cartesian coordinates. The induction motors stator windings are usually in star or delta connection, then the source term of the magnetic field is explicitly an applied line voltage or implicitly the magnetizing current. The squirrel rotor cage is formed by massive conductive bars short-circuited at their ends through massive and conductive end-rings. Mathematically, the squirrel rotor cage appear as a polyphases circuits modelled by the same way that the stator windings but with affecting a zero voltage for each adjacent bars with theirs end-rings portion.

Finite Element Method Applied to the Modelling and Analysis of Induction Motors 205

The theory of electromagnetic electrical machines modeling is described by the time-space differential Maxwell's equations where the displacement current are neglected because of the low frequency oft he supply source (Joao, et al., 2003; Binns, et al.,1994; Arkkio, 1987):

> *<sup>B</sup> <sup>E</sup> t*

Moreover the electric and magnetic fields quantities are related with the material properties

 <sup>2</sup> *H BB* 

In the frequency domain and time-dependence with taking into account the eddy current, through (2) and (3) the electric field *E* and the magnetic flux density *B* are expressed using

> *<sup>A</sup> <sup>r</sup> E U t*

Two types of conductors are considered in the field model parts. A solid conductor corresponds to a massive part of conductive material in the computational domain, whereas a stranded conductor models and thereby assumes the current to be homogeneously distributed along the cross-section of the coil. The positive or negative direction of the

To formulate the magnetic field problem, we consider a two-dimensional domain partitioned into electrically conducting and non-conducting regions as shown in Fig.1. This

conducting regions are the cross-sections of stranded stator windings conductors *<sup>s</sup>* and

domain represents for instance the cross-section of an induction motor with length *L*

Stranded stator conductors

Solid conductors rotor bars

the magnetic vector potential *A* and scalar electric potential *<sup>r</sup> U* , such as :

*r m* *H J* (1)

*B* 0 (3)

(4.a)

(4.b)

the electric conductivity, *H* is the magnetic

(5)

(7)

. The

*B A* (6)

(2)

**2. Magnetic field native equations** 

expressed by the following constitutive relations:

*J E*

*B* ist he magnetic reluctivity,

current is fixed by the unit vector *d* 1 , as follow.

 

*s cn n n*

*<sup>A</sup> <sup>U</sup>*

 

*N I <sup>d</sup> <sup>S</sup> <sup>J</sup>*

> *t*

 

field, and *J* the conduction current density.

Where <sup>2</sup> 

Generally, permittivity and conductivity can be considered as constants, however the magnetic reluctivity of the core ferromagnetic materials depend on the magnetic flux density intensity which is implicitly fixed by the voltage excitation or currents level at each step time of the motor operating. This magnetic flux density-reluctivity non linear dependence is take into account in the model by the classical iterative Newton-Raphson method. The finite element formulation of the non-linear transient coupled magnetic fieldelectric circuits of induction motors model leads to an algebraic differential equations system. The solution process requires firstly a major loop concerning the time-discretization using the effectiveness Cranck-Nicholson scheme, and secondly for each step time we have to unsure the minor loop convergence of the Newton-Raphson algorithm for determining the appropriate magnetic reluctivity values.

The time stepping finite magnetic field-electric circuits coupled model is sequentially coupled with the mechanical equation of the rotor motion. The interaction between stator and rotor flux densities generate an electromagnetic torque responsible of the motion. Since the physical position of the moving part of the induction motors will change at each time step, the finite element coefficients matrix are consequently changes. The unknown mechanical position of the moving part can be found after solving the mechanical motional equation by the fourth order Rung-Kutta method. To take the movement into account, several strategies have been proposed, the boundary integral method, the air-gap-element, and the connecting meshes through the sliding line, moving band, and the Lagrange multipliers or nodal interpolation techniques (Dreher, et al., 1996).

A particularly elegant and accurate method is that due to (Abdel-Razek, et al., 1982) named the Air-Gap Element (AGE). The air gap element consist on the coupling between the meshes of the stator and rotor through the unmeshed air-gap band. The air-gap appears such as a multi-nodes finite element (Macro-Element) where it corresponding Laplace equation solution leads to an analytical expression of the magnetic vector potential. The combination between the magnetic vector potential of the air gap interfaces leads to a macro-element matrix. At each displacement step the rotor movement is simulated through only new computation of the air-gap element matrix, then the rotor implicitly moved without any changes on the motor mesh topologies. In addition, since the magnetic vector potential is derived from the field analytical solution in the air-gap, the magnetic flux density can be directly deduced permitting an accurate calculation of the electromagnetic torque using the Maxwell stress tensor method.

The magnetic-electric model obtained from the strong coupling of electric circuit equations of stator windings and polyphases rotor squirrel cage and the magnetic vector potential diffusion equations of the magnetic field, are solved using the nodal based finite element method with step-by-step algorithm. Finally, the magnetic vector potential, stator windings currents and bars voltages differences are the unknown variables. The studied simulation concerns different operating modes such as electrical transients where the speed is constant for no-load and nominal conditions, and the general no-load and loaded electro-magnetomechanical transient mode. Despite complex mathematical background, the simply and detailed presentation of the model offers an important aid for students, teachers and industrial employers for understanding the basis in simulation of electrical machines and particularly induction motors.

#### **2. Magnetic field native equations**

204 Numerical Modelling

Generally, permittivity and conductivity can be considered as constants, however the magnetic reluctivity of the core ferromagnetic materials depend on the magnetic flux density intensity which is implicitly fixed by the voltage excitation or currents level at each step time of the motor operating. This magnetic flux density-reluctivity non linear dependence is take into account in the model by the classical iterative Newton-Raphson method. The finite element formulation of the non-linear transient coupled magnetic fieldelectric circuits of induction motors model leads to an algebraic differential equations system. The solution process requires firstly a major loop concerning the time-discretization using the effectiveness Cranck-Nicholson scheme, and secondly for each step time we have to unsure the minor loop convergence of the Newton-Raphson algorithm for determining

The time stepping finite magnetic field-electric circuits coupled model is sequentially coupled with the mechanical equation of the rotor motion. The interaction between stator and rotor flux densities generate an electromagnetic torque responsible of the motion. Since the physical position of the moving part of the induction motors will change at each time step, the finite element coefficients matrix are consequently changes. The unknown mechanical position of the moving part can be found after solving the mechanical motional equation by the fourth order Rung-Kutta method. To take the movement into account, several strategies have been proposed, the boundary integral method, the air-gap-element, and the connecting meshes through the sliding line, moving band, and the Lagrange

A particularly elegant and accurate method is that due to (Abdel-Razek, et al., 1982) named the Air-Gap Element (AGE). The air gap element consist on the coupling between the meshes of the stator and rotor through the unmeshed air-gap band. The air-gap appears such as a multi-nodes finite element (Macro-Element) where it corresponding Laplace equation solution leads to an analytical expression of the magnetic vector potential. The combination between the magnetic vector potential of the air gap interfaces leads to a macro-element matrix. At each displacement step the rotor movement is simulated through only new computation of the air-gap element matrix, then the rotor implicitly moved without any changes on the motor mesh topologies. In addition, since the magnetic vector potential is derived from the field analytical solution in the air-gap, the magnetic flux density can be directly deduced permitting an accurate calculation of the electromagnetic

The magnetic-electric model obtained from the strong coupling of electric circuit equations of stator windings and polyphases rotor squirrel cage and the magnetic vector potential diffusion equations of the magnetic field, are solved using the nodal based finite element method with step-by-step algorithm. Finally, the magnetic vector potential, stator windings currents and bars voltages differences are the unknown variables. The studied simulation concerns different operating modes such as electrical transients where the speed is constant for no-load and nominal conditions, and the general no-load and loaded electro-magnetomechanical transient mode. Despite complex mathematical background, the simply and detailed presentation of the model offers an important aid for students, teachers and industrial employers for understanding the basis in simulation of electrical machines and

the appropriate magnetic reluctivity values.

torque using the Maxwell stress tensor method.

particularly induction motors.

multipliers or nodal interpolation techniques (Dreher, et al., 1996).

The theory of electromagnetic electrical machines modeling is described by the time-space differential Maxwell's equations where the displacement current are neglected because of the low frequency oft he supply source (Joao, et al., 2003; Binns, et al.,1994; Arkkio, 1987):

$$
\nabla \times \mathbf{H} = \mathbf{J} \tag{1}
$$

$$
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \tag{2}
$$

$$\nabla \cdot B = 0 \tag{3}$$

Moreover the electric and magnetic fields quantities are related with the material properties expressed by the following constitutive relations:

$$
\Delta H = \nu \left( B^2 \right) \cdot B \tag{4.a}
$$

$$\mathbf{J} = \boldsymbol{\sigma} \cdot \mathbf{E} \tag{4.9}$$

Where <sup>2</sup> *B* ist he magnetic reluctivity, the electric conductivity, *H* is the magnetic field, and *J* the conduction current density.

In the frequency domain and time-dependence with taking into account the eddy current, through (2) and (3) the electric field *E* and the magnetic flux density *B* are expressed using the magnetic vector potential *A* and scalar electric potential *<sup>r</sup> U* , such as :

$$E = -\frac{\partial A}{\partial t} + \nabla \mathcal{U}^r \tag{5}$$

$$B = \nabla \times A \tag{6}$$

Two types of conductors are considered in the field model parts. A solid conductor corresponds to a massive part of conductive material in the computational domain, whereas a stranded conductor models and thereby assumes the current to be homogeneously distributed along the cross-section of the coil. The positive or negative direction of the current is fixed by the unit vector *d* 1 , as follow.

$$J = \begin{cases} d\frac{N\_{cn}I\_n^s}{S\_n} & \text{Stranded stator conductors} \\ -\sigma \frac{\partial A}{\partial t} + \sigma \left(\nabla U\_m'\right) & \text{Solid conductors rotor bars} \end{cases} \tag{7}$$

To formulate the magnetic field problem, we consider a two-dimensional domain partitioned into electrically conducting and non-conducting regions as shown in Fig.1. This domain represents for instance the cross-section of an induction motor with length *L* . The conducting regions are the cross-sections of stranded stator windings conductors *<sup>s</sup>* and

Finite Element Method Applied to the Modelling and Analysis of Induction Motors 207

The delta and star connection (see Fig. 2 and Fig.3) are the two commonly used ways to connect the stator windings. In the delta connection the potential differences induced in the stator windings are equal to the line voltages. In the star connection with neutral point, the

potential differences of the stator windings are equal to the phase voltages.

**3.1 Electric circuits equations of the stator windings** 

Fig. 2. Stator windings in delta connection.

Fig. 3. Stator windings in star connection.

The three phases stator circuit equations are in matrix form:

 *<sup>s</sup> s s ss s*

*U t E t RI t L*

*end*

*dI t*

*dt* (9)

solid conductors *b* of the rotor bars, the non-conducting ferromagnetic region *core* , and the air gap region by *air* .

Fig. 1. Geometrical configuration of induction motors

To develop the mathematical model in term of magnetic vector potential in the three-phase induction motor, it is assumed that the magnetic field lies in the cross-sectional twodimensional (x, y) plane. Hence, only the z-component of the induced current and the magnetic vector potential can be considered. It also assumes that magnetic material of the motor cores is non-linearly isotropic. The magnetic property of the laminated iron cores is modelled by Marrocco approximation of the recluctivity (Brauer, et al, 1985; Hecht, et al., 1990), which is a single-valued nonlinear function of the flux density *B*, thus exclude the effect of magnetic hysteresis from the analysis.

The fundamental equations obtained from (1)-(6) and describing the time-space variation of the magnetic vector potential with the component (0,0, ( , , )) *A A x <sup>z</sup> y t* has the following form

$$\frac{\partial}{\partial \mathbf{x}} \left( \nu(\mathcal{B}^2) \frac{\partial A\_z(\mathbf{x}, y, t)}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial y} \left( \nu(\mathcal{B}^2) \frac{\partial A\_z(\mathbf{x}, y, t)}{\partial y} \right) = \mathbf{-} d \frac{N\_{\text{cn}} I\_n^s}{S\_n} + \sigma \left( -\frac{\partial A\_z(\mathbf{x}, y, t)}{\partial t} + \frac{\mathbf{U}\_m^r}{L\_\delta} \right) \tag{8}$$

In the model of an electrical machine, the magnetic field due to the currents in the coils. However, it is often more appropriate to model the feeding circuit as a voltage source, which leads to the combined solution of the magnetic field and circuit equations. The stator phase windings are generally modelled as filamentary conductors, and the rotor bars are modelled as solid conductors with eddy currents.

#### **3. Electric circuits equations model**

The computational model of the induction motors can be greatly improved by coupling the circuit equations of the stator and rotor windings with the two-dimensional field equation (8). In the circuit equations, the dependence between current and voltage is solved and the circuit quantities are coupled with the magnetic field by means of flux linkage. Also, the end-windings outside the core region are modelled by including an additional inductance in the circuit model (Hecht, et al., 1990; Kanerva, 2005; Piriou, et al., 1990).

#### **3.1 Electric circuits equations of the stator windings**

206 Numerical Modelling

solid conductors *b* of the rotor bars, the non-conducting ferromagnetic region *core* , and

To develop the mathematical model in term of magnetic vector potential in the three-phase induction motor, it is assumed that the magnetic field lies in the cross-sectional twodimensional (x, y) plane. Hence, only the z-component of the induced current and the magnetic vector potential can be considered. It also assumes that magnetic material of the motor cores is non-linearly isotropic. The magnetic property of the laminated iron cores is modelled by Marrocco approximation of the recluctivity (Brauer, et al, 1985; Hecht, et al., 1990), which is a single-valued nonlinear function of the flux density *B*, thus exclude the

The fundamental equations obtained from (1)-(6) and describing the time-space variation of the magnetic vector potential with the component (0,0, ( , , )) *A A x <sup>z</sup> y t* has the following

2 2 , , , , , , ( ) ( ) -

 

In the model of an electrical machine, the magnetic field due to the currents in the coils. However, it is often more appropriate to model the feeding circuit as a voltage source, which leads to the combined solution of the magnetic field and circuit equations. The stator phase windings are generally modelled as filamentary conductors, and the rotor bars are

The computational model of the induction motors can be greatly improved by coupling the circuit equations of the stator and rotor windings with the two-dimensional field equation (8). In the circuit equations, the dependence between current and voltage is solved and the circuit quantities are coupled with the magnetic field by means of flux linkage. Also, the end-windings outside the core region are modelled by including an additional inductance in

the circuit model (Hecht, et al., 1990; Kanerva, 2005; Piriou, et al., 1990).

*A xyt A xyt N I A xyt <sup>U</sup> B Bd x x y y S tL*

*s r zz z cn n <sup>m</sup>*

*n*

 

(8)

the air gap region by *air* .

Fig. 1. Geometrical configuration of induction motors

effect of magnetic hysteresis from the analysis.

modelled as solid conductors with eddy currents.

**3. Electric circuits equations model** 

form

The delta and star connection (see Fig. 2 and Fig.3) are the two commonly used ways to connect the stator windings. In the delta connection the potential differences induced in the stator windings are equal to the line voltages. In the star connection with neutral point, the potential differences of the stator windings are equal to the phase voltages.

Fig. 2. Stator windings in delta connection.

Fig. 3. Stator windings in star connection.

The three phases stator circuit equations are in matrix form:

$$
\Delta I^s \left( t \right) = E^s \left( t \right) + R^s I^s \left( t \right) + L\_{end}^s \frac{dI^s \left( t \right)}{dt} \tag{9}
$$

Finite Element Method Applied to the Modelling and Analysis of Induction Motors 209

In a cage rotor, each rotor bar requires its own equation. In time variation, the potential

*m rm r m j b*

Integration of the current density in a rotor bar over its cross section *Sb* gives the total currents of the *th m* bar. When constant conductivity and uniform cross section area *Sb* are assumed in the bar and the end-bar, *N N b b* unit end-bars self inductance *Lbe* , and resistance *Rbe* matrices are included, the above equation (13) for each bar can be expressed by (15) and (17). All the rotor bars are connected by short-circuit rings in both ends of the rotor core (16). This is taken into account by defining the end-ring unit resistance matrix and

 

*A U RI R N d m N t*

0 Otherwise

*b*

*dt*

*dt*

*rb b <sup>m</sup> m be m be <sup>m</sup> dI U RI L U*

*sc sc sc <sup>m</sup> m sc m sc dI U RI L*

1,...,

(13)

*m*

(15)

(16)

(14)

1

*j*

*Nd*

*br r j*

1 if x, <sup>y</sup> belongs to rotor bar

Fig. 4. Electric circuit configuration of squirrel rotor cage.

difference induced in the *th m* rotor bar is given by:

*r m*

 

Where *Rr* is *N N b b* unit matrix.

the end-ring unit inductance matrix.

$$E\_n^s(t) = N\_s \left(\frac{L\_{\mathcal{S}} N\_{cn}}{S\_n}\right) \sum\_{n=1}^{N\_{c\_n}} \left[ \iint\_{\Omega\_n^+} \left(\frac{\partial A}{\partial t}\right) d\Omega - \iint\_{\Omega\_n^-} \left(\frac{\partial A}{\partial t}\right) d\Omega \right] \qquad n = A, B, C \tag{10}$$

Where A,B,C denote the three stator phase, *<sup>s</sup>* and *<sup>s</sup>* are respectively, the cross-sectional areas of the "go" and "return" side of the phase conductors. The column vectors of the potential differences of the stator windings with their currents and electromotive force are detailed as follows:

$$\begin{aligned} \boldsymbol{U}^{s}\left(\boldsymbol{t}\right) &= \begin{bmatrix} \boldsymbol{U}^{s}\_{A}\left(\boldsymbol{t}\right) \\ \boldsymbol{U}^{s}\_{B}\left(\boldsymbol{t}\right) \\ \boldsymbol{U}^{s}\_{C}\left(\boldsymbol{t}\right) \end{bmatrix} \quad , \ \boldsymbol{E}^{s}\left(\boldsymbol{t}\right) &= \begin{bmatrix} \boldsymbol{E}^{s}\_{A}\left(\boldsymbol{t}\right) \\ \boldsymbol{E}^{s}\_{B}\left(\boldsymbol{t}\right) \\ \boldsymbol{E}^{s}\_{C}\left(\boldsymbol{t}\right) \end{bmatrix} \quad , \ \boldsymbol{I}^{s}\left(\boldsymbol{t}\right) &= \begin{bmatrix} \boldsymbol{I}^{s}\_{A}\left(\boldsymbol{t}\right) \\ \boldsymbol{I}^{s}\_{B}\left(\boldsymbol{t}\right) \\ \boldsymbol{I}^{s}\_{C}\left(\boldsymbol{t}\right) \end{bmatrix}, \\\ \boldsymbol{R}^{s} &= \begin{pmatrix} \boldsymbol{R}^{s}\_{A} & \boldsymbol{0} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{R}^{s}\_{B} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{R}^{s}\_{\circ} \end{bmatrix}, \ \boldsymbol{L}^{s}\_{\text{end}} = \begin{pmatrix} \boldsymbol{L}^{s}\_{\text{end}} & \boldsymbol{0} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{L}^{s}\_{\text{end}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{L}^{s}\_{\text{end}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{L}^{s}\_{\text{end}} \end{aligned}$$

When the stator windings has star connection with non-connected neutral star point (see Fig. 3), only two from the three phase currents are independent variables, and the third is determined by an additional constraint which unsure a zero sequence of the phases currents *s ss C AB I II* . For this reason the connectivity matrix is formed:

$$\begin{bmatrix} K \end{bmatrix} = \begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & -1 \end{bmatrix} \tag{11}$$

The line voltages *<sup>s</sup> V* and loops currents 1,2 *s J* containing the two independent currents, are formed in the following way:

$$K \begin{bmatrix} \mathcal{U}\_{\rm AN}^{s} \\ \mathcal{U}\_{\rm BN}^{s} \\ \mathcal{U}\_{\rm CN}^{s} \end{bmatrix} = \begin{bmatrix} \mathbf{1} & \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} & \mathbf{0} \end{bmatrix} \begin{Bmatrix} \mathcal{U}\_{\rm AB}^{s} \\ \mathcal{U}\_{\rm BC}^{s} \\ \mathcal{U}\_{\rm CA}^{s} \end{Bmatrix} = \mathcal{Q}^{s} \begin{Bmatrix} \mathcal{V}^{s} \end{Bmatrix} \tag{12.a}$$

$$\left\{ I^{s} \right\} = \left[ K \right]^{tr} \begin{Bmatrix} I\_1^{s} \\ I\_2^{s} \end{Bmatrix} = \left[ K \right]^{tr} \begin{Bmatrix} I\_A^{s} \\ I\_B^{s} \end{Bmatrix} = \left[ K \right]^{tr} \left\{ I\_{1,2}^{s} \right\} \tag{12.b}$$

#### **3.2 Electric circuits equations of the rotor cage**

A network of the non-skewed rotor cage is shown in Fig. 4. For normal operating frequencies (50 or 60 Hz), the inductive component of the inter-bar impedance can be neglected. Two adjacent bars are connected by the end-ring resistances and inductances (Arkkio, 1987; Benali, 1997; Ho, et al., 2000).

*s s n n*

*S tt*

*L N A A Et N <sup>d</sup> d n ABC*

areas of the "go" and "return" side of the phase conductors. The column vectors of the potential differences of the stator windings with their currents and electromotive force are

> *s s B s C*

*Et Et*

and *<sup>s</sup>*

 

*s end s s end end*

*L L L*

*s A*

*E t*

 

*E t*

When the stator windings has star connection with non-connected neutral star point (see Fig. 3), only two from the three phase currents are independent variables, and the third is determined by an additional constraint which unsure a zero sequence of the phases currents

> 10 1 01 1 *<sup>K</sup>*

> > *s*

010

*K U U QV*

*s s s s tr tr A tr s s*

A network of the non-skewed rotor cage is shown in Fig. 4. For normal operating frequencies (50 or 60 Hz), the inductive component of the inter-bar impedance can be neglected. Two adjacent bars are connected by the end-ring resistances and inductances

*J I I K K KJ J I* 

*BN BC s s CN CA*

*s s ss*

*B*

*s s AN AB*

 

*U U*

*U U*

2

*A*

,

 

 00 0 0

*s s B s C*

0 0

*B*

*It It*

 

,,

are respectively, the cross-sectional

 

,

(11)

(12.a)

(12.b)

*s A*

*I t*

*I t*

*C*

*s end*

*J* containing the two independent currents, are

1,2

*L*

(10)

1

*cn*

*N*

 

*s A*

*U t*

*B s C*

*U t* 

*s A s s*

*R R R* 0 0 0 0 0 0

*s C*

*R*

*B*

*C AB I II* . For this reason the connectivity matrix is formed:

<sup>110</sup>

<sup>1</sup>

 

,

,

*n n*

detailed as follows:

*s ss*

*<sup>s</sup> cn n s*

Where A,B,C denote the three stator phase, *<sup>s</sup>*

The line voltages *<sup>s</sup> V* and loops currents 1,2

**3.2 Electric circuits equations of the rotor cage** 

(Arkkio, 1987; Benali, 1997; Ho, et al., 2000).

formed in the following way:

*s s*

*Ut Ut*

Fig. 4. Electric circuit configuration of squirrel rotor cage.

In a cage rotor, each rotor bar requires its own equation. In time variation, the potential difference induced in the *th m* rotor bar is given by:

$$\mathbf{U}\_{m}^{b} = \mathbf{R}\_{r}\mathbf{I}\_{m}^{r} + \mathbf{R}\_{r} \iint\_{\Omega} \beta\_{m}^{r} \sum\_{j=1}^{N\_{d}} \left(\mathbf{N}\_{j} \frac{\partial A\_{j}}{\partial t}\right) d\Omega \qquad\qquad m = \mathbf{1}, \ldots, N\_{b} \tag{13}$$

$$
\beta\_m' = \begin{cases} 1 & \text{if (x,y) belongs to rotor bar } m \\ 0 & \text{Otherwise} \end{cases} \tag{14}
$$

Where *Rr* is *N N b b* unit matrix.

Integration of the current density in a rotor bar over its cross section *Sb* gives the total currents of the *th m* bar. When constant conductivity and uniform cross section area *Sb* are assumed in the bar and the end-bar, *N N b b* unit end-bars self inductance *Lbe* , and resistance *Rbe* matrices are included, the above equation (13) for each bar can be expressed by (15) and (17). All the rotor bars are connected by short-circuit rings in both ends of the rotor core (16). This is taken into account by defining the end-ring unit resistance matrix and the end-ring unit inductance matrix.

$$
\mathcal{L}I\_m^r = R\_{bv}I\_m^b + L\_{bv}\frac{dI\_m^b}{dt} + \mathcal{L}I\_m^b \tag{15}
$$

$$
\Delta I\_m^{sc} = R\_{sc} I\_m^{sc} + L\_{sc} \frac{dI\_m^{sc}}{dt} \tag{16}
$$

Finite Element Method Applied to the Modelling and Analysis of Induction Motors 211

Several methods can be used for the numerical solution of the magnetic field equation (8), such as reluctance networks, the boundary element method, the finite difference method or the finite element method. In this work, the numerical analysis is based on the finite element method. The two-dimensional geometry is covered by a finite element mesh, consisting of first-order triangular elements. If possible, the cross section of the electrical machine is divided in *Ns* symmetry sectors, from which only one is modelled by the finite element method and symmetry constraints are set on the periodic or anti-periodic boundary (Nougier, 1999; Binns, et al., 1994; Ho, et al., 1997). The magnetic vector potential can be approximated as the sum of the element shape functions times and nodal potential values:

1

*Nnodes z j z j j*

, , , ,,

(22)

*A*

*t*

*A xyt N xy A xyt* 

Where *Nnodes* is the total nodes number of the finite element mesh, *N xy <sup>j</sup>* , the shape

The numerical field equation is derived by Galerkin's method, where (8) is multiplied by shape functions and integrated over the whole finite element mesh with substituting the magnetic vector potential approximation (22). The last line integral term of the formulation

<sup>2</sup>

*i j j ij*

*B N N A NN d*

*s r s s i m m i nn*

> *nnt AGE AGE j j*

> > *n*

The problems in the analysis of the electrical machine are almost non-linearly isotropic due to the presence of ferromagnetic materials. The magnetic permeability is non-homogeneous and will be a function of the local magnetic field which is unknown at the start of the problem. The permeability is low at very low flux densities, rises quickly as the flux density increases and then decreases in the saturation region. As the permeability is unavoidably contained in all of the element stiffness matrices, an iterative process must be used to keep correcting the permeability until it consistent with the field solution. The Newton-Raphson iterative technique is used for the analysis of the non-linear problem (Brauer, et al., 1985;

At the beginning, an unsaturated value of permeability is assigned for each element of the mesh. When solving the problem, the magnitude of the flux density in each element is computed and the magnetic reluctivities are corrected to be consistent with the computed

*N A*

 

 *NUNId*

(23)

*d*

*Nnodes <sup>j</sup>*

1

*j o*

function, and *A xyt z j* , , is the magnetic vector potential of the node *j* .

(23) correspond to the air-gap contribution due to the rotor movement.

1

*j*

*Nnodes*

1

*j*

Joao, et al.,2003; Neagoe, et al., 1994).

$$I\_m^r = \sigma \left( \iint\_{\Omega\_\delta} \left( -\frac{\partial A}{\partial t} + \frac{U\_m^r}{L\_\delta} \right) d\Omega \right) \tag{17}$$

From Kirchhoff's second law applied to the rotor cage electric circuit (Fig.4), a relation between the potential difference and currents of bars and end-ring are obtained such as:

$$\mathcal{D}\mathcal{U}^{\rm sc} = \mathcal{M} \cdot \mathcal{U}^{\rm \prime} \tag{18.a}$$

$$I^b = M^{tr} \cdot I^{sc} \tag{18.b}$$

 Where *M* ist he rotor cage connection matrix, and *h* is the periodicity factor *h* (+1 if periodic and -1 if non-periodic).

$$M = \begin{bmatrix} 1 & 0 & \dots & \dots & \dots \\ 0 & 1 & \dots & \dots & \dots \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & 0 \\ \cdot & \cdot & \cdot & 0 & 1 \end{bmatrix} + \begin{bmatrix} 0 & 0 & \dots & \dots & \dots \\ -1 & 0 & \dots & \dots & \dots \\ \cdot & -1 & \dots & \dots & \dots \\ \cdot & \cdot & \cdot & \cdot & 0 \\ \cdot & \cdot & \cdot & \cdot & -1 & 0 \end{bmatrix} + \begin{bmatrix} 0 & 0 & \dots & \dots & -h \\ 0 & 0 & \dots & \dots & \dots \\ \cdot & \cdot & \cdot & \dots & \dots \\ \cdot & \cdot & \cdot & \dots & 0 \\ \cdot & \cdot & \cdot & 0 & 0 \end{bmatrix} \tag{19}$$

#### **4. Magnetic field – Electric circuits coupling**

#### **4.1 Time stepping finite element formulation of the non-linear magnetic field model**

In the electric machine model, the magnetic field in the iron core, windings and air gap is solved by the two-dimensional finite element code and coupled with the voltage equations of the stator and rotor windings. The model is based on the direct coupling, which means that magnetic field equations and electrical circuit equations are solved simultaneously by time-stepping approach with handling magnetic non-linearities using Newton-Raphson iterative algorithm.

In the time stepping formulation, the derivative of the vector potential, stator windings currents and bars voltages are approximated by first-order difference ratios:

$$\frac{\partial}{\partial t} \begin{Bmatrix} A \\ I\_n^s \\ I\_m^s \end{Bmatrix} = \frac{1}{\Delta t} \begin{Bmatrix} A \\ I\_n^s \\ I\_m^r \end{Bmatrix}\_{k+1} - \begin{Bmatrix} A \\ I\_n^s \\ \mathcal{U}\_m^r \end{Bmatrix}\_k \tag{20}$$

The time discretization is performed by using the Crank-Nicholson sheme as:

$$\begin{Bmatrix} A \\ I\_n^s \\ \left\{ \boldsymbol{I}\_m^r \right\}\_{k+1} \end{Bmatrix}\_{k+1} = \frac{\mathbf{1}}{2} \begin{Bmatrix} \frac{\vec{c}A}{\vec{c}t} \\ \frac{\vec{c}U\_n^\*}{\vec{c}t} \\ \frac{\vec{c}U\_m^r}{\vec{c}t} \end{Bmatrix}\_{k+1} + \frac{\vec{c}U\_n^\*}{\vec{c}t} \begin{Bmatrix} A \\ I\_n^s \\ \frac{\vec{c}U\_m^\*}{\vec{c}t} \end{Bmatrix}\_k \end{Bmatrix} \Delta t + \begin{Bmatrix} A \\ I\_n^s \\ \boldsymbol{I}\_n^r \end{Bmatrix}\_k \tag{21}$$

 

From Kirchhoff's second law applied to the rotor cage electric circuit (Fig.4), a relation between the potential difference and currents of bars and end-ring are obtained such as:

2 *sc <sup>b</sup> U MU* (18.a)

Where *M* ist he rotor cage connection matrix, and *h* is the periodicity factor *h* (+1 if

10.. . 0 0 . . . 00.. . 01. . . 1 0 . . . 00. . . . . . . . . 1. . . . . . . . . ...0 . . . . 0 . ... 0 . . .01 . . . 10 . . .0 0

**4.1 Time stepping finite element formulation of the non-linear magnetic field model**  In the electric machine model, the magnetic field in the iron core, windings and air gap is solved by the two-dimensional finite element code and coupled with the voltage equations of the stator and rotor windings. The model is based on the direct coupling, which means that magnetic field equations and electrical circuit equations are solved simultaneously by time-stepping approach with handling magnetic non-linearities using Newton-Raphson

In the time stepping formulation, the derivative of the vector potential, stator windings

1 *ss s nn n sr r mm m k k*

*s s n n r r m m*

*A A t t s s I I n n t t r r U U m m k k t t k k*

*A A I t I U U*

 

*UU U*

*AA A II I*

 

1

currents and bars voltages are approximated by first-order difference ratios:

*t t*

The time discretization is performed by using the Crank-Nicholson sheme as:

1 2

<sup>1</sup> <sup>1</sup>

*t L*

(17)

*I MI* (18.b)

*h*

(19)

(20)

(21)

*b <sup>r</sup> <sup>r</sup> <sup>m</sup> <sup>m</sup> <sup>A</sup> <sup>U</sup> I d*

*b tr sc*

**4. Magnetic field – Electric circuits coupling** 

periodic and -1 if non-periodic).

*M*

iterative algorithm.

Several methods can be used for the numerical solution of the magnetic field equation (8), such as reluctance networks, the boundary element method, the finite difference method or the finite element method. In this work, the numerical analysis is based on the finite element method. The two-dimensional geometry is covered by a finite element mesh, consisting of first-order triangular elements. If possible, the cross section of the electrical machine is divided in *Ns* symmetry sectors, from which only one is modelled by the finite element method and symmetry constraints are set on the periodic or anti-periodic boundary (Nougier, 1999; Binns, et al., 1994; Ho, et al., 1997). The magnetic vector potential can be approximated as the sum of the element shape functions times and nodal potential values:

$$A\_z(\mathbf{x}, y, t) = \sum\_{j=1}^{N\_{\text{mid}}} N\_j(\mathbf{x}, y) \cdot A\_{zj}(\mathbf{x}, y, t) \tag{22}$$

Where *Nnodes* is the total nodes number of the finite element mesh, *N xy <sup>j</sup>* , the shape function, and *A xyt z j* , , is the magnetic vector potential of the node *j* .

The numerical field equation is derived by Galerkin's method, where (8) is multiplied by shape functions and integrated over the whole finite element mesh with substituting the magnetic vector potential approximation (22). The last line integral term of the formulation (23) correspond to the air-gap contribution due to the rotor movement.

$$\begin{aligned} \iint \sum\_{j=1}^{N\_{\text{sub}}} \left[ \nu \left( \boldsymbol{\mathcal{B}}^{2} \right) \left( \nabla N\_{i} \right) \cdot \left( \nabla N\_{j} \right) \left\{ A\_{j} \right\} + \left( \sigma \boldsymbol{N}\_{i} N\_{j} \frac{\partial \left\{ A\_{j} \right\}}{\partial t} \right) \right] d\Omega + \\ \iint \sum\_{j=1}^{N\_{\text{sub}}} \left( -\sigma \boldsymbol{N}\_{i} \left( \boldsymbol{\mathcal{B}}\_{m}^{s} \boldsymbol{L}\_{m}^{\prime} \right) - \boldsymbol{N}\_{i} \left( \boldsymbol{\mathcal{B}}\_{n}^{s} \boldsymbol{I}\_{n}^{\prime} \right) \right) \right] d\Omega = \\ \oint \boldsymbol{\mathcal{O}}\_{o} \left[ \frac{\partial \sum\_{j=1}^{m \text{int}} \boldsymbol{N}\_{j}^{\prime G \text{E}} \boldsymbol{A}\_{j}^{\prime G \text{E}}}{\partial n} \right] d\Gamma \end{aligned} (23)$$

The problems in the analysis of the electrical machine are almost non-linearly isotropic due to the presence of ferromagnetic materials. The magnetic permeability is non-homogeneous and will be a function of the local magnetic field which is unknown at the start of the problem. The permeability is low at very low flux densities, rises quickly as the flux density increases and then decreases in the saturation region. As the permeability is unavoidably contained in all of the element stiffness matrices, an iterative process must be used to keep correcting the permeability until it consistent with the field solution. The Newton-Raphson iterative technique is used for the analysis of the non-linear problem (Brauer, et al., 1985; Joao, et al.,2003; Neagoe, et al., 1994).

At the beginning, an unsaturated value of permeability is assigned for each element of the mesh. When solving the problem, the magnitude of the flux density in each element is computed and the magnetic reluctivities are corrected to be consistent with the computed

Finite Element Method Applied to the Modelling and Analysis of Induction Motors 213

After the substitution of the approximation (21) in (26), the voltages equations of the *th n*

1 1 <sup>1</sup>

*U U A A I I <sup>R</sup> NL N d I IL*

 

 1 1 <sup>1</sup> <sup>1</sup> , *s s s sTs <sup>s</sup> i k nk <sup>k</sup> <sup>n</sup> <sup>k</sup> <sup>k</sup> <sup>A</sup> I KD A G KK J KD A*

> 1,2 <sup>1</sup> *sTs s s s n n k kk H KK J C V V*

> > *e s se D Nd ij n j*

*Rt L <sup>G</sup>*

*Rt L <sup>H</sup>*

 3 3 2 *s s s*

 

 

*<sup>t</sup> C Q N L*

 

By undertaking the same way as the stator windings, after applying Crank-Nicholson scheme, equations (13), (16) and (17) of the voltage equations of the rotor cage becomes:

*Nd j j b b rr r k k m m rm m r m k k kk <sup>j</sup>*

*U U RI I R d*

3 3

3 3

**4.3 Time stepping finite element formulation of the rotor cage equations** 

 <sup>1</sup> 1 1 <sup>1</sup>

2 *s s <sup>s</sup> end*

2 *s s <sup>s</sup> end*

2

2

*s*

*N L*

 

*s*

*N L*

*s s s s <sup>N</sup> <sup>s</sup> n n j j n n k k <sup>s</sup> k k s ss k k sz n j n n end k k <sup>j</sup>*

*t t*

1,2 <sup>1</sup>

 

(31.a)

(31.b)

(31.c)

(31.d)

*A A*

*t*

(32)

(29)

(30)

(28)

*Rt L <sup>t</sup> KD A KK J QV V N L N L*

*s s <sup>s</sup> end T s ss s <sup>k</sup> n n <sup>k</sup> k k s s*

<sup>1</sup> <sup>1</sup> 2 2

The voltages equations (28) are expressed in matrix form as follow:

1 1 <sup>1</sup> 1,2 <sup>1</sup>

*s s s s <sup>s</sup> end T s i k nk <sup>k</sup> <sup>k</sup> <sup>s</sup>*

Equation (29) can be written under this following form:

The different matrix components of (30) are:

1 1 2 2

*Rt L A I KD A KK J N L*

2 2 2

*d*

<sup>2</sup> , 2

phase of the stator windings becomes:

values of the flux density. The problem is then solved again using the new values. This process is continued till a satisfactory result is obtained when the difference between the actual solution and the previous one is smaller than a pre-specified value. The equations for the time-stepping simulation are derived by adding the equations from two successive steps together and replacing the derivatives with expressions (20) and (21). Using this approach, the magnetic vector potential integrals formulation (23) are formed for each node in the finite element mesh. Following, a residual vector *<sup>f</sup>* is obtained after the finite element discretization, and the *th i* element of the residual vector is:

 <sup>1</sup> 11 1 <sup>1</sup> <sup>1</sup> <sup>2</sup> , , () *<sup>n</sup> k N f r s <sup>i</sup> k n nk k ij ij j <sup>k</sup> <sup>j</sup> AU I A N N NN A d t* 1 1 <sup>1</sup> , , - *Nb r r s s i m m i jn k k <sup>m</sup> n ABC N U N Id L* (24) <sup>1</sup> <sup>1</sup> <sup>2</sup> ( ) *Nn k i j ij j <sup>k</sup> <sup>j</sup> A N N NN A d t* 1 , , 1 - *Nb nnt r r s s AGE AGE i m m i nn o jj k k <sup>m</sup> n ABC <sup>j</sup> N U N I d NAd L n* 

In matrix form, equation (24) can be written as follows:

$$\begin{aligned} \mathfrak{M}\_{i}^{s} \left( A\_{k+1} \mathcal{U}\_{nk+1}^{r} I\_{nk+1}^{s} \right) &= \left[ S \left( A\_{k+1} \right) + M \left( A\_{k+1} \right) + S^{\rm AGE} \left( A\_{k+1} \right) \right] \left\{ A\_{k+1} \right\} + \left( D^{r} \right)^{T} \mathcal{U}\_{k+1}^{r} \\ &+ \left( D^{sT} K^{T} \right) \left( I\_{1,2}^{s} \right)\_{k+1} + \left[ S \left( A\_{k} \right) + M \left( A\_{k} \right) + S^{\rm AGE} \left( A\_{k} \right) \right] \left\{ A\_{k} \right\} + \left( D^{r} \right) \mathcal{U}\_{k}^{r} + \left( D^{sT} K^{T} \right) \left( I\_{1,2}^{s} \right)\_{k} \end{aligned} \tag{25}$$

#### **4.2 Time stepping finite element formulation of the stator windings equations**

The same approximation (20), and (22) is also applied to the winding equations (9) and (10). The resulting equations of the average value of the potential difference at the time steps *k* and *k+1* is used to approximate the true potential difference as:

$$\left. \mathbf{U}\_{n}^{s} \right|\_{k+1} = \left( \mathbf{R}^{s} I\_{n}^{k} + \mathbf{L}\_{end}^{s} \frac{d \mathbf{I}\_{n}^{s}}{dt} \right)\_{k+1} + N\_{s} \mathbf{I}\_{z} \left. \int\_{\Omega} \boldsymbol{\beta}\_{n}^{s} \frac{\partial \mathbf{A}}{\partial t} \right|\_{k+1} d\Omega \tag{26.a}$$

$$\left. \delta L^s\_n \right|\_k = \left( R^s I\_n^s + L\_{end}^s \frac{dI\_n^s}{dt} \right) \bigg|\_{k} + N\_s L\_z \int\_{\Omega} \mathcal{J}\_n^s \frac{\partial A}{\partial t} \bigg|\_{k} d\Omega \tag{26.b}$$

$$
\mathcal{J}\_n^s = \frac{N\_{cn}}{S\_n} \begin{bmatrix} -1 & \text{Negatively oriented coil} \\ +1 & \text{Positively oriented coil} \\ 0 & \text{Otherwise} \end{bmatrix} \tag{27}
$$

values of the flux density. The problem is then solved again using the new values. This process is continued till a satisfactory result is obtained when the difference between the actual solution and the previous one is smaller than a pre-specified value. The equations for the time-stepping simulation are derived by adding the equations from two successive steps together and replacing the derivatives with expressions (20) and (21). Using this approach, the magnetic vector potential integrals formulation (23) are formed for each node in the finite element mesh. Following, a residual vector *<sup>f</sup>* is obtained after the finite element

<sup>1</sup> 11 1 <sup>1</sup> <sup>1</sup>

 

> 1 1 <sup>1</sup> , , -

> > <sup>1</sup> <sup>1</sup>

*A N N NN A d t* 

*Nb nnt r r s s AGE AGE*

 

(25)

*k i j ij j <sup>k</sup> <sup>j</sup>*

1 , , 1

 1 11 1 1 1 1 <sup>1</sup> ,, *<sup>T</sup> <sup>s</sup> r s AGE r r i k nk nk A U I SA MA S A A D U <sup>k</sup> <sup>k</sup> k k <sup>k</sup>*

 1,2 1,2 <sup>1</sup> *sT T s AGE r r sT T s k k kk k <sup>k</sup> <sup>k</sup> D K J SA MA S A A D U D K J*

The same approximation (20), and (22) is also applied to the winding equations (9) and (10). The resulting equations of the average value of the potential difference at the time steps *k*

> 

*dt t*

*dt t*

1 Negatively oriented coil 1 Positively orriented coil 0 Otherwise

(26.a)

(26.b)

*i m m i nn o jj k k <sup>m</sup> n ABC <sup>j</sup> N U N I d NAd*

 

 

*r r s s i m m i jn k k <sup>m</sup> n ABC N U N Id*

*t* 

(24)

(27)

*<sup>i</sup> k n nk k ij ij j <sup>k</sup> <sup>j</sup> AU I A N N NN A d*

*i* element of the residual vector is:

<sup>2</sup> , , () *<sup>n</sup>*

 

<sup>2</sup> ( )

**4.2 Time stepping finite element formulation of the stator windings equations** 

*<sup>s</sup> s sk s <sup>n</sup> <sup>s</sup> <sup>n</sup> n end sz n <sup>k</sup> <sup>k</sup> <sup>k</sup> dI <sup>A</sup> U RI L NL d*

*<sup>s</sup> s ss s <sup>n</sup> <sup>s</sup> <sup>n</sup> n end sz n <sup>k</sup> <sup>k</sup> <sup>k</sup> dI <sup>A</sup> U RI L NL d*

*L n*

*N*

*Nb*

 *L*

*Nn*

discretization, and the *th*


*k*

*f r s*

In matrix form, equation (24) can be written as follows:

and *k+1* is used to approximate the true potential difference as:

*<sup>s</sup> cn <sup>n</sup> n N S*

<sup>1</sup> <sup>1</sup> <sup>1</sup>

 

 After the substitution of the approximation (21) in (26), the voltages equations of the *th n* phase of the stator windings becomes:

$$\frac{\left.\left\|\boldsymbol{L}\_{n}^{s}\right\|\_{k+1} + \left.\boldsymbol{L}\_{n}^{s}\right\|\_{k}}{2} = N\_{s}\boldsymbol{L}\_{z}\left[\left.\boldsymbol{\beta}\_{n}^{s}\sum\_{j=1}^{N\_{d}}\boldsymbol{N}\_{j}\right|\frac{\left.\boldsymbol{A}\_{j}\right|\_{k+1} + \left.\boldsymbol{A}\_{j}\right|\_{k}}{\Delta t}\right]d\Omega + \left.\frac{\left.\boldsymbol{R}^{s}}{2}\left(\left.\boldsymbol{I}\_{n}^{s}\right|\_{k+1} + \left.\boldsymbol{I}\_{n}^{s}\right|\_{k}\right) + \left.\boldsymbol{I}\_{cmd}^{s}\frac{\left.\boldsymbol{I}\_{n}^{s}\right|\_{k+1} + \left.\boldsymbol{I}\_{n}^{s}\right|\_{k}}{\Delta t}\right) \tag{28}$$

The voltages equations (28) are expressed in matrix form as follow:

$$\begin{split} \mathfrak{M}\_{l}^{s} \left( A\_{k+1}, I\_{nk+1}^{s} \right) &= \left( \mathbf{K} \mathbf{D}^{s} \right) A\_{k+1} + \left( -\frac{\mathbf{R}^{s} \Delta t + 2 \mathbf{L}\_{end}^{s}}{2 \mathbf{N}\_{s} \mathbf{L}\_{\delta}} \right) \mathbf{K} \mathbf{K}^{T} \left( I\_{1,2}^{s} \right)\_{k+1} - \\ & \left( \mathbf{K} \mathbf{D}^{s} \right) A\_{k} + \left( -\frac{\mathbf{R}^{s} \Delta t - 2 \mathbf{L}\_{end}^{s}}{2 \mathbf{N}\_{s} \mathbf{L}\_{\delta}} \right) \mathbf{K} \mathbf{K}^{T} \left( I\_{1,2}^{s} \right)\_{k} + \left( \frac{\Delta t}{2 \mathbf{N}\_{s} \mathbf{L}\_{\delta}} \right) \mathbf{Q}^{s} \left[ \left( V\_{n}^{s} \right)\_{k+1} + \left( V\_{n}^{s} \right)\_{k} \right] \end{split} \tag{29}$$

Equation (29) can be written under this following form:

$$\begin{aligned} \mathfrak{R}\_i^s \left( A\_{k+1}, I\_{nk+1}^s \right) &= \left( \mathrm{KD}^s \right) A\_{k+1} + \left( \mathrm{G}^s \mathrm{KK}^T \right) \left( I\_n^s \right)\_{k+1} - \left( \mathrm{KD}^s \right) A\_k \\ &+ \left( \mathrm{H}^s \mathrm{KK}^T \right) \left( I\_{1,2}^s \right)\_k + \left( \mathrm{C}^s \right) \left[ \left( V\_n^s \right)\_{k+1} + \left( V\_n^s \right)\_k \right] \end{aligned} \tag{30}$$

The different matrix components of (30) are:

$$D\_{ij}^s = -\iint\_{\Omega'} \left(\mathcal{J}\_n^s \cdot \mathbf{N}\_j\right) d\Omega' \tag{31.a}$$

$$\left[\left.G^{s}\right]\_{\left(3\times3\right)} = -\left(\frac{R^{s}\Delta t + 2L\_{end}^{s}}{2N\_{s}L\_{\delta}}\right) \tag{31.b}$$

$$\left[\left.H^{s}\right]\_{\text{(3\times 3)}} = -\left(\frac{R^{s}\Delta t - 2L\_{end}^{s}}{2N\_{s}L\_{\mathcal{S}}}\right) \tag{31.c}$$

$$
\left[\mathbb{C}^s\right]\_{\text{(3\times3)}} = \left(\frac{\Delta t}{2N\_s L\_\delta}\right) \mathbb{Q}^s \tag{31.d}
$$

#### **4.3 Time stepping finite element formulation of the rotor cage equations**

By undertaking the same way as the stator windings, after applying Crank-Nicholson scheme, equations (13), (16) and (17) of the voltage equations of the rotor cage becomes:

$$\frac{1}{2} \left( \left. U\_m^b \right|\_{k+1} + \left. U\_m^b \right|\_k \right) = \frac{1}{2} R\_r \left( \left. I\_m^r \right|\_{k+1} + \left. I\_m^r \right|\_k \right) + \left. R\_r \int\_{\Omega} \beta\_m^r \sigma \left\{ \sum\_{j=1}^{N\_d} \frac{A\_{\phantom{i}j}}{\Delta t} \bigg|\_{k+1} - A\_{\phantom{i}j} \bigg|\_{k} \right\} d\Omega \tag{32}$$

Finite Element Method Applied to the Modelling and Analysis of Induction Motors 215

method, a final algebraic system of equations for the nonlinear time-stepping simulation of

*q q <sup>f</sup> <sup>q</sup> r s tr tr <sup>q</sup> r s tr <sup>q</sup> <sup>k</sup> <sup>m</sup> k k <sup>k</sup> <sup>k</sup>*

<sup>1</sup> <sup>1</sup> 1,2 1 1 <sup>1</sup> <sup>1</sup> 1

 

1

0 ,

*k ij*

1

 

*j j*

*A*

*A* 

 

(37)

<sup>1</sup> 1 1

1,2 <sup>1</sup> <sup>1</sup> 1,2 <sup>1</sup>

*<sup>k</sup> <sup>k</sup> <sup>k</sup>*

(38.a)

(38.b)

(38.c)

*m m <sup>k</sup> k k*

*q q r r <sup>r</sup> r r <sup>q</sup>*

, , ( )

*D C U A U*

*K D <sup>G</sup> <sup>J</sup> A J*

*<sup>q</sup> <sup>N</sup> <sup>j</sup> AGE AGE <sup>k</sup>*

*AU J P A D D KK <sup>A</sup>*

<sup>0</sup> ,

Where *P* is the Jacobian matrix system expressed through the following matrices elements

 () <sup>1</sup> *AGE S S A N Nd ij* 

*<sup>n</sup>*

*ij ij ij ij i j*

*PS S J S S N Nd*

<sup>2</sup> *Mij NN d i j <sup>t</sup>* 

For modelling the movement in rotating electrical machines using the Air-Gap-Element (AGE), the space discretised domain is commonly split up into two subdomains, a stator *stator* , and rotor *rotor* meshed domains, and the unmeshed air gap with *AGE* boundary

*s s <sup>q</sup> <sup>s</sup> <sup>q</sup> s s <sup>q</sup>*

the electrical machine is obtained such as:

(Joao, et al., 2003; Arkkio, 1987; Benali, 1997):

**5. Mechanical model and movement simulation** 

**5.1 Movement simulation technique** 

Fig. 5. Air-Gap Element (macro-element).

(Fig. 5).

$$\frac{1}{2} \left( \left. \boldsymbol{L}\_{m}^{\boldsymbol{r}} \right|\_{k+1} + \left. \boldsymbol{L}\_{m}^{\boldsymbol{r}} \right|\_{k} \right) = \frac{R\_{\boldsymbol{r}}}{2} \left( \left. \boldsymbol{I}\_{m}^{\boldsymbol{r}} \right|\_{k+1} + \left. \boldsymbol{I}\_{m}^{\boldsymbol{r}} \right|\_{k} \right) + \left. \boldsymbol{L}\_{be} \frac{\left( \left. \boldsymbol{I}\_{m}^{\boldsymbol{r}} \right|\_{k+1} + \left. \boldsymbol{I}\_{m}^{\boldsymbol{r}} \right|\_{k} \right)}{\Delta t} + \frac{1}{2} \left( \left. \boldsymbol{L}\_{m}^{\boldsymbol{b}} \right|\_{k+1} + \left. \boldsymbol{L}\_{m}^{\boldsymbol{b}} \right|\_{k} \right) \tag{33.a}$$

$$\frac{1}{2} \left( \left. U\_{m}^{\rm sc} \right|\_{k+1} + \left. U\_{m}^{\rm sc} \right|\_{k} \right) = \left( \frac{R\_{\rm sc}}{2} + \frac{L\_{\rm sc}}{\Delta t} \right) \left( \left. I\_{m}^{\rm sc} \right|\_{k+1} + \left. I\_{m}^{\rm sc} \right|\_{k} \right) \tag{33.b}$$

The combination of the expressions (32) and (33) with the end-rings voltages and currents (18), lead to the matrix form of the unified loops voltages equations in the rotor cage expressed as:

$$\Re\left(A\_{k+1}, \mathcal{U}\_{mk+1}^{r}\right) = \left(\left.D^{r}\right)A\_{k+1} + \left(\mathcal{C}^{r}\right)\left(\mathcal{U}\_{m}^{r}\right)\_{k+1} + \left.\left(-\mathcal{D}^{r}\right)A\_{k} + \left(\mathcal{C}^{r}\right)\left(\mathcal{U}\_{m}^{r}\right)\_{k} + \left(\mathcal{G}^{r}\right)\left(\mathcal{U}\_{m}^{r}\right)\_{k}\right.\tag{34}$$

$$\begin{aligned} \left[\mathbf{C}^{\tau}\right]\_{\left(N\_{b}\times N\_{b}\right)} &= \frac{\Delta t}{2L\_{\delta}R\_{b}}\times \left[I\_{N\_{b}\times N\_{b}} + \frac{R\_{b}}{2}\right] \left[\left(R\_{\mathrm{sc}} + 2\frac{L\_{\mathrm{sc}}}{\Delta t}\right)I\_{\left(N\_{b}\times N\_{b}\right)} + \left(R\_{\mathrm{he}} + 2\frac{L\_{\mathrm{he}}}{\Delta t}\right)M\_{b}\right]^{-1}M\_{b} \end{aligned} \tag{35.a}$$

$$\begin{aligned} \mathbf{C}^{\tau} &= \frac{\Delta t}{2L\_{\delta}}I\_{N\_{b}\times N\_{b}} - \frac{\Delta t}{2L\_{\delta}}\left[\left(R\_{\mathrm{sc}} + 2\frac{L\_{\mathrm{sc}}}{\Delta t}\right)I + \left(R\_{\mathrm{he}} + 2\frac{L\_{\mathrm{he}}}{\Delta t}\right)M\_{b}\right]^{-1} \\ &\quad \times \left[\left(R\_{\mathrm{sc}} - 2\frac{L\_{\mathrm{sc}}}{\Delta t}\right)I + \left(R\_{\mathrm{he}} - 2\frac{L\_{\mathrm{he}}}{\Delta t}\right)M\_{b}\right] \end{aligned} \tag{35.b}$$

*t t*

22 *sc be sc be <sup>b</sup>*

$$D\_{ij}^r = -\frac{\sigma}{L\_\delta} \int\_{\Omega} \left(\mathcal{J}\_i^r \cdot \mathcal{N}\_j\right) d\Omega \tag{35.c}$$

Where *tr Mb M M* is the auxiliary connection matrix.

#### **4.4 Full magnetic field – Electric circuits coupling model**

Combining equation (25), (30) and (34) a system of coupled equations is obtained:

 1 1 1 1 1 1,2 <sup>1</sup> ( ) 0 0 0 0 *tr tr <sup>q</sup> AGE r s tr <sup>q</sup> k <sup>q</sup> rr r m s s <sup>q</sup> <sup>s</sup> k tr tr <sup>q</sup> AGE r s tr <sup>q</sup> k r r m s s S M A S D D KK A DC U K D G J S M A S D D KK A DC U K D H* <sup>1</sup> 0 - *<sup>q</sup> <sup>r</sup> r r <sup>m</sup> <sup>k</sup> <sup>q</sup> <sup>s</sup> ss s <sup>n</sup> n n k k G I <sup>I</sup> CV V* (36)

Because of the non linearity of the core material, the stiffness matrix [S] depends on the nodal values of the magnetic vector potential. After applied the Newton Raphson iteration method, a final algebraic system of equations for the nonlinear time-stepping simulation of the electrical machine is obtained such as:

$$
\begin{bmatrix}
\begin{bmatrix}
\boldsymbol{P}(\boldsymbol{A}\_{k+1}^{q}) & \left[\boldsymbol{D}^{\boldsymbol{r}}\right]^{\mathrm{tr}} & \left[\boldsymbol{D}^{\boldsymbol{s}}\right]^{\mathrm{tr}}\operatorname{KK}^{\mathrm{tr}} \\
\left[\boldsymbol{D}^{\boldsymbol{r}}\right] & \left[\boldsymbol{C}^{\boldsymbol{r}}\right] & \boldsymbol{0} \\
\boldsymbol{K}\left[\boldsymbol{D}^{\boldsymbol{s}}\right] & \boldsymbol{0} & \left[\boldsymbol{G}^{\boldsymbol{s}}\right]
\end{bmatrix}
\begin{bmatrix}
\boldsymbol{\Delta}\boldsymbol{A}\_{k+1}^{q+1} \\
\boldsymbol{\Delta}\left(\boldsymbol{U}\_{m}^{\boldsymbol{r}}\right)\_{k+1}^{q+1} \\
\boldsymbol{\Delta}\left(\boldsymbol{J}\_{1,2}^{q}\right)\_{k+1}^{q+1}
\end{bmatrix} = \begin{bmatrix}
\mathsf{R}^{\boldsymbol{f}}\left(\boldsymbol{A}\_{k+1}^{q}\left(\boldsymbol{U}\_{m}^{\boldsymbol{r}}\right)\_{k+1}^{q}\left(\boldsymbol{I}\_{1,2}^{\boldsymbol{s}}\right)\_{k+1}^{q}\right) \\
& \mathsf{R}^{\boldsymbol{r}}\left(\boldsymbol{A}\_{k+1}^{q}\left(\boldsymbol{U}\_{m}^{\boldsymbol{r}}\right)\_{k+1}\right) \\
& \mathsf{R}^{\boldsymbol{s}}\left(\boldsymbol{A}\_{k+1}^{q}\left(\boldsymbol{I}\_{1,2}^{\boldsymbol{s}}\right)\_{k+1}^{q}\right)
\end{bmatrix}\tag{37}$$

Where *P* is the Jacobian matrix system expressed through the following matrices elements (Joao, et al., 2003; Arkkio, 1987; Benali, 1997):

$$S\_{ij} = S^{AGE} + \iint\_{\Omega} \nu(A\_{k+1}) (\nabla N\_i) \big(\nabla N\_j\big) d\Omega \tag{38.a}$$

$$P\_{ij} = S^{AGE+} S\_{ij} + I\_{ij} = S^{AGE} + S\_{ij} \quad + \iint \sum\_{\Omega}^{N\_v} \left| \frac{\partial \nu \left( A\_j \right)\_k^q}{\partial A\_j} \right| \left( \nabla N\_i \right) \cdot \left( \nabla N\_j \right) d\Omega \tag{38.b}$$

$$M\_{ij} = \iint \frac{2\sigma}{\Delta t} N\_i N\_j \, d\Omega \tag{38.c}$$

#### **5. Mechanical model and movement simulation**

#### **5.1 Movement simulation technique**

214 Numerical Modelling

 <sup>1</sup> 1 1 1

1 1

The combination of the expressions (32) and (33) with the end-rings voltages and currents (18), lead to the matrix form of the unified loops voltages equations in the rotor cage expressed as:

 11 1 <sup>1</sup> , *r r r r r r r rr k mk <sup>k</sup> <sup>m</sup> <sup>k</sup> <sup>m</sup> <sup>m</sup> <sup>k</sup> k k <sup>A</sup> U DA C U DA C U G I* (34)

*N N sc N N be b b N N <sup>b</sup> <sup>t</sup> RL L <sup>C</sup> I R I R MM L R t t*

2 2

 

*N N sc be b*

*sc sc sc sc sc sc m m m m k k k k R L U U I I <sup>t</sup>* 

*m m r r rr <sup>r</sup> k k b b m m m m be m m k k kk k k I I <sup>R</sup> UU II L U U*

*r r*

*t* 

2 2

(35.c)

(33.a)

1

(35.a)

(35.b)

(36)

1

*G I*


*<sup>q</sup> <sup>s</sup> ss s <sup>n</sup> n n k k*

*<sup>I</sup> CV V*

0

<sup>1</sup>

*<sup>m</sup> <sup>k</sup>*

(33.b)

1 1 2 2 <sup>2</sup>

2 2

2 2 *b b*

Where *tr Mb M M* is the auxiliary connection matrix.

1

( )

*k*

**4.4 Full magnetic field – Electric circuits coupling model** 

0

*K D G J*

*S M A S D D KK A*

0

*r r*

*k*

2 2 *b b b b b b*

*r b sc be*

*r sc be*

 

*t t L L G I R IR M L Lt t*

> 22 *sc be sc be <sup>b</sup> L L R IR M t t*

> > *r r Dij i j N d L*

*m*

1

1 1,2 <sup>1</sup>

*m*

Because of the non linearity of the core material, the stiffness matrix [S] depends on the nodal values of the magnetic vector potential. After applied the Newton Raphson iteration

*k*

Combining equation (25), (30) and (34) a system of coupled equations is obtained:

0

*tr tr <sup>q</sup> AGE r s tr <sup>q</sup>*

*DC U*

1 1

*<sup>q</sup> rr r*

*s s <sup>q</sup> <sup>s</sup>*

*tr tr <sup>q</sup> AGE r s tr <sup>q</sup>*

*DC U*

0

*s s*

*K D H*

*S M A S D D KK A*

1

For modelling the movement in rotating electrical machines using the Air-Gap-Element (AGE), the space discretised domain is commonly split up into two subdomains, a stator *stator* , and rotor *rotor* meshed domains, and the unmeshed air gap with *AGE* boundary (Fig. 5).

Fig. 5. Air-Gap Element (macro-element).

Finite Element Method Applied to the Modelling and Analysis of Induction Motors 217

The theory of the previous paragraphs is applied for the simulation of an induction motor in different operating modes. Two cases are studied, the first one concern the electric transient state simulation of induction motor while considering constant speed, and the second one treat the general electromagnetic-mechanical transient state simulation. The solution process of the electromagnetic-mechanical non-linear transient model is summurised by the

**6. Simulation results and discussions** 

following chart (see Fig.6).

**6.1 Algorithm of Non-linear step by step finite element solver** 

Fig. 6. General algorithm for induction machine numerical analysis.

In a general step-by-step solution of the magnetic field in rotating electrical machines, the stator and rotor magnetic field equations are expressed in their own coordinate systems. The solutions of both fields equations are matched to each other in the air-gap. The rotor is rotated at each time step by an angle corresponding to the mechanical angular frequency, this means that a new finite element mesh in the air gap has to be constructed. The basic form of the air gap element matrix general terms is given by the expression:

$$A(a,b,r,\theta) = \sum\_{j=1}^{m\text{ell}} A\_i \left[ \frac{1}{2} a\_{0j} + \sum\_{r=1}^{n} \left[ a\_{r\bar{\eta}} \cos(\lambda\_r \theta) + b\_{\bar{\eta}} \sin(\lambda\_r \theta) \right] \right] \\ = \sum\_{j=1}^{m\text{t}} \left( N\_j^{AGE} \left( a,b,r,\theta \right) \right) \cdot A\_j^{\Gamma\_{A\Sigma}} \tag{39}$$

Where <sup>0</sup> ,,, *<sup>j</sup> rj rj <sup>r</sup> a ab* are the Fourier's expansion coefficient which depends on the air-gap nodes coordinates locations, *nnt* is the total numbers of the air-gap nodes.

The movement simulation is take into account through the expression (39) while computing the associated matrix of the air-gap element given by the line integral term defined in the formulation (24). The air-gap-element matrix is given as follow:

$$\oint\_{\Gamma^{ACE}} \nu\_o \sum\_{j=1}^{\text{mt}} \left\{ A\_j^{AGE} \right\} \frac{\partial}{\partial \mathfrak{n}} \left( N\_j^{AGE} \left( r, \theta \right) \right) d\Gamma^{AGE} = \left[ S^{AGE} \right] \left\{ A\_j^{AGE} \right\} \tag{40}$$

For a complete development of the air gap element, the reader are invited to detailed implementation given in (Abdel-Razek, et al., 1982; Joao, et al., 2003).

#### **5.2 Mechanical equations and torque computation**

In a general case the magnetic field and electric circuits equations are coupled to the rotor mechanical equation through the electromagnetic torque. This includes the interaction between mechanical and electromagnetic quantities (Ho, et al., 2000). The mechanical differential system equations of speed and angular displacement is given as follow:

$$
\frac{d}{dt} \begin{bmatrix} \alpha \\ \theta \end{bmatrix} = \begin{bmatrix} -\frac{f}{J\_m} & 0 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} \alpha \\ \theta \end{bmatrix} + \begin{bmatrix} \frac{1}{J\_m} (C\_{em}(t) - C\_{load}) \\ 0 \end{bmatrix} \tag{41}
$$

At each step time, the computed electromagnetic torque is introduced in the mechanical model (41) solved using the fourth order Rung- Kutta method to get the rotor angular displacement and speed. The electromagnetic torque is computed from Maxwell's Tensor as a function of radial and tangential components of the magnetic flux density:

$$\mathbf{C}\_{cm} = \frac{\operatorname{pr}^2 \mathbf{L}\_{\delta}}{\mu\_0} \int\_{\theta\_1}^{\theta\_2} B\_r B\_\theta d\theta \tag{42}$$

Where *r* is the rotor external radius, and *p* the number of poles pairs. The magnetic flux density components , *B B <sup>r</sup>* are computed in the air gap boundaries through the derivatives expression of the analytical magnetic vector potential shape functions (39).

#### **6. Simulation results and discussions**

216 Numerical Modelling

In a general step-by-step solution of the magnetic field in rotating electrical machines, the stator and rotor magnetic field equations are expressed in their own coordinate systems. The solutions of both fields equations are matched to each other in the air-gap. The rotor is rotated at each time step by an angle corresponding to the mechanical angular frequency, this means that a new finite element mesh in the air gap has to be constructed. The basic

<sup>0</sup>

(39)

are the Fourier's expansion coefficient which depends on the air-gap

*N abr A*

*i j rj r rj r j j*

  *AGE*

(41)

form of the air gap element matrix general terms is given by the expression:

nodes coordinates locations, *nnt* is the total numbers of the air-gap nodes.

*n*

implementation given in (Abdel-Razek, et al., 1982; Joao, et al., 2003).

formulation (24). The air-gap-element matrix is given as follow:

1

**5.2 Mechanical equations and torque computation** 

*j*

*AGE*

density components , *B B <sup>r</sup>*

*Aabr A a a b*

Where <sup>0</sup> ,,, *<sup>j</sup> rj rj <sup>r</sup> a ab*

1 1 1 <sup>1</sup> (,,, ) cos( ) sin( ) ,,, <sup>2</sup>

*j r j*

 

*nntl nnt AGE*

The movement simulation is take into account through the expression (39) while computing the associated matrix of the air-gap element given by the line integral term defined in the

 

*nnt AGE AGE AGE AGE AGE oj j j*

*A N rd S A*

 

For a complete development of the air gap element, the reader are invited to detailed

In a general case the magnetic field and electric circuits equations are coupled to the rotor mechanical equation through the electromagnetic torque. This includes the interaction between mechanical and electromagnetic quantities (Ho, et al., 2000). The mechanical

1 0 0

At each step time, the computed electromagnetic torque is introduced in the mechanical model (41) solved using the fourth order Rung- Kutta method to get the rotor angular displacement and speed. The electromagnetic torque is computed from Maxwell's Tensor as

2

2

0 *em r pr L <sup>C</sup> BBd* 

Where *r* is the rotor external radius, and *p* the number of poles pairs. The magnetic flux

derivatives expression of the analytical magnetic vector potential shape functions (39).

1

(42)

are computed in the air gap boundaries through the

*m m <sup>f</sup> <sup>d</sup> CtC*

 

 

differential system equations of speed and angular displacement is given as follow:

*<sup>J</sup> <sup>J</sup> dt*

a function of radial and tangential components of the magnetic flux density:

,

<sup>1</sup> <sup>0</sup> ( )

*em load*

(40)

#### **6.1 Algorithm of Non-linear step by step finite element solver**

The theory of the previous paragraphs is applied for the simulation of an induction motor in different operating modes. Two cases are studied, the first one concern the electric transient state simulation of induction motor while considering constant speed, and the second one treat the general electromagnetic-mechanical transient state simulation. The solution process of the electromagnetic-mechanical non-linear transient model is summurised by the following chart (see Fig.6).

Fig. 6. General algorithm for induction machine numerical analysis.

Finite Element Method Applied to the Modelling and Analysis of Induction Motors 219

Fig. 8. Finite Element Mesh with (AGE).

respectively for no load and nominal conditions.

voltages.

**6.3 Electromagnetic transient operating condition with constant speed** 

Results of this part concern the transient electromagnetic state simulation, at no load and nominal operation modes while the speed is considered constant. The electromagnetic transient is simulated while considering that the motor is operating in steady state with constant speed. Stator and rotor meshes are coupled by air-gap-element matrix. The constant speed value is equal to 1495 tr/mn and 1348 tr/mn respectively for the no-load and nominal modes. Since the mechanical phenomena is not considered, the speed is constant and the rotor displacement is not taken into account. The air-gap-element matrix is calculated only once. At each step time, the algebraic system (37) corresponded to Newton-Raphson algorithm is iteratively solved in order to get the magnetic permeability value. The latest is then used to establish the algebraic system (36) which the solution lead to the values of the magnetic vector potential, stator windings currents and the rotor bars

The stator windings currents wave forms corresponded to the electromagnetic transient state of the no-load and nominal conditions are respectively shown in Fig. 9 and Fig. 10. We note that, in agreement with the theory a high starting currents are obtained which decreases quickly because of the small electromagnetic durations. Electromagnetic torque for the no-load and nominal electromagnetic transient operations is given by the Fig. 11. After a brief transient duration the torque is stabilized at 4.5 N.m and 37.3 N.m values

#### **6.2 Presentation of the studied induction motor**

The considered simulated system to apply the model of the present works is an three phase induction motor Leroy Sommer (Mezani, 2004). The poles number is four, the rated power, the efficiency, and voltage are respectively 5.5KW, 84.26% and 380V. The stator windings and the rotor cage (bars-end rings) are made respectively with cooper and aluminium materials. More detailed characteristics of the motor are presented in Table. 1.


Table 1. Studied induction motor geometrical and physical datas.

The stator and rotor slots geometrical dimensions are detailed in the following Fig. 7. To reduce the computational time due to nodes number of the finite element mesh and the geometrical complexity, usually the electrical machines models are created on the smallest symmetrical part of the machine. The (Fig. 8) shows the finite element mesh of the motor studied domain where only ¼ of the motors. The mesh containing 3204 nodes and 5623 first order triangular element is obtained using the Matlab PdeTool mesh automatic generator. We note that the air-gap between the stator and rotor meshes is not meshed and coupled together through the air-gap matrix (40). Homogeneous Dirichlet boundary condition is imposed for the external and internal motor radius, and anti-periodic ones at the other boundaries.

Fig. 7. Gometrical dimensions of the motor slots.

Fig. 8. Finite Element Mesh with (AGE).

218 Numerical Modelling

The considered simulated system to apply the model of the present works is an three phase induction motor Leroy Sommer (Mezani, 2004). The poles number is four, the rated power, the efficiency, and voltage are respectively 5.5KW, 84.26% and 380V. The stator windings and the rotor cage (bars-end rings) are made respectively with cooper and aluminium

materials. More detailed characteristics of the motor are presented in Table. 1.

Table 1. Studied induction motor geometrical and physical datas.

Fig. 7. Gometrical dimensions of the motor slots.

boundaries.

Geometrical components Values (mm) Physical components Values Stator core external diameter 168.0 Rated nominal current 11.62A Stator core internal diameter 130.0 Nominal torque 37 N.m Rotor core external diameter 109.2 Power factor 0.865

Rotor core inernal diameter 66.4 Moment of inertia 0.014 (Kg.m2) Axial length 160.0 Friction coefficient 0.011 (1/ms) Stator conductors per slot 19 Slip 4.13% Number of stator slot 48 slots Stator phase resistance 1.4 Number of rotor slot 24 bars Stator phase inductance 0.2 mH

The stator and rotor slots geometrical dimensions are detailed in the following Fig. 7. To reduce the computational time due to nodes number of the finite element mesh and the geometrical complexity, usually the electrical machines models are created on the smallest symmetrical part of the machine. The (Fig. 8) shows the finite element mesh of the motor studied domain where only ¼ of the motors. The mesh containing 3204 nodes and 5623 first order triangular element is obtained using the Matlab PdeTool mesh automatic generator. We note that the air-gap between the stator and rotor meshes is not meshed and coupled together through the air-gap matrix (40). Homogeneous Dirichlet boundary condition is imposed for the external and internal motor radius, and anti-periodic ones at the other

**6.2 Presentation of the studied induction motor** 

#### **6.3 Electromagnetic transient operating condition with constant speed**

Results of this part concern the transient electromagnetic state simulation, at no load and nominal operation modes while the speed is considered constant. The electromagnetic transient is simulated while considering that the motor is operating in steady state with constant speed. Stator and rotor meshes are coupled by air-gap-element matrix. The constant speed value is equal to 1495 tr/mn and 1348 tr/mn respectively for the no-load and nominal modes. Since the mechanical phenomena is not considered, the speed is constant and the rotor displacement is not taken into account. The air-gap-element matrix is calculated only once. At each step time, the algebraic system (37) corresponded to Newton-Raphson algorithm is iteratively solved in order to get the magnetic permeability value. The latest is then used to establish the algebraic system (36) which the solution lead to the values of the magnetic vector potential, stator windings currents and the rotor bars voltages.

The stator windings currents wave forms corresponded to the electromagnetic transient state of the no-load and nominal conditions are respectively shown in Fig. 9 and Fig. 10. We note that, in agreement with the theory a high starting currents are obtained which decreases quickly because of the small electromagnetic durations. Electromagnetic torque for the no-load and nominal electromagnetic transient operations is given by the Fig. 11. After a brief transient duration the torque is stabilized at 4.5 N.m and 37.3 N.m values respectively for no load and nominal conditions.

Finite Element Method Applied to the Modelling and Analysis of Induction Motors 221

Fig. 11. Electromagnetic torque at no-load and nominal modes.

position is calculated at each displacement step.

**6.4 Electromagnetic-mechanical transient operating with direct start** 

Results of this part concern the transient electromagnetic-mechanical general simulation, at no load and load direct start operation modes. The solution process detailed by the Fig.6 is summarized by the following steps. Firstly, the algebraic equation system (37) is solved to get the magnetic reluctivity associated to the voltage level at each step time. Secondly, the algebraic equation system (36) is solved, which permit us to know the stator windings currents, the rotor bar voltages, and the magnetic vector potential which lead to deduce the magnetic flux density, and permit the computation of the electromagnetic torque. The latest is introduced in the mechanical model, which solution leads to the speed and angular displacement of the rotor. Since the mechanical phenomena is considered, the rotor displacement is taken into account, the air-gap-element matrix corresponded to each rotor

The motor simulations concerns a direct start loaded condition with load torque of 10 N.m. The stator windings currents wave forms are given by the Fig. 12 and Fig. 13. Motor angular

speed, and electromagnetic torque are given by the Fig. 14, and Fig. 15, respectively.

Fig. 9. Stator currents in no-load mode.

Fig. 10. Stator currents nominal mode.

Fig. 9. Stator currents in no-load mode.

Fig. 10. Stator currents nominal mode.

Fig. 11. Electromagnetic torque at no-load and nominal modes.

#### **6.4 Electromagnetic-mechanical transient operating with direct start**

Results of this part concern the transient electromagnetic-mechanical general simulation, at no load and load direct start operation modes. The solution process detailed by the Fig.6 is summarized by the following steps. Firstly, the algebraic equation system (37) is solved to get the magnetic reluctivity associated to the voltage level at each step time. Secondly, the algebraic equation system (36) is solved, which permit us to know the stator windings currents, the rotor bar voltages, and the magnetic vector potential which lead to deduce the magnetic flux density, and permit the computation of the electromagnetic torque. The latest is introduced in the mechanical model, which solution leads to the speed and angular displacement of the rotor. Since the mechanical phenomena is considered, the rotor displacement is taken into account, the air-gap-element matrix corresponded to each rotor position is calculated at each displacement step.

The motor simulations concerns a direct start loaded condition with load torque of 10 N.m. The stator windings currents wave forms are given by the Fig. 12 and Fig. 13. Motor angular speed, and electromagnetic torque are given by the Fig. 14, and Fig. 15, respectively.

Fig. 12. Stator windings currents for direct start loaded motor.

Finite Element Method Applied to the Modelling and Analysis of Induction Motors 223

Fig. 13. Steady state stator windings currents for loaded direct start motor.

Fig. 12. Stator windings currents for direct start loaded motor.

Fig. 13. Steady state stator windings currents for loaded direct start motor.

Finite Element Method Applied to the Modelling and Analysis of Induction Motors 225

From the transient stator currents results given by the Fig. 12 we note a high starting currents which reach to nominal steady state values after average half periods such as given by the Fig. 13. The several ascillation of the currents transients behavior is due to the strong electromagnetic and mechanical interaction through theirs corresponded time constants.For the rotor speed given by the Fig.14, we note that after some modulations at the motor start, gthe speed increase linearely till it steady state values according to mechanical first order differential equation. The Fig. 15 of electromagnetic torque show that after some periods oft

This chapter goal is to present a detailed finite element method use to solved partial differential equation of electromagnetic phenomena occurred in induction motor. The magnetic field equation expressed in term of magnetic vector potential strongly coupled with the electric circuits equations of the stator windings and rotor cage are solved using the nodal based finite element process. The resulting nonlinear time-dependent algebraic differential equations system obtained from the finite element formulation is solved using step-by-step numerical integration based on the Crank-Nicholson scheme, combined to the Newton-Raphson iterative process for handling the magnetic material non-linearity. The electromagnetic and mechanic interaction is considered firstly by the computation of the electromagnetic torque by the Maxwell stress tensor responsible of the rotor displacement, and secondly by solving the mechanical motional equation to get the new rotor angular position. Since the motor is meshed only once, the rotor movement is taking into account by the macro-element method which lead to an air-gap matrix of the movement. The validation of the model is performed through simulation of an induction motor in no-load and loaded direct start operating modes. The numerical results are in good agreement with corresponding results appearing in the recent literature. The contribution of this work can be applied to analyze a large class of electrical machines, and offers an important support for students, teachers and industrial employers for understanding the basis of numerical

Abdel-Razek, A.; Coulomb, J.L.; Féliachi, M. & Sabonnadière, J.C. (1982), Conception of an

Arkkio, A. (1987), Analysis of induction motors based on the numerical solution of the

Benali, B. (1997), Contribution à la modélisation des systèmes électrotechniques à l'aide des

University of Sciences and Technology of Lille, France.

Air-Gap Element for the Dynamic Analysis of the Electromagnetic Field in Electric Machines, *IEEE Transaction On Magnetics,* Vol.18, No.2, (March 1982), pp. 655-659,

magnetic field and circuit equations. PhD Dissertation, Helsinki University of

formulations en potentiel : Application à la machine asynchrone*,* Doctorat thesis,

he magnetic duration the torque reach to it steady state value of average 37.3 N.m.

**7. Conclusion** 

modelling of electrical machines.

ISSN 0018-9464.

Technology, Sweeden.

**8. References** 

Fig. 14. Speed for loaded direct start motor.

Fig. 15. Electromagnetic torque for loaded direct stat motor.

From the transient stator currents results given by the Fig. 12 we note a high starting currents which reach to nominal steady state values after average half periods such as given by the Fig. 13. The several ascillation of the currents transients behavior is due to the strong electromagnetic and mechanical interaction through theirs corresponded time constants.For the rotor speed given by the Fig.14, we note that after some modulations at the motor start, gthe speed increase linearely till it steady state values according to mechanical first order differential equation. The Fig. 15 of electromagnetic torque show that after some periods oft he magnetic duration the torque reach to it steady state value of average 37.3 N.m.

### **7. Conclusion**

224 Numerical Modelling

Fig. 14. Speed for loaded direct start motor.

Fig. 15. Electromagnetic torque for loaded direct stat motor.

This chapter goal is to present a detailed finite element method use to solved partial differential equation of electromagnetic phenomena occurred in induction motor. The magnetic field equation expressed in term of magnetic vector potential strongly coupled with the electric circuits equations of the stator windings and rotor cage are solved using the nodal based finite element process. The resulting nonlinear time-dependent algebraic differential equations system obtained from the finite element formulation is solved using step-by-step numerical integration based on the Crank-Nicholson scheme, combined to the Newton-Raphson iterative process for handling the magnetic material non-linearity. The electromagnetic and mechanic interaction is considered firstly by the computation of the electromagnetic torque by the Maxwell stress tensor responsible of the rotor displacement, and secondly by solving the mechanical motional equation to get the new rotor angular position. Since the motor is meshed only once, the rotor movement is taking into account by the macro-element method which lead to an air-gap matrix of the movement. The validation of the model is performed through simulation of an induction motor in no-load and loaded direct start operating modes. The numerical results are in good agreement with corresponding results appearing in the recent literature. The contribution of this work can be applied to analyze a large class of electrical machines, and offers an important support for students, teachers and industrial employers for understanding the basis of numerical modelling of electrical machines.

#### **8. References**


**Part 3** 

**Mechanics and Materials** 


**Part 3** 

**Mechanics and Materials** 

226 Numerical Modelling

Binns, K.J.; Lawrenson, P.J, & Trowbridge, C.W. (1994), The Analytical and Numerical

Brauer, J.R. ; Ruehl, J.J, & Hirtenfelder, F. (1985), Coupled nonlinear electromagnetic and

Dreher, T.; Perrin-Bit, R.; Meunier, G. & Coulomb, J.L. (1996), A 3D finite element

Hecht, F.; Marrocco, A.; Piriou, F. & Abdel-Razek,A. (1990), Modélisation des systèmes

Ho, S.L.; Li, H.L.; Fu, W.N. & Wong, H.C. (2000), A novel approach to circuit-field-torque

Ho, S.L., & Fu, W.N. (1997), A comprehensive approach to the solution of direct-coupled

Joao, P.; Bastos, A. & Sadowski, N. (2003), Electromagnetic Modeling by Finite Element Methods, Marcel Dekker Inc, (Ed.), ISBN 0824742699, New York, United States. Kanerva, S. (2005), Simulation of electrical machines, circuits and control systems using

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**11** 

Nadia Bhuiyan *Concordia University* 

*Canada* 

**Numerical Evaluation of Product** 

The management of new product development (NPD) processes is a continual challenge facing organizations that develop complex, innovative products. While market trends are forcing shorter product development times in order to meet time-to-market (TTM) goals, companies are trying to develop mechanisms to streamline their NPD processes. One approach that has provided much success towards achieving shorter TTM is concurrent engineering (Winner et al., 1988; Clark and Fujimoto, 1991; Blackburn, 1991; Wheelwright and Clark, 1992; Smith and Reinersten, 1991). Concurrent engineering (CE) can broadly be defined as the integration of inter-related functions at the outset of the product development process in order to minimize risk and reduce effort downstream in the process, and to better meet customers' needs (Winner et al., 1988). Multi-functional teams, concurrency of product/process development, integration tools, information technologies, and process coordination are among the elements that enable CE to improve the performance of the product development process (Blackburn, 1991). The traditional NPD process suffers many setbacks. This process evolves in a sequential fashion, where phases follow one another serially, each one dominated by a single functional role. There is little or no crosscommunication among various functions, and information generated from one activity gets handed off to the next only after its completion. The commonly encountered problems with this type of process are increased downstream effort, process span time, i.e., the start to

In order to study and evaluate the performance of CE and sequential NPD processes, a new approach is used based on an existing mathematical technique called the expected payoff method, which is the basis of decision theory. Under this framework, the mathematics which describe the micro-processes, such as information sharing between team members and overlapping of activities, and their relationships with the macroprocess performance in terms of expected payoff (where the macro-process is the overall development process), are described. Network diagrams are presented as a formalism for expressing product development processes. The fundamental concept of the model is based on the premise that team members make decisions or choose actions that maximize the payoff (utility or usefulness) that their actions bring to the team. Team members must obtain, process, and communicate information to one another to make decisions that will

**1. Introduction** 

finish time of the process, and costs.

optimize their performance.

**Development Processes** 

## **Numerical Evaluation of Product Development Processes**

Nadia Bhuiyan *Concordia University Canada* 

#### **1. Introduction**

The management of new product development (NPD) processes is a continual challenge facing organizations that develop complex, innovative products. While market trends are forcing shorter product development times in order to meet time-to-market (TTM) goals, companies are trying to develop mechanisms to streamline their NPD processes. One approach that has provided much success towards achieving shorter TTM is concurrent engineering (Winner et al., 1988; Clark and Fujimoto, 1991; Blackburn, 1991; Wheelwright and Clark, 1992; Smith and Reinersten, 1991). Concurrent engineering (CE) can broadly be defined as the integration of inter-related functions at the outset of the product development process in order to minimize risk and reduce effort downstream in the process, and to better meet customers' needs (Winner et al., 1988). Multi-functional teams, concurrency of product/process development, integration tools, information technologies, and process coordination are among the elements that enable CE to improve the performance of the product development process (Blackburn, 1991). The traditional NPD process suffers many setbacks. This process evolves in a sequential fashion, where phases follow one another serially, each one dominated by a single functional role. There is little or no crosscommunication among various functions, and information generated from one activity gets handed off to the next only after its completion. The commonly encountered problems with this type of process are increased downstream effort, process span time, i.e., the start to finish time of the process, and costs.

In order to study and evaluate the performance of CE and sequential NPD processes, a new approach is used based on an existing mathematical technique called the expected payoff method, which is the basis of decision theory. Under this framework, the mathematics which describe the micro-processes, such as information sharing between team members and overlapping of activities, and their relationships with the macroprocess performance in terms of expected payoff (where the macro-process is the overall development process), are described. Network diagrams are presented as a formalism for expressing product development processes. The fundamental concept of the model is based on the premise that team members make decisions or choose actions that maximize the payoff (utility or usefulness) that their actions bring to the team. Team members must obtain, process, and communicate information to one another to make decisions that will optimize their performance.

Numerical Evaluation of Product Development Processes 231

overlapping can reduce rework effects, but at the cost of communication time. They also use

Yassine et al. (1999) have studied the CE problem of overlapping activities through a decision analytic framework. Using a probabilistic model consisting of an upstream activity and a downstream activity, their methodology finds the optimal overlapping policy based on the study of independent, dependent, and interdependent activities, described as the information structure of a process. A schedule of when to transfer information based on the information structures can fall under one of three categories: sequential, partial overlapping, and concurrent. Sequential transfer of information takes place for dependent activities. Partial overlapping can take place for either dependent or interdependent activities. In both cases, however, the information exchange/transfer must appropriately minimize the risk of downstream rework in the event of a change in the upstream activity. A concurrent schedule can take place when the activities are independent; since neither requires

Ha and Porteus (1995) developed a simple model that proposes the optimal policy for the frequency and timing of progress reviews in an overlapped process. The authors study two overlapped, interdependent activities, an upstream design activity and a downstream process activity. In contrast to sequentially dependent activities, the nature of interdependent activities requires team members to communicate frequently. They develop a dynamic program that shows that, in order for overlapped activities to be beneficial, the design activity must be accompanied by progress reviews to minimize the risk of downstream rework and thus span time, and to improve quality. However, these gains are only achieved at the expense of the time and cost spent on communication. Therefore, the frequency of communication or progress reviews must be balanced with the value gained from having them. The optimal policy of reviews minimizes span time by providing sufficient information at the right time, helping to identify potential design

Different methods to address the problem of overlapping have been suggested in the literature. This research contributes to the existing work by introducing a methodology

An information processing view of organizations, and thus of product development, is assumed in this research. From this perspective, the product development process must go through a set of decision-making processes to transform information inputs into information outputs, which are used to develop tangible outputs, i.e., the end product(s) (Clark and Fujimoto, 1991; Galbraith, 1973). Therefore, the focus of the models is on the flow of information as it evolves from the beginning to the end of the development process, making

Product development can be defined as the process of undertaking all the activities and processing the information required to develop a concept for a product up to the product's market introduction. NPD processes may vary from one organization to the next, and as such, there is no one standard process agreed to by all. However, the general steps required

information from the other to proceed, they may be executed in parallel.

based on decision theory to study the performance of processes.

the relationships between development activities more readily apparent.

the concepts of evolution and sensitivity.

problems early.

**3. NPD processes** 

This chapter is organized as follows. Section 2 discusses the existing literature, and highlights the contributions of the research. Section 3 explains the characteristics of NPD processes. In Section 4, the expected payoff method is described and the results of the mathematical analysis are presented. The results are detailed in Section 5, and in Section 6, conclusions and paths for future research are presented.

#### **2. Literature review**

In this section, a review of the relevant theoretical and analytical research is presented. Krishnan et al. (1997) developed a deterministic model based on properties of the design process that help to determine when and how two development activities should be overlapped (Figure 1). These properties are defined as 'upstream information evolution' and 'downstream iteration sensitivity'. The former is the rate at which upstream information converges to a final solution, and the information is modeled as an interval that gets refined over time. Sensitivity describes how vulnerable the downstream activity is to any changes in the upstream information, and is defined by the time needed by the downstream activity to incorporate the changes, which represents rework. Different patterns of information exchange between two activities, represented by the arrows in the diagram, are studied.

Fig. 1. Krishnan et al.'s model.

The authors address the overlapping problem by studying how values of the two properties determine the extent to which overlapping is appropriate between the dependent activities, A and B, and consequently how the span time is affected. Various overlapping policies between the upstream and downstream activities are examined based on varying the values of these two properties, and an integer program is developed to minimize span time.

Loch and Terwiesch (1998) have developed an analytical model of CE that considers the overlapping of two sequentially dependent activities, an upstream product design activity, and a downstream process design activity. The authors study the trade-offs between the downstream activity using upstream preliminary information to overlap activities, and the corresponding delay this might cause in terms of downstream rework. They suggest that when engineering changes (EC) arise during the product design, this poses the risk of redoing the overlapped work of the downstream activity, and this can be significant if the dependency between the two activities is high. They propose that communication during

This chapter is organized as follows. Section 2 discusses the existing literature, and highlights the contributions of the research. Section 3 explains the characteristics of NPD processes. In Section 4, the expected payoff method is described and the results of the mathematical analysis are presented. The results are detailed in Section 5, and in Section 6,

In this section, a review of the relevant theoretical and analytical research is presented. Krishnan et al. (1997) developed a deterministic model based on properties of the design process that help to determine when and how two development activities should be overlapped (Figure 1). These properties are defined as 'upstream information evolution' and 'downstream iteration sensitivity'. The former is the rate at which upstream information converges to a final solution, and the information is modeled as an interval that gets refined over time. Sensitivity describes how vulnerable the downstream activity is to any changes in the upstream information, and is defined by the time needed by the downstream activity to incorporate the changes, which represents rework. Different patterns of information exchange between two activities, represented by the arrows in the diagram, are studied.

**Upstream A**

**Downstream B**

Level of overlap between A and B

Impact of preliminary information on downstream activity: rework

Span Time

The authors address the overlapping problem by studying how values of the two properties determine the extent to which overlapping is appropriate between the dependent activities, A and B, and consequently how the span time is affected. Various overlapping policies between the upstream and downstream activities are examined based on varying the values

Loch and Terwiesch (1998) have developed an analytical model of CE that considers the overlapping of two sequentially dependent activities, an upstream product design activity, and a downstream process design activity. The authors study the trade-offs between the downstream activity using upstream preliminary information to overlap activities, and the corresponding delay this might cause in terms of downstream rework. They suggest that when engineering changes (EC) arise during the product design, this poses the risk of redoing the overlapped work of the downstream activity, and this can be significant if the dependency between the two activities is high. They propose that communication during

of these two properties, and an integer program is developed to minimize span time.

conclusions and paths for future research are presented.

Upstream Evolution of information from an interval to a final point

> Information exchange/transfer between A and B

Fig. 1. Krishnan et al.'s model.

**2. Literature review** 

overlapping can reduce rework effects, but at the cost of communication time. They also use the concepts of evolution and sensitivity.

Yassine et al. (1999) have studied the CE problem of overlapping activities through a decision analytic framework. Using a probabilistic model consisting of an upstream activity and a downstream activity, their methodology finds the optimal overlapping policy based on the study of independent, dependent, and interdependent activities, described as the information structure of a process. A schedule of when to transfer information based on the information structures can fall under one of three categories: sequential, partial overlapping, and concurrent. Sequential transfer of information takes place for dependent activities. Partial overlapping can take place for either dependent or interdependent activities. In both cases, however, the information exchange/transfer must appropriately minimize the risk of downstream rework in the event of a change in the upstream activity. A concurrent schedule can take place when the activities are independent; since neither requires information from the other to proceed, they may be executed in parallel.

Ha and Porteus (1995) developed a simple model that proposes the optimal policy for the frequency and timing of progress reviews in an overlapped process. The authors study two overlapped, interdependent activities, an upstream design activity and a downstream process activity. In contrast to sequentially dependent activities, the nature of interdependent activities requires team members to communicate frequently. They develop a dynamic program that shows that, in order for overlapped activities to be beneficial, the design activity must be accompanied by progress reviews to minimize the risk of downstream rework and thus span time, and to improve quality. However, these gains are only achieved at the expense of the time and cost spent on communication. Therefore, the frequency of communication or progress reviews must be balanced with the value gained from having them. The optimal policy of reviews minimizes span time by providing sufficient information at the right time, helping to identify potential design problems early.

Different methods to address the problem of overlapping have been suggested in the literature. This research contributes to the existing work by introducing a methodology based on decision theory to study the performance of processes.

#### **3. NPD processes**

An information processing view of organizations, and thus of product development, is assumed in this research. From this perspective, the product development process must go through a set of decision-making processes to transform information inputs into information outputs, which are used to develop tangible outputs, i.e., the end product(s) (Clark and Fujimoto, 1991; Galbraith, 1973). Therefore, the focus of the models is on the flow of information as it evolves from the beginning to the end of the development process, making the relationships between development activities more readily apparent.

Product development can be defined as the process of undertaking all the activities and processing the information required to develop a concept for a product up to the product's market introduction. NPD processes may vary from one organization to the next, and as such, there is no one standard process agreed to by all. However, the general steps required

Numerical Evaluation of Product Development Processes 233

Compared to a sequential approach, CE can decrease span time at the expense of increased interdependencies between activities (sequential to reciprocal). To handle the increased interdependencies, close intensive coordination is required through functional participation.

Figure 3 shows an overlapped CE process. Note that information flows are more frequent than in the sequential case, and they are also bi-directional. Major milestones exist at the same gates as before, and each phase is made up of activities, not shown in the figure.

C - DEFINITION

In this section, a mathematical approach is described to measure the performance of processes, namely, sequential and CE processes, through the study of macro- and microvariables. The macro-variable is the expected payoff, while the micro-variables are team interaction and level of overlap. The concepts of information processing and decisionmaking are presented as the basis of this framework. An information processing view is assumed, so that processes are studied through the way in which several team members perform activities such as acquiring, communicating, and processing information in order to make decisions, which in turn organizes the way activities are executed. Processes and corresponding team activities are modeled via networks of interconnected elements. These elements transform inputs into outputs, and represent people, machines, or other real-world objects. Each network realizes an output which is a measure of process performance, and is used to evaluate and compare processes. The methodology is based on the expected payoff method, a technique used in decision theory. It is applied in the calculation of a simple

The principle of the expected payoff method has been applied mainly in the field of economics, management science, and in certain areas of artificial intelligence, with respect to decision-making. In this field, economists study 'the best use of available (limited) resources' (Marschak and Radner, 1972). There has been no use of this method in the evaluation of CE in new product development processes. In an organizational environment, teams are also concerned with making the best use of alternatives or limited resources. The interested reader can find several readings in the literature on the principle of utility theory and its various applications (Marschak and Radner, 1972; Fishburn, 1970; Marschak, 1959; Marschak, 1954; von Neumann and Morgenstern, 1943). The framework developed in this part of the research will compare a simple model of a sequential process to a CE process,

D - IMPLEMENTATION

However, this may increase effort.

Fig. 3. CE development process.

**4. Expected payoff method** 

A - CONCEPT

B - DEVELOPMENT

model of both a sequential and a CE process, and the results are compared.

and evaluate the two in terms of the total expected payoff.

in a product development process are fundamentally similar (Ulrich and Eppinger 2011). The NPD process defined in this study is a generic one which outlines the major steps in product development. It is a summary of the common phases and activities used in many instances in the literature as well as in the case study, and as such, it is a reasonably accepted approach to representing the product development process (Schilling and Will, 1998; Nihtila, 1999; Eastman, 1980).

The NPD process, shown in Figure 2, begins with the development of a concept for a marketable product (Phase A). In this phase, market requirements are determined, new ideas are generated, screened for economic and technical feasibility, and one is selected. In Phase B, 'Definition', a set of specifications to make the product is defined, and the product architecture is developed. Phase C, 'Development', consists of detailed design, physical prototyping, and testing. Finally, in Phase D, 'Implementation', the product volume is ramped up in manufacturing and launched onto the market.

Fig. 2. A schematic diagram for a general stage-gate process with Phases A, B, C, D.

Figure 2 is an example of the traditional NPD process, where the phases are performed sequentially one after the other. Between phases, a one-way dependence is assumed, that is, the downstream phase depends on information generated by the upstream phase, but not vice-versa. This is represented by the uni-directional arrows between phases.

In an NPD process, the relationship between product and process design is mutually interdependent (Tian et al., 1998). This means that the information generated by one or more functions poses contingencies for others, thus, the parameters of the product and the process should be considered simultaneously (Adler 1995). Therefore a higher degree of coordination is required to manage more people collaborating on interdependent activities. In a sequential process, this interdependence is ignored; a dependent relationship is assumed, and this leads to unplanned coordination downstream. While better management of interdependencies does lead to shortened span time as compared to the sequential process, the price is higher cost of upstream effort.

CE uses two main mechanisms to reduce the span time for NPD processes: 1) increased information sharing from the start of a project (functional participation), and 2) overlapping of phases and activities. In a CE process, functional participation takes place through the formation of a team consisting of a representative from each of the functions that contribute to the development of a product. The goal is to make downstream activities easier to perform by releasing preliminary information to them early in the process to allow for overlap of activities. However, due to uncertainty in the early stages of an NPD process, the release of incomplete information to downstream functions may potentially introduce the need for rework should there be a change in upstream information. Thus, potential risks must be carefully examined to ensure that added time and effort are kept to a minimum (Krishnan et al., 1997).

in a product development process are fundamentally similar (Ulrich and Eppinger 2011). The NPD process defined in this study is a generic one which outlines the major steps in product development. It is a summary of the common phases and activities used in many instances in the literature as well as in the case study, and as such, it is a reasonably accepted approach to representing the product development process (Schilling and Will,

The NPD process, shown in Figure 2, begins with the development of a concept for a marketable product (Phase A). In this phase, market requirements are determined, new ideas are generated, screened for economic and technical feasibility, and one is selected. In Phase B, 'Definition', a set of specifications to make the product is defined, and the product architecture is developed. Phase C, 'Development', consists of detailed design, physical prototyping, and testing. Finally, in Phase D, 'Implementation', the product volume is

> Preliminary Review

Fig. 2. A schematic diagram for a general stage-gate process with Phases A, B, C, D.

vice-versa. This is represented by the uni-directional arrows between phases.

Figure 2 is an example of the traditional NPD process, where the phases are performed sequentially one after the other. Between phases, a one-way dependence is assumed, that is, the downstream phase depends on information generated by the upstream phase, but not

In an NPD process, the relationship between product and process design is mutually interdependent (Tian et al., 1998). This means that the information generated by one or more functions poses contingencies for others, thus, the parameters of the product and the process should be considered simultaneously (Adler 1995). Therefore a higher degree of coordination is required to manage more people collaborating on interdependent activities. In a sequential process, this interdependence is ignored; a dependent relationship is assumed, and this leads to unplanned coordination downstream. While better management of interdependencies does lead to shortened span time as compared to the sequential

CE uses two main mechanisms to reduce the span time for NPD processes: 1) increased information sharing from the start of a project (functional participation), and 2) overlapping of phases and activities. In a CE process, functional participation takes place through the formation of a team consisting of a representative from each of the functions that contribute to the development of a product. The goal is to make downstream activities easier to perform by releasing preliminary information to them early in the process to allow for overlap of activities. However, due to uncertainty in the early stages of an NPD process, the release of incomplete information to downstream functions may potentially introduce the need for rework should there be a change in upstream information. Thus, potential risks must be carefully examined to ensure that added time and effort are kept to a minimum

**C** DEVELOPMENT

Reject End

Critical Design Review

**D** IMPLEMEN-TATION

Project

1998; Nihtila, 1999; Eastman, 1980).

Accept/

**A**  CONCEPT

ramped up in manufacturing and launched onto the market.

**B** DEFINITION

process, the price is higher cost of upstream effort.

(Krishnan et al., 1997).

Compared to a sequential approach, CE can decrease span time at the expense of increased interdependencies between activities (sequential to reciprocal). To handle the increased interdependencies, close intensive coordination is required through functional participation. However, this may increase effort.

Figure 3 shows an overlapped CE process. Note that information flows are more frequent than in the sequential case, and they are also bi-directional. Major milestones exist at the same gates as before, and each phase is made up of activities, not shown in the figure.

Fig. 3. CE development process.

#### **4. Expected payoff method**

In this section, a mathematical approach is described to measure the performance of processes, namely, sequential and CE processes, through the study of macro- and microvariables. The macro-variable is the expected payoff, while the micro-variables are team interaction and level of overlap. The concepts of information processing and decisionmaking are presented as the basis of this framework. An information processing view is assumed, so that processes are studied through the way in which several team members perform activities such as acquiring, communicating, and processing information in order to make decisions, which in turn organizes the way activities are executed. Processes and corresponding team activities are modeled via networks of interconnected elements. These elements transform inputs into outputs, and represent people, machines, or other real-world objects. Each network realizes an output which is a measure of process performance, and is used to evaluate and compare processes. The methodology is based on the expected payoff method, a technique used in decision theory. It is applied in the calculation of a simple model of both a sequential and a CE process, and the results are compared.

The principle of the expected payoff method has been applied mainly in the field of economics, management science, and in certain areas of artificial intelligence, with respect to decision-making. In this field, economists study 'the best use of available (limited) resources' (Marschak and Radner, 1972). There has been no use of this method in the evaluation of CE in new product development processes. In an organizational environment, teams are also concerned with making the best use of alternatives or limited resources. The interested reader can find several readings in the literature on the principle of utility theory and its various applications (Marschak and Radner, 1972; Fishburn, 1970; Marschak, 1959; Marschak, 1954; von Neumann and Morgenstern, 1943). The framework developed in this part of the research will compare a simple model of a sequential process to a CE process, and evaluate the two in terms of the total expected payoff.

Numerical Evaluation of Product Development Processes 235

outcomes also depend on external factors out of the decision-maker's control, which can be called the environment, represented by the variable x. Since an outcome depends on both the action taken and the environment, the outcome function can now be expressed as r = ρ (x, a). Because x is uncertain, the outcome variable r given a is also said to be uncertain. The decision problem now is made up of a set X of alternative states of the environment x, a set A of all possible actions a, a set R of all possible outcomes r, and an outcome function ρ from

The problem of choosing among alternative actions can be generalized by saying that individuals choose among rules of actions or strategies, rather than from a set of possible actions alone. In an organization, rules of action play a very big role in contingency planning, where team members must decide in advance how they will respond to incoming information. This is obviously important in making economic decisions because individuals must be ready to act as soon as they can (e.g. stock brokers). In the context of an engineering firm, if a task is to design and develop a new product and get it to market as quickly as possible, the designers make use of incoming information as soon as they receive it, and they must additionally decide upon how much of it should be transferred to downstream

An action can now be described as a = α (y), where α is the decision function, and y is the information which will be obtained in the future. It should be noted that the information y is not the same as the variable x, the state of the environment, which describes information already received. The expression says that an action a depends upon the information y

Information can be generated and obtained by team members through various means, such as through observation, communication, and/or computation. There are two sets of information available to the decision-maker: one is the set X of all possible states of the environment, and the other is the set Y of all possible information signals. An information signal y is a partition of the environment X. The information structure η is the partitioning of X into different signals of y. Therefore, a signal y will correspond to each x in X. An information structure is thus defined as y = η (x). Any partition of X can be viewed as a way to describe the states of the environment. As an example, suppose a marketing manager can offer a customer a product in small, medium, or large. The customer wants either small or medium. The marketing manager must make a decision about which size to choose, which will impact the design of the product, and the information relevant to his decision is the size small or medium. The set X of all possible sizes is thus partitioned into one subset, small, and another subset, medium, and this partitioning defines the information structure, η.

The expected payoff method is based on the premise that every individual has preferences as to how to prioritize a list of alternatives due to personal beliefs or interests (assuming that the individual is consistent). Preferences can be described by the ranking of alternatives

X x A to r, giving the outcome of each state-action pair, r = ρ (x, a).

**4.2.2 Decision rules** 

functions, and when.

received.

**4.2.3 Information** 

**4.3 Expected payoff** 

#### **4.1 Methodology**

The approach assumes that individuals in a team work towards achieving common goals with common interests and beliefs, within the constraints of their work, all of which guide their behavior. Given the complexities of such a situation, the problem is allocating appropriate information at the right time, such that team members can make the 'right' decisions which serve to accomplish their common goals (Marschak and Radner, 1972). This chapter will describe the means by which the activities of teams can be described, as well the mathematical analysis which can evaluate team performance, namely, through the use of the expected payoff method. The expected utility or payoff of an action measures the usefulness that an action brings to a person. By combining this with probability theory, decision theory helps a person determine that the action which maximizes his or her expected payoff over all possible actions (from this point forward, for simplicity, the term 'his' will be understood to include 'his/her'). The development of the expected payoff function will be described in detail. Processes can be explicitly represented through network diagrams that illustrate the activities that team members must perform, the inter-relatedness of activities through information requirements, and the communication required among team members (Figure 4). A network realizes a response function, or outcome function, which is based on the actions of the individuals in the organization, and these actions affect the outcome or expected payoff. Among various possible network configurations, the network with the greatest expected payoff is considered optimal.

Fig. 4. Expected payoff conceptual model.

An upcoming section describes how network diagrams are constructed, as well as the mathematical tools which compute and evaluate the networks to obtain the total expected payoff.

#### **4.2 Mathematical model: Definition of fundamental quantities**

In the following sections, the fundamental quantities of the expected payoff model are defined mathematically.

#### **4.2.1 Actions and outcomes**

Faced with a set of alternatives, the decision made by the decision-maker is called his action, a. An action, or decision, can have more than one outcome (or result or consequence). This is denoted as r = ρ (a), where ρ is the outcome function of the action taken. The possible outcomes also depend on external factors out of the decision-maker's control, which can be called the environment, represented by the variable x. Since an outcome depends on both the action taken and the environment, the outcome function can now be expressed as r = ρ (x, a). Because x is uncertain, the outcome variable r given a is also said to be uncertain. The decision problem now is made up of a set X of alternative states of the environment x, a set A of all possible actions a, a set R of all possible outcomes r, and an outcome function ρ from X x A to r, giving the outcome of each state-action pair, r = ρ (x, a).

#### **4.2.2 Decision rules**

234 Numerical Modelling

The approach assumes that individuals in a team work towards achieving common goals with common interests and beliefs, within the constraints of their work, all of which guide their behavior. Given the complexities of such a situation, the problem is allocating appropriate information at the right time, such that team members can make the 'right' decisions which serve to accomplish their common goals (Marschak and Radner, 1972). This chapter will describe the means by which the activities of teams can be described, as well the mathematical analysis which can evaluate team performance, namely, through the use of the expected payoff method. The expected utility or payoff of an action measures the usefulness that an action brings to a person. By combining this with probability theory, decision theory helps a person determine that the action which maximizes his or her expected payoff over all possible actions (from this point forward, for simplicity, the term 'his' will be understood to include 'his/her'). The development of the expected payoff function will be described in detail. Processes can be explicitly represented through network diagrams that illustrate the activities that team members must perform, the inter-relatedness of activities through information requirements, and the communication required among team members (Figure 4). A network realizes a response function, or outcome function, which is based on the actions of the individuals in the organization, and these actions affect the outcome or expected payoff. Among various possible network configurations, the network with the

Network Modeling Method

*Inputs Output* 

Micro-level interactions

**4.2 Mathematical model: Definition of fundamental quantities** 

*Action a* 

Overlapping

An upcoming section describes how network diagrams are constructed, as well as the mathematical tools which compute and evaluate the networks to obtain the total expected

In the following sections, the fundamental quantities of the expected payoff model are

Faced with a set of alternatives, the decision made by the decision-maker is called his action, a. An action, or decision, can have more than one outcome (or result or consequence). This is denoted as r = ρ (a), where ρ is the outcome function of the action taken. The possible

Response Function

Macro-Variable - Expected Payoff

Performance Indicator

**4.1 Methodology** 

greatest expected payoff is considered optimal.

Micro-variables - Overlapping - Team interaction

defined mathematically.

**4.2.1 Actions and outcomes** 

payoff.

Development Process

Fig. 4. Expected payoff conceptual model.

*Team members*

The problem of choosing among alternative actions can be generalized by saying that individuals choose among rules of actions or strategies, rather than from a set of possible actions alone. In an organization, rules of action play a very big role in contingency planning, where team members must decide in advance how they will respond to incoming information. This is obviously important in making economic decisions because individuals must be ready to act as soon as they can (e.g. stock brokers). In the context of an engineering firm, if a task is to design and develop a new product and get it to market as quickly as possible, the designers make use of incoming information as soon as they receive it, and they must additionally decide upon how much of it should be transferred to downstream functions, and when.

An action can now be described as a = α (y), where α is the decision function, and y is the information which will be obtained in the future. It should be noted that the information y is not the same as the variable x, the state of the environment, which describes information already received. The expression says that an action a depends upon the information y received.

#### **4.2.3 Information**

Information can be generated and obtained by team members through various means, such as through observation, communication, and/or computation. There are two sets of information available to the decision-maker: one is the set X of all possible states of the environment, and the other is the set Y of all possible information signals. An information signal y is a partition of the environment X. The information structure η is the partitioning of X into different signals of y. Therefore, a signal y will correspond to each x in X. An information structure is thus defined as y = η (x). Any partition of X can be viewed as a way to describe the states of the environment. As an example, suppose a marketing manager can offer a customer a product in small, medium, or large. The customer wants either small or medium. The marketing manager must make a decision about which size to choose, which will impact the design of the product, and the information relevant to his decision is the size small or medium. The set X of all possible sizes is thus partitioned into one subset, small, and another subset, medium, and this partitioning defines the information structure, η.

#### **4.3 Expected payoff**

The expected payoff method is based on the premise that every individual has preferences as to how to prioritize a list of alternatives due to personal beliefs or interests (assuming that the individual is consistent). Preferences can be described by the ranking of alternatives

Numerical Evaluation of Product Development Processes 237

The expected payoff now depends on the decision function α and the information structure η, and on the factors over which the decision-maker has no control, namely ω, and Φ. The information structure η is assumed to be under the control of the team member; each member has the ability to observe and partition the information into the subsets needed for his activity. The individual has more than one pair (η, α) available, and he will choose the one that maximizes U. This expression is the measure that describes the performance of the various processes through the evaluation of actions under uncertainty. It is used to evaluate the process network diagrams to be developed in upcoming sections. The optimal process structure will be that which maximizes the expected payoff of the network under certain

The payoff function can be expressed as a quadratic function of the team action variables. Although this is an approximation, it is useful. The quadratic function is one that has been used to describe many real-life phenomena, such as in economics for the law of diminishing returns. The concave quadratic function describes the expected payoff function in that there is a point that is optimum, i.e., the maximum point, and before and after this point, the value of the payoff decreases. The use of functions of orders higher than two is very complex and difficult to solve, and a linear function is neither sufficient nor appropriate to describe the present phenomena in detail since it is not expected that the payoff function continuously increases or decreases. Also, since the goal here is to make comparisons between two process structures, the relative comparisons do not require the payoff function to be exact. Taking the case of two members in a team, 1 and 2, where each must make a

This particular form of the quadratic function is similar to the one used by Marschak and Radner (1972), with some of the coefficients chosen to simplify calculations. In the above expression, Q measures the interaction between a1 and a2, the action variables of team members 1 and 2, respectively, and must be between zero and one. The interaction is one of the micro-variables of the process model. For M action variables, if the second derivative of the expected payoff function exists, then a measure of the interaction between the action variables i and j is ∂2ω /∂ai∂aj. In other words, it measures "the degree to which a change in action j influences the effect of a change in action i on the payoff for given values of the other action variables and of x" (Marschak and Radner, 1972, p.101). The functions η1 (x) and η2

The assumption that the payoff is quadratic gives meaning to the variances and the correlations of the information variables. Normal distributions are fully described by their means and variances. The variance can help gauge uncertainty, as it takes the difference between the maximum and minimum values of x. For multivariate distributions, the correlation coefficient describes the degree of statistical interdependence between variables (above, r describes the correlation between two variables). Due to the interdependencies in processes, it is often important to understand how one variable affects another. Whether this

u = -a12 – a22 + 2Q a1 a2 - 2η1 (x) a1 - 2η2 (x) a2 (5)

conditions, given the probability distribution of the states of the environment.

**4.3.1 Expected payoff as a quadratic function** 

decision, then the quadratic payoff function can be chosen as:

(x) are related to the information structure.

(the use of functions of x will be suppressed for simplicity in the future).

according to some subjective probability distribution, consistent with what the person believes will happen. Under uncertainty, each of the alternatives is an action which may result in one or more outcomes, as discussed.

In this sense, the term 'utility' refers to the usefulness an action brings to an individual. For the order of preference given to all actions, each position can be assigned a single number representing the utility of each, or a person's desirability of the occurrence of an event, thus capturing his preferences. The probability of each action occurring is represented by the subjective probability assignments. The expected utility of an action is therefore the sum of the utilities of its various possible outcomes, weighted by the probability of each outcome's occurrence.

Given these basic definitions described in the previous section, for a set R of alternative outcomes r1… rN, if Zi(a) denotes the event that an action a results in the outcome ri (since ri = ρ (x, a)), then the "expected utility" for an action a is:

$$
\Omega\left(\mathbf{a};\mathbf{p},\mathbf{n},\boldsymbol{\upsilon}\right) = \Sigma\left\|\mathbf{u}\right\|\mathbf{n}\left[\text{Zi(a)}\right] \tag{1}
$$

where:

π = subjective probability function = utility function.

The left-hand side of the expected utility function in (1) shows that the expected utility depends only on the decision-maker's action, given the functions ρ, π, and , which describe the factors which are out of his control. The individual's actions are under his control, and his goal is to choose the action which maximizes the corresponding expected utility. In the utility function (r) in (1), r can be replaced to obtain the new payoff function ω: (r) = [ρ (x, a)] ≡ ω (x, a). The expression in (1) can be further simplified by stating that, given the set X of alternative states of the environment x, the probability of the state x can be written as Φ (x) = π ({x}), where Φ is the probability density (or mass) function, and x is assumed to be a random variable that is normally distributed. This expression is, in other words, the probability of the set X consisting of the single element x denoted by {x}. The expected utility function in (1) can be re-written as:

$$\mathfrak{Q}\left(\mathbf{a};\mathbf{a},\mathbf{0}\right) \equiv \mathbf{E} \text{ oz}\left(\mathbf{x},\mathbf{a}\right) = \Sigma \text{ oz}\left(\mathbf{x},\mathbf{a}\right) \Phi\left(\mathbf{x}\right) \tag{2}$$

The expression in (2) can now be called the expected payoff of the action a, where the expected utility depends on the decision-maker's action only, and where ω and Φ describe the factors uncontrolled by the decision-maker. Though the utility and the probability functions may be thought of as being controllable, it is assumed for simplicity that they are not, and that they are treated as givens of the problem. By replacing the actions by decision rules, and introducing the information structure into the equation, the payoff function can be re-written as:

$$\alpha \circ (\mathbf{x}, \mathbf{a}) = \alpha \begin{bmatrix} \mathbf{x}, \mathbf{a} \ (\mathbf{y}) \end{bmatrix} = \alpha \begin{bmatrix} (\mathbf{x}, \mathbf{a} \ (\mathbf{r} \ (\mathbf{x})) \end{bmatrix} \tag{3}$$

From this, the expected payoff becomes:

$$\mathbf{U} = \Sigma \text{ o } \left[ \left( \mathbf{x}, \mathbf{a} \left( \mathfrak{n} \left( \mathfrak{n} \right) \right) \right] \Phi \left( \mathbf{x} \right) \equiv \mathfrak{Q} \left( \mathfrak{n}, \mathfrak{a}; \mathfrak{o}, \mathfrak{Q} \right) \tag{4}$$

according to some subjective probability distribution, consistent with what the person believes will happen. Under uncertainty, each of the alternatives is an action which may

In this sense, the term 'utility' refers to the usefulness an action brings to an individual. For the order of preference given to all actions, each position can be assigned a single number representing the utility of each, or a person's desirability of the occurrence of an event, thus capturing his preferences. The probability of each action occurring is represented by the subjective probability assignments. The expected utility of an action is therefore the sum of the utilities of its various possible outcomes, weighted by the probability of each outcome's

Given these basic definitions described in the previous section, for a set R of alternative outcomes r1… rN, if Zi(a) denotes the event that an action a results in the outcome ri (since

The left-hand side of the expected utility function in (1) shows that the expected utility depends only on the decision-maker's action, given the functions ρ, π, and , which describe the factors which are out of his control. The individual's actions are under his control, and his goal is to choose the action which maximizes the corresponding expected utility. In the utility function (r) in (1), r can be replaced to obtain the new payoff function ω: (r) = [ρ (x, a)] ≡ ω (x, a). The expression in (1) can be further simplified by stating that, given the set X of alternative states of the environment x, the probability of the state x can be written as Φ (x) = π ({x}), where Φ is the probability density (or mass) function, and x is assumed to be a random variable that is normally distributed. This expression is, in other words, the probability of the set X consisting of the single element x denoted by {x}. The expected utility

The expression in (2) can now be called the expected payoff of the action a, where the expected utility depends on the decision-maker's action only, and where ω and Φ describe the factors uncontrolled by the decision-maker. Though the utility and the probability functions may be thought of as being controllable, it is assumed for simplicity that they are not, and that they are treated as givens of the problem. By replacing the actions by decision rules, and introducing the information structure into the equation, the payoff function can

(a; ρ, π, ) = ( ri) π [Zi(a)] (1)

Ω (a; ω, Φ) ≡ E ω (x, a) = Σ ω (x, a) Φ (x) (2)

ω (x, a) = ω [x, α (y)] = ω [(x, α (η (x))] (3)

U = Σ ω [(x, α (η (x))] Φ (x) ≡ Ω (η, α; ω, Φ) (4)

result in one or more outcomes, as discussed.

π = subjective probability function

function in (1) can be re-written as:

From this, the expected payoff becomes:

ri = ρ (x, a)), then the "expected utility" for an action a is:

occurrence.

where:

= utility function.

be re-written as:

The expected payoff now depends on the decision function α and the information structure η, and on the factors over which the decision-maker has no control, namely ω, and Φ. The information structure η is assumed to be under the control of the team member; each member has the ability to observe and partition the information into the subsets needed for his activity. The individual has more than one pair (η, α) available, and he will choose the one that maximizes U. This expression is the measure that describes the performance of the various processes through the evaluation of actions under uncertainty. It is used to evaluate the process network diagrams to be developed in upcoming sections. The optimal process structure will be that which maximizes the expected payoff of the network under certain conditions, given the probability distribution of the states of the environment.

#### **4.3.1 Expected payoff as a quadratic function**

The payoff function can be expressed as a quadratic function of the team action variables. Although this is an approximation, it is useful. The quadratic function is one that has been used to describe many real-life phenomena, such as in economics for the law of diminishing returns. The concave quadratic function describes the expected payoff function in that there is a point that is optimum, i.e., the maximum point, and before and after this point, the value of the payoff decreases. The use of functions of orders higher than two is very complex and difficult to solve, and a linear function is neither sufficient nor appropriate to describe the present phenomena in detail since it is not expected that the payoff function continuously increases or decreases. Also, since the goal here is to make comparisons between two process structures, the relative comparisons do not require the payoff function to be exact. Taking the case of two members in a team, 1 and 2, where each must make a decision, then the quadratic payoff function can be chosen as:

$$\mathbf{u} = \mathbf{-a12} - \mathbf{a22} + 2\mathbf{Q} \text{ a1 } \mathbf{a2} - 2\mathbf{p1} \text{ (x) } \mathbf{a1} - 2\mathbf{p2} \text{ (x) } \mathbf{a2} \tag{5}$$

(the use of functions of x will be suppressed for simplicity in the future).

This particular form of the quadratic function is similar to the one used by Marschak and Radner (1972), with some of the coefficients chosen to simplify calculations. In the above expression, Q measures the interaction between a1 and a2, the action variables of team members 1 and 2, respectively, and must be between zero and one. The interaction is one of the micro-variables of the process model. For M action variables, if the second derivative of the expected payoff function exists, then a measure of the interaction between the action variables i and j is ∂2ω /∂ai∂aj. In other words, it measures "the degree to which a change in action j influences the effect of a change in action i on the payoff for given values of the other action variables and of x" (Marschak and Radner, 1972, p.101). The functions η1 (x) and η2 (x) are related to the information structure.

The assumption that the payoff is quadratic gives meaning to the variances and the correlations of the information variables. Normal distributions are fully described by their means and variances. The variance can help gauge uncertainty, as it takes the difference between the maximum and minimum values of x. For multivariate distributions, the correlation coefficient describes the degree of statistical interdependence between variables (above, r describes the correlation between two variables). Due to the interdependencies in processes, it is often important to understand how one variable affects another. Whether this

Numerical Evaluation of Product Development Processes 239

All the discussion up until now has involved the static case of the team decision problem, but time can be incorporated into the various concepts. If one team member's action at time t (t = 1,...T) is ai(t), and x(t) is the state of the world at time t, then the team action variable

a = [a1(1),…an(1),...an(T)],

x = [x(1),...x(T)].

ai(t)= ai[ηi (x,t ),t ]. An important assumption of this situation is that for actions that are spaced apart in time, the larger the time difference, the less the interaction between those actions. Therefore, it is assumed for simplicity that the actions that are distant in time need less coordination than those that are closer together. The payoff function with no interaction is thus additive in

ω (x, a) = Σ ωt [x, a (t)] (for t=1,…T).

Networks can be used as a powerful tool to represent and evaluate the structure of a process, and more specifically, the structure of information flow and work patterns in a team. A network can be defined as a system of interconnected elements, all of which work together to produce a desired output. A network consists of the following basic




Each element in a network has an input which is transformed into or transferred as an output; the message of this output then feeds into one or more downstream elements. Elements are connected to one another through the input and output arcs, which carry information messages to and from elements. Messages coming from the environment (i.e., external to the organization) are called observations. Messages from one element to another

communication tool, etc., in the process of performing an activity.

**4.3.4 Consideration of time** 

and the state of the world is:

time, and can be expressed as:

**4.4 Design of network models** 

knowledge or expertise, etc.).

either to another element or to the environment.

components:

For yi (t) = ηi (x, t ), the action variable becomes:

becomes:

is the correlation between action variables or the environment variables, it is reasonable that these correlations may affect the information structure and/or probability of occurrence chosen by the organizer. Another simplification is the normalizing assumption, where each variable is considered to be measured from its mean, so that m = 0, and E (m) = 0. There is no loss of meaning since this is simply a coordinate transformation. In this case, the correlation coefficient becomes:

r = r12 = Ex1x2/s1s2, or

Ex1x2 = r \* s1s2.

These properties of probability distributions will be useful in solving the expected payoff functions and determining the macro-variables of interest, discussed in upcoming sections.

#### **4.3.2 Multi-person teams**

For n members in a team, then there will also be n information structures and n decision rules. Each member i chooses an action ai from set Ai of all possible alternatives. The payoff function can be written as:

$$\mathbf{u = a (x, a1 \,\text{a2}\dots)}$$

where u is now the utility to the team (and to each of its members). Although ai is the action variable controlled by the ith member, ai itself can be an m-tuple of many distinct variables, each controlled by the ith member. If there is no interaction among the action variables, then the payoff is said to be additive, and the form of the expression becomes:

$$\text{oo (x, a)} = \Sigma \text{ a i (x, aj)}$$

If, however, there are interactions among action variables, then the quadratic function must include an extra term to express this interaction, namely Q, as before.

#### **4.3.3 Team decision functions and information functions**

In a single person team, the person's action is related to the decision function through a = α(y). For a multi-person team, there are now n decision functions, α = (α1 … αn) and ai = α1( yi). The same decision rules as before can be applied for a team. The joint action of the team members is a = (a1,… an), and y = (y1 , ... yn) is the team information, so there are n decision functions, and the team decision rule can then be denoted as α = (α1, α2,…. αn). The same expression for an action a = α(y) for a single person team is also applicable for teams, keeping in mind what each term means individually. The information structure for each team member can be expressed as yi = ηi (x), and for the team, the information structure is η = (η1,... η n). Then, for y = η (x), and a = α (y), a = α [η (x)] applies for the team action. The payoff of the team can be written, as before:

$$\mathbf{u} = \mathbf{a} \text{ (x, a1, a2, \dots)} = \mathbf{a} \text{ (x, a1 [n] 1 (x))}, \dots \text{ } \mathbf{a} \text{ (n [n (x)])} = \mathbf{a} \text{ (x, a [n] (x))}.$$

and the expected payoff of the team is:

$$\mathbf{E}\left(\mathbf{u}\right) = \mathbf{Q}\left(\mathbf{r}\right)\mathbf{a}\right) = \mathbf{E}\left(\mathbf{a}\left(\mathbf{x}, \mathbf{a}\left[\mathbf{r}\right]\right)\right) \tag{6}$$

#### **4.3.4 Consideration of time**

238 Numerical Modelling

is the correlation between action variables or the environment variables, it is reasonable that these correlations may affect the information structure and/or probability of occurrence chosen by the organizer. Another simplification is the normalizing assumption, where each variable is considered to be measured from its mean, so that m = 0, and E (m) = 0. There is no loss of meaning since this is simply a coordinate transformation. In this case, the

r = r12 = Ex1x2/s1s2, or

Ex1x2 = r \* s1s2. These properties of probability distributions will be useful in solving the expected payoff functions and determining the macro-variables of interest, discussed in upcoming sections.

For n members in a team, then there will also be n information structures and n decision rules. Each member i chooses an action ai from set Ai of all possible alternatives. The payoff

u = ω (x, a1 ,a2 ...) where u is now the utility to the team (and to each of its members). Although ai is the action variable controlled by the ith member, ai itself can be an m-tuple of many distinct variables, each controlled by the ith member. If there is no interaction among the action variables, then

ω (x, a) = Σ ω i (x, ai) If, however, there are interactions among action variables, then the quadratic function must

In a single person team, the person's action is related to the decision function through a = α(y). For a multi-person team, there are now n decision functions, α = (α1 … αn) and ai = α1( yi). The same decision rules as before can be applied for a team. The joint action of the team members is a = (a1,… an), and y = (y1 , ... yn) is the team information, so there are n decision functions, and the team decision rule can then be denoted as α = (α1, α2,…. αn). The same expression for an action a = α(y) for a single person team is also applicable for teams, keeping in mind what each term means individually. The information structure for each team member can be expressed as yi = ηi (x), and for the team, the information structure is η = (η1,... η n). Then, for y = η (x), and a = α (y), a = α [η (x)] applies for the team

u = ω (x, a1 ,a2 ,…) = ω (x, α 1 [η1 (x)], ... α n [ηn (x)]) = ω (x, α [η (x)],

E (u) = Ω (η, α) = E (ω (x, α [η (x)]) (6)

the payoff is said to be additive, and the form of the expression becomes:

include an extra term to express this interaction, namely Q, as before.

**4.3.3 Team decision functions and information functions** 

action. The payoff of the team can be written, as before:

and the expected payoff of the team is:

correlation coefficient becomes:

**4.3.2 Multi-person teams** 

function can be written as:

All the discussion up until now has involved the static case of the team decision problem, but time can be incorporated into the various concepts. If one team member's action at time t (t = 1,...T) is ai(t), and x(t) is the state of the world at time t, then the team action variable becomes:

$$\mathbf{a} = [\mathbf{a}1(1), \dots \text{an}(1), \dots \text{an}(\mathbf{T})]\_{\prime}$$

and the state of the world is:

$$\mathbf{x} = [\mathbf{x}(1), \dots, \mathbf{x}(T)].$$

For yi (t) = ηi (x, t ), the action variable becomes:

$$\text{ai}(\mathbf{t}) \mathbf{=} \text{ai}[\mathbf{r} \mathbf{j} \ (\mathbf{x}, \mathbf{t}) \mathbf{j}, \mathbf{t}].$$

An important assumption of this situation is that for actions that are spaced apart in time, the larger the time difference, the less the interaction between those actions. Therefore, it is assumed for simplicity that the actions that are distant in time need less coordination than those that are closer together. The payoff function with no interaction is thus additive in time, and can be expressed as:

$$
\text{co (x, a)} = \Sigma \text{ ot [x, a (t)] (for t=1,...T)}.
$$

#### **4.4 Design of network models**

Networks can be used as a powerful tool to represent and evaluate the structure of a process, and more specifically, the structure of information flow and work patterns in a team. A network can be defined as a system of interconnected elements, all of which work together to produce a desired output. A network consists of the following basic components:


Each element in a network has an input which is transformed into or transferred as an output; the message of this output then feeds into one or more downstream elements. Elements are connected to one another through the input and output arcs, which carry information messages to and from elements. Messages coming from the environment (i.e., external to the organization) are called observations. Messages from one element to another

Numerical Evaluation of Product Development Processes 241

The time distribution and spatial distribution of members in a team must be separated. For teams in a dynamic environment, networks are divided into time periods by several elements based on the structure of information flow, which is illustrated by elements broken down into intermediate stages or actions. Figures 6 and 7 show the network diagrams of

ε ε ε ε ε ε

 1 2 3 4 T1 T2 T3 T4

ε ε ε

product. This action is sent out to the environment, i.e., the customer.

ε ε ε

In the context of NPD processes, as an example, an element i can be a designer who receives information ε from the environment, say from the customer. An action a can be the release of design specifications from the designer to another element, say the manufacturing resource. This resource then uses information from this action as well as information from observations through personal experience and/or company databases for example, transforms the new combined information, and takes an action, such as manufacturing the

A simple model is designed in this section using network diagrams and its evaluation using the expected payoff method will be studied. The main purpose here is to compare the

 1 2 3 4 5 6 T1' T2'

 0 1 2 3 4 5 � B01 B02 � � � � � � B13 � � � � � B23 � � � � � � B34 B35 B40 � � � � � B50 � � � � �

Table 1. Information dependencies.

possible sequential and CE processes.

Fig. 6. Sequential network diagram.

Fig. 7. CE network diagram.

**4.5 Application to models** 

are communication, while messages going out into the environment are called actions. In the context of an organization, networks can be used to represent processes from an information processing point of view. Once all intermediate elements have been completed, a final action(s) is issued, which signals project completion. Networks can be organized according to time structure.

#### **4.4.1 Connections between elements**

The connections between elements in a network can be described in the form of a square array. For each element i and j, the set of all possible messages that can be sent from i to j, is denoted by Bij. Any messages that come from outside, that is, from the environment, are described by the set Zi, and Ei which denotes the set of all possible values of noise coming from the environment to element i. This noise can be information that is observed from outside the organization, such as customer input, best practices, etc. The messages sent out to the environment are defined as the action variables, a = (a1, … an), for n actions, where a is the team action variable.

The set Bi0 denotes the set of all possible messages from element i to the environment. This set will consist either of the Cartesian product of some sets Aj, where for each j, Aj is the set of all possible values that action variable aj can take, or it will be empty since not all elements will have an action as an output. The set B0i is symmetric to this set, and it represents the set of all possible messages from the environment outside to an element i, which is the Cartesian product of Zi and Ei. Therefore, the set Bi of possible alternative output messages of element i is denoted by Bi = Π Bij (j=0...m). For m elements, the set B́i of combined messages from other elements to i is given by B́i = Π Bki (k=0...m). The transformation of each element i is expressed through the task function βi = (βi0,..., βim), which transforms each input message into an output message. The set Bii is empty as it is assumed that messages will not be sent from an element to itself. Figure 5 below shows an example of a simple network diagram. The corresponding square array consisting of the sets Bij in Table 1 illustrates message transfers between elements in the figure. The symbol Φ denotes an empty set.

Fig. 5. Simple network diagram.


Table 1. Information dependencies.

are communication, while messages going out into the environment are called actions. In the context of an organization, networks can be used to represent processes from an information processing point of view. Once all intermediate elements have been completed, a final action(s) is issued, which signals project completion. Networks can be organized according

The connections between elements in a network can be described in the form of a square array. For each element i and j, the set of all possible messages that can be sent from i to j, is denoted by Bij. Any messages that come from outside, that is, from the environment, are described by the set Zi, and Ei which denotes the set of all possible values of noise coming from the environment to element i. This noise can be information that is observed from outside the organization, such as customer input, best practices, etc. The messages sent out to the environment are defined as the action variables, a = (a1, … an), for n actions, where a

The set Bi0 denotes the set of all possible messages from element i to the environment. This set will consist either of the Cartesian product of some sets Aj, where for each j, Aj is the set of all possible values that action variable aj can take, or it will be empty since not all elements will have an action as an output. The set B0i is symmetric to this set, and it represents the set of all possible messages from the environment outside to an element i, which is the Cartesian product of Zi and Ei. Therefore, the set Bi of possible alternative output messages of element i is denoted by Bi = Π Bij (j=0...m). For m elements, the set B́i of combined messages from other elements to i is given by B́i = Π Bki (k=0...m). The transformation of each element i is expressed through the task function βi = (βi0,..., βim), which transforms each input message into an output message. The set Bii is empty as it is assumed that messages will not be sent from an element to itself. Figure 5 below shows an example of a simple network diagram. The corresponding square array consisting of the sets Bij in Table 1 illustrates message transfers between elements in the figure. The symbol Φ

1 2

3

4 5

to time structure.

is the team action variable.

denotes an empty set.

Fig. 5. Simple network diagram.

**4.4.1 Connections between elements** 

The time distribution and spatial distribution of members in a team must be separated. For teams in a dynamic environment, networks are divided into time periods by several elements based on the structure of information flow, which is illustrated by elements broken down into intermediate stages or actions. Figures 6 and 7 show the network diagrams of possible sequential and CE processes.

Fig. 6. Sequential network diagram.

Fig. 7. CE network diagram.

In the context of NPD processes, as an example, an element i can be a designer who receives information ε from the environment, say from the customer. An action a can be the release of design specifications from the designer to another element, say the manufacturing resource. This resource then uses information from this action as well as information from observations through personal experience and/or company databases for example, transforms the new combined information, and takes an action, such as manufacturing the product. This action is sent out to the environment, i.e., the customer.

#### **4.5 Application to models**

A simple model is designed in this section using network diagrams and its evaluation using the expected payoff method will be studied. The main purpose here is to compare the

Numerical Evaluation of Product Development Processes 243

Thus the maximum expected payoff is calculated in each period, which are the added up to give the total maximum expected payoff. In other words, the expected value of the

Rework is not modeled in either the sequential or CE process. Though this is a simplifying assumption not characteristic of most NPD processes, the model is presented in basic form, with the intent of bringing out some essential features of the expected payoff method.

The simple network shown below in Figure 8 will be defined here to illustrate how a network diagram is evaluated in terms of its gross expected payoff ('gross' since the cost of a network is not considered). The network is assumed to be in one time period, reflecting a single action. In cases when there is more than one element, each element in the network can be evaluated separately in terms of its expected payoff, and the total expected payoff is simply the sum of the individual ones. Actions taken at different times can be considered to

Figure 8 illustrates an action. Element 1 has η1 = x as input variable, where η1 is a random variable dependent upon the state of nature x (which is suppressed). The state variable observed by the team member at element 1 is processed, which also receives some information ε from outside (qualified as information such as team member's personal expertise), which, for simplicity, is considered to be a constant. This information is processed, and an action a1 is taken, which is a function of the inputs to the element. The information ε combined with the information µ1 is additive. With a single action, the payoff

a1 = 1 (1 + )

ω = -a12 + 2a1x (6)

ά (x) = x (7)

Ω = E (x + ε)2 = s2 (8)

maximum payoffs is equal to the total maximum expected payoffs for each period.

be corresponding to different team members.

1 = x

Fig. 8. Action taken in one time period.

function is chosen as a quadratic in one input variable, in the form:

gives the following expected value of the maximum payoff:

Taking the derivative of (6) with respect to a1 and setting it equal to zero gives:

ω΄ (a1) = -2a1 + 2x = 0, and solving for a1gives the best decision function, denoted by ά:

which is the optimal decision. The second derivative of (6) is negative (-2), ensuring a maximum point, so plugging (7) back into (6) and taking the expected value of the payoff

where s2 is the variance of x. The decision function in equation (7) has a distribution of possible decisions, which implies that multiple choices can be made. Assuming that this distribution is normal, then, equation (8) shows that the payoff is equal to the variance. This

relative differences between process structures in terms of the expected payoff. In the evaluation that follows, the expected payoff function is equation (5), repeated here:

$$\mathbf{u} = \mathbf{-a12} - \mathbf{a22} + 2\mathbf{Q} \,\mathbf{a1} \,\mathbf{a2} - 2\mathbf{p1} \,\mathbf{a1} \,\mathbf{-2p2} \,\mathbf{a2} \tag{5}$$

Recall that the coefficient "Q" in (5) denotes the interaction, specifically in this case between two overlapping activities occurring in the same time period. This function evaluates every step of performing work, and also evaluates the different processes as a whole (i.e., each intermediate step is evaluated using this function, as well as the overall structure of the network).

#### **4.5.1 Assumptions of the model**

In order to compare the two processes at the same level, some assumptions must be made to ensure consistency. First, both of the processes begin with the same input information variable ηi which is a random variable dependent upon x. Thus, the first member of each process begins by observing the same information that is coming from the environment. Another assumption in the model is that the members inside the organization not only receive information from other sources (i.e., other elements or the environment), they also contribute to the processing of their work through the use of their own expertise, which is denoted in the models by ε as an input into each element (Howard, 1966). However, this is considered as being a special state of the set X of information from the environment despite the fact that it comes from the element itself. Therefore, during the evolution of the activity, not only does the information that a member receives get processed, but also because each member is contributing his own knowledge and expertise, this pooled information adds value to the activity, which results in an increase in the expected payoff.

Earlier, it was mentioned that the choice of the payoff function as a quadratic equation is appropriate since a quadratic function has a maximum point. This assumption is important and must be re-stated. Furthermore, since the expected payoff is the measure being used to compare relative process performance, an absolute measure is unnecessary, so the problem of defining a specific and accurate form of the function can be avoided.

The cost of a network is not considered in the models. Marschak and Radner (1972) did not include this important factor explicitly in their decision functions, although they acknowledge its importance. There is a cost associated with decision-making, with how information is obtained, with team organization, etc. In the context of this research, cost was not chosen as a parameter of the models, however, since cost can help in assessing the tradeoffs of one process design over another.

The sequential and CE process networks are created as sequences of single-period decision problems (see Figures 9 and 10), where the interaction between periods is assumed to be zero, i.e., Q=0, as previously discussed in Section 4.4.4. Thus, interaction is assumed to be zero across periods, though there is interaction within time periods. Therefore, it is assumed that the total payoff function is additive in time. In each time period, optimal decisions are made. This difference in interactions addresses the case for which the sum of the maximum expected payoffs is equal to maximum of the sum of expected payoffs. In other words:

Max Σ E ω (x, a) = Σ Max E ω (x, a) = Σ E ω max (x, a).

relative differences between process structures in terms of the expected payoff. In the

Recall that the coefficient "Q" in (5) denotes the interaction, specifically in this case between two overlapping activities occurring in the same time period. This function evaluates every step of performing work, and also evaluates the different processes as a whole (i.e., each intermediate step is evaluated using this function, as well as the overall structure of the

In order to compare the two processes at the same level, some assumptions must be made to ensure consistency. First, both of the processes begin with the same input information variable ηi which is a random variable dependent upon x. Thus, the first member of each process begins by observing the same information that is coming from the environment. Another assumption in the model is that the members inside the organization not only receive information from other sources (i.e., other elements or the environment), they also contribute to the processing of their work through the use of their own expertise, which is denoted in the models by ε as an input into each element (Howard, 1966). However, this is considered as being a special state of the set X of information from the environment despite the fact that it comes from the element itself. Therefore, during the evolution of the activity, not only does the information that a member receives get processed, but also because each member is contributing his own knowledge and expertise, this pooled information adds

Earlier, it was mentioned that the choice of the payoff function as a quadratic equation is appropriate since a quadratic function has a maximum point. This assumption is important and must be re-stated. Furthermore, since the expected payoff is the measure being used to compare relative process performance, an absolute measure is unnecessary, so the problem

The cost of a network is not considered in the models. Marschak and Radner (1972) did not include this important factor explicitly in their decision functions, although they acknowledge its importance. There is a cost associated with decision-making, with how information is obtained, with team organization, etc. In the context of this research, cost was not chosen as a parameter of the models, however, since cost can help in assessing the trade-

The sequential and CE process networks are created as sequences of single-period decision problems (see Figures 9 and 10), where the interaction between periods is assumed to be zero, i.e., Q=0, as previously discussed in Section 4.4.4. Thus, interaction is assumed to be zero across periods, though there is interaction within time periods. Therefore, it is assumed that the total payoff function is additive in time. In each time period, optimal decisions are made. This difference in interactions addresses the case for which the sum of the maximum expected payoffs is equal to maximum of the sum of expected payoffs. In other words:

Max Σ E ω (x, a) = Σ Max E ω (x, a) = Σ E ω max (x, a).

value to the activity, which results in an increase in the expected payoff.

of defining a specific and accurate form of the function can be avoided.

u = - a12 – a22 + 2Q a1 a2 - 2η1 a1 - 2η2 a2 (5)

evaluation that follows, the expected payoff function is equation (5), repeated here:

network).

**4.5.1 Assumptions of the model** 

offs of one process design over another.

Thus the maximum expected payoff is calculated in each period, which are the added up to give the total maximum expected payoff. In other words, the expected value of the maximum payoffs is equal to the total maximum expected payoffs for each period.

Rework is not modeled in either the sequential or CE process. Though this is a simplifying assumption not characteristic of most NPD processes, the model is presented in basic form, with the intent of bringing out some essential features of the expected payoff method.

The simple network shown below in Figure 8 will be defined here to illustrate how a network diagram is evaluated in terms of its gross expected payoff ('gross' since the cost of a network is not considered). The network is assumed to be in one time period, reflecting a single action. In cases when there is more than one element, each element in the network can be evaluated separately in terms of its expected payoff, and the total expected payoff is simply the sum of the individual ones. Actions taken at different times can be considered to be corresponding to different team members.

1 = x a1 = 1 (1 + )

Fig. 8. Action taken in one time period.

Figure 8 illustrates an action. Element 1 has η1 = x as input variable, where η1 is a random variable dependent upon the state of nature x (which is suppressed). The state variable observed by the team member at element 1 is processed, which also receives some information ε from outside (qualified as information such as team member's personal expertise), which, for simplicity, is considered to be a constant. This information is processed, and an action a1 is taken, which is a function of the inputs to the element. The information ε combined with the information µ1 is additive. With a single action, the payoff function is chosen as a quadratic in one input variable, in the form:

$$\mathbf{a} = \mathbf{-a12} + \mathbf{2a1x} \tag{6}$$

Taking the derivative of (6) with respect to a1 and setting it equal to zero gives:

ω΄ (a1) = -2a1 + 2x = 0, and solving for a1gives the best decision function, denoted by ά:

$$
\dot{\mathbf{x}} \tag{7}
$$

which is the optimal decision. The second derivative of (6) is negative (-2), ensuring a maximum point, so plugging (7) back into (6) and taking the expected value of the payoff gives the following expected value of the maximum payoff:

$$\mathbf{Q} = \mathbf{E} \ (\mathbf{x} + \mathbf{e}) \mathbf{2} = \mathbf{s} \mathbf{2} \tag{8}$$

where s2 is the variance of x. The decision function in equation (7) has a distribution of possible decisions, which implies that multiple choices can be made. Assuming that this distribution is normal, then, equation (8) shows that the payoff is equal to the variance. This

Numerical Evaluation of Product Development Processes 245

A similar procedure as above applies to each time period, up until time period six. The total

Ω = Σ (i = 1…6) Ω i = Ω 1+ Ω 2 + Ω 3 + Ω 4 + Ω 5 + Ω 6

If it is assumed for simplicity that the variance for each information structure is the same,

The concurrent engineering diagram shown in Figure 10 is an appropriate modification of the sequential engineering network. It takes into account the two teams of three members each, but this time with a few added features. The two teams' activities are now overlapping in time periods T2 and T3. These two teams now are also communicating with each other through the transfer of information denoted by the arrows between overlapped activities. There is interaction between the two members from both teams in the same two time periods. Since rework is not modeled, overlapping part of the six time periods in the sequential process gives the resulting four time periods in the CE process. The main

comparison of interest at this point is the difference between expected payoffs.

ε ε ε

ε ε ε

 1 2 3 4 T1 T2 T3 T4

Ω TOT = s02 + s12 + s22 + s32 + s42 + s52 = 6s2 (11)

Ω 2 = s12 (10)

As in time period 1, member 2 receives output x from element 1, giving the payoff:

= s02 + s12 + s22 + s32 + s42 + s52

where s02 = E(x + ε) 2. 2. Second Period T2:

where s12 = E(x + ε + ε) 2.

Expected Value of the Maximum Payoff:

then the total expected payoff becomes:

Fig. 10. CE network diagram.

**4.5.3 Concurrent engineering network diagram** 

means that in making a decision, i.e., reducing possible choices to a single value, the payoff is equal to the value of the reduction of uncertainty of information. It is reasonable to conclude that the larger the variance, i.e., the more uncertain the decision, then, the more benefit (payoff) there is in making a decision.

#### **4.5.2 Sequential engineering network diagram**

The sequential engineering network diagram is illustrated in Figure 9, and consists of six time periods, T1 to T6, which represent the division of the sequential work done by six different functional team members. Team member 1 receives complete information represented by the state variable x, and then uses this information, along with his own expertise represented by ε, to complete his activity. At the end of his activity, he sends complete information to the downstream activity, which is again processed by the second team member. The output of this activity is a message sent to the next team member, etc.

Fig. 9. Sequential network diagram.

Because of the assumption of no cross-functional communication in a sequential process, there is no interaction between team members, and the communication of information is assumed to be 'over-the-wall', thus even if there is some interaction, it is assumed to be so weak that it is negligible.

Evaluation of the Network

1. First Period T1:

Similar to the example above, team member 1 receives complete information, and every member within the network contributes his special technical knowledge to the flow, denoted by ε. As before, the payoff function is (6) and the expected payoff is (9):

$$\mathbf{a} \mathbf{a} = \mathbf{-a12} + \mathbf{2a1x} \tag{6}$$

$$\mathbf{Q}\mathbf{1} = \mathbf{s}\mathbf{0}\mathbf{2} \tag{9}$$

where s02 = E(x + ε) 2.

244 Numerical Modelling

means that in making a decision, i.e., reducing possible choices to a single value, the payoff is equal to the value of the reduction of uncertainty of information. It is reasonable to conclude that the larger the variance, i.e., the more uncertain the decision, then, the more

The sequential engineering network diagram is illustrated in Figure 9, and consists of six time periods, T1 to T6, which represent the division of the sequential work done by six different functional team members. Team member 1 receives complete information represented by the state variable x, and then uses this information, along with his own expertise represented by ε, to complete his activity. At the end of his activity, he sends complete information to the downstream activity, which is again processed by the second team member. The output of this activity is a message sent to the next team member, etc.

ε ε ε ε ε ε

Because of the assumption of no cross-functional communication in a sequential process, there is no interaction between team members, and the communication of information is assumed to be 'over-the-wall', thus even if there is some interaction, it is assumed to be so

Similar to the example above, team member 1 receives complete information, and every member within the network contributes his special technical knowledge to the flow,

denoted by ε. As before, the payoff function is (6) and the expected payoff is (9):

a3 = 3{ + 2[ + η3]} = η<sup>4</sup>

3

ω = -a12 + 2a1x (6)

Ω1 = s02 (9)

a2 = 2{ + 1[ + η2]} = η<sup>3</sup>

1 2

a1 = 1[ + η1] = η<sup>2</sup>

T1 T2 T3 T4 T5 T6

benefit (payoff) there is in making a decision.

Fig. 9. Sequential network diagram.

weak that it is negligible. Evaluation of the Network

1. First Period T1:

η1(x) = x

**4.5.2 Sequential engineering network diagram** 

2. Second Period T2:

As in time period 1, member 2 receives output x from element 1, giving the payoff:

$$
\Omega \, \mathfrak{Z} = \mathfrak{s} 1 \mathfrak{Z} \tag{10}
$$

where s12 = E(x + ε + ε) 2.

A similar procedure as above applies to each time period, up until time period six. The total Expected Value of the Maximum Payoff:

$$\begin{aligned} \mathfrak{Q} &= \Sigma \begin{pmatrix} \mathbf{i} = \mathbf{1} \dots \mathbf{6} \end{pmatrix} \mathfrak{Q} \ \mathbf{i} = \mathfrak{Q} \ \mathbf{1} + \mathfrak{Q} \ \mathbf{2} + \mathfrak{Q} \ \mathbf{3} + \mathfrak{Q} \ \mathbf{4} + \mathfrak{Q} \ \mathbf{5} + \mathfrak{Q} \ \mathbf{6} \\ &= \mathfrak{s} \mathbf{0} \mathbf{2} + \mathfrak{s} \mathbf{1} \mathbf{2} + \mathfrak{s} \mathbf{2} \mathbf{2} + \mathfrak{s} \mathbf{3} \mathbf{2} + \mathfrak{s} \mathbf{4} \mathbf{2} + \mathfrak{s} \mathbf{5} \mathbf{2} \end{aligned}$$

If it is assumed for simplicity that the variance for each information structure is the same, then the total expected payoff becomes:

$$\text{Q TOT} = \text{s02} + \text{s12} + \text{s22} + \text{s32} + \text{s42} + \text{s52} = \text{6s2} \tag{11}$$

#### **4.5.3 Concurrent engineering network diagram**

The concurrent engineering diagram shown in Figure 10 is an appropriate modification of the sequential engineering network. It takes into account the two teams of three members each, but this time with a few added features. The two teams' activities are now overlapping in time periods T2 and T3. These two teams now are also communicating with each other through the transfer of information denoted by the arrows between overlapped activities. There is interaction between the two members from both teams in the same two time periods. Since rework is not modeled, overlapping part of the six time periods in the sequential process gives the resulting four time periods in the CE process. The main comparison of interest at this point is the difference between expected payoffs.

Fig. 10. CE network diagram.

Numerical Evaluation of Product Development Processes 247

ά2 = [-1/(1-Q2)] \* 2 + [-Q/(1-Q2)] \* 4

ά4 = [-Q/(1-Q2)] \* μ 2 + [-1/(1-Q2)] \* 4

s12 = E [x + ε + ε] 2

s22 = E [x + ε + ε + ε]2

r12 = [E (x + ε + ε)(x + ε + ε + ε)]/ s1 s2

3 5

Performing the same calculations as in T2 gives the following best decision functions:

Plugging (18) and (19) into (13) gives the payoff for time period T2:

where r12 is the correlation coefficient and Q is the interaction.

2[2 + ]

a3 = 3 [ + 2(2 + )]

and the expected payoff is:

3. Third Period T3:

a2 = 2 [2 + ]

The payoff function is:

where:

where:

= [-1/(1-Q2)] \* [x + ε + ε] + [-Q/(1-Q2)] \* [x + ε + ε + ε] (18)

= [-Q/(1-Q2)] \* [x + ε + ε] + [-1/(1-Q2)] \* [x + ε + ε + ε] (19)

ω = [ 22 + 2Q 2 4 + 42]/ [1-Q2], (20)

Ω 2 = Eω = [s12 + 2Qr12 s1 s2 + s22]/ [1-Q2] (21)

ω = -a32 – a52 + 2Q a3 a5 – 2 3 a3 – 2 5 a5 (22)

3 = ε + ά 2 = [-1/(1-Q2)] \* [x + ε + ε] + [-Q/(1-Q2)] \* [x + ε + ε + ε] + ε (23)

5 = ε + ά 2 = [-Q/(1-Q2)] \* [x + ε + ε] + [-1/(1-Q2)] \* [x + ε + ε + ε] + ε (24)

a4 = 4[ + 2(2 + )]

Solving for a2 and a4 yields the following best decision functions for T2 period actions:

Q a2 - a4 = 4 (17)

#### Evaluation of the Network

1. First Period T1:

$$\pi^{\mathfrak{u}} = [(\mathfrak{x})\_{\mathfrak{l}} + \varepsilon]\_{\mathfrak{v}} = \underbrace{\sum\_{\mathfrak{l}} \underbrace{\sum\_{\mathfrak{l}}^{\mathfrak{a}}}\_{\mathfrak{l} \pmod{\mathfrak{v}}}}\_{\text{x} = [\mathfrak{x}]\_{\mathfrak{l}} \text{ y}} \qquad \text{x} = (\mathfrak{x})\_{\mathfrak{l}} \text{ y}$$

Again, it is assumed that member 1 obtains complete information x, that is the information structure η1(x)=x. Also, every member within the network contributes his special technical knowledge to the processing and transferring of information. The output α[x] = a1 is determined as before. Choosing u1= ω (x, a1) = -2a12 + 2a1[x + ε], the best decision function is:

$$
\dot{\mathbf{d}} \cdot [\mathbf{x}] = \mathbf{x} + \varepsilon
$$

and the expected value of the maximum payoff for the first time period is:

$$
\Omega \text{ 1} = \text{s } \Omega \text{ 2} \tag{12}
$$

where s02 = E[x + ε]

2. Second Period T2:

The payoff function is:

$$\mathbf{a}(\mathbf{x}, \mathbf{a}) = \mathbf{-a}2\mathbf{2} - \mathbf{a}4\mathbf{2} + 2\mathbf{Q} \text{ a} \mathbf{2} \text{ a} \mathbf{4} - \mathbf{2} \text{ } \text{\(\mathbf{2} \times \mathbf{2} - \mathbf{2} \text{ } \text{\(\mathbf{4}\)}\)}\tag{13}$$

Taking the first derivative of the payoff function first with respect to a2 and a4 setting each equal to zero:

$$2\text{ ба} / \text{ ба} \, 2 = \text{-} 2\text{a} \\ 2 + \text{2Q} \text{ а} \\ 4 - \text{2 } \text{ η } 2 = 0 \tag{14}$$

$$\text{Cao/}\,\text{ða} = \text{-} 2\text{a} \mathbf{4} + \text{2Q} \text{ a} \mathbf{2} \text{ - } \text{2} \text{ } \text{η } \text{4} = \text{0} \tag{15}$$

gives the following system of equations:

$$\mathbf{a} \cdot \mathbf{a}\mathbf{2} + \mathbf{Q} \cdot \mathbf{a}\mathbf{4} = \mathbf{\eta} \cdot \mathbf{2} \tag{16}$$

$$\mathbf{Q} \text{ a}\mathbf{2} \text{ - } \mathbf{a}4 = \mathbf{\eta} \text{ 4} \tag{17}$$

Solving for a2 and a4 yields the following best decision functions for T2 period actions:

$$\begin{aligned} \text{id2} &= \text{[-1/(1-Q2)]} \, ^\ast \eta \, 2 + \text{[-Q/(1-Q2)]} \, ^\ast \eta \, 4 \\\\ &= \text{[-1/(1-Q2)]} \, ^\ast \left[ \text{x} + \text{c} + \text{e} \right] + \text{[-Q/(1-Q2)]} \, ^\ast \left[ \text{x} + \text{c} + \text{e} + \text{e} \right] \end{aligned} \tag{18}$$

$$\text{id4} = \left[\text{-Q}/(\text{1-Q2})\right]^\* \text{\tiny\text{\tiny{}^\*}\text{\tiny{}^\*} + \left[\text{-1}/(\text{1-Q2})\right]^\* \text{\tiny{}^\*} \text{\tiny{}^\*} \text{\tiny{}^\*} \tag{19}$$

$$\mathbf{x} = \begin{bmatrix} \text{-Q}/(\text{1-Q2}) \end{bmatrix} \* \begin{bmatrix} \text{x} + \text{z} + \text{z} \end{bmatrix} + \begin{bmatrix} \text{-1}/(\text{1-Q2}) \end{bmatrix} \* \begin{bmatrix} \text{x} + \text{z} + \text{z} + \text{z} \end{bmatrix} \tag{19}$$

Plugging (18) and (19) into (13) gives the payoff for time period T2:

$$\mathbf{u} = [\eta \,\, 22 + 2\mathbf{Q} \,\, \eta \,\, 2 \,\, \eta \,\, 4 + \eta \,\, 42] / \,\, [1\text{-Q}\,\, 2] \tag{20}$$

and the expected payoff is:

$$\text{Q.2 = Eo} = \left[\text{s12 + 2Qr12 s1 s2 + s22}\right] / \left[\text{1-Q2}\right] \tag{21}$$

where:

246 Numerical Modelling

1(x) = x 1 a1 = 1[ + 1(x)] = 2

Again, it is assumed that member 1 obtains complete information x, that is the information structure η1(x)=x. Also, every member within the network contributes his special technical knowledge to the processing and transferring of information. The output α[x] = a1 is determined as before. Choosing u1= ω (x, a1) = -2a12 + 2a1[x + ε], the best decision function

ά [x] = x + ε

Ω 1 = s 02 (12)

a4 = 4[ + 2(2 + )]

ω(x, a) = -a22 – a42 + 2Q a2 a4 – 2 2 a2 – 2 4 a4 (13)

δω/δa2 = -2a2 + 2Q a4 - 2 2 = 0 (14)

δω/δa4 = -2a4 + 2Q a2 - 2 4 = 0 (15)


and the expected value of the maximum payoff for the first time period is:

2 4

a2 = 2 [2 + ]

Taking the first derivative of the payoff function first with respect to a2 and a4 setting each

2[2 + ]

Evaluation of the Network

1. First Period T1:

where s02 = E[x + ε] 2. Second Period T2:

2

The payoff function is:

gives the following system of equations:

equal to zero:

is:

$$\begin{array}{l} \text{s12} = \text{E } \left[ \chi + \varepsilon + \varepsilon \right] \text{ 2} \\\\ \text{s22} = \text{E } \left[ \chi + \varepsilon + \varepsilon + \varepsilon \right] \text{2} \\\\ \text{r12} = \left[ \text{E } (\chi + \varepsilon + \varepsilon)(\chi + \varepsilon + \varepsilon + \varepsilon) \right] / \text{ s1 s2} \end{array}$$

where r12 is the correlation coefficient and Q is the interaction.

3. Third Period T3:

$$\sum\_{\{\mathbf{a}\_2 = \alpha\_2 \mid \eta\_2 + \varepsilon\}\_{\mathbf{a}} = \mathbf{a}} \underbrace{\epsilon \sum\_{\iota} \epsilon}\_{\begin{pmatrix} \mathbf{a}\_2 + \mathfrak{a}\_1 \\ \mathbf{a}\_2 \end{pmatrix} \smile \sum\_{\iota} \mathbf{a}\_2 \llcorner \mathfrak{a}\_1 \mathbf{a}\_2 \mathbf{a}\_1} \underbrace{\epsilon \mathbf{a}\_2 \llcorner \mathfrak{a}\_1 \mathbf{a}\_2 \mathbf{a}\_1}\_{\begin{pmatrix} \mathbf{a}\_2 + \mathfrak{a}\_1 \mathbf{a}\_2 \\ \mathbf{a}\_2 + \mathfrak{a}\_1 \mathbf{a}\_2 \end{pmatrix} \llcorner \mathbf{a}\_1}$$

The payoff function is:

$$\mathbf{a} = \mathbf{-a}\mathbf{3}\mathbf{2} - \mathbf{a}\mathbf{5}\mathbf{2} + \mathbf{2}\mathbf{Q}\mathbf{a}\mathbf{3}\mathbf{a}\mathbf{5} - \mathbf{2}\mathbf{\eta}\mathbf{3}\mathbf{a}\mathbf{3} - \mathbf{2}\mathbf{\eta}\mathbf{5}\mathbf{a}\mathbf{5} \tag{22}$$

where:

$$\mathbf{u}\cdot\mathbf{v}\cdot\mathbf{s} = \mathbf{c} + \mathbf{d}\cdot\mathbf{2} = \left[\mathbf{-1}/(\mathbf{1}\cdot\mathbf{Q}2)\right]^\ast\left[\mathbf{x}+\mathbf{c}+\mathbf{e}\right] + \left[\mathbf{-Q}/(\mathbf{1}\cdot\mathbf{Q}2)\right]^\ast\left[\mathbf{x}+\mathbf{c}+\mathbf{e}+\mathbf{e}\right] + \mathbf{e} \tag{23}$$

$$\mathbf{v}\cdot\mathbf{5} = \varepsilon + \dot{\mathbf{d}}\cdot\mathbf{2} = \begin{bmatrix} \mathbf{-Q}/(\mathbf{1}\cdot\mathbf{Q}2) \end{bmatrix}^\* \begin{bmatrix} \mathbf{x} + \varepsilon + \varepsilon \end{bmatrix} + \begin{bmatrix} \mathbf{-l}/(\mathbf{1}\cdot\mathbf{Q}2) \end{bmatrix}^\* \begin{bmatrix} \mathbf{x} + \varepsilon + \varepsilon + \varepsilon \end{bmatrix} + \varepsilon \tag{24}$$

Performing the same calculations as in T2 gives the following best decision functions:

where:

**5. Results** 

Total Expected Value of the Maximum Payoff:

where the coefficients A, B, C are:

below for each of the two processes:

then Figure 11 depicts the resulting curves for each process.

 

6s2

Fig. 11. Expected payoff vs interaction.

Numerical Evaluation of Product Development Processes 249

s52= E [ε + ά 5]2

Ω = Σ (i = 1...4) Ω i = Ω 1+ Ω 2 + Ω 3 + Ω 4

= s02 + [s12 + 2Qr s1 s2 + s22]/ [1-Q2] + [s32 + 2Qr34s3 s4 + s42]/ [1-Q2] + s52

= [s02+ s52] + [s12 + s22 + s32 + s42]/ [1-Q2] + [2Q(r12 s1 s2 + r34s3 s4]/ [1-Q2]

Ω TOT = A + B/[1-Q2] + CQ/[1-Q2]

A = s02+ s52

B = s12 + s22 + s32 + s42

C = 2(r12s1 s2 + r34s3 s4)

From the calculations in the previous section, the total expected payoffs are summarized

Sequential: Ω TOT = s02 + s12 + s22 + s32 + s42 + s52 (11)

CE: Ω TOT = A + B/[1-Q2] + CQ/[1-Q2] (32) where the coefficients A, B, and C are as before. The equation for the expected value of the maximum payoff for the sequential process is a constant with respect to Q, while for the CE process it is polynomial in Q. If it is assumed for simplicity that all variances are equal and the correlation coefficients are equal to zero, i.e., the information variables are independent,

0 1 Interaction Q

Sequential

CE

Ω 4 = s52 (31)

(32)

$$
\dot{\mathbf{c}} = [-1/(1 \text{-Q2})] \ast \eta \, \mathbf{3} + [\text{-Q/(1-Q2)}] \ast \eta \, \mathbf{5}
$$

$$
= [-1/(1 \text{-Q2})] \left[ -1/(1 \text{-Q2}) \right] \ast \left[ \mathbf{x} + \varepsilon + \varepsilon \right] + [\text{-Q/(1-Q2)}] \ast \left( \mathbf{x} + \varepsilon + \varepsilon + \varepsilon \right) + \varepsilon \right. \tag{25}
$$

$$
\left[ -\text{Q/(1-Q2)} \right] \left[ -\text{Q/(1-Q2)} \right] \ast \left[ \mathbf{x} + \varepsilon + \varepsilon \right] + [\text{-1/(1-Q2)}] \ast \left( \mathbf{x} + \varepsilon + \varepsilon + \varepsilon \right) + \varepsilon
$$

$$
\text{d}\mathbf{5} = \left[ -\text{Q/(1-Q2)} \right] \ast \left[ \mathbf{n} \right] \odot \left[ -\text{1/(1-Q2)} \right] \ast \left[ \mathbf{x} + \varepsilon + \varepsilon \right] + [\text{-Q/(1-Q2)}] \ast \left( \mathbf{x} + \varepsilon + \varepsilon + \varepsilon \right) + \varepsilon \right. \tag{26}
$$

$$
\left[ -\text{1/(1-Q2)} \right] \left[ -\text{Q/(1-Q2)} \right] \left[ \mathbf{x} + \varepsilon + \varepsilon \right] + [\text{-1/(1-Q2)}] \ast \left[ \mathbf{x} + \varepsilon + \varepsilon + \varepsilon \right] + \varepsilon
$$

The expected payoff is:

$$
\Delta \mathcal{B} = \left[ \text{s} \mathcal{Q} + 2 \text{Qr} \text{34s} \mathcal{B} \text{ s} + \text{s} \text{42} \right] / \left[ \text{1-Q2} \right] \tag{27}
$$

where:

$$\begin{aligned} \text{s32} &= \text{E} \left[ \text{-1/(1-Q2) [x + \varepsilon + \varepsilon] + [\text{-Q/(1-Q2)}] \* (\chi + \varepsilon + \varepsilon + \varepsilon) + \varepsilon] 2 \right] \\\\ \text{s42} &= \text{E} \left[ \text{-Q/(1-Q2) [x + \varepsilon + \varepsilon] + [\text{-1/(1-Q2)}] \* (\chi + \varepsilon + \varepsilon + \varepsilon) + \varepsilon] 2 \right] \\\\ &\text{r34} = \text{E} \left[ \eta \text{3 } \eta \text{ 5} \right] / \text{s3 s4} \end{aligned}$$

4. Fourth Period T4:

The payoff function is chosen as:

$$\text{rad} = \text{-a}\text{62} + \text{2a}\text{6\text{ }\eta\text{ }6} \tag{28}$$

where:

6 = ε + ά 5

$$\begin{aligned} &= \left( \text{-Q} / (\text{1-Q2}) \left( \left[ \text{-1/} (\text{1-Q2}) \right] \left[ \text{x} + \text{c} + \text{e} \right] + \left[ \text{-Q} / (\text{1-Q2}) \right] \left( \text{x} + \text{c} + \text{e} + \text{e} + \text{e} \right) + \text{e} \right) + \varepsilon \\ &+ \left( \text{-1/} (\text{1-Q2}) \left( \left[ \text{-Q/} (\text{1-Q2}) \right] \left[ \text{x} + \text{c} + \text{e} \right] + \left[ \text{-1/} (\text{1-Q2}) \right] \left( \text{x} + \text{c} + \text{e} + \text{e} \right) + \text{e} \right) + \varepsilon \right) \end{aligned} \tag{29}$$

The best decision function is:

$$
\dot{\mathfrak{a}} \ \mathfrak{G} = \mathfrak{n} \ \mathfrak{G} \tag{30}
$$

Therefore the expected value of the maximum payoff is:

$$
\Omega \, 4 = \text{s} \' 2 \tag{31}
$$

where:

248 Numerical Modelling

= [-1/(1-Q2)] {-1/(1-Q2)] \* [x + ε + ε] + [-Q/(1-Q2)] \* (x + ε + ε + ε)+ ε} + (25)

= [-Q/(1-Q2)] {-1/(1-Q2)] [x + ε + ε] + [-Q/(1-Q2)] \* (x + ε + ε + ε) + ε } + (26)

Ω 3 = [s32 + 2Qr34s3 s4 + s42]/ [1-Q2] (27)

ω = -a62 + 2a6 6 (28)

ά 6 = 6 (30)

ά3 = [-1/(1-Q2)] \* 3 + [-Q/(1-Q2)] \* 5

[-Q/(1-Q2)] {-Q/(1-Q2)] \* [x + ε + ε] + [-1/(1-Q2)] \* (x + ε + ε + ε) + ε}

ά5 = [-Q/(1-Q2)] \* 3 + [-1/(1-Q2)] \* 5

[-1/(1-Q2)] {-Q/(1-Q2)] [x + ε + ε] + [-1/(1-Q2)] \* (x + ε + ε + ε) + ε }

s32 = E[-1/(1-Q2) [x + ε + ε] + [-Q/(1-Q2)] \* (x + ε + ε + ε) + ε]2

s42 = E[-Q/(1-Q2) [x + ε + ε] + [-1/(1-Q2)] \* (x + ε + ε + ε) + ε]2

r34 = E[3 5]/ s3 s4

6 a5 = 5 a6 = <sup>6</sup>

6 = ε + ά 5

+ (-1/(1-Q2) {[-Q/(1-Q2)] [x + ε + ε] + [-1/(1-Q2)] (x + ε + ε + ε) + ε] + ε} (29)

= (-Q/(1-Q2) {[-1/(1-Q2)] [x + ε + ε] + [-Q/(1-Q2)] (x + ε + ε + ε + ε) + ε] + ε

The expected payoff is:

4. Fourth Period T4:

The payoff function is chosen as:

The best decision function is:

Therefore the expected value of the maximum payoff is:

where:

where:

$$\mathbf{s}\mathbf{5} \mathbf{2} = \mathrm{E}\left[\boldsymbol{\varepsilon} + \dot{\mathbf{a}}\,\,\mathbf{5}\right] \mathbf{2}$$

Total Expected Value of the Maximum Payoff:

$$\Omega = \Sigma \text{ (i = 1...4) } \Omega \text{ i = } \Omega \text{ 1+ } \Omega \text{ 2+ } \Omega \text{ 3+ } \Omega \text{ 4+ } \Omega$$

$$\begin{aligned} \text{s} &= \text{s}02 + [\text{s}12 + 2\text{Qr s} \, \text{s} 2 + \text{s}22] / \, [\text{1-Q2}] + [\text{s}32 + 2\text{Qr} \, \text{34s} \, \text{s}4 + \text{s}42] / \, [\text{1-Q2}] + \text{s}52 \\\\ \text{s} &= [\text{s}02 + \text{s}52] + [\text{s}12 + \text{s}22 + \text{s}32 + \text{s}42] / \, [\text{1-Q2}] + [\text{2Q}(\text{r}12 \, \text{s} \, \text{1} \, \text{s} 2 + \text{r}3 \, \text{s}4 \, \text{s} 4] / \, [\text{1-Q2}] \end{aligned} \tag{32}$$

$$\text{Q TOT} = \text{A} + \text{B}/[1\text{-Q}2] + \text{CQ}/[1\text{-Q}2]$$

where the coefficients A, B, C are:

$$\mathbf{A} = \mathbf{s}\mathbf{0}\mathbf{2} + \mathbf{s}\mathbf{5}\mathbf{2}$$

$$\mathbf{B} = \mathbf{s}\mathbf{1}\mathbf{2} + \mathbf{s}\mathbf{2}\mathbf{2} + \mathbf{s}\mathbf{3}\mathbf{2} + \mathbf{s}\mathbf{4}\mathbf{2}$$

$$\mathbf{C} = \mathbf{2}(\mathbf{r}\mathbf{1}\mathbf{2}\mathbf{s}\mathbf{1}\mathbf{s}\mathbf{2} + \mathbf{r}\mathbf{3}\mathbf{4}\mathbf{s}\mathbf{3}\mathbf{s}\mathbf{4})$$

#### **5. Results**

From the calculations in the previous section, the total expected payoffs are summarized below for each of the two processes:

$$\text{Sequential:}\\\text{s\u021}\\\text{TOT} = \text{s\u02} + \text{s12} + \text{s22} + \text{s32} + \text{s42} + \text{s52} \tag{11}$$

$$\text{CE:} \tag{1} \\ \text{TOT} = \text{A} + \text{B}/[1\text{-Q2}] + \text{CQ/[1-Q2]} \tag{32}$$

where the coefficients A, B, and C are as before. The equation for the expected value of the maximum payoff for the sequential process is a constant with respect to Q, while for the CE process it is polynomial in Q. If it is assumed for simplicity that all variances are equal and the correlation coefficients are equal to zero, i.e., the information variables are independent, then Figure 11 depicts the resulting curves for each process.

Fig. 11. Expected payoff vs interaction.

Numerical Evaluation of Product Development Processes 251

 Sequential CE

0 Q

0.6

The curve for the CE process shows how the expected value of the maximum payoff changes with interaction, showing that as team interaction increases, the expected payoff increases as well. For this particular case, it was found that a sequential process has a higher expected payoff when the interaction is lower than 0.6, and a CE process has a higher payoff for values of interaction greater than 0.6. In other words, the results show that when actions in a CE process highly influence one another, i.e., the interaction is higher than 0.6, then a CE process is more valuable in terms of expected value of the maximum payoff. If the interaction between action variables is not strong, i.e., less than 0.6, then the sequential

In conclusion, the expected payoff method from decision theory provided some initial results in the comparison of a sequential and CE process. From the mathematical derivation presented in this chapter, comparing equation (11) to (32) shows that a CE process is always more valuable than a sequential process in terms of expected payoff. In most instances in reality, however, a sequential process has some benefit. Under the conditions when this holds true, the sequential process has a higher total expected payoff when the interaction intensity is low, while CE is better than a sequential process for high

The expected payoff method is presented here as a very simple introduction to studying NPD processes. A more elaborate and detailed development is in progress (Kong and Thomson, 2001), the results of which are expected to provide a major contribution to the

Some avenues for future research are now discussed. In the comparison of sequential and CE processes, it is assumed that there is no interaction between team members in the sequential process, thus emulating the 'over-the-wall' approach, where team members throw information over an invisible wall. In practice however, there exist interactions among members (or departments) of a team, though they may be very weak. Future work should consider this. It was stated that the expected payoff method assumes that individuals in a team work towards achieving common goals with common interests and beliefs within

existing body of work in studying organizational processes and their coordination.

1

This analysis is now illustrated in Figure 12.

Fig. 12. Expected payoff vs interaction.

interaction.

**6. Conclusion** 

8.25

6

process is sufficient, and superior in terms of expected payoff.

 

This analysis shows that a CE process is always better than a sequential one in terms of expected value of the maximum payoff. This is contradictory to practical observations of both processes. This is due to the fact that the analytical model oversimplifies the sequential process, whereby it is assumed that there is virtually no interaction between phases, and that information is 'thrown over the wall' from one function to another. Under this assumption, there is no interaction, which naturally results in an expected payoff that is independent of the interaction, Q, thus giving a constant. Additionally, it is also assumed in the modeling process that the contribution of each member's specialized information is the same for both the sequential and CE processes. This results in the total expected payoff for the sequential process being always lower than that of CE. Again, this assumption is not consistent with practical observations. In order to make the analysis more meaningful, some further assumptions should be made with regards to team members in a sequential process as compared to a CE process.

In some practical situations, a sequential process can be better than a CE process (Krishnan et al., 1997). When this is true, in the modeling process it is reasonable to assume that for a sequential process, every team member's knowledge and information is sufficient to allow him to finish his activity independently. In fact, it may even be argued that in a sequential process, the amount and types of information that functional members must possess is greater than members in a cross-functional team, which allows them to finish their activity independently. They must possess not only information about their own specialization, but they must also have, to some extent, information about other functions as well. After all, a designer will not design a product which requires milling if the company does not own a milling machine. In contrast, in a CE process, it can be assumed that members on a crossfunctional team do not need to possess as much information about other functions since sharing of information will occur naturally as a consequence of teamwork, in which case it is reasonable to assume that more work is required to obtain information. Therefore, the variance of knowledge and information measured by s2 is assumed to be larger for members in a sequential process than for the same members who would work in the overlapped periods in a CE process. This implies that the lack of information or knowledge by members in a CE process can be compensated by the exchange of information in the overlapped periods.

Given this assumption, the straight line in Figure 11 would move up the y-axis, while the CE curve would remain the same. This would create a point of intersection between the two curves, indicating that, for a given point of interaction, one process will be superior to the other in terms of expected payoff. For simplicity, it is assumed that for the CE process, all variances are equal to 1, and that the correlation coefficients are equal to 0. It can be further assumed that team members 2, 3, 4, and 5 in a sequential process have a variance that is slightly higher than the same members in a CE process, who, as explained above, exchange information during the overlapped periods. For simplicity, the variance for the sequential members' information is taken to be one-quarter higher than that of the CE members' information i.e., si2 (sequential) = 1.25 si2 (CE). Plugging these values back into 11 and 32, the total expected payoffs are:


This analysis is now illustrated in Figure 12.

250 Numerical Modelling

This analysis shows that a CE process is always better than a sequential one in terms of expected value of the maximum payoff. This is contradictory to practical observations of both processes. This is due to the fact that the analytical model oversimplifies the sequential process, whereby it is assumed that there is virtually no interaction between phases, and that information is 'thrown over the wall' from one function to another. Under this assumption, there is no interaction, which naturally results in an expected payoff that is independent of the interaction, Q, thus giving a constant. Additionally, it is also assumed in the modeling process that the contribution of each member's specialized information is the same for both the sequential and CE processes. This results in the total expected payoff for the sequential process being always lower than that of CE. Again, this assumption is not consistent with practical observations. In order to make the analysis more meaningful, some further assumptions should be made with regards to team members in a sequential process

In some practical situations, a sequential process can be better than a CE process (Krishnan et al., 1997). When this is true, in the modeling process it is reasonable to assume that for a sequential process, every team member's knowledge and information is sufficient to allow him to finish his activity independently. In fact, it may even be argued that in a sequential process, the amount and types of information that functional members must possess is greater than members in a cross-functional team, which allows them to finish their activity independently. They must possess not only information about their own specialization, but they must also have, to some extent, information about other functions as well. After all, a designer will not design a product which requires milling if the company does not own a milling machine. In contrast, in a CE process, it can be assumed that members on a crossfunctional team do not need to possess as much information about other functions since sharing of information will occur naturally as a consequence of teamwork, in which case it is reasonable to assume that more work is required to obtain information. Therefore, the variance of knowledge and information measured by s2 is assumed to be larger for members in a sequential process than for the same members who would work in the overlapped periods in a CE process. This implies that the lack of information or knowledge by members in a CE process can be compensated by the exchange of information in the

Given this assumption, the straight line in Figure 11 would move up the y-axis, while the CE curve would remain the same. This would create a point of intersection between the two curves, indicating that, for a given point of interaction, one process will be superior to the other in terms of expected payoff. For simplicity, it is assumed that for the CE process, all variances are equal to 1, and that the correlation coefficients are equal to 0. It can be further assumed that team members 2, 3, 4, and 5 in a sequential process have a variance that is slightly higher than the same members in a CE process, who, as explained above, exchange information during the overlapped periods. For simplicity, the variance for the sequential members' information is taken to be one-quarter higher than that of the CE members' information i.e., si2 (sequential) = 1.25 si2 (CE). Plugging these values back into 11 and 32,

as compared to a CE process.

overlapped periods.

the total expected payoffs are:

Sequential: Ω TOT = 8.25

CE: Ω TOT = 2 + 4/[1-Q2]

Fig. 12. Expected payoff vs interaction.

The curve for the CE process shows how the expected value of the maximum payoff changes with interaction, showing that as team interaction increases, the expected payoff increases as well. For this particular case, it was found that a sequential process has a higher expected payoff when the interaction is lower than 0.6, and a CE process has a higher payoff for values of interaction greater than 0.6. In other words, the results show that when actions in a CE process highly influence one another, i.e., the interaction is higher than 0.6, then a CE process is more valuable in terms of expected value of the maximum payoff. If the interaction between action variables is not strong, i.e., less than 0.6, then the sequential process is sufficient, and superior in terms of expected payoff.

In conclusion, the expected payoff method from decision theory provided some initial results in the comparison of a sequential and CE process. From the mathematical derivation presented in this chapter, comparing equation (11) to (32) shows that a CE process is always more valuable than a sequential process in terms of expected payoff. In most instances in reality, however, a sequential process has some benefit. Under the conditions when this holds true, the sequential process has a higher total expected payoff when the interaction intensity is low, while CE is better than a sequential process for high interaction.

#### **6. Conclusion**

The expected payoff method is presented here as a very simple introduction to studying NPD processes. A more elaborate and detailed development is in progress (Kong and Thomson, 2001), the results of which are expected to provide a major contribution to the existing body of work in studying organizational processes and their coordination.

Some avenues for future research are now discussed. In the comparison of sequential and CE processes, it is assumed that there is no interaction between team members in the sequential process, thus emulating the 'over-the-wall' approach, where team members throw information over an invisible wall. In practice however, there exist interactions among members (or departments) of a team, though they may be very weak. Future work should consider this. It was stated that the expected payoff method assumes that individuals in a team work towards achieving common goals with common interests and beliefs within

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the constraints of their work, all of which guide their behavior. For a CE team, this is conceivable in the sense that any 'team' usually works together to achieve some goal, and a cross-functional team, ideally, works towards the common project goals of being on time, and within budget. However, in practice there is tension between meeting project and functional goals, as team members have project-specific goals, but also have departmental obligations to fulfill. The same assumption is debatable for a sequential process, where functional teams in different activities tend to have differing goals. For example, in isolation, a designer's goal is to create a product design without much concern for the production process that will build it. Similarly, a marketing manager's goal is to get customers to buy the company product without much concern for how the product will be made. This is partly due to the fact that functional goals are tied to functional rewards. Taking into account this divergence of beliefs would require further analysis into economic and organization theory where individuals' actions are based on self-interest. A more detailed description of the influence of time on the payoff function must be developed. Presently, it is assumed that interaction between action variables at different times is weaker the farther apart they are in time. However, if there is interaction between actions at different times, the payoff function will not be additive in time. The sequential process will have constraints which link actions that are distant in time, and can no longer be evaluated as a series of single-period problems, in which interaction is so weak that it does not exist. Most activities are not deterministic in a product development process. In fact, many situations arise where a stochastic relation between activities apply. A commonly occurring phenomenon is the failure of one or more activities, which consequently require rework. Rework loops in the network diagrams must be expressed to incorporate this very important characteristic of development processes. The function of rework in a network is to prevent the expected payoff from reaching a maximum when an activity is reworked, though a maximum can be reached after a few iterations but at a greater cost. The measure of the expected utility of a network has always been considered in its gross form, that is, without any consideration for the cost of the network. In reality, obtaining information can be very costly, and though one network may be superior to another in terms of the expected payoff, the cost of that network may not justify its use. This concept should also be incorporated into the models.

#### **7. Acknowledgment**

The author is grateful to Dr. Vincent Thomson and Dr. Linghua Kong for their guidance and constructive advice during this research.

#### **8. References**


the constraints of their work, all of which guide their behavior. For a CE team, this is conceivable in the sense that any 'team' usually works together to achieve some goal, and a cross-functional team, ideally, works towards the common project goals of being on time, and within budget. However, in practice there is tension between meeting project and functional goals, as team members have project-specific goals, but also have departmental obligations to fulfill. The same assumption is debatable for a sequential process, where functional teams in different activities tend to have differing goals. For example, in isolation, a designer's goal is to create a product design without much concern for the production process that will build it. Similarly, a marketing manager's goal is to get customers to buy the company product without much concern for how the product will be made. This is partly due to the fact that functional goals are tied to functional rewards. Taking into account this divergence of beliefs would require further analysis into economic and organization theory where individuals' actions are based on self-interest. A more detailed description of the influence of time on the payoff function must be developed. Presently, it is assumed that interaction between action variables at different times is weaker the farther apart they are in time. However, if there is interaction between actions at different times, the payoff function will not be additive in time. The sequential process will have constraints which link actions that are distant in time, and can no longer be evaluated as a series of single-period problems, in which interaction is so weak that it does not exist. Most activities are not deterministic in a product development process. In fact, many situations arise where a stochastic relation between activities apply. A commonly occurring phenomenon is the failure of one or more activities, which consequently require rework. Rework loops in the network diagrams must be expressed to incorporate this very important characteristic of development processes. The function of rework in a network is to prevent the expected payoff from reaching a maximum when an activity is reworked, though a maximum can be reached after a few iterations but at a greater cost. The measure of the expected utility of a network has always been considered in its gross form, that is, without any consideration for the cost of the network. In reality, obtaining information can be very costly, and though one network may be superior to another in terms of the expected payoff, the cost of that network

may not justify its use. This concept should also be incorporated into the models.

The author is grateful to Dr. Vincent Thomson and Dr. Linghua Kong for their guidance and

Adler, P.S. (1995). Interdepartmental Interdependence and Coordination: The Case of the Design/Manufacturing Interface, *Organizational Science*, Vol.6, No.2, pp. 147-167. Blackburn J. (1991) New Product Development: The New Time Wars, in J. Blackburn (ed.),

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Time-Based Competition: The Next Battleground in *American Manufacturing*, (pp.

**7. Acknowledgment** 

**8. References** 

constructive advice during this research.


**12** 

*Poland* 

**Numerical Modelling of Steel** 

Marcin Hojny and Miroslaw Glowacki *AGH University of Science and Technology, Krakow* 

**Deformation at Extra-High Temperatures** 

Due to the globalized energy crisis and high consciousness of environmental protection in last year's, more and more products and new technology that put stress on energy preservation and environmental protection are being developed. The integrated casting and rolling technologies are newest and very efficient ways of hot strip production. Only few companies all over the world are able to manage such processes. Among them one can mention the plant located in Cremona Italy which develops the Arvedi Steel Technologies – new methods of steel strip manufacturing. They are called ISP (Inline Strip Production) and AST (Arvedi Steel Technologies) processes and are characterized by very high temperature allowed at the mill entry. The instant rolling of slabs which leave the casting machine allows

The rolling equipment for the Inline Strip Production process consist of cast rolling machine, liquid core reduction equipment, high reduction mill, inductive heater, Cremona coiling station, descaler, traditional finishing mill and the cooling zone. The initial mould strip thickness is 74 mm and is reduced to 55 mm during liquid core reduction process. The region of maximum strip temperature for a high reduction mill is placed in the strip centre and varies from 1220oC to 1375oC depending on the casting speed. The main benefits of the

very low level of heating energy consumption which drops to 0 when the casting speed

inverse (in comparison to traditional rolling) temperature gradient in the cross-section

 low level of installed mill power in high reduction mill (3 rolling stands with 0.5, 0.6 and 0.8 MW) providing reduction from 55 to 12.5 mm by the strip width of 1300 mm.

the utilization of the heat stored in the strips during inline casting.

compact rolling equipment layout – total rolling line is only 170 m long,

good product quality – 1 mm strip with best shape and microstructure,

of the strip, which is very useful for the rolling process,

**1. Introduction** 

technology are:

very low investment costs,

no need of tunnel furnace,

is over around 0.14 m/s,

up to 20 times lower water consumption,

Yassine, A.; Chelst, K.R. and Falkenburg D. R. (1999). A Decision Analytic Framework for Evaluating Concurrent Engineering, *IEEE Transactions on Engineering Management*, Vol.46, No.2, pp. 144-157.

## **Numerical Modelling of Steel Deformation at Extra-High Temperatures**

Marcin Hojny and Miroslaw Glowacki *AGH University of Science and Technology, Krakow Poland* 

#### **1. Introduction**

254 Numerical Modelling

Yassine, A.; Chelst, K.R. and Falkenburg D. R. (1999). A Decision Analytic Framework for

Vol.46, No.2, pp. 144-157.

Evaluating Concurrent Engineering, *IEEE Transactions on Engineering Management*,

Due to the globalized energy crisis and high consciousness of environmental protection in last year's, more and more products and new technology that put stress on energy preservation and environmental protection are being developed. The integrated casting and rolling technologies are newest and very efficient ways of hot strip production. Only few companies all over the world are able to manage such processes. Among them one can mention the plant located in Cremona Italy which develops the Arvedi Steel Technologies – new methods of steel strip manufacturing. They are called ISP (Inline Strip Production) and AST (Arvedi Steel Technologies) processes and are characterized by very high temperature allowed at the mill entry. The instant rolling of slabs which leave the casting machine allows the utilization of the heat stored in the strips during inline casting.

The rolling equipment for the Inline Strip Production process consist of cast rolling machine, liquid core reduction equipment, high reduction mill, inductive heater, Cremona coiling station, descaler, traditional finishing mill and the cooling zone. The initial mould strip thickness is 74 mm and is reduced to 55 mm during liquid core reduction process. The region of maximum strip temperature for a high reduction mill is placed in the strip centre and varies from 1220oC to 1375oC depending on the casting speed. The main benefits of the technology are:


Numerical Modelling of Steel Deformation at Extra-High Temperatures 257

The mathematical and experimental modelling of mushy steel deformation is an innovative topic regarding the very high temperature range deformation processes. Tracing the related papers published in the past ten years, one can find many papers regarding experimental results (Kang & Yoon, 1997; Koc et al., 1996; Kop et al., 2003) and modelling (Modigell et al., 2004; Hufschmidt et al., 2004) for non-ferrous metals tests. The papers deal mainly with tixotrophy. The first results regarding steel deformation at extra high temperature were presented in the past few years (Jing et al., 2005; Jin et al., 2002). The situation is caused by the very high level of steel liquidus and solidus temperatures in comparison with nonferrous metals. The deformation tests for non-ferrous metals are much easier. The rising abilities of thermo-mechanical simulators enable steel sample testing and as a result both computer simulation and improvement of new rolling technologies, similar to ISP and AST

The chapter sheds light on these problems. It focuses on the axial-symmetrical computer model, which ensures the possibility of its experimental verification with the help of

The numerical analysis of deformation of samples having liquid phase in their central parts shows extremely high strain inhomogeneity requiring application of hybrid analyticalnumerical solution of the problem (Hojny & Glowacki, 2008, 2009, 2011). A coupled thermalmechanical mathematical model was developed for simulation of plastic behaviour of such a species. The model is dedicated to modelling processes, which require high accuracy of resulting parameter fields. Analytical solutions of both incompressibility and mass conservation (for the mushy zone) conditions are important parts of the model. Analytical condition eliminates problems with unintentional specimen volume changes caused by application of numerical methods. The existing, physical changes of steel density in the mushy zone have influence on real variations of controlled volume. On the other hand numerical errors can be a source of volume loss which cause interference with real changes. This effect is very undesirable in modelling of thermal-mechanical behaviour of steel in

**2. Thermo-mechanical model of steel deformation in semi-solid state** 

temperature range characteristic for the transformation of state of aggregation.

tensile strength, which is very low within discussed temperature range.

The mathematical model of the process consists of two main parts – mechanical and thermal – both of them supported by density changes model. The mechanical part is responsible for the strain, strain rate and stress distribution in a controlled volume. The stress is substantial due to shrinkage and plastic deformation. It can cause cracks when the stress exceeds the ultimate

Thermal solution has crucial influence on simulation results, since the temperature has strong effect on remaining variables, especially if the specimen temperature is close to solidus line. Plastic flow of solid and mushy materials, stress distribution and density changes are relevant to the temperature field particularly for deformation of body which consist of both solid and semi-solid regions. The temperature field is a result of solution of Fourier-Kirchhoff equation. The combined Hankel's boundary conditions have been

processes.

physical simulation.

**2.1 Thermal model** 

The AST (Arvedi Steel Technologies) technology is a result of further development of ISP to a real endless process. The main difference between these two technologies is the absence of the heating equipment in case of AST. The whole reduction process is running in one rolling mill consisting of 5 or 7 stands, which can reduce the strip thickness from 55÷70 mm to 0.8 mm. AST is the most compact hot strip production process using oscillating mould technology with excellent efficiency and cost. The total equipment length is 70 to 80 m including casting machine at the front and final coolers at the rear end. The maximal temperature of the strip occurs in central region of its cross-section and varies from 1340oC to 1420oC according to the casting speed. It suggests that the central region of the strand subjected to the rolling is still mushy.

Both the technologies mentioned above ensure huge reduction of rolling forces, very high product quality and low investment costs and their details are usually classified. The main goal of the mentioned new technologies is to significantly lower the rolling forces and to reach very favourable temperature field inside the plate in comparison with traditional processes. However certain problems specific to such a metal treatment arise. The central part of the material is still mushy. This results in changes in material density and occurrence of characteristic temperatures, which have great influence on plastic behaviour of the material. A vital problem is also the lack of data regarding material's thermal and mechanical properties and significant changes of density.

The material behaviour above the solidus line is strongly temperature-dependent. There are a few characteristic temperature values between solidus and liquidus. The Nil Strength Temperature (NST) is the temperature level at which material strength drops to zero while the steel is being heated above the solidus temperature. Another temperature associated with NST is the Strength Recovery Temperature (SRT), which is specific to cooling. At this temperature the material regains strength greater than 0.5 N/mm2. Nil Ductility Temperature (NDT) represents the temperature at which the heated steel loses its ductility. The Ductility Recovery Temperature (DRT) is the temperature at which the ductility of the material (characterised by reduction of area) reaches 5% while it is being cooled. Over this temperature the plastic deformation is not allowed at any stress tensor configuration.

Very important for plastic behaviour of steel is also its density. It varies with temperature and depends on the cooling rate. The solidification process causes non-uniform density distribution in the controlled volume resulting in non-uniform deformation and heat conduction. There are three main factors causing density changes: solid phase formation, thermal shrinkage and movement of liquid particles inside the solid skeleton. The density plays an important role in both mechanical and thermal solutions.

The most important steel property having crucial influence on metal flow paths is the strainstress relationship. It is not easy to run the isothermal tests that could be the source of the computation of yield stress function parameters for temperature range close to solidus line. Keeping constant temperature during the whole experiment course is difficult. There are also some difficulties with interpretation of tests results. Lack of good methods of particular metal flow simulation and significant inhomogeneity in strain distribution in the deformation zone lead to weak accuracy of standard FEM solutions.

The AST (Arvedi Steel Technologies) technology is a result of further development of ISP to a real endless process. The main difference between these two technologies is the absence of the heating equipment in case of AST. The whole reduction process is running in one rolling mill consisting of 5 or 7 stands, which can reduce the strip thickness from 55÷70 mm to 0.8 mm. AST is the most compact hot strip production process using oscillating mould technology with excellent efficiency and cost. The total equipment length is 70 to 80 m including casting machine at the front and final coolers at the rear end. The maximal temperature of the strip occurs in central region of its cross-section and varies from 1340oC to 1420oC according to the casting speed. It suggests that the central region of the strand

Both the technologies mentioned above ensure huge reduction of rolling forces, very high product quality and low investment costs and their details are usually classified. The main goal of the mentioned new technologies is to significantly lower the rolling forces and to reach very favourable temperature field inside the plate in comparison with traditional processes. However certain problems specific to such a metal treatment arise. The central part of the material is still mushy. This results in changes in material density and occurrence of characteristic temperatures, which have great influence on plastic behaviour of the material. A vital problem is also the lack of data regarding material's thermal and

The material behaviour above the solidus line is strongly temperature-dependent. There are a few characteristic temperature values between solidus and liquidus. The Nil Strength Temperature (NST) is the temperature level at which material strength drops to zero while the steel is being heated above the solidus temperature. Another temperature associated with NST is the Strength Recovery Temperature (SRT), which is specific to cooling. At this temperature the material regains strength greater than 0.5 N/mm2. Nil Ductility Temperature (NDT) represents the temperature at which the heated steel loses its ductility. The Ductility Recovery Temperature (DRT) is the temperature at which the ductility of the material (characterised by reduction of area) reaches 5% while it is being cooled. Over this temperature the plastic deformation is not allowed at any stress tensor

Very important for plastic behaviour of steel is also its density. It varies with temperature and depends on the cooling rate. The solidification process causes non-uniform density distribution in the controlled volume resulting in non-uniform deformation and heat conduction. There are three main factors causing density changes: solid phase formation, thermal shrinkage and movement of liquid particles inside the solid skeleton. The density

The most important steel property having crucial influence on metal flow paths is the strainstress relationship. It is not easy to run the isothermal tests that could be the source of the computation of yield stress function parameters for temperature range close to solidus line. Keeping constant temperature during the whole experiment course is difficult. There are also some difficulties with interpretation of tests results. Lack of good methods of particular metal flow simulation and significant inhomogeneity in strain distribution in the

subjected to the rolling is still mushy.

configuration.

mechanical properties and significant changes of density.

plays an important role in both mechanical and thermal solutions.

deformation zone lead to weak accuracy of standard FEM solutions.

The mathematical and experimental modelling of mushy steel deformation is an innovative topic regarding the very high temperature range deformation processes. Tracing the related papers published in the past ten years, one can find many papers regarding experimental results (Kang & Yoon, 1997; Koc et al., 1996; Kop et al., 2003) and modelling (Modigell et al., 2004; Hufschmidt et al., 2004) for non-ferrous metals tests. The papers deal mainly with tixotrophy. The first results regarding steel deformation at extra high temperature were presented in the past few years (Jing et al., 2005; Jin et al., 2002). The situation is caused by the very high level of steel liquidus and solidus temperatures in comparison with nonferrous metals. The deformation tests for non-ferrous metals are much easier. The rising abilities of thermo-mechanical simulators enable steel sample testing and as a result both computer simulation and improvement of new rolling technologies, similar to ISP and AST processes.

The chapter sheds light on these problems. It focuses on the axial-symmetrical computer model, which ensures the possibility of its experimental verification with the help of physical simulation.

### **2. Thermo-mechanical model of steel deformation in semi-solid state**

The numerical analysis of deformation of samples having liquid phase in their central parts shows extremely high strain inhomogeneity requiring application of hybrid analyticalnumerical solution of the problem (Hojny & Glowacki, 2008, 2009, 2011). A coupled thermalmechanical mathematical model was developed for simulation of plastic behaviour of such a species. The model is dedicated to modelling processes, which require high accuracy of resulting parameter fields. Analytical solutions of both incompressibility and mass conservation (for the mushy zone) conditions are important parts of the model. Analytical condition eliminates problems with unintentional specimen volume changes caused by application of numerical methods. The existing, physical changes of steel density in the mushy zone have influence on real variations of controlled volume. On the other hand numerical errors can be a source of volume loss which cause interference with real changes. This effect is very undesirable in modelling of thermal-mechanical behaviour of steel in temperature range characteristic for the transformation of state of aggregation.

The mathematical model of the process consists of two main parts – mechanical and thermal – both of them supported by density changes model. The mechanical part is responsible for the strain, strain rate and stress distribution in a controlled volume. The stress is substantial due to shrinkage and plastic deformation. It can cause cracks when the stress exceeds the ultimate tensile strength, which is very low within discussed temperature range.

#### **2.1 Thermal model**

Thermal solution has crucial influence on simulation results, since the temperature has strong effect on remaining variables, especially if the specimen temperature is close to solidus line. Plastic flow of solid and mushy materials, stress distribution and density changes are relevant to the temperature field particularly for deformation of body which consist of both solid and semi-solid regions. The temperature field is a result of solution of Fourier-Kirchhoff equation. The combined Hankel's boundary conditions have been

Numerical Modelling of Steel Deformation at Extra-High Temperatures 259

functional (3), because numerical solution of both the mentioned conditions generates a lot of local minima and leads to wide flat neighbourhood of the global optimum. The accuracy of the proposed hybrid solution is also much better because of negligible volume loss caused by numerical errors which is very important for materials with changing density. Fully numerical solution shows lower accuracy contrary to the proposed analyticalnumerical one. For solid regions of the sample the incompressibility condition is satisfactory

> <sup>0</sup> *rr z vv v rr z*

where *vr* and *vz* are the radial and longitudinal velocity field components in cylindrical

<sup>1</sup> <sup>0</sup> *rr z vv v*

Density distribution is one of the most important properties of the mushy steel which undergo the deformation. Its changes have influence on both the mechanical and thermal parts of the presented model. The knowledge concerning effective density distribution is very important for modelling of deformation of porous and mushy materials. Density

Total density changes can be calculated according to the Darcy method which can be

 1 div *<sup>s</sup> <sup>l</sup> ss ll l l ss s ll l*

the controlled volume. Solution of equation (7) requires further time and computer memory resources. Nevertheless another way of taking density into consideration is possible due to temperature dependency of this quantity. In order to avoid additional problems with solution of equation (7) density changes were calculated according to an empirical model taking into consideration the experimental data. The model is slightly less accurate but such a method makes the solution much easier. The density changes of the investigated steels were computed using commercial JMatPro software. Figure 1 presents an example graph of density versus temperature dependency drawn for steel having 0.41% carbon content.

> 

*X T X X Xv X X*

are fraction and linear expansion coefficients, respectively. Indexes *l* and *s* 

 

*rr z* 

changes of liquid, solid-liquid and solid materials are ruled by three phenomena:

*l*

(5)

 

(7)

 

is the time variable, *T* is the temperature distribution in

(6)

*, z*. For the mushy zone equation (5) is replaced by the mass

is the time variable.

 

and in cylindrical coordinate system it has been described with an equation:

coordinate system *r,* 

**2.3 Density changes** 


 

denote the liquid and solid phases,

where *X* and


where  is the temporary material density and


 

formulated in a form of differential equation:

conservation condition, which takes a form:

adopted for the model (Glowacki, 2005). The Fourier-Kirchhoff equation in cylindrical coordinate system is written as follows:

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\mathbf{k}\_r\frac{\partial T}{\partial r}\right) + \frac{1}{r}\frac{\partial}{\partial \theta}\left(\frac{1}{r}\mathbf{k}\_\theta\frac{\partial T}{\partial \theta}\right) + \frac{\partial}{\partial z}\left(k\_z\frac{\partial T}{\partial z}\right) + Q = \rho c\_p\frac{\partial T}{\partial \tau} \tag{1}$$

where *r, , z* are cylindrical coordinate system, *T* is the temperature distribution in the controlled volume, is the time variable, *k* denotes the heat conduction coefficient (or coefficients matrix in case of thermal inhomogeneity), *Q* represents the rate of heat generation (or consumption) due to the transformation of the aggregation state, plastic work done and due to electric current flow in case of resistance heating of the sample. Finally, *cp* describes the specific heat. The solution of equation (1) has to satisfy appropriate boundary conditions. The Hankel's boundary conditions can be written in form of differential equation (Glowacki, 2005):

$$k\frac{\partial^{\gamma}T}{\partial^{\gamma}n} + \alpha(T - T\_0) + q = 0\tag{2}$$

In equation (2) *T0* is the distribution of the border temperature, *q* describes the heat flux through the boundary of the deformation zone, is the heat transfer coefficient, *n* is a vector which is normal to the boundary surface.

#### **2.2 Mechanical model with analytically controlled compressibility**

As mentioned above the mechanical part is responsible for calculation of the strain, strain rate and stress distribution in the deformation zone which consist of solid and semi-solid regions. The analysis of mathematical models which can be applied for strain field calculations for semi-solid steel deformation process has proved good predictive ability of a rigid-plastic model of metal flow. The model is completed with numerical solution of Navier stress equilibrium equations in order to satisfy all the requirements leading to stress field. Rigid-plastic model was selected due to its very good accuracy with reference to strain field during the hot deformation and sufficient correctness of calculated deviatoric part of stress field. Moreover, the elastic part of stress tensor components is very low at temperatures close to solidus line and practically can be neglected in calculations of strain distribution. Classical rigid-plastic solutions are based on optimisation of following power functional:

$$\int \left[ \upsilon(r, z) \right] = \mathcal{W}\_{\sigma} + \mathcal{W}\_{\lambda} + \mathcal{W}\_{t} \tag{3}$$

where *W* is the plastic deformation power, *W* the penalty for the departure from either incompressibility or mass conservation conditions and *Wt* the friction power. In presented solution the second part of functional (3) is missing and both incompressibility and mass conservation conditions are given in analytical form constraining the velocity field components. The functional takes the following shape:

$$\int \left[ v \left( r, z \right) \right] = \mathcal{W}\_{\sigma} + \mathcal{W}\_{t} \tag{4}$$

where *v* describes the velocity field distribution in the deformation zone. In case of functional (4) the optimisation procedure is much more convergent than the one concerning functional (3), because numerical solution of both the mentioned conditions generates a lot of local minima and leads to wide flat neighbourhood of the global optimum. The accuracy of the proposed hybrid solution is also much better because of negligible volume loss caused by numerical errors which is very important for materials with changing density. Fully numerical solution shows lower accuracy contrary to the proposed analyticalnumerical one. For solid regions of the sample the incompressibility condition is satisfactory and in cylindrical coordinate system it has been described with an equation:

$$\frac{\partial v\_r}{\partial r} + \frac{v\_r}{r} + \frac{\partial v\_z}{\partial z} = 0 \tag{5}$$

where *vr* and *vz* are the radial and longitudinal velocity field components in cylindrical coordinate system *r, , z*. For the mushy zone equation (5) is replaced by the mass conservation condition, which takes a form:

$$\frac{\partial v\_r}{\partial r} + \frac{v\_r}{r} + \frac{\partial v\_z}{\partial z} - \frac{1}{\rho} \frac{\partial \rho}{\partial \tau} = 0 \tag{6}$$

where is the temporary material density and is the time variable.

#### **2.3 Density changes**

258 Numerical Modelling

adopted for the model (Glowacki, 2005). The Fourier-Kirchhoff equation in cylindrical

coefficients matrix in case of thermal inhomogeneity), *Q* represents the rate of heat generation (or consumption) due to the transformation of the aggregation state, plastic work done and due to electric current flow in case of resistance heating of the sample. Finally, *cp* describes the specific heat. The solution of equation (1) has to satisfy appropriate boundary conditions. The Hankel's boundary conditions can be written in form of differential

<sup>0</sup> ( )0 *<sup>T</sup> k TT q*

In equation (2) *T0* is the distribution of the border temperature, *q* describes the heat flux

As mentioned above the mechanical part is responsible for calculation of the strain, strain rate and stress distribution in the deformation zone which consist of solid and semi-solid regions. The analysis of mathematical models which can be applied for strain field calculations for semi-solid steel deformation process has proved good predictive ability of a rigid-plastic model of metal flow. The model is completed with numerical solution of Navier stress equilibrium equations in order to satisfy all the requirements leading to stress field. Rigid-plastic model was selected due to its very good accuracy with reference to strain field during the hot deformation and sufficient correctness of calculated deviatoric part of stress field. Moreover, the elastic part of stress tensor components is very low at temperatures close to solidus line and practically can be neglected in calculations of strain distribution. Classical rigid-plastic solutions are based on optimisation of following power functional:

> \* , *<sup>t</sup> J vrz W W W*

incompressibility or mass conservation conditions and *Wt* the friction power. In presented solution the second part of functional (3) is missing and both incompressibility and mass conservation conditions are given in analytical form constraining the velocity field

\* , *<sup>t</sup> J vrz W W*

where *v* describes the velocity field distribution in the deformation zone. In case of functional (4) the optimisation procedure is much more convergent than the one concerning

 

*n*

**2.2 Mechanical model with analytically controlled compressibility** 

is the plastic deformation power, *W*

components. The functional takes the following shape:

∂� ���

*, z* are cylindrical coordinate system, *T* is the temperature distribution in the

∂�

is the time variable, *k* denotes the heat conduction coefficient (or

∂�� � � � ���

(2)

is the heat transfer coefficient, *n* is a vector

(3)

the penalty for the departure from either

(4)

∂�

∂� (1)

coordinate system is written as follows:

through the boundary of the deformation zone,

which is normal to the boundary surface.

∂� ∂�� � <sup>1</sup> � ∂ ∂� � 1 � k� ∂� ∂�� � <sup>∂</sup>

1 � ∂ ∂� ��k�

where *r,* 

where *W*

equation (Glowacki, 2005):

controlled volume,

Density distribution is one of the most important properties of the mushy steel which undergo the deformation. Its changes have influence on both the mechanical and thermal parts of the presented model. The knowledge concerning effective density distribution is very important for modelling of deformation of porous and mushy materials. Density changes of liquid, solid-liquid and solid materials are ruled by three phenomena:


Total density changes can be calculated according to the Darcy method which can be formulated in a form of differential equation:

$$\frac{\partial \hat{\boldsymbol{\rho}}}{\partial \tau} = \left(\rho\_s \mathbf{X}\_s + \rho\_l \mathbf{X}\_l\right) \left(\frac{\rho\_s}{\rho\_l} - 1\right) \frac{\partial \mathbf{X}\_l}{\partial \tau} + \rho\_l \mathbf{X}\_l \text{div}\,\mathbf{v} + \left(\beta\_s \rho\_s \mathbf{X}\_s + \beta\_l \rho\_l \mathbf{X}\_l\right) \frac{\partial \mathbf{T}}{\partial \tau} \tag{7}$$

where *X* and are fraction and linear expansion coefficients, respectively. Indexes *l* and *s*  denote the liquid and solid phases, is the time variable, *T* is the temperature distribution in the controlled volume. Solution of equation (7) requires further time and computer memory resources. Nevertheless another way of taking density into consideration is possible due to temperature dependency of this quantity. In order to avoid additional problems with solution of equation (7) density changes were calculated according to an empirical model taking into consideration the experimental data. The model is slightly less accurate but such a method makes the solution much easier. The density changes of the investigated steels were computed using commercial JMatPro software. Figure 1 presents an example graph of density versus temperature dependency drawn for steel having 0.41% carbon content.

Numerical Modelling of Steel Deformation at Extra-High Temperatures 261

The last module (DSS/Post module) is dedicated to the visualisation of the numerical results and printing the final reports. In the current version the possibility of visualization was significant improved. The main are: shading options using OpenGL mode (2D and 3D) and

Fig. 3. The Post-processor of the newest version of Def\_Semi\_Solid system (contour option).

The less visible but powerful heart of the system is of course the solver. The finite element code dedicated to the axial-symmetrical compression/tension tests has been developed. The solution is based on the thermo-mechanical approach with density changes described in the previous section. The second software used during theoretical work was commercial code JMatPro. This program calculates a wide range of materials properties for alloys and steels. Using JMatPro we can make calculations for stable and metastable phase equilibrium, solidification behaviour and properties, thermo-physical and physical properties, phase

Fig. 2. View of DSS program setup window (local version).

possibility make a full contour map (2D and 3D) as shown in Figure 3.

Fig. 1. The density versus temperature curve for C45 grade steel.

More details about presenting thermo-mechanical model, numerical methods and techniques will be described in the next chapter of this book.

#### **3. Numerical systems – Def\_Semi\_Solid and JMatPro**

In aim to allow easy working with the Gleeble® 3800 simulator a user friendly system called Def\_Semi\_Solid v.5.0 was developed in the Department of Computer Science and Modeling of the Faculty of Metals Engineering and Computer Science in Industry, AGH. The numerical part of the program was developed in FORTRAN/C++ language, which guaranties fast computation and the graphical interface was written using visual version of C++ language, taking advantage of its object oriented character. This approach has sufficient usability both in Windows and Linux based systems. The newest version of Def\_Semi\_Solid system is equipped with full automatic installation unit (Figure 2) and new graphical interface. It allows the computer aided testing of mechanical properties of steels at very high temperature using Gleeble® 3800 physical simulators to avoid problems which arise by traditional testing procedures. The first module allows the establishment of new projects or working with previously existing ones. The integral parts of each project are: input data for a specific compression/tension test as well as the results of measurements and optimization. In the current version of the program the module permits application of a number of database engines (among other standard MSAcces, dBASE IV and Paradox 7-8 for PC-based systems) and allows the implementation of material databases and procedures of automatic data verification. The next module (the solver) gives user the possibility of managing the working conditions of the simulation process. The inverse analysis can be turned off or on using this part of the system.

More details about presenting thermo-mechanical model, numerical methods and

In aim to allow easy working with the Gleeble® 3800 simulator a user friendly system called Def\_Semi\_Solid v.5.0 was developed in the Department of Computer Science and Modeling of the Faculty of Metals Engineering and Computer Science in Industry, AGH. The numerical part of the program was developed in FORTRAN/C++ language, which guaranties fast computation and the graphical interface was written using visual version of C++ language, taking advantage of its object oriented character. This approach has sufficient usability both in Windows and Linux based systems. The newest version of Def\_Semi\_Solid system is equipped with full automatic installation unit (Figure 2) and new graphical interface. It allows the computer aided testing of mechanical properties of steels at very high temperature using Gleeble® 3800 physical simulators to avoid problems which arise by traditional testing procedures. The first module allows the establishment of new projects or working with previously existing ones. The integral parts of each project are: input data for a specific compression/tension test as well as the results of measurements and optimization. In the current version of the program the module permits application of a number of database engines (among other standard MSAcces, dBASE IV and Paradox 7-8 for PC-based systems) and allows the implementation of material databases and procedures of automatic data verification. The next module (the solver) gives user the possibility of managing the working conditions of the simulation process. The inverse analysis can be turned off or on

Fig. 1. The density versus temperature curve for C45 grade steel.

techniques will be described in the next chapter of this book.

using this part of the system.

**3. Numerical systems – Def\_Semi\_Solid and JMatPro** 

Fig. 2. View of DSS program setup window (local version).

The last module (DSS/Post module) is dedicated to the visualisation of the numerical results and printing the final reports. In the current version the possibility of visualization was significant improved. The main are: shading options using OpenGL mode (2D and 3D) and possibility make a full contour map (2D and 3D) as shown in Figure 3.

Fig. 3. The Post-processor of the newest version of Def\_Semi\_Solid system (contour option).

The less visible but powerful heart of the system is of course the solver. The finite element code dedicated to the axial-symmetrical compression/tension tests has been developed. The solution is based on the thermo-mechanical approach with density changes described in the previous section. The second software used during theoretical work was commercial code JMatPro. This program calculates a wide range of materials properties for alloys and steels. Using JMatPro we can make calculations for stable and metastable phase equilibrium, solidification behaviour and properties, thermo-physical and physical properties, phase

Numerical Modelling of Steel Deformation at Extra-High Temperatures 263

Figure 6 shows the shape of the testing samples and locations of thermocouples used during

Material tests in the semi-solid state should be carried out in as isothermal conditions as possible due to the very high sensitivity of material rheology on small changes of temperature (Hojny et al., 2009). The basic reason for uneven temperature distribution inside samples on the Gleeble® simulator is their contact with cooper handles during

Fig. 7. The handle with short contact zone (sample - handle) used during experiments

section the example results of the melting and deformation procedure are presented.

The estimated liquidus and solidus temperature of the investigated steels are: 1495 oC and 1410 oC, respectively for C45, 1513 oC and 1464 oC, respectively for S355J2G3So. In the next

Thermal solution has crucial influence on simulation results, since the temperature has strong effect on remaining. The resistance heating processes cause non-uniform distribution

stage 1: the sample was prepared (e.g. mounting thermocouples, die selection),

Fig. 6. Samples used for the experiments. TC2, TC3 and TC4 thermocouples.

In all cases, experiments ran according to schedule:

experiments.

resistance heating (Figure 7).

(called "hot handle").

**4.1 Melting procedure** 

stage 2: melting procedure of the sample was realized, stage 3: at the end the deformation process was done.

transformations. JMatPro includes a Java based user interface, with calculation modules using C/C++, and will run under any Windows and under Linux system.

#### **4. Computer aided experimental procedure**

The steel grades subjected to series of experiments in Institute for Ferrous Metallurgy in Gliwice, Poland using Gleeble 3800® thermo-mechanical simulator (Figure 4) were the C45 (0.45% C) and S355J2G3So (0.11% C).

Fig. 4. The standard Gleeble equipment allowing deformation is semi-solid state.

The essential aim of the investigation was reconstruction of both temperature changes and strain evolution on specimen exposed to simultaneous deformation and solidification. The analysis of metal flow in subsequent regions of the sample deformation zone requires adequate methods. Classical techniques of interpretation of results of compression testing procedures fail due to significant barrelling of the sample which is inevitable at any temperature close to solidus level. The developed user friendly dedicated FEM solution (Figure 5) with variable density based on the hybrid model described in previous sections is the core of strain-stress curves calculation system.

Fig. 5. The Def\_Semi\_Solid system as a feedback unit with Gleeble 3800 simulator.

In all cases, experiments ran according to schedule:

262 Numerical Modelling

transformations. JMatPro includes a Java based user interface, with calculation modules

The steel grades subjected to series of experiments in Institute for Ferrous Metallurgy in Gliwice, Poland using Gleeble 3800® thermo-mechanical simulator (Figure 4) were the C45

Fig. 4. The standard Gleeble equipment allowing deformation is semi-solid state.

Fig. 5. The Def\_Semi\_Solid system as a feedback unit with Gleeble 3800 simulator.

The essential aim of the investigation was reconstruction of both temperature changes and strain evolution on specimen exposed to simultaneous deformation and solidification. The analysis of metal flow in subsequent regions of the sample deformation zone requires adequate methods. Classical techniques of interpretation of results of compression testing procedures fail due to significant barrelling of the sample which is inevitable at any temperature close to solidus level. The developed user friendly dedicated FEM solution (Figure 5) with variable density based on the hybrid model described in previous sections is

using C/C++, and will run under any Windows and under Linux system.

**4. Computer aided experimental procedure** 

the core of strain-stress curves calculation system.

(0.45% C) and S355J2G3So (0.11% C).

stage 1: the sample was prepared (e.g. mounting thermocouples, die selection), stage 2: melting procedure of the sample was realized, stage 3: at the end the deformation process was done.

Figure 6 shows the shape of the testing samples and locations of thermocouples used during experiments.

Fig. 6. Samples used for the experiments. TC2, TC3 and TC4 thermocouples.

Material tests in the semi-solid state should be carried out in as isothermal conditions as possible due to the very high sensitivity of material rheology on small changes of temperature (Hojny et al., 2009). The basic reason for uneven temperature distribution inside samples on the Gleeble® simulator is their contact with cooper handles during resistance heating (Figure 7).

Fig. 7. The handle with short contact zone (sample - handle) used during experiments (called "hot handle").

The estimated liquidus and solidus temperature of the investigated steels are: 1495 oC and 1410 oC, respectively for C45, 1513 oC and 1464 oC, respectively for S355J2G3So. In the next section the example results of the melting and deformation procedure are presented.

#### **4.1 Melting procedure**

Thermal solution has crucial influence on simulation results, since the temperature has strong effect on remaining. The resistance heating processes cause non-uniform distribution

Numerical Modelling of Steel Deformation at Extra-High Temperatures 265

<sup>1</sup>

<sup>2</sup> ,,, ,, *F Q T Q r z T T r z T dT c m*

In Figure 10 the comparison between experimental and theoretical temperature versus time curves is presented for steering thermocouple (mounting locations of thermocouples shown

Fig. 10. Comparison between the experimental and theoretical time-temperature curves

Fig. 11. The temperature change versus time for hot handle (final stage of physical

In the final stage of physical simulation for different holding time, the temperature difference between core (TC3 thermocouple) of the sample and the surface (TC4 thermocouple) was analyzed. In all cases the core temperature was higher than surface temperature, for example, difference between core of the sample and surface was about 40 oC for test at 1380 oC (Figure 11).

during initial heating and final compression at 1400°C.

simulation right before deformation at 1380 oC ).

(8)

0

where 

in Figure 6),

is the time variable.

of temperature inside heated materials especially in longitudinal section of the sample. In the case of the semi-solid steels, such distribution gives significant differences in the microstructure and rheology. The thermo-physical properties of this steel, necessary in calculations, were determined using *JMatPro* software. This software determines this properties on the basis of the chemical composition. The example main properties of C45 grade steel used in calculations are presented in Figure 8 and Figure 9.

Fig. 8. Specific heat versus temperature for the investigated steel (C45).

Fig. 9. Thermal conductivity versus temperature for the investigated steel (C45).

In case of each physical and computer simulation samples were heated to 1430 oC and after holding at constant temperature the sample was cooled to nominal deformation temperature. The heat generated (*Q*) due to direct current flow was calculated using inverse approach. The objective function (*F*) was defined as a norm of discrepancies between calculated (*Tc*) and measured (*Tm*) temperatures (only for indication steering thermocouples TC4) according to the following equation:

$$F(Q) = \int\_{z\_0}^{z} \left[ \left( T\_c \left( Q, r, z, T \right) - T\_m \left( r, z, T \right) \right)^2 \right] dT \tag{8}$$

where is the time variable.

264 Numerical Modelling

of temperature inside heated materials especially in longitudinal section of the sample. In the case of the semi-solid steels, such distribution gives significant differences in the microstructure and rheology. The thermo-physical properties of this steel, necessary in calculations, were determined using *JMatPro* software. This software determines this properties on the basis of the chemical composition. The example main properties of C45

grade steel used in calculations are presented in Figure 8 and Figure 9.

Fig. 8. Specific heat versus temperature for the investigated steel (C45).

Fig. 9. Thermal conductivity versus temperature for the investigated steel (C45).

TC4) according to the following equation:

In case of each physical and computer simulation samples were heated to 1430 oC and after holding at constant temperature the sample was cooled to nominal deformation temperature. The heat generated (*Q*) due to direct current flow was calculated using inverse approach. The objective function (*F*) was defined as a norm of discrepancies between calculated (*Tc*) and measured (*Tm*) temperatures (only for indication steering thermocouples In Figure 10 the comparison between experimental and theoretical temperature versus time curves is presented for steering thermocouple (mounting locations of thermocouples shown in Figure 6),

Fig. 10. Comparison between the experimental and theoretical time-temperature curves during initial heating and final compression at 1400°C.

In the final stage of physical simulation for different holding time, the temperature difference between core (TC3 thermocouple) of the sample and the surface (TC4 thermocouple) was analyzed. In all cases the core temperature was higher than surface temperature, for example, difference between core of the sample and surface was about 40 oC for test at 1380 oC (Figure 11).

Fig. 11. The temperature change versus time for hot handle (final stage of physical simulation right before deformation at 1380 oC ).

Numerical Modelling of Steel Deformation at Extra-High Temperatures 267

(calculated core temperature equal 1417oC, measured core temperature equal 1420oC.) Finally, the micro and macrostructure of the tested samples was investigated. Figure 13 show example microstructure of the tested samples right before deformation for middle and

Fig. 13. Microstructure of the middle/border of the sample right before deformation.

simulation of resistance heating of samples using Gleeble® 3800 physical simulator.

Microstructure of the cooled samples consist of pearlite (the darkest phase), bainit (grey phase mainly near the borders of grains) and the bright ferrite (Figure 13). It is result of such phase composition the wide zone of melting and almost twice smaller speed of cooling in the case of samples warmed in "hot" handles. Figure 14 show example macrostructure in the cross-sections of the tested samples right before deformation and calculated core temperature. One can observe that for analyzed temperatures liquid phase particle exist in the central part of the sample. The comparison between experimental results and numerical show that mathematical model of resistance heating right reflect back the physical

Variant with hot handle. Magnification: 400x.

border of the heating zone.

The numerical simulation confirmed results obtained during experimental parts. In the Figure 12 temperature distributions in the cross section of the sample tested at temperature 1380 oC are presented for 3 and 6 seconds of heating and final distribution right before deformation.

Fig. 12. Distribution of temperature in the cross section of the sample tested at temperature 1380 oC after time heating a) 3 seconds, b) 6 seconds, c) right before deformation.

The one can observe, major temperature gradient between contact surface die-sample. The difference between experimental and theoretical core temperatures for hot handles was 3oC

The numerical simulation confirmed results obtained during experimental parts. In the Figure 12 temperature distributions in the cross section of the sample tested at temperature 1380 oC are presented for 3 and 6 seconds of heating and final distribution right before deformation.

Fig. 12. Distribution of temperature in the cross section of the sample tested at temperature

The one can observe, major temperature gradient between contact surface die-sample. The difference between experimental and theoretical core temperatures for hot handles was 3oC

1380 oC after time heating a) 3 seconds, b) 6 seconds, c) right before deformation.

(calculated core temperature equal 1417oC, measured core temperature equal 1420oC.) Finally, the micro and macrostructure of the tested samples was investigated. Figure 13 show example microstructure of the tested samples right before deformation for middle and border of the heating zone.

Fig. 13. Microstructure of the middle/border of the sample right before deformation. Variant with hot handle. Magnification: 400x.

Microstructure of the cooled samples consist of pearlite (the darkest phase), bainit (grey phase mainly near the borders of grains) and the bright ferrite (Figure 13). It is result of such phase composition the wide zone of melting and almost twice smaller speed of cooling in the case of samples warmed in "hot" handles. Figure 14 show example macrostructure in the cross-sections of the tested samples right before deformation and calculated core temperature. One can observe that for analyzed temperatures liquid phase particle exist in the central part of the sample. The comparison between experimental results and numerical show that mathematical model of resistance heating right reflect back the physical simulation of resistance heating of samples using Gleeble® 3800 physical simulator.

Numerical Modelling of Steel Deformation at Extra-High Temperatures 269

*<sup>p</sup> A BT* 

easy to construct isothermal experiments for temperatures higher than 1300oC. Several serious experimental problems arise. First of all, keeping so high temperature constant during the whole experimental procedure is extremely difficult. There are also severe difficulties concerning interpretation of the measurement results. The significant inhomogeneity in the strain distribution in the deformation zone and distortion of the central part of the sample lead to poor accuracy of the stress field calculated using traditional methods, which are good for lower temperatures. The only possibility to have good coefficients of strain-stress formula is the inverse method (Glowacki & Hojny, 2009). The long calculation time, which is usual by this kind of analysis requires sometimes parallel computation. The application such a method is

The objective function was defined as a norm of discrepancies between calculated (*Fc*) and measured (*Fm*) loads in a number of subsequent stages of the compression according to the

1

The theoretical forces *Fc* were calculated with the help of sophisticated FEM solver facilitating accurate computation of strain, stress and temperature fields for materials with variable density. The example comparison between the calculated and measured loads are

*i x FF* 

*<sup>n</sup> c m i i*

presented in Figure 15-17, showing quite good agreement between both loads.

Fig. 15. Comparison between measured and predicted loads at temperature 1350oC.

(9)



2

**<sup>φ</sup>** (10)

 

exp( ) *n m*

where *A, B, n, m* are material constant, *T* - temperature,

considered for the future application.

following equation:

Fig. 14. Macrostructure of the middle of the sample right before deformation. Variant with hot handle. Magnification: 10x.

#### **4.2 Deformation procedure**

In the next stage of the experimental part the compression tests were done. During experiments die displacement, force and temperature changes in the heating zone are recorded. Parallel, the computer simulations were realized in order to obtain optimal value parameters of process. The strain-stress curves were described by following equation:

Fig. 14. Macrostructure of the middle of the sample right before deformation. Variant with

In the next stage of the experimental part the compression tests were done. During experiments die displacement, force and temperature changes in the heating zone are recorded. Parallel, the computer simulations were realized in order to obtain optimal value parameters of process. The strain-stress curves were described by following equation:

hot handle. Magnification: 10x.

**4.2 Deformation procedure** 

$$
\sigma\_p = \Lambda \boldsymbol{\varepsilon}^n \boldsymbol{\varepsilon}^m \exp(-BT) \tag{9}
$$

where *A, B, n, m* are material constant, *T* - temperature, - strain and - strain rate. It is not easy to construct isothermal experiments for temperatures higher than 1300oC. Several serious experimental problems arise. First of all, keeping so high temperature constant during the whole experimental procedure is extremely difficult. There are also severe difficulties concerning interpretation of the measurement results. The significant inhomogeneity in the strain distribution in the deformation zone and distortion of the central part of the sample lead to poor accuracy of the stress field calculated using traditional methods, which are good for lower temperatures. The only possibility to have good coefficients of strain-stress formula is the inverse method (Glowacki & Hojny, 2009). The long calculation time, which is usual by this kind of analysis requires sometimes parallel computation. The application such a method is considered for the future application.

The objective function was defined as a norm of discrepancies between calculated (*Fc*) and measured (*Fm*) loads in a number of subsequent stages of the compression according to the following equation:

$$\mathfrak{sp}\left(\mathfrak{x}\right) = \sum\_{i=1}^{n} \left[ F\_i^c - F\_i^m \right]^2 \tag{10}$$

The theoretical forces *Fc* were calculated with the help of sophisticated FEM solver facilitating accurate computation of strain, stress and temperature fields for materials with variable density. The example comparison between the calculated and measured loads are presented in Figure 15-17, showing quite good agreement between both loads.

Fig. 15. Comparison between measured and predicted loads at temperature 1350oC.

Numerical Modelling of Steel Deformation at Extra-High Temperatures 271

Fig. 18. Stress-strain curves at several strain rate levels for temperature 1400oC.

Using previously developed curves, example simulations of compression of cylindrical samples with mushy zone have been performed. The results of the tests demonstrate the possibilities of the computer system. For all series of tests the simulations were done using short contact zone between the sample and simulator jaws. The deformation zone had the initial height of 62.5 mm. The radius of the sample was 5 mm. An example specimen was melted at 1430oC, and then after cooling temperature deformed at demanding temperature. The first variant at 1430oC, the second variant at 1425oC. During the tests each sample was subjected to 10 mm reduction of height. The final temperature for variant no. 1 and no. 2 is shown in Figure 19. The temperature distribution for the both variants is similar. Taking into account the value of core temperature for variant no. 2 one can state the existence of the mushy zone in the sample centre. The analysis of the effective strain fields for specimens no. 1 and no. 2, which are presented in Figure 20 show that influence of density variations on the metal flow scheme is not very significant, although small differences are clearly visible. The analysis of the strain shows maximal values of strain in the central region of the sample. In Figure 21 mean stress distribution is presented. The initial temperature distribution has great influence on the stress field in the deformation zone. The inhomogeneity of the strain field also leads to stress generation. The analysis of the mean stress (Figure 21) shows the stress concentration near the upper surface and in the centre of the sample. In accordance with the earlier conjectures for sample no. 1 the stress level is significantly higher than for sample no. 2. The material properties which have been used in the model are not very accurate because of difficulties in experiment. The existing inhomogeneity of deformation causes problems concerning the calculation of right stress values. The good analysis of the results of experiments needs computer programs to simulate plastic deformation of steel samples. The created model, which takes into account

**4.3 Verification of the computer system – Example results** 

the variable density, can be helpful for the discussed purposes.

Fig. 16. Comparison between measured and predicted loads at temperature 1400oC.

Fig. 17. Comparison between measured and predicted loads at temperature 1425oC.

The coefficients obtained during inverse analysis allow the construction of stress-strain curves, which are presented in Figure 18.

Fig. 16. Comparison between measured and predicted loads at temperature 1400oC.

Fig. 17. Comparison between measured and predicted loads at temperature 1425oC.

curves, which are presented in Figure 18.

The coefficients obtained during inverse analysis allow the construction of stress-strain

Fig. 18. Stress-strain curves at several strain rate levels for temperature 1400oC.

#### **4.3 Verification of the computer system – Example results**

Using previously developed curves, example simulations of compression of cylindrical samples with mushy zone have been performed. The results of the tests demonstrate the possibilities of the computer system. For all series of tests the simulations were done using short contact zone between the sample and simulator jaws. The deformation zone had the initial height of 62.5 mm. The radius of the sample was 5 mm. An example specimen was melted at 1430oC, and then after cooling temperature deformed at demanding temperature. The first variant at 1430oC, the second variant at 1425oC. During the tests each sample was subjected to 10 mm reduction of height. The final temperature for variant no. 1 and no. 2 is shown in Figure 19. The temperature distribution for the both variants is similar. Taking into account the value of core temperature for variant no. 2 one can state the existence of the mushy zone in the sample centre. The analysis of the effective strain fields for specimens no. 1 and no. 2, which are presented in Figure 20 show that influence of density variations on the metal flow scheme is not very significant, although small differences are clearly visible. The analysis of the strain shows maximal values of strain in the central region of the sample. In Figure 21 mean stress distribution is presented. The initial temperature distribution has great influence on the stress field in the deformation zone. The inhomogeneity of the strain field also leads to stress generation. The analysis of the mean stress (Figure 21) shows the stress concentration near the upper surface and in the centre of the sample. In accordance with the earlier conjectures for sample no. 1 the stress level is significantly higher than for sample no. 2. The material properties which have been used in the model are not very accurate because of difficulties in experiment. The existing inhomogeneity of deformation causes problems concerning the calculation of right stress values. The good analysis of the results of experiments needs computer programs to simulate plastic deformation of steel samples. The created model, which takes into account the variable density, can be helpful for the discussed purposes.

Numerical Modelling of Steel Deformation at Extra-High Temperatures 273

Fig. 21. Example results of computer simulation of deformation at 1350°C (left) and 1425°C

The calculated and experimental (Figure 22) shapes of the sample allow a rough verification of rheological model. For verification of the computer system two comparative criteria were


Figures 23 and 24 show example application of the 1st and 2nd criterion, respectively. The


figures confirm quite good agreement between theoretical and experimental results.

Fig. 22. Final shape of the sample after deformation at 1350oC, 1400oC and 1425oC.

(right): final mean stress distribution in the cross-section of the sample.

used:

subjected to the deformation.

Fig. 19. Example results of computer simulation of deformation at 1350°C (left) and 1425°C (right): final temperature distribution in the cross-section of the sample.

Fig. 20. Example results of computer simulation of deformation at 1350°C (left) and 1425°C (right): final strain distribution in the cross-section of the sample.

Fig. 19. Example results of computer simulation of deformation at 1350°C (left) and 1425°C

Fig. 20. Example results of computer simulation of deformation at 1350°C (left) and 1425°C

(right): final strain distribution in the cross-section of the sample.

(right): final temperature distribution in the cross-section of the sample.

Fig. 21. Example results of computer simulation of deformation at 1350°C (left) and 1425°C (right): final mean stress distribution in the cross-section of the sample.

The calculated and experimental (Figure 22) shapes of the sample allow a rough verification of rheological model. For verification of the computer system two comparative criteria were used:


Figures 23 and 24 show example application of the 1st and 2nd criterion, respectively. The figures confirm quite good agreement between theoretical and experimental results.

Fig. 22. Final shape of the sample after deformation at 1350oC, 1400oC and 1425oC.

Numerical Modelling of Steel Deformation at Extra-High Temperatures 275

Computer aided testing of steel samples deformation at coexistence liquid and solid phase requires resolving a number of problems. One of them is the difficulty in determination of material thermal and mechanical properties, such as: coefficients of heat transfer and other thermal properties, diagrams of density changes, which is dependent on temperature, etc. The main problem is the interpretation of compression tests results leading to strain – stress curves. The presented model with incompressibility condition in analytical form allows the simulation of the deformation of material with mushy zone avoiding volume loss, which cause problems with density. The presented Def\_Semi\_Solid program is a unique tool, which can be very helpful and may enable the right interpretation of results of very high temperature tests. The paper has shown its predictive ability regarding: temperature, shape and size of the deformation zone. The focus of attention were mechanical properties of investigated steel and specific character of theoretical model applied to the analysis. One can observe that the compression tests interpretation was possible only due to application of

The work has been supported by the Ministry of Science and Higher Education Grant N

Glowacki M. (2005). The mathematical modelling of thermo-mechanical processing of steel

Glowacki M.; Hojny M. (2009). Inverse analysis applied for determination of strain-stress

Hojny M.; Glowacki M. (2008). Computer modelling of deformation of steel samples with

Hojny M.; Glowacki M. (2009). The physical and computer modeling of plastic deformation

Hojny M.; Glowacki M.; Malinowski Z. (2009). Computer aided methodology of strain-stress

Hojny M.; Glowacki M. (2011). Modeling of Strain-Stress Relationship for Carbon Steel

*Materials and Processes*, Vol. 28, No. 4, pp. 245–252, ISSN: 0334-6455

*Engineering*, Vol. 17, No.2,pp. 159–174, ISSN: 1741-5977

Vol. 131, No. 4, pp. 041003-1–041003-7, ISSN: 0094-4289

during multi-pass shape rolling, *Journal of Materials Processing Technology*, Vol. 168,

curves for steel deformed in semi-solid state, *Inverse Problems in Science and* 

mushy zone, *Steel Research International*, Vol. 79, No. 11, pp. 868-874,ISSN: 1611-

of low carbon steel in semisolid state, *Journal of Engineering Materials and Technology*,

curve construction for steels deformed at extra high temperature, *High Temperature* 

Deformed at Temperature Exceeding Hot Rolling Range, *Journal of Engineering Materials and Technology*, Vol. 133, No. 2, pp. 021008-1–021008-7, ISSN: 0094-4289 Hufschmidt M.; Modigell M.; Petera L. (2004). Two-Phase Simulations as a Development

Tool for Thixoforming Processes. *Steel Research International*, Vol. 75, No.3, pp. 513–

right model and implementation of the inverse analysis.

No.2, pp. 336–343, ISSN: 0924-0136

N508 410637 and partially N N508 585539.

**5. Conclusions** 

**6. Acknowledgments** 

**7. References** 

3683

518, ISSN: 1611-3683

Fig. 23. The comparison of the measured and calculated maximal diameters of the sample – experiments between 1350°C and 1425°C.

Fig. 24. The comparison between the measured and calculated length of zone which was not subjected to the deformation – Experiments between 1350°C and 1425°C.

#### **5. Conclusions**

274 Numerical Modelling

Fig. 23. The comparison of the measured and calculated maximal diameters of the sample –

Fig. 24. The comparison between the measured and calculated length of zone which was not

subjected to the deformation – Experiments between 1350°C and 1425°C.

experiments between 1350°C and 1425°C.

Computer aided testing of steel samples deformation at coexistence liquid and solid phase requires resolving a number of problems. One of them is the difficulty in determination of material thermal and mechanical properties, such as: coefficients of heat transfer and other thermal properties, diagrams of density changes, which is dependent on temperature, etc. The main problem is the interpretation of compression tests results leading to strain – stress curves. The presented model with incompressibility condition in analytical form allows the simulation of the deformation of material with mushy zone avoiding volume loss, which cause problems with density. The presented Def\_Semi\_Solid program is a unique tool, which can be very helpful and may enable the right interpretation of results of very high temperature tests. The paper has shown its predictive ability regarding: temperature, shape and size of the deformation zone. The focus of attention were mechanical properties of investigated steel and specific character of theoretical model applied to the analysis. One can observe that the compression tests interpretation was possible only due to application of right model and implementation of the inverse analysis.

#### **6. Acknowledgments**

The work has been supported by the Ministry of Science and Higher Education Grant N N508 410637 and partially N N508 585539.

#### **7. References**


**13** 

*Poland* 

Miroslaw Glowacki

*AGH University of Science and Technology* 

**Inverse Analysis Applied to Mushy Steel** 

Integrated casting and rolling technologies are most recent and very efficient way of hot strip production. More and more companies all over the world are able to manage such processes. The mentioned technologies ensure huge reduction of rolling costs, very high product quality and low investment costs. Computer simulation is of vital importance to the development of "know how" theory for these processes. The lack of publications concerning mechanical properties and behaviour of steels simultaneously subjected to both plastic deformation and solidification was the inspiration for the investigation. This also necessitated the development of an appropriate mathematical model of mushy-steel deformation. The contribution summarizes the results of the author's recent theoretical research concerning the computer simulation of mushy steel published in recent years in well-known journals and book chapters [Glowacki, 2006; Glowacki at al., 2010; Glowacki &

Hojny, 2006, 2009; Hojny & Glowacki, 2008, 2009a, 2009b, 2011; Hojny at al., 2009].

As an example of a company providing the integrated casting and rolling technologies one can mention the plant located in Cremona, Italy which develops the new methods of steel strip manufacturing. They are called Inline Strip Production (ISP) and Arvedi Steel Technology (AST) processes and are characterized by very high temperature allowed at the mill entry. The instant rolling of slabs which leave the casting machine allows for the utilization of the heat stored in the strips during inline casting. Both the mentioned technologies ensure huge reduction of rolling forces and their details are usually classified. The development of "know how" theory for the semi-solid steel rolling technology requires numerical modelling. The development of appropriate mathematical models is limited by the lack of thermal and mechanical properties concerning mushy steels deformation in temperature range which is close to solidus line. The work presented in the current contribution is an attempt to cover the gap providing a proposition of a hybrid numericalanalytical model of semi-solid steel deformation. The mathematical modelling of steel deformation in semi-solid state, as well as experimental work in this field, are innovative topics regarding the very high temperature range deformation processes. Tracing the related papers published in the past 10 years one can find many dealings with experimental results for non-ferrous metals tests (Kang & Yoon, 1997; Koc at al., 1996; Kopp at al., 2003; Sang-

**1. Introduction** 

**Rheological Properties Testing Using** 

**Hybrid Numerical-Analytical Model** 


### **Inverse Analysis Applied to Mushy Steel Rheological Properties Testing Using Hybrid Numerical-Analytical Model**

Miroslaw Glowacki *AGH University of Science and Technology Poland* 

#### **1. Introduction**

276 Numerical Modelling

Jing Y.L.; Sumio S.; Jun Y. (2005). Microstructural evolution and flow stress of semi-solid

Jin S. D.; Hwan O.K. (2002). Phase-field modelling of the thermo-mechanical properties of carbon steels. *Acta Materialia*, Vol. 50, No. 9, pp. 2259-6454, ISSN: 1359-6454 Kang, C.G.; Yoon, J.H. (1997). A finite-element analysis on the upsetting process of semi-

Koc, M.; Vazquez V.; Witulski T.; Altan T. (1996). Application of the finite element method

Kopp R.; Choi J.; Neudenberger D. (2003). Simple compression test and simulation of an Sn–

Modigell M.; Pape L.; Hufschmidt M. (2004). The rheological behaviour of metallic

396-406, ISSN: 0924-0136

pp. 76-84, ISSN: 0924-0136

135, No. 3, pp. 317-323, ISSN: 0924-0136

0924-0136

type 304 stainless steel. *Journal of Materials Processing Technology*, Vol. 161, No. 3, pp.

solid aluminium material. *Journal of Materials Processing Technology*, Vol. 66, No.3,

to predict material flow and defects in the semi-solid forging of A356 aluminium alloys. *Journal of Materials Processing Technology*, Vol. 59, No. 2, pp. 106-112, ISSN:

15% Pb alloy in the semi-solid state. *Journal of Materials Processing Technology*, Vol.

suspensions. *Steel Research International*, Vol. 75, No.3, pp. 506–512, ISSN: 1611-3683

Integrated casting and rolling technologies are most recent and very efficient way of hot strip production. More and more companies all over the world are able to manage such processes. The mentioned technologies ensure huge reduction of rolling costs, very high product quality and low investment costs. Computer simulation is of vital importance to the development of "know how" theory for these processes. The lack of publications concerning mechanical properties and behaviour of steels simultaneously subjected to both plastic deformation and solidification was the inspiration for the investigation. This also necessitated the development of an appropriate mathematical model of mushy-steel deformation. The contribution summarizes the results of the author's recent theoretical research concerning the computer simulation of mushy steel published in recent years in well-known journals and book chapters [Glowacki, 2006; Glowacki at al., 2010; Glowacki & Hojny, 2006, 2009; Hojny & Glowacki, 2008, 2009a, 2009b, 2011; Hojny at al., 2009].

As an example of a company providing the integrated casting and rolling technologies one can mention the plant located in Cremona, Italy which develops the new methods of steel strip manufacturing. They are called Inline Strip Production (ISP) and Arvedi Steel Technology (AST) processes and are characterized by very high temperature allowed at the mill entry. The instant rolling of slabs which leave the casting machine allows for the utilization of the heat stored in the strips during inline casting. Both the mentioned technologies ensure huge reduction of rolling forces and their details are usually classified.

The development of "know how" theory for the semi-solid steel rolling technology requires numerical modelling. The development of appropriate mathematical models is limited by the lack of thermal and mechanical properties concerning mushy steels deformation in temperature range which is close to solidus line. The work presented in the current contribution is an attempt to cover the gap providing a proposition of a hybrid numericalanalytical model of semi-solid steel deformation. The mathematical modelling of steel deformation in semi-solid state, as well as experimental work in this field, are innovative topics regarding the very high temperature range deformation processes. Tracing the related papers published in the past 10 years one can find many dealings with experimental results for non-ferrous metals tests (Kang & Yoon, 1997; Koc at al., 1996; Kopp at al., 2003; Sang-

Inverse Analysis Applied to Mushy Steel

allowed at any stress tensor configuration.

some problems with the interpretation of tests results.

modelling of plastic behaviour of semi-solid steels.

Rheological Properties Testing Using Hybrid Numerical-Analytical Model 279

behaviour and limit plastic deformation. The Nil Strength Temperature (NST) is the temperature level at which material strength drops to zero while the steel is being heated above the solidus temperature. Another temperature associated with NST is the Strength Recovery Temperature (SRT). At this temperature the cooled material regains strength greater than 0.5 N/mm2. Nil Ductility Temperature (NDT) represents the temperature at which the heated steel loses its ductility. The Ductility Recovery Temperature (DRT) is the temperature at which the ductility of the material (characterised by reduction of area) reaches 5% while it is being cooled. Over this temperature the plastic deformation is not

Significant changes of density and lack of data regarding material's thermal and mechanical properties are vital problems of the modelling. They have great influence on steel rheology and heat transfer. An issue of great importance is the lack of strain-stress relationships, which in the temperature range above 1400 °C strongly depend on the density and are very temperature sensitive. It is not easy to run isothermal tests that could be the source of the computation of yield stress function parameters for such high temperatures. There are also

Density is very important for plastic behaviour of mushy steel plates. It varies with temperature and depends on the cooling rate. The solidification process causes non-uniform density distribution in the controlled volume resulting in non-uniform deformation and heat conduction. There are three main factors causing density changes: solid phase formation, thermal shrinkage and movement of liquid particles inside the solid skeleton.

The contribution sheds some light on the physical problems but it focuses on the axialsymmetrical computer model, which ensures the right simulation of mushy steel samples deformation reflecting the physical requirements. The presented model fills the gap in

Testing of steels at temperature higher than 1400 °C is difficult due to deformation instability and risk of sample damage during experiment. Such experiments do not assure the strain homogeneity and cannot be interpreted using traditional methods. Appropriate interpretation of the results is possible only with the help of a computer aided engineering system. The contribution reports a new model underlying such a system developed by the author's team. Together with GLEEBLE physical simulator equipped with high temperature

The numerical solver is the less visible yet very powerful kernel of the system. It is based on a thermal-mechanical model with variable density. The mechanical part of the model is a hybrid variational solution with analytical mass conservation condition constraining the velocity field components. The accuracy of the proposed solution is very good due to negligible volume loss guaranteed by the analytical form of the mass conservation condition. This is important for materials with variable density and is not captured by classical solutions. Analytical condition eliminates problems with unintentional specimen volume changes caused by application of numerical methods. The existing, physical changes of steel density in the mushy zone have influence on real variations of controlled volume.

The density plays an important role in both mechanical and thermal solutions.

**3. Hybrid numerical-analytical model of mushy steel deformation** 

module the code allows for investigation of properties of semi-solid steel.

Yong at al., 2001; Zhao at al., 2006). The first results regarding steel deformation at extra high temperature were presented during last few years (Li, 2005; Seol, 1999, 2002). Most of the problems concerning semi-solid steel testing are caused by the very high level of steel liquidus and solidus temperatures in comparison with non-ferrous metals. The deformation tests for non-ferrous metals are much easier. The rising abilities of thermo-mechanical simulators enable investigation of steel samples and as a result both computer simulation and the development of new, very high temperature rolling technologies like Arvedi ISP and AST processes. The lack of mathematical models describing the steel behaviour in the last phase of solidification with simultaneous plastic deformation was the inspiration of the investigation described in the proposed book chapter.

The main goal of the chapter is to present problems of theoretical work leading to the development of a methodology of very high temperature testing of steel samples while their central parts are still mushy. In such conditions the deformation of samples is strongly inhomogeneous and all the well-known methods of yield stress curve examination fail due to significant barrelling of the sample. Although the investigation concerned both physical tests and dedicated simulation system, the author sacrifices the contribution to the hybrid model which is the heart of the system. With the help of inverse analysis it allows for the right interpretation of deformation tests providing data regarding the mushy steel rheological properties.

#### **2. Physical basis and characteristic features of steel deformed at very high temperature**

The rolling equipment for the ISP process allows for reduction of initial mould strip thickness from 74 mm to 55 mm during liquid core reduction process. The region of maximum strip temperature for a high reduction mill is located in the strip centre and varies from 1220 °C to 1375 °C depending on the casting speed. The main benefits of the technology are: inverse temperature gradient, good product quality, very low level of heating energy consumption, up to 20 times lower water consumption in comparison to traditional rolling, low level of installed mill power, compact rolling equipment layout, no need for tunnel furnace and very low investment costs. The AST technology is a result of further development of ISP into a real endless process and the benefits of its application are even greater. The whole reduction process is running in one rolling mill consisting of 5 or 7 stands, which can reduce the strip thickness from 55÷70 mm to 0.8 mm. The maximum temperature of the strip occurs in central region of its cross-section and varies from 1340 °C to 1420 °C depending on the casting speed. This suggests that the central region of the strand subjected to the rolling is still mushy.

The main benefits of the new very high temperature technologies are significantly lower rolling forces and very favourable temperature field inside the steel plate. However, certain problems arise which are specific for this kind of metal treatment. The central parts of slabs are mushy and the solidification is not yet finished while the deformation is in progress. This results in changes in material density and occurrence of characteristic temperatures having great influence on the plastic behaviour of the material (Senk, 2000; Suzuki, 1988). The nil strength temperature (NST), strength recovery temperature (SRT), nil ductility temperature (NDT) and ductility recovery temperature (DRT) have effect on steel plastic

Yong at al., 2001; Zhao at al., 2006). The first results regarding steel deformation at extra high temperature were presented during last few years (Li, 2005; Seol, 1999, 2002). Most of the problems concerning semi-solid steel testing are caused by the very high level of steel liquidus and solidus temperatures in comparison with non-ferrous metals. The deformation tests for non-ferrous metals are much easier. The rising abilities of thermo-mechanical simulators enable investigation of steel samples and as a result both computer simulation and the development of new, very high temperature rolling technologies like Arvedi ISP and AST processes. The lack of mathematical models describing the steel behaviour in the last phase of solidification with simultaneous plastic deformation was the inspiration of the

The main goal of the chapter is to present problems of theoretical work leading to the development of a methodology of very high temperature testing of steel samples while their central parts are still mushy. In such conditions the deformation of samples is strongly inhomogeneous and all the well-known methods of yield stress curve examination fail due to significant barrelling of the sample. Although the investigation concerned both physical tests and dedicated simulation system, the author sacrifices the contribution to the hybrid model which is the heart of the system. With the help of inverse analysis it allows for the right interpretation of deformation tests providing data regarding the mushy steel

**2. Physical basis and characteristic features of steel deformed at very high** 

The rolling equipment for the ISP process allows for reduction of initial mould strip thickness from 74 mm to 55 mm during liquid core reduction process. The region of maximum strip temperature for a high reduction mill is located in the strip centre and varies from 1220 °C to 1375 °C depending on the casting speed. The main benefits of the technology are: inverse temperature gradient, good product quality, very low level of heating energy consumption, up to 20 times lower water consumption in comparison to traditional rolling, low level of installed mill power, compact rolling equipment layout, no need for tunnel furnace and very low investment costs. The AST technology is a result of further development of ISP into a real endless process and the benefits of its application are even greater. The whole reduction process is running in one rolling mill consisting of 5 or 7 stands, which can reduce the strip thickness from 55÷70 mm to 0.8 mm. The maximum temperature of the strip occurs in central region of its cross-section and varies from 1340 °C to 1420 °C depending on the casting speed. This suggests that the central region of the strand

The main benefits of the new very high temperature technologies are significantly lower rolling forces and very favourable temperature field inside the steel plate. However, certain problems arise which are specific for this kind of metal treatment. The central parts of slabs are mushy and the solidification is not yet finished while the deformation is in progress. This results in changes in material density and occurrence of characteristic temperatures having great influence on the plastic behaviour of the material (Senk, 2000; Suzuki, 1988). The nil strength temperature (NST), strength recovery temperature (SRT), nil ductility temperature (NDT) and ductility recovery temperature (DRT) have effect on steel plastic

investigation described in the proposed book chapter.

rheological properties.

subjected to the rolling is still mushy.

**temperature** 

behaviour and limit plastic deformation. The Nil Strength Temperature (NST) is the temperature level at which material strength drops to zero while the steel is being heated above the solidus temperature. Another temperature associated with NST is the Strength Recovery Temperature (SRT). At this temperature the cooled material regains strength greater than 0.5 N/mm2. Nil Ductility Temperature (NDT) represents the temperature at which the heated steel loses its ductility. The Ductility Recovery Temperature (DRT) is the temperature at which the ductility of the material (characterised by reduction of area) reaches 5% while it is being cooled. Over this temperature the plastic deformation is not allowed at any stress tensor configuration.

Significant changes of density and lack of data regarding material's thermal and mechanical properties are vital problems of the modelling. They have great influence on steel rheology and heat transfer. An issue of great importance is the lack of strain-stress relationships, which in the temperature range above 1400 °C strongly depend on the density and are very temperature sensitive. It is not easy to run isothermal tests that could be the source of the computation of yield stress function parameters for such high temperatures. There are also some problems with the interpretation of tests results.

Density is very important for plastic behaviour of mushy steel plates. It varies with temperature and depends on the cooling rate. The solidification process causes non-uniform density distribution in the controlled volume resulting in non-uniform deformation and heat conduction. There are three main factors causing density changes: solid phase formation, thermal shrinkage and movement of liquid particles inside the solid skeleton. The density plays an important role in both mechanical and thermal solutions.

The contribution sheds some light on the physical problems but it focuses on the axialsymmetrical computer model, which ensures the right simulation of mushy steel samples deformation reflecting the physical requirements. The presented model fills the gap in modelling of plastic behaviour of semi-solid steels.

### **3. Hybrid numerical-analytical model of mushy steel deformation**

Testing of steels at temperature higher than 1400 °C is difficult due to deformation instability and risk of sample damage during experiment. Such experiments do not assure the strain homogeneity and cannot be interpreted using traditional methods. Appropriate interpretation of the results is possible only with the help of a computer aided engineering system. The contribution reports a new model underlying such a system developed by the author's team. Together with GLEEBLE physical simulator equipped with high temperature module the code allows for investigation of properties of semi-solid steel.

The numerical solver is the less visible yet very powerful kernel of the system. It is based on a thermal-mechanical model with variable density. The mechanical part of the model is a hybrid variational solution with analytical mass conservation condition constraining the velocity field components. The accuracy of the proposed solution is very good due to negligible volume loss guaranteed by the analytical form of the mass conservation condition. This is important for materials with variable density and is not captured by classical solutions. Analytical condition eliminates problems with unintentional specimen volume changes caused by application of numerical methods. The existing, physical changes of steel density in the mushy zone have influence on real variations of controlled volume.

Inverse Analysis Applied to Mushy Steel

where ��� is the Kronecker delta.

1 � ∂ ∂� ����

Rheological Properties Testing Using Hybrid Numerical-Analytical Model 281

������ 0 0 0 ��� 0

Furthermore for a thermally isotropic material ��� ������ ������ � � �� and tensor of the heat

The temperature of samples compressed in axially-symmetric process can be determined by solving the appropriate form of Fourier-Kirchhoff equation. Here the equation will be expressed in the cylindrical coordinate system, which is a natural choice for the

The assumption of axial symmetry can be considered appropriate for the tensile and compression tests of steel in semi-solid state in all physically stable cases. It is invalid only for failed experiments. The symmetry simplifies the model by implying identical temperature distribution at any axial sample cross-section. This results in the equation:

Equation (5) can be further simplified if the heat properties of the medium are assumed isotropic. By calculating the differentials in equation (5) and using equation (6) we get the following form of Fourier-Kirchhoff equations for isotropic, axially-symmetric heat flow:

∂��� � � � ���

Equation (7) needs to be solved with appropriate initial and boundary conditions. The initial conditions relate to cases of non-stationary heat exchange. Most solutions use Cauchy condition which assume the known a priori temperature distribution at time ��: ���������. In a particular (but often adopted) case the temperature is assumed to be constant

Boundary conditions have more complex nature and relate to all cases of heat transfer and describe the spatial aspect of the heat exchange. The considered continuous medium changes its temperature though convection, radiation, conduction, or a combination of these phenomena. Theoretical solutions of the problem are generally subject to one or more boundary conditions. Combined Hankel's boundary conditions have been adopted for the presented model. The conditions for axially-symmetrical problem can be written in form of

∂�

∂��

∂�

∂� ���

∂�

∂�� � � � ���

∂� � 0 (6)

∂� (7)

transformation can be written in the index notation can be as:

cylindrically-shaped samples. It takes following differential form:

∂� ∂�� � <sup>1</sup> � ∂ ∂� � 1 � �� ∂� ∂�� � <sup>∂</sup>

� �∂�� ∂�� � 1 � ∂� ∂� �

throughout the considered area ������ � �� � �����.

a differential equation:

0 0���� (3)

�� � ���� (4)

∂�

∂� (5)

On the other hand numerical errors can be a source of volume loss which interferes with real changes. This effect is very undesirable in modelling of thermal-mechanical behaviour of steel in temperature range characteristic for the (transformation of state of aggregation).

The mentioned mechanical and thermal parts of the mathematical model of the process are supported by a third one, i.e. the density changes model. The mechanical part is responsible for the strain, strain rate and stress distribution in a controlled volume.

#### **4. Thermal part of the model**

Heat exchange between solid metal and environment, and its flow inside the metal is controlled by a number of factors. During phase change two additional phenomena have to be taken into account. Note that in the process of deformation of steel at temperature of liquid to solid phase transition there are two sources of heat changes. On the one hand heat is generated due to the state transformation. On the other hand it is secreted as a result of plastic deformation. In addition, steel density variations also cause changes of body temperature.

Thermal solution has a major impact on simulation results, since the temperature has strong effect on remaining variables. This is especially evident if the specimen temperature is close to solidus line when the body consist of both solid and semi-solid regions. In such case the affected phenomena are: plastic flow of solid and mushy materials, stress evolution and density changes. The theoretical temperature field is a solution of Fourier-Kirchhoff equation with appropriate boundary conditions.

The most general form of the Fourier-Kirchhoff equation in any coordinate system can be written in operator form as follows:

$$\nabla^T(\Lambda \nabla T) + Q = c\_p \rho \left(\mathbf{v}^T \nabla T + \frac{\partial T}{\partial \mathbf{r}}\right) \tag{1}$$

where ܶ is the temperature distribution in the controlled volume and denotes the symmetrical second order tensor called heat transformation tensor. In case of thermal inhomogeneity the whole tensor has to be considered. ܳ represents the rate of heat generation (or consumption) due to the phase transformation, due to plastic work done and due to electric current flow (resistance heating of the sample is usually applied). Finally ܿ describes the specific heat, ߩ the steel density, the velocity vector of specimen particles and ߬ the elapsed time. The heat transformation tensor consists of a set of anisotropic heat transformation coefficients and can be given in a form:

$$
\mathbf{A} = \begin{pmatrix}
\lambda\_{\rm xx} & \lambda\_{\rm xy} & \lambda\_{\rm xz} \\
\lambda\_{\rm yx} & \lambda\_{\rm yy} & \lambda\_{\rm yz} \\
\lambda\_{\rm zx} & \lambda\_{\rm xy} & \lambda\_{\rm zz}
\end{pmatrix} \tag{2}
$$

In the case of anisotropic bodies, the solution is carried out locally, and the axes of coordinate system are oriented in accordance with the principal directions of the thermal conductivity. In this case all off-diagonal components of the heat transformation tensor are zeros (ߣ ൌ Ͳ, ്݆݅) and equation (2) becomes:

$$
\mathbf{A} = \begin{pmatrix}
\lambda\_{xx} & 0 & 0 \\
0 & \lambda\_{yy} & 0 \\
0 & 0 & \lambda\_{zz}
\end{pmatrix} \\ \tag{3}
$$

Furthermore for a thermally isotropic material ��� ������ ������ � � �� and tensor of the heat transformation can be written in the index notation can be as:

$$
\Lambda\_{lj} = \lambda \delta\_{lj} \tag{4}
$$

where ��� is the Kronecker delta.

280 Numerical Modelling

On the other hand numerical errors can be a source of volume loss which interferes with real changes. This effect is very undesirable in modelling of thermal-mechanical behaviour of steel in temperature range characteristic for the (transformation of state of aggregation). The mentioned mechanical and thermal parts of the mathematical model of the process are supported by a third one, i.e. the density changes model. The mechanical part is responsible

Heat exchange between solid metal and environment, and its flow inside the metal is controlled by a number of factors. During phase change two additional phenomena have to be taken into account. Note that in the process of deformation of steel at temperature of liquid to solid phase transition there are two sources of heat changes. On the one hand heat is generated due to the state transformation. On the other hand it is secreted as a result of plastic deformation. In addition, steel density variations also cause changes of body

Thermal solution has a major impact on simulation results, since the temperature has strong effect on remaining variables. This is especially evident if the specimen temperature is close to solidus line when the body consist of both solid and semi-solid regions. In such case the affected phenomena are: plastic flow of solid and mushy materials, stress evolution and density changes. The theoretical temperature field is a solution of Fourier-Kirchhoff

The most general form of the Fourier-Kirchhoff equation in any coordinate system can be

where ܶ is the temperature distribution in the controlled volume and denotes the symmetrical second order tensor called heat transformation tensor. In case of thermal inhomogeneity the whole tensor has to be considered. ܳ represents the rate of heat generation (or consumption) due to the phase transformation, due to plastic work done and due to electric current flow (resistance heating of the sample is usually applied). Finally ܿ describes the specific heat, ߩ the steel density, the velocity vector of specimen particles and ߬ the elapsed time. The heat transformation tensor consists of a set of anisotropic heat

> ௫௭ߣ ௫௬ߣ ௫௫ߣ ௬௭ߣ ௬௬ߣ ௬௫ߣ ௭௭ߣ ௭௬ߣ ௭௫ߣ

In the case of anisotropic bodies, the solution is carried out locally, and the axes of coordinate system are oriented in accordance with the principal directions of the thermal conductivity. In this case all off-diagonal components of the heat transformation tensor are

߲ܶ

߲߬<sup>൰</sup> (1)

ቍ (2)

ܶ൬ ߩܿൌܳ ሻܶߘ ሺ்ߘ

ൌቌ

for the strain, strain rate and stress distribution in a controlled volume.

**4. Thermal part of the model** 

equation with appropriate boundary conditions.

transformation coefficients and can be given in a form:

zeros (ߣ ൌ Ͳ, ്݆݅) and equation (2) becomes:

written in operator form as follows:

temperature.

The temperature of samples compressed in axially-symmetric process can be determined by solving the appropriate form of Fourier-Kirchhoff equation. Here the equation will be expressed in the cylindrical coordinate system, which is a natural choice for the cylindrically-shaped samples. It takes following differential form:

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\lambda\_r\frac{\partial T}{\partial r}\right) + \frac{1}{r}\frac{\partial}{\partial \theta}\left(\frac{1}{r}\lambda\_\theta\frac{\partial T}{\partial \theta}\right) + \frac{\partial}{\partial z}\left(\lambda\_z\frac{\partial T}{\partial z}\right) + Q = \rho c\_p\frac{\partial T}{\partial \tau} \tag{5}$$

The assumption of axial symmetry can be considered appropriate for the tensile and compression tests of steel in semi-solid state in all physically stable cases. It is invalid only for failed experiments. The symmetry simplifies the model by implying identical temperature distribution at any axial sample cross-section. This results in the equation:

$$\frac{\partial T}{\partial \theta} = 0 \tag{6}$$

Equation (5) can be further simplified if the heat properties of the medium are assumed isotropic. By calculating the differentials in equation (5) and using equation (6) we get the following form of Fourier-Kirchhoff equations for isotropic, axially-symmetric heat flow:

$$
\lambda \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r} \frac{\partial T}{\partial r} + \frac{\partial^2 T}{\partial z^2} \right) + Q = \rho c\_p \frac{\partial T}{\partial \tau} \tag{7}
$$

Equation (7) needs to be solved with appropriate initial and boundary conditions. The initial conditions relate to cases of non-stationary heat exchange. Most solutions use Cauchy condition which assume the known a priori temperature distribution at time ��: ���������. In a particular (but often adopted) case the temperature is assumed to be constant throughout the considered area ������ � �� � �����.

Boundary conditions have more complex nature and relate to all cases of heat transfer and describe the spatial aspect of the heat exchange. The considered continuous medium changes its temperature though convection, radiation, conduction, or a combination of these phenomena. Theoretical solutions of the problem are generally subject to one or more boundary conditions. Combined Hankel's boundary conditions have been adopted for the presented model. The conditions for axially-symmetrical problem can be written in form of a differential equation:

$$
\lambda r \frac{\partial T}{\partial n} + a(T - T\_0) + q = 0 \tag{8}
$$

$$T = T(r, z, \tau) \tag{9}$$


$$
\lambda \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r} \frac{\partial T}{\partial r} + \frac{\partial^2 T}{\partial z^2} \right) + Q = 0 \tag{10}
$$

$$\chi = \int\_{V} f\left(\mathbf{r}, \mathbf{z}, T, T\_r, T\_\mathbf{z}\right) \, dV + \int\_{S} \left( qT + \frac{1}{2} a (T - T\_0)^2 \right) \, dS \tag{11}$$

$$T\_r = \frac{\partial T}{\partial r}; \qquad T\_\mathbf{z} = \frac{\partial T}{\partial \mathbf{z}}\tag{12}$$

$$\delta \chi = \int\_{V} \left( \frac{\partial f}{\partial T} \delta T + \frac{\partial f}{\partial T\_r} \delta T\_r + \frac{\partial f}{\partial T\_z} \delta T\_z \right) dV + \int\_{S} [q \delta T + a(T - T\_0) \cdot \delta T] dS \tag{13}$$

$$\delta \chi = \int\_{V} \delta T \left[ \frac{\partial f}{\partial T} - \frac{\partial}{\partial r} \left( \frac{\partial f}{\partial T\_r} \right) - \frac{\partial}{\partial \mathbf{z}} \left( \frac{\partial f}{\partial T\_\mathbf{z}} \right) \right] dV + \int\_{S} \delta T \left( q + a(T - T\_0) + l\_r \frac{\partial f}{\partial T\_r} + l\_\mathbf{z} \frac{\partial f}{\partial T\_\mathbf{z}} \right) dS \tag{14}$$

$$\frac{\partial}{\partial r}\left(\frac{\partial f}{\partial T\_r}\right) + \frac{\partial}{\partial z}\left(\frac{\partial f}{\partial T\_z}\right) - \frac{\partial f}{\partial T} = 0 \tag{15}$$

$$l\_{\chi} \frac{\partial f}{\partial T\_{\chi}} + l\_{z} \frac{\partial f}{\partial T\_{z}} + q + a(T - T\_{0}) = 0 \tag{16}$$

$$f = r \left[ \frac{1}{2} \lambda (T\_r^2 + T\_z^2) - QT \right] \tag{17}$$

$$\begin{aligned} \lambda \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r} T\_r + \frac{\partial^2 T}{\partial z^2} \right) + Q &= 0\\ \lambda r \frac{\partial T}{\partial n} + q + a(T - T\_0) &= 0 \end{aligned} \tag{18}$$

$$\chi = \int\_{V} r \left\{ \frac{1}{2} \lambda \left[ \left( \frac{\partial T}{\partial r} \right)^2 + \left( \frac{\partial T}{\partial \mathbf{z}} \right)^2 \right] - QT \right\} dV + \int\_{S} \left( qT + \frac{1}{2} a (T - T\_0)^2 \right) \, dS \tag{19}$$

$$T(r, z) = \mathbf{n}^T(r, z) \,\mathbf{T} \tag{20}$$

$$\chi = \int\_{V} r \left\{ \frac{1}{2} \lambda \left[ \left( \frac{\partial \mathbf{n}^{T}}{\partial r} \mathbf{T} \right)^{2} + \left( \frac{\partial \mathbf{n}^{T}}{\partial z} \mathbf{T} \right)^{2} \right] - Q \mathbf{n}^{T} \mathbf{T} \right\} dV + \int\_{S} \left( q \mathbf{n}^{T} \mathbf{T} + \frac{1}{2} a (\mathbf{n}^{T} \mathbf{T} - T\_{0})^{2} \right) dS \tag{21}$$

$$\frac{\partial \chi}{\partial \mathbf{T}} = \int\_{V} r \left[ \lambda \mathbf{T}^{T} \left( \frac{\partial \mathbf{n}}{\partial r} \frac{\partial \mathbf{n}^{T}}{\partial r} + \frac{\partial \mathbf{n}}{\partial z} \frac{\partial \mathbf{n}^{T}}{\partial z} \right) - \mathbf{Q} \mathbf{n}^{T} \right] dV + \int\_{S} \left( q \mathbf{n}^{T} + a (\mathbf{T}^{T} \mathbf{n} - T\_{0}) \mathbf{n}^{T} \right) dS \tag{22}$$

$$\mathbf{H}\mathbf{T} + \mathbf{p} = \mathbf{0} \tag{23}$$

$$\begin{aligned} \mathbf{H} &= \int\_{V} r\lambda \left( \frac{\partial \mathbf{n}}{\partial r} \frac{\partial \mathbf{n}^{T}}{\partial r} + \frac{\partial \mathbf{n}}{\partial z} \frac{\partial \mathbf{n}^{T}}{\partial z} \right) dV + \int\_{S} \alpha \mathbf{n} \mathbf{n}^{T} dS\\ \mathbf{p} &= -\int\_{V} rQ\mathbf{n} \cdot dV - \int\_{S} (\alpha T\_{0} - q) \mathbf{n} \cdot dS \end{aligned} \tag{24}$$

$$\begin{split} \chi = \int\_{V} r \left\{ \frac{1}{2} \lambda \left[ \left( \frac{\partial \mathbf{n}^{T}}{\partial r} \mathbf{T} \right)^{2} + \left( \frac{\partial \mathbf{n}^{T}}{\partial \mathbf{z}} \mathbf{T} \right)^{2} \right] - \left[ Q - \rho c\_{p} \frac{\partial}{\partial \tau} (\mathbf{n}^{T} \mathbf{T}) \right] \mathbf{n}^{T} \mathbf{T} \right\} dV + \\ \quad + \int\_{S} \left( q \mathbf{n}^{T} \mathbf{T} + \frac{1}{2} a (\mathbf{n}^{T} \mathbf{T} - T\_{0})^{2} \right) dS \end{split} \tag{25}$$

$$\begin{split} \frac{\partial \chi}{\partial \mathbf{T}} = \int\_{V} r \left[ \lambda \mathbf{T}^{T} \left( \frac{\partial \mathbf{n}}{\partial r} \frac{\partial \mathbf{n}^{T}}{\partial r} + \frac{\partial \mathbf{n}}{\partial z} \frac{\partial \mathbf{n}^{T}}{\partial z} \right) - \left( Q - \rho c\_{p} \frac{\partial \mathbf{T}^{T}}{\partial \tau} \mathbf{n} \right) \mathbf{n}^{T} \right] dV + \\ \quad + \int\_{S} (q \mathbf{n}^{T} + \alpha (\mathbf{T}^{T} \mathbf{n} - T\_{0}) \mathbf{n}^{T}) dS \end{split} \tag{26}$$

$$\mathbf{H}\mathbf{T} + \mathbf{C}\frac{\partial \mathbf{T}}{\partial \tau} + \mathbf{p} = \mathbf{0} \tag{27}$$

$$\mathbf{C} = \int\_{\nu} \rho c\_p \mathbf{n} \, \mathbf{n}^T dV \tag{28}$$

$$
\overline{\mathbf{H}}\mathbf{T}\_{l+1} + \overline{\mathbf{p}} = \mathbf{0} \tag{29}
$$

$$\begin{aligned} \overline{\mathbf{H}} &= \left( 2\mathbf{H} + \frac{3}{\Delta \tau} \mathbf{C} \right) \\ \overline{\mathbf{p}} &= \left( \mathbf{H} - \frac{3}{\Delta \tau} \mathbf{C} \right) \mathbf{T}\_l + 3\mathbf{p} \end{aligned} \tag{30}$$


Inverse Analysis Applied to Mushy Steel

The functional takes the following shape:

with variable density.

leakage of liquid phase.

solely on the ܹሶ

Rheological Properties Testing Using Hybrid Numerical-Analytical Model 287

functional (34). It is used in most solutions and has a significant share of total power. Even when the iterative process approaches the end, this power component is still significant, especially if the convergence of the optimization procedures is insufficient. In case of discretization of the deformation area (e.g. using the finite element method) if one focuses

number of possible directions of movement of discretization nodes providing the volume preservation of the deformation zone. Each of these solutions creates a local optimum for ܹሶ

power and thus for the entire functional (34). This makes it difficult to optimize because of lack of uniform direction of fall of total power which leads to global optimum. The material density fluctuation causes further optimization difficulties, resulting from additional replacement of incompressibility condition with a full condition of mass conservation.

The proposed solution requires high accuracy in ensuring the incompressibility condition for the solid material or mass conservation condition for the semi-solid areas. This approach stems from the fact that the errors resulting from the breach of these conditions can be treated as a volume change caused by the steel density variation in the semi-solid zone. High accuracy solution is required also due to large differences in yield stress for the individual subareas of the deformation zone. In the discussed temperature range they appear due to even slight fluctuations in temperature. In presented solution the second component of functional (34) is left out and mass conservation condition is given in analytical form constraining the radial (ݒሻand longitudinal ሺݒ௭ሻvelocity field components.

ܬሾሿ ൌ ܹሶ

In case of functional (35) the numerical optimisation procedure converges faster than the one for functional (34) due to the reduced number of velocity field parameters (only radial components are optimisation parameters) and the lack of numerical form of mass conservation condition. The accuracy of the proposed hybrid solution is higher also due to negligible volume loss caused by numerical errors which is very important for materials

As mentioned before the solution of the problem is a velocity field in cylindrical coordinate system in axial-symmetrical state of deformation. Optimization of metal flow velocity field in the deformation zone of semi-variational problem requires the formulation according to equation (35). The radial velocity distribution ݒሺݎǡ ߠǡ ݖሻ and the longitudinal one ݒ௭ሺݎǡ ߠǡ ݖሻ are so complex that such wording in the global coordinate system poses considerable difficulties. These difficulties are the result of the mutual dependence of these velocities. Therefore the basic formulation will be written for the local cylindrical coordinate system ܱݖߠݎ with a view to the future discretization of deformation area using one of the dedicated methods. In addition one will find that the deformation of cylindrical samples is characterized by axial symmetry. As demonstrated by experimental studies conducted using semi-solid samples the symmetry may be disturbed only as a result of unexpected

Such experiments, however, are regarded as unsuccessful and not subject to numerical analysis. Establishment of the axial symmetry, which except in cases of physical instability can be considered valid also for the process of compression or tensile test of semi-solid

<sup>ఙ</sup> ܹሶ

<sup>௧</sup> (35)

<sup>ఒ</sup> a number of possible optimal solutions appear. They are related to a

<sup>ఒ</sup>, which occurs in

ఒ

A further problem specific to the variable density continuum is power ܹሶ

$$
\sigma\_{lj} - \frac{1}{3} \sigma\_{kk} \delta\_{lj} = \frac{2}{3} \frac{\sigma\_p}{\varepsilon\_l} \,\,\varepsilon\_{lj} \tag{31}
$$

Rigid-plastic model was selected due to its very good accuracy at the strain field during the hot deformation and sufficient correctness of calculated deviatoric part of the stress field. Moreover, the elastic part of each stress tensor component is very low at temperatures close to solidus line and can in practice be neglected in calculations of strain distribution. The limits for plastic metal behavior are defined according to Huber-Mises-Hencky yield criterion:

$$
\sigma\_{lj}\sigma\_{lj} = \text{ 2 } \left(\frac{\sigma\_p}{\sqrt{3}}\right)^2 \tag{32}
$$

In equations (31) and (32) ��� denotes the stress tensor components, ��� represents the mean stress, ��� is the Kronecker delta, �� indicates the yield stress, �� � is the effective strain rate, and �� �� denotes strain rate tensor components. The components are given by an equation:

$$\varepsilon\_{lj} = \frac{1}{2} \{\nabla\_l \upsilon\_j + \nabla\_j \upsilon\_l\} \tag{33}$$

In cylindrical coordinate system ���� the solution is a vector velocity field defined by the distribution of three coordinates � � ���� ��� ���. The field is a result of optimization of a power functional, which can be written in general form as the sum of power necessary to run the main physical phenomena related to plastic deformation. Due to the axial-symmetry of the sample the velocity field the circumferential component of the velocity field can be neglected and the functional is usually formulated as:

$$J[\mathfrak{v}] = \mathcal{W} = \mathcal{W}\_{\sigma} + \dot{\mathcal{W}}\_{\lambda} + \dot{\mathcal{W}}\_{f} \tag{34}$$

Component �� � occurring in equation (34) represents the plastic deformation power, �� � is the power which is a penalty for the departure from mass conservation condition, �� � denotes the friction power and � � ���� ��� describes the reduced velocity field distribution.

Rigid-plastic formulation of metal deformation problem requires the condition of mass conservation in the deformation zone. In case of solids and liquids with a constant density, this condition can be simplified to the incompressibility condition. Such a condition is generally satisfied with sufficient accuracy during the optimization of functional (34). In most solutions a slight, but noticeable loss of volume is observed. The loss is caused by incomplete fulfilment of the incompressibility condition imposed on the solution in numerical form. It is negligible in case of traditional computer simulation of deformation processes although in some embodiments more accurate methods are used to restore the volume of metal subjected to the deformation. Unlike this case the density of semi-solid materials varies during the deformation process and these changes result in a physically reasonable change in the volume of a body having constant mass. The size of the volume loss due to numerical errors is comparable with changes caused by fluctuation in the density of the material.

<sup>3</sup> ������ � <sup>2</sup>

Rigid-plastic model was selected due to its very good accuracy at the strain field during the hot deformation and sufficient correctness of calculated deviatoric part of the stress field. Moreover, the elastic part of each stress tensor component is very low at temperatures close to solidus line and can in practice be neglected in calculations of strain distribution. The limits for plastic metal behavior are defined according to Huber-Mises-Hencky yield

������ �2���

In equations (31) and (32) ��� denotes the stress tensor components, ��� represents the mean

�� denotes strain rate tensor components. The components are given by an equation:

In cylindrical coordinate system ���� the solution is a vector velocity field defined by the distribution of three coordinates � � ���� ��� ���. The field is a result of optimization of a power functional, which can be written in general form as the sum of power necessary to run the main physical phenomena related to plastic deformation. Due to the axial-symmetry of the sample the velocity field the circumferential component of the velocity field can be

� � ��

the power which is a penalty for the departure from mass conservation condition, ��

denotes the friction power and � � ���� ��� describes the reduced velocity field distribution. Rigid-plastic formulation of metal deformation problem requires the condition of mass conservation in the deformation zone. In case of solids and liquids with a constant density, this condition can be simplified to the incompressibility condition. Such a condition is generally satisfied with sufficient accuracy during the optimization of functional (34). In most solutions a slight, but noticeable loss of volume is observed. The loss is caused by incomplete fulfilment of the incompressibility condition imposed on the solution in numerical form. It is negligible in case of traditional computer simulation of deformation processes although in some embodiments more accurate methods are used to restore the volume of metal subjected to the deformation. Unlike this case the density of semi-solid materials varies during the deformation process and these changes result in a physically reasonable change in the volume of a body having constant mass. The size of the volume loss due to numerical errors is comparable with changes caused by fluctuation in the density

� occurring in equation (34) represents the plastic deformation power, ��

� � ��

3 �� �� � ��

√3 � � �� (31)

� is the effective strain rate,

� (34)

<sup>2</sup> ����� � ����� (33)

(32)

� is

�

��� � <sup>1</sup>

stress, ��� is the Kronecker delta, �� indicates the yield stress, ��

neglected and the functional is usually formulated as:

�� �� � <sup>1</sup>

���� � �� � ��

criterion:

and ��

Component ��

of the material.

A further problem specific to the variable density continuum is power ܹሶ <sup>ఒ</sup>, which occurs in functional (34). It is used in most solutions and has a significant share of total power. Even when the iterative process approaches the end, this power component is still significant, especially if the convergence of the optimization procedures is insufficient. In case of discretization of the deformation area (e.g. using the finite element method) if one focuses solely on the ܹሶ <sup>ఒ</sup> a number of possible optimal solutions appear. They are related to a number of possible directions of movement of discretization nodes providing the volume preservation of the deformation zone. Each of these solutions creates a local optimum for ܹሶ ఒ power and thus for the entire functional (34). This makes it difficult to optimize because of lack of uniform direction of fall of total power which leads to global optimum. The material density fluctuation causes further optimization difficulties, resulting from additional replacement of incompressibility condition with a full condition of mass conservation.

The proposed solution requires high accuracy in ensuring the incompressibility condition for the solid material or mass conservation condition for the semi-solid areas. This approach stems from the fact that the errors resulting from the breach of these conditions can be treated as a volume change caused by the steel density variation in the semi-solid zone. High accuracy solution is required also due to large differences in yield stress for the individual subareas of the deformation zone. In the discussed temperature range they appear due to even slight fluctuations in temperature. In presented solution the second component of functional (34) is left out and mass conservation condition is given in analytical form constraining the radial (ݒሻand longitudinal ሺݒ௭ሻvelocity field components. The functional takes the following shape:

$$J[\boldsymbol{\nu}] = \dot{\boldsymbol{W}}\_{\sigma} + \dot{\boldsymbol{W}}\_{t} \tag{35}$$

In case of functional (35) the numerical optimisation procedure converges faster than the one for functional (34) due to the reduced number of velocity field parameters (only radial components are optimisation parameters) and the lack of numerical form of mass conservation condition. The accuracy of the proposed hybrid solution is higher also due to negligible volume loss caused by numerical errors which is very important for materials with variable density.

As mentioned before the solution of the problem is a velocity field in cylindrical coordinate system in axial-symmetrical state of deformation. Optimization of metal flow velocity field in the deformation zone of semi-variational problem requires the formulation according to equation (35). The radial velocity distribution ݒሺݎǡ ߠǡ ݖሻ and the longitudinal one ݒ௭ሺݎǡ ߠǡ ݖሻ are so complex that such wording in the global coordinate system poses considerable difficulties. These difficulties are the result of the mutual dependence of these velocities. Therefore the basic formulation will be written for the local cylindrical coordinate system ܱݖߠݎ with a view to the future discretization of deformation area using one of the dedicated methods. In addition one will find that the deformation of cylindrical samples is characterized by axial symmetry. As demonstrated by experimental studies conducted using semi-solid samples the symmetry may be disturbed only as a result of unexpected leakage of liquid phase.

Such experiments, however, are regarded as unsuccessful and not subject to numerical analysis. Establishment of the axial symmetry, which except in cases of physical instability can be considered valid also for the process of compression or tensile test of semi-solid

$$
\dot{W}\_{\sigma} = \int\_{\nu} \sigma\_{l} \varepsilon\_{l} \, dV \tag{36}
$$

$$
\varepsilon\_l = \sqrt{\frac{2}{3} \varepsilon\_{lj} \varepsilon\_{lj}}\tag{37}
$$

$$
\begin{pmatrix}
\frac{\partial v\_r}{\partial r} & 0 & \frac{1}{2}\frac{\partial v\_r}{\partial z} + \frac{1}{2}\frac{\partial v\_z}{\partial r} \\
0 & \frac{v\_r}{r} & 0 \\
\frac{1}{2}\frac{\partial v\_r}{\partial z} + \frac{1}{2}\frac{\partial v\_z}{\partial r} & 0 & \frac{\partial v\_z}{\partial z}
\end{pmatrix}
\tag{38}
$$

$$
\dot{W}\_t = \int\_S m \frac{\sigma\_p}{\sqrt{3}} \|\overline{\nu}\| \, d\mathcal{S} \tag{39}
$$

$$
\mathcal{V}\mathfrak{v} = \mathfrak{0} \tag{40}
$$

$$
\nabla \boldsymbol{\nu} - \frac{1}{\rho} \frac{\partial \rho}{\partial t} = \begin{array}{c} \text{0} \\ \end{array} \tag{41}
$$

$$\frac{\partial v\_r}{\partial r} + \frac{v\_r}{r} + \frac{\partial v\_z}{\partial z} = 0 \tag{42}$$

$$\frac{\partial v\_r}{\partial r} + \frac{v\_r}{r} + \frac{\partial v\_z}{\partial z} - \frac{1}{\rho} \frac{\partial \rho}{\partial \tau} = 0 \tag{43}$$

$$v\_x = -\int \left(\frac{\partial v\_r}{\partial r} + \frac{v\_r}{r} - \frac{1}{\rho} \frac{\partial \rho}{\partial \tau}\right) dz \tag{44}$$

$$\begin{aligned} \xi(r,z) &= \frac{r}{R\_m} \\ \eta(r,z) &= \frac{2z - g(r) - f(r)}{g(r) - f(r)} \end{aligned} \tag{45}$$

$$\psi\_r(\xi,\eta) = \frac{1}{2} \frac{r\upsilon\_0}{g(r) - f(r)} \left( 1 + \frac{\partial \psi(\xi,\eta)}{\partial \eta} \right) \tag{46}$$

$$\upsilon\_{\mathbf{z}}(\xi,\eta) = \int \left[\frac{1}{\rho}\frac{\partial\rho}{\partial t} - \frac{\partial\upsilon\_{r}(\xi,\eta)}{\partial r} - \frac{\upsilon\_{r}(\xi,\eta)}{r}\right] d\mathbf{z} \tag{47}$$

$$\frac{\partial \upsilon\_r}{\partial r} = \frac{\upsilon\_0}{2} \left[ \frac{\partial}{\partial r} \left( \frac{r}{g - f} \right) \left( 1 + \frac{\partial \psi}{\partial \eta} \right) + \frac{r}{g - f} \frac{\partial}{\partial r} \left( 1 + \frac{\partial \psi}{\partial \eta} \right) \right] \tag{48}$$

$$\frac{\partial v\_r}{\partial r} = \frac{v\_0}{2(g-f)} \left( 1 + \frac{\partial \psi}{\partial \eta} + \xi \frac{\partial^2 \psi}{\partial \xi \ \partial \eta} \right) - \frac{r v\_0 \left( \frac{\partial g}{\partial r} - \frac{\partial f}{\partial r} \right)}{2(g-f)^2} \left[ 1 + \frac{\partial \psi}{\partial \eta} + \frac{\partial^2 \psi}{\partial \eta^2} \left( \frac{\frac{\partial g}{\partial r} + \frac{\partial f}{\partial r}}{\frac{\partial g}{\partial r} - \frac{\partial f}{\partial r}} + \eta \right) \right] \tag{49}$$

$$\frac{\partial \upsilon\_{z}}{\partial z} = -\frac{\partial \upsilon\_{r}}{\partial r} - \frac{\upsilon\_{r}}{r} + \frac{1}{\rho} \frac{\partial \rho}{\partial \tau} = \frac{r \upsilon\_{0} \left(\frac{\partial g}{\partial r} - \frac{\partial f}{\partial r}\right)}{2(g - f)^{2}} \left[1 + \frac{\partial \psi}{\partial \eta} + \frac{\partial^{2} \psi}{\partial \eta^{2}} \left(\frac{\frac{\partial g}{\partial r} + \frac{\partial f}{\partial r}}{\frac{\partial g}{\partial r} - \frac{\partial f}{\partial r}} + \eta\right)\right] - \tag{50}$$

$$-\frac{\upsilon\_{0}}{2(g - f)} \left(1 + \frac{\partial \psi}{\partial \eta} + \frac{1}{2} \xi \frac{\partial^{2} \psi}{\partial \xi \partial \eta}\right) + \frac{1}{\rho} \frac{\partial \rho}{\partial \tau}$$

$$\begin{split} \boldsymbol{\nu} = -\frac{\upsilon\_0}{4} \Bigg\{ 2 \left( \eta + \frac{g + f}{g - f} + \psi \right) + \xi \frac{\partial \psi}{\partial \xi} - \frac{\left( \frac{\partial g}{\partial r} - \frac{\partial f}{\partial r} \right)}{g - f} \Bigg[ \eta + (1 - r)\psi + r \left( \frac{\frac{\partial g}{\partial r} + \frac{\partial f}{\partial r}}{\frac{\partial g}{\partial r} - \frac{\partial f}{\partial r}} + \eta \right) \frac{\partial \psi}{\partial \eta} \Bigg] \Bigg\} \\\\ + \frac{\eta(g - f) + g + f}{2\rho} \frac{\partial \rho}{\partial \tau} \end{split} (51)$$


$$\frac{\partial \rho}{\partial \tau} = \frac{\partial \rho\_p}{\partial \tau} + \frac{\partial \rho\_f}{\partial \tau} + \frac{\partial \rho\_t}{\partial \tau} \tag{52}$$

$$\frac{\partial \rho\_p}{\partial \tau} = \left[\rho\_s (1 - X\_l) + \rho\_l X\_l\right] \left(\frac{\rho\_s}{\rho\_l} - 1\right) \frac{\partial X\_l}{\partial \tau} \tag{53}$$

$$\frac{\partial \rho\_f}{\partial \tau} = \rho\_l X\_l \left(\frac{\partial v\_r}{\partial r} + \frac{v\_r}{r} + \frac{\partial v\_z}{\partial z}\right) \tag{54}$$

$$\frac{\partial \rho\_t}{\partial \tau} = \left[\beta\_s \rho\_s (1 - X\_l) + \beta\_l \rho\_l X\_l\right] \frac{\partial T}{\partial \tau} \tag{55}$$


$$\rho = \frac{7850}{(1+\Delta l)^3}; \left[\frac{\text{kg}}{\text{m}^3}\right] \tag{56}$$

$$
\Delta l = 0.004 \left( \frac{T + 273}{1000} \right)^2
$$

$$\rho = \frac{7897}{(1+\Delta l)^3}; \left[\frac{\text{kg}}{\text{m}^3}\right] \tag{57}$$

Inverse Analysis Applied to Mushy Steel

published in (Mizukami at al., 2002).

appropriate phases with respect to solidus temperature.

Rheological Properties Testing Using Hybrid Numerical-Analytical Model 295

Fig. 1. Density of MC1 grade steel as a function of temperature (left) and density changes of its liquid phase as a function of temperature increase (right). Plots are based on data

Subsequent charts presented in Figure 2 show changes in density of ߜ i ߛ phases, respectively. Both of them are functions of undercooling temperature οܶఋ and οܶఊ of

Fig. 2. Density changes of steel phase ߜ) left) and ߛ) right) of MC1 steel grade as a function of

The presented graphs were used to develop analytical dependencies, which describe changes in the density of steel during the transformation of state of aggregation. In the region of coexistence of ߜ and ߛ phases density was estimated using the additivity rule. The correctness of this approximation was verified by comparing the theoretical results with

temperature increase – based on data published in (Mizukami at al., 2002).

those which were obtained from the measured values.

The �� parameter from (57) is calculated as:

$$
\Delta l = -0.00358 + 0.00947 \frac{T + 273}{1000} + 0.0103 \left(\frac{T + 273}{1000}\right)^2 - 0.00298 \left(\frac{T + 273}{1000}\right)^3
$$

Similar dependence can be used for high-alloy steels. In this case it is necessary to modify the equation (56) in a manner appropriate for the particular steel grade. Thus, for temperature range which is proper for traditional process of steel hot deformation the calculation of changes in density seems to be pretty simple. Such temperatures are characteristic for certain sample areas.

Otherwise presents itself the problem for higher temperature ranges, where the deformation occurs during the simultaneous metal solidification. Here the density variations may be significant. For purposes of the current mathematical model an approach proposed by Mizukami was used (Mizukami at al., 2002). For carbon steels containing no other elements the density changes are functions of temperature. Steels were tested with a wide range of carbon content, which ranges from 0.005% to 0.56% by mass. The authors develop tests for typical steels having chemical composition expressed in% by mass given in Table. 1.

Left side of Figure 1 shows the change in density for MC1 grade steel as a function of temperature. For steel changing its states of aggregation some plots of density in the various phases of the transformation process has been developed. The right side of Figure 1 shows the course of the changes in the density of liquid phase as a function of ��� – undercooling temperature with respect to the liquidus line. Changes in density are presented in relation to the base density of 7060 ��������.


Table 1. Chemical composition (mass %) of typical steels tested by authors of (Mizukami at al., 2002).

<sup>1000</sup> � 0�010� �� � ���

Similar dependence can be used for high-alloy steels. In this case it is necessary to modify the equation (56) in a manner appropriate for the particular steel grade. Thus, for temperature range which is proper for traditional process of steel hot deformation the calculation of changes in density seems to be pretty simple. Such temperatures are

Otherwise presents itself the problem for higher temperature ranges, where the deformation occurs during the simultaneous metal solidification. Here the density variations may be significant. For purposes of the current mathematical model an approach proposed by Mizukami was used (Mizukami at al., 2002). For carbon steels containing no other elements the density changes are functions of temperature. Steels were tested with a wide range of carbon content, which ranges from 0.005% to 0.56% by mass. The authors develop tests for typical steels having chemical composition expressed

Left side of Figure 1 shows the change in density for MC1 grade steel as a function of temperature. For steel changing its states of aggregation some plots of density in the various phases of the transformation process has been developed. The right side of Figure 1 shows the course of the changes in the density of liquid phase as a function of ��� – undercooling temperature with respect to the liquidus line. Changes in density are presented in relation to

Steel ULC LC MC1 MC2 HC

[C] 0.005 0.040 0.110 0.140 0.550

[Si] 0.010 0.040 0.100 0.160 0.150

[Mn] 0.120 0.190 0.480 0.540 0.910

[P] 0.014 0.026 0.020 0.016 0.021

[S] 0.003 0.006 0.008 0.003 0.001

Table 1. Chemical composition (mass %) of typical steels tested by authors of (Mizukami at

<sup>1000</sup> �

�

� 0�00��� �� � ���

<sup>1000</sup> �

�

� � ���

The �� parameter from (57) is calculated as:

�� � �0�00��� � 0�00���

characteristic for certain sample areas.

in% by mass given in Table. 1.

the base density of 7060 ��������.

al., 2002).

Fig. 1. Density of MC1 grade steel as a function of temperature (left) and density changes of its liquid phase as a function of temperature increase (right). Plots are based on data published in (Mizukami at al., 2002).

Subsequent charts presented in Figure 2 show changes in density of ߜ i ߛ phases, respectively. Both of them are functions of undercooling temperature οܶఋ and οܶఊ of appropriate phases with respect to solidus temperature.

Fig. 2. Density changes of steel phase ߜ) left) and ߛ) right) of MC1 steel grade as a function of temperature increase – based on data published in (Mizukami at al., 2002).

The presented graphs were used to develop analytical dependencies, which describe changes in the density of steel during the transformation of state of aggregation. In the region of coexistence of ߜ and ߛ phases density was estimated using the additivity rule. The correctness of this approximation was verified by comparing the theoretical results with those which were obtained from the measured values.

Inverse Analysis Applied to Mushy Steel

**7.1 Characteristic temperature levels** 

region is higher than for quasi-static processes.

**7.2 Yield stress functions** 

subsections.

subsequent regions of the sample deformation zone.

Rheological Properties Testing Using Hybrid Numerical-Analytical Model 297

analysis of metal flow in subsequent regions of the sample deformation zone requires adequate methods. Classical techniques of interpretation of results of compression testing procedures fail due to significant samples barrelling which is inevitable at any temperature close to solidus level and which requires right analysis of metal flow in

A number of steel grades were subjected to series of experiments in Institute for Ferrous Metallurgy in Gliwice, Poland using GLEEBLE 3800 simulator. Example results of examination of two steels are reported in the current contribution. The first one is the 18G2A grade steel having 0.16% of carbon and the second was the S355J2G3So grade with 0.11% of carbon content. The essential aim of the investigation was the reconstruction of both temperature changes and strain evolution on specimen exposed to simultaneous deformation and solidification. The inverse procedure has been reported in (Glowacki & Hojny, 2009). Example results of inverse analysis are shortly described in succeeding

As mentioned before, apart from the liquidus and solidus temperatures, four other temperature levels are characteristic for the mushy steel behaviour. All the levels split the liquidus-solidus range into intervals. The most important for the extra high temperature rolling process design is the nil ductility temperature (NDT). The plastic deformation of a steel specimen is possible only below the NDT temperature. The temperature levels have to be calculated according to results of series of difficult experiments which are not a subject of the current paper. For carbon steels with the carbon content of around 0.1 % the equilibrium liquidus and solidus temperature levels are 1523°C and 1482°C, respectively and the NDT is 1420°C. One must note that the last one is a conventional temperature of a sample surface (indicated during experimental procedure). The maximum and minimum temperatures in the sample's central cross-section may differ by 60- 70 °C. The equilibrium liquidus and solidus temperatures for 18G2A grade steel are 1513°C and 1465°C, respectively. The measured mean value of NDT temperature of the steel falls into the range of 1420°C÷1425°C. The NDT is related to the temperature at which the last liquid phase particles existing in the central part of the sample disappear in static processes. It has been observed that for temperatures higher than NDT a remainder of liquid phase still exist in the central part of the sample (Hojny & Glowacki, 2009a). For dynamic cooling and deformation processes in some regions of the sample the remainder of liquid phase can be observed at temperatures lower than NDT because the difference between sample surface and its central

The well-known Voce formula (Voce, 1955) was adopted for the description of the shape of yield stress function. Figure 3 presents four subsequent stages of an example compression test at higher sample surface temperature, i.e. 1425°C for the quasi-static process. One can observe that the experiment was successful (no metal outflow) and the deformation of the

sample was realised despite the significant barrelling of the sample.

The density of carbon steels depends on the temperature and the existing fraction of liquid phase. The effect of alloying elements (except of coal) on the density of each steel phase is small, although the concentration of these components significantly affect the fraction of the phases. The density of each phase is calculated according to the following equations:

$$\begin{aligned} \rho\_l &= 7, 02 - 5, 50 \cdot 10^{-4} \,\Delta T\_l \\ \rho\_\delta &= 7, 27 + 3, 07 \cdot 10^{-4} \,\Delta T\_\delta \\ \rho\_Y &= 7, 41 + 4, 80 \cdot 10^{-4} \,\Delta T\_Y \end{aligned} \tag{58}$$

In equation (58) ��, �� and �� indicate densities of liquid steel and its � and � phases, respectively. The density in the regions of occurrence of several phases simultaneously is given by the following equations:

$$\begin{aligned} \rho\_{l+\delta} &= \rho\_l^0 + \Delta \rho\_{l/\delta} \cdot X\_{\delta} \\ \rho\_{l+\mathbf{y}} &= \rho\_l^0 + \Delta \rho\_{l/\mathbf{y}} \cdot X\_{\mathbf{y}} \\ \rho\_{l+\delta+\mathbf{y}} &= \rho\_l^0 + \rho\_{l/\delta} \cdot f\_{\delta} + \Delta \rho\_{l/\mathbf{y}} \cdot f\_{\mathbf{y}} \end{aligned} \tag{59}$$

where �� � denotes the density of the liquid phase for temperature discrepancy ���, ����� and ����� are density differences between � and � phases for temperature drop from liquidus to solidus level, X� and X� are the fractions of � i � phases in surrounding liquid phase, respectively. and finally �� i �� are relative fractions of � i � phases. The density of ��� phase was estimated according to relationship:

$$
\rho\_{\delta+\gamma} = \rho\_{\delta} \cdot X\_{\delta} + \rho\_{\gamma} \cdot X\_{\gamma} \tag{60}
$$

#### **7. Mushy steel flow stress curves development**

The subsequent part of the chapter deals with the computation of mushy steel flow stress curves based on the developed mathematical model which helps to avoid interpretational problems occurring in traditional testing procedures. Proper interpretation of the experimental results is possible with the help of appropriate computer aided testing system. Such a user friendly dedicated computer system with variable density has been developed (Glowacki & Hojny, 2009; Hojny & Glowacki, 2009a). The system codename called *Def\_Semi\_Solid* is a result of theoretical research conducted in a team lead by the chapter author with the financial support of grants awarded by Polish Committee of Scientific Research. The system in itself is not a subject of the chapter and its details are not discussed. The program was developed using an object oriented technique and is compatible with both Windows and Unix based platforms.

During experiments a few quantities were recorded. Among them the most important are GLEEBLE jaws displacement, force and temperature. This is a start point for the inverse analysis. The system calculates the shape and size of the deformation zone and strain and stress fields as well as optimal values of flow stress curve parameters. The model described in the previous section allows for the comparison of theoretical and experimental results for non-uniform temperature field. Isothermal tests in the temperature range over 1400 °C are impossible even using sophisticated equipment like GLEEBLE simulator. The presented model is a solution to the experimental problems. The analysis of metal flow in subsequent regions of the sample deformation zone requires adequate methods. Classical techniques of interpretation of results of compression testing procedures fail due to significant samples barrelling which is inevitable at any temperature close to solidus level and which requires right analysis of metal flow in subsequent regions of the sample deformation zone.

A number of steel grades were subjected to series of experiments in Institute for Ferrous Metallurgy in Gliwice, Poland using GLEEBLE 3800 simulator. Example results of examination of two steels are reported in the current contribution. The first one is the 18G2A grade steel having 0.16% of carbon and the second was the S355J2G3So grade with 0.11% of carbon content. The essential aim of the investigation was the reconstruction of both temperature changes and strain evolution on specimen exposed to simultaneous deformation and solidification. The inverse procedure has been reported in (Glowacki & Hojny, 2009). Example results of inverse analysis are shortly described in succeeding subsections.

#### **7.1 Characteristic temperature levels**

296 Numerical Modelling

The density of carbon steels depends on the temperature and the existing fraction of liquid phase. The effect of alloying elements (except of coal) on the density of each steel phase is small, although the concentration of these components significantly affect the fraction of the

> �� � ���� � ���� � ���� ��� �� � ���� � ���� � ���� ��� �� � ���� � ���� � ���� ���

In equation (58) ��, �� and �� indicate densities of liquid steel and its � and � phases, respectively. The density in the regions of occurrence of several phases simultaneously is

� � ����� � ��

� � ����� � ��

����� are density differences between � and � phases for temperature drop from liquidus to solidus level, X� and X� are the fractions of � i � phases in surrounding liquid phase, respectively. and finally �� i �� are relative fractions of � i � phases. The density of ���

The subsequent part of the chapter deals with the computation of mushy steel flow stress curves based on the developed mathematical model which helps to avoid interpretational problems occurring in traditional testing procedures. Proper interpretation of the experimental results is possible with the help of appropriate computer aided testing system. Such a user friendly dedicated computer system with variable density has been developed (Glowacki & Hojny, 2009; Hojny & Glowacki, 2009a). The system codename called *Def\_Semi\_Solid* is a result of theoretical research conducted in a team lead by the chapter author with the financial support of grants awarded by Polish Committee of Scientific Research. The system in itself is not a subject of the chapter and its details are not discussed. The program was developed using an object oriented technique and is compatible with both

During experiments a few quantities were recorded. Among them the most important are GLEEBLE jaws displacement, force and temperature. This is a start point for the inverse analysis. The system calculates the shape and size of the deformation zone and strain and stress fields as well as optimal values of flow stress curve parameters. The model described in the previous section allows for the comparison of theoretical and experimental results for non-uniform temperature field. Isothermal tests in the temperature range over 1400 °C are impossible even using sophisticated equipment like GLEEBLE simulator. The presented model is a solution to the experimental problems. The

� � ���� � �� � ����� � ��

� denotes the density of the liquid phase for temperature discrepancy ���, ����� and

���� � �� � �� � �� � �� (60)

(58)

(59)

phases. The density of each phase is calculated according to the following equations:

���� � ��

���� � ��

������ � ��

given by the following equations:

phase was estimated according to relationship:

Windows and Unix based platforms.

**7. Mushy steel flow stress curves development** 

where ��

As mentioned before, apart from the liquidus and solidus temperatures, four other temperature levels are characteristic for the mushy steel behaviour. All the levels split the liquidus-solidus range into intervals. The most important for the extra high temperature rolling process design is the nil ductility temperature (NDT). The plastic deformation of a steel specimen is possible only below the NDT temperature. The temperature levels have to be calculated according to results of series of difficult experiments which are not a subject of the current paper. For carbon steels with the carbon content of around 0.1 % the equilibrium liquidus and solidus temperature levels are 1523°C and 1482°C, respectively and the NDT is 1420°C. One must note that the last one is a conventional temperature of a sample surface (indicated during experimental procedure). The maximum and minimum temperatures in the sample's central cross-section may differ by 60- 70 °C. The equilibrium liquidus and solidus temperatures for 18G2A grade steel are 1513°C and 1465°C, respectively. The measured mean value of NDT temperature of the steel falls into the range of 1420°C÷1425°C. The NDT is related to the temperature at which the last liquid phase particles existing in the central part of the sample disappear in static processes. It has been observed that for temperatures higher than NDT a remainder of liquid phase still exist in the central part of the sample (Hojny & Glowacki, 2009a). For dynamic cooling and deformation processes in some regions of the sample the remainder of liquid phase can be observed at temperatures lower than NDT because the difference between sample surface and its central region is higher than for quasi-static processes.

#### **7.2 Yield stress functions**

The well-known Voce formula (Voce, 1955) was adopted for the description of the shape of yield stress function. Figure 3 presents four subsequent stages of an example compression test at higher sample surface temperature, i.e. 1425°C for the quasi-static process. One can observe that the experiment was successful (no metal outflow) and the deformation of the sample was realised despite the significant barrelling of the sample.

Inverse Analysis Applied to Mushy Steel

**8. Conclusions** 

Rheological Properties Testing Using Hybrid Numerical-Analytical Model 299

Fig. 4. Flow stress vs. strain at several temperature levels for 18G2A steel grade deformed

Example results of investigation of S355J2G3So grade steel are presented in Figure 5. The investigation procedures were analogous to those applied in case of18G2A grade steel.

Fig. 5. Stress-strain curves at several temperature levels from the range of 1400-1450°C (S355J2G3So grade steel, quasi-static process – strain rate 1 s-1 (Glowacki & Hojny, 2010).

Modelling of deformation of steel samples with mushy zone requires resolving several problems which are characteristic for the temperature range close to the solidus level. Some of the problems are independent of the strain and stress state of the material and are similar for both axial-symmetrical and three dimensional cases. The computation of characteristic temperatures and temperature-dependent sudden changes of steel plastic properties require advanced methods of computer simulation. The most important property for material

during quasi-static process – strain rate 1 s-1 (Glowacki & Hojny, 2010).

Fig. 3. Four stages of the deformation process ran at temperature 1425°C for the quasi–static deformation process. The figure presents the central part of the sample.

Due to significant strain inhomogeneity inverse analysis is the only method allowing for appropriate calculation of coefficients of yield stress functions at any temperature higher than NDT. The objective function of the analysis was defined as a norm of discrepancies between calculated (ܨ (and measured (ܨ (loads in a number of subsequent stages of the compression according to the following equation:

$$\varphi(\mathbf{x}) = \sum\_{l=1}^{n} (F\_l^c - F\_l^m)^2 \tag{61}$$

where ݊ is the number of subsequent intervals of stress versus strain curve. The theoretical forces ܨ were calculated with the help of sophisticated numerical solver being the implementation of the model which was described in this chapter. Due to the very low level of recorded stresses the experimental curves obtained from the GLEEBLE machine are noisy. Before the application of inverse analysis they were smoothed using Fast Fourier Transformation (FFT) algorithm.

The final shape of the curves for 18G2A and S355J2G3So grade steels after interpretation using inverse analysis are presented in figures 8 and 16, respectively. Figure 4 summarise the results of calculation of example coefficients of Voce formula for 18G2A grade steel which was deformed in a quasi-static process. The effective strain inside the deformation zone varied from 0 to 0.6 and the effective strain rate reached its maximum value of 2.9 s-1 in final stage of the deformation process. The presented curves are plotted using the calculated coefficients of Voce curve for temperature levels observed in the sam ples' crosssections and for strain rate equal to 1 *s*-1.

Fig. 3. Four stages of the deformation process ran at temperature 1425°C for the quasi–static

Due to significant strain inhomogeneity inverse analysis is the only method allowing for appropriate calculation of coefficients of yield stress functions at any temperature higher than NDT. The objective function of the analysis was defined as a norm of discrepancies between calculated (ܨ (and measured (ܨ (loads in a number of subsequent stages of the

where ݊ is the number of subsequent intervals of stress versus strain curve. The theoretical forces ܨ were calculated with the help of sophisticated numerical solver being the implementation of the model which was described in this chapter. Due to the very low level of recorded stresses the experimental curves obtained from the GLEEBLE machine are noisy. Before the application of inverse analysis they were smoothed using Fast Fourier

The final shape of the curves for 18G2A and S355J2G3So grade steels after interpretation using inverse analysis are presented in figures 8 and 16, respectively. Figure 4 summarise the results of calculation of example coefficients of Voce formula for 18G2A grade steel which was deformed in a quasi-static process. The effective strain inside the deformation zone varied from 0 to 0.6 and the effective strain rate reached its maximum value of 2.9 s-1 in final stage of the deformation process. The presented curves are plotted using the calculated coefficients of Voce curve for temperature levels observed in the sam ples' cross-

ܨ െ ሻ <sup>ଶ</sup>

ୀଵ (61)

߮ሺݔሻ ൌ σ ሺܨ

deformation process. The figure presents the central part of the sample.

compression according to the following equation:

Transformation (FFT) algorithm.

sections and for strain rate equal to 1 *s*-1.

Fig. 4. Flow stress vs. strain at several temperature levels for 18G2A steel grade deformed during quasi-static process – strain rate 1 s-1 (Glowacki & Hojny, 2010).

Example results of investigation of S355J2G3So grade steel are presented in Figure 5. The investigation procedures were analogous to those applied in case of18G2A grade steel.

Fig. 5. Stress-strain curves at several temperature levels from the range of 1400-1450°C (S355J2G3So grade steel, quasi-static process – strain rate 1 s-1 (Glowacki & Hojny, 2010).

#### **8. Conclusions**

Modelling of deformation of steel samples with mushy zone requires resolving several problems which are characteristic for the temperature range close to the solidus level. Some of the problems are independent of the strain and stress state of the material and are similar for both axial-symmetrical and three dimensional cases. The computation of characteristic temperatures and temperature-dependent sudden changes of steel plastic properties require advanced methods of computer simulation. The most important property for material

Inverse Analysis Applied to Mushy Steel

83-60958-59-9

9824205-4-4, USA

475–483, ISSN 1733-3490

7, ISSN 0094-4289

pp. 76-84, ISSN 0924-0136

1611-3683

501-504, ISSN 0924-0136

Publishing House Akapit, Krakow Poland

Rheological Properties Testing Using Hybrid Numerical-Analytical Model 301

Glowacki, M. (1996). Finite element three-dimensional modelling of the solidification of a

Glowacki, M. (1998). Thermal-mechanical–microstruktural model of shape rolling. *Dissertations and Monographs*, Vol. 76, AGH Publishing, Krakow Poland, ISSN 0867-6631 Głowacki, M. (2002) Possibilities of mathematical modeling of deformation of samples with

pp. 1151-1162, Orlando USA, September 1, 2002, ISBN 978-1-886362-62-8 Glowacki, M. (2006). Mathematical modelling of deformation of steel samples with mushy

Glowacki, M. & Hojny, M. (2006). Development of a computer system for high temperature

Glowacki, M. & Hojny, M. (2010). Investigation of mushy steel rheological properties at

Glowacki, M., Hojny, M. & Jędrzejczyk, D. (2010). Hybrid analytical-numerical system of

Hojny, M. & Glowacki, M. (2008). Computer modelling of deformation of steel samples with

Hojny, M. & Glowacki, M. (2009a) The methodology of strain – stress curves determination

Hojny, M. & Glowacki, (2009b) The physical and computer modelling of plastic deformation of

Kang, C.G. & Yoon, J.H. (1997). A finite-element analysis on the upsetting process of semi-

*Foundry Processes*, pp. 145-156, Krakow Poland, November 22-24, 2006 Glowacki, M. & Hojny, M. (2009). Inverse analysis applied for determination of strain-stress

*Engineering*, Vol.17, No. 2, pp. 159–174, ISSN 1741-5977

metal forming charge, *Journal of Materials Processing Technology*, Vol. 60, No. 1-4, pp.

mushy zone, *Proceedings of 44th Mechanical Working and Steel Processing Conference*,

zone, In: *Research in Polish metallurgy at the beginning of XXI century, Committee of Metallurgy of the Polish Academy of Science*, K. Swiatkowski, (Ed.), 305-324,

steel deformation testing procedure, *Proceedings of Simulation, Design and Control of* 

curves for steel deformed in semi-solid state, *Inverse Problems in Science and* 

temperatures close to solidus level, In: *Polish metallurgy 2006–2010 in time of the worldwide economic crisis , Committee of Metallurgy of the Polish Academy of Science*, K. Swiatkowski, (Ed.), 193-212, Publishing House Akapit, Krakow, Poland, ISBN 978-

mushy steel deformation. In : *Recent studeis in meshless & other novel computational methods*, B. Sarler & S.N. Atluri, (Eds.), pp. 35-54, Tech Science Press, ISBN-10 0-

mushy zone, *Steel Research International*, vol. 79, No. 11, (2008), pp. 868-874, ISSN

for steel in semi-solid state, *Archives of Metallurgy and Materials*, Vol. 54, No. 2, pp.

low carbon steel in semi-solid state, *Transactions of the ASME, Journal of Engineering Materials and Technology*, Vol. 131 No. 4, pp. 041003-1–041003-7, ISSN 0094-4289 Hojny, M., Glowacki, M. & Malinowski Z. (2009), Computer aided methodology of strain-

stress curve construction for steels deformed at extra high temperature, *High Temperature Materials and Processes*, Vol. 28, No. 4, pp. 245–252, ISSN 0334-6455 Hojny, M. & Glowacki, M. (2011). Modeling of Strain-Stress Relationship for Carbon Steel

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solid aluminum material, *Journal of Materials Processing Technology*, Vol. 66, No. 1-3,

plastic behaviour is the yield stress function describing strain-stress curve. The proposed analytical model allows computation of such kind relationships.

The chapter has been dedicated to hybrid numerical-analytical model of semi-solid steel behaviour under plastic deformation. Application of inverse analysis and the proposed model allows for the testing of rheological properties of steels at temperature higher than 1400 °C. The results of the research are crucial for a unique computer system allowing for proper interpretation of the results of very high temperature compression tests. The classical interpretation of such results is improper due to strong strain inhomogeneity. The developed system is a tool to overcome many interpretational problems allowing for the computation of the appropriate shape and parameters of yield-stress curves. The curves have crucial influence on the results of computer simulation of semi-solid steel deformation.

The model presented in the current contribution is an axial-symmetrical one. The author have run further investigations leading to the development of a fully three-dimensional model of integrated casting and rolling processes as well as the soft reduction process, that is a part of strip casting technology. Like the model presented in the hereby chapter the spatial one also focuses on three main aspects: thermal, mechanical and density changes. Further intention of the research is the development of fully three dimensional model of mushy steel behaviour during rolling of plates with mushy region. The model will be useful for technologists working on the development of an integrated casting and rolling process. It is the most recent technology of sheet steel production, which is very profitable and requires extremely low energy consumption – very important for steel and automotive industries.

The compression tests carried out have shown good predictive ability of the proposed solution. They show that the flow stress above the NDT is strongly temperature and strain rate dependent. Low carbon steels, having carbon content of 0.11% and 0.16%, have been investigated in wide temperature range and strain rate. Example results of the experimental work were presented delivering a set of equations describing rheological behaviour of the investigated steels. The presented model and experimental procedure requires further investigation leading to the improvement of the solution and modelling additional phenomena accompanying the simultaneous deformation and solidification processes.

#### **9. Acknowledgments**

The work has been supported by the Polish Ministry of Science and Higher Education - Grant No. N N508 585539.

#### **10. References**


plastic behaviour is the yield stress function describing strain-stress curve. The proposed

The chapter has been dedicated to hybrid numerical-analytical model of semi-solid steel behaviour under plastic deformation. Application of inverse analysis and the proposed model allows for the testing of rheological properties of steels at temperature higher than 1400 °C. The results of the research are crucial for a unique computer system allowing for proper interpretation of the results of very high temperature compression tests. The classical interpretation of such results is improper due to strong strain inhomogeneity. The developed system is a tool to overcome many interpretational problems allowing for the computation of the appropriate shape and parameters of yield-stress curves. The curves have crucial influence on the results of computer simulation of semi-solid steel deformation. The model presented in the current contribution is an axial-symmetrical one. The author have run further investigations leading to the development of a fully three-dimensional model of integrated casting and rolling processes as well as the soft reduction process, that is a part of strip casting technology. Like the model presented in the hereby chapter the spatial one also focuses on three main aspects: thermal, mechanical and density changes. Further intention of the research is the development of fully three dimensional model of mushy steel behaviour during rolling of plates with mushy region. The model will be useful for technologists working on the development of an integrated casting and rolling process. It is the most recent technology of sheet steel production, which is very profitable and requires extremely low

analytical model allows computation of such kind relationships.

energy consumption – very important for steel and automotive industries.

problems. Wiley, New York USA, ISBN 0471181935

New York USA, ISBN 987-1-4398-0247-2

PWN, Warszawa, ISBN 8301009764

ISBN 978-0-7506-6638-2

0821807722.

**9. Acknowledgments** 

Grant No. N N508 585539.

**10. References** 

The compression tests carried out have shown good predictive ability of the proposed solution. They show that the flow stress above the NDT is strongly temperature and strain rate dependent. Low carbon steels, having carbon content of 0.11% and 0.16%, have been investigated in wide temperature range and strain rate. Example results of the experimental work were presented delivering a set of equations describing rheological behaviour of the investigated steels. The presented model and experimental procedure requires further investigation leading to the improvement of the solution and modelling additional phenomena accompanying the simultaneous deformation and solidification processes.

The work has been supported by the Polish Ministry of Science and Higher Education -

Adhikari S.K. (1998). Variational principles for the numerical solution of scattering

Bower, A.F. (2010) Applied mechanics and solids, CRC Press – Taylor & Francis Group,

Chakrabarty, J. (2006). Theory of plasticity. Elsevier Butterworth-Heinemann, Oxford UK,

Evans, L.C. (1998), Partial Differential Equations, American Mathematical Society, ISBN

Findaeisen, W., Szymanowski, J., Wierzbicki, A (1980). Theory and optimization methods.


**1. Introduction** 

used;

materials;

long construction periods; Unknown construction sequence;

Unknown existing damage in the structure.

employed herein to model masonry structures.

masonry failure modes, according to Sutcliffe et *al*., 2001.

**14** 

*France* 

Marwan Al-Heib

**Distinct Element Method Applied** 

Masonry structures have specific aspects and different numerical approaches are available for studying their behavior. The analysis of masonry constructions is a complex task (Lourenco, 2002), especially under special loads and when the soil-structure interaction

Difficult and expensive characterization of the mechanical properties of the materials

Large variability of mechanical properties, due to workmanship and use of natural

Significant changes in the core and constitution of structural elements, associated with

In addition, under the different loading conditions, many experimental studies have shown that joints or interfaces are the weakest zones of masonry structures. Figure 1 shows some

Several methods and computational tools are available (Massart et al, 2005) for the assessment of the mechanical behavior of old constructions. The empirical approaches and the Eurocode (6) recommendations are generally satisfactory for engineers. The methods resort to different theories or approaches, resulting in: different levels of complexity (from simple graphical methods and hand calculations to complex mathematical formulations and large systems of non-linear equations), different availability for the practitioner (from readily available in any consulting engineer office to scarcely available in a few researchoriented institutions and large consulting offices), different time requirements (from a few seconds of computer time to a few days of processing) and, of course, different costs. Three approaches (Figure 2) are generally employed by engineers and researchers to model the masonry element: equivalent medium, discontinuous medium using continuous numerical approach (finite element and boundary element methods) and discontinuous medium using distinct element approach (distinct element method). The distinct element code will be

becomes essential for studying the real behavior. Usually, salient aspects are:

**on Old Masonry Structures** 

*Ineris – Ecole des Mines de Nancy, Parc de Saurupt* 


## **Distinct Element Method Applied on Old Masonry Structures**

Marwan Al-Heib

*Ineris – Ecole des Mines de Nancy, Parc de Saurupt France* 

### **1. Introduction**

302 Numerical Modelling

Koc, M., Vazquez, V., Witulski, T. & Altan, T. (1996). Application of the finite element

Kopp, R., Choi, J. & Neudenberger D. (2003). Simple compression test and simulation of an

Leader, J. J. (2004). Numerical Analysis and Scientific Computation. Addison Wesley, Boston

Li, J.Y., Sugiyama, S. & Yanagimoto, J. (2005), Microstructural evolution and flow stress of

Malinowski, Z. (1986). Analysis of upsetting process based on velocity fields, PhD thesis,

Malinowski, Z. (1997). Effect of heat generation on flow stress deformation based on the

Malinowski Z. (2005). Numerical models in metal forming and heat transfer. Wyd. AGH

Mizukami, H., Yamanaka, A. & Watanabe, T. (2002). Prediction of density of carbon steels,

Nocedal J. & Wright S.J. (2006). Numerical Optimization. Springer-Verlag, Berlin Germany.

Pinchover, Y. & Rubinstein, J. (2005). An Introduction to Partial Differential Equations, New

Polyanin, A.D. & Zaitsev, V.F. (2004). Handbook of Nonlinear Partial Differential Equations,

Sang-Yong, L., Jung-Hwan L. & Young-Seon L. (2001). Characterization of Al 7075 alloys

*Materials Processing Technology*, Vol. 111, No. 1-3, pp. 42-47, ISSN 0924-0136 Seol, D.J., Won, Y.M., Yeo, T., Oh, K.H., Park, J.K. & Yim, C.H. (1999). High Temperature

Seol, D.J., Oh, K.H., Cho, J.W., Lee, J.E., Yoon, U.S. (2002). Phase-field modelling of the

Senk, D., Hagemann, F., Hammer, B., Kopp, R., Schmitz, H.P. & Schmitz, W. (2000).

Suzuki, H.G., Nishimura, S. & Yamaguchi S. (1988). Physical simulation of the continuous

Voce, E. (1955). A Practical Strain Hardening Function, *Metallurgia*, vol. 51, 1955, pp. 219-226,

Zhao Y.Q., Wu W.L. & Chang H. (2006). Research on microstructure and mechanical

Zienkiewicz, O.C., Taylor, R. L. & Zhu, J.Z. (2005). The Finite Element Method: Its Basis and Fundamentals, Elsevier Butterworth-Heinemann, Oxford UK, ISBN 0-7506-6320-0

*Con- tinuous Casting*, pp. 166-191, Canmet Canada, May 2-4, 1988

*and Engineering*, Vol. 416, No. 1-2, pp. 181-186, ISSN 0921-5093

after cold working and heating in the semi-solid temperature range. *Journal of* 

Deformation Behavior of Carbon Steel in the Austenite and δ-Ferrite Regions, *ISIJ* 

thermo-mechanical properties of carbon steels, Acta Materialia, Vol. 50, No. 9, pp.

Umformen und Kühlen von direktgegossenem, Stahlband, *Stahl und Eisen*, Vol. 120,

casting of steels, *Proceedings of Physical Simulation of Welding, Hot Forming and* 

properties of a new α + Ti2Cu alloy after semi-solid deformation, *Materials Science* 

Publishing, Krakow Poland, in Polish, ISBN: 83-89388-98-7

*ISIJ International*, Vol. 42, No. 4, pp. 375-384, ISSN 0915-1559

Boca Raton: Chapman & Hall/CRC Press, ISBN 1584883553.

York: Cambridge University Press, ISBN 0521848865.

*International*, Vol. 39, No. 1, pp. 91-98, ISSN 0915-1559

112, ISSN 0924-0136

Vol. 135, No. 2-3, pp. 317-323, ISSN 0924-0136

Massachusetts, ISBN 978-0-201-73499-7

161, No. 3, pp. 396-406, ISSN 0924-0136

AGH Krakow Poland, in Polish

459-467, ISSN 1239-2325

ISBN 0-387-30303-0

2259-2268, ISSN 1359-6454

ISSN 0141-8602

No. 6, pp. 65-69, ISSN 0340-4803

method to predict material flow and defects in the semi-solid forging of A356 aluminum alloys, *Journal of Materials Processing Technology*, Vol. 59, No. 4, pp. 106-

Sn–15% Pb alloy in the semi-solid state, *Journal of Materials Processing Technology*,

semi-solid type 304 stainless steel, *Journal of Materials Processing Technology*, Vol.

axially symmetric compression test. Metallurgy & Foundry Engineering, 23, 1997,

Masonry structures have specific aspects and different numerical approaches are available for studying their behavior. The analysis of masonry constructions is a complex task (Lourenco, 2002), especially under special loads and when the soil-structure interaction becomes essential for studying the real behavior. Usually, salient aspects are:


In addition, under the different loading conditions, many experimental studies have shown that joints or interfaces are the weakest zones of masonry structures. Figure 1 shows some masonry failure modes, according to Sutcliffe et *al*., 2001.

Several methods and computational tools are available (Massart et al, 2005) for the assessment of the mechanical behavior of old constructions. The empirical approaches and the Eurocode (6) recommendations are generally satisfactory for engineers. The methods resort to different theories or approaches, resulting in: different levels of complexity (from simple graphical methods and hand calculations to complex mathematical formulations and large systems of non-linear equations), different availability for the practitioner (from readily available in any consulting engineer office to scarcely available in a few researchoriented institutions and large consulting offices), different time requirements (from a few seconds of computer time to a few days of processing) and, of course, different costs. Three approaches (Figure 2) are generally employed by engineers and researchers to model the masonry element: equivalent medium, discontinuous medium using continuous numerical approach (finite element and boundary element methods) and discontinuous medium using distinct element approach (distinct element method). The distinct element code will be employed herein to model masonry structures.

Distinct Element Method Applied on Old Masonry Structures 305

A discontinuous medium is distinguished from a continuous medium by the existence of interfaces or contacts between the discrete bodies that comprise the system. Discrete methods can be categorized both by the way they represent contacts and by the way they

In the second group (using a hard-contact approach), interpenetration is regarded as nonphysical, and algorithms are used to prevent any interpenetration of the two bodies that form a contact. The discrete (or distinct) element methods fall within the general classification of discontinuous analysis techniques. Originally used to model jointed and fractured rock masses (Tzamtzis et al, 2004), they were developed for the analysis of structures composed of particles or blocks and are especially suitable for problems in which a significant part of the deformation is accounted for by relative motion between blocks. *Masonry provides a natural application for these techniques*, as the deformation and failure modes of these structures are strongly dependent on the role of the joints. This approach is well suited for collapse analysis, and may thus provide support for studies of safety

Two main features of the discrete element method (DEM) led to its use for the analysis of masonry structures. One is the allowance for large displacements and rotations between blocks, including their complete detachment. The other is the automatic detection of new contacts as the calculation progresses. Block material may be assumed rigid or deformable. Concerning masonry blocks, they are generally bonded by a lime or cement mortar. The model does not take the thickness of the mortar into account. Many numerical works have been performed for modeling masonry structures with the discrete elements methods (Verdel, 1994, Lemos, 1998). These studies looked essentially to the dynamic solicitation on

In discrete element models, the representation of the interface between blocks relies on sets of point contacts (Figure 4 and Figure 6). Adjacent blocks can touch along a common edge

assessment, namely of historical stone masonry structures under earthquakes.

represent the discrete bodies in the numerical formulation.

Fig. 3. Distinct element method and principal aspects.

dams and historic buildings.

**2.2 Masonry joint modeling** 

Fig. 1. Masonry failure modes a- direct tensile cracking of joint, b-sliding along joint ccracking of unit and joint, diagonal tensile cracking of units e-compressive failure due to mortar militancy (Idris et al, 2009).

Fig. 2. Different approaches to model the behavior of masonry structures (Idris et al, 2009).

Two case studies will be presented in this chapter. The first case study concerns the simulation of the behavior of an underground structure of old tunnel supported by masonry of stone elements. The second case study concerns particularly the behavior of a masonry wall under the effect of an underground excavation (tunnel, mine, soil settlement, etc.).

### **2. Distinct element method**

#### **2.1 Description and background**

A numerical model must represent two types of mechanical behavior in a discontinuous system: (1) behavior of the discontinuities; and (2) behavior of the solid material. In addition, the model must recognize the existence of contacts or interfaces between the discrete bodies that comprise the system. Numerical methods are divided into two groups according to the way in which they treat behavior in the normal direction of motion at contacts. In the first group (using a soft-contact approach), a finite normal stiffness is taken to represent the measurable stiffness that exists at a contact or joint.

The distinct element method was presented for the first time by Cundall in Nancy (1971), it considers the medium as an assembly of distinct rigid blocs that are linked together by joints. One can distinguish between rigid blocs and deformable blocs. Deformable blocs can be studied using the difference element method.

**b c**

**d e**

Fig. 1. Masonry failure modes a- direct tensile cracking of joint, b-sliding along joint ccracking of unit and joint, diagonal tensile cracking of units e-compressive failure due to

Fig. 2. Different approaches to model the behavior of masonry structures (Idris et al, 2009).

Two case studies will be presented in this chapter. The first case study concerns the simulation of the behavior of an underground structure of old tunnel supported by masonry of stone elements. The second case study concerns particularly the behavior of a masonry wall under the effect of an underground excavation (tunnel, mine, soil settlement, etc.).

A numerical model must represent two types of mechanical behavior in a discontinuous system: (1) behavior of the discontinuities; and (2) behavior of the solid material. In addition, the model must recognize the existence of contacts or interfaces between the discrete bodies that comprise the system. Numerical methods are divided into two groups according to the way in which they treat behavior in the normal direction of motion at contacts. In the first group (using a soft-contact approach), a finite normal stiffness is taken

The distinct element method was presented for the first time by Cundall in Nancy (1971), it considers the medium as an assembly of distinct rigid blocs that are linked together by joints. One can distinguish between rigid blocs and deformable blocs. Deformable blocs can

to represent the measurable stiffness that exists at a contact or joint.

be studied using the difference element method.

mortar militancy (Idris et al, 2009).

**a**

**2. Distinct element method 2.1 Description and background**  A discontinuous medium is distinguished from a continuous medium by the existence of interfaces or contacts between the discrete bodies that comprise the system. Discrete methods can be categorized both by the way they represent contacts and by the way they represent the discrete bodies in the numerical formulation.

Fig. 3. Distinct element method and principal aspects.

In the second group (using a hard-contact approach), interpenetration is regarded as nonphysical, and algorithms are used to prevent any interpenetration of the two bodies that form a contact. The discrete (or distinct) element methods fall within the general classification of discontinuous analysis techniques. Originally used to model jointed and fractured rock masses (Tzamtzis et al, 2004), they were developed for the analysis of structures composed of particles or blocks and are especially suitable for problems in which a significant part of the deformation is accounted for by relative motion between blocks. *Masonry provides a natural application for these techniques*, as the deformation and failure modes of these structures are strongly dependent on the role of the joints. This approach is well suited for collapse analysis, and may thus provide support for studies of safety assessment, namely of historical stone masonry structures under earthquakes.

Two main features of the discrete element method (DEM) led to its use for the analysis of masonry structures. One is the allowance for large displacements and rotations between blocks, including their complete detachment. The other is the automatic detection of new contacts as the calculation progresses. Block material may be assumed rigid or deformable. Concerning masonry blocks, they are generally bonded by a lime or cement mortar. The model does not take the thickness of the mortar into account. Many numerical works have been performed for modeling masonry structures with the discrete elements methods (Verdel, 1994, Lemos, 1998). These studies looked essentially to the dynamic solicitation on dams and historic buildings.

#### **2.2 Masonry joint modeling**

In discrete element models, the representation of the interface between blocks relies on sets of point contacts (Figure 4 and Figure 6). Adjacent blocks can touch along a common edge

Distinct Element Method Applied on Old Masonry Structures 307

 τ = jks us(2) Where jks, jkn are joint shear stiffness and normal stiffness and us and un are shear displacement and normal displacement of joint. The value of jkn will depends on the contact area ratio between the two joint surfaces and the relevant properties of the joint filing material, if present (Souley, 1993). The value of jks depends on the roughness of the joint surface, which can be determined by the distribution, amplitude, and inclination of the asperities on the friction along the joint, the cohesion due to interlocking, and the strength of the filing material, if present. Figure 5 shows the evolution of joint behavior under normal and shear loads. Figure 6 resume a joint proprieties model for the distinct elements method. The following parameters are used to define the mechanical behavior of the contacts: the normal stiffness (kn), shear stiffness (ks), friction angle (), cohesion (c) and tensile strength (Rt). To approximate a displacement-weakening response, the Coulomb slip model with

Fig. 7. Elasto-plastic Mohr-Coulomb joint model, code (jcoh: joint cohesion, jfric: joint

The algorithm of the calculation method for the discrete element method (DEM) must take into account the fact that the geometry of the system, as well as the number and type of contacts between the discrete bodies, may change during the analysis. In the discrete element method the structural analysis, both static and dynamic, is based on explicit algorithms. Among the most important capabilities of DEM that make it very suitable for masonry structures could be mentioned: the ability to simulate progressive failure associated with crack propagation; the capability of simulating large displacements/rotations between blocks; the fact that contact points are updated automatically as block motion occurs and the fact that the problem of

The calculations performed in DEM alternate between applications of a force and displacement law at all contacts and Newton's second law at all blocks. The forcedisplacement law is used to find contact forces from known (and fixed) displacements.

friction angle and jtens joint tensile strength) (Itasca, 2000, UDEC).

interlocking is overcome by automatically rounding the corners.

The shear stress increment is calculated as:

residual strength (Figure 7) is used.

**2.3 Calculation method and algorithm** 

σn = jkn un(1)

=coh+ n tan fric (3)

segment or at discrete points where a corner meets an edge or another corner. At each contact, the mechanical interaction between blocks is represented by a force (stress), resolved into a normal (Fn or n) and a shear (Fs or ) component. Contact displacements are defined as the relative displacement between two blocks at the contact point. In the elastic range, contact forces and displacements are related through the contact stiffness parameters (normal and shear).

Fig. 4. Coulomb slip model with residual strength (shear and normal behavior).

Fig. 5. Interface model code (jkn: joint normal stiffness, jks joint shear stiffness, jcoh: joint cohesion, jfric: joint friction angle and jtens joint tensile strength) (Idris et al, 2009).

Fig. 6. Joint behavior (respectively) under normal and shear loads.

The mechanical behavior of joints is described as follows, (Itasca, 2000):

 The response to normal loading is expressed by the normal stiffness, jkn and normal displacement un:

$$\text{Aɗn} \equiv \text{jkn } \text{Aun} \tag{1}$$

The shear stress increment is calculated as:

306 Numerical Modelling

segment or at discrete points where a corner meets an edge or another corner. At each contact, the mechanical interaction between blocks is represented by a force (stress), resolved into a normal (Fn or n) and a shear (Fs or ) component. Contact displacements are defined as the relative displacement between two blocks at the contact point. In the elastic range, contact forces and displacements are related through the contact stiffness parameters

Fig. 6. Joint behavior (respectively) under normal and shear loads.

Plastic phase

Normal displacements (un)

displacement un:

Detailed Masonry structure

Elastic phase

jks

Normal stress (

n)

The mechanical behavior of joints is described as follows, (Itasca, 2000):

Fig. 4. Coulomb slip model with residual strength (shear and normal behavior).

Mortar Interfaces

Fig. 5. Interface model code (jkn: joint normal stiffness, jks joint shear stiffness, jcoh: joint cohesion, jfric: joint friction angle and jtens joint tensile strength) (Idris et al, 2009).

Masonry micro model Joint properties model

jks

Shear stress ()

K

Elastic phase

Shear displacements (us)

Plastic phase

block

block K

The response to normal loading is expressed by the normal stiffness, jkn and normal

(normal and shear).

$$
\Delta \mathbf{r} = \text{jks} \,\Lambda \,\text{us} \tag{2}
$$

Where jks, jkn are joint shear stiffness and normal stiffness and us and un are shear displacement and normal displacement of joint. The value of jkn will depends on the contact area ratio between the two joint surfaces and the relevant properties of the joint filing material, if present (Souley, 1993). The value of jks depends on the roughness of the joint surface, which can be determined by the distribution, amplitude, and inclination of the asperities on the friction along the joint, the cohesion due to interlocking, and the strength of the filing material, if present. Figure 5 shows the evolution of joint behavior under normal and shear loads. Figure 6 resume a joint proprieties model for the distinct elements method.

The following parameters are used to define the mechanical behavior of the contacts: the normal stiffness (kn), shear stiffness (ks), friction angle (), cohesion (c) and tensile strength (Rt). To approximate a displacement-weakening response, the Coulomb slip model with residual strength (Figure 7) is used.

Fig. 7. Elasto-plastic Mohr-Coulomb joint model, code (jcoh: joint cohesion, jfric: joint friction angle and jtens joint tensile strength) (Itasca, 2000, UDEC).

#### **2.3 Calculation method and algorithm**

The algorithm of the calculation method for the discrete element method (DEM) must take into account the fact that the geometry of the system, as well as the number and type of contacts between the discrete bodies, may change during the analysis. In the discrete element method the structural analysis, both static and dynamic, is based on explicit algorithms. Among the most important capabilities of DEM that make it very suitable for masonry structures could be mentioned: the ability to simulate progressive failure associated with crack propagation; the capability of simulating large displacements/rotations between blocks; the fact that contact points are updated automatically as block motion occurs and the fact that the problem of interlocking is overcome by automatically rounding the corners.

The calculations performed in DEM alternate between applications of a force and displacement law at all contacts and Newton's second law at all blocks. The forcedisplacement law is used to find contact forces from known (and fixed) displacements.

$$\frac{du}{dt} = \frac{F}{m} \tag{4}$$

$$\frac{d\,\dot{u}}{dt} = \frac{u^{(t + \frac{\Delta t}{2})} - u^{(t - \frac{\Delta t}{2})}}{\Delta t} \tag{5}$$

$$
\dot{\mathfrak{u}}^{\left(\mathfrak{t} + \frac{\Delta t}{2}\right)} = \dot{\mathfrak{u}}^{\left(\mathfrak{t} - \frac{\Delta t}{2}\right)} + \frac{F^{\left(\mathfrak{t}\right)}}{m} \Delta t \tag{6}
$$

$$u^{\left(t+\Delta t\right)} = u^t + \dot{u}^{\left(t+\frac{\Delta t}{2}\right)\Delta t} \tag{7}$$

$$\dot{\mathfrak{u}}\_{l}^{\left(t+\frac{\Delta t}{2}\right)} = \dot{\mathfrak{u}}\_{l}^{\left(t-\frac{\Delta t}{2}\right)} + \dot{\left(\frac{\Sigma F\_{l}^{\left(t\right)}}{m} + g\_{l}\right)} \Delta t \tag{8}$$

$$\dot{\theta}\_l^{\left(\mathfrak{t} + \frac{\Delta t}{2}\right)} = \dot{\theta}\_l^{\left(\mathfrak{t} - \frac{\Delta t}{2}\right)} + (\frac{\Sigma^M l^{\left(\mathfrak{t}\right)}}{l})\Delta t \tag{9}$$

$$\mathbf{x}\_{l}^{(\mathbf{t}+\Delta t)} = \mathbf{x}\_{l}^{(\mathbf{t})} + \dot{\mathbf{u}}\_{l}^{(\mathbf{t}+\frac{\Delta t}{2})} \Delta t \tag{10}$$

$$
\theta\_l^{(t+\Delta t)} = \theta\_l^{(t)} + \theta\_l^{(t+\frac{\Delta t}{x})} \Delta t \tag{11}
$$

$$F\_l = F\_l^x + F\_l^c + F\_l^l \tag{12}$$

$$F\_l^{(2)} = \int\_{\mathcal{C}} \ \sigma\_{l\bar{l}} \ \ n\_{\bar{l}} \ d\_{\bar{s}} \tag{13}$$

Distinct Element Method Applied on Old Masonry Structures 311

The study focuses on the evolution of masonry support joint mechanical behavior in built tunnels over time. This study is carried out with the help of the experimental design strategy and numerical modeling by the well-known Universal Distinct Element Code

A tunnel masonry structure is a discontinuous medium consisting of blocks bonded to each other by mortar; in addition, such a structure forms an interface with the surrounding soil. The Distinct Element Method (DEM) is a suitable technique for modeling these structures. By means of the Universal Distinct Element Code (UDEC), a simplified micro-model of an ancient tunnel has been derived (Idris et al., 2009), (Figure. 10). The representative model is positioned at a shallow depth of 20 m. The masonry-supporting section consists of a regular rectangular and square limestone blocks (Figure. 10). Masonry blocks are bonded by lime mortar. The masonry support thickness is 80 cm and the sidewall height amounts to 3 m. By taking into account model section symmetry, only half of each set-up needed to be modeled.

The soil surrounding the tunnel consists of a homogeneous mix of clay and sand (Verdel and Bigarre, 1999). Table 1 lists the basic mechanical properties assigned to the surrounding soil, masonry and masonry joints, based on the work by Verdel and Bigarre (1999), Hoek

> **0.4 m 0.2 m**

**0.8 m 0.5 m**

**0.5 m 0.5 m**

**2 m**

**4 m**

Fig. 9. Illustration and examples of degradations of masonry tunnels.

Fig. 10. Old tunnel model using Universal Distinct Element Code UDEC.

(2000) and Janssen (1997).

(UDEC).

after each time step, the strain state of each zone is known. The program then needs to know the stress in each zone in order to proceed to the next time step. The stress is uniquely defined by the stress-strain model whether it is a linearly elastic relation or a complex, nonlinear and post-peak strength model.

The basic failure model for blocks is the Mohr-Coulomb failure criterion with a nonassociated flow rule. Other nonlinear plasticity models available in *Distinct Element codes (UDEC, 3DEC)* are the Drucker-Prager failure criterion, the ubiquitous joint model and strain-softening models for both shear and volumetric (collapse) yield.

#### **2.4 Data limited problems and numerical modeling recommendations**

It is necessary when data are limited to measure the quality and the quantity of available data in order to help the understanding of the problem to be solved. In soil-structure interaction and in many other branches of engineering geology, this is a category with limited data available. In this case, it is necessary before starting to be clear on why we are building the model; a good conceptual model can lead to savings in time and money on field tests that are better designed, and then a first model to identify with realistic data and then analyzing the mechanism of the problem, the visualization and the analysis help to understand the behavior of the model. Once we have learned all we can from the simple model or models, then a more complex model can be used. We apply this methodology in this paper.

#### **3. Tunnel masonry structure support proposed models**

The first case study (Idris et al, 2009) concerns the simulation of the behavior of an underground structure (Figure 9): old tunnel supported by masonry of stone elements, through this example, we insist on the importance of discontinuities behavior and their characterization (friction angle and cohesion). A majority of the world's tunnels are currently more than 100 years old; these would all be considered as ancient infrastructure. Old tunnels are often supported by a masonry structure. The type of masonry support or lining depends upon utilization of the high compressive strength in the stones, which explains the vaulted section shape of old tunnels supported by masonry.

Old underground constructions, especially tunnels, display specific characters regarding behavioral evolution over time. Infrastructure environment, surrounding ground and used construction materials all contribute to this evolution. Apart from the environment and evolution in surrounding soil and in the absence of an effective isolation system for such underground structures, subsoil water can easily penetrate the masonry joints and circulate within. Over time and in the presence of other aggressive ambient factors, several physical, chemical and biological processes may develop inside the masonry structure; this phenomenon and its impact are collectively called the tunnel-ageing phenomenon. One impact is the alteration in mechanical properties of construction materials (masonry structure composed of blocks and mortar). As a result, various types of disorders appear inside old tunnels (Figure 9); these would include: longitudinal or transverse structural cracks, convergence and partial masonry collapse. The instability of old tunnels depends on the interaction between soil, tunnel support (blocks and mortar).

after each time step, the strain state of each zone is known. The program then needs to know the stress in each zone in order to proceed to the next time step. The stress is uniquely defined by the stress-strain model whether it is a linearly elastic relation or a complex,

The basic failure model for blocks is the Mohr-Coulomb failure criterion with a nonassociated flow rule. Other nonlinear plasticity models available in *Distinct Element codes (UDEC, 3DEC)* are the Drucker-Prager failure criterion, the ubiquitous joint model and

It is necessary when data are limited to measure the quality and the quantity of available data in order to help the understanding of the problem to be solved. In soil-structure interaction and in many other branches of engineering geology, this is a category with limited data available. In this case, it is necessary before starting to be clear on why we are building the model; a good conceptual model can lead to savings in time and money on field tests that are better designed, and then a first model to identify with realistic data and then analyzing the mechanism of the problem, the visualization and the analysis help to understand the behavior of the model. Once we have learned all we can from the simple model or models, then a more complex model can be used. We apply this

The first case study (Idris et al, 2009) concerns the simulation of the behavior of an underground structure (Figure 9): old tunnel supported by masonry of stone elements, through this example, we insist on the importance of discontinuities behavior and their characterization (friction angle and cohesion). A majority of the world's tunnels are currently more than 100 years old; these would all be considered as ancient infrastructure. Old tunnels are often supported by a masonry structure. The type of masonry support or lining depends upon utilization of the high compressive strength in the stones, which

Old underground constructions, especially tunnels, display specific characters regarding behavioral evolution over time. Infrastructure environment, surrounding ground and used construction materials all contribute to this evolution. Apart from the environment and evolution in surrounding soil and in the absence of an effective isolation system for such underground structures, subsoil water can easily penetrate the masonry joints and circulate within. Over time and in the presence of other aggressive ambient factors, several physical, chemical and biological processes may develop inside the masonry structure; this phenomenon and its impact are collectively called the tunnel-ageing phenomenon. One impact is the alteration in mechanical properties of construction materials (masonry structure composed of blocks and mortar). As a result, various types of disorders appear inside old tunnels (Figure 9); these would include: longitudinal or transverse structural cracks, convergence and partial masonry collapse. The instability of old tunnels depends on

strain-softening models for both shear and volumetric (collapse) yield.

**3. Tunnel masonry structure support proposed models** 

explains the vaulted section shape of old tunnels supported by masonry.

the interaction between soil, tunnel support (blocks and mortar).

**2.4 Data limited problems and numerical modeling recommendations** 

nonlinear and post-peak strength model.

methodology in this paper.

The study focuses on the evolution of masonry support joint mechanical behavior in built tunnels over time. This study is carried out with the help of the experimental design strategy and numerical modeling by the well-known Universal Distinct Element Code (UDEC).

A tunnel masonry structure is a discontinuous medium consisting of blocks bonded to each other by mortar; in addition, such a structure forms an interface with the surrounding soil. The Distinct Element Method (DEM) is a suitable technique for modeling these structures. By means of the Universal Distinct Element Code (UDEC), a simplified micro-model of an ancient tunnel has been derived (Idris et al., 2009), (Figure. 10). The representative model is positioned at a shallow depth of 20 m. The masonry-supporting section consists of a regular rectangular and square limestone blocks (Figure. 10). Masonry blocks are bonded by lime mortar. The masonry support thickness is 80 cm and the sidewall height amounts to 3 m. By taking into account model section symmetry, only half of each set-up needed to be modeled.

Fig. 9. Illustration and examples of degradations of masonry tunnels.

The soil surrounding the tunnel consists of a homogeneous mix of clay and sand (Verdel and Bigarre, 1999). Table 1 lists the basic mechanical properties assigned to the surrounding soil, masonry and masonry joints, based on the work by Verdel and Bigarre (1999), Hoek (2000) and Janssen (1997).

Fig. 10. Old tunnel model using Universal Distinct Element Code UDEC.

Distinct Element Method Applied on Old Masonry Structures 313

The first experimental design represents the evolution of joint cohesion, joint tensile strength, and joint friction angle. To evaluate the influence of each chosen factor on masonry structure behavior, it proved necessary to observe significant changes in model behavior

Two significant response factors were detected; herein is the cumulated length of open joints and the cumulated length of joints at limiting friction (slip joints). Open joint means that the induced tension stress is greater than joint tension strength and slip joint means the shear

For this purpose, a complete factorial design was proposed; this three-level design is written as Kn factorial design (with K = 3: the studied factor number, n: level number). This nomenclature means that three factors are considered, each one at three distinct levels (Barrentine, 1999). Consequently, a complete factorial design with 27 experiments was proposed. Table 2 contains all of the experimental results (i.e. changed experimental factors and observed responses). In all simulations, soil and masonry blocks properties have been

Param. Unit Value Param. Unit Value Param. Unit Value M Kg/m3 1900 M Kg/m3 2000 jkn GPa/m 5 E MPa 100 E MPa 10000 jks GPa/m 2 0.3 0,3 jcoh MPa 1 C MPa 0.1 C MPa 6 jfric ° 40 ° 30 ° 60 jtens MPa 0

M: Volumic mass; E: Young modulus; : Poisson's ratio; C: Cohesion; : Friction angle; T: Tensile strength; jkn, jks: Normal, tangential joint stiffness; jcoh: Joint cohesion; jfric: joint friction angle; jtens: joint tensile

Figure 11 and Figure 12 provide some selected results, which show the influence of joint parameters on the length of open and slip joint evolution on mechanical behavior of

Fig. 11. A sample of numerical simulation results; the observed response is the total length

Joint fill material proprieties (cohesion, tensile strength, friction angle).

strength is less than the induced shear stress using Mohr-Coulomb criteria.

Surrounding ground Masonry Masonry joints

once factor values had been changed.

given the unchanged values shown in Table 2.

Tr MPa 0.10 Tr MPa 3

Table 2. Characterization of soil, blocs and joints of masonry.

strength

masonry support structure.

of open joints (n<JRT).


M: Volumic mass; E: Young modulus; : Poisson's ratio; C: Cohesion; : Friction angle; T: Tensile strength;

jkn, jks: Normal, tangential joint stiffness; jcoh: Joint cohesion; jfric: joint friction angle; jtens: joint tensile strength

Table 1. Mechanical properties choice of surrounding ground and masonry.

Calculations were carried out in plane strain: the soil and masonry follow a perfect elastoplastic Mohr–Coulomb plasticity criterion. The calculation step was proceeded by two main stages: model consolidation in the initial stress condition prior to tunnel excavation; and tunnel excavation and simultaneous installation of masonry support. The calculation could then be continued until reaching model equilibrium.

#### **3.1 Ageing simulation of masonry joints in built tunnels**

This part of our study sought to understand the evolution in masonry joints behavior over time and evaluate the influence of certain masonry joint mechanical properties on the behavior of masonry old tunnels and the surrounding ground. To simplify the simulation of the complex ageing phenomenon, the strategy has consisted of utilizing experimental designs and response surfaces in combination with various data analyses in order to identify the most powerful experimental factors influencing joint masonry structure behavior over time. A factorial experiment entails a statistical study in which each observation is categorized according to more than one factor. Such an experimental set-up makes it possible to study the effect of each factor on the response variable, while requiring fewer observations than when conducting separate experiments for each factor independently. It also allows studying the effect of the interaction between factors on the response variable (Barrentine, 1999).

In this study, two different experimental designs were proposed to simulate the evolution of mechanical properties of joints. The first experimental design explains the evolution of joints filling material properties and the second explains the evolution of ratio (normal stiffness/ shear stiffness).

#### **3.2 First proposed experimental design**

Many factors may influence joint mechanical behavior parameters. In order to predict the 'potential behavior' of the joints under loading, three distinct joint parameters must be introduced into the analysis (Goodman et *al.*, 1968):


*Param. Unit Value Param. Unit Value Param. Unit Value M* Kg/m3 1900 *M* Kg/m3 2000 *jkn* GPa/m 150 E MPa 200 E MPa 6000 jks GPa/m 69.7 0.3 0.2 jcoh MPa 1.2 C MPa 0.50 C MPa 3 jfric ° 25 ° 20 ° 30 jtens MPa 0.4

M: Volumic mass; E: Young modulus; : Poisson's ratio; C: Cohesion; : Friction angle; T: Tensile

Table 1. Mechanical properties choice of surrounding ground and masonry.

jkn, jks: Normal, tangential joint stiffness; jcoh: Joint cohesion; jfric: joint friction angle; jtens: joint tensile

Calculations were carried out in plane strain: the soil and masonry follow a perfect elastoplastic Mohr–Coulomb plasticity criterion. The calculation step was proceeded by two main stages: model consolidation in the initial stress condition prior to tunnel excavation; and tunnel excavation and simultaneous installation of masonry support. The calculation could

This part of our study sought to understand the evolution in masonry joints behavior over time and evaluate the influence of certain masonry joint mechanical properties on the behavior of masonry old tunnels and the surrounding ground. To simplify the simulation of the complex ageing phenomenon, the strategy has consisted of utilizing experimental designs and response surfaces in combination with various data analyses in order to identify the most powerful experimental factors influencing joint masonry structure behavior over time. A factorial experiment entails a statistical study in which each observation is categorized according to more than one factor. Such an experimental set-up makes it possible to study the effect of each factor on the response variable, while requiring fewer observations than when conducting separate experiments for each factor independently. It also allows studying the

effect of the interaction between factors on the response variable (Barrentine, 1999).

In this study, two different experimental designs were proposed to simulate the evolution of mechanical properties of joints. The first experimental design explains the evolution of joints filling material properties and the second explains the evolution of ratio (normal stiffness/

Many factors may influence joint mechanical behavior parameters. In order to predict the 'potential behavior' of the joints under loading, three distinct joint parameters must be

 The unit stiffness across the joint, jkn, which characterizes the elastic phase behavior; The unit stiffness along the joint, jks, which characterizes the elastic phase behavior;

Surrounding ground Masonry Masonry joints

*Tr* MPa 0.10 T MPa 1

then be continued until reaching model equilibrium.

**3.1 Ageing simulation of masonry joints in built tunnels** 

strength;

strength

shear stiffness).

**3.2 First proposed experimental design** 

introduced into the analysis (Goodman et *al.*, 1968):

Joint fill material proprieties (cohesion, tensile strength, friction angle).

The first experimental design represents the evolution of joint cohesion, joint tensile strength, and joint friction angle. To evaluate the influence of each chosen factor on masonry structure behavior, it proved necessary to observe significant changes in model behavior once factor values had been changed.

Two significant response factors were detected; herein is the cumulated length of open joints and the cumulated length of joints at limiting friction (slip joints). Open joint means that the induced tension stress is greater than joint tension strength and slip joint means the shear strength is less than the induced shear stress using Mohr-Coulomb criteria.

For this purpose, a complete factorial design was proposed; this three-level design is written as Kn factorial design (with K = 3: the studied factor number, n: level number). This nomenclature means that three factors are considered, each one at three distinct levels (Barrentine, 1999). Consequently, a complete factorial design with 27 experiments was proposed. Table 2 contains all of the experimental results (i.e. changed experimental factors and observed responses). In all simulations, soil and masonry blocks properties have been given the unchanged values shown in Table 2.


M: Volumic mass; E: Young modulus; : Poisson's ratio; C: Cohesion; : Friction angle; T: Tensile strength; jkn, jks: Normal, tangential joint stiffness; jcoh: Joint cohesion; jfric: joint friction angle; jtens: joint tensile strength

Table 2. Characterization of soil, blocs and joints of masonry.

Figure 11 and Figure 12 provide some selected results, which show the influence of joint parameters on the length of open and slip joint evolution on mechanical behavior of masonry support structure.

Fig. 11. A sample of numerical simulation results; the observed response is the total length of open joints (n<JRT).

Distinct Element Method Applied on Old Masonry Structures 315

Fig. 14. Response surfaces for relating the total length of joints at limiting friction with the data set (jcoh: joint cohesion, jtens: joint tensile strength and jfric: joint friction angle).

**jtens (MPa) jcoh (MPa) jfric(°) jfric(°) jtens (MPa) jcoh (MPa)**

0,8

0,1 0,2 0,3 0,4 0,5 0,6 0,7

> 0,2 0,3 0,4 0,5 0,6 0,7 0,8

0,9 Jc

The determination of jkn and jks is a complex operation and many authors propose experimental studies. The second experimental design expresses just the evolution of the

The (jkn/jks) ratio may exceed 100 (Souley, 1993), so this experimental design suggests that the ratio (jkn/jks) changes between 2 and 100, where jkn remains constant and only jks changes its value. It proved necessary to observe significant changes in model behavior once the (jkn / jks) ratio had been changed. The observed response in this step of the study is the total length of shear displacements detected on the tunnel masonry support section after every modeling test. Table 3 provides the detailed experimental design and the obtained results. This experimental design contains 12 modeling tests. Figure 15 provides some selected results, which show the influence of (jkn/jks) ratio evolution on mechanical

Figure 16 shows the relations between (jkn/jks) ratio and observed response of the cumulated length of joint shear displacements. On figure 16, we can distinguish a remarkable rapid increase in the total length of joint shear displacements when the (jkn/jks)

ratio (jkn/jks) to evaluate its influence on masonry structure behavior, where:

<sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10121416182022</sup> <sup>24</sup> <sup>26</sup>

**3.4 Second proposed experimental design** 

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

°

 jkn is joint normal stiffness; jks is joint shear stiffness.

0,2 0,3 0,4 0,5 0,6 0,7 0,8

behavior of masonry support structure.

Fig. 15. Some numerical simulation results examples.

ratio varies between 2 and 20.

Fig. 12. A sample of numerical simulation results, the observed response is the total length of joints at limiting friction (slip joints).

#### **3.3 Response surface analysis**

To summarize the experimental design results, 3D graphical response surfaces were generated, where the predicted responses (cumulated length of open joints and cumulated length at limiting friction) were indicated by a plane surface distance that relates every pair of modified factors (Kresic, 1997). These response surfaces yield a graphical indication of the reliability of results obtained; they also make it possible to compare dual influences from the studied factors and to observe possible interactions between them (Figure. 13 and Figure. 14).

Fig. 13. Response surfaces for relating the total length of open joints with the data set (jcoh: joint cohesion, jtens: joint tensile strength and jfric: joint friction angle).

Response surface (Figure. 14) analysis highlights a number of key points:

The joint cohesion has a higher influence than joint tensile strength on the cumulated length of open joints.


Fig. 12. A sample of numerical simulation results, the observed response is the total length

jcoh: 0.5 MPa jten: 0.25 MPa jfric : 5 °

**Test 18 Test 18** 

To summarize the experimental design results, 3D graphical response surfaces were generated, where the predicted responses (cumulated length of open joints and cumulated length at limiting friction) were indicated by a plane surface distance that relates every pair of modified factors (Kresic, 1997). These response surfaces yield a graphical indication of the reliability of results obtained; they also make it possible to compare dual influences from the studied factors and to observe possible interactions between them (Figure. 13 and Figure.

Fig. 13. Response surfaces for relating the total length of open joints with the data set

The joint cohesion has a higher influence than joint tensile strength on the cumulated length

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

0,2 0,3 0,4 0,5 0,6 0,7 0,8

 **300 250 200** 

<sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> <sup>22</sup> <sup>24</sup> <sup>26</sup>

**jfric (°) jtens (MPa) jcoh (MPa)**

jcoh: 0.2 MPa jten: 0.25 MPa jfric : 5 °

**Test 27 Test 27** 

 The evolution in two factors together (joint cohesion and joint friction angle) exerts remarkable influence on masonry block mechanical behavior (i.e. on the cumulated length of open joints) which means that these two parameters have an important

The difficulty is in comparing the influence of joint cohesion with that of joint friction

(jcoh: joint cohesion, jtens: joint tensile strength and jfric: joint friction angle).

101214161820222426

Response surface (Figure. 14) analysis highlights a number of key points:

4 6 8

**jfric (°)**

interaction influence on the observed response.

angle on the mechanical behavior of masonry.

of joints at limiting friction (slip joints).

**Test 1 Test 1** 

jcoh: 0.8 MPa jten: 0.75 MPa jfric : 25 °

**3.3 Response surface analysis** 

<sup>450</sup>**<sup>350</sup>**

**jcoh (MPa) jtens (MPa)**

14).

of open joints.

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 0,2 0,3 0,4 0,5 0,6 0,7 0,8

**Total length of open joints (cm)**

Fig. 14. Response surfaces for relating the total length of joints at limiting friction with the data set (jcoh: joint cohesion, jtens: joint tensile strength and jfric: joint friction angle).

#### **3.4 Second proposed experimental design**

The determination of jkn and jks is a complex operation and many authors propose experimental studies. The second experimental design expresses just the evolution of the ratio (jkn/jks) to evaluate its influence on masonry structure behavior, where:


The (jkn/jks) ratio may exceed 100 (Souley, 1993), so this experimental design suggests that the ratio (jkn/jks) changes between 2 and 100, where jkn remains constant and only jks changes its value. It proved necessary to observe significant changes in model behavior once the (jkn / jks) ratio had been changed. The observed response in this step of the study is the total length of shear displacements detected on the tunnel masonry support section after every modeling test. Table 3 provides the detailed experimental design and the obtained results. This experimental design contains 12 modeling tests. Figure 15 provides some selected results, which show the influence of (jkn/jks) ratio evolution on mechanical behavior of masonry support structure.

Figure 16 shows the relations between (jkn/jks) ratio and observed response of the cumulated length of joint shear displacements. On figure 16, we can distinguish a remarkable rapid increase in the total length of joint shear displacements when the (jkn/jks) ratio varies between 2 and 20.

Fig. 15. Some numerical simulation results examples.

Distinct Element Method Applied on Old Masonry Structures 317

An analysis of results indicated that the three studied masonry joint mechanical factors have significant influence on masonry mechanical behavior expressed by the total length of open joints. Joint cohesion is the most important factor, then joint tensile strength and finally joint friction angle. Only the interaction of joint cohesion and joint friction angle have significant influence on the total length of open joints. For the total length of joints at limiting friction, only joint cohesion and joint friction angle have a significant influence and joint cohesion remains the most important factor. These two factors have a significant interaction influence

The second proposed experiment design expressed the evolution of the ratio jkn/jks. The detected significant response was the total length of shear displacements along masonry joints. Results showed that shear displacements increase according to jkn/jks ratio increase. The total length of shear displacements evolution follows two types of relations as a function of the jkn/jks ratio; firstly it follows a non-linear relation according to certain jkn/jks values (jkn/jks=[2-20]), then it changes (increases slightly) behavior according to a linear model.

The second case study concerns in particular the behavior of a masonry wall under the effect of an underground excavation (tunnel, mine, soil settlement, etc.). The main objective is to verify and to improve the comprehension of masonry structure behavior using the

The excavation of tunnels and underground mines modifies the initial stress distribution and induces displacements on the ground surface (Peck, 1964, Standing, 2008, Al Heib, 2008). Figure 17 diagrammatically shows the surface subsidence trough above an advancing tunnel. The surface subsides and structures can be damaged due to the induced strains (Figure 17). For 'Greenfield sites', i.e. those without the presence of buildings or subsurface structures, the shape of this trough transverse to the axis of the tunnel closely approximates to a normal Gaussian distribution curve - an idealization which has considerable mathematical

advantages. The subsidence trough consists of vertical and horizontal displacements.

Fig. 17. Surface settlement above an advancing tunnel (Standing and Burland, 2008).

on the mechanical behavior of tunnel masonry support.

**4. Impact of underground movement on a masonry wall** 

numerical modeling approach by distinct element method.

Z0: depth of the tunnel, Smax: maximum subsidence

After that, the relation between the (jkn/jks) ratio and the cumulated length of joint shear displacements seems to be a linear relation and the cumulated length of joint shear displacements seems to increase slightly with the rise of the (jkn/jks) ratio more than 20. We can extract from the previous analysis that the (jkn /jks) ratio increases and directly influences the behavior of masonry support structure in old tunnels by the increase of shear displacement along affected joints.

#### **3.5 Influence of masonry joint proprieties evolution on the surrounding ground of tunnel**

The evolution of mechanical joint parameters (jcoh, jten, jfric, jkn and jks) did not have any significant influence on the mechanical behavior of the surrounding soil. The significant evolutions concern only the masonry joint behavior. Generally, in old tunnels, the use of the masonry support is strong enough, the models inspired from real built tunnels, with 80 cm of masonry support which have very strong support. Masonry structure is mainly loaded in compression due to its vaulted section shape. This massive support may explain the absence of block mechanical parameters that influence the behavior of surrounding soil (Idris et *al.*, 2008).

NLR: Non Lenar Relation LR: Linear Relation, Jkn : Joint normal stiffness, Jks : Joint shear stiffness Fig. 16. Graphical presentation of shear displacements as a function of the jkn/jks ratio.

#### **3.6 Conclusion**

The study concerned the behavior of masonry tunnel structures due to the ageing phenomena by using numerical modeling and experimental design. A first experimental design was proposed to simulate ageing effects of old tunnel behavior; a complete factorial experimental design, which expresses the evolution of three selected masonry joints mechanical properties, was then forwarded. The factors selected for the present study were: masonry joint cohesion, joint tensile strength and joint friction angle. All experimental design tests were modeled by means of the distinct element method. Two significant responses were detected, they are respectively: the total length of open joints; the total length of joints at limiting friction (slip joints).

After that, the relation between the (jkn/jks) ratio and the cumulated length of joint shear displacements seems to be a linear relation and the cumulated length of joint shear displacements seems to increase slightly with the rise of the (jkn/jks) ratio more than 20. We can extract from the previous analysis that the (jkn /jks) ratio increases and directly influences the behavior of masonry support structure in old tunnels by the increase of shear

**3.5 Influence of masonry joint proprieties evolution on the surrounding ground of** 

NLR: Non Lenar Relation LR: Linear Relation, Jkn : Joint normal stiffness, Jks : Joint shear stiffness Fig. 16. Graphical presentation of shear displacements as a function of the jkn/jks ratio.

The study concerned the behavior of masonry tunnel structures due to the ageing phenomena by using numerical modeling and experimental design. A first experimental design was proposed to simulate ageing effects of old tunnel behavior; a complete factorial experimental design, which expresses the evolution of three selected masonry joints mechanical properties, was then forwarded. The factors selected for the present study were: masonry joint cohesion, joint tensile strength and joint friction angle. All experimental design tests were modeled by means of the distinct element method. Two significant responses were detected, they are respectively: the total length of open joints; the total

The evolution of mechanical joint parameters (jcoh, jten, jfric, jkn and jks) did not have any significant influence on the mechanical behavior of the surrounding soil. The significant evolutions concern only the masonry joint behavior. Generally, in old tunnels, the use of the masonry support is strong enough, the models inspired from real built tunnels, with 80 cm of masonry support which have very strong support. Masonry structure is mainly loaded in compression due to its vaulted section shape. This massive support may explain the absence of block mechanical parameters that influence the behavior of surrounding soil (Idris et *al.*, 2008).

displacement along affected joints.

**tunnel** 

**3.6 Conclusion** 

length of joints at limiting friction (slip joints).

An analysis of results indicated that the three studied masonry joint mechanical factors have significant influence on masonry mechanical behavior expressed by the total length of open joints. Joint cohesion is the most important factor, then joint tensile strength and finally joint friction angle. Only the interaction of joint cohesion and joint friction angle have significant influence on the total length of open joints. For the total length of joints at limiting friction, only joint cohesion and joint friction angle have a significant influence and joint cohesion remains the most important factor. These two factors have a significant interaction influence on the mechanical behavior of tunnel masonry support.

The second proposed experiment design expressed the evolution of the ratio jkn/jks. The detected significant response was the total length of shear displacements along masonry joints. Results showed that shear displacements increase according to jkn/jks ratio increase. The total length of shear displacements evolution follows two types of relations as a function of the jkn/jks ratio; firstly it follows a non-linear relation according to certain jkn/jks values (jkn/jks=[2-20]), then it changes (increases slightly) behavior according to a linear model.

#### **4. Impact of underground movement on a masonry wall**

The second case study concerns in particular the behavior of a masonry wall under the effect of an underground excavation (tunnel, mine, soil settlement, etc.). The main objective is to verify and to improve the comprehension of masonry structure behavior using the numerical modeling approach by distinct element method.

The excavation of tunnels and underground mines modifies the initial stress distribution and induces displacements on the ground surface (Peck, 1964, Standing, 2008, Al Heib, 2008). Figure 17 diagrammatically shows the surface subsidence trough above an advancing tunnel. The surface subsides and structures can be damaged due to the induced strains (Figure 17). For 'Greenfield sites', i.e. those without the presence of buildings or subsurface structures, the shape of this trough transverse to the axis of the tunnel closely approximates to a normal Gaussian distribution curve - an idealization which has considerable mathematical advantages. The subsidence trough consists of vertical and horizontal displacements.

Z0: depth of the tunnel, Smax: maximum subsidence

Fig. 17. Surface settlement above an advancing tunnel (Standing and Burland, 2008).

Distinct Element Method Applied on Old Masonry Structures 319

continuous and simplified 2D numerical modeling. The building was modeled by weightless, elastic beams. Franzius (2004) presented the results of an extensive parametric study using 3D FE analysis. The relative stiffness expressions were introduced to relate the stiffness of a surface structure to the stiffness of the ground; they defined the relative

and �<sup>∗</sup> � ��

��� � ��

(14)

�<sup>∗</sup> � �� ��� � ���

Fig. 19. Transfer surface movement to the surface structure (Deck et al, 2003).

Son and Cording (2007) had evaluated the stiffness for a masonry building using the distinct element code UDEC, their studies were limited to evaluating the influence of Young modulus and shear modulus for different value parameters and taking into account the presence of windows and doors. Giorgia Giardina et al (2008), studied masonry wall damaging due to tunneling; they confirm, using finite elements, that the Greenfield approach is too conservative, and that the total interaction (coupling) approach is more realistic but needs more time and energy to obtain the results. The masonry wall is generally very vulnerable due to the compression strain and shear strain as shown in the pictures. The failure generally appears along the mortar joints around openings corresponding to weakness zones of the walls. To study the effect of soil-structure interaction and the role of the stiffness due to the horizontal strain, we consider the numerical modeling approach.

The present work is focused on the behavior of a masonry wall under horizontal strain. In addition to the evaluation of the stiffness of the masonry wall, the objective of the numerical

bending and axial stiffness respectively:

**4.3 Numerical model description** 

The amount of damage caused by subsidence depends upon the magnitude and the type of ground movements, structural factors and geological factors (Burland, 1995, Standing and Burland, 2008). The magnitude of ground movements on the structure are governed by its location and orientation in relation to the underground workings and the depth of underground cavities (Figure 18).

Fig. 18. Damages induced by compression strain due to soil subsidence (Deck et al., 2003).

#### **4.1 Damage due to horizontal strain**

The different types of movements can affect structures in different ways (Figure 19). Vertical subsidence may affect tall buildings (local tilt) and long buildings (deferential settlements). Horizontal extension and compression strain, tilt and curvature are the causes of the most commonly seen type of subsidence damages. Damage to buildings is generally caused by differential horizontal movements (horizontal strain) and the concavity and convexity of the subsidence profile. The extension horizontal strain is characterized by the fracturing of the masonry and the compression strain is characterized by squeezing-in of voids (doors and windows). Unlike settlements, there are fewer case histories where horizontal movements have been measured. The maximal horizontal displacement depends on the soil behavior and the geometry of the excavations; it generally equals 40% of the maximal vertical displacement (Lake et al, 1998). Horizontal displacements can be differentiated to give the horizontal strain εh at any location on the ground surface.

#### **4.2 Building damage assessment and soil-structure interaction**

It is clear from the above, considering tunnel construction and underground excavation, even with Greenfield conditions, that precise prediction of ground movements due to ground excavation is not realistic. However, it is possible, *for non-stiff buildings*, to make reasonable estimates of the likely range of movements provided excavation is carried out under the control of suitably qualified and experienced engineers and highlighted by numerical and physical modeling (Standing and Burland, 2008).

Building deformation and its potential damage caused by subsidence in urban areas has a major impact on the planning and construction process of any underground excavation. The use of Greenfield subsidence parameters to determine the level of the damages appears as a conservative approach; it can lead to costly projects. Potts and Addenbrooke (1997) and others (Dimmok et al, 2008) introduce design charts for tunnels to consider the influence of the buildings own stiffness, thus leading to more realistic predictions of induced deformations. The approach introduced by Potts and Addenbrooke was based on

The amount of damage caused by subsidence depends upon the magnitude and the type of ground movements, structural factors and geological factors (Burland, 1995, Standing and Burland, 2008). The magnitude of ground movements on the structure are governed by its location and orientation in relation to the underground workings and the depth of

Fig. 18. Damages induced by compression strain due to soil subsidence (Deck et al., 2003).

The different types of movements can affect structures in different ways (Figure 19). Vertical subsidence may affect tall buildings (local tilt) and long buildings (deferential settlements). Horizontal extension and compression strain, tilt and curvature are the causes of the most commonly seen type of subsidence damages. Damage to buildings is generally caused by differential horizontal movements (horizontal strain) and the concavity and convexity of the subsidence profile. The extension horizontal strain is characterized by the fracturing of the masonry and the compression strain is characterized by squeezing-in of voids (doors and windows). Unlike settlements, there are fewer case histories where horizontal movements have been measured. The maximal horizontal displacement depends on the soil behavior and the geometry of the excavations; it generally equals 40% of the maximal vertical displacement (Lake et al, 1998). Horizontal displacements can be differentiated to give the

It is clear from the above, considering tunnel construction and underground excavation, even with Greenfield conditions, that precise prediction of ground movements due to ground excavation is not realistic. However, it is possible, *for non-stiff buildings*, to make reasonable estimates of the likely range of movements provided excavation is carried out under the control of suitably qualified and experienced engineers and highlighted by

Building deformation and its potential damage caused by subsidence in urban areas has a major impact on the planning and construction process of any underground excavation. The use of Greenfield subsidence parameters to determine the level of the damages appears as a conservative approach; it can lead to costly projects. Potts and Addenbrooke (1997) and others (Dimmok et al, 2008) introduce design charts for tunnels to consider the influence of the buildings own stiffness, thus leading to more realistic predictions of induced deformations. The approach introduced by Potts and Addenbrooke was based on

underground cavities (Figure 18).

**4.1 Damage due to horizontal strain** 

horizontal strain εh at any location on the ground surface.

**4.2 Building damage assessment and soil-structure interaction** 

numerical and physical modeling (Standing and Burland, 2008).

continuous and simplified 2D numerical modeling. The building was modeled by weightless, elastic beams. Franzius (2004) presented the results of an extensive parametric study using 3D FE analysis. The relative stiffness expressions were introduced to relate the stiffness of a surface structure to the stiffness of the ground; they defined the relative bending and axial stiffness respectively:

and �<sup>∗</sup> � ��

�<sup>∗</sup> � ��

�

��� ��� ��� ��

(14)

�

Fig. 19. Transfer surface movement to the surface structure (Deck et al, 2003).

Son and Cording (2007) had evaluated the stiffness for a masonry building using the distinct element code UDEC, their studies were limited to evaluating the influence of Young modulus and shear modulus for different value parameters and taking into account the presence of windows and doors. Giorgia Giardina et al (2008), studied masonry wall damaging due to tunneling; they confirm, using finite elements, that the Greenfield approach is too conservative, and that the total interaction (coupling) approach is more realistic but needs more time and energy to obtain the results. The masonry wall is generally very vulnerable due to the compression strain and shear strain as shown in the pictures. The failure generally appears along the mortar joints around openings corresponding to weakness zones of the walls. To study the effect of soil-structure interaction and the role of the stiffness due to the horizontal strain, we consider the numerical modeling approach.

#### **4.3 Numerical model description**

The present work is focused on the behavior of a masonry wall under horizontal strain. In addition to the evaluation of the stiffness of the masonry wall, the objective of the numerical

Distinct Element Method Applied on Old Masonry Structures 321

a- Single bloc Masonry blocs

c- equivalent elastic parameters

*\** 100

I: Volumetric mass; E: Young modulus;

of the masonry wall.

Fig. 21. Assumptions for determining the relative stiffness of the masonry wall.

Continue Wall Masonry wall

Table 3 lists the parameters assigned to the wall and the estimated axial and flexion stiffness

Table 3. Wall stiffness calculation according to the Potts and Addenbrooke approach.

*\** 21.56

*Parameter Unit Value Parameter Unit Value I m<sup>4</sup> 32e-4 I m<sup>4</sup> 32e-4 A m<sup>2</sup> 0.125 A m<sup>2</sup> 0.125 E* MPa 10000 *E* MPa 2000 EI MNm2 32 EI MNm2 2156 Es MPa 100 Es MPa 100 B m 10 B m 10 °1/m 83 °1/m 17

modeling is to quantify the movement transfer from the soil to the masonry structure. The class of the wall damage depends on the characterization of the wall: geometry and proprieties. The 3D numerical models use the distinct elements code (3DEC). The model presents an individual wall of unreinforced masonry; the wall is under only the effect of the horizontal displacements and dead load. The horizontal displacements were applied on the boundary of soil (model). The masonry wall dimensions are: 10 m length, 5 m high and 25 cm thick. The wall consists of masonry units of 50 cm \* 25 cm \* 25 cm (reference). Two configurations were studied with and without windows. The model has two parts: the soil and the wall, the soil behavior is considered as elastic-plastic, homogenous and isotropic. The plastic criterion (failure) is the Mohr-Coulomb defined by cohesion and friction angle,

Fig. 20. 3DEC numerical model using distinct element method for ISS.

The behavior assumption of masonry units is as elastic and isotropic material. The mortar between blocks corresponds to vertical and horizontal joints. The behavior of the joints is considered as elastic-plastic and the plastic criterion is Mohr-Coulomb which is defined by the friction angle, the cohesion and the tensile strength. The table presents the priorities of soil, blocs and joints of masonry. Table 1 lists the basic mechanical properties assigned to the soil, the masonry and the masonry joints; they are based on the estimations from different international publications. The characterization of soil and building are very close to those used by Potts and Addenbrooke. The calculation was divided into two main stages: the stage of the consolidation to obtain the initial stress condition and the application of the horizontal displacements on the lateral boundary surface. The calculation could then be continued until reaching model equilibrium. Two directions of displacement were studied. The first one corresponds to in-plane solicitation and the second one corresponds to outplane. The boundary condition of the horizontal displacement induced in the soil a compression horizontal strain in Greenfield equal to 8 mm/m.

#### **4.4 Evaluation of relative stiffness of the wall**

Two assumptions were employed to quantify the relative stiffness according to the relation introduced by Potts and Addenbrooke (1996): the first one considers the wall as a continuous single bloc and the second assumption considers the wall as an assembly of masonry units (Figure 21). The following relations (Figure C) determine the equivalent Young modulus (E1, E2) and the equivalent shear modulus (G12) taking into account the geometrical and mechanical parameters of discontinuities for the second assumption. The second assumption reduces the stiffness of the wall.

modeling is to quantify the movement transfer from the soil to the masonry structure. The class of the wall damage depends on the characterization of the wall: geometry and proprieties. The 3D numerical models use the distinct elements code (3DEC). The model presents an individual wall of unreinforced masonry; the wall is under only the effect of the horizontal displacements and dead load. The horizontal displacements were applied on the boundary of soil (model). The masonry wall dimensions are: 10 m length, 5 m high and 25 cm thick. The wall consists of masonry units of 50 cm \* 25 cm \* 25 cm (reference). Two configurations were studied with and without windows. The model has two parts: the soil and the wall, the soil behavior is considered as elastic-plastic, homogenous and isotropic. The plastic criterion (failure) is the Mohr-Coulomb defined by cohesion and friction angle,

Fig. 20. 3DEC numerical model using distinct element method for ISS.

compression horizontal strain in Greenfield equal to 8 mm/m.

**4.4 Evaluation of relative stiffness of the wall** 

second assumption reduces the stiffness of the wall.

The behavior assumption of masonry units is as elastic and isotropic material. The mortar between blocks corresponds to vertical and horizontal joints. The behavior of the joints is considered as elastic-plastic and the plastic criterion is Mohr-Coulomb which is defined by the friction angle, the cohesion and the tensile strength. The table presents the priorities of soil, blocs and joints of masonry. Table 1 lists the basic mechanical properties assigned to the soil, the masonry and the masonry joints; they are based on the estimations from different international publications. The characterization of soil and building are very close to those used by Potts and Addenbrooke. The calculation was divided into two main stages: the stage of the consolidation to obtain the initial stress condition and the application of the horizontal displacements on the lateral boundary surface. The calculation could then be continued until reaching model equilibrium. Two directions of displacement were studied. The first one corresponds to in-plane solicitation and the second one corresponds to outplane. The boundary condition of the horizontal displacement induced in the soil a

Two assumptions were employed to quantify the relative stiffness according to the relation introduced by Potts and Addenbrooke (1996): the first one considers the wall as a continuous single bloc and the second assumption considers the wall as an assembly of masonry units (Figure 21). The following relations (Figure C) determine the equivalent Young modulus (E1, E2) and the equivalent shear modulus (G12) taking into account the geometrical and mechanical parameters of discontinuities for the second assumption. The

c- equivalent elastic parameters

Fig. 21. Assumptions for determining the relative stiffness of the masonry wall.



I: Volumetric mass; E: Young modulus;

Table 3. Wall stiffness calculation according to the Potts and Addenbrooke approach.

Distinct Element Method Applied on Old Masonry Structures 323

The second configuration corresponds to the introduction of a window in the centre of the masonry wall, which surface corresponds to 16% of the wall surface. The presence of a void in the wall modifies the distribution of the displacements and the localization of the cracks (Figure 24). The horizontal strain increases and it is equal to 2.65 mm/m instead of 1.4 mm/m, the transferred strain is equal to 33% of soil strain. The stiffness of the wall decreases due to the window, the density of cracks increases also, we observe new cracks in the upper part of the wall above the window zone due to tension stresses. The numerical

Fig. 23a. Horizontal and vertical displacements of the wall.

Fig. 23b. Failure of the joints creating inclined and vertical cracks.

results are very close to the in-situ observations due to the subsidence.

 a- Horizontal displacement b- Failure localization Fig. 24. Behavior of masonry wall under the effect of ground subsidence due to the

The third and the fourth configurations correspond to the horizontal displacement applied perpendicularly to the wall (out-plane). Figure 25 presents the mechanism of the wall

underground excavation – results obtained by 3DEC numerical model.

Observing the charts which were introduced by Potts and Addenbrooke (Figure 22), the studying wall is stiff enough and the transfer of soil deformations to the wall is small, may be equal to zero for all structure positions from the position of the underground excavation (e/B). The adopted approach by Potts and Addenbrooke is very simplified for a masonry wall and the realistic behavior can be suspected as completely different from the above conclusions according to in-situ observations.

Fig. 22. Chart for determining the strain (ε and D) transfer from soil to structures function of relative axial α and flexion ρ stiffness.

#### **4.5 Results analysis**

The analysis of results will be limited to the behavior of the masonry wall due to the effect of the horizontal displacements (horizontal strain). We looked for the state of joints (mortar): elastic, open or sliding; and the horizontal and vertical displacements of the wall. We compare the horizontal strains of the masonry wall to the horizontal strain of the soil under Greenfield conditions.

The first configuration corresponds to in-plane solicitation without windows. The Figure 23 presents the horizontal and the vertical displacements of the wall. The horizontal displacement direction corresponds to the direction of the solicitation and the vertical displacement direction is oriented toward the top. The distribution of displacements depends on the localization of the cracks due to the failure of vertical joints. A principal discontinuity was formed near the second part of the wall; the principal discontinuity is associated with three or four cracks (joints). The dip of discontinuity is about 45° to 60°. The variation of the horizontal strain decreases from the bottom to the top of the wall. The maximum horizontal strain is 1.4 mm/m. The transfer of the horizontal strain from soil to the wall is equal to 17.5%.

Observing the charts which were introduced by Potts and Addenbrooke (Figure 22), the studying wall is stiff enough and the transfer of soil deformations to the wall is small, may be equal to zero for all structure positions from the position of the underground excavation (e/B). The adopted approach by Potts and Addenbrooke is very simplified for a masonry wall and the realistic behavior can be suspected as completely different from the above

Fig. 22. Chart for determining the strain (ε and D) transfer from soil to structures function of

The analysis of results will be limited to the behavior of the masonry wall due to the effect of the horizontal displacements (horizontal strain). We looked for the state of joints (mortar): elastic, open or sliding; and the horizontal and vertical displacements of the wall. We compare the horizontal strains of the masonry wall to the horizontal strain of the soil under

The first configuration corresponds to in-plane solicitation without windows. The Figure 23 presents the horizontal and the vertical displacements of the wall. The horizontal displacement direction corresponds to the direction of the solicitation and the vertical displacement direction is oriented toward the top. The distribution of displacements depends on the localization of the cracks due to the failure of vertical joints. A principal discontinuity was formed near the second part of the wall; the principal discontinuity is associated with three or four cracks (joints). The dip of discontinuity is about 45° to 60°. The variation of the horizontal strain decreases from the bottom to the top of the wall. The maximum horizontal strain is 1.4 mm/m. The transfer of the horizontal strain from soil to

conclusions according to in-situ observations.

relative axial α and flexion ρ stiffness.

**4.5 Results analysis** 

Greenfield conditions.

the wall is equal to 17.5%.

Fig. 23a. Horizontal and vertical displacements of the wall.

Fig. 23b. Failure of the joints creating inclined and vertical cracks.

The second configuration corresponds to the introduction of a window in the centre of the masonry wall, which surface corresponds to 16% of the wall surface. The presence of a void in the wall modifies the distribution of the displacements and the localization of the cracks (Figure 24). The horizontal strain increases and it is equal to 2.65 mm/m instead of 1.4 mm/m, the transferred strain is equal to 33% of soil strain. The stiffness of the wall decreases due to the window, the density of cracks increases also, we observe new cracks in the upper part of the wall above the window zone due to tension stresses. The numerical results are very close to the in-situ observations due to the subsidence.

a- Horizontal displacement b- Failure localization

Fig. 24. Behavior of masonry wall under the effect of ground subsidence due to the underground excavation – results obtained by 3DEC numerical model.

The third and the fourth configurations correspond to the horizontal displacement applied perpendicularly to the wall (out-plane). Figure 25 presents the mechanism of the wall

Distinct Element Method Applied on Old Masonry Structures 325

A ground horizontal strain, in Greenfield conditions, equaling 8 mm/m can induce severe damage on structures according to different prediction methods without the effect of soil-

The damages can be determined by the chart of Boscardin and Cording (1989) and Potts and Addenbrooke (1996) and Table 5. In this case, damage must be very severs. The results of numerical modeling using 3D distinct elements code highlights this point and shows this evaluation must be improved taking into acount the interaction between soil and structure and the behavior of joints. The numerical results are strongly close to the physical modeling

The study focused on the behavior of the masonry wall due to horizontal deformation of the soil; the study also focused on soil-structure interaction and the movement of transfer of soil structure using numerical modeling. A 3D numerical model of the masonry wall was made to simulate the behavior. The study examined the observance of the level and location of damage due to horizontal deformation of the soil; particular attention was granted to the location of cracks based on the direction of solicitation and the existing window. Different configurations have been calculated and analyzed horizontal and vertical movements.

One principal inclined shear crack and many incomplete shear cracks associated with open cracks, with the direction of horizontal

Horizontal crack at 1.5 m from the base and distortion and tilt of

Inclined shear cracks associated with open cracks, with the direction of horizontal

displacement. Up the window, inclined cracks in the opposite

in the number of cracks

Distortion of the wall associated to the fall of certain blocks, increase

displacement.

direction

< 0.5 for negligible degradations. 0.5 to 0.75 for very light damage. 0.75 to 1.5 for light damage. 1.5 to 3 for moderate damage.

> 3 for severe damage.

transversal section

Ux (mm) Uy (mm) Uz (mm) Localization of damages

structure interaction (Deck et al., 2003, table 5).

In-plane 34 31 4.9

Out-plane 18 56 -154

window 36 7

window 11.5 62 144

Limits horizontal strain mm/m

Table 4. Displacements and damage localization of masonry wall.

results obtained by Cox (Cox, 1980) and in situ observations (Deck, 2002).

Table 5. Classes of degradation according to horizontal strain.

In-plane –

Out-plane -

**5. Conclusion** 

deformation, the displacement of the bottom part of the wall is the opposite of the displacement of the upper part. The behavior of the wall is influenced by the direction of the solicitation and the stiffness of the wall. A horizontal open crack was created by the impact of horizontal displacement point C), the localization of this crack is at 1 m from the base of the horizontal strain between points A and C. It is equal to 100 mm/m, it is much greater than the applied horizontal strain (8 mm/m) on the ground, and in the second part of the wall, and the strain is less than the soil horizontal strain.

Fig. 25. Behavior of masonry wall under the effect of ground subsidence due to the underground excavation – results obtained by 3DEC numerical model.

The fourth configuration takes into account the presence of the window. The presence of the window increases the number of cracks very strongly. Some blocks lose their contacts because of the free surface and can fall.

Fig. 26. Behavior of masonry wall under the effect of ground subsidence due to the underground excavation – results obtained by 3DEC numerical model.

Table 4 resumes and compares the values of horizontal displacements (Ux and Zu) and vertical displacement (Uy) and the localization damages according to the behavior of vertical and horizontal joints. Figure 27 presents an interpretation of the wall behavior under the influence of the horizontal strain. The wall is influenced by the ground displacements. This solicitation of the wall by the horizontal strain reveals traction failures in the joints of masonry blocks. These failures can form oblique cracks in the wall along vertical and horizontal mortar.

deformation, the displacement of the bottom part of the wall is the opposite of the displacement of the upper part. The behavior of the wall is influenced by the direction of the solicitation and the stiffness of the wall. A horizontal open crack was created by the impact of horizontal displacement point C), the localization of this crack is at 1 m from the base of the horizontal strain between points A and C. It is equal to 100 mm/m, it is much greater than the applied horizontal strain (8 mm/m) on the ground, and in the second part of the

wall, and the strain is less than the soil horizontal strain.

a- Horizontal displacement b- Failure localization

underground excavation – results obtained by 3DEC numerical model.

because of the free surface and can fall.

vertical and horizontal mortar.

Fig. 25. Behavior of masonry wall under the effect of ground subsidence due to the

Fig. 26. Behavior of masonry wall under the effect of ground subsidence due to the

Table 4 resumes and compares the values of horizontal displacements (Ux and Zu) and vertical displacement (Uy) and the localization damages according to the behavior of vertical and horizontal joints. Figure 27 presents an interpretation of the wall behavior under the influence of the horizontal strain. The wall is influenced by the ground displacements. This solicitation of the wall by the horizontal strain reveals traction failures in the joints of masonry blocks. These failures can form oblique cracks in the wall along

underground excavation – results obtained by 3DEC numerical model.

The fourth configuration takes into account the presence of the window. The presence of the window increases the number of cracks very strongly. Some blocks lose their contacts A ground horizontal strain, in Greenfield conditions, equaling 8 mm/m can induce severe damage on structures according to different prediction methods without the effect of soilstructure interaction (Deck et al., 2003, table 5).


Table 4. Displacements and damage localization of masonry wall.

The damages can be determined by the chart of Boscardin and Cording (1989) and Potts and Addenbrooke (1996) and Table 5. In this case, damage must be very severs. The results of numerical modeling using 3D distinct elements code highlights this point and shows this evaluation must be improved taking into acount the interaction between soil and structure and the behavior of joints. The numerical results are strongly close to the physical modeling results obtained by Cox (Cox, 1980) and in situ observations (Deck, 2002).


Table 5. Classes of degradation according to horizontal strain.

### **5. Conclusion**

The study focused on the behavior of the masonry wall due to horizontal deformation of the soil; the study also focused on soil-structure interaction and the movement of transfer of soil structure using numerical modeling. A 3D numerical model of the masonry wall was made to simulate the behavior. The study examined the observance of the level and location of damage due to horizontal deformation of the soil; particular attention was granted to the location of cracks based on the direction of solicitation and the existing window. Different configurations have been calculated and analyzed horizontal and vertical movements.

Distinct Element Method Applied on Old Masonry Structures 327

Al Heib M. (2008). State Of The Art Of The Prediction Methods Of Short And Long-Term

Barrentine, Larry B., (1999). An introduction to design of experiments: a simplified

Boscardin, M.D. and Cording, E.G. (1989). Building response to excavation-induced

Burland, J.B. (1995). Assessment of risk of damage to buildings due to tunneling and

Cox D. W. (1980).- Modeling stochastic behaviour using the friction table with examples of

Cundall, P. A., (1971), A computer model for simulating progressive large scale movements

Deck O., Al Heib M., and Homand F. (2003). Taking the soil-structure interaction into

Dimmock, P.S. and Mair R.J. (2008). Effect of building stiffness on tunneling-induced ground movement, Tunneling and Underground Space Technology, 23: 438-450. Eurocode 6, (1996). Calcul des ouvrages en maçonnerie – Partie 4: Méthode de calcul

Franzius, J.N., Potts, D.M., Burland, J.B., (2006). The response of surface structures to tunnel

Giorgia Giardina, Max A.N. Hendriks, Jan G. Rots (2008). Numerical analyses of tunnel-

Goodman, R. E., Taylor, R. L., and Brekke, T. L., (1968). A Model for the Mechanics of

Itasca, 2000. UDEC Universal Distinct Elements Code Manual. Continuously yielding joint

Idris J., Al Heib M., Verdel T. (2009). Masonry joints mechanical behaviour evolution in built

Janssen, H.J.M., (1997). Structural masonry. In: Balkema, A.A. (Ed.), Numerical studies with

Hoek, E., (2000). Practical Rock engineering. Rock Mass Properties. rocscience.com on line,

simplifies pour les ouvrages non armée, NBN EN (1996-3) (ANB).

construction, Proc. Inst. Civ. Eng. Geotech. Eng., 159 (1), 3–17.

model. Itasca Consulting Group Inc., pp. 1–16.

Underground Space Technology, 24 (2009) 617–626.

Hugo Fournier. 2008 Nova Science Publishers, Inc. pp 53-76.

settlement, *Journal of Geotechnical Engineering*, ASCE, 115(1): 1-21.

Rock Mechanics (Nancy, France, 1971), 1, Paper No. II-8.

Ground Movements (Subsidence And Sinkhole) For The Mines In France. In: Coal Geology Research Progress. ISBN: 978-1-60456-596-6. Editor: Thomas Michel and

approach. Experiments with Three Factors. The American Society for Quality, pp.

excavations. Invited special lecture, in: Proc. 1st Int. Conf. Earthquake Geotechnical

cracked brickwork and subsidence. Proc. of the 2nd international conference on ground movements *and structures*, Cardiff (Avril 1980), Edité par GEDDES J. D.,

in blocky rock systems', in Proc. of the Symposium of the International Society of

account in assessing the loading of a structure in a mining area. Engineering

induced settlement damage to a masonry wall. 7th fib PhD Symposium in

Jointed Rock, Journal of the Soil Mechanics and Foundations Div., ASCE, Vol. 94,

tunnels. Analysis by numerical modelling and experimental design, Tunnelling and

UDEC. Centre for civil engineering research and codes. Roterdam, Netherlands,

**6. References** 

27–35 (Chapter 3).

Engineering, IS-Tokyo, 1189-1201.

Pentech press, pp. 307-328.

Structure 25:435-448.

Stuttgart, Germany

No SM3, pp. 637-659.

ISBN 90 5410680 8, pp. 96–106.

Chapter 11, pp. 161–203.

Dh: horizontal displacement, Dv: vertical displacement, εh: horizontal deformation

Fig. 27. Analyze of soil structure interaction due to the ground movement.

The resulting analysis indicated that a masonry wall formed by masonry units is more sensitive to horizontal strain than a continuous structure idealized by a beam. We can conclude that the theory of beams and the Potts and Addenbrooke chart underestimate the impact of horizontal strain. The out-plane horizontal strain can seriously damage the masonry wall, with the introduction of a main horizontal crack, the level of damage in this case is more important than the in-plane horizontal solicitation. The presence of a window decreases the stiffness and increases the damage to the wall.

In conclusion, numerical modeling using the distinct element method is an original tool to improve the comprehension of wall damaging; the obtained results are very close to in-situ observations and can supplement an advancing progress for the evaluation of masonry structures.

Fig. 28. Effect of horizontal strain on masonry wall using physical modeling (Cox, 1980).

#### **6. References**

326 Numerical Modelling

Masonry

Dh: horizontal displacement, Dv: vertical displacement, εh: horizontal deformation Fig. 27. Analyze of soil structure interaction due to the ground movement.

h=Dh (left) /L= const.= 8 mm/m

decreases the stiffness and increases the damage to the wall.

structures.

Dh (left) fix = 40

Free surface

Soil

The resulting analysis indicated that a masonry wall formed by masonry units is more sensitive to horizontal strain than a continuous structure idealized by a beam. We can conclude that the theory of beams and the Potts and Addenbrooke chart underestimate the impact of horizontal strain. The out-plane horizontal strain can seriously damage the masonry wall, with the introduction of a main horizontal crack, the level of damage in this case is more important than the in-plane horizontal solicitation. The presence of a window

Dv=0

cm Dh (right) =0

DhSol > Dhstr

Crack

In conclusion, numerical modeling using the distinct element method is an original tool to improve the comprehension of wall damaging; the obtained results are very close to in-situ observations and can supplement an advancing progress for the evaluation of masonry

Fig. 28. Effect of horizontal strain on masonry wall using physical modeling (Cox, 1980).


**15** 

*Czech Republic* 

**Phenomenological Modelling of Cyclic Plasticity** 

The stress-strain behaviour of metals under a cyclic loading is very miscellaneous and needs an individual approach for different metallic materials. There are many different models that have been developed for the case of cyclic plasticity. This chapter will address only so called phenomenological models, which are based purely on the observed behaviour of materials. The second section of this chapter describes the main experimental observations of cyclic plasticity for metals. Material models development for the correct description of particular phenomenon of cyclic plasticity is complicated by such effects as cyclic hardening/softening and cyclic creep (also called ratcheting). Effect of cyclic hardening/softening corresponds to hardening or softening of material response, more accurately to decreasing/increasing resistance to deformation of material subjected to cyclic loading. Some materials show very strong cyclic softening/hardening (stainless steels, copper, etc.), others less pronounced (medium carbon steels). The material can show cyclic hardening/softening behaviour during force controlled or strain controlled loading. On the contrary, the cyclic creep phenomenon can arise only under force controlled loading. The cyclic creep can be defined as accumulation of any plastic strain component with increasing number of cycles and can influence the fatigue life of mechanical parts due to the exhaustion of plastic ability of material earlier than the

The third section of this chapter deals with the cyclic plasticity models included in the most popular Finite Element packages (Ansys, Abaqus, MSC.Nastran/Marc). A particular attention is paid to the calibration of classical nonlinear kinematic hardening models. Stressstrain behaviour of materials may be significantly different for proportional and non proportional loading, i.e. loading which leads to the rotation of principal stresses. In case of stainless steels an additional hardening occurs under non proportional loading. This additional hardening is investigated mostly under tension/torsion loading using the circular, elliptical, cross, star and other loading path shapes. Classical cyclic plasticity models implemented in the commercial Finite Element software are not able to describe well the non proportional hardening and correct prediction of multiaxial ratcheting is also problematic. This problem can be solved by implementation of more complex cyclic plasticity model to a FE code. As a conclusion there are briefly summarized phenomenological modelling theories of ratcheting. The main attention is focused on the most progressive group of cyclic plasticity models with a one yield surface only. The

initiation of fatigue crack caused by low-cycle fatigue is started.

**1. Introduction** 

Radim Halama1, Josef Sedlák and Michal Šofer1

*1Centre of Excellence IT4Innovations Department of Mechanics of Materials VŠB-Technical University of Ostrava* 


### **Phenomenological Modelling of Cyclic Plasticity**

Radim Halama1, Josef Sedlák and Michal Šofer1

*1Centre of Excellence IT4Innovations Department of Mechanics of Materials VŠB-Technical University of Ostrava Czech Republic* 

#### **1. Introduction**

328 Numerical Modelling

Lourenço, Paulo B., (2002). Guidelines for the analysis of historical masonry structures.

Kresic, N., (1997). Quantitative Solution in Hydrology and Groundwater Modelling, Inverse

Lake L.M., Rankin W.J. and Hawley J. (1992). Prediction and effects of ground movements

Massart T.J., Peerlings R.H.J., Geers M.G.D., Gottcheiner S., (2005). Mesoscopic modeling of

Lemos, J. V. (1998). Discrete Element Modeling of the Seismic Behavior of Stone Masonry

Peck, R. (1969). Deep excavations and tunnelling in soft ground, Proceedings of the 7th

Potts, D.M. and Addenbrooke, T.I. (1997). A structure's influence on tunneling-induced

Souley, M., (1993). Modelling of jointed rock masses by distinct element method, influence

Son M. and Cording E. J. (2007). Evaluation of building stiffness for building response

Standing, J. and Burland, J.B. (2008). Impact of underground works on existing

Sutcliffe D.J., Yu H.S., Page A.W., (2001). Lower bound limit analysis of unrienforced

Tzamtzis A.D. and Asteris P.G., (2004). FE Analysis of Complex Discontinuous and Jointed

Verdel T., (1994). La méthode des éléments distincts (DEM) : un outil pour évaluer les

Verdel, T. and Bigarre, P. (1999). Modélisation de tunnels anciens avec le logiciel UDEC,

thesis of (Institut national polytechnique de Lorraine), pp.75-136.

infrastructure, Invited lecture in Post-Mining, France, 1-39.

Electronic Journal of Structural Engineering*,* pp. 75-92.

Rapport INERIS, Société SIMECSOL, pp. 1-12.

masonry shear walls, Computers and structures, pp. 1295-1312.

caused by tunnelling in soft ground beneath urban areas. CIRIA funders report

failure in brick masonry accounting for three-dimensional effects. Engineering

Arches," in Computer Methods in Structural Masonry 4 (4th International Symposium, Florence, Italy, September 1997), pp. 220-227, G. N. Pande et al., Ed. London: E&FN Spon.National Coal Board 1975. Subsidence Engineers Handbook.

International Conference on Soil Mechanics Foundation Engineering, Mexico,

ground movements. Proc. Instn. Civil Engrs., Geotechnical Engineering, 125 : 109-

of the discontinuities constitutive laws upon the stability excavation. Doctoral

analysis to excavation-induced ground movements. Journal of Geotechnical

Structural Systems (Part 1: Presentation of the Method - A State-of-the-Art Review).

risques d'instabilités des monuments en maçonnerie. Applications à des cas égyptiens. Actes du Septième Congrès de l'Association Internationale de Géologie de l'Ingénieur, Lisbonne, Septembre 1994, pp 3551-3560, Ed BALKEMA (ISBN 90 54

Distance to a Power Method. Lewis Publishers, pp.121–123.

University of Minho, Guimarães, Portugal

Fracture Mechanics 72 (2005) 1238–1253

National Coal Board Production Dept., U.K.

engineering. ASCE/August 2007 (995-1002).

CP/5 129p.

3:225-290.

10 503 8)

125.

The stress-strain behaviour of metals under a cyclic loading is very miscellaneous and needs an individual approach for different metallic materials. There are many different models that have been developed for the case of cyclic plasticity. This chapter will address only so called phenomenological models, which are based purely on the observed behaviour of materials. The second section of this chapter describes the main experimental observations of cyclic plasticity for metals. Material models development for the correct description of particular phenomenon of cyclic plasticity is complicated by such effects as cyclic hardening/softening and cyclic creep (also called ratcheting). Effect of cyclic hardening/softening corresponds to hardening or softening of material response, more accurately to decreasing/increasing resistance to deformation of material subjected to cyclic loading. Some materials show very strong cyclic softening/hardening (stainless steels, copper, etc.), others less pronounced (medium carbon steels). The material can show cyclic hardening/softening behaviour during force controlled or strain controlled loading. On the contrary, the cyclic creep phenomenon can arise only under force controlled loading. The cyclic creep can be defined as accumulation of any plastic strain component with increasing number of cycles and can influence the fatigue life of mechanical parts due to the exhaustion of plastic ability of material earlier than the initiation of fatigue crack caused by low-cycle fatigue is started.

The third section of this chapter deals with the cyclic plasticity models included in the most popular Finite Element packages (Ansys, Abaqus, MSC.Nastran/Marc). A particular attention is paid to the calibration of classical nonlinear kinematic hardening models. Stressstrain behaviour of materials may be significantly different for proportional and non proportional loading, i.e. loading which leads to the rotation of principal stresses. In case of stainless steels an additional hardening occurs under non proportional loading. This additional hardening is investigated mostly under tension/torsion loading using the circular, elliptical, cross, star and other loading path shapes. Classical cyclic plasticity models implemented in the commercial Finite Element software are not able to describe well the non proportional hardening and correct prediction of multiaxial ratcheting is also problematic. This problem can be solved by implementation of more complex cyclic plasticity model to a FE code. As a conclusion there are briefly summarized phenomenological modelling theories of ratcheting. The main attention is focused on the most progressive group of cyclic plasticity models with a one yield surface only. The

Phenomenological Modelling of Cyclic Plasticity 331

relates to softening/hardening of material response or decreasing/increasing of resistance against material deformation under cyclic loading. Its intensity usually decrease with number of cycles until the saturated state is reached. During uniaxial cyclic loading, the condition is characterized by closed hysteresis loop. Transient responses in initial cycles caused by cyclic hardening/softening under plastic strain control and stress control are

Fig. 2. Uniaxial fatigue test material response: Cyclic softening a) and cyclic hardening b) under plastic strain controlled loading and cyclic hardening c) and cyclic softening d) under

Some materials show very strong cyclic softening/hardening (stainless steels, cooper, etc.) some less obvious (structural steels). There can be also notable cyclic hardening in certain cycles range and in the remaining lifetime cyclical softening. Properties of cyclic hardening/softening don't depend only on material microstructure, but also on loading amplitude or more generally on previous strain history. Such transient behaviour of material makes accurate stress-strain modelling more difficult. There is very often mentioned possibility of transient stress-strain behaviour estimation according to its strength limit and yield limit ratio, but also very simple hypothesis is used, claiming that

hard material cyclically softens whereas soft material cyclically hardens.

shown at the Fig.2.

stress controlled loading.

AbdelKarim-Ohno model is also described, which gives very good prediction of ratcheting under uniaxial as well as multiaxial loading.

Comparison of the AbdelKarim-Ohno model and classical models is presented through simulations in the fourth section. Numerical analyses were performed for various uniaxial and multiaxial loading cases of specimen made from the R7T wheel steel. It is shown that classical models can also get sufficient ratcheting prediction when are correctly calibrated.

#### **2. Experimental facts**

Good understanding the nature of particular effects of cyclic plasticity plays a key role in the phenomenological modelling. Main findings from this chapter will be useful for a reader in the field of understanding the calibration of cyclic plasticity models and for correct numerical analysis results evaluation.

#### **2.1 Bauschinger's effect**

Bauschinger's effect is a basic and well known phenomenon of cyclic plasticity. It describes the fact that due to uniaxial loading of a specimen above yield limit in one direction the limit of elasticity in the opposite direction is reduced. As an example can serve the stress-strain curve corresponding to the first cycle of strain controlled low cycle fatigue test of the steel ST52 (see Fig.1). If the yield limit is marked as *<sup>Y</sup>*, then the material during unloading from maximal axial stress state *<sup>1</sup>* behaves elastically up to the point, where the difference between maximal and immediate stress *1* – *2* is equal to the double of yield limit 2*Y*.

Fig. 1. Presentation of Bauschinger's effect.

#### **2.2 Cyclic hardening/softening**

Results of the micro structural changes in the beginning stage of cyclic loading are changes of physical properties and stress response in the material. Cyclic softening/hardening effect

AbdelKarim-Ohno model is also described, which gives very good prediction of ratcheting

Comparison of the AbdelKarim-Ohno model and classical models is presented through simulations in the fourth section. Numerical analyses were performed for various uniaxial and multiaxial loading cases of specimen made from the R7T wheel steel. It is shown that classical models can also get sufficient ratcheting prediction when are correctly calibrated.

Good understanding the nature of particular effects of cyclic plasticity plays a key role in the phenomenological modelling. Main findings from this chapter will be useful for a reader in the field of understanding the calibration of cyclic plasticity models and for correct

Bauschinger's effect is a basic and well known phenomenon of cyclic plasticity. It describes the fact that due to uniaxial loading of a specimen above yield limit in one direction the limit of elasticity in the opposite direction is reduced. As an example can serve the stress-strain curve corresponding to the first cycle of strain controlled low cycle fatigue test of the steel

Results of the micro structural changes in the beginning stage of cyclic loading are changes of physical properties and stress response in the material. Cyclic softening/hardening effect

*1* –  *<sup>Y</sup>*, then the material during unloading from

*Y*.

*<sup>1</sup>* behaves elastically up to the point, where the difference

*2* is equal to the double of yield limit 2

under uniaxial as well as multiaxial loading.

**2. Experimental facts** 

**2.1 Bauschinger's effect** 

maximal axial stress state

numerical analysis results evaluation.

ST52 (see Fig.1). If the yield limit is marked as

between maximal and immediate stress

Fig. 1. Presentation of Bauschinger's effect.

**2.2 Cyclic hardening/softening** 

relates to softening/hardening of material response or decreasing/increasing of resistance against material deformation under cyclic loading. Its intensity usually decrease with number of cycles until the saturated state is reached. During uniaxial cyclic loading, the condition is characterized by closed hysteresis loop. Transient responses in initial cycles caused by cyclic hardening/softening under plastic strain control and stress control are shown at the Fig.2.

Fig. 2. Uniaxial fatigue test material response: Cyclic softening a) and cyclic hardening b) under plastic strain controlled loading and cyclic hardening c) and cyclic softening d) under stress controlled loading.

Some materials show very strong cyclic softening/hardening (stainless steels, cooper, etc.) some less obvious (structural steels). There can be also notable cyclic hardening in certain cycles range and in the remaining lifetime cyclical softening. Properties of cyclic hardening/softening don't depend only on material microstructure, but also on loading amplitude or more generally on previous strain history. Such transient behaviour of material makes accurate stress-strain modelling more difficult. There is very often mentioned possibility of transient stress-strain behaviour estimation according to its strength limit and yield limit ratio, but also very simple hypothesis is used, claiming that hard material cyclically softens whereas soft material cyclically hardens.

Phenomenological Modelling of Cyclic Plasticity 333

The Figure 5 illustrates the basic types of loading in the stress space. The tensioncompression and simple shear belongs to the category of proportional loading, because there is no change of principal stress directions. This group also includes multi-axialloading in which the stress tensor components change proportionally. Non-proportional loading can be therefore defined as a loading that does not meet the specified condition, and is generally

**2.4 Non-proportional hardening** 

characterized by the loading path in the form of curve (Fig. 5).

Fig. 5. Loading paths for non-proportional and proportional loading.

loading path. Thereafter we can express stress amplitude

Fig.6.

and the material parameter of additional hardening is define as

Conception of non-proportional hardening represents material hardening as a result of nonproportional loading. Most often it is investigated under tension-compression/torsion loading. Generally, the non-proportional hardening depends on material and shape of

where ap is the equivalent stress amplitude under proportional loading, whereas the influence of loading path shape in a cycle is involved in the non proportional parameter

where the quantity an is the maximum value of von Mises equivalent stresses under nonproportional deformation (circular path). The equivalent stress amplitude is the radius of the minimum circle that circumscribes the loading path in the deviatoric stress space, see

a()=(1+)ap , (1)

=an / ap - 1 (2)

From upper peaks of several hysteresis loops corresponding to half lifetime is possible to obtain cyclic strain curve (Fig.3), which is often used in engineering computations.

Fig. 3. Cyclic stress-strain curve of ST52 steel.

#### **2.3 Non-masing behaviour**

A material obeys Masing behaviour when the upper branches of hysteresis loops with different strain ranges after alignment in lower peaks overlap. More accurately, in the ideal case, single solid curve is created. From microscopic point of view Masing behaviour indicates stable microstructure in fatigue process. Most steel materials haven't Masing behaviour. Some engineering materials show Masing behaviour under certain testing conditions (Jiang & Zhang, 2008). As can be seen from the Fig.4, where the upper branches of hysteresis loops of the investigated material are displayed, the non-Masing behaviour is dependent on the amplitude of plastic strain *ap*.

Fig. 4. Non-Masing's Behaviour of ST52 steel and schematic representation of Masing's Behaviour.

#### **2.4 Non-proportional hardening**

332 Numerical Modelling

From upper peaks of several hysteresis loops corresponding to half lifetime is possible to

A material obeys Masing behaviour when the upper branches of hysteresis loops with different strain ranges after alignment in lower peaks overlap. More accurately, in the ideal case, single solid curve is created. From microscopic point of view Masing behaviour indicates stable microstructure in fatigue process. Most steel materials haven't Masing behaviour. Some engineering materials show Masing behaviour under certain testing conditions (Jiang & Zhang, 2008). As can be seen from the Fig.4, where the upper branches of hysteresis loops of the investigated material are displayed, the non-Masing behaviour is

> *ap*.

Axial stress [MPa]

Axial plastic strain

Fig. 4. Non-Masing's Behaviour of ST52 steel and schematic representation of Masing's

obtain cyclic strain curve (Fig.3), which is often used in engineering computations.

Fig. 3. Cyclic stress-strain curve of ST52 steel.

dependent on the amplitude of plastic strain

**2.3 Non-masing behaviour** 

Behaviour.

The Figure 5 illustrates the basic types of loading in the stress space. The tensioncompression and simple shear belongs to the category of proportional loading, because there is no change of principal stress directions. This group also includes multi-axialloading in which the stress tensor components change proportionally. Non-proportional loading can be therefore defined as a loading that does not meet the specified condition, and is generally characterized by the loading path in the form of curve (Fig. 5).

Fig. 5. Loading paths for non-proportional and proportional loading.

Conception of non-proportional hardening represents material hardening as a result of nonproportional loading. Most often it is investigated under tension-compression/torsion loading. Generally, the non-proportional hardening depends on material and shape of loading path. Thereafter we can express stress amplitude

$$
\sigma\_{\mathbf{a}}(\Phi) = (1 + \alpha \Phi) \,\,\sigma\_{\mathbf{a}}\mathbf{p} \,,\,\tag{1}
$$

where ap is the equivalent stress amplitude under proportional loading, whereas the influence of loading path shape in a cycle is involved in the non proportional parameter and the material parameter of additional hardening is define as

$$\alpha \equiv \sigma\_\mathbf{a}^\mathbf{n} \;/\ \sigma\_\mathbf{a} \mathbf{\bar{p}} \; - \; \mathbf{1} \tag{2}$$

where the quantity an is the maximum value of von Mises equivalent stresses under nonproportional deformation (circular path). The equivalent stress amplitude is the radius of the minimum circle that circumscribes the loading path in the deviatoric stress space, see Fig.6.

Phenomenological Modelling of Cyclic Plasticity 335

compression/torsion loading or the biaxial ratcheting caused by internal/external pressure with simultaneous cyclic tension-compression, bending or torsion. The ratcheting strain corresponds to the stress component with non-zero mean stress. The typical example is thinwalled tube subjected to internal (external) pressure and cyclic axial tension (Fig.8c,d). For pure symmetrical bending case (a) it was experimentally observed, that the cross-section becomes more and more oval with increasing number of cycles. This process is then

Fig. 8. Ratcheting of a piping component due to a) pure bending, b) bending and external pressure, c) external pressure and push-pull and d) internal pressure and push-pull.

As a next sample results of the fatigue test realised under tension – compression and torsion can serve (Fig.9). The test simulates ratcheting of shear strain, which occurs in surface layer subjected to rolling/sliding contact loading and was proposed by (McDowell, 1995). In the both axes force control was used. For measuring the axial and shear strain during the fatigue test two strain gauges rosette HBM RY3x3/120 were glued to the specimen. The pulsating torque leads to the accumulation of shear strain in the direction of applied torsional moment. The tested material R7T steel becomes almost elastic in initial cycles and then shows significant softening behaviour. After two hundred of loading cycles steady state is

strengthened, when the external pressure is applied too (b).

reached and the ratcheting rate is constant.

Fig. 9. Ratcheting of shear strain in the McDowell´s tension/torsion test.

A lot of rail and wheel steels show the decreasing ratcheting rate with the increasing number of cycles, which complicates accurate modelling of ratcheting effect. Ratcheting makes also life prediction of fatigue crack initiation difficult as well because the material

Fig. 6. Definition of equivalent stress amplitude under non-proportional loading.

Non-proportional hardening of FCC alloys pertinents to the stacking fault energy. For strain controlled 90° out-of-phase loading (circular path) it was found out, that the material parameter of non proportional strain hardening is higher for materials with lower value of the stacking fault energy (Doquet & Clavel, 1996).

#### **2.5 Ratcheting**

In an uniaxial test under load controll with non-zero mean stress m the accumulation of axial plastic strain can occur cycle by cycle. This effect is called cyclic creep or ratcheting, see Fig.7. The uniaxial ratcheting is characterised by an open hysteresis loop and it is a result of different nonlinear behaviour of the material in tension and compression. The accumulation of plastic strain in initial cycles depends on the cyclic hardening/softening behaviour.

Fig. 7. Scheme of uniaxial ratcheting and influence of hardening/softening behaviour.

Generally, the ratcheting effect can be described as an accumulation of any component of strain tensor with increasing number of cycles. From the practical point of view the research of ratcheting, which occurs under multiaxial stress states, is also very important. There has been investigated mainly the multiaxial ratcheting under combined tension-

Fig. 6. Definition of equivalent stress amplitude under non-proportional loading.

the stacking fault energy (Doquet & Clavel, 1996).

**2.5 Ratcheting** 

Non-proportional hardening of FCC alloys pertinents to the stacking fault energy. For strain controlled 90° out-of-phase loading (circular path) it was found out, that the material parameter of non proportional strain hardening is higher for materials with lower value of

In an uniaxial test under load controll with non-zero mean stress m the accumulation of axial plastic strain can occur cycle by cycle. This effect is called cyclic creep or ratcheting, see Fig.7. The uniaxial ratcheting is characterised by an open hysteresis loop and it is a result of different nonlinear behaviour of the material in tension and compression. The accumulation of plastic strain in initial cycles depends on the cyclic hardening/softening behaviour.

Fig. 7. Scheme of uniaxial ratcheting and influence of hardening/softening behaviour.

Generally, the ratcheting effect can be described as an accumulation of any component of strain tensor with increasing number of cycles. From the practical point of view the research of ratcheting, which occurs under multiaxial stress states, is also very important. There has been investigated mainly the multiaxial ratcheting under combined tensioncompression/torsion loading or the biaxial ratcheting caused by internal/external pressure with simultaneous cyclic tension-compression, bending or torsion. The ratcheting strain corresponds to the stress component with non-zero mean stress. The typical example is thinwalled tube subjected to internal (external) pressure and cyclic axial tension (Fig.8c,d). For pure symmetrical bending case (a) it was experimentally observed, that the cross-section becomes more and more oval with increasing number of cycles. This process is then strengthened, when the external pressure is applied too (b).

Fig. 8. Ratcheting of a piping component due to a) pure bending, b) bending and external pressure, c) external pressure and push-pull and d) internal pressure and push-pull.

As a next sample results of the fatigue test realised under tension – compression and torsion can serve (Fig.9). The test simulates ratcheting of shear strain, which occurs in surface layer subjected to rolling/sliding contact loading and was proposed by (McDowell, 1995). In the both axes force control was used. For measuring the axial and shear strain during the fatigue test two strain gauges rosette HBM RY3x3/120 were glued to the specimen. The pulsating torque leads to the accumulation of shear strain in the direction of applied torsional moment. The tested material R7T steel becomes almost elastic in initial cycles and then shows significant softening behaviour. After two hundred of loading cycles steady state is reached and the ratcheting rate is constant.

Fig. 9. Ratcheting of shear strain in the McDowell´s tension/torsion test.

A lot of rail and wheel steels show the decreasing ratcheting rate with the increasing number of cycles, which complicates accurate modelling of ratcheting effect. Ratcheting makes also life prediction of fatigue crack initiation difficult as well because the material

Phenomenological Modelling of Cyclic Plasticity 337

is composed of the plastic strain tensor *<sup>p</sup>* **ε** and the elastic strain tensor *<sup>e</sup>* **ε** . The second

where *<sup>e</sup>* **D** is the elastic stiffness matrix and the symbol "**:"** is contraction, i.e. using Einstein

In the uniaxial case, the development of irreversible deformation occurs due to crossing the yield limit σY. Under multiaxial stress state it is necessary to consider an appropriate yield

**sa sa** 0 , *Y R*

isotropic internal variable. The contraction operation "**:"** in (5) can be expressed again in terms of Einstein summation convention *d*=*bijcij*. Now it is necessary to answer the question: When happens a change of plastic strain increment? If the point representing the current stress state lies on the yield surface it can be supposed that this point do not leave the yield surface, the so called consistency condition *f = 0* must be valid. In case of active loading

*f f* 0, *= 0* and *<sup>f</sup>*

the plastic deformation development is directed by the associated plastic flow rule

 

*<sup>p</sup> <sup>3</sup> <sup>f</sup> d= d 2* 

**ε**

In this concept of single yield surface, the kinematic hardening rule

*ee e <sup>p</sup>* **σ** *= := :* **D ε D ε ε** , (4)

: 0 *d*

**s a**

**n**

in (7) corresponds to the equivalent plastic strain increment

<sup>3</sup> **ε ε** . (8)

*p p d = g , ,d ,dp, ,etc.* **a a ε ε n** (9)

*<sup>p</sup> dY = h R,dp, , , ,etc.* **σ a ε** (10)

**σ**

**<sup>σ</sup>** , *<sup>f</sup> <sup>3</sup> = = 2 Y* 

*p p dp d d* <sup>2</sup> :

play an essential role for the robustness of stress-strain model response. When the both hardening rules are used we speak about mixed hardening. Transient effects from initial cycles (cyclical hardening/softening), non-proportional hardening, ratcheting and other

**σ**

*<sup>Y</sup>* (5)

, (6)

, (7)

*<sup>Y</sup>*and *R* is the

, **a** is the deviatoric part of back-stress ,

consideration is that stresses and elastic strains are subjected to Hook's law

condition. For metallic materials the von Mises condition is mostly used

*<sup>3</sup> f= : Y <sup>2</sup>*

which states the centre position for the yield surface with the initial size

summation convention *dij*=B*ijklckl*.

where the plastic multiplier *d*

and the isotropic hardening rule

where **s** is the deviatoric part of stress tensor

could fail due to the fatigue or to the accumulation of a critical unidirectional plastic strain (ratcheting failure).

#### **2.6 Other effects in cyclic plasticity**

From the theory of elasticity and strength it is well known that the yield locus of ductile materials can be described by an ellipse in the diagram shear stress - normal stress. However, through experiments carried out under uniaxial loading was found (Williams & Svensson, 1971), that if the specimen is loaded by torsion prior to tensile test, then the yield locus (yield surface) has deformed shape. The anisotropy is usually neglected in cyclic plasticity modelling. All of the reported effects of cyclic plasticity are dependent on temperature. With increasing temperature is also strengthened the influence of strain rate on the material response.

#### **3. Constitutive modelling**

Basically, cyclic plasticity models can be devided into these groups:

```
Overlay models (Besseling, 1958) 
Single surface models (Armstrong&Frederick, 1966) 
Multisurface models (Mroz, 1967) 
Two-surface models (Dafalias&Popov, 1976) 
Endochronicmodels (Valanis, 1971) 
Models with yield surface distortion (Kurtyka, 1988)
```
Due to its wide popularity and robustness we focus only to the group of models with a single yield surface based on various evolution equations for internal variables.

#### **3.1 Basics of incremental theory of plasticity**

The elastoplasticity theory is based on the observations found in the case of uniaxial loading (Fig.10). The rate-independent material's behaviour model includes the additive rule, i.e. the total strain tensor

Fig. 10. Decomposition of total strain under uniaxial loading.

could fail due to the fatigue or to the accumulation of a critical unidirectional plastic strain

From the theory of elasticity and strength it is well known that the yield locus of ductile materials can be described by an ellipse in the diagram shear stress - normal stress. However, through experiments carried out under uniaxial loading was found (Williams & Svensson, 1971), that if the specimen is loaded by torsion prior to tensile test, then the yield locus (yield surface) has deformed shape. The anisotropy is usually neglected in cyclic plasticity modelling. All of the reported effects of cyclic plasticity are dependent on temperature. With increasing temperature is also strengthened the influence of strain rate on

Due to its wide popularity and robustness we focus only to the group of models with a

The elastoplasticity theory is based on the observations found in the case of uniaxial loading (Fig.10). The rate-independent material's behaviour model includes the additive rule, i.e. the

*<sup>e</sup> <sup>p</sup>* **εε ε** *= +* (3)

single yield surface based on various evolution equations for internal variables.

Basically, cyclic plasticity models can be devided into these groups:

Single surface models (Armstrong&Frederick, 1966)

Models with yield surface distortion (Kurtyka, 1988)

**3.1 Basics of incremental theory of plasticity** 

Fig. 10. Decomposition of total strain under uniaxial loading.

Two-surface models (Dafalias&Popov, 1976)

(ratcheting failure).

the material response.

total strain tensor

**3. Constitutive modelling** 

Overlay models (Besseling, 1958)

Multisurface models (Mroz, 1967)

Endochronicmodels (Valanis, 1971)

**2.6 Other effects in cyclic plasticity** 

is composed of the plastic strain tensor *<sup>p</sup>* **ε** and the elastic strain tensor *<sup>e</sup>* **ε** . The second consideration is that stresses and elastic strains are subjected to Hook's law

$$\mathbf{u} = \mathbf{D}^{\varepsilon} : \mathbf{c}^{\varepsilon} = \mathbf{D}^{\varepsilon} : \left(\mathbf{c} - \mathbf{c}^{\mathcal{V}}\right) \,, \tag{4}$$

where *<sup>e</sup>* **D** is the elastic stiffness matrix and the symbol "**:"** is contraction, i.e. using Einstein summation convention *dij*=B*ijklckl*.

In the uniaxial case, the development of irreversible deformation occurs due to crossing the yield limit σY. Under multiaxial stress state it is necessary to consider an appropriate yield condition. For metallic materials the von Mises condition is mostly used

$$f = \sqrt{\frac{3}{2}(\mathbf{s} - \mathbf{a}) \cdot (\mathbf{s} - \mathbf{a})} \ -Y = 0 \ , \ Y = \sigma\_Y + R \tag{5}$$

where **s** is the deviatoric part of stress tensor , **a** is the deviatoric part of back-stress , which states the centre position for the yield surface with the initial size *<sup>Y</sup>*and *R* is the isotropic internal variable. The contraction operation "**:"** in (5) can be expressed again in terms of Einstein summation convention *d*=*bijcij*. Now it is necessary to answer the question: When happens a change of plastic strain increment? If the point representing the current stress state lies on the yield surface it can be supposed that this point do not leave the yield surface, the so called consistency condition *f = 0* must be valid. In case of active loading

$$f = 0, \quad \dot{f} = 0 \quad \text{and} \quad \frac{\partial f}{\partial \mathbf{0}} \colon d\sigma \ge 0 \,\, \,\tag{6}$$

the plastic deformation development is directed by the associated plastic flow rule

$$d\mathbf{e}^{\mathcal{P}} = \sqrt{\frac{3}{2}} d\boldsymbol{\lambda} \frac{\partial f}{\partial \mathbf{o}} \; \prime \; \frac{\partial f}{\partial \mathbf{o}} = \sqrt{\frac{3}{2}} \frac{\mathbf{s} - \mathbf{a}}{Y} = \mathbf{n} \; \prime \tag{7}$$

where the plastic multiplier *d*in (7) corresponds to the equivalent plastic strain increment

$$dp = \sqrt{\frac{2}{3}d\mathbf{e}^p : d\mathbf{e}^p} \,. \tag{8}$$

In this concept of single yield surface, the kinematic hardening rule

$$d\mathbf{a} = \mathcal{g}\left(\mathbf{a}, \dot{\mathbf{e}}^p, d\dot{\mathbf{e}}^p, dp, \mathbf{n}, \text{etc.}\right) \tag{9}$$

and the isotropic hardening rule

$$dY = h\left(R, dp, \mathbf{o}, \mathbf{a}, \dot{\mathbf{e}}^p, \text{etc.}\right) \tag{10}$$

play an essential role for the robustness of stress-strain model response. When the both hardening rules are used we speak about mixed hardening. Transient effects from initial cycles (cyclical hardening/softening), non-proportional hardening, ratcheting and other

Phenomenological Modelling of Cyclic Plasticity 339

uniaxial loading independently of mean stress value. Unfortunately, the mean stress

The important work, leading to the introduction of nonlinearity in the kinematic hardening rule, was the research report of Armstrong and Frederick (1966). In their model the memory

> *<sup>p</sup> d = Cd dp* 2 3 *α εα*

where *C* and are material parameters. Their physical meaning will be explained for pushpull loading. The quantity *dp* is an increment of accumulated plastic strain, which is

> *p p dp = dd* <sup>2</sup> :

Considering initially isotropic homogenous material, Von-Misses condition can be again

 *<sup>Y</sup> f =* <sup>3</sup> :

For the uniaxial loading case, the von Mises yield condition becomes to the simpler form

*<sup>Y</sup> f =* 

Similarly we can modify nonlinear kinematic hardening rule if we will consider only

*p p d Cd d*

*p pp p <sup>p</sup> d Cd d Cd d C d*

*<sup>d</sup> <sup>d</sup>*

 

 

0 0

 

*C*

1 to dispose of the absolute value

*p*

*sa sa*

  2

deviatoric part of the equation (11) taking into account plastic incompressibility. Then the nonlinear kinematic hardening rule leads to the differential equation

> 

 

and integrate to get the equation for backstress evolution

where *a* is a deviator of backstress and *s* is deviator of stress tensor .

3

(11)

(13)

(15)

(16)

*ε ε* (12)

0 (14)

 

(17)

relaxation effect cannot be described too.

term is added to the Prager rule

expressed as follows

used as follows

Now, we can use a multiplier

separate variables

**3.3 Armstrong-Frederick kinematic hardening model** 

effects of cyclic plasticity can be described by a suitable superposition of kinematic and isotropic hardening rules.

In the case of cyclic loading a kinematic hardening rule should always be included in the plasticity model, otherwise the Bauschinger's effect cannot be correctly described.

Fig. 11.Von Mises yield function with mixed hardening in the deviatoric plane.

The kinematic hardening rule is important in terms of the needs of capturing the cyclical response of the material. Description of the classical kinematic hardening rules and the resulting cyclic plasticity models is contained in next four subsections. Their availability in selected commercial software based on finite element method is given in Table 1.


Table 1.Occurrence of cyclic plasticity models in some popular FE software.

#### **3.2 Bilinear and multilinear kinematic hardening models**

There are two bilinear kinematic hardening rules coded in the most popular FE software, suggested by Prager (1953) and Ziegler (1959). The models predict the same response for von Mises material (Ottosen&Ristinmaa, 2005) and for uniaxial loading their response is bilinear. The models show no ratcheting under uniaxial loading and tend to plastic shakedown for a biaxial history of loading. The nonlinearity in stress-strain behaviour can be introduced by a multisurface model, when each surface represents a constant work hardening modulus in the stress space (Mroz, 1967).

Besseling in 1958 introduced a multilinear overlay model, which has a physical meaning and does not use any notion of surfaces. The Besseling model predicts plastic shakedown for

effects of cyclic plasticity can be described by a suitable superposition of kinematic and

In the case of cyclic loading a kinematic hardening rule should always be included in the

plasticity model, otherwise the Bauschinger's effect cannot be correctly described.

Fig. 11.Von Mises yield function with mixed hardening in the deviatoric plane.

selected commercial software based on finite element method is given in Table 1.

(Prager) <sup>x</sup>

(*M*max=5)

Table 1.Occurrence of cyclic plasticity models in some popular FE software.

*Armstrong-Frederick* x x x x

There are two bilinear kinematic hardening rules coded in the most popular FE software, suggested by Prager (1953) and Ziegler (1959). The models predict the same response for von Mises material (Ottosen&Ristinmaa, 2005) and for uniaxial loading their response is bilinear. The models show no ratcheting under uniaxial loading and tend to plastic shakedown for a biaxial history of loading. The nonlinearity in stress-strain behaviour can be introduced by a multisurface model, when each surface represents a constant work

Besseling in 1958 introduced a multilinear overlay model, which has a physical meaning and does not use any notion of surfaces. The Besseling model predicts plastic shakedown for

x

*Bilinear* <sup>x</sup>

*Multilinear* <sup>x</sup>

*Chaboche* <sup>x</sup>

hardening modulus in the stress space (Mroz, 1967).

**3.2 Bilinear and multilinear kinematic hardening models** 

The kinematic hardening rule is important in terms of the needs of capturing the cyclical response of the material. Description of the classical kinematic hardening rules and the resulting cyclic plasticity models is contained in next four subsections. Their availability in

**Kinematic hardening Ansys 13 Abaqus MSC.Marc MSC. Nastran** 

x (Ziegler)

(*M*max=3) - -

(Besseling) - - -

x (Ziegler)

isotropic hardening rules.

uniaxial loading independently of mean stress value. Unfortunately, the mean stress relaxation effect cannot be described too.

#### **3.3 Armstrong-Frederick kinematic hardening model**

The important work, leading to the introduction of nonlinearity in the kinematic hardening rule, was the research report of Armstrong and Frederick (1966). In their model the memory term is added to the Prager rule

$$
\Delta d \mathbf{a} = \frac{2}{3} \mathbf{C} d \mathbf{e}\_p - \chi \mathbf{a} dp \tag{11}
$$

where *C* and are material parameters. Their physical meaning will be explained for pushpull loading. The quantity *dp* is an increment of accumulated plastic strain, which is expressed as follows

$$dp = \sqrt{\frac{2}{3} \, d\mathfrak{e}\_p : d\mathfrak{e}\_p} \tag{12}$$

Considering initially isotropic homogenous material, Von-Misses condition can be again used as follows

$$f = \sqrt{\frac{3}{2}(s-a) : (s-a)} - \sigma\_Y \tag{13}$$

where *a* is a deviator of backstress and *s* is deviator of stress tensor .

For the uniaxial loading case, the von Mises yield condition becomes to the simpler form

$$f = \left| \sigma - \alpha \right| - \sigma\_Y = 0 \tag{14}$$

Similarly we can modify nonlinear kinematic hardening rule if we will consider only deviatoric part of the equation (11) taking into account plastic incompressibility.

Then the nonlinear kinematic hardening rule leads to the differential equation

$$
d\alpha = \mathbb{C}d\varepsilon\_p - \gamma \alpha \left| d\varepsilon\_p \right| \tag{15}
$$

Now, we can use a multiplier 1 to dispose of the absolute value

$$\text{rad}\alpha = \text{Cd}\varepsilon\_p - \gamma \alpha \left| d\varepsilon\_p \right| = \text{Cd}\varepsilon\_p - \gamma \alpha \gamma \nu \, d\varepsilon\_p = \left( \text{C} - \gamma \alpha \gamma \nu \right) d\varepsilon\_p \tag{16}$$

separate variables

$$\int\_{a\_0}^{a} \frac{da}{C - \gamma \alpha \nu} = \int\_{c\_0}^{c} d\varepsilon\_p \tag{17}$$

and integrate to get the equation for backstress evolution

Phenomenological Modelling of Cyclic Plasticity 341

max *= e* min

 

min *= e* max

 

 *=* 

) similarly

Fig. 13. Initial conditions for the backstress and plastic strain.

For the compression ( 1

strain amplitude respectively.

should be satisfied

isotropic hardening)

After substitution of (24) to the equation (25) we get

*C C <sup>p</sup> ap*

 

*C C <sup>p</sup> ap*

*a Y ap C*

where tanh(x) is the hyperbolic tangent function and a, ap are stress amplitude and plastic

For a proper ratcheting description, the condition of equality for computed and experimentally stated ratcheting strain rate for opened stabilized hysteresis loop (Fig.7)

*pFEM = pEXP*

According to (Chaboche & Lemaitre, 1990), for Armstrong-Frederick kinematic hardening rule the plastic strain increment per cycle can be written as follows (with absence of

 

 

 tanh 

 

  (24)

(25)

(26)

(27)

$$
\alpha = \nu \frac{\mathbb{C}}{\gamma} + \left(\alpha\_0 - \nu \frac{\mathbb{C}}{\gamma}\right) e^{-\nu \gamma \left(\varepsilon\_r - \varepsilon\_{r0}\right)} \tag{18}
$$

Therefore, the relation for stress is given by yield condition

$$
\sigma = \alpha + \psi \sigma\_Y \tag{19}
$$

For tension 1 and considering zeros initial values of plastic strain and backstress this equation is given

$$
\sigma = \sigma\_Y + \frac{C}{\mathcal{Y}} \mathbf{(1} - e^{-\mathcal{Y}\_p}) \tag{20}
$$

Now, we can investigate the limit values of the nonlinear function and its first derivation to get a concept about influence of parameters *C* and on stress – strain response of the Armstrong-Frederick model

$$\lim\_{x\_r \to 0} \mathbf{C} e^{-\gamma x\_r} = \mathbf{C} \tag{21}$$

$$\lim\_{\sigma\_p \to \infty} \sigma\_Y + \frac{\mathbb{C}}{\mathcal{Y}} \mathbf{1} - e^{-\gamma x\_p} \stackrel{\mathcal{C}}{=} \sigma\_Y + \frac{\mathcal{C}}{\mathcal{Y}} \tag{22}$$

Fig. 12. Properties of the nonlinear kinematic hardening model of Armstrong/Frederick.

Described nonlinear kinematic hardening model allows to correctly capture Bauschinger effect and even behavior by nonsymmetrical loading in tension-compression. The large advantage of Armstrong-Frederick model is its easy implementation and the mentioned nonlinear behavior of the model. On the other hand, the model can not descibe the hysteresis loop shape precisely.

For the case of cyclic loading the parameters *Y*, *C* and should be estimated from the cyclic strain curve. It is possible to determine the equation corresponding to the cyclic curve of Armstrong-Frederick model by application of equation (18) for the upper branch and the bottom branch of hysteresis loop. For tension 1 is valid and we have

0

 

 

 *=* 

*Y C = e* 1

*p*

*p*

0 lim

Fig. 12. Properties of the nonlinear kinematic hardening model of Armstrong/Frederick.

Described nonlinear kinematic hardening model allows to correctly capture Bauschinger effect and even behavior by nonsymmetrical loading in tension-compression. The large advantage of Armstrong-Frederick model is its easy implementation and the mentioned nonlinear behavior of the model. On the other hand, the model can not descibe the

strain curve. It is possible to determine the equation corresponding to the cyclic curve of Armstrong-Frederick model by application of equation (18) for the upper branch and the

*Y*, *C* and

is valid and we have

Now, we can investigate the limit values of the nonlinear function and its first derivation to

*p*

*Ce = C*

*<sup>p</sup>*

*Y Y C C* lim 1 *e =* 

 

Therefore, the relation for stress is given by yield condition

get a concept about influence of parameters *C* and

For tension

equation is given

Armstrong-Frederick model

hysteresis loop shape precisely.

For the case of cyclic loading the parameters

bottom branch of hysteresis loop. For tension 1

*C C p p <sup>e</sup>* <sup>0</sup>

   

1 and considering zeros initial values of plastic strain and backstress this

*<sup>p</sup>*

 

(22)

(18)

*<sup>Y</sup>* (19)

(20)

on stress – strain response of the

(21)

should be estimated from the cyclic

$$
\alpha\_{\text{max}} = \frac{\text{C}}{\mathcal{Y}} + \left(\alpha\_{\text{min}} - \frac{\text{C}}{\mathcal{Y}}\right) e^{-\mathcal{I}\left(\varepsilon\_{\text{p}} - \varepsilon\_{\text{ap}}\right)} \tag{24}
$$

For the compression ( 1 ) similarly

$$\alpha\_{\min} = -\frac{\mathbf{C}}{\mathcal{Y}} + \left(\alpha\_{\max} + \frac{\mathbf{C}}{\mathcal{Y}}\right) e^{\mathcal{I}\left(\varepsilon\_r - \varepsilon\_{qr}\right)} \tag{25}$$

After substitution of (24) to the equation (25) we get

$$
\sigma\_a = \sigma\_Y + \frac{C}{\mathcal{Y}} \tanh\left(\mathcal{Y} \sigma\_{a\mathcal{Y}}\right) \tag{26}
$$

where tanh(x) is the hyperbolic tangent function and a, ap are stress amplitude and plastic strain amplitude respectively.

Fig. 13. Initial conditions for the backstress and plastic strain.

For a proper ratcheting description, the condition of equality for computed and experimentally stated ratcheting strain rate for opened stabilized hysteresis loop (Fig.7) should be satisfied

$$
\delta \mathfrak{E}\_{pFEM} = \delta \mathfrak{E}\_{pEXP} \tag{27}
$$

According to (Chaboche & Lemaitre, 1990), for Armstrong-Frederick kinematic hardening rule the plastic strain increment per cycle can be written as follows (with absence of isotropic hardening)

Phenomenological Modelling of Cyclic Plasticity 343

*i a Y M p i i <sup>C</sup> = eC* 1

1

When the cyclic strain curve of the investigated material is not available, it is possible to use for the calibration of the model also large saturated hysteresis loop. Based on the

*M*

relationship (31), considering tension ( 1

1

*i i*

*<sup>C</sup> <sup>=</sup>* ( ) 0 0

*i*

*<sup>C</sup> = eC* <sup>2</sup> ( ( ))

In the Chaboche model the parameter *γ<sup>M</sup>* influences ratcheting (provided that the last backstress part has the lowest value of the parameter γ*i*) and is chosen to be small (up to γ*M=10*). For the case of γ*M= 0* ratcheting cannot occur. However, after particular number of cycles the stabilized hysteresis loop will be formed (the Chaboche model tends to plastic shakedown) as it is clear from the graph at the Fig.16. For many materials such behavior does not correspond with reality and during numerical modeling, constant deformation increment can be achieved with aim of suitable choice of parameter *γM .* We can also provide the relation

1 2

 *<sup>i</sup> i p ap Y p*

*p M*

Thus, with suitable choice of *γ<sup>M</sup>* we get very good model for uniaxial ratcheting prediction (ratcheting with steady state only). In case of non-proportional loading the Chaboche model with three backstresses (*M*=3) considered in Fig.14 and Fig.15 drastically over predicts

Fig. 14. Properties of constants of Chaboche nonlinear kinematic hardening model (case *M*=3).

*m p <sup>C</sup> <sup>=</sup>* , <sup>2</sup> 

 

,

*i i*

*M*

ratcheting as has been shown by other researchers (Bari & Hassan, 2000).

1

we can get for the upper branch of the hysteresis loop this expression

 

published elsewhere (Chaboche and Nouailhas 1989).

*i p*

 

*p ap*

   

) and these initial values (see Fig15)

3

 

(36)

(34)

(35)

(37)

$$\delta \varepsilon\_{pFEM} = \frac{1}{\mathcal{Y}} \cdot \ln \left| \frac{\left(\frac{C}{\mathcal{Y}}\right)^2 - \left(\sigma\_{\min} + \sigma\_Y\right)^2}{\left(\frac{C}{\mathcal{Y}}\right)^2 - \left(\sigma\_{\max} + \sigma\_Y\right)^2} \right| \tag{28}$$

where *<sup>p</sup>* is measured between upper peaks of two hysteresis loops as can be seen in Fig.7.

#### **3.4 Chaboche kinematic hardening model**

Very important improvement was the proposal of nonlinear kinematic hardening model by Chaboche (1979), which eliminated Armstrong-Frederick model disadvantages by creating a backstress through superposition of *M* parts

$$a = \sum\_{i=1}^{M} a^{(i)} \tag{29}$$

whereas for each part the evolution equation of Armstrong and Frederick is used

$$
tau^{(i)} = \frac{2}{3} \mathbf{C}\_i d\varepsilon\_p - \boldsymbol{\gamma}\_i a^{(i)} dp \tag{30}
$$

where *Ci* and γ*i* are material parameters.

Due to the usage of Armstrong-Frederick evolution law we can directly write the expression for static strain curve

$$
\sigma = \psi \sigma\_Y + \sum\_{i=1}^{M} \psi \frac{\mathbf{C}\_i}{\mathcal{Y}\_i} + \left( \alpha\_0^{(i)} - \psi \frac{\mathbf{C}\_i}{\mathcal{Y}\_i} \right) e^{-\psi \mathcal{Y}\_i \left( \varepsilon\_p - \varepsilon\_{p0} \right)} \tag{31}
$$

and for cyclic strain curve

$$
\sigma\_a = \sigma\_Y + \sum\_{i=1}^{M} \frac{\mathbb{C}\_i}{\mathcal{I}\_i} \tanh\left(\mathcal{I}\_i \varepsilon\_{ap}\right) \tag{32}
$$

The quality of cyclic strain curve description is adequate in the case of Chaboche model with the three evolution parts.

Thanks to the similar properties of functions tanh(x) and 1-exp(-x), including its derivatives, it is possible to use the same approach for parameter estimation from the static even cyclic strain curve. Parameters should be determined for example by a nonlinear least-squares method. It is useful to consider Prager's rule for the last backstress part (γ*M= 0*). The parameter influence ratcheting and mean stress relaxation effects. Therefore, we can use this approximation function for cyclic and static strain curves respectively

$$
\sigma\_a = \sigma\_Y + \sum\_{i=1}^{M-1} \frac{\mathbb{C}\_i}{\mathcal{I}\_i} \tanh\left(\boldsymbol{\gamma}\_i \boldsymbol{\varepsilon}\_{ap}\right) + \mathbb{C}\_M \boldsymbol{\varepsilon}\_{ap} \tag{33}
$$

2

2

*C*

*C*

Very important improvement was the proposal of nonlinear kinematic hardening model by Chaboche (1979), which eliminated Armstrong-Frederick model disadvantages by creating a

> *M i*

*i a= a*( ) 1

*i i ip i da = C d a dp* ( ) <sup>2</sup> ( )

 

Due to the usage of Armstrong-Frederick evolution law we can directly write the expression

*i i í*

 *<sup>M</sup> i a Y i ap i i*

 

 

*i i i C C = e* <sup>0</sup> ( ) ( ) 0

tanh

The quality of cyclic strain curve description is adequate in the case of Chaboche model with

Thanks to the similar properties of functions tanh(x) and 1-exp(-x), including its derivatives, it is possible to use the same approach for parameter estimation from the static even cyclic strain curve. Parameters should be determined for example by a nonlinear least-squares method. It is useful to consider Prager's rule for the last backstress part (γ*M= 0*). The parameter influence ratcheting and mean stress relaxation effects. Therefore, we can use this

> *<sup>M</sup> i a Y i ap M ap*

 

(33)

*<sup>C</sup> = C* 1

*i i*

tanh

1

 

whereas for each part the evolution equation of Armstrong and Frederick is used

3

*M*

1

 

*<sup>C</sup> <sup>=</sup>* 1

*Y*

approximation function for cyclic and static strain curves respectively

 

*pFEM*

**3.4 Chaboche kinematic hardening model** 

backstress through superposition of *M* parts

where *Ci* and γ*i* are material parameters.

for static strain curve

and for cyclic strain curve

the three evolution parts.

where *<sup>p</sup>* 

*=*

<sup>1</sup> ln

min

 

is measured between upper peaks of two hysteresis loops as can be seen in Fig.7.

max

*Y*

 

  2

2

(29)

(30)

*ip p*

(31)

(32)

 

(28)

*Y*

$$
\sigma\_d = \sigma\_Y + \sum\_{i=1}^{M-1} \frac{\mathbf{C}\_i}{\mathcal{I}\_i} \left( \mathbf{1} - e^{-\gamma\_i \varepsilon\_p} \right) + \mathbf{C}\_M \varepsilon\_p \tag{34}
$$

When the cyclic strain curve of the investigated material is not available, it is possible to use for the calibration of the model also large saturated hysteresis loop. Based on the relationship (31), considering tension ( 1 ) and these initial values (see Fig15)

$$
\alpha\_0^{(i)} = -\frac{C\_i}{\gamma\_i}, \varepsilon\_{p0} = -\varepsilon\_{ap} \tag{35}
$$

we can get for the upper branch of the hysteresis loop this expression

$$
\sigma = \sigma\_Y + \sum\_{i=1}^{2} \frac{\mathbf{C}\_i}{\mathcal{I}\_i} \left( 1 - 2e^{\left( -\gamma\_i \varepsilon\_p - (-\varepsilon\_{\eta p}) \right)} \right) + \mathbf{C}\_3 \varepsilon\_p \tag{36}
$$

In the Chaboche model the parameter *γ<sup>M</sup>* influences ratcheting (provided that the last backstress part has the lowest value of the parameter γ*i*) and is chosen to be small (up to γ*M=10*). For the case of γ*M= 0* ratcheting cannot occur. However, after particular number of cycles the stabilized hysteresis loop will be formed (the Chaboche model tends to plastic shakedown) as it is clear from the graph at the Fig.16. For many materials such behavior does not correspond with reality and during numerical modeling, constant deformation increment can be achieved with aim of suitable choice of parameter *γM .* We can also provide the relation

$$\gamma\_M = \frac{\delta \varepsilon\_p \cdot \mathbb{C}\_M}{2 \cdot \sigma\_m \cdot \Delta \varepsilon\_p} \, ^\prime \tag{37}$$

published elsewhere (Chaboche and Nouailhas 1989).

Thus, with suitable choice of *γ<sup>M</sup>* we get very good model for uniaxial ratcheting prediction (ratcheting with steady state only). In case of non-proportional loading the Chaboche model with three backstresses (*M*=3) considered in Fig.14 and Fig.15 drastically over predicts ratcheting as has been shown by other researchers (Bari & Hassan, 2000).

Fig. 14. Properties of constants of Chaboche nonlinear kinematic hardening model (case *M*=3).

Phenomenological Modelling of Cyclic Plasticity 345

*<sup>b</sup> <sup>p</sup> R=R e* (1 )

which follows from integration of equation (39) under following assumption: Change of

Fig. 17. Evolution of isotropic internal variable *R* for the nonlinear isotropic hardening rule.

After Chaboche (1979) there were designed many evolution equations for better ratcheting prediction in the category of nonlinear kinematic hardening rules, but mostly based on the Chaboche superposition of several backstress parts. Because of the large number of theories we choose for their presentation form of table, which contains links to the original publications (Table 2). Presented group of cyclic plasticity models of Chaboche type, which

*M*

*i d d* ( ) 1

3 **a ε**

*i i p i i i d C d dp* ( ) <sup>2</sup> () ()

Authors do not guarantee completeness of the set of theories. There are also some new models and approaches. Very progressive are so called models with yield surface distortion, for example Vincent (2004), which are able to model the anisotropy induced by previous

*i*

variable *R* from zero to *R*

The material parameter *R*

where *C*i,

considers the backstress to be defined by *M* parts

i are material parameters, *d*

plastic deformation (see chapter 2.6).

of accumulated plastic strain causing dynamic recovery of

can be generalized considering the evolution equation in the form

cyclic strain curve of particular material.

**3.6 Other cyclic plasticity models** 

constant *b* are briefly described in (Chaboche & Lemaitre, 1990).

(39)

and *p* from zero to *p*. Other possibilities how to identify the

can be determined, for example, by comparison of static and

**a a** (40)

p is plastic strain increment and *dp*(i) is the increment

(i).

**a** (41)

Fig. 15. Scheme for use of the hysteresis loop to identify parameters of Chaboche model.

Fig. 16. Influence of parameter *γ2* on ratcheting response of the Chaboche model (*M*=2).

#### **3.5 Mixed hardening models**

Most of materials exhibit Masing and cyclic softening/hardening behaviour, which can be described by superposition of isotropic hardening to a kinematic hardening rule. In this case the size of yield surface *Y* is expressed with aim of initial value of *Y* and isotropic variable *R*, which is usually dependent on the accumulated plastic deformation.

If we would like to describe cyclic softening/hardening, it is convenient to use simple evolutionary equation, leading to nonlinear isotropic hardening rule

$$dR \equiv b(R\_{\oplus} - R)dp\tag{38}$$

where *R*, *b* are material parameters and *dp* is an increment of accumulated plastic strain. The value of constant *b*, which determines the rate of stabilization of the hysteresis loop in case of loading with constant strain amplitude (Fig.17), can be determined directly from following equation

Fig. 15. Scheme for use of the hysteresis loop to identify parameters of Chaboche model.

Fig. 16. Influence of parameter *γ2* on ratcheting response of the Chaboche model (*M*=2).

*R*, which is usually dependent on the accumulated plastic deformation.

evolutionary equation, leading to nonlinear isotropic hardening rule

Most of materials exhibit Masing and cyclic softening/hardening behaviour, which can be described by superposition of isotropic hardening to a kinematic hardening rule. In this case the size of yield surface *Y* is expressed with aim of initial value of *Y* and isotropic variable

If we would like to describe cyclic softening/hardening, it is convenient to use simple

The value of constant *b*, which determines the rate of stabilization of the hysteresis loop in case of loading with constant strain amplitude (Fig.17), can be determined directly from

, *b* are material parameters and *dp* is an increment of accumulated plastic strain.

*dR = b R R dp* ( ) (38)

**3.5 Mixed hardening models** 

where *R*

following equation

$$R = R\_{\psi} (1 - e^{-b \cdot p}) \tag{39}$$

which follows from integration of equation (39) under following assumption: Change of variable *R* from zero to *R* and *p* from zero to *p*. Other possibilities how to identify the constant *b* are briefly described in (Chaboche & Lemaitre, 1990).

Fig. 17. Evolution of isotropic internal variable *R* for the nonlinear isotropic hardening rule.

The material parameter *R* can be determined, for example, by comparison of static and cyclic strain curve of particular material.

#### **3.6 Other cyclic plasticity models**

After Chaboche (1979) there were designed many evolution equations for better ratcheting prediction in the category of nonlinear kinematic hardening rules, but mostly based on the Chaboche superposition of several backstress parts. Because of the large number of theories we choose for their presentation form of table, which contains links to the original publications (Table 2). Presented group of cyclic plasticity models of Chaboche type, which considers the backstress to be defined by *M* parts

$$d\mathbf{a} = \sum\_{i=1}^{M} d\mathbf{a}^{(i)}\tag{40}$$

can be generalized considering the evolution equation in the form

$$d\mathbf{a}^{(i)} = \frac{2}{3} \mathbf{C}\_i d\mathbf{e}^p - \boldsymbol{\gamma}\_i \mathbf{a}^{(i)} d\boldsymbol{p}^{(i)} \tag{41}$$

where *C*i, i are material parameters, *d*p is plastic strain increment and *dp*(i) is the increment of accumulated plastic strain causing dynamic recovery of (i).

Authors do not guarantee completeness of the set of theories. There are also some new models and approaches. Very progressive are so called models with yield surface distortion, for example Vincent (2004), which are able to model the anisotropy induced by previous plastic deformation (see chapter 2.6).

Phenomenological Modelling of Cyclic Plasticity 347

It is obvious that many theories differ very little. For correct description of ratcheting in proportional and non-proportional loading more and more authors introduced a nonproportional parameter, which enables simultaneous correct description of uniaxial and multiaxial ratcheting (Chen & Jiao 2004; Chen, 2005; Halama, 2008). Significant improvements of prediction capability can be reached by using memory surfaces (Jiang & Sehitoglu, 1996 and Döring, 2003). Presented models should be compared in terms of nonlinearity, established for each backstress in the case of uniaxial loading. The Fig.18

Common values of parameters are considered for models Ohno-Wang-II and AbdelKarim-Ohno in the Fig.18. Both models lead to the Ohno-Wang model I at a certain choice of parameters affecting ratcheting. In the case of the AbdelKarim-Ohno model this occurs

I, if *mi*= ∞ for all *i*. Thus, the nonlinearity in the Fig.18 is weak for common parameters

Ohno-Wang model I. For determination of these material parameters we can use again either cyclic strain curve of the material (Ohno-Wang, 1993), (AbdelKarim-Ohno, 2000), or a

*<sup>i</sup>* = 0 for all *i,* see Fig.19. Ohno-Wang model II corresponds to the Ohno-Wang model

*<sup>i</sup>*<0.2) and we can therefore use the same

i (*i* = 1 , ...,*M*) as in the case of multilinear

compares four basic hardening rules.

(Ohno-Wang II: *mi*>>1, AbdelKarim-Ohno:

procedure for estimation of the parameters *C*i,

stabilised hysteresis loop (Bari & Hassan, 2000).

Fig. 18. Nonlinearity introduced in four basic cyclic plasticity models.

when 

Table 2. Overview of some kinematic hardening rules.

Table 2. Overview of some kinematic hardening rules.

It is obvious that many theories differ very little. For correct description of ratcheting in proportional and non-proportional loading more and more authors introduced a nonproportional parameter, which enables simultaneous correct description of uniaxial and multiaxial ratcheting (Chen & Jiao 2004; Chen, 2005; Halama, 2008). Significant improvements of prediction capability can be reached by using memory surfaces (Jiang & Sehitoglu, 1996 and Döring, 2003). Presented models should be compared in terms of nonlinearity, established for each backstress in the case of uniaxial loading. The Fig.18 compares four basic hardening rules.

Common values of parameters are considered for models Ohno-Wang-II and AbdelKarim-Ohno in the Fig.18. Both models lead to the Ohno-Wang model I at a certain choice of parameters affecting ratcheting. In the case of the AbdelKarim-Ohno model this occurs when *<sup>i</sup>* = 0 for all *i,* see Fig.19. Ohno-Wang model II corresponds to the Ohno-Wang model I, if *mi*= ∞ for all *i*. Thus, the nonlinearity in the Fig.18 is weak for common parameters (Ohno-Wang II: *mi*>>1, AbdelKarim-Ohno: *<sup>i</sup>*<0.2) and we can therefore use the same procedure for estimation of the parameters *C*i, i (*i* = 1 , ...,*M*) as in the case of multilinear Ohno-Wang model I. For determination of these material parameters we can use again either cyclic strain curve of the material (Ohno-Wang, 1993), (AbdelKarim-Ohno, 2000), or a stabilised hysteresis loop (Bari & Hassan, 2000).

Fig. 18. Nonlinearity introduced in four basic cyclic plasticity models.

Phenomenological Modelling of Cyclic Plasticity 349

For a description of the stress strain behaviour the AbdelKarim-Ohno model and two classical models of cyclic plasticity were chosen - Armstrong-Frederick model and Chaboche model with two backstress parts (*M*=2). The AbdelKarim-Ohno model was implemented to the ANSYS program via user subroutine. Material parameters used in simulations are listed in the tables below, except elastic constants (Young modulus E=180000MPa and Poisson´s

> *Y*=500 MPa, *C*=108939, =2.5 *R*=-250MPa, *b*=30

*Y*=500 MPa, *C*1=264156, 1=873, *C*2=20973, 2=1 *R*=-320MPa, *b*=30

*<sup>Y</sup>σ = 200MPa C = 3 0,130770,36290,32420,12940,18350MPa 1 6* 1060 *1 6 γ = 5 2020,980,520,255,3* 884, *= 0. = 0* <sup>0</sup>

 5, 0.5, .14

Plasticitymodel Material parameters

Fig. 21. Saturated uniaxial hysteresis loop and its prediction by AF and CHAB models.

**Total strain**

E CH AF

CHAB

AF

EXPERIMENT

Cyclic plasticity models were calibrated using saturated hysteresis loop from the test with strain range of 1.5% (Fig.21) and a uniaxial ratcheting test. The calibration procedure used for AbdelKarim-Ohno model was described in the paper Halama (2008). The results of ratcheting prediction gained from simulations of the low cycle fatigue test with nonzero mean stress (case D: 500 cycles with σm = 40MPa and σa = 500MPa) are shown for all three

ratio =0.3).

Armstrong-Frederick and Voce rule (AF)

Chabocheand Voce rule (CHAB)

AbdelKarim-Ohno (AKO)

material models in Fig. 22.

Table 3. Material parameters of used cyclic plasticity models.

‐600



0


200

**Stress [MPa]**

400

600

Fig. 19. Influence of ratcheting parameter on the response of AbdelKarim-Ohno model.

As shown in Fig.19, appropriate choice of parameter gives desired ratcheting rate. Considering the only one parameter for ratcheting *i* =and its evolution by equation

$$d\mu \equiv \left\| \left( \mu\_{\oplus} - \mu \right) dp \right\|\tag{42}$$

transient effects in initial cycles can be described too.

#### **4. Ratcheting simulations for a wheel steel**

There was realized a set of low-cycle fatigue tests of specimen made from R7T wheel steel at the Czech Technical University in Prague. The specimens were subjected to tensioncompression and tension/torsion on the test machine MTS 858 MiniBionix. All tests were force controlled. More detailed description of experiments was reported elsewhere (Halama, 2009). Four cases considered for simulations in this book are shown in Fig.20.

Fig. 20. The scheme of four realized loading paths.

Fig. 19. Influence of ratcheting parameter on the response of AbdelKarim-Ohno model.

*d =* 

2009). Four cases considered for simulations in this book are shown in Fig.20.

<sup>a</sup>

x

x

<sup>a</sup> <sup>a</sup>

x

Considering the only one parameter for ratcheting

transient effects in initial cycles can be described too.

**4. Ratcheting simulations for a wheel steel** 

Fig. 20. The scheme of four realized loading paths.

<sup>a</sup>

As shown in Fig.19, appropriate choice of parameter gives desired ratcheting rate.

 

There was realized a set of low-cycle fatigue tests of specimen made from R7T wheel steel at the Czech Technical University in Prague. The specimens were subjected to tensioncompression and tension/torsion on the test machine MTS 858 MiniBionix. All tests were force controlled. More detailed description of experiments was reported elsewhere (Halama,

*i* =

and its evolution by equation

xm

xa

x

*ω dp* (42)

For a description of the stress strain behaviour the AbdelKarim-Ohno model and two classical models of cyclic plasticity were chosen - Armstrong-Frederick model and Chaboche model with two backstress parts (*M*=2). The AbdelKarim-Ohno model was implemented to the ANSYS program via user subroutine. Material parameters used in simulations are listed in the tables below, except elastic constants (Young modulus E=180000MPa and Poisson´s ratio =0.3).


Table 3. Material parameters of used cyclic plasticity models.

Cyclic plasticity models were calibrated using saturated hysteresis loop from the test with strain range of 1.5% (Fig.21) and a uniaxial ratcheting test. The calibration procedure used for AbdelKarim-Ohno model was described in the paper Halama (2008). The results of ratcheting prediction gained from simulations of the low cycle fatigue test with nonzero mean stress (case D: 500 cycles with σm = 40MPa and σa = 500MPa) are shown for all three material models in Fig. 22.

Fig. 21. Saturated uniaxial hysteresis loop and its prediction by AF and CHAB models.

Phenomenological Modelling of Cyclic Plasticity 351

The models CHAB and AF contain the nonlinear isotropic hardening rule (39), which enables to describe cyclic softening in initial cycles, see Fig.22. On the other hand, ratcheting prediction is better in the case of AbdelKarim-Ohno model. The same conclusion we have for simulations of the last loading case (case B: 500 cycles with σa = 490MPa and a = 170MPa, 250 cycles with σa = 490MPa and a = 115MPa, 250 cycles with σa = 490MPa and a =

Fig. 24. Comparison of multiaxial ratcheting predictions with experiment (case B).

Background of the particular effects in cyclic plasticity of metals explained in the second section makes possible to understand well described incremental theory of plasticity and main features of cyclic plasticity models of Chaboche type. There have been shown interesting results of fatigue test simulations with emphasis on cyclic creep (ratcheting) prediction. It can be concluded from the results of simulations of the section 4 that used combined hardening model of Chaboche with two backstress parts can fairly well predicts the trend of accumulation of plastic deformation (ratcheting) for uniaxial and multiaxial loading cases, even under non-proportional loading, in comparison with the experimental observations of the R7T wheel steel. Indeed, the AbdelKarim-Ohno model gives better

prediction of ratcheting for all cases than Armstrong-Frederick and Chaboche model.

215MPa) as can be seen at the Fig. 24.

**5. Conclusion** 

Fig. 22. Comparison of uniaxial ratcheting predictions with experiment (case D).

The results of multiaxial ratcheting predictions corresponding to simulations of the low cycle fatigue test performed under tension/torsion non-proportional (case A: 150 cycles with σa = 125MPa and a = 300MPa) and proportional loading (case C: 100 cycles with σa = 225MPa and a = 65MPa) are displayed in Fig. 23.

Fig. 23. Comparison of multiaxial ratcheting predictions with experiment (cases A and C).

Fig. 22. Comparison of uniaxial ratcheting predictions with experiment (case D).

225MPa and a = 65MPa) are displayed in Fig. 23.

The results of multiaxial ratcheting predictions corresponding to simulations of the low cycle fatigue test performed under tension/torsion non-proportional (case A: 150 cycles with σa = 125MPa and a = 300MPa) and proportional loading (case C: 100 cycles with σa =

Fig. 23. Comparison of multiaxial ratcheting predictions with experiment (cases A and C).

The models CHAB and AF contain the nonlinear isotropic hardening rule (39), which enables to describe cyclic softening in initial cycles, see Fig.22. On the other hand, ratcheting prediction is better in the case of AbdelKarim-Ohno model. The same conclusion we have for simulations of the last loading case (case B: 500 cycles with σa = 490MPa and a = 170MPa, 250 cycles with σa = 490MPa and a = 115MPa, 250 cycles with σa = 490MPa and a = 215MPa) as can be seen at the Fig. 24.

Fig. 24. Comparison of multiaxial ratcheting predictions with experiment (case B).

#### **5. Conclusion**

Background of the particular effects in cyclic plasticity of metals explained in the second section makes possible to understand well described incremental theory of plasticity and main features of cyclic plasticity models of Chaboche type. There have been shown interesting results of fatigue test simulations with emphasis on cyclic creep (ratcheting) prediction. It can be concluded from the results of simulations of the section 4 that used combined hardening model of Chaboche with two backstress parts can fairly well predicts the trend of accumulation of plastic deformation (ratcheting) for uniaxial and multiaxial loading cases, even under non-proportional loading, in comparison with the experimental observations of the R7T wheel steel. Indeed, the AbdelKarim-Ohno model gives better prediction of ratcheting for all cases than Armstrong-Frederick and Chaboche model.

Phenomenological Modelling of Cyclic Plasticity 353

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#### **6. Acknowledgment**

This chapter has been elaborated in the framework of the IT4Innovations Centre of Excellence project, reg. no. CZ.1.05/1.1.00/02.0070 supported by Operational Programme 'Research and Development for Innovations' funded by Structural Funds of the European Union and state budget of the Czech Republic. Authors would like to appreciate the assistance of BONATRANS GROUP a.s. Bohumín company during the preparation and realization of experiments.

#### **7. References**


This chapter has been elaborated in the framework of the IT4Innovations Centre of Excellence project, reg. no. CZ.1.05/1.1.00/02.0070 supported by Operational Programme 'Research and Development for Innovations' funded by Structural Funds of the European Union and state budget of the Czech Republic. Authors would like to appreciate the assistance of BONATRANS GROUP a.s. Bohumín company during the preparation and

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**6. Acknowledgment** 

realization of experiments.

**7. References** 

905.

10.


**1. Introduction**

models is the following initial value problem

(*k*)

*Dα*

<sup>∗</sup> *<sup>y</sup>*(*t*) = *<sup>J</sup>*

<sup>∗</sup> *<sup>y</sup>*(*t*) = *<sup>f</sup>*(*t*, *<sup>y</sup>*(*t*)), *<sup>y</sup>*(*k*)

*J*

*Dα*

the Caputo sense, defined by

where *α* ∈ (0, ∞), *y*

follows

Fractional calculus, which has almost the same history as classic calculus, did not attract enough attention for a long time. However, in recent decades, fractional calculus and fractional differential equations become more and more popular because of its powerful potential applications. A large number of new differential equations (models) that involve fractional calculus are developed. These models have been applied successfully, e.g., in mechanics (theory of viscoelasticity), biology (modelling of polymers and proteins), chemistry (modelling the anomalous diffusion behavior of Brownian particles), electrical engineering (electromagnetic waves) etc (Bouchaud & Georges, 1990; Hilfer, 2000; Kirchner et al., 2000; Metzler & Klafter, 2000; Zaslavsky, 2002; Mainrdi, 2008). Meanwhile, some rich fractional dynamical behavior which reflect the inherent nature of realistic physical systems are observed. In short, fractional calculus and fractional differential equations have played more and more important role in almost all the scientific fields. One of the most important fractional

(*x*0) = *y*

Γ(*n* − *α*)

here *<sup>n</sup>* :<sup>=</sup> �*α*� is the smallest integer not less than *<sup>α</sup>*, *<sup>y</sup>*(*n*)(*x*) is the classical *<sup>n</sup>*th-order derivative of *y*(*x*) and for *μ* > 0,*J<sup>μ</sup>* is the *μ*-order Riemann-Liouville integral operator expressed as

> *t* 0

The use of Caputo derivative in above equation is partly because of the convenience to specify the initial conditions. Since the initial conditions are expressed in terms of values of the unknown function and its integer-order derivatives which have clear physical meaning (Podlubny, 1999; 2002). However, from pure mathematical viewpoint, the Riemann-Liouville derivative is more welcome and many earlier research papers use it instead of Caputo derivative (Podlubny, 1999). In general, specifying some additional conditions is necessary

<sup>0</sup> can be any real numbers, and *<sup>D</sup><sup>α</sup>*

*<sup>n</sup>*−*αDny*(*t*) = <sup>1</sup>

*<sup>μ</sup>y*(*t*) = <sup>1</sup>

Γ(*μ*)

(*k*)

 *t* 0

*y*(*τ*) (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)1−*<sup>μ</sup>* *y*(*n*)(*τ*)

**Numerical Schemes for Fractional** 

*School of Mathematics and Statistics, Lanzhou University, Lanzhou* 

**Ordinary Differential Equations** 

Weihua Deng and Can Li

*People's Republic of China* 

*dτ*.

<sup>0</sup> , *k* = 0, 1, ··· , �*α*� − 1, (1)

<sup>∗</sup> denotes the fractional derivative in

**16**

(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)1+*α*−*<sup>n</sup> <sup>d</sup>τ*, (2)

Yaguchi, M. & Takahashi, Y. (2005). Ratcheting of viscoplastic material with cyclic softening, part 2: Application of constitutive models. *International Journal of Plasticity* 21, p. 835–860.

Ziegler, H. (1959). A modification of Prager's hardening rule. *Quart. Appl. Math.* 17, p. 55-65.

### **Numerical Schemes for Fractional Ordinary Differential Equations**

Weihua Deng and Can Li

*School of Mathematics and Statistics, Lanzhou University, Lanzhou People's Republic of China* 

#### **1. Introduction**

354 Numerical Modelling

Yaguchi, M. & Takahashi, Y. (2005). Ratcheting of viscoplastic material with cyclic softening,

Ziegler, H. (1959). A modification of Prager's hardening rule. *Quart. Appl. Math.* 17, p. 55-65.

835–860.

part 2: Application of constitutive models. *International Journal of Plasticity* 21, p.

Fractional calculus, which has almost the same history as classic calculus, did not attract enough attention for a long time. However, in recent decades, fractional calculus and fractional differential equations become more and more popular because of its powerful potential applications. A large number of new differential equations (models) that involve fractional calculus are developed. These models have been applied successfully, e.g., in mechanics (theory of viscoelasticity), biology (modelling of polymers and proteins), chemistry (modelling the anomalous diffusion behavior of Brownian particles), electrical engineering (electromagnetic waves) etc (Bouchaud & Georges, 1990; Hilfer, 2000; Kirchner et al., 2000; Metzler & Klafter, 2000; Zaslavsky, 2002; Mainrdi, 2008). Meanwhile, some rich fractional dynamical behavior which reflect the inherent nature of realistic physical systems are observed. In short, fractional calculus and fractional differential equations have played more and more important role in almost all the scientific fields. One of the most important fractional models is the following initial value problem

$$D\_\*^
u y(t) = f(t, y(t)), \quad y^{(k)}(\\\mathbf{x}\_0) = y\_0^{(k)}, \quad k = 0, 1, \cdots, \lceil n \rceil - 1,\tag{1}$$

where *α* ∈ (0, ∞), *y* (*k*) <sup>0</sup> can be any real numbers, and *<sup>D</sup><sup>α</sup>* <sup>∗</sup> denotes the fractional derivative in the Caputo sense, defined by

$$D\_\*^{\alpha}y(t) = J^{n-\alpha}D^ny(t) = \frac{1}{\Gamma(n-\alpha)} \int\_0^t \frac{y^{(n)}(\tau)}{(t-\tau)^{1+\alpha-n}}d\tau,\tag{2}$$

here *<sup>n</sup>* :<sup>=</sup> �*α*� is the smallest integer not less than *<sup>α</sup>*, *<sup>y</sup>*(*n*)(*x*) is the classical *<sup>n</sup>*th-order derivative of *y*(*x*) and for *μ* > 0,*J<sup>μ</sup>* is the *μ*-order Riemann-Liouville integral operator expressed as follows

$$J^{\mu}y(t) = \frac{1}{\Gamma(\mu)} \int\_0^t \frac{y(\tau)}{(t-\tau)^{1-\mu}} d\tau.$$

The use of Caputo derivative in above equation is partly because of the convenience to specify the initial conditions. Since the initial conditions are expressed in terms of values of the unknown function and its integer-order derivatives which have clear physical meaning (Podlubny, 1999; 2002). However, from pure mathematical viewpoint, the Riemann-Liouville derivative is more welcome and many earlier research papers use it instead of Caputo derivative (Podlubny, 1999). In general, specifying some additional conditions is necessary

numerical examples to illustrate the performance of our numerical schemes. Conclusions are

Numerical Schemes for Fractional Ordinary Differential Equations 357

We can see that the fractional derivative (2) is an operator depending on the past states of the process *y*(*t*) (see Fig 1). However, for *t* � 0 and 0 < *α* < 1, the behavior of *y*(*t*) far

from upper terminal suggests that the original fractional derivative can possibly be replaced by the fractional derivative with a moving lower terminal (Podlubny, 1999). This means that the history of the process *y*(*t*) can be approached only over a fixed period of recent history. Inspired by this kind of idea, Podlubny in (Podlubny, 1999) introduces a *fixed integral length memory principle* for Riemann-Liouville derivative (see Fig 2). He shows that the truncation

}

error gives *<sup>E</sup>* <sup>&</sup>lt; *ML*−*α*/Γ(<sup>1</sup> <sup>−</sup> *<sup>α</sup>*) with a fixed integral length *<sup>L</sup>*. To accelerate the computation without significant loss of accuracy, in (Ford & Simpson, 2001) Ford and Simpson present a *short-memory principle* for Caputo derivative. In (Deng, 2007b) we apprehend the short memory principle from a new viewpoint, and then it is closer to reality and extends the effective range of short memory principle from *α* ∈ (0, 1) to *α* ∈ (0, 2). In view of the scaling property of fractional integral, the nested meshes are possibly used. And the *nested meshes* can

[0.*tn*]=[0, *tn* <sup>−</sup> *<sup>p</sup>m*<sup>T</sup> ] <sup>∪</sup> [*tn* <sup>−</sup> *<sup>p</sup>m*<sup>T</sup> , *tn* <sup>−</sup> *<sup>p</sup>m*−1<sup>T</sup> ] ∪··· [*tn* <sup>−</sup> *<sup>p</sup>*2<sup>T</sup> , *tn* <sup>−</sup> *<sup>p</sup>*<sup>T</sup> ] <sup>∪</sup> [*tn* <sup>−</sup> *<sup>p</sup>*<sup>T</sup> , *tn*], (3)

where <sup>T</sup> <sup>=</sup> *<sup>ω</sup>h*, *<sup>h</sup>* <sup>∈</sup> **<sup>R</sup>**+, *<sup>m</sup>*, *<sup>ω</sup>*, *<sup>p</sup>* <sup>∈</sup> **<sup>N</sup>** and *<sup>p</sup>m*T ≤ *tn* <sup>&</sup>lt; *<sup>p</sup>m*+1<sup>T</sup> . Denote *Mh* <sup>=</sup> {*hn*, *<sup>n</sup>* <sup>∈</sup> **<sup>N</sup>**}

The problem of determining *y*(*t*) by means of the information of *initial value y*0, is called *initial value problem*. As the classic theory of ordinary differential equation, if a function *y*(*t*) satisfies the initial value problem (1) pointwisely, then *y*(*t*) is called the *solution of the fractional differential equation* (1). For the existence and uniqueness of the solutions of fractional differential equations, one can see (Podlubny, 1999). But the explicit formula of the solution *y*(*t*) can't usually be obtained in spite of we can prove the existence of solution. The most

 ( ) *tL t D yt <sup>a</sup>* -

*<sup>t</sup>* denote

\* *D yt*( ) *<sup>a</sup>*

the Riemann-Liouville derivative operator.

and *l*1, *l*<sup>2</sup> ∈ **N**, *l*<sup>1</sup> > *l*2, then *Ml*2*<sup>h</sup>* ⊃ *Ml*1*h*.

**3. Predictor-corrector schemes**

be produced by splitting the integral interval [0, *tn*] into

0 *t* }

*the* " " *past of y t*( ) Fig. 1. The fractional derivative operating on the "past" of *y*(*t*) (the red part).

> <sup>0</sup> ( ) *D yt <sup>t</sup> a*

0 *t*

*t L* -

Fig. 2. The "short-memory" principle with "fixed memory length" *L*(< *t*), where <sup>0</sup>*D<sup>α</sup>*

given in the last section.

**2. Short memory principle**

to make sure the discussed equation has a unique solution. These additional conditions, in many situations, describe some properties of the solution at the initial time (Heymans & Podlubny, 2005), the fractional derivative does not have convenient used physical meaning (there are already some progress in the geometric and the physical interpretation of the fractional calculus (Podlubny, 2002) and the physical interpretation of the initial conditions in terms of the Riemann-Liouville fractional derivatives of the unknown function has also been discussed in (Podlubny, 2002)).

Just like the classic calculus and differential equations, the theories of fractional differentials, integrals and differential equations have been developing. With the development of the theories of fractional calculus, many research monographs are published, e.g., (Oldham & Spanier, 1974; Podlubny, 1999; Samko et al., 1993). In the literatures, several analytical methodologies, such as, Laplace transform, Mellin transform, Fourier transform, are restored to obtain the analytical solutions of the fractional equations by many authors (Metzler & Klafter, 2000; Podlubny, 1999; Samko et al., 1993; Zaslavsky, 2002; Mainrdi, 2008), however, similar to treating classical differential equations, they can mainly deal with linear fractional differential equation with constant coefficients. Usually, for nonlinear systems these techniques do not work. So in many cases the more reasonable option is to find its numerical solution. As is well known, the difficulty of solving fractional differential equations is essentially because fractional calculus are non-local operators. This non-local property means that the next state of a system not only depends on its current state but also on its historical states starting from the initial time. This property is closer to reality and is the main reason why fractional calculus has become more and more useful and popular. In other words, this non-local property is good for modeling reality, but a challenge for numerical computations. Much effort has been devoted during the recent years to the numerical investigations of fractional calculus and fractional dynamics of (1) (Lubich, 1985; 1986; Podlubny, 1999). More recently, Diethelm et al successfully presented the numerical approximation of (1) using Adams-type predictor-corrector approach (Diethelm & Ford, 2002a) and the detailed error analysis of this method was given in (Diethelm et al., 2004). The convergent order of Diethelm's predictor-corrector approach was proved to be min(2, 1+ *α*) (Diethelm et al., 2004). Because of the non-local property of the fractional derivatives, the arithmetic complexity of their algorithm with step size *h* is *O*(*h*−2), whereas a comparable algorithm for a classical initial value problem only gives rise to *O*(*h*−1). To improve the accuracy and reduce the arithmetic complexity, some techniques such as the Richardson extrapolation, short memory principle and corresponding mixed numerical schemes are developed. In (Deng, 2007a), we present an improved version of the predictor-corrector algorithm with the accuracy increased to min(2, 1 + 2*α*) and half of the computational cost is reduced comparing to the original one in (Diethelm et al., 2004). Furthermore, we apprehend the short memory principle from a new viewpoint (Deng, 2007b); after using the nested meshes presented in (Ford & Simpson, 2001) and combining the short memory principle and the predictor-corrector approach, we minimize the computational complexity to *O*(*h*−<sup>1</sup> log(*h*−1)) at preserving the order of accuracy 2.

This chapter briefly reviews the recent development of the predictor-corrector approach for fractional dynamic systems. The plan of this chapter is as follows. In Section 2, we briefly discuss the short memory principle and the nested meshes. In Section 3, the predictor-corrector schemes and its improved versions are presented, meanwhile the convergent order and arithmetic complexity are also proposed. In Section 4, we provide two numerical examples to illustrate the performance of our numerical schemes. Conclusions are given in the last section.

#### **2. Short memory principle**

2 Will-be-set-by-IN-TECH

to make sure the discussed equation has a unique solution. These additional conditions, in many situations, describe some properties of the solution at the initial time (Heymans & Podlubny, 2005), the fractional derivative does not have convenient used physical meaning (there are already some progress in the geometric and the physical interpretation of the fractional calculus (Podlubny, 2002) and the physical interpretation of the initial conditions in terms of the Riemann-Liouville fractional derivatives of the unknown function has also

Just like the classic calculus and differential equations, the theories of fractional differentials, integrals and differential equations have been developing. With the development of the theories of fractional calculus, many research monographs are published, e.g., (Oldham & Spanier, 1974; Podlubny, 1999; Samko et al., 1993). In the literatures, several analytical methodologies, such as, Laplace transform, Mellin transform, Fourier transform, are restored to obtain the analytical solutions of the fractional equations by many authors (Metzler & Klafter, 2000; Podlubny, 1999; Samko et al., 1993; Zaslavsky, 2002; Mainrdi, 2008), however, similar to treating classical differential equations, they can mainly deal with linear fractional differential equation with constant coefficients. Usually, for nonlinear systems these techniques do not work. So in many cases the more reasonable option is to find its numerical solution. As is well known, the difficulty of solving fractional differential equations is essentially because fractional calculus are non-local operators. This non-local property means that the next state of a system not only depends on its current state but also on its historical states starting from the initial time. This property is closer to reality and is the main reason why fractional calculus has become more and more useful and popular. In other words, this non-local property is good for modeling reality, but a challenge for numerical computations. Much effort has been devoted during the recent years to the numerical investigations of fractional calculus and fractional dynamics of (1) (Lubich, 1985; 1986; Podlubny, 1999). More recently, Diethelm et al successfully presented the numerical approximation of (1) using Adams-type predictor-corrector approach (Diethelm & Ford, 2002a) and the detailed error analysis of this method was given in (Diethelm et al., 2004). The convergent order of Diethelm's predictor-corrector approach was proved to be min(2, 1+ *α*) (Diethelm et al., 2004). Because of the non-local property of the fractional derivatives, the arithmetic complexity of their algorithm with step size *h* is *O*(*h*−2), whereas a comparable algorithm for a classical initial value problem only gives rise to *O*(*h*−1). To improve the accuracy and reduce the arithmetic complexity, some techniques such as the Richardson extrapolation, short memory principle and corresponding mixed numerical schemes are developed. In (Deng, 2007a), we present an improved version of the predictor-corrector algorithm with the accuracy increased to min(2, 1 + 2*α*) and half of the computational cost is reduced comparing to the original one in (Diethelm et al., 2004). Furthermore, we apprehend the short memory principle from a new viewpoint (Deng, 2007b); after using the nested meshes presented in (Ford & Simpson, 2001) and combining the short memory principle and the predictor-corrector approach, we minimize the computational complexity to *O*(*h*−<sup>1</sup> log(*h*−1)) at preserving the

This chapter briefly reviews the recent development of the predictor-corrector approach for fractional dynamic systems. The plan of this chapter is as follows. In Section 2, we briefly discuss the short memory principle and the nested meshes. In Section 3, the predictor-corrector schemes and its improved versions are presented, meanwhile the convergent order and arithmetic complexity are also proposed. In Section 4, we provide two

been discussed in (Podlubny, 2002)).

order of accuracy 2.

We can see that the fractional derivative (2) is an operator depending on the past states of the process *y*(*t*) (see Fig 1). However, for *t* � 0 and 0 < *α* < 1, the behavior of *y*(*t*) far

Fig. 1. The fractional derivative operating on the "past" of *y*(*t*) (the red part).

from upper terminal suggests that the original fractional derivative can possibly be replaced by the fractional derivative with a moving lower terminal (Podlubny, 1999). This means that the history of the process *y*(*t*) can be approached only over a fixed period of recent history. Inspired by this kind of idea, Podlubny in (Podlubny, 1999) introduces a *fixed integral length memory principle* for Riemann-Liouville derivative (see Fig 2). He shows that the truncation

Fig. 2. The "short-memory" principle with "fixed memory length" *L*(< *t*), where <sup>0</sup>*D<sup>α</sup> <sup>t</sup>* denote the Riemann-Liouville derivative operator.

error gives *<sup>E</sup>* <sup>&</sup>lt; *ML*−*α*/Γ(<sup>1</sup> <sup>−</sup> *<sup>α</sup>*) with a fixed integral length *<sup>L</sup>*. To accelerate the computation without significant loss of accuracy, in (Ford & Simpson, 2001) Ford and Simpson present a *short-memory principle* for Caputo derivative. In (Deng, 2007b) we apprehend the short memory principle from a new viewpoint, and then it is closer to reality and extends the effective range of short memory principle from *α* ∈ (0, 1) to *α* ∈ (0, 2). In view of the scaling property of fractional integral, the nested meshes are possibly used. And the *nested meshes* can be produced by splitting the integral interval [0, *tn*] into

$$\{0.t\_n\} = [0, t\_n - p^m \mathcal{T}] \cup [t\_n - p^m \mathcal{T}, t\_n - p^{m-1} \mathcal{T}] \cup \dots \cup [t\_n - p^2 \mathcal{T}, t\_n - p \mathcal{T}] \cup [t\_n - p \mathcal{T}, t\_n] \tag{3}$$

where <sup>T</sup> <sup>=</sup> *<sup>ω</sup>h*, *<sup>h</sup>* <sup>∈</sup> **<sup>R</sup>**+, *<sup>m</sup>*, *<sup>ω</sup>*, *<sup>p</sup>* <sup>∈</sup> **<sup>N</sup>** and *<sup>p</sup>m*T ≤ *tn* <sup>&</sup>lt; *<sup>p</sup>m*+1<sup>T</sup> . Denote *Mh* <sup>=</sup> {*hn*, *<sup>n</sup>* <sup>∈</sup> **<sup>N</sup>**} and *l*1, *l*<sup>2</sup> ∈ **N**, *l*<sup>1</sup> > *l*2, then *Ml*2*<sup>h</sup>* ⊃ *Ml*1*h*.

#### **3. Predictor-corrector schemes**

The problem of determining *y*(*t*) by means of the information of *initial value y*0, is called *initial value problem*. As the classic theory of ordinary differential equation, if a function *y*(*t*) satisfies the initial value problem (1) pointwisely, then *y*(*t*) is called the *solution of the fractional differential equation* (1). For the existence and uniqueness of the solutions of fractional differential equations, one can see (Podlubny, 1999). But the explicit formula of the solution *y*(*t*) can't usually be obtained in spite of we can prove the existence of solution. The most

where

and

and

*yh*(*tn*+1) =

(8) as (Deng, 2007a)

� *tn*<sup>+</sup><sup>1</sup> 0

recast as � *tn* 0

where

*aj*,*n*+<sup>1</sup> =

⎧ ⎪⎨

*<sup>n</sup>α*+<sup>1</sup> <sup>−</sup> (*<sup>n</sup>* <sup>−</sup> *<sup>α</sup>*)(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)*α*, if *<sup>j</sup>* <sup>=</sup> 0, (*<sup>n</sup>* <sup>−</sup> *<sup>j</sup>* <sup>+</sup> <sup>2</sup>)*α*+<sup>1</sup> <sup>−</sup> <sup>2</sup>(*<sup>n</sup>* <sup>−</sup> *<sup>j</sup>* <sup>+</sup> <sup>1</sup>)*α*+<sup>1</sup> + (*<sup>n</sup>* <sup>−</sup> *<sup>j</sup>*)*α*<sup>+</sup>1, if 1 <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> *<sup>n</sup>*, 1, if *j* = *n* + 1

Numerical Schemes for Fractional Ordinary Differential Equations 359

*<sup>α</sup>* [(*<sup>n</sup>* <sup>+</sup> <sup>1</sup> <sup>−</sup> *<sup>j</sup>*)*<sup>α</sup>* <sup>−</sup> (*<sup>n</sup>* <sup>−</sup> *<sup>j</sup>*)*α*], if 0 <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> *<sup>n</sup>* <sup>+</sup> 1.

1 Γ(*α*)

*n* ∑ *j*=0

*<sup>h</sup>* (*tn*+1)) + *<sup>h</sup><sup>α</sup>*

Γ(2 + *α*)

� *tn*<sup>+</sup><sup>1</sup> *tn*

*α*(*α* + 1)

*n* ∑ *j*=0

�*bj*,*n*+<sup>1</sup> *f*(*tj*, *yh*(*tj*)), (15)

*bj*,*n*+<sup>1</sup> *f*(*tj*, *yh*(*tj*)) (11)

*aj*,*n*+<sup>1</sup> *f*(*tj*, *yh*(*tj*)).

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*g*(*tn*)*dτ*, (13)

�*bj*,*n*+1*g*(*tj*) (14)

(12)

*n* ∑ *j*=0

Then the predictor and corrector formulae for solving (7) are given, respectively, by

<sup>Γ</sup>(<sup>2</sup> <sup>+</sup> *<sup>α</sup>*) *<sup>f</sup>*(*tn*+1, *<sup>y</sup><sup>P</sup>*

The approximation accuracy of the scheme (11)-(12) is *O*(*h*min{2,1+*α*}) (Diethelm et al., 2004). Now we make some improvements for the scheme (11)-(12). We modify the approximation of

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*g*�*n*(*τ*)*d<sup>τ</sup>* <sup>+</sup>

where *<sup>g</sup>*�*<sup>n</sup>* is the piecewise linear interpolation for *<sup>g</sup>* with nodes and knots chosen at *tj*, *<sup>j</sup>* <sup>=</sup> 0, 1, 2, ··· , *n*. Then using the standard quadrature technique, the right hand of (13) can be

*aj*,*n*<sup>+</sup>1, if 0 ≤ *j* ≤ *n* − 1,

The new predictor-corrector approach (15) and (12) has the numerical accuracy *O*(*h*min{2,1+2*α*}) (the detailed analysis is left in Section 3.2). Obviously half of the

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*g*(*tn*)*d<sup>τ</sup>* <sup>=</sup> *<sup>h</sup><sup>α</sup>*

<sup>2</sup>*α*+<sup>1</sup> <sup>−</sup> 1, if *<sup>j</sup>* <sup>=</sup> *<sup>n</sup>*. if *<sup>n</sup>* <sup>&</sup>gt; 0, *b*0,1 = *α* + 1, if *n* = 0.

> *n* ∑ *j*=0

*hα* Γ(2 + *α*)

⎪⎩

*yP*

*t k n*+1 *<sup>k</sup>*! *<sup>y</sup>* (*k*) <sup>0</sup> +

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*g*(*τ*)*d<sup>τ</sup>* <sup>≈</sup>

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*g*�*n*(*τ*)*d<sup>τ</sup>* <sup>+</sup>

*yP*

�*bj*,*n*+<sup>1</sup> =

*<sup>h</sup>* (*tn*+1) =

⎧ ⎪⎨ �

�*α*�−1 ∑ *k*=0

*t k n*+1 *<sup>k</sup>*! *<sup>y</sup>* (*k*) <sup>0</sup> +

⎪⎩

�*α*�−1 ∑ *k*=0

*bj*,*n*+<sup>1</sup> <sup>=</sup> *<sup>h</sup><sup>α</sup>*

*<sup>h</sup>* (*tn*+1) =

�*α*�−1 ∑ *k*=0

*t k n*+1 *<sup>k</sup>*! *<sup>y</sup>* (*k*) <sup>0</sup> +

*hα*

� *tn* 0

> � *tn*<sup>+</sup><sup>1</sup> *tn*

Hence, this algorithm for the predictor step can be improved as (Deng, 2007b)

reasonable way is to use numerical methods, and the obtained solution is called the *numerical approximation of the solution of the differential equations* (1) and denoted by *yh* in the following sections.

#### **3.1 Numerical schemes**

In this section, we show the predictor-corrector schemes of (1). Using the Laplace transform formula for the Caputo fractional derivative (Podlubny, 1999)

$$\mathcal{L}\left\{D\_{\ast}^{\mathfrak{a}}y(t)\right\} = s^{\mathfrak{a}}Y(s) - \sum\_{k=0}^{n-1} s^{\mathfrak{a}-k-1} y^{(k)}(0), \ n - 1 < \mathfrak{a} \le n. \tag{4}$$

From (1), we have

$$s^{\mathfrak{a}}Y(s) - \sum\_{k=0}^{n-1} s^{\mathfrak{a}-k-1} y^{(k)}(0) = F(s, Y(s)),\tag{5}$$

or

$$Y(s) = s^{-\alpha} F(s, Y(s)) + \sum\_{k=0}^{n-1} s^{-k-1} y^{(k)}(0),\tag{6}$$

where *F*(*s*,*Y*(*s*)) = L(*f*(*t*, *y*(*t*))). Applying the inverse Laplace transform gives

$$y(t) = \sum\_{k=0}^{\lceil \mathfrak{a} \rceil - 1} y\_0^{(k)} \frac{t^k}{k!} + \frac{1}{\Gamma(\mathfrak{a})} \int\_0^t (t - \tau)^{\mathfrak{a} - 1} f(\tau, y(\tau)) d\tau,\tag{7}$$

where the fact

$$\mathcal{L}\{J^{\mu}y(t)\} = \mathcal{L}\{\frac{1}{\Gamma(\mu)}\int\_{0}^{t} \frac{y(\tau)}{(t-\tau)^{1-\mu}}d\tau\} = \mathcal{L}\{\frac{t^{\mu-1}}{\Gamma(\mu)} \* y(t)\} = s^{-\mu}Y(s),$$

and

$$
\mathcal{L}\left\{{}^{\mu-1}\right\} = s^{-\mu}\Gamma(\mu).
$$

are used. The approximation is based on the equivalent form of the Volterra integral equation (7). A fractional Adams-predictor-corrector approach was firstly developed by Diethelm et al (Diethelm et al., 2002b) to numerically solve the problem (7). Using the standard quadrature techniques for the integral in (7), denote *g*(*τ*) = *f*(*τ*, *y*(*τ*)), the integral is replaced by the trapezoidal quadrature formula at point *tn*+<sup>1</sup>

$$\int\_{0}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} g(\tau) d\tau \approx \int\_{0}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} \tilde{g}\_{n+1}(\tau) d\tau,\tag{8}$$

where *<sup>g</sup><sup>n</sup>*+<sup>1</sup> is the piecewise linear interpolation of *<sup>g</sup>* with nodes and knots chosen at *tj*, *<sup>j</sup>* <sup>=</sup> 0, 1, 2, ··· , *n* + 1. After some elementary calculations, the right hand side of (8) gives

$$\int\_0^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} \tilde{g}\_{n+1}(\tau) d\tau = \frac{h^a}{a(a+1)} \sum\_{j=0}^{n+1} a\_{j, n+1} g(t\_j),\tag{9}$$

where the uniform mesh is used and *h* is the stepsize. And if we use the product rectangle rule, the right hand of (8) can be written as

$$\int\_{0}^{t\_{n+1}} (t\_{n+1} - \tau)^{\alpha - 1} \widetilde{g}\_{n+1}(\tau) d\tau = \sum\_{j=0}^{n} b\_{j, n+1} g(t\_j),\tag{10}$$

where

4 Will-be-set-by-IN-TECH

reasonable way is to use numerical methods, and the obtained solution is called the *numerical approximation of the solution of the differential equations* (1) and denoted by *yh* in the following

In this section, we show the predictor-corrector schemes of (1). Using the Laplace transform

*<sup>α</sup>*−*k*−1*y*(*k*)

*n*−1 ∑ *k*=0 *s* <sup>−</sup>*k*−1*y*(*k*)

 *t* 0

> *dτ* <sup>=</sup> <sup>L</sup> *t μ*−1 <sup>Γ</sup>(*μ*) <sup>∗</sup> *<sup>y</sup>*(*t*)

<sup>−</sup>*μ*Γ(*μ*).

*α*(*α* + 1)

*n* ∑ *j*=0 *n*+1 ∑ *j*=0

(0), *n* − 1 < *α* ≤ *n*. (4)

(0), (6)

<sup>−</sup>*μY*(*s*),

(0) = *F*(*s*,*Y*(*s*)), (5)

(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*dτ*, (7)

= *s*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*<sup>g</sup><sup>n</sup>*+1(*τ*)*dτ*, (8)

*aj*,*n*+1*g*(*tj*), (9)

*bj*,*n*+1*g*(*tj*), (10)

*n*−1 ∑ *k*=0 *s*

*<sup>α</sup>*−*k*−1*y*(*k*)

<sup>−</sup>*αF*(*s*,*Y*(*s*)) +

*y*(*τ*) (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)1−*<sup>μ</sup>*

*<sup>μ</sup>*−<sup>1</sup> = *s*

are used. The approximation is based on the equivalent form of the Volterra integral equation (7). A fractional Adams-predictor-corrector approach was firstly developed by Diethelm et al (Diethelm et al., 2002b) to numerically solve the problem (7). Using the standard quadrature techniques for the integral in (7), denote *g*(*τ*) = *f*(*τ*, *y*(*τ*)), the integral is replaced by the

> *tn*<sup>+</sup><sup>1</sup> 0

where *<sup>g</sup><sup>n</sup>*+<sup>1</sup> is the piecewise linear interpolation of *<sup>g</sup>* with nodes and knots chosen at *tj*, *<sup>j</sup>* <sup>=</sup>

where the uniform mesh is used and *h* is the stepsize. And if we use the product rectangle

0, 1, 2, ··· , *n* + 1. After some elementary calculations, the right hand side of (8) gives

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*<sup>g</sup><sup>n</sup>*+1(*τ*)*d<sup>τ</sup>* <sup>=</sup>

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*<sup>g</sup><sup>n</sup>*+1(*τ*)*d<sup>τ</sup>* <sup>=</sup> *<sup>h</sup><sup>α</sup>*

where *F*(*s*,*Y*(*s*)) = L(*f*(*t*, *y*(*t*))). Applying the inverse Laplace transform gives

formula for the Caputo fractional derivative (Podlubny, 1999)

= *s*

*Y*(*s*) = *s*

�*α*�−1 ∑ *k*=0 *y* (*k*) 0 *t k k*! + 1 Γ(*α*)

 1 Γ(*μ*)  *t* 0

> L *t*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*g*(*τ*)*d<sup>τ</sup>* <sup>≈</sup>

*s <sup>α</sup>Y*(*s*) <sup>−</sup>

*y*(*t*) =

<sup>=</sup> <sup>L</sup>

trapezoidal quadrature formula at point *tn*+<sup>1</sup>

 *tn*<sup>+</sup><sup>1</sup> 0

> *tn*<sup>+</sup><sup>1</sup> 0

rule, the right hand of (8) can be written as

 *tn*<sup>+</sup><sup>1</sup> 0

*<sup>α</sup>Y*(*s*) <sup>−</sup>

*n*−1 ∑ *k*=0 *s*

sections.

**3.1 Numerical schemes**

From (1), we have

where the fact

and

L *J <sup>μ</sup>y*(*t*)

or

L *Dα* <sup>∗</sup> *<sup>y</sup>*(*t*)

$$a\_{j,n+1} = \begin{cases} n^{a+1} - (n-\alpha)(n+1)^a, & \text{if } \quad j=0, \\ (n-j+2)^{a+1} - 2(n-j+1)^{a+1} + (n-j)^{a+1}, & \text{if } \quad 1 \le j \le n, \\ 1, & \text{if } \quad j=n+1 \end{cases}$$

and

$$b\_{j,n+1} = \frac{h^a}{a} [(n+1-j)^a - (n-j)^a], \quad \text{if} \quad 0 \le j \le n+1.$$

Then the predictor and corrector formulae for solving (7) are given, respectively, by

$$y\_h^P(t\_{n+1}) = \sum\_{k=0}^{\lceil \alpha \rceil - 1} \frac{t\_{n+1}^k}{k!} y\_0^{(k)} + \frac{1}{\Gamma(\alpha)} \sum\_{j=0}^n b\_{j, n+1} f(t\_j, y\_h(t\_j)) \tag{11}$$

and

$$y\_h(t\_{n+1}) = \sum\_{k=0}^{\lceil \mathfrak{a} \rceil - 1} \frac{t\_{n+1}^k}{k!} y\_0^{(k)} + \frac{h^a}{\Gamma(2+a)} f(t\_{n+1}, y\_h^P(t\_{n+1})) + \frac{h^a}{\Gamma(2+a)} \sum\_{j=0}^n a\_{j, n+1} f(t\_j, y\_h(t\_j)). \tag{12}$$

The approximation accuracy of the scheme (11)-(12) is *O*(*h*min{2,1+*α*}) (Diethelm et al., 2004).

Now we make some improvements for the scheme (11)-(12). We modify the approximation of (8) as (Deng, 2007a)

$$\int\_{0}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} g(\tau) d\tau \approx \int\_{0}^{t\_n} (t\_{n+1} - \tau)^{a-1} \widetilde{g}\_{\mathbb{R}}(\tau) d\tau + \int\_{t\_n}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} g(t\_{\mathbb{R}}) d\tau,\tag{13}$$

where *<sup>g</sup>*�*<sup>n</sup>* is the piecewise linear interpolation for *<sup>g</sup>* with nodes and knots chosen at *tj*, *<sup>j</sup>* <sup>=</sup> 0, 1, 2, ··· , *n*. Then using the standard quadrature technique, the right hand of (13) can be recast as

$$\int\_{0}^{t\_n} (t\_{n+1} - \tau)^{a-1} \widetilde{g}\_{\mathfrak{A}}(\tau) d\tau + \int\_{t\_n}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} g(t\_n) d\tau = \frac{h^a}{a(a+1)} \sum\_{j=0}^n \widetilde{b}\_{j, n+1} g(t\_j) \tag{14}$$

where

$$
\widetilde{b}\_{j,n+1} = \begin{cases}
\begin{cases}
a\_{j,n+1}, & \text{if } & 0 \le j \le n-1, \\
2^{n+1} - 1, & \text{if } & j = n.
\end{cases} & \text{if } \quad n > 0, \\
b\_{0,1} = n + 1, & \text{if } \quad n = 0.
\end{cases}
$$

Hence, this algorithm for the predictor step can be improved as (Deng, 2007b)

$$y\_h^P(t\_{n+1}) = \sum\_{k=0}^{\lceil \alpha \rceil - 1} \frac{t\_{n+1}^k}{k!} y\_0^{(k)} + \frac{h^a}{\Gamma(2+a)} \sum\_{j=0}^n \widetilde{b}\_{j, n+1} f(t\_j, y\_h(t\_j)),\tag{15}$$

The new predictor-corrector approach (15) and (12) has the numerical accuracy *O*(*h*min{2,1+2*α*}) (the detailed analysis is left in Section 3.2). Obviously half of the

In fact, if *α* ∈ (0, 2) the integration kernel of (20) fades faster when the time history becomes

Numerical Schemes for Fractional Ordinary Differential Equations 361

*f*(*τ*, *y*(*τ*))*dτ*

*f*(*τ*, *y*(*τ*))*dτ*

*f*(*τ*, *y*(*τ*))*dτ*

*f*(*τ*, *y*(*τ*))*dτ*

*<sup>m</sup>*(*τ*) ∈ (*tm*−1, *tm*+1), *<sup>m</sup>* = 1, 2, ··· , *<sup>n</sup>*. Obviously, we can see that the integration (20)'s kernel (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> decays (algebraically) by the order 2 <sup>−</sup> *<sup>α</sup>* when *<sup>α</sup>* <sup>∈</sup> (0, 2), but in (Ford & Simpson, 2001) the integral kernel (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> decays with the order 1 <sup>−</sup> *<sup>α</sup>* when *α* ∈ (0, 1). This is the main reason why we can extend the range of the *short memory principle* of fractional differential equations from *α* ∈ (0, 1) to *α* ∈ (0, 2). For the nested mashes defined by (3), we can take the step length *h* in the integral [*tn* − *p*T , *tn*] and in the subsequent intervals [*tn* <sup>−</sup> *<sup>p</sup>*2<sup>T</sup> , *tn* <sup>−</sup> *<sup>p</sup>*<sup>T</sup> ], [*tn* <sup>−</sup> *<sup>p</sup>*3<sup>T</sup> , *tn* <sup>−</sup> *<sup>p</sup>*2<sup>T</sup> ], ··· , [*tn* <sup>−</sup> *<sup>p</sup>m*<sup>T</sup> , *tn* <sup>−</sup> *<sup>p</sup>m*−1<sup>T</sup> ], [*t*0, *tn* <sup>−</sup> *<sup>p</sup>m*<sup>T</sup> ], step lengths *ph*, *<sup>p</sup>*2*h*, ··· , *<sup>p</sup>m*−1*h*, *<sup>p</sup>mh* are used, respectively. Noting that (*tn* <sup>−</sup> *<sup>p</sup>m*<sup>T</sup> ) <sup>−</sup> 0 can't be divided by *<sup>p</sup>mh*, so the integral in the interval [0, *<sup>l</sup>*] (*<sup>l</sup>* = (*tn* <sup>−</sup> *<sup>p</sup>m*<sup>T</sup> ) − �(*tn* <sup>−</sup> *<sup>p</sup>m*<sup>T</sup> )/(*pmh*)� · (*pmh*)) is ignored, it does not destroy the computational accuracy in general. We will pay our attention to the numerical approximation by using *short memory principle* in the integration

With the similar approximation of (13), the product trapezoidal quadrature formula is applied to replace the integral of (19), where nodes *tj*, *j* = *n*, *n* + 1 are taken with respect to the weighted function (*tn*+<sup>1</sup> − ·)*α*−<sup>1</sup> for the first integral and nodes *tj*, *<sup>j</sup>* <sup>=</sup> 0, 1, ··· , *<sup>n</sup>* are used w.r.t. the weighted function (*tn*+<sup>1</sup> − ·)*α*−<sup>1</sup> <sup>−</sup> (*tn* − ·)*α*−<sup>1</sup> for the second integral, and it yields

*<sup>α</sup> <sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*)) + *<sup>f</sup>*(*tn*+1, *<sup>y</sup>*(*tn*+1))

 *tn*<sup>+</sup><sup>1</sup> *tn*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*

*<sup>n</sup>*+1(*τ*, *y*(*τ*))*dτ*

, *α* ∈ (1, ∞) (22)

*f*(*τ*, *y*(*τ*))*dτ*

1 Γ(*α*)(*α* − 1)

*f*(*τ*, *y*(*τ*))*dτ* (21)

<sup>2</sup> (*τ*))*α*−<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>*

 *tn*−<sup>1</sup> *tn*−<sup>2</sup> (*z*∗

longer, more concretely,

Γ(*α*)(*α* − 1)

Γ(*α*)(*α* − 1)

1 Γ(*α*)(*α* − 1)

1 Γ(*α*)(*α* − 1)

Γ(*α*)(*α* − 1)

1 Γ(*α*)(*α* − 1)

(20) in the following part.

 *tn*<sup>+</sup><sup>1</sup> *tn*

> <sup>=</sup> *<sup>h</sup><sup>α</sup> α*(*α* + 1)

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup>

 *tn*<sup>+</sup>1−*<sup>τ</sup> tn*−*τ*

 *tn*<sup>+</sup>1−*<sup>τ</sup> tn*−*τ*

 *tn*<sup>+</sup>1−*<sup>τ</sup> tn*−*τ*

> *tn*<sup>+</sup>1−*<sup>τ</sup> tn*−*τ*

 *t*<sup>2</sup> *t*1

 *tn*<sup>+</sup>1−*<sup>τ</sup> tn*−*τ*

> *t*<sup>2</sup> *t*1 (*z*∗

*zα*−2*dz*

*zα*−2*dz*

*zα*−2*dz*

*zα*−2*dz*

<sup>1</sup> (*τ*))*α*−<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* <sup>+</sup>

*<sup>n</sup>*(*τ*))*α*−<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*dτ*,

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* <sup>≈</sup>

*zα*−2*dz*

*<sup>n</sup>*−1(*τ*))*α*−<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>*

 *tn* 0

 *tn tn*−<sup>1</sup>

 *tn*−<sup>1</sup> *tn*−<sup>2</sup>

1 Γ(*α*)(*α* − 1)

1 Γ(*α*)(*α* − 1)

 *t*<sup>1</sup> *t*0

> *tn tn*−<sup>1</sup> (*z*∗

 *t*<sup>1</sup> *t*0 (*z*∗

 *tn* 0 

<sup>=</sup> <sup>1</sup>

<sup>=</sup> <sup>1</sup>

+ ··· +

<sup>=</sup> <sup>1</sup>

+ ··· +

1 Γ(*α*)

+

+

+

where *z*∗

computational cost can be reduced, for 0 < *α* 1, if we reformulate (15) and (12) as

$$y\_h^P(t\_{n+1}) = \begin{cases} y\_0 + \frac{h^a}{\Gamma(a+1)} f(y\_h(t\_0), t\_0), & \text{if } n = 0, \\\\ y\_0 + \frac{h^a}{\Gamma(a+2)} \cdot (2^{a+1} - 1) \cdot f(y\_h(t\_n), t\_n) \\ + \frac{h^a}{\Gamma(a+2)} \sum\_{j=0}^{n-1} a\_{j, n+1} f(y\_h(t\_j), t\_j), & \text{if } n \gg 1 \end{cases} \tag{16}$$

and

$$\begin{cases} \begin{aligned} y\_0 + \frac{h^a}{\Gamma(a+2)} \left( f(y\_h^P(t\_1), t\_1) + \mathfrak{a} \cdot f(y\_h(t\_0), t\_0) \right), & \text{if } \mathfrak{a} = \mathfrak{0}, \\\\ \begin{aligned} \vdots \end{aligned} \end{cases} \end{cases} \tag{17}$$

$$y\_h(t\_{n+1}) = \begin{cases} y\_0 + \frac{h^a}{\Gamma(a+2)} \left( f(y\_h^P(t\_{n+1}), t\_{n+1}) + (2^{a+1} - 2) \cdot f(y\_h(t\_n), t\_n) \right) & \text{if } n \gg 1. \\\ + \frac{h^a}{\Gamma(a+2)} \sum\_{j=0}^{n-1} a\_{j, n+1} f(y\_h(t\_j), t\_j), & \text{if } n \gg 1. \end{cases}$$

The arithmetic complexity of the above two predictor-corrector schemes ((11)-(12), and (16)-(17)) is *O*(*h*−2), where *h* is step size.

For further reducing the computational cost, we understand the short memory principle from a new viewpoint, and then go to design the predictor-corrector scheme (Deng, 2007b). For *α* ∈ (0, 1) and *α* ∈ (1, ∞), we rewrite (7) as, respectively,

$$y(t\_{n+1}) = y(t\_n) + \frac{1}{\Gamma(a)} \int\_{t\_n}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} f(\tau, y(\tau)) d\tau$$

$$+ \frac{1}{\Gamma(a)} \int\_0^{t\_n} \left( (t\_{n+1} - \tau)^{a-1} - (t\_n - \tau)^{a-1} \right) f(\tau, y(\tau)) d\tau, \quad a \in (0, 1) \tag{18}$$

and

$$y(t\_{n+1}) = \sum\_{k=1}^{\lceil a \rceil - 1} \frac{y\_0^{(k)}}{k!} (t\_{n+1}^k - t\_n^k) + y(t\_n) + \frac{1}{\Gamma(a)} \int\_{t\_n}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} f(\tau, y(\tau)) d\tau$$

$$+ \frac{1}{\Gamma(a)} \int\_0^{t\_n} ((t\_{n+1} - \tau)^{a-1} - (t\_n - \tau)^{a-1}) f(\tau, y(\tau)) d\tau, \quad a \in (1, \infty). \tag{19}$$

By observation of (18) and (19), we can see that the non-local property of *D<sup>α</sup>* <sup>∗</sup> induces the term

$$\frac{1}{\Gamma(\mathfrak{a})} \int\_0^{t\_n} \left( (t\_{n+1} - \tau)^{\mathfrak{a}-1} - (t\_n - \tau)^{\mathfrak{a}-1} \right) f(\tau, y(\tau)) d\tau. \tag{20}$$

6 Will-be-set-by-IN-TECH

<sup>Γ</sup>(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>) *<sup>f</sup>*(*yh*(*t*0), *<sup>t</sup>*0), if *<sup>n</sup>* <sup>=</sup> 0,

�

if *n* 1

, if *n* = 0,

�

*f*(*τ*, *y*(*τ*))*dτ*, *α* ∈ (0, 1) (18)

*f*(*τ*, *y*(*τ*))*dτ*, *α* ∈ (1, ∞). (19)

*f*(*τ*, *y*(*τ*))*dτ*. (20)

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>*

<sup>∗</sup> induces the term

if *n* 1.

(16)

(17)

<sup>Γ</sup>(*<sup>α</sup>* <sup>+</sup> <sup>2</sup>) · (2*α*+<sup>1</sup> <sup>−</sup> <sup>1</sup>) · *<sup>f</sup>*(*yh*(*tn*), *tn*)

*<sup>h</sup>* (*t*1), *t*1) + *α* · *f*(*yh*(*t*0), *t*0)

The arithmetic complexity of the above two predictor-corrector schemes ((11)-(12), and

For further reducing the computational cost, we understand the short memory principle from a new viewpoint, and then go to design the predictor-corrector scheme (Deng, 2007b). For

*<sup>n</sup>*) + *<sup>y</sup>*(*tn*) + <sup>1</sup>

*aj*,*n*+<sup>1</sup> *f*(*yh*(*tj*), *tj*),

*<sup>h</sup>* (*tn*+1), *tn*+1)+(2*α*+<sup>1</sup> <sup>−</sup> <sup>2</sup>) · *<sup>f</sup>*(*yh*(*tn*), *tn*)

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>*

Γ(*α*)

� *tn*<sup>+</sup><sup>1</sup> *tn*

computational cost can be reduced, for 0 < *α* 1, if we reformulate (15) and (12) as

*n*−1 ∑ *j*=0

*aj*,*n*+<sup>1</sup> *f*(*yh*(*tj*), *tj*),

� *tn*<sup>+</sup><sup>1</sup> *tn*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1�

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1�

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1�

By observation of (18) and (19), we can see that the non-local property of *D<sup>α</sup>*

*hα*

*hα*

*hα* Γ(*α* + 2)

*yP*

⎧

*y*<sup>0</sup> +

*y*<sup>0</sup> +

+

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

+ 1 Γ(*α*)

*y*(*tn*+1) =

+ 1 Γ(*α*)

and

and

*yh*(*tn*+1) =

*<sup>h</sup>* (*tn*+1) =

⎧

*y*<sup>0</sup> +

*y*<sup>0</sup> +

� *f*(*y<sup>P</sup>*

� *f*(*y<sup>P</sup>*

*n*−1 ∑ *j*=0

*α* ∈ (0, 1) and *α* ∈ (1, ∞), we rewrite (7) as, respectively,

Γ(*α*)

*<sup>y</sup>*(*tn*+1) = *<sup>y</sup>*(*tn*) + <sup>1</sup>

� *tn* 0 �

> �*α*�−1 ∑ *k*=1

� *tn* 0 �

1 Γ(*α*) � *tn* 0 �

*y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k*

+

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

*hα* Γ(*α* + 2)

*hα* Γ(*α* + 2)

*hα* Γ(*α* + 2)

(16)-(17)) is *O*(*h*−2), where *h* is step size.

In fact, if *α* ∈ (0, 2) the integration kernel of (20) fades faster when the time history becomes longer, more concretely,

1 Γ(*α*) *tn* 0 (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *f*(*τ*, *y*(*τ*))*dτ* <sup>=</sup> <sup>1</sup> Γ(*α*)(*α* − 1) *tn* 0 *tn*<sup>+</sup>1−*<sup>τ</sup> tn*−*τ zα*−2*dz f*(*τ*, *y*(*τ*))*dτ* <sup>=</sup> <sup>1</sup> Γ(*α*)(*α* − 1) *tn tn*−<sup>1</sup> *tn*<sup>+</sup>1−*<sup>τ</sup> tn*−*τ zα*−2*dz f*(*τ*, *y*(*τ*))*dτ* + 1 Γ(*α*)(*α* − 1) *tn*−<sup>1</sup> *tn*−<sup>2</sup> *tn*<sup>+</sup>1−*<sup>τ</sup> tn*−*τ zα*−2*dz f*(*τ*, *y*(*τ*))*dτ* + ··· + 1 Γ(*α*)(*α* − 1) *t*<sup>2</sup> *t*1 *tn*<sup>+</sup>1−*<sup>τ</sup> tn*−*τ zα*−2*dz f*(*τ*, *y*(*τ*))*dτ* + 1 Γ(*α*)(*α* − 1) *t*<sup>1</sup> *t*0 *tn*<sup>+</sup>1−*<sup>τ</sup> tn*−*τ zα*−2*dz f*(*τ*, *y*(*τ*))*dτ* (21) <sup>=</sup> <sup>1</sup> Γ(*α*)(*α* − 1) *tn tn*−<sup>1</sup> (*z*∗ <sup>1</sup> (*τ*))*α*−<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* <sup>+</sup> 1 Γ(*α*)(*α* − 1) *tn*−<sup>1</sup> *tn*−<sup>2</sup> (*z*∗ <sup>2</sup> (*τ*))*α*−<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* + ··· + 1 Γ(*α*)(*α* − 1) *t*<sup>2</sup> *t*1 (*z*∗ *<sup>n</sup>*−1(*τ*))*α*−<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* + 1 Γ(*α*)(*α* − 1) *t*<sup>1</sup> *t*0 (*z*∗ *<sup>n</sup>*(*τ*))*α*−<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*dτ*,

where *z*∗ *<sup>m</sup>*(*τ*) ∈ (*tm*−1, *tm*+1), *<sup>m</sup>* = 1, 2, ··· , *<sup>n</sup>*. Obviously, we can see that the integration (20)'s kernel (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> decays (algebraically) by the order 2 <sup>−</sup> *<sup>α</sup>* when *<sup>α</sup>* <sup>∈</sup> (0, 2), but in (Ford & Simpson, 2001) the integral kernel (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> decays with the order 1 <sup>−</sup> *<sup>α</sup>* when *α* ∈ (0, 1). This is the main reason why we can extend the range of the *short memory principle* of fractional differential equations from *α* ∈ (0, 1) to *α* ∈ (0, 2). For the nested mashes defined by (3), we can take the step length *h* in the integral [*tn* − *p*T , *tn*] and in the subsequent intervals [*tn* <sup>−</sup> *<sup>p</sup>*2<sup>T</sup> , *tn* <sup>−</sup> *<sup>p</sup>*<sup>T</sup> ], [*tn* <sup>−</sup> *<sup>p</sup>*3<sup>T</sup> , *tn* <sup>−</sup> *<sup>p</sup>*2<sup>T</sup> ], ··· , [*tn* <sup>−</sup> *<sup>p</sup>m*<sup>T</sup> , *tn* <sup>−</sup> *<sup>p</sup>m*−1<sup>T</sup> ], [*t*0, *tn* <sup>−</sup> *<sup>p</sup>m*<sup>T</sup> ], step lengths *ph*, *<sup>p</sup>*2*h*, ··· , *<sup>p</sup>m*−1*h*, *<sup>p</sup>mh* are used, respectively. Noting that (*tn* <sup>−</sup> *<sup>p</sup>m*<sup>T</sup> ) <sup>−</sup> 0 can't be divided by *<sup>p</sup>mh*, so the integral in the interval [0, *<sup>l</sup>*] (*<sup>l</sup>* = (*tn* <sup>−</sup> *<sup>p</sup>m*<sup>T</sup> ) − �(*tn* <sup>−</sup> *<sup>p</sup>m*<sup>T</sup> )/(*pmh*)� · (*pmh*)) is ignored, it does not destroy the computational accuracy in general. We will pay our attention to the numerical approximation by using *short memory principle* in the integration (20) in the following part.

With the similar approximation of (13), the product trapezoidal quadrature formula is applied to replace the integral of (19), where nodes *tj*, *j* = *n*, *n* + 1 are taken with respect to the weighted function (*tn*+<sup>1</sup> − ·)*α*−<sup>1</sup> for the first integral and nodes *tj*, *<sup>j</sup>* <sup>=</sup> 0, 1, ··· , *<sup>n</sup>* are used w.r.t. the weighted function (*tn*+<sup>1</sup> − ·)*α*−<sup>1</sup> <sup>−</sup> (*tn* − ·)*α*−<sup>1</sup> for the second integral, and it yields

$$\int\_{t\_n}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} f(\tau, y(\tau)) d\tau \approx \int\_{t\_n}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} \widetilde{f}\_{n+1}(\tau, y(\tau)) d\tau$$

$$= \frac{h^a}{a(a+1)} (af(t\_n, y(t\_n)) + f(t\_{n+1}, y(t\_{n+1}))), \quad a \in (1, \infty) \tag{22}$$

∑*n*

� *tn* 0 �

≈

+ *m*−1 ∑ *i*=1

where

*bj*,*pi* ,*<sup>n</sup>* =

*yP*

*bj*,*p*0,*<sup>n</sup>* =

� � *tn*

<sup>=</sup> *<sup>h</sup><sup>α</sup> α*(*α* + 1)

<sup>+</sup> (*pmh*)*<sup>α</sup> α*(*α* + 1)

*tn*−*pωh*

(*pi h*)*<sup>α</sup> α*(*α* + 1)

⎧ ⎪⎪⎪⎨

⎪⎪⎪⎩

and for *i* = 1, 2, ··· , *m*,

(*ω*+*j*−1+1/*p<sup>i</sup>*

(1)

<sup>+</sup> (*pmh*)*<sup>α</sup> α*(*α* + 1)

*<sup>ω</sup>* <sup>−</sup> <sup>1</sup> <sup>+</sup> 1/*p<sup>i</sup>*

*ω*)*α*+<sup>1</sup> + ((*p<sup>i</sup>*

(*pi h*)*<sup>α</sup> α*(*α* + 1)

+2(*ω*+*j*)*α*+1+(*ω*+*j*+1+1/*p<sup>i</sup>*

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

*<sup>h</sup>* (*tn*+1) = *y*

(*pi*

+(*p<sup>i</sup>*

+ *m*−1 ∑ *i*=1

+ *m*−1 ∑ *i*=1

*n* ∑ *j*=*n*−*pω*

The predictor and corrector formulae based on the analytical formula (19) is fully described by (28) and (29) with the weighted �*aj*,*<sup>n</sup>* defined by (24). It is apparent that the same term

Numerical Schemes for Fractional Ordinary Differential Equations 363

*<sup>j</sup>*=<sup>0</sup> �*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*)) exists in both predictor (28) and corrector formulae (29), we just need to

In order to reduce the computational cost, in conjunction with the nested meshes memory principle (3), decompose the integral and still use the product trapezoidal quadrature formula

��

(*pω*+1)*α*(*α*−*pω*)+(*pω*)*α*(2*pω*−*α*−1)−(*pω*−1)*α*<sup>+</sup>1, if *<sup>j</sup>* <sup>=</sup> *<sup>n</sup>* <sup>−</sup> *<sup>p</sup>ω*,

<sup>2</sup>*α*+<sup>1</sup> <sup>−</sup> *<sup>α</sup>* <sup>−</sup> 3, if *<sup>j</sup>* <sup>=</sup> *<sup>n</sup>*

<sup>−</sup>(*<sup>ω</sup>* <sup>+</sup> <sup>1</sup>)*α*+<sup>1</sup> <sup>−</sup> ((1/*p<sup>i</sup>* <sup>+</sup> *<sup>ω</sup>*)*<sup>α</sup>* <sup>−</sup> *<sup>ω</sup>α*)(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>), if *<sup>j</sup>* <sup>=</sup> 0,

*<sup>ω</sup>* <sup>−</sup> <sup>1</sup>)*α*+<sup>1</sup> <sup>−</sup> (*p<sup>i</sup>*

<sup>Γ</sup>(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>) *<sup>f</sup>*(*tn*, *yh*(*tn*)) + *<sup>h</sup><sup>α</sup>*

,*<sup>n</sup> <sup>f</sup>*(*tn* <sup>−</sup> *<sup>p</sup><sup>i</sup>*

)*α*+1−(*ω*+*j*−1)*α*+1−2(*ω*+*j*+1/*p<sup>i</sup>*

)*<sup>α</sup>* <sup>−</sup> (*p<sup>i</sup>*

Employing above analysis we obtain the following predictor-corrector algorithm

*bj*,*pi*

(*<sup>ω</sup>* <sup>+</sup> *<sup>j</sup>*)*h*, *<sup>y</sup>*(*tn* <sup>−</sup> *<sup>p</sup><sup>i</sup>*

<sup>−</sup>(*<sup>n</sup>* <sup>−</sup> *<sup>j</sup>* <sup>−</sup> <sup>1</sup>)*α*<sup>+</sup>1, if *<sup>n</sup>* <sup>−</sup> *<sup>p</sup><sup>ω</sup>* <sup>+</sup> <sup>1</sup> <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> *<sup>n</sup>* <sup>−</sup> 1,

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1�

(*<sup>ω</sup>* <sup>+</sup> *<sup>j</sup>*)*h*))�

*bj*,*p*0,*<sup>n</sup> f*(*tj*, *y*(*tj*)) (30)

*bj*,*pm*,*<sup>n</sup> <sup>f</sup>*(*tn* <sup>−</sup> *<sup>p</sup>m*(*<sup>ω</sup>* <sup>+</sup> *<sup>j</sup>*)*h*, *<sup>y</sup>*(*tn* <sup>−</sup> *<sup>p</sup>m*(*<sup>ω</sup>* <sup>+</sup> *<sup>j</sup>*)*h*)), (31)

)*α*<sup>+</sup>1,

*α*(*α* + 1)

*bj*,*pm*,*<sup>n</sup> <sup>f</sup>*(*tn* <sup>−</sup> *<sup>p</sup>m*(*<sup>ω</sup>* <sup>+</sup> *<sup>j</sup>*)*h*, *<sup>y</sup>*(*tn* <sup>−</sup> *<sup>p</sup>m*(*<sup>ω</sup>* <sup>+</sup> *<sup>j</sup>*)*h*)) (32)

(*<sup>ω</sup>* <sup>+</sup> *<sup>j</sup>*)*h*, *<sup>y</sup>*(*tn* <sup>−</sup> *<sup>p</sup><sup>i</sup>*

)*α*+<sup>1</sup>

*<sup>ω</sup>*)*α*)(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>), if *<sup>j</sup>* = (*<sup>p</sup>* <sup>−</sup> <sup>1</sup>)*ω*.

*n* ∑ *j*=*n*−*pω*

(*<sup>ω</sup>* <sup>+</sup> *<sup>j</sup>*)*h*))�

*bj*,*p*0,*<sup>n</sup> f*(*tj*, *y*(*tj*))

*ω* + 1/*p<sup>i</sup>*

)*α*+1−(*ω*+*j*+1)*α*<sup>+</sup>1, if 1 <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> (*<sup>p</sup>* <sup>−</sup> <sup>1</sup>)*<sup>ω</sup>* <sup>−</sup> 1,

*f* �

*<sup>n</sup>*(*τ*, *y*(*τ*))*dτ*

*f*(*τ*, *y*(*τ*))*dτ*

� *tn*−*pmω<sup>h</sup>*

compute one times at each predictor-corrector iteration step.

at each subinterval but with different step lengths.

� *tn*−*p<sup>i</sup> ωh*

*tn*−*pi*<sup>+</sup>1*ω<sup>h</sup>*

*bj*,*pi*

+

0

,*<sup>n</sup> <sup>f</sup>*(*tn* <sup>−</sup> *<sup>p</sup><sup>i</sup>*

(*<sup>n</sup>* <sup>−</sup> *<sup>j</sup>* <sup>+</sup> <sup>2</sup>)*α*+<sup>1</sup> <sup>−</sup> <sup>3</sup>(*<sup>n</sup>* <sup>−</sup> *<sup>j</sup>* <sup>+</sup> <sup>1</sup>)*α*+<sup>1</sup> <sup>+</sup> <sup>3</sup>(*<sup>n</sup>* <sup>−</sup> *<sup>j</sup>*)*α*+<sup>1</sup>

<sup>−</sup>(1/*p<sup>i</sup>* <sup>+</sup> *<sup>ω</sup>*)*α*+<sup>1</sup> <sup>+</sup> *<sup>ω</sup>α*+<sup>1</sup> + (1/*p<sup>i</sup>* <sup>+</sup> *<sup>ω</sup>* <sup>+</sup> <sup>1</sup>)*α*+<sup>1</sup>

)*α*+<sup>1</sup> <sup>−</sup> (*p<sup>i</sup>*

<sup>0</sup> · *<sup>h</sup>* <sup>+</sup> *yh*(*tn*) + *<sup>h</sup><sup>α</sup>*

*ω* + 1/*p<sup>i</sup>*

� (*p*−1)*<sup>ω</sup>* ∑ *j*=0

�*n*/*p<sup>m</sup>*−*ω*�−<sup>1</sup> ∑ *j*=0

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1�

� (*p*−1)*<sup>ω</sup>* ∑ *j*=0

�*n*/*p<sup>m</sup>*−*ω*�−<sup>1</sup> ∑ *j*=0

and

$$\int\_{0}^{t\_n} \left( (t\_{n+1} - \tau)^{a-1} - (t\_n - \tau)^{a-1} \right) f(\tau, y(\tau)) d\tau$$

$$\approx \int\_{0}^{t\_n} \left( (t\_{n+1} - \tau)^{a-1} - (t\_n - \tau)^{a-1} \right) \tilde{f}\_n(\tau, y(\tau)) d\tau$$

$$= \frac{h^a}{\pi(a+1)} \sum\_{j=0}^n \tilde{a}\_{j,n} f(t\_j, y(t\_j)) \tag{23}$$

where

$$
\tilde{a}\_{j,n} = \begin{cases}
(n+1)^{a+1}(n-n) + n^a(2n-a-1) - (n-1)^{a+1}, & \text{if } j=0, \\
(n-j+2)^{a+1} - 3(n-j+1)^{a+1} + 3(n-j)^{a+1} - (n-j-1)^{a+1}, & \text{if } 1 \le j \le n-1, \\
2^{a+1} - n - 3, & \text{if } \quad j=n.
\end{cases}
\tag{24}
$$

For the first integral of (18) or (19), the product rectangle formula is used

$$\int\_{0}^{t\_n} (t\_{n+1} - \tau)^{a-1} f(\tau, y(\tau)) d\tau \approx \int\_{0}^{t\_n} (t\_{n+1} - \tau)^{a-1} f(t\_n, y(t\_n)) d\tau = \frac{h^a}{a} f(t\_n, y(t\_n)). \tag{25}$$

Combing (22)-(25), for *α* ∈ (1, 2), the new predictor-corrector approach gives

$$y\_h^P(t\_{n+1}) = y\_0^{(1)} \cdot h + y\_h(t\_n) + \frac{h^a}{\Gamma(a+1)} f(t\_n, y\_h(t\_n))$$

$$+ \frac{h^a}{\Gamma(a+2)} \sum\_{j=0}^n \tilde{a}\_{j,n} f(t\_j, y\_h(t\_j)), \quad a \in (1,2) \tag{26}$$

and

$$y\_h(t\_{n+1}) = y\_0^{(1)} \cdot h + y\_h(t\_n) + \frac{h^a}{\Gamma(a+2)} \left( \alpha f(t\_n, y\_h(t\_n)) \right)$$

$$+ f(t\_{n+1}, y\_h^P(t\_{n+1})) + \frac{h^a}{\Gamma(a+2)} \sum\_{j=0}^n \tilde{a}\_{j,n} f(t\_j, y\_h(t\_j)), \quad a \in (1,2). \tag{27}$$

For *α* ∈ (2, ∞), the predictor-corrector approach is

$$\begin{split} y\_h^P(t\_{n+1}) &= \sum\_{k=1}^{\lceil \mathfrak{a} \rceil - 1} \frac{y\_0^{(k)}}{k!} (t\_{n+1}^k - t\_n^k) + y\_h(t\_n) \\ &+ \frac{h^\mathfrak{a}}{\Gamma(\mathfrak{a}+1)} f(t\_n, y\_h(t\_n)) + \frac{h^\mathfrak{a}}{\Gamma(\mathfrak{a}+2)} \sum\_{j=0}^n \widetilde{a}\_{j,n} f(t\_j, y\_h(t\_j)), \quad \mathfrak{a} \in (2, \infty) \end{split} \tag{28}$$

and

$$y\_h(t\_{n+1}) = \sum\_{k=1}^{\lceil \alpha \rceil - 1} \frac{y\_0^{(k)}}{k!} (t\_{n+1}^k - t\_n^k) + y\_h(t\_n) + \frac{h^{\alpha}}{\Gamma(\alpha + 2)} (\alpha f(t\_n, y\_h(t\_n)) + 1)$$

$$f(t\_{n+1}, y\_h^P(t\_{n+1}))) + \frac{h^{\alpha}}{\Gamma(\alpha + 2)} \sum\_{j=0}^n \tilde{a}\_{j,n} f(t\_j, y\_h(t\_j)), \quad \alpha \in (2, \infty). \tag{29}$$

The predictor and corrector formulae based on the analytical formula (19) is fully described by (28) and (29) with the weighted �*aj*,*<sup>n</sup>* defined by (24). It is apparent that the same term ∑*n <sup>j</sup>*=<sup>0</sup> �*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*)) exists in both predictor (28) and corrector formulae (29), we just need to compute one times at each predictor-corrector iteration step.

In order to reduce the computational cost, in conjunction with the nested meshes memory principle (3), decompose the integral and still use the product trapezoidal quadrature formula at each subinterval but with different step lengths.

$$\begin{split} & \int\_{0}^{t\_{n}} \left( (t\_{n+1} - \tau)^{\kappa-1} - (t\_{n} - \tau)^{\kappa-1} \right) f(\tau, y(\tau)) d\tau \\ & \approx \left( \int\_{t\_{n} - p\omega h}^{t\_{n}} + \sum\_{i=1}^{m-1} \int\_{t\_{n} - p^{i} + i\omega h}^{t\_{n} - p^{i}\omega h} + \int\_{0}^{t\_{n} - p^{m}\omega h} \right) ((t\_{n+1} - \tau)^{\kappa-1} - (t\_{n} - \tau)^{\kappa-1}) \tilde{f}\_{n}(\tau, y(\tau)) d\tau \\ & = \frac{h^{\kappa}}{a(a+1)} \sum\_{j=n-p\omega}^{n} b\_{j, p^{j}, n} f(t\_{j}, y(t\_{j})) \\ & + \sum\_{i=1}^{m-1} \frac{(p^{i}h)^{\kappa}}{a(a+1)} \left( \sum\_{j=0}^{(p-1)\omega} b\_{j, p^{i}, n} f(t\_{n} - p^{i}(\omega + j)h, y(t\_{n} - p^{i}(\omega + j)h)) \right) \\ & + \frac{(p^{m}h)^{\kappa}}{a(a+1)} \sum\_{j=0}^{\lceil n/p^{m} - \omega\rceil - 1} b\_{j, p^{n}, n} f(t\_{n} - p^{m}(\omega + j)h, y(t\_{n} - p^{m}(\omega + j)h)), \end{split} \tag{30}$$

where

8 Will-be-set-by-IN-TECH

*f*(*τ*, *y*(*τ*))*dτ*

*<sup>n</sup>*(*τ*, *y*(*τ*))*dτ*

�*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *<sup>y</sup>*(*tj*)) (23)

<sup>Γ</sup>(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>) *<sup>f</sup>*(*tn*, *yh*(*tn*))

�*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*)), *<sup>α</sup>* <sup>∈</sup> (1, 2) (26)

�*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*)), *<sup>α</sup>* <sup>∈</sup> (1, 2). (27)

�*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*)), *<sup>α</sup>* <sup>∈</sup> (2, <sup>∞</sup>) (28)

*α f*(*tn*, *yh*(*tn*))+

(24)

*<sup>α</sup> <sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*)). (25)

*f* �

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1�

*n* ∑ *j*=0

For the first integral of (18) or (19), the product rectangle formula is used

� *tn* 0

Combing (22)-(25), for *α* ∈ (1, 2), the new predictor-corrector approach gives

(1)

*n* ∑ *j*=0

<sup>0</sup> · *<sup>h</sup>* <sup>+</sup> *yh*(*tn*) + *<sup>h</sup><sup>α</sup>*

*hα* Γ(*α* + 2)

+

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1�

(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)*α*+1(*<sup>α</sup>* <sup>−</sup> *<sup>n</sup>*) + *<sup>n</sup>α*(2*<sup>n</sup>* <sup>−</sup> *<sup>α</sup>* <sup>−</sup> <sup>1</sup>) <sup>−</sup> (*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)*α*<sup>+</sup>1, if *<sup>j</sup>* <sup>=</sup> 0,

<sup>2</sup>*α*+<sup>1</sup> <sup>−</sup> *<sup>α</sup>* <sup>−</sup> 3, if *<sup>j</sup>* <sup>=</sup> *<sup>n</sup>*.

(*<sup>n</sup>* <sup>−</sup> *<sup>j</sup>* <sup>+</sup> <sup>2</sup>)*α*+<sup>1</sup> <sup>−</sup> <sup>3</sup>(*<sup>n</sup>* <sup>−</sup> *<sup>j</sup>* <sup>+</sup> <sup>1</sup>)*α*+<sup>1</sup> <sup>+</sup> <sup>3</sup>(*<sup>n</sup>* <sup>−</sup> *<sup>j</sup>*)*α*+<sup>1</sup> <sup>−</sup> (*<sup>n</sup>* <sup>−</sup> *<sup>j</sup>* <sup>−</sup> <sup>1</sup>)*α*<sup>+</sup>1, if 1 <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> *<sup>n</sup>* <sup>−</sup> 1,

<sup>0</sup> · *<sup>h</sup>* <sup>+</sup> *yh*(*tn*) + *<sup>h</sup><sup>α</sup>*

Γ(*α* + 2)

*<sup>n</sup>*) + *yh*(*tn*)

*n* ∑ *j*=0

*n* ∑ *j*=0

*<sup>n</sup>*) + *yh*(*tn*) + *<sup>h</sup><sup>α</sup>*

Γ(*α* + 2)

�

�*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*)), *<sup>α</sup>* <sup>∈</sup> (2, <sup>∞</sup>). (29)

Γ(*α* + 2)

*n* ∑ *j*=0 �

*α f*(*tn*, *yh*(*tn*))

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*))*d<sup>τ</sup>* <sup>=</sup> *<sup>h</sup><sup>α</sup>*

and

where

⎧ ⎪⎨

⎪⎩

� *tn* 0

and

and

�*aj*,*<sup>n</sup>* <sup>=</sup>

� *tn* 0 �

> ≈ � *tn* 0 �

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* <sup>≈</sup>

*yh*(*tn*+1) = *y*

+*f*(*tn*+1, *y<sup>P</sup>*

*yP*

+

*<sup>h</sup>* (*tn*+1) =

*hα*

*yh*(*tn*+1) =

*f*(*tn*+1, *y<sup>P</sup>*

For *α* ∈ (2, ∞), the predictor-corrector approach is

�*α*�−1 ∑ *k*=1

*yP*

+

(1)

*<sup>h</sup>* (*tn*+1))�

*y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k*

<sup>Γ</sup>(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>) *<sup>f</sup>*(*tn*, *yh*(*tn*)) + *<sup>h</sup><sup>α</sup>*

�*α*�−1 ∑ *k*=1

*<sup>h</sup>* (*tn*+1))�

*y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k*

+

*hα* Γ(*α* + 2)

*<sup>h</sup>* (*tn*+1) = *y*

*hα* Γ(*α* + 2)

<sup>=</sup> *<sup>h</sup><sup>α</sup> α*(*α* + 1)

$$b\_{j,p^0,n} = \begin{cases} (p\omega+1)^a (n-p\omega) + (p\omega)^a (2p\omega - n - 1) - (p\omega - 1)^{a+1}, & \text{if } \quad j = n - p\omega, \\ (n-j+2)^{a+1} - 3(n-j+1)^{a+1} + 3(n-j)^{a+1} \\ -(n-j-1)^{a+1}, & \text{if } \quad n - p\omega + 1 \le j \le n - 1, \\ 2^{a+1} - a - 3, & \text{if } \quad j = n \end{cases}$$

and for *i* = 1, 2, ··· , *m*,

$$b\_{j,p',n} = \begin{cases} -(1/p^i + \omega)^{a+1} + \omega^{a+1} + (1/p^i + \omega + 1)^{a+1} \\ -(\omega + 1)^{a+1} - ((1/p^i + \omega)^a - \omega^a)(a+1), & \text{if } j = 0, \\ (\omega + j - 1 + 1/p^i)^{a+1} - (\omega + j - 1)^{a+1} - 2(\omega + j + 1/p^i)^{a+1}, \\ + 2(\omega + j)^{a+1} + (\omega + j + 1 + 1/p^i)^{a+1} - (\omega + j + 1)^{a+1}, & \text{if } & 1 \le j \le (p-1)\omega - 1, \\ (p^i \omega - 1 + 1/p^i)^{a+1} - (p^i \omega - 1)^{a+1} - (p^i \omega + 1/p^i)^{a+1} \\ + (p^i \omega)^{a+1} + ((p^i \omega + 1/p^i)^a - (p^i \omega)^a)(a+1), & \text{if } & j = (p-1)\omega. \end{cases}$$

Employing above analysis we obtain the following predictor-corrector algorithm

$$y\_h^p(t\_{n+1}) = y\_0^{(1)} \cdot h + y\_h(t\_n) + \frac{h^a}{\Gamma(a+1)} f(t\_{h^a} y\_h(t\_n)) + \frac{h^a}{a(a+1)} \sum\_{j=n-p\omega}^n b\_{j,p^0,n} f(t\_j, y(t\_j))$$

$$+ \sum\_{i=1}^{m-1} \frac{(p^i h)^a}{a(a+1)} \left( \sum\_{j=0}^{(p-1)\omega} b\_{j,p^i,n} f(t\_n - p^i(\omega + j)h, y(t\_n - p^i(\omega + j)h)) \right)$$

$$+ \frac{(p^m h)^a}{a(a+1)} \sum\_{j=0}^{\lceil n/p^m - \omega \rceil -1} b\_{j,p^n,n} f(t\_n - p^m(\omega + j)h, y(t\_n - p^m(\omega + j)h)) \tag{32}$$

With the similar method, we can obtain the following lemma.

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*g*(*τ*)*d<sup>τ</sup>* <sup>−</sup>

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup>

 *tn*<sup>+</sup><sup>1</sup> *tn*

<sup>≤</sup> �*<sup>∂</sup> <sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))/*∂τ*�<sup>∞</sup> Γ(*α*)

<sup>=</sup> �*<sup>∂</sup> <sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))/*∂τ*�<sup>∞</sup> Γ(*α*)

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup>

*Proof*. According to the property of linear interpolation polynomials,

where *f* [*τ*, *tn*, *tn*+1] is second divided differences. And using the fact

*α*+1

*α*+2 *<sup>n</sup>*−*j*) <sup>−</sup> *<sup>h</sup>*

The error of the product rectangle rule is given as

 *tn*<sup>+</sup><sup>1</sup> *tn*

 1 Γ(*α*)

 *tn*<sup>+</sup><sup>1</sup> *tn*

*f*(*τ*, *y*(*τ*)) − *f*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1

(*t α*+1

*α*(*α* + 1)(*α* + 2)

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1(*<sup>τ</sup>* <sup>−</sup> *tj*)(*<sup>τ</sup>* <sup>−</sup> *tj*<sup>+</sup>1)*d<sup>τ</sup>*

*<sup>n</sup>*−*j*+1*tn*−*<sup>j</sup>* <sup>−</sup> *<sup>t</sup>*

(*t α*+2 *<sup>n</sup>*−*j*+<sup>1</sup> <sup>−</sup> *<sup>t</sup>*  *tn* 0

Numerical Schemes for Fractional Ordinary Differential Equations 365

**Lemma 3.** *Suppose that ∂ f*(*τ*, *y*(*τ*))/*∂t* ∈ *C*[0, *t*)*, for some suitable t, then we have*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup>

 *tn*<sup>+</sup><sup>1</sup> *tn*

<sup>≤</sup> *<sup>C</sup>αhα*+<sup>1</sup> where *<sup>C</sup><sup>α</sup>* <sup>=</sup> �*<sup>∂</sup> <sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))/*∂τ*�<sup>∞</sup>

*f*(*τ*, *y*(*τ*)) − *f*

**Lemma 4.** *Suppose that <sup>∂</sup>*<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))/*∂*2*<sup>τ</sup>* <sup>∈</sup> *<sup>C</sup>*[0, *<sup>t</sup>*)*, for some suitable t, then we have*

1 *α*(*α* + 1)

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*<sup>g</sup><sup>n</sup>*(*τ*)*d<sup>τ</sup>*

*<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*)) <sup>−</sup> *<sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*))

*hα*+<sup>1</sup>

(*tj* − *tn*+<sup>1</sup> + *tn*+<sup>1</sup> − *τ*)(*tj*<sup>+</sup><sup>1</sup> − *tn*+<sup>1</sup> + *tn*+<sup>1</sup> − *τ*)

*tn*−*j*+1*tn*−*j*(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn*−*<sup>j</sup>* <sup>+</sup> *tn*−*j*+1)(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*<sup>α</sup>* + (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*+1

(*α* + 1)(*α* + 2)

(*t α*+1 *<sup>n</sup>*−*j*+<sup>1</sup> <sup>+</sup> *<sup>t</sup>*

*<sup>n</sup>*−*<sup>j</sup> tn*−*j*+1) + <sup>1</sup>

*α*(*α* + 1)

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1(*<sup>τ</sup>* <sup>−</sup> *tn*)*d<sup>τ</sup>*

*<sup>n</sup>*+1(*τ*, *<sup>y</sup>*(*τ*))

*<sup>n</sup>*+1(*τ*, *y*(*τ*)) = *f* [*τ*, *tn*, *tn*+1](*τ* − *tn*)(*τ* − *tn*+1), (40)

(*t α*+2 *<sup>n</sup>*−*<sup>j</sup>* <sup>−</sup> *<sup>t</sup>*

*α*+1

*dτ* 

*<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*)) <sup>−</sup> *<sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*))

 

*dτ* 

> *dτ*

<sup>Γ</sup>(*<sup>α</sup>* <sup>+</sup> <sup>2</sup>) . (38)

 *dτ*

*<sup>n</sup>*−*j*) for all *<sup>j</sup>* <sup>≥</sup> 0, (41)

*α*+2 *<sup>n</sup>*−*j*+1)

<sup>≤</sup> *<sup>C</sup>αh*2, (36)

<sup>≤</sup> *<sup>C</sup>αhα*<sup>+</sup>1, (37)

<sup>≤</sup> *<sup>C</sup>αhα*<sup>+</sup>2, (39)

*dτ*

**Lemma 2.** *Suppose g* <sup>∈</sup> *<sup>C</sup>*2[0, *<sup>T</sup>*]*, tn* 0

*where C<sup>α</sup> only depends on α.*

 1 Γ(*α*)

*where C<sup>α</sup> only depends on α.*

 1 Γ(*α*)

*where C<sup>α</sup> only depends on α.*

 *tj*<sup>+</sup><sup>1</sup> *tj*

> = *tj*<sup>+</sup><sup>1</sup> *tj*

> = *tj*<sup>+</sup><sup>1</sup> *tj*

<sup>=</sup> <sup>1</sup> *α*(*α* + 1)

<sup>=</sup> <sup>2</sup>

*Proof*.

and

$$\begin{split} y\_h(t\_{n+1}) &= y\_0^{(1)} \cdot h + y\_h(t\_n) + \frac{h^a}{\Gamma(a+2)} \left( \alpha f(t\_n, y\_h(t\_n)) \right) \\ &+ f(t\_{n+1}, y\_h^p(t\_{n+1})) \Big) + \frac{h^a}{a(a+1)} \sum\_{j=-p,\omega}^n b\_{j,p^0,n} f(t\_j, y(t\_j)) \\ &+ \sum\_{i=1}^{m-1} \frac{(p^i h)^a}{a(a+1)} \left( \sum\_{j=0}^{(p-1)\omega} b\_{j,p^i,n} f(t\_n - p^i(\omega+j)h, y(t\_n - p^i(\omega+j)h)) \right) \\ &+ \frac{(p^m h)^a}{a(a+1)} \sum\_{j=0}^{\lceil n/p^m - \omega \rceil -1} b\_{j,p^n,n} f(t\_n - p^m(\omega+j)h, y(t\_n - p^m(\omega+j)h)). \end{split}$$

In this case, the nested meshes predictor-corrector algorithm has the computational cost of *<sup>O</sup>*(*h*−1*log*(*h*−1)) for *<sup>α</sup>* <sup>∈</sup> (1, 2).

#### **3.2 Error analysis and convergent order**

In this section, we make the local truncation error and convergent order analysis for the improved predictor-corrector approaches (16)-(17), (26)-(29), and (32)-(33). First we present several lemmas which will be used later.

**Lemma 1.** *(Diethelm et al., 2004) Suppose g* <sup>∈</sup> *<sup>C</sup>*2[0, *<sup>T</sup>*]*,*

$$\left| \int\_{0}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} g(\tau) d\tau - \int\_{0}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} \widetilde{g}\_{n+1}(\tau) d\tau \right| \leq C\_{a} h^{2} \tag{34}$$

*where C<sup>α</sup> only depends on α.*

*Proof*. �

� � �

� *tn*<sup>+</sup><sup>1</sup> 0 (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1� *<sup>g</sup>*(*τ*) <sup>−</sup> *<sup>g</sup>*�*n*+1(*τ*) � *dτ* � � � � <sup>≤</sup> �*g*���<sup>∞</sup> 2 *n*+1 ∑ *j*=1 � *tj tj*−<sup>1</sup> (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1(*tj* <sup>−</sup> *<sup>τ</sup>*)(*<sup>τ</sup>* <sup>−</sup> *tj*−1)*d<sup>τ</sup>* <sup>=</sup> �*g*���∞*hα*+<sup>2</sup> 2*α*(*α* + 1) *n*+1 ∑ *j*=1 � (*n*−*j*+2)*α*+1+(*n*−*j*+1)*α*+1<sup>+</sup> 2 *α* + 2 � (*n*−*j*+2)*α*+2−(*n*−*j*+1)*α*+2� � <sup>=</sup> �*g*���∞*hα*+<sup>2</sup> 2*α*(*α* + 1) *n*+1 ∑ *j*=1 � (*j* + 1)*α*+<sup>1</sup> + *j <sup>α</sup>*+<sup>1</sup> + 2 *α* + 2 � <sup>1</sup> <sup>−</sup> (*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)*α*+2� � <sup>=</sup> <sup>−</sup>�*g*���∞*hα*+<sup>2</sup> 2*α*(*α* + 1) � 2 � *n*+1 1 *<sup>τ</sup>α*+1*d<sup>τ</sup>* <sup>−</sup> *n*+1 ∑ *j*=1 � (*j* + 1)*α*+<sup>1</sup> + *j α*+1� � ≤ ⎧ ⎨ ⎩ �*g*���∞*hα*+<sup>2</sup> <sup>24</sup> <sup>∑</sup>*n*+<sup>1</sup> *<sup>j</sup>*=<sup>1</sup> *j <sup>α</sup>*−1, if *α* < 1, �*g*���∞*hα*+<sup>2</sup> <sup>24</sup> <sup>∑</sup>*n*+<sup>1</sup> *<sup>j</sup>*=<sup>1</sup> (*<sup>j</sup>* <sup>+</sup> <sup>1</sup>)*α*−1, if *<sup>α</sup>* <sup>≥</sup> 1, ≤ � �*g*���∞*hα*+<sup>2</sup> 24 � *<sup>n</sup>*+<sup>1</sup> <sup>0</sup> *<sup>τ</sup>α*−1*dτ*, if *<sup>α</sup>* <sup>&</sup>lt; 1, �*g*���∞*hα*+<sup>2</sup> 24 � *<sup>n</sup>*+<sup>2</sup> <sup>2</sup> *<sup>τ</sup>α*−1*dτ*, if *<sup>α</sup>* <sup>≥</sup> 1, <sup>≤</sup> *<sup>C</sup>αh*2. (35)

With the similar method, we can obtain the following lemma.

**Lemma 2.** *Suppose g* <sup>∈</sup> *<sup>C</sup>*2[0, *<sup>T</sup>*]*,*

$$\left| \int\_{0}^{t\_n} (t\_{n+1} - \tau)^{n-1} g(\tau) d\tau - \int\_{0}^{t\_n} (t\_{n+1} - \tau)^{n-1} \widetilde{g}\_n(\tau) d\tau \right| \le \mathbb{C}\_{\mathfrak{a}} h^2 \tag{36}$$

*where C<sup>α</sup> only depends on α.*

The error of the product rectangle rule is given as

**Lemma 3.** *Suppose that ∂ f*(*τ*, *y*(*τ*))/*∂t* ∈ *C*[0, *t*)*, for some suitable t, then we have*

 1 Γ(*α*) *tn*<sup>+</sup><sup>1</sup> *tn* (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*)) <sup>−</sup> *<sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*)) *dτ* <sup>≤</sup> *<sup>C</sup>αhα*<sup>+</sup>1, (37)

*where C<sup>α</sup> only depends on α.*

*Proof*.

10 Will-be-set-by-IN-TECH

Γ(*α* + 2)

,*<sup>n</sup> <sup>f</sup>*(*tn* <sup>−</sup> *<sup>p</sup><sup>i</sup>*

In this case, the nested meshes predictor-corrector algorithm has the computational cost of

In this section, we make the local truncation error and convergent order analysis for the improved predictor-corrector approaches (16)-(17), (26)-(29), and (32)-(33). First we present

> � *tn*<sup>+</sup><sup>1</sup> 0

�

*n* ∑ *j*=*n*−*pω*

*α f*(*tn*, *yh*(*tn*))

*bj*,*p*0,*<sup>n</sup> f*(*tj*, *y*(*tj*))

(*<sup>ω</sup>* <sup>+</sup> *<sup>j</sup>*)*h*, *<sup>y</sup>*(*tn* <sup>−</sup> *<sup>p</sup><sup>i</sup>*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*g*�*n*+1(*τ*)*d<sup>τ</sup>*

2 *α* + 2 �

<sup>1</sup> <sup>−</sup> (*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)*α*+2�

*α*+1� �

� *<sup>n</sup>*+<sup>1</sup>

� *<sup>n</sup>*+<sup>2</sup>

*bj*,*pm*,*<sup>n</sup> <sup>f</sup>*(*tn* <sup>−</sup> *<sup>p</sup>m*(*<sup>ω</sup>* <sup>+</sup> *<sup>j</sup>*)*h*, *<sup>y</sup>*(*tn* <sup>−</sup> *<sup>p</sup>m*(*<sup>ω</sup>* <sup>+</sup> *<sup>j</sup>*)*h*)). (33)

(*ω* + *j*)*h*))

� � � �

(*n*−*j*+2)*α*+2−(*n*−*j*+1)*α*+2�

<sup>0</sup> *<sup>τ</sup>α*−1*dτ*, if *<sup>α</sup>* <sup>&</sup>lt; 1,

<sup>2</sup> *<sup>τ</sup>α*−1*dτ*, if *<sup>α</sup>* <sup>≥</sup> 1,

�

<sup>≤</sup> *<sup>C</sup>αh*2, (34)

�

�

and

*yh*(*tn*+1) = *y*

+*f*(*tn*+1, *y<sup>P</sup>*

<sup>+</sup> (*pmh*)*<sup>α</sup> α*(*α* + 1)

**3.2 Error analysis and convergent order**

several lemmas which will be used later.

**Lemma 1.** *(Diethelm et al., 2004) Suppose g* <sup>∈</sup> *<sup>C</sup>*2[0, *<sup>T</sup>*]*,*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*g*(*τ*)*d<sup>τ</sup>* <sup>−</sup>

*<sup>g</sup>*(*τ*) <sup>−</sup> *<sup>g</sup>*�*n*+1(*τ*)

(*j* + 1)*α*+<sup>1</sup> + *j*

*<sup>τ</sup>α*+1*d<sup>τ</sup>* <sup>−</sup>

*<sup>α</sup>*−1, if *α* < 1,

*<sup>j</sup>*=<sup>1</sup> (*<sup>j</sup>* <sup>+</sup> <sup>1</sup>)*α*−1, if *<sup>α</sup>* <sup>≥</sup> 1,

� *dτ* � � � �

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1(*tj* <sup>−</sup> *<sup>τ</sup>*)(*<sup>τ</sup>* <sup>−</sup> *tj*−1)*d<sup>τ</sup>*

(*n*−*j*+2)*α*+1+(*n*−*j*+1)*α*+1<sup>+</sup>

*<sup>α</sup>*+<sup>1</sup> +

*n*+1 ∑ *j*=1 �

2 *α* + 2 �

≤

<sup>≤</sup> *<sup>C</sup>αh*2. (35)

(*j* + 1)*α*+<sup>1</sup> + *j*

� �*g*���∞*hα*+<sup>2</sup> 24

�*g*���∞*hα*+<sup>2</sup> 24

(*pi h*)*<sup>α</sup> α*(*α* + 1)

+ *m*−1 ∑ *i*=1

*<sup>O</sup>*(*h*−1*log*(*h*−1)) for *<sup>α</sup>* <sup>∈</sup> (1, 2).

� � � �

*where C<sup>α</sup> only depends on α.*

*Proof*. � � � �

� *tn*<sup>+</sup><sup>1</sup> 0

<sup>≤</sup> �*g*���<sup>∞</sup> 2

<sup>=</sup> �*g*���∞*hα*+<sup>2</sup> 2*α*(*α* + 1)

<sup>=</sup> �*g*���∞*hα*+<sup>2</sup> 2*α*(*α* + 1)

<sup>=</sup> <sup>−</sup>�*g*���∞*hα*+<sup>2</sup> 2*α*(*α* + 1)

> �*g*���∞*hα*+<sup>2</sup> <sup>24</sup> <sup>∑</sup>*n*+<sup>1</sup> *<sup>j</sup>*=<sup>1</sup> *j*

�*g*���∞*hα*+<sup>2</sup> <sup>24</sup> <sup>∑</sup>*n*+<sup>1</sup>

≤

⎧ ⎨ ⎩ � *tn*<sup>+</sup><sup>1</sup> 0

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1�

*n*+1 ∑ *j*=1

� *tj tj*−<sup>1</sup>

> *n*+1 ∑ *j*=1

> > *n*+1 ∑ *j*=1

> > > � 2 � *n*+1 1

�

�

(1)

*<sup>h</sup>* (*tn*+1))�

<sup>0</sup> · *<sup>h</sup>* <sup>+</sup> *yh*(*tn*) + *<sup>h</sup><sup>α</sup>*

*hα α*(*α* + 1)

*bj*,*pi*

+

� (*p*−1)*<sup>ω</sup>* ∑ *j*=0

�*n*/*p<sup>m</sup>*−*ω*�−<sup>1</sup> ∑ *j*=0

$$\left|\frac{1}{\Gamma(\alpha)} \int\_{t\_n}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} \left( f(\tau, y(\tau)) - f(t\_n, y(t\_n)) \right) d\tau \right| $$

$$\leq \frac{||\partial f(\tau, y(\tau)) / \partial \tau||\_{\infty}}{\Gamma(\alpha)} \int\_{t\_n}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} (\tau - t\_n) d\tau $$

$$= \frac{||\partial f(\tau, y(\tau)) / \partial \tau||\_{\infty}}{\Gamma(\alpha)} \frac{1}{a(\alpha + 1)} h^{\alpha + 1} $$

$$\leq \mathbb{C}\_{\mathfrak{d}} h^{\alpha + 1} \quad \text{where} \quad \mathbb{C}\_{\mathfrak{k}} = \frac{||\partial f(\tau, y(\tau)) / \partial \tau||\_{\infty}}{\Gamma(\alpha + 2)}. \tag{38}$$

**Lemma 4.** *Suppose that <sup>∂</sup>*<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))/*∂*2*<sup>τ</sup>* <sup>∈</sup> *<sup>C</sup>*[0, *<sup>t</sup>*)*, for some suitable t, then we have*

$$\left| \frac{1}{\Gamma(\mathfrak{a})} \int\_{t\_{\mathfrak{u}}}^{t\_{\mathfrak{u}+1}} (t\_{\mathfrak{u}+1} - \tau)^{\mathfrak{a}-1} (f(\tau, y(\tau)) - \tilde{f}\_{\mathfrak{u}+1}(\tau, y(\tau))) d\tau \right| \le \mathsf{C}\_{\mathfrak{a}} h^{\mathfrak{a}+2} \,\mathrm{s} \tag{39}$$

*where C<sup>α</sup> only depends on α.*

*Proof*. According to the property of linear interpolation polynomials,

$$f(\tau, y(\tau)) - \tilde{f}\_{n+1}(\tau, y(\tau)) = f[\tau, t\_n, t\_{n+1}](\tau - t\_n)(\tau - t\_{n+1}),\tag{40}$$

$$f(\tau, \dots, \tau, \dots, \tau, \dots, \dots) = \tau \tau \dots \tau \dots \dots \tag{40}$$

where *f* [*τ*, *tn*, *tn*+1] is second divided differences. And using the fact

$$\begin{split} &\int\_{t\_{j}}^{t\_{j+1}} (t\_{n+1} - \tau)^{a-1} (\tau - t\_{j}) (\tau - t\_{j+1}) d\tau \\ &= \int\_{t\_{j}}^{t\_{j+1}} (t\_{n+1} - \tau)^{a-1} \Big[ (t\_{j} - t\_{n+1} + t\_{n+1} - \tau) (t\_{j+1} - t\_{n+1} + t\_{n+1} - \tau) \Big] d\tau \\ &= \int\_{t\_{j}}^{t\_{j+1}} \Big[ t\_{n-j+1} t\_{n-j} (t\_{n+1} - \tau)^{a-1} - (t\_{n-j} + t\_{n-j+1}) (t\_{n+1} - \tau)^{a} + (t\_{n+1} - \tau)^{a+1} \Big] d\tau \\ &= \frac{1}{a(a+1)} (t\_{n-j+1}^{a+1} t\_{n-j} - t\_{n-j}^{a+1} t\_{n-j+1}) + \frac{1}{(a+1)(a+2)} (t\_{n-j}^{a+2} - t\_{n-j+1}^{a+2}) \\ &= \frac{2}{a(a+1)(a+2)} (t\_{n-j+1}^{a+2} - t\_{n-j}^{a+2}) - \frac{h}{a(a+1)} (t\_{n-j+1}^{a+1} + t\_{n-j}^{a+1}) \quad \text{for all} \quad j \ge 0, \tag{41} \end{split}$$

**Lemma 6.** *(Diethelm et al., 2004) Assume that the solution of the initial value problem satisfies*

where *<sup>z</sup>*<sup>∗</sup> <sup>∈</sup> [*tn*, *tn*+1], *<sup>z</sup>*∗∗ <sup>∈</sup> [*z*∗, *tn*+1] <sup>⊂</sup> [*tn*, *tn*+1], *<sup>z</sup>*∗∗ <sup>∈</sup> [*tn*, *<sup>z</sup>*∗] <sup>⊂</sup> [*tn*, *tn*+1], *<sup>z</sup>*∗∗∗ <sup>∈</sup> [*<sup>z</sup>*∗∗, *<sup>z</sup>*∗∗] <sup>⊂</sup>

 

Numerical Schemes for Fractional Ordinary Differential Equations 13

Numerical Schemes for Fractional Ordinary Differential Equations 367

*α*(*α* + 1)

*<sup>α</sup>*+<sup>1</sup> *<sup>n</sup>* <sup>−</sup> <sup>2</sup>(*z*∗)*α*+1) *<sup>α</sup>*(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>α</sup>* <sup>+</sup> <sup>2</sup>) <sup>+</sup>

*<sup>α</sup>*+<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> <sup>−</sup> (*z*∗)*α*+1) <sup>−</sup> ((*z*∗)*α*+<sup>1</sup> <sup>−</sup> *<sup>t</sup>*

+

*hα*+<sup>2</sup> (*α* + 1)(*α* + 2)

<sup>≤</sup> *<sup>C</sup><sup>α</sup>* · *<sup>h</sup>*min{*α*+2,3}, (44)

(*t <sup>α</sup>*+<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> <sup>+</sup> *<sup>t</sup>*

*<sup>α</sup>*(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>) <sup>+</sup>

*hα*+<sup>2</sup> (*α* + 1)(*α* + 2)

> 

*<sup>α</sup>*+<sup>1</sup> *<sup>n</sup>* ) + *<sup>h</sup>α*+<sup>2</sup>

 

*hα*+<sup>2</sup> (*α* + 1)(*α* + 2)

*<sup>α</sup>*+<sup>1</sup> *<sup>n</sup>* )

  (*α* + 1)(*α* + 2)

*hα*+<sup>2</sup> (*α* + 1)(*α* + 2)

(mean value theorem)

 .

(mean value theorem)

 

(mean value theorem)

*n*+1 ∑ *k*=0

(*z*∗∗∗)*α*−<sup>1</sup> <sup>−</sup> <sup>1</sup>

*n*+1 ∑ *k*=0

*bj*,*n*+1*D<sup>α</sup>*

*aj*,*n*+1*D<sup>α</sup>*

 *dτ* ≤ �*g*� �∞

<sup>∗</sup> *<sup>y</sup>*(*tj*) ≤ *C*1*t γ*1

(*α* + 1)(*α* + 2)

<sup>∗</sup> *<sup>y</sup>*(*tj*) ≤ *C*2*t γ*2

*<sup>n</sup>*+1*hδ*<sup>1</sup> (45)

*<sup>n</sup>*+1*hδ*<sup>2</sup> , (46)

, (48)

<sup>=</sup> *<sup>O</sup>*(*hq*), (47)

<sup>∗</sup> *<sup>y</sup>*(*t*) <sup>∈</sup> *<sup>C</sup>*2[0, *<sup>T</sup>*] *for some suitable T, then*

*hα*+<sup>1</sup> *α*(*α* + 1)

<sup>∗</sup> *<sup>y</sup>*(*τ*)*d<sup>τ</sup>* −

<sup>∗</sup> *<sup>y</sup>*(*τ*)*d<sup>τ</sup>* −

*y*(*tj*) − *yh*(*tj*)

*the convergent order of our algorithm with the predictor and corrector formulae (16) and (17) gives*

*g*(*τ*) − *g*(*tn*)

**Theorem 2.** *When <sup>α</sup>* <sup>&</sup>gt; <sup>1</sup>*, if <sup>∂</sup>*<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))/*∂*2*<sup>τ</sup>* <sup>∈</sup> *<sup>C</sup>*[0, *<sup>t</sup>*) *for some suitable t, then the local truncation error of our algorithm with the predictor and corrector formulae (26)-(27) (α* ∈ (1, 2)*) and (28)-(29) (<sup>α</sup>* <sup>∈</sup> (2, <sup>∞</sup>)*) is O*(*h*3)*, and the convergent order is 2, i.e.,* max*j*=0,1,··· ,*n*+<sup>1</sup> <sup>|</sup>*y*(*tj*) <sup>−</sup> *yh*(*tj*)<sup>|</sup> <sup>=</sup> *<sup>O</sup>*(*h*2)*.*

*with some γ*1, *γ*<sup>2</sup> ≥ 0 *and δ*1, *δ*<sup>2</sup> ≥ 0*. Then, for some suitably chosen T* ≥ 0*, we have*

max 0≤*j*≤*N* 

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup>

 

<sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup>

<sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup>

<sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup>

<sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup>

<sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup>

[*tn*, *tn*+1] and

<sup>2</sup>Γ(*α*) ·

<sup>2</sup>Γ(*α*) ·

<sup>2</sup>Γ(*α*) ·

<sup>2</sup>Γ(*α*) ·

<sup>2</sup>Γ(*α*) ·

 

 

 

 

 

−2(*t*

−*h*(*t*

−*h*((*t*

*<sup>α</sup>*+<sup>2</sup> *<sup>n</sup>* <sup>−</sup> *<sup>t</sup>*

*<sup>α</sup>*+<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> <sup>+</sup> *<sup>t</sup>*

−*h*2((*z*∗∗)*<sup>α</sup>* <sup>−</sup> (*<sup>z</sup>*∗∗)*α*) *α*

<sup>−</sup> *<sup>h</sup>*3((*z*∗∗∗)*α*−<sup>1</sup> <sup>+</sup>

*<sup>α</sup>*+<sup>2</sup> *<sup>n</sup>*+1) *<sup>α</sup>*(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>α</sup>* <sup>+</sup> <sup>2</sup>) <sup>−</sup> *<sup>h</sup>*

 

*and*

*Proof*. Noting

 *tn*<sup>+</sup><sup>1</sup> 0

 *tn*<sup>+</sup><sup>1</sup> 0

*with q* = min{*δ*<sup>1</sup> + *α*, *δ*2} *and N* = �*T*/*h*�*.*

 

 *tn*<sup>+</sup><sup>1</sup> *tn*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*D<sup>α</sup>*

*<sup>C</sup><sup>α</sup>* <sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup>

<sup>2</sup>Γ(*α*) ·

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1*D<sup>α</sup>*

**Theorem 1.** *For the fractional initial problem (1), if D<sup>α</sup>*

and applying lemmas 2, 4, 6, we have the above result.

max*j*=0,1,··· ,*n*+<sup>1</sup> <sup>|</sup>*y*(*tj*) <sup>−</sup> *yh*(*tj*)<sup>|</sup> <sup>=</sup> *<sup>O</sup>*(*h*min{2,1+2*α*})*.*

we have

$$\begin{split} & \left| \frac{1}{\Gamma(\alpha)} \int\_{t\_n}^{t\_{n+1}} (t\_{n+1} - \tau)^{\alpha - 1} (f(\tau, y(\tau)) - \tilde{f}\_{n+1}(\tau, y(\tau))) d\tau \right| \\ & \quad = \left| \frac{1}{\Gamma(\alpha)} \int\_{t\_n}^{t\_{n+1}} (t\_{n+1} - \tau)^{\alpha - 1} f[\tau, t\_n, t\_{n+1}] (\tau - t\_n) (\tau - t\_{n+1}) d\tau \right| \\ & \quad = \left| \frac{[\tilde{f}\_n, t\_n, t\_{n+1}]}{\Gamma(\alpha)} \right| \cdot \int\_{t\_n}^{t\_{n+1}} (t\_{n+1} - \tau)^{\alpha - 1} (\tau - t\_n) (\tau - t\_{n+1}) d\tau \\ & \quad = \left| \frac{f''(\eta, y(\eta))}{2\Gamma(\alpha)} \right| \cdot \int\_{t\_n}^{t\_{n+1}} (t\_{n+1} - \tau)^{\alpha - 1} (\tau - t\_n) (\tau - t\_{n+1}) d\tau \\ & \quad = \left| \frac{f''(\eta, y(\eta))}{2\Gamma(\alpha)} \right| \cdot \frac{\hbar^{n+2}}{(\alpha + 1)(\alpha + 2)} \quad \text{(taking } j = n \text{ in (41))} \\ & \quad \leq C\_{\hbar} \hbar^{n+2} \quad \text{where} \quad C\_{\hbar} = \frac{\|f''(\eta, y(\eta))\|\_{\infty}}{2\Gamma(\alpha)(\alpha + 1)(\alpha + 2)}. \tag{42} \end{split}$$

Here *ξ*, *η* ∈ [*tn*, *tn*+1] and in the above equalities the second integral mean value theorem and the properties of second divided differences are used.

**Lemma 5.** *Suppose that <sup>∂</sup>*<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))/*∂*2*<sup>τ</sup>* <sup>∈</sup> *<sup>C</sup>*[0, *<sup>t</sup>*)*, for some suitable t, then we have*

$$\left| \frac{1}{\Gamma(a)} \int\_0^{t\_n} \left( (t\_{n+1} - \tau)^{a-1} - (t\_n - \tau)^{a-1} \right) \left( f(\tau, y(\tau)) - \tilde{f}\_n(\tau, y(\tau)) \right) d\tau \right| \le C\_a \mathfrak{l}^{\min\{a+2, 3\}}, \text{ (43)}$$

*where C<sup>α</sup> only depends on α.*

*Proof*. The idea of this lemma's proof is similar to the above lemmas, namely

 1 Γ(*α*) *tn* 0 (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1 *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*)) <sup>−</sup> *<sup>f</sup> <sup>n</sup>*(*τ*, *y*(*τ*)) *dτ* <sup>≤</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup> <sup>2</sup>Γ(*α*) · *n*−1 ∑ *j*=0 *tj*<sup>+</sup><sup>1</sup> *tj* (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1 (*τ* − *tj*)(*τ* − *tj*<sup>+</sup>1)*dτ* <sup>≤</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup> <sup>2</sup>Γ(*α*) · *n*−1 ∑ *j*=0 *tj*<sup>+</sup><sup>1</sup> *tj* (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1(*<sup>τ</sup>* <sup>−</sup> *tj*)(*<sup>τ</sup>* <sup>−</sup> *tj*<sup>+</sup>1)*d<sup>τ</sup>* − *tj*<sup>+</sup><sup>1</sup> *tj* (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1(*<sup>τ</sup>* <sup>−</sup> *tj*)(*<sup>τ</sup>* <sup>−</sup> *tj*<sup>+</sup>1)*d<sup>τ</sup>* <sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup> <sup>2</sup>Γ(*α*) · *n*−1 ∑ *j*=0 2 *α*(*α* + 1)(*α* + 2) (*t α*+2 *<sup>n</sup>*−*j*+<sup>1</sup> <sup>−</sup> *<sup>t</sup> α*+2 *<sup>n</sup>*−*j*) <sup>−</sup> *<sup>h</sup> α*(*α* + 1) (*t α*+1 *<sup>n</sup>*−*j*+<sup>1</sup> <sup>+</sup> *<sup>t</sup> α*+1 *<sup>n</sup>*−*j*) <sup>−</sup> <sup>2</sup> *α*(*α* + 1)(*α* + 2) (*t α*+2 *<sup>n</sup>*−*<sup>j</sup>* <sup>−</sup> *<sup>t</sup> α*+2 *<sup>n</sup>*−*j*−1) + *<sup>h</sup> α*(*α* + 1) (*t α*+1 *<sup>n</sup>*−*<sup>j</sup>* <sup>+</sup> *<sup>t</sup> α*+1 *<sup>n</sup>*−*j*−1) (using(41)) <sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup> <sup>2</sup>Γ(*α*) · *n*−1 ∑ *j*=0 −2(2*t α*+2 *<sup>n</sup>*−*<sup>j</sup>* <sup>−</sup> *<sup>t</sup> α*+2 *<sup>n</sup>*−*j*+<sup>1</sup> <sup>−</sup> *<sup>t</sup> α*+2 *<sup>n</sup>*−*j*−1) *<sup>α</sup>*(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>α</sup>* <sup>+</sup> <sup>2</sup>) <sup>−</sup> *<sup>h</sup> α*(*α* + 1) (*t α*+1 *<sup>n</sup>*−*j*+<sup>1</sup> <sup>−</sup> *<sup>t</sup> α*+1 *<sup>n</sup>*−*j*) <sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup> <sup>2</sup>Γ(*α*) · −2(*t <sup>α</sup>*+<sup>2</sup> *<sup>n</sup>* <sup>−</sup> *<sup>t</sup> <sup>α</sup>*+<sup>2</sup> *<sup>n</sup>*+<sup>1</sup> <sup>−</sup> *<sup>t</sup> α*+2 <sup>1</sup> ) *<sup>α</sup>*(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>α</sup>* <sup>+</sup> <sup>2</sup>) <sup>−</sup> *<sup>h</sup> α*(*α* + 1) (*t <sup>α</sup>*+<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> <sup>+</sup> *<sup>t</sup> <sup>α</sup>*+<sup>1</sup> *<sup>n</sup>* <sup>−</sup> *<sup>t</sup> α*+1 <sup>1</sup> ) 

<sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup> <sup>2</sup>Γ(*α*) · −2(*t <sup>α</sup>*+<sup>2</sup> *<sup>n</sup>* <sup>−</sup> *<sup>t</sup> <sup>α</sup>*+<sup>2</sup> *<sup>n</sup>*+1) *<sup>α</sup>*(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>α</sup>* <sup>+</sup> <sup>2</sup>) <sup>−</sup> *<sup>h</sup> α*(*α* + 1) (*t <sup>α</sup>*+<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> <sup>+</sup> *<sup>t</sup> <sup>α</sup>*+<sup>1</sup> *<sup>n</sup>* ) + *<sup>h</sup>α*+<sup>2</sup> (*α* + 1)(*α* + 2) <sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup> <sup>2</sup>Γ(*α*) · −*h*(*t <sup>α</sup>*+<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> <sup>+</sup> *<sup>t</sup> <sup>α</sup>*+<sup>1</sup> *<sup>n</sup>* <sup>−</sup> <sup>2</sup>(*z*∗)*α*+1) *<sup>α</sup>*(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>α</sup>* <sup>+</sup> <sup>2</sup>) <sup>+</sup> *hα*+<sup>2</sup> (*α* + 1)(*α* + 2) (mean value theorem) <sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup> <sup>2</sup>Γ(*α*) · −*h*((*t <sup>α</sup>*+<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> <sup>−</sup> (*z*∗)*α*+1) <sup>−</sup> ((*z*∗)*α*+<sup>1</sup> <sup>−</sup> *<sup>t</sup> <sup>α</sup>*+<sup>1</sup> *<sup>n</sup>* ) *<sup>α</sup>*(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>) <sup>+</sup> *hα*+<sup>2</sup> (*α* + 1)(*α* + 2) <sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup> <sup>2</sup>Γ(*α*) · −*h*2((*z*∗∗)*<sup>α</sup>* <sup>−</sup> (*<sup>z</sup>*∗∗)*α*) *α* + *hα*+<sup>2</sup> (*α* + 1)(*α* + 2) (mean value theorem) <sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup> <sup>2</sup>Γ(*α*) · <sup>−</sup> *<sup>h</sup>*3((*z*∗∗∗)*α*−<sup>1</sup> <sup>+</sup> *hα*+<sup>2</sup> (*α* + 1)(*α* + 2) (mean value theorem) <sup>≤</sup> *<sup>C</sup><sup>α</sup>* · *<sup>h</sup>*min{*α*+2,3}, (44)

where *<sup>z</sup>*<sup>∗</sup> <sup>∈</sup> [*tn*, *tn*+1], *<sup>z</sup>*∗∗ <sup>∈</sup> [*z*∗, *tn*+1] <sup>⊂</sup> [*tn*, *tn*+1], *<sup>z</sup>*∗∗ <sup>∈</sup> [*tn*, *<sup>z</sup>*∗] <sup>⊂</sup> [*tn*, *tn*+1], *<sup>z</sup>*∗∗∗ <sup>∈</sup> [*<sup>z</sup>*∗∗, *<sup>z</sup>*∗∗] <sup>⊂</sup> [*tn*, *tn*+1] and *<sup>C</sup><sup>α</sup>* <sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup> <sup>2</sup>Γ(*α*) · (*z*∗∗∗)*α*−<sup>1</sup> <sup>−</sup> <sup>1</sup> (*α* + 1)(*α* + 2) .

**Lemma 6.** *(Diethelm et al., 2004) Assume that the solution of the initial value problem satisfies*

$$\left| \int\_{0}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} D\_{\ast}^{a} y(\tau) d\tau - \sum\_{k=0}^{n+1} \tilde{b}\_{j,n+1} D\_{\ast}^{a} y(t\_j) \right| \leq C\_1 t\_{n+1}^{\gamma\_1} h^{\delta\_1} \tag{45}$$

*and*

12 Will-be-set-by-IN-TECH

366 Numerical Modelling

*f*(*τ*, *y*(*τ*)) − *f*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>* [*τ*, *tn*, *tn*+1](*<sup>τ</sup>* <sup>−</sup> *tn*)(*<sup>τ</sup>* <sup>−</sup> *tn*+1)*d<sup>τ</sup>*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1(*<sup>τ</sup>* <sup>−</sup> *tn*)(*<sup>τ</sup>* <sup>−</sup> *tn*+1)*d<sup>τ</sup>*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1(*<sup>τ</sup>* <sup>−</sup> *tn*)(*<sup>τ</sup>* <sup>−</sup> *tn*+1)*d<sup>τ</sup>*

*α*+2 *<sup>n</sup>*−*j*) <sup>−</sup> *<sup>h</sup>*

*α*+1 *<sup>n</sup>*−*j*−1)

*α*(*α* + 1)

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1(*<sup>τ</sup>* <sup>−</sup> *tj*)(*<sup>τ</sup>* <sup>−</sup> *tj*<sup>+</sup>1)*d<sup>τ</sup>*

 

> (*t α*+2 *<sup>n</sup>*−*j*+<sup>1</sup> <sup>−</sup> *<sup>t</sup>*

> > (*t α*+1 *<sup>n</sup>*−*<sup>j</sup>* <sup>+</sup> *<sup>t</sup>*

*α*+2 *<sup>n</sup>*−*j*−1) *<sup>α</sup>*(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>α</sup>* <sup>+</sup> <sup>2</sup>) <sup>−</sup> *<sup>h</sup>*

*α*(*α* + 1)

*<sup>α</sup>*+<sup>2</sup> *<sup>n</sup>*+<sup>1</sup> <sup>−</sup> *<sup>t</sup>*

*α*+2 *<sup>n</sup>*−*j*+<sup>1</sup> <sup>−</sup> *<sup>t</sup>*

*α*+2 <sup>1</sup> ) *<sup>α</sup>*(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>α</sup>* <sup>+</sup> <sup>2</sup>) <sup>−</sup> *<sup>h</sup>*

*<sup>n</sup>*(*τ*, *y*(*τ*))

*<sup>n</sup>*(*τ*, *y*(*τ*))

*dτ* 

> *dτ*

*α*(*α* + 1)

 

*α*(*α* + 1)

(*t <sup>α</sup>*+<sup>1</sup> *<sup>n</sup>*+<sup>1</sup> <sup>+</sup> *<sup>t</sup>*

(*τ* − *tj*)(*τ* − *tj*<sup>+</sup>1)*dτ*

(*t α*+1 *<sup>n</sup>*−*j*+<sup>1</sup> <sup>+</sup> *<sup>t</sup>*

(using(41))

(*t α*+1 *<sup>n</sup>*−*j*+<sup>1</sup> <sup>−</sup> *<sup>t</sup>*

*<sup>α</sup>*+<sup>1</sup> *<sup>n</sup>* <sup>−</sup> *<sup>t</sup>*

*α*+1 <sup>1</sup> ) 

(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>α</sup>* <sup>+</sup> <sup>2</sup>) (taking *<sup>j</sup>* <sup>=</sup> *<sup>n</sup>* in (41))

2Γ(*α*)(*α* + 1)(*α* + 2)

Here *ξ*, *η* ∈ [*tn*, *tn*+1] and in the above equalities the second integral mean value theorem and

*<sup>n</sup>*+1(*τ*, *<sup>y</sup>*(*τ*))

*dτ* 

> 

. (42)

<sup>≤</sup> *<sup>C</sup>αh*min{*α*+2,3}, (43)

 

*α*+1 *<sup>n</sup>*−*j*)

*α*+1 *<sup>n</sup>*−*j*) 

we have

 1 Γ(*α*)

> 1 Γ(*α*)

> > − *tj*<sup>+</sup><sup>1</sup> *tj*

 *tn* 0 

*where C<sup>α</sup> only depends on α.*

<sup>≤</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup>

<sup>≤</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup>

<sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup>

*α*(*α* + 1)(*α* + 2)

<sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup>

<sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup>

<sup>−</sup> <sup>2</sup>

<sup>2</sup>Γ(*α*) ·

<sup>2</sup>Γ(*α*) ·

<sup>2</sup>Γ(*α*) ·

<sup>2</sup>Γ(*α*) ·

<sup>2</sup>Γ(*α*) ·

 

 

 

(*t α*+2 *<sup>n</sup>*−*<sup>j</sup>* <sup>−</sup> *<sup>t</sup>*

> 

> 

*n*−1 ∑ *j*=0

−2(*t*

*n*−1 ∑ *j*=0

*n*−1 ∑ *j*=0

(*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1(*<sup>τ</sup>* <sup>−</sup> *tj*)(*<sup>τ</sup>* <sup>−</sup> *tj*<sup>+</sup>1)*d<sup>τ</sup>*

*n*−1 ∑ *j*=0

 *tj*<sup>+</sup><sup>1</sup> *tj*

 *tj*<sup>+</sup><sup>1</sup> *tj*

 *tn* 0   1 Γ(*α*)

> = 1 Γ(*α*)

= 

= 

=   *tn*<sup>+</sup><sup>1</sup> *tn*

> *tn*<sup>+</sup><sup>1</sup> *tn*

*f* [*ξ*, *tn*, *tn*+1] Γ(*α*)

*f* ��(*η*, *y*(*η*)) 2Γ(*α*)

*f* ��(*η*, *y*(*η*)) 2Γ(*α*)

the properties of second divided differences are used.

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup>

 ·

 ·

   *tn*<sup>+</sup><sup>1</sup> *tn*

 *tn*<sup>+</sup><sup>1</sup> *tn*

· *<sup>h</sup>α*+<sup>2</sup>

<sup>≤</sup> *<sup>C</sup>αhα*+<sup>2</sup> where *<sup>C</sup><sup>α</sup>* <sup>=</sup> � *<sup>f</sup>* ��(*η*, *<sup>y</sup>*(*η*))�<sup>∞</sup>

**Lemma 5.** *Suppose that <sup>∂</sup>*<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))/*∂*2*<sup>τ</sup>* <sup>∈</sup> *<sup>C</sup>*[0, *<sup>t</sup>*)*, for some suitable t, then we have*

*Proof*. The idea of this lemma's proof is similar to the above lemmas, namely

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1 *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*)) <sup>−</sup> *<sup>f</sup>*

2

*α*+2

−2(2*t*

*<sup>α</sup>*+<sup>2</sup> *<sup>n</sup>* <sup>−</sup> *<sup>t</sup>*

*α*(*α* + 1)(*α* + 2)

*<sup>n</sup>*−*j*−1) + *<sup>h</sup>*

*α*+2 *<sup>n</sup>*−*<sup>j</sup>* <sup>−</sup> *<sup>t</sup>*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1 *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*)) <sup>−</sup> *<sup>f</sup>*

$$\left| \int\_{0}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} D\_{\*}^{a} y(\tau) d\tau - \sum\_{k=0}^{n+1} a\_{j,n+1} D\_{\*}^{a} y(t\_j) \right| \leq C\_{2} t\_{n+1}^{\gamma\_2} h^{\delta\_2},\tag{46}$$

*with some γ*1, *γ*<sup>2</sup> ≥ 0 *and δ*1, *δ*<sup>2</sup> ≥ 0*. Then, for some suitably chosen T* ≥ 0*, we have*

$$\max\_{0 \le j \le N} \left| y(t\_j) - y\_h(t\_j) \right| = O(h^q), \tag{47}$$

*with q* = min{*δ*<sup>1</sup> + *α*, *δ*2} *and N* = �*T*/*h*�*.*

**Theorem 1.** *For the fractional initial problem (1), if D<sup>α</sup>* <sup>∗</sup> *<sup>y</sup>*(*t*) <sup>∈</sup> *<sup>C</sup>*2[0, *<sup>T</sup>*] *for some suitable T, then the convergent order of our algorithm with the predictor and corrector formulae (16) and (17) gives* max*j*=0,1,··· ,*n*+<sup>1</sup> <sup>|</sup>*y*(*tj*) <sup>−</sup> *yh*(*tj*)<sup>|</sup> <sup>=</sup> *<sup>O</sup>*(*h*min{2,1+2*α*})*.*

*Proof*. Noting

$$\left| \int\_{t\_n}^{t\_{n+1}} (t\_{n+1} - \tau)^{a-1} (\boldsymbol{g}(\tau) - \boldsymbol{g}(t\_n)) d\tau \right| \le \|\boldsymbol{g}'\|\_{\infty} \frac{h^{a+1}}{a(a+1)},\tag{48}$$

and applying lemmas 2, 4, 6, we have the above result.

**Theorem 2.** *When <sup>α</sup>* <sup>&</sup>gt; <sup>1</sup>*, if <sup>∂</sup>*<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))/*∂*2*<sup>τ</sup>* <sup>∈</sup> *<sup>C</sup>*[0, *<sup>t</sup>*) *for some suitable t, then the local truncation error of our algorithm with the predictor and corrector formulae (26)-(27) (α* ∈ (1, 2)*) and (28)-(29) (<sup>α</sup>* <sup>∈</sup> (2, <sup>∞</sup>)*) is O*(*h*3)*, and the convergent order is 2, i.e.,* max*j*=0,1,··· ,*n*+<sup>1</sup> <sup>|</sup>*y*(*tj*) <sup>−</sup> *yh*(*tj*)<sup>|</sup> <sup>=</sup> *<sup>O</sup>*(*h*2)*.*

and the similar idea to above proof, its sketch proof is given as

Γ(*α* + 2)

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1

Γ(*α* + 2)

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1

when *α* > 1, so the convergent order is 2.

*underlying quadrature rule on which it is based.*

*<sup>n</sup>*) + *<sup>y</sup>*(*tn*) + <sup>1</sup>

*n* ∑ *j*=0

*n* ∑ *j*=0

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* <sup>−</sup> *<sup>h</sup><sup>α</sup>*

Γ(*α*)

*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*))

Numerical Schemes for Fractional Ordinary Differential Equations 369

 *tn*<sup>+</sup><sup>1</sup> *tn*

*f*(*τ*, *y*(*τ*))*dτ*

*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*))

*<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*)) <sup>−</sup> *<sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*))

We have proved that the local truncation error of our algorithm (26)-(27) and (28)-(29) is *O*(*h*3)

**Lemma 7.** *(Ford Simpson [6,Theorem 1]) The nested mesh scheme preserves the order of the*

**Theorem 3.** *When <sup>α</sup>* <sup>&</sup>gt; <sup>1</sup>*, if <sup>∂</sup>*<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))/*∂*2*<sup>τ</sup>* <sup>∈</sup> *<sup>C</sup>*[0, *<sup>t</sup>*) *for some suitable t, then the local truncation error of our algorithm with the predictor and corrector formulae (32)-(33) (<sup>α</sup>* <sup>∈</sup> (1, 2)*) is O*(*h*3) *and*

The comparison of the local truncation error, convergent order and arithmetic complexity for different predictor-corrector schemes are presented in the following table. So far, our

In this section we present two numerical examples to illustrate the performance of our proposed predictor-corrector schemes. We show the order of convergence in the the absolute

Because of Theorem 2, Lemma 7, and the analysis in above section, we have

convergence results are obtained under the smoothness assumptions of *D<sup>α</sup>*

obviously depend on the smoothness properties of the solution *y*(*t*) (Deng, 2010).

*the convergent order is 2, i.e.,* max*j*=0,1,··· ,*n*+<sup>1</sup> <sup>|</sup>*y*(*tj*) <sup>−</sup> *yh*(*tj*)<sup>|</sup> <sup>=</sup> *<sup>O</sup>*(*h*2)*.*

 

 −

<sup>Γ</sup>(*<sup>α</sup>* <sup>+</sup> <sup>2</sup>) *<sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*))

*<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* <sup>−</sup> *<sup>h</sup><sup>α</sup>*

*dτ* 

*<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* <sup>−</sup> *<sup>h</sup><sup>α</sup>*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>*

 �*α*�−<sup>1</sup> ∑ *k*=1

Γ(*α* + 2)

Γ(*α* + 2)

*y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k <sup>n</sup>*) + *y*(*tn*)

*n* ∑ *j*=0

*n* ∑ *j*=0 *aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*))

*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*))

<sup>∗</sup> *<sup>y</sup>*(*t*) and *<sup>f</sup>* , which

 

 

+

= 

+ 1 Γ(*α*)

+

= 

+ 1 Γ(*α*)

≤ 1 Γ(*α*)

+ 1 Γ(*α*)

*y*(*tn*+1) −

*hα*

 �*α*�−<sup>1</sup> ∑ *k*=1

*hα*

 1 Γ(*α*)

 *tn* 0 

 �*α*�−<sup>1</sup> ∑ *k*=1

> *y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k*

*y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k <sup>n</sup>*) + *y*(*tn*)

<sup>Γ</sup>(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>) *<sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*)) + *<sup>h</sup><sup>α</sup>*

<sup>Γ</sup>(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>) *<sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*)) + *<sup>h</sup><sup>α</sup>*

 *tn*<sup>+</sup><sup>1</sup> *tn*

 *tn* 0 

 *tn*<sup>+</sup><sup>1</sup> *tn*

 *tn* 0 

≤···≤ *Ch*min{*α*+1,3}.

**4. Numerical results**

*Proof*. This proof will be used based on mathematical induction. In view of the given initial condition, the induction basis *j* = 0 is presupposed, it has convergent order 2. Now, assuming that the convergent order is 2 for *j* = 0, 1, ··· , *k*, *k* ≤ *n*, we have the local truncation error

 *y*(*tn*+1) − �*α*�−<sup>1</sup> ∑ *k*=1 *y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k <sup>n</sup>*) + *y*(*tn*) + *hα* Γ(*α* + 2) *α f*(*tn*, *y*(*tn*)) + *f*(*tn*+1, *y<sup>P</sup> <sup>h</sup>* (*tn*+1)) + *hα* Γ(*α* + 2) *n* ∑ *j*=0 *aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*)) = �*α*�−<sup>1</sup> ∑ *k*=1 *y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k <sup>n</sup>*) + *<sup>y</sup>*(*tn*) + <sup>1</sup> Γ(*α*) *tn*<sup>+</sup><sup>1</sup> *tn* (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* + 1 Γ(*α*) *tn* 0 (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *f*(*τ*, *y*(*τ*))*dτ* − �*α*�−<sup>1</sup> ∑ *k*=1 *y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k <sup>n</sup>*) + *y*(*tn*) + *hα* Γ(*α* + 2) *α f*(*tn*, *y*(*tn*)) + *f*(*tn*+1, *y<sup>P</sup> <sup>h</sup>* (*tn*+1)) + *hα* Γ(*α* + 2) *n* ∑ *j*=0 *aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*)) = 1 Γ(*α*) *tn*<sup>+</sup><sup>1</sup> *tn* (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* <sup>−</sup> *<sup>h</sup><sup>α</sup>* Γ(*α* + 2) *<sup>α</sup> <sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*)) + *<sup>f</sup>*(*tn*+1, *<sup>y</sup>*(*tn*+1)) + *hα* Γ(*α* + 2) *<sup>f</sup>*(*tn*+1, *<sup>y</sup>*(*tn*+1)) <sup>−</sup> *<sup>f</sup>*(*tn*+1, *<sup>y</sup><sup>P</sup> <sup>h</sup>* (*tn*+1)) + 1 Γ(*α*) *tn* 0 (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* <sup>−</sup> *<sup>h</sup><sup>α</sup>* Γ(*α* + 2) *n* ∑ *j*=0 *aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *<sup>y</sup>*(*tj*)) + *h<sup>α</sup>* Γ(*α* + 2) *n* ∑ *j*=0 *aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *<sup>y</sup>*(*tj*)) <sup>−</sup> *<sup>h</sup><sup>α</sup>* Γ(*α* + 2) *n* ∑ *j*=0 *aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*)) .

Then we have

$$\begin{aligned} & \left| y(t\_{n+1}) - \left\{ \sum\_{k=1}^{\lceil a \rceil -1} \frac{y\_0^{(k)}}{k!} (t\_{n+1}^k - t\_n^k) + y(t\_n) \right. \right. \\ & \left. + \frac{h^a}{\Gamma(a+2)} \left( a f(t\_n, y(t\_n)) + f(t\_{n+1}, y\_h^P(t\_{n+1})) \right) + \frac{h^a}{\Gamma(a+2)} \sum\_{j=0}^n \tilde{a}\_{j,n} f(t\_j, y\_h(t\_j)) \right\} \right| \\ & \leq C\_1 h^{a+2} + \frac{aL}{\Gamma(a+2)} h^{a+\min\{a+1, 3\}} + \frac{aL}{\Gamma(a+2)} h^{a+\min\{a+2, 3\}} + \left| \left( -\frac{1}{2} h^a + (z\_\*)^{a-1} h \right) \right| Lh^2 \\ & \leq C h^3 . \end{aligned}$$

where *z*<sup>∗</sup> ∈ (*tn*, *tn*+1), Lemmas 4 and 5 in the above proof are used, and we also utilize the result <sup>|</sup>*y*(*tn*+1) <sup>−</sup> *<sup>y</sup><sup>P</sup> <sup>h</sup>* (*tn*+1)<sup>|</sup> <sup>=</sup> *<sup>O</sup>*(*h*min{*α*+1,3}) which can be proved by using Lemmas 4 and 5 14 Will-be-set-by-IN-TECH

*Proof*. This proof will be used based on mathematical induction. In view of the given initial condition, the induction basis *j* = 0 is presupposed, it has convergent order 2. Now, assuming that the convergent order is 2 for *j* = 0, 1, ··· , *k*, *k* ≤ *n*, we have the local truncation error

*<sup>h</sup>* (*tn*+1))

Γ(*α*)

*<sup>h</sup>* (*tn*+1))

+

*f*(*τ*, *y*(*τ*))*dτ*

+

*<sup>h</sup>* (*tn*+1))

*n* ∑ *j*=0 Γ(*α* + 2)

 *tn*<sup>+</sup><sup>1</sup> *tn*

*hα* Γ(*α* + 2)

> −

*hα* Γ(*α* + 2)

*<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* <sup>−</sup> *<sup>h</sup><sup>α</sup>*

*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*))

*hα* Γ(*α* + 2)

*hα*+min{*α*+2,3} +

*<sup>h</sup>* (*tn*+1)<sup>|</sup> <sup>=</sup> *<sup>O</sup>*(*h*min{*α*+1,3}) which can be proved by using Lemmas 4 and 5

*n* ∑ *j*=0

> − 1 2

*n* ∑ *j*=0

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>*

 �*α*�−<sup>1</sup> ∑ *k*=1

Γ(*α* + 2)

 .

*n* ∑ *j*=0 *aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*))

*y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k <sup>n</sup>*) + *y*(*tn*)

*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*))

*<sup>α</sup> <sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*)) + *<sup>f</sup>*(*tn*+1, *<sup>y</sup>*(*tn*+1))

*n* ∑ *j*=0

*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*))

 

*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *<sup>y</sup>*(*tj*))

 

*<sup>h</sup><sup>α</sup>* + (*z*∗)*α*−1*<sup>h</sup>*

 *Lh*<sup>2</sup>

 

+

= 

+ 1 Γ(*α*)

+

= 

+

+ 1 Γ(*α*)

+

 

+

*y*(*tn*+1) −

*hα* Γ(*α* + 2)

> �*α*�−<sup>1</sup> ∑ *k*=1

*hα* Γ(*α* + 2)

> 1 Γ(*α*)

*hα* Γ(*α* + 2)

 *h<sup>α</sup>* Γ(*α* + 2)

Then we have

*y*(*tn*+1) −

*hα* Γ(*α* + 2)

<sup>≤</sup> *<sup>C</sup>*1*hα*+<sup>2</sup> <sup>+</sup>

result <sup>|</sup>*y*(*tn*+1) <sup>−</sup> *<sup>y</sup><sup>P</sup>*

<sup>≤</sup> *Ch*3,

 *tn* 0 

 �*α*�−<sup>1</sup> ∑ *k*=1

*y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k*

 *tn* 0 

> *n* ∑ *j*=0

 �*α*�−<sup>1</sup> ∑ *k*=1

*y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k <sup>n</sup>*) + *y*(*tn*)

*αL* Γ(*α* + 2)

 *tn*<sup>+</sup><sup>1</sup> *tn*

*y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k <sup>n</sup>*) + *y*(*tn*)

*α f*(*tn*, *y*(*tn*)) + *f*(*tn*+1, *y<sup>P</sup>*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup>

*α f*(*tn*, *y*(*tn*)) + *f*(*tn*+1, *y<sup>P</sup>*

*<sup>f</sup>*(*tn*+1, *<sup>y</sup>*(*tn*+1)) <sup>−</sup> *<sup>f</sup>*(*tn*+1, *<sup>y</sup><sup>P</sup>*

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup>

*aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *<sup>y</sup>*(*tj*)) <sup>−</sup> *<sup>h</sup><sup>α</sup>*

*α f*(*tn*, *y*(*tn*)) + *f*(*tn*+1, *y<sup>P</sup>*

*hα*+min{*α*+1,3} +

*<sup>n</sup>*) + *<sup>y</sup>*(*tn*) + <sup>1</sup>

(*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* <sup>−</sup> *<sup>h</sup><sup>α</sup>*

Γ(*α* + 2)

*<sup>h</sup>* (*tn*+1))

*αL* Γ(*α* + 2)

where *z*<sup>∗</sup> ∈ (*tn*, *tn*+1), Lemmas 4 and 5 in the above proof are used, and we also utilize the

+

and the similar idea to above proof, its sketch proof is given as

 *y*(*tn*+1) − �*α*�−<sup>1</sup> ∑ *k*=1 *y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k <sup>n</sup>*) + *y*(*tn*) + *hα* <sup>Γ</sup>(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>) *<sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*)) + *<sup>h</sup><sup>α</sup>* Γ(*α* + 2) *n* ∑ *j*=0 *aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*)) = �*α*�−<sup>1</sup> ∑ *k*=1 *y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k <sup>n</sup>*) + *<sup>y</sup>*(*tn*) + <sup>1</sup> Γ(*α*) *tn*<sup>+</sup><sup>1</sup> *tn* (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* + 1 Γ(*α*) *tn* 0 (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1 *f*(*τ*, *y*(*τ*))*dτ* − �*α*�−<sup>1</sup> ∑ *k*=1 *y* (*k*) 0 *<sup>k</sup>*! (*<sup>t</sup> k <sup>n</sup>*+<sup>1</sup> − *t k <sup>n</sup>*) + *y*(*tn*) + *hα* <sup>Γ</sup>(*<sup>α</sup>* <sup>+</sup> <sup>1</sup>) *<sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*)) + *<sup>h</sup><sup>α</sup>* Γ(*α* + 2) *n* ∑ *j*=0 *aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*)) = 1 Γ(*α*) *tn*<sup>+</sup><sup>1</sup> *tn* (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* <sup>−</sup> *<sup>h</sup><sup>α</sup>* <sup>Γ</sup>(*<sup>α</sup>* <sup>+</sup> <sup>2</sup>) *<sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*)) + 1 Γ(*α*) *tn* 0 (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1 *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* <sup>−</sup> *<sup>h</sup><sup>α</sup>* Γ(*α* + 2) *n* ∑ *j*=0 *aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*)) ≤ 1 Γ(*α*) *tn*<sup>+</sup><sup>1</sup> *tn* (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1 *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*)) <sup>−</sup> *<sup>f</sup>*(*tn*, *<sup>y</sup>*(*tn*)) *dτ* + 1 Γ(*α*) *tn* 0 (*tn*+<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*tn* <sup>−</sup> *<sup>τ</sup>*)*α*−1 *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))*d<sup>τ</sup>* <sup>−</sup> *<sup>h</sup><sup>α</sup>* Γ(*α* + 2) *n* ∑ *j*=0 *aj*,*<sup>n</sup> <sup>f</sup>*(*tj*, *yh*(*tj*)) ≤···≤ *Ch*min{*α*+1,3}.

We have proved that the local truncation error of our algorithm (26)-(27) and (28)-(29) is *O*(*h*3) when *α* > 1, so the convergent order is 2.

**Lemma 7.** *(Ford Simpson [6,Theorem 1]) The nested mesh scheme preserves the order of the underlying quadrature rule on which it is based.*

Because of Theorem 2, Lemma 7, and the analysis in above section, we have

**Theorem 3.** *When <sup>α</sup>* <sup>&</sup>gt; <sup>1</sup>*, if <sup>∂</sup>*<sup>2</sup> *<sup>f</sup>*(*τ*, *<sup>y</sup>*(*τ*))/*∂*2*<sup>τ</sup>* <sup>∈</sup> *<sup>C</sup>*[0, *<sup>t</sup>*) *for some suitable t, then the local truncation error of our algorithm with the predictor and corrector formulae (32)-(33) (<sup>α</sup>* <sup>∈</sup> (1, 2)*) is O*(*h*3) *and the convergent order is 2, i.e.,* max*j*=0,1,··· ,*n*+<sup>1</sup> <sup>|</sup>*y*(*tj*) <sup>−</sup> *yh*(*tj*)<sup>|</sup> <sup>=</sup> *<sup>O</sup>*(*h*2)*.*

The comparison of the local truncation error, convergent order and arithmetic complexity for different predictor-corrector schemes are presented in the following table. So far, our convergence results are obtained under the smoothness assumptions of *D<sup>α</sup>* <sup>∗</sup> *<sup>y</sup>*(*t*) and *<sup>f</sup>* , which obviously depend on the smoothness properties of the solution *y*(*t*) (Deng, 2010).

#### **4. Numerical results**

In this section we present two numerical examples to illustrate the performance of our proposed predictor-corrector schemes. We show the order of convergence in the the absolute

*h α* = 0.1 Order *α* = 0.5 Order *α* = 0.9 Order 1/10 3.64×10−<sup>1</sup> - 3.55×10−<sup>2</sup> - 1.07×10−<sup>2</sup> - 1/20 1.70×10−<sup>1</sup> 1.10 8.79×10−<sup>3</sup> 2.01 2.31×10−<sup>3</sup> 2.21 1/40 7.13×10−<sup>2</sup> 1.26 2.16×10−<sup>3</sup> 2.03 5.21×10−<sup>4</sup> 2.15 1/80 2.88×10−<sup>2</sup> 1.31 5.31×10−<sup>4</sup> 2.02 1.22×10−<sup>4</sup> 2.09 1/160 1.15×10−<sup>2</sup> 1.32 1.31×10−<sup>4</sup> 2.02 2.94×10−<sup>5</sup> 2.06 1/320 4.64×10−<sup>3</sup> 1.31 3.24×10−<sup>5</sup> 2.02 7.18×10−<sup>6</sup> 2.03 1/640 1.88×10−<sup>3</sup> 1.30 8.03×10−<sup>6</sup> 2.01 1.77×10−<sup>6</sup> 2.01 TOC 1.20 2.00 2.00 Table 2. Absolute errors and convergence orders for predictor-corrector schemes (16)-(17) at

Numerical Schemes for Fractional Ordinary Differential Equations 371

*h α* = 1.25 Order *α* = 1.5 Order *α* = 1.85 Order 1/10 8.48×10−<sup>3</sup> - 8.58×10−<sup>3</sup> - 9.04×10−<sup>3</sup> - 1/20 2.03×10−<sup>3</sup> 2.06 2.12×10−<sup>3</sup> 2.02 2.25×10−<sup>3</sup> 2.00 1/40 5.00×10−<sup>4</sup> 2.02 5.28×10−<sup>4</sup> 2.00 5.63×10−<sup>4</sup> 2.00 1/80 1.24×10−<sup>4</sup> 2.01 1.32×10−<sup>4</sup> 2.00 1.41×10−<sup>4</sup> 2.00 1/160 3.10×10−<sup>5</sup> 2.00 3.30×10−<sup>5</sup> 2.00 3.52×10−<sup>5</sup> 2.00 1/320 7.75×10−<sup>6</sup> 2.00 8.24×10−<sup>6</sup> 2.00 8.79×10−<sup>6</sup> 2.00 1/640 1.94×10−<sup>6</sup> 2.00 2.06×10−<sup>6</sup> 2.00 2.20×10−<sup>6</sup> 2.00 TOC 2.00 2.00 2.00 Table 3. Absolute errors and convergence orders for predictor-corrector schemes (16)-(17) at

Table 4 shows the numerical errors at time *t* = 1 between the exact solution and the predictor-corrector schemes (16)-(17) for (53) with different *α* ∈ (0, 1) for various step sizes. Again we numerically verify that the rate of convergence for predictor-corrector schemes (16)-(17) is *O*(*τ*min{2,1+2*α*}). From Table 4, it can be seen that both the convergence orders

*h α* = 0.1 Order *α* = 0.5 Order *α* = 1.25 Order 1/10 1.04×10−<sup>1</sup> - 9.27×10−<sup>3</sup> - 7.96×10−<sup>3</sup> - 1/20 4.66×10−<sup>2</sup> 1.16 2.29×10−<sup>3</sup> 2.02 2.08×10−<sup>3</sup> 1.94 1/40 1.87×10−<sup>2</sup> 1.32 5.87×10−<sup>4</sup> 1.96 5.53×10−<sup>4</sup> 1.91 1/80 7.39×10−<sup>3</sup> 1.34 1.56×10−<sup>4</sup> 1.91 1.49×10−<sup>4</sup> 1.89 1/160 2.94×10−<sup>3</sup> 1.33 4.31×10−<sup>5</sup> 1.86 4.07×10−<sup>5</sup> 1.87 1/320 1.18×10−<sup>3</sup> 1.31 1.23×10−<sup>5</sup> 1.81 1.12×10−<sup>5</sup> 1.86 TOC 1.20 2.00 2.00 Table 4. Absolute errors and convergence orders for predictor-corrector schemes (16)-(17) at

time *t* = 1 with different 0 < *α* < 1.

time *t* = 1 with different 1 < *α* < 2.

time *t* = 1 with different *α*.


Table 1. Comparison of different predictor-corrector schemes.

error. And the convergence order is measured by

$$\text{Order} = \log\_2\left(\frac{error(h)}{error(h/2)}\right)$$

where *error*(*h*) is the absolute error |*y*(*t*) − *yh*(*t*)| with the step *h*.

**Example 1.** *Consider the following fractional differential equation*

$$D\_\*^{\mathfrak{a}}y(t) = \frac{\Gamma(5)}{\Gamma(5-\mathfrak{a})}t^{4-\mathfrak{a}} - y(t) + t^4, \quad \mathfrak{a} \in (0,2), \tag{49}$$

*with the initial conditions*

$$y(0) = 0, \ a \in (0, 1), \tag{50}$$

*or*

$$y(0) = 0, \ y'(0) = 0, \ \mathfrak{a} \in (1, 2), \tag{51}$$

*note that the exact solution to this problem is*

$$y(t) = t^4.\tag{52}$$

The numerical results for the predictor-corrector scheme (16)-(17) at time *t* = 1 with different steps and different *α* are reported in Tables 2 and 3. Table 1 shows the numerical errors at time *t* = 1 between the exact solution and the numerical solution of the predictor-corrector schemes (16)-(17) for (49) with different *α* ∈ (0, 1) for various step sizes. It is seen that the rate of convergence of the numerical results is of order *O*(*τ*min{2,1+2*α*}) for the predictor-corrector schemes (16)-(17). Tables 2 and 3 show the ratio of the error reduction as the grid is refined. The last row states the theoretical orders of convergence (abbreviated as TOC), which are the results we theoretically prove in the above theorems. From Tables 2 and 3, it can be noted that the numerical results are in excellent agreement with the theoretical ones given in Theorem 1.

**Example 2.** *Now, consider the following fractional differential equation (Diethelm et al., 2004)*

$$D\_{\*}^{\mathfrak{a}}y(t) = \begin{cases} \frac{2}{\Gamma(3-\mathfrak{a})}t^{2-\mathfrak{a}} - \frac{1}{\Gamma(2-\mathfrak{a})}t^{1-\mathfrak{a}} - y(t) + t^2 - t, & \text{if} \quad 0 < \mathfrak{a} < 1, \\\\ \frac{2}{\Gamma(3-\mathfrak{a})}t^{2-\mathfrak{a}} - y(t) + t^2 - t, & \text{if} \quad \mathfrak{a} \ge 1. \end{cases} \tag{53}$$

*with the initial conditions*

$$y(0) = 0, \ a \in (0, 1), \tag{54}$$

*or*

$$y(0) = 0, \ y'(0) = -1, \ a \in (1, 2), \tag{55}$$

*the exact solution to this problem is*

$$y(t) = t^2 - t.\tag{56}$$

16 Will-be-set-by-IN-TECH

Schemes Local truncation error Convergent order Arithmetic complexity

� *error*(*h*) *error*(*h*/2)

<sup>4</sup>−*<sup>α</sup>* <sup>−</sup> *<sup>y</sup>*(*t*) + *<sup>t</sup>*

�

*y*(0) = 0, *α* ∈ (0, 1), (50)

(0) = 0, *α* ∈ (1, 2), (51)

4. (52)

<sup>2</sup> <sup>−</sup> *<sup>t</sup>*, *if* <sup>0</sup> <sup>&</sup>lt; *<sup>α</sup>* <sup>&</sup>lt; 1,

(53)

4, *<sup>α</sup>* <sup>∈</sup> (0, 2), (49)

(11),(12) *O*(*h*3) *O*(*h*min{2,1+*α*}) *O*(*h*−2) (16),(17) *O*(*h*3) *O*(*h*min{2,1+2*α*}) *O*(*h*−2) (26)-(29) *O*(*h*3) *O*(*h*2) *O*(*h*−2) (32),(33) *O*(*h*3) *O*(*h*2) *O*(*h*−<sup>1</sup> log(*h*−1))

Order = log2

Γ(5 − *α*)

*y*(0) = 0, *y*�

*t*

*y*(*t*) = *t*

The numerical results for the predictor-corrector scheme (16)-(17) at time *t* = 1 with different steps and different *α* are reported in Tables 2 and 3. Table 1 shows the numerical errors at time *t* = 1 between the exact solution and the numerical solution of the predictor-corrector schemes (16)-(17) for (49) with different *α* ∈ (0, 1) for various step sizes. It is seen that the rate of convergence of the numerical results is of order *O*(*τ*min{2,1+2*α*}) for the predictor-corrector schemes (16)-(17). Tables 2 and 3 show the ratio of the error reduction as the grid is refined. The last row states the theoretical orders of convergence (abbreviated as TOC), which are the results we theoretically prove in the above theorems. From Tables 2 and 3, it can be noted that the numerical results are in excellent agreement with the theoretical ones given in Theorem 1.

**Example 2.** *Now, consider the following fractional differential equation (Diethelm et al., 2004)*

*y*(*t*) = *t*

<sup>1</sup>−*<sup>α</sup>* <sup>−</sup> *<sup>y</sup>*(*t*) + *<sup>t</sup>*

<sup>2</sup> <sup>−</sup> *<sup>t</sup>*, *if <sup>α</sup>* <sup>≥</sup> 1.

*y*(0) = 0, *α* ∈ (0, 1), (54)

(0) = −1, *α* ∈ (1, 2), (55)

<sup>2</sup> <sup>−</sup> *<sup>t</sup>*. (56)

<sup>Γ</sup>(2−*α*)*<sup>t</sup>*

<sup>2</sup>−*<sup>α</sup>* <sup>−</sup> <sup>1</sup>

*y*(0) = 0, *y*�

<sup>2</sup>−*<sup>α</sup>* <sup>−</sup> *<sup>y</sup>*(*t*) + *<sup>t</sup>*

Table 1. Comparison of different predictor-corrector schemes.

where *error*(*h*) is the absolute error |*y*(*t*) − *yh*(*t*)| with the step *h*. **Example 1.** *Consider the following fractional differential equation*

<sup>∗</sup> *<sup>y</sup>*(*t*) = <sup>Γ</sup>(5)

error. And the convergence order is measured by

*Dα*

*note that the exact solution to this problem is*

*with the initial conditions*

*Dα* <sup>∗</sup> *<sup>y</sup>*(*t*) =

*the exact solution to this problem is*

*with the initial conditions*

*or*

⎧ ⎪⎪⎨

2 <sup>Γ</sup>(3−*α*)*<sup>t</sup>*

2 <sup>Γ</sup>(3−*α*)*<sup>t</sup>*

⎪⎪⎩

*or*


Table 2. Absolute errors and convergence orders for predictor-corrector schemes (16)-(17) at time *t* = 1 with different 0 < *α* < 1.


Table 3. Absolute errors and convergence orders for predictor-corrector schemes (16)-(17) at time *t* = 1 with different 1 < *α* < 2.

Table 4 shows the numerical errors at time *t* = 1 between the exact solution and the predictor-corrector schemes (16)-(17) for (53) with different *α* ∈ (0, 1) for various step sizes. Again we numerically verify that the rate of convergence for predictor-corrector schemes (16)-(17) is *O*(*τ*min{2,1+2*α*}). From Table 4, it can be seen that both the convergence orders


Table 4. Absolute errors and convergence orders for predictor-corrector schemes (16)-(17) at time *t* = 1 with different *α*.

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Table 5. Error behavior versus the variation of *p* and T (the definition of *p* and T are given in (3)) at time *t* = 50 with exact (analytical) value *y*(50) = 2450, fractional order *α* = 1.5, step length *h* = 1/80.

and the errors of the improved predictor-corrector scheme (16)-(17) are improved significantly compared with the Tables 3 and 4 given in (Diethelm et al., 2004).

Table 5 shows the computing values, the absolute numerical errors, and the relative numerical errors by using the scheme (32) and (33) to compute Example 2 for different values of *p* and T defined in (3). According to the numerical results we can see that the computing errors are generally acceptable in engineering when the computational cost is greatly minimized, especially the computing error is not sensitive to the value of *p*. On the other hand, this numerical example also illuminates that the algorithm is numerically stable.

#### **5. Conclusions**

We briefly review the numerical techniques for efficiently solving fractional ordinary differential equations. The possible improvements for the fractional predictor-corrector algorithm are presented. Even though the algorithm is designed for scalar fractional ordinary differential equation, it can be easily extended to the systems. In fact, based on this algorithm, we have simulated the dynamics of fractional systems (Deng, 2007c;d). In addition, the fractional predictor-corrector algorithm combining with the scheme of generating stochastic variables also works well for the stochastic fractional ordinary differential equations (Deng & Barkai, 2009).

#### **6. Acknowledgments**

This research was partly supported by the Program for New Century Excellent Talents in University under Grant No. NCET-09-0438, the National Natural Science Foundation of China under Grant No. 10801067, and the Fundamental Research Funds for the Central Universities under Grant No. lzujbky-2010-63 and No. lzujbky-2012-k26.

#### **7. References**

18 Will-be-set-by-IN-TECH

*p* T Computing value Absolute errors Relative error (%) 1 2448.8 1.2 0.0489 4 2465.6 15.6 0.6367 3 2466.0 16.0 0.6530 2 2432.7 18.0 0.7347 4 2467.2 17.2 0.7020 3 2467.0 17.0 0.6939 2 2466.1 16.1 0.6571 4 2469.3 19.3 0.7878 3 2469.1 19.1 0.7796 2 2467.0 17.0 0.6939

Table 5. Error behavior versus the variation of *p* and T (the definition of *p* and T are given in (3)) at time *t* = 50 with exact (analytical) value *y*(50) = 2450, fractional order *α* = 1.5, step

and the errors of the improved predictor-corrector scheme (16)-(17) are improved significantly

Table 5 shows the computing values, the absolute numerical errors, and the relative numerical errors by using the scheme (32) and (33) to compute Example 2 for different values of *p* and T defined in (3). According to the numerical results we can see that the computing errors are generally acceptable in engineering when the computational cost is greatly minimized, especially the computing error is not sensitive to the value of *p*. On the other hand, this

We briefly review the numerical techniques for efficiently solving fractional ordinary differential equations. The possible improvements for the fractional predictor-corrector algorithm are presented. Even though the algorithm is designed for scalar fractional ordinary differential equation, it can be easily extended to the systems. In fact, based on this algorithm, we have simulated the dynamics of fractional systems (Deng, 2007c;d). In addition, the fractional predictor-corrector algorithm combining with the scheme of generating stochastic variables also works well for the stochastic fractional ordinary differential equations (Deng &

This research was partly supported by the Program for New Century Excellent Talents in University under Grant No. NCET-09-0438, the National Natural Science Foundation of China under Grant No. 10801067, and the Fundamental Research Funds for the Central Universities

compared with the Tables 3 and 4 given in (Diethelm et al., 2004).

under Grant No. lzujbky-2010-63 and No. lzujbky-2012-k26.

numerical example also illuminates that the algorithm is numerically stable.

length *h* = 1/80.

**5. Conclusions**

Barkai, 2009).

**6. Acknowledgments**


Hilfer, H. (2000). *Applications of Fractional Calculus in Physics*, World Scientific Press, Singapore.


**17** 

*México* 

**Biorthogonal Decomposition for Wide-Area** 

Characterization of spatial and temporal changes in the dynamic pattern that arise when a wide-area system is subjected to a perturbation becomes a significant problem of great theoretical and practical importance. The computation time required to solve large analytical models might become prohibitive for practical systems. Thus, to reduce the complexity of the problem, several simplifications have been commonly used which may result in a poor characterization of global system behaviour. Therefore, a great deal of attention has been paid to identify and to characterize oscillatory activity in large interconnected systems through use of wide-area monitoring schemes such as global positioning systems (GPS) based in multiple phasor measurements units (PMUs) (Messina, et al., 2010). When simultaneously measured responses throughout an interconnected system are available, modal behaviour should be extracted using correlation techniques rather that individual analysis of the system response. This provides a global picture on the system behaviour and enables statistical characterization of the observed phenomena. The problem of selecting the most significant modes is of considerable interest and it has been studied intensively for several researchers (Esquivel & Messina, 2008; Hannachi, et al., 2007; Hasselmann, 1988; Holmes, et al., 1996; Kwasniok, 1996, 2007). Statistical models have been widely used in many engineering and science applications for the analysis of space-time varying system response from measured data (Aubry, et al., 1990; Dankowicz, et al., 1996; Delsole, 2001;Lezama et al., 2009; Messina, et al., 2010, 2011; Spletzer, et al., 2010); i.e., unsteady fluid flow (Terradas, et al., 2004), turbulence (Hannachi, et al. 2007; Leonardi, et al., 2002; Susanto, et al., 1997; Toh, 1987), optimal control (Wallaschek, 1988), structural dynamics (Feeny & Kappagantu, 1998; Han & Feeny, 2003; Holmes, et al., 1996; Marrifield & Guza,1990; Oey, 2007), heat transfer (Barnett, 1983; Kaihatu, et al., 1997) and system identification have been reported (Esquivel, et al., 2009; Feeny, 2008; Hasselmann, 1988; Horel, 1984; Kwasniok, 1996, 2007). These methodologies use statistical techniques such as, empirical orthogonal function (EOF) (Esquivel & Messina, 2008), principal interaction pattern (PIP) (Achatz, et al., 1995), principal oscillation pattern (POP) (Hasselmann, 1988),

**1. Introduction** 

**Wave Motion Monitoring Using** 

P. Esquivel1, D. Cabuto1, V. Sanchez2 and F. Chan2

*1Technological Institute of Tepic, Electrical and Electronics Engineering Division, Nayarit, 2University of Quintana Roo, Sciences and Engineering Division, Quintana Roo* 

**Statistical Models** 


### **Biorthogonal Decomposition for Wide-Area Wave Motion Monitoring Using Statistical Models**

P. Esquivel1, D. Cabuto1, V. Sanchez2 and F. Chan2 *1Technological Institute of Tepic, Electrical and Electronics Engineering Division, Nayarit, 2University of Quintana Roo, Sciences and Engineering Division, Quintana Roo México* 

#### **1. Introduction**

20 Will-be-set-by-IN-TECH

374 Numerical Modelling

Zaslavsky, G.M. (2002). Chaos, fractional kinetic, and anomalous transport, *Phys. Rep.*, Vol.

Mainrdi, F. (2008). *Fractional calculus and waves in linear viscoelasticity: an introduction to*

*mathematical models*. Imperial College Press, London.

371, No. 6, pp. 461-580.

Characterization of spatial and temporal changes in the dynamic pattern that arise when a wide-area system is subjected to a perturbation becomes a significant problem of great theoretical and practical importance. The computation time required to solve large analytical models might become prohibitive for practical systems. Thus, to reduce the complexity of the problem, several simplifications have been commonly used which may result in a poor characterization of global system behaviour. Therefore, a great deal of attention has been paid to identify and to characterize oscillatory activity in large interconnected systems through use of wide-area monitoring schemes such as global positioning systems (GPS) based in multiple phasor measurements units (PMUs) (Messina, et al., 2010). When simultaneously measured responses throughout an interconnected system are available, modal behaviour should be extracted using correlation techniques rather that individual analysis of the system response. This provides a global picture on the system behaviour and enables statistical characterization of the observed phenomena. The problem of selecting the most significant modes is of considerable interest and it has been studied intensively for several researchers (Esquivel & Messina, 2008; Hannachi, et al., 2007; Hasselmann, 1988; Holmes, et al., 1996; Kwasniok, 1996, 2007). Statistical models have been widely used in many engineering and science applications for the analysis of space-time varying system response from measured data (Aubry, et al., 1990; Dankowicz, et al., 1996; Delsole, 2001;Lezama et al., 2009; Messina, et al., 2010, 2011; Spletzer, et al., 2010); i.e., unsteady fluid flow (Terradas, et al., 2004), turbulence (Hannachi, et al. 2007; Leonardi, et al., 2002; Susanto, et al., 1997; Toh, 1987), optimal control (Wallaschek, 1988), structural dynamics (Feeny & Kappagantu, 1998; Han & Feeny, 2003; Holmes, et al., 1996; Marrifield & Guza,1990; Oey, 2007), heat transfer (Barnett, 1983; Kaihatu, et al., 1997) and system identification have been reported (Esquivel, et al., 2009; Feeny, 2008; Hasselmann, 1988; Horel, 1984; Kwasniok, 1996, 2007). These methodologies use statistical techniques such as, empirical orthogonal function (EOF) (Esquivel & Messina, 2008), principal interaction pattern (PIP) (Achatz, et al., 1995), principal oscillation pattern (POP) (Hasselmann, 1988),

Biorthogonal Decomposition for Wide-Area Wave Motion Monitoring Using Statistical Models 377

considered from a statistical perspective (Aubry, et al., 1990; Dankowicz, et al., 1996; Spletzer, et al., 2010). This approach provides an efficient and accurate way to compute standing and propagating features of general nonstationary processes identifying important information for the analysis of dynamical phenomena such as seismic wave components recorded from earthquakes. Moreover, this may lead to greater understanding of the oscillatory activity in interconnected systems. The method allows the introduction of several measures that define moving features in space-time varying fields as: spatial amplitude and phase function, temporal amplitude and phase function, spatial and temporal energy, wave number, angular frequency and average phase speed (Barnett, 1983; Esquivel & Messina, 2008; Susanto, et al. 1997; Terradas, et al., 2004; Hannachi, et al., 2007). The method developed is general and could be applied without loss of generality to measured or simulated data. As an illustrative case, the method is applied to a synthetic example; additionally, data recorded from GPS-based multiple phasor measurements units from a real event of seismic wave components recorded during a submarine earthquake are used to study the practical applicability of the method

Empirical orthogonal function (EOF) analysis is a procedure for extracting a basis for a modal decomposition from an ensemble of signals in multidimensional measurements. A very appealing property of the basis is its optimality. Among all possible decompositions of a random field, the EOF analysis is the most efficient in the sense that for a given number of modes, the projection on the subspace used for modelling the random field will on average contain the most energy possible. Although EOF analysis has been regularly applied to nonlinear problems (Marrifield & Guza, 1990; Susanto, et al., 1997; Toh, 1987; Kaihatu, et al., 1997), it is essential to underline that it is a linear technique and that it is optimal only with respect to other linear representations. Empirical orthogonal function analysis, also known as proper orthogonal decomposition (POD) and Karhunen-Loève transform was introduced by (Kosambi, 1943). It is also worth pointing out that EOF analysis is closely related to principal component analysis (PCA) introduced by (Hotelling, 1933). For a detailed historical review of POD or PCA, the reader is referred to (Barnett, 1983; Hasselmann, 1988;

be a zero mean random field on a domain Ω. In practice, the field is sampled at a finite number of pints in time. Then, at time *tk*, the system displays a snapshot *u*(*x*,*tk*) which is a continuous function of *x* in Ω. The aim of the EOF analysis is to find the most persistent structure among the ensemble of *N* snapshots. More precisely, assume that **X**(*xj*,*tk*), *j*=1,...,*n* and *k*=1,...,*N* denotes a sequence of observations on some domain *x*єΩ where *x* is a vector of spatial variables, and *tk* is the time at which the observations are made. The method of EOF analysis, both spatial and time-dependent, is a specification of the general theory of expansion of random functions (random fields or random processes) in a series of some deterministic (nonrandom) functions with random uncorrelated coefficients (Feeny & Kappagantu, 1998). The essential idea of the proper orthogonal decomposition is to generate

*u*(*x*,*t*) (1)

to characterize spatio-temporal behaviour in wide-area systems.

Hostelling, 1933; Horel, 1984; Kosambi, 1943; Toh, 1987).

**2.1 Theoretical development** 

Let

optimal persistent pattern (OPP) (DelSol, 2001), and the canonical correlation analysis (CCA) (Kwasniok, 2007) that capture various forms of spatio-temporal variability. Among other approaches, empirical orthogonal functions (EOFs) have been used since the mid-1970s for the identification of space-time dynamic systems. More recently, these techniques have gained wide popularity in applications related to wide-area data analysis and reduced-order modelling of various physical processes or models (Messina, et al. 2010; Spletzer, et al., 2010). Underlying issues of these techniques, such as the estimation and localization of propagating and standing features that may be associated with observed or measured data and their applications to space-time varying processes do not seem to be recognized or, at least, they have not been reported. This fact motivates the derivation of a model based on statistical techniques to identify the behaviour of multivariate processes such as the seismic wave propagation components that surge during an earthquake which involve variability over both space and time. These processes may contain moving patterns, travelling waves of different spatial scales and temporal frequencies that are proposed to identify in our study using complex EOF analysis.

#### **2. Theoretical fundamentals of empirical orthogonal functions**

The conventional analysis of empirical orthogonal functions is primarily a method of compressing of time and space variability of a data set into the lower possible number of spatial patterns. Each one of these patterns is composed of standing modes of variability and modulated by a time function. The conventional formulation of EOF analysis involves a set of optimal basis which is forced to approach the original field with modes at infinite frequency. In this section is shown that this requirement reduces the ability in the conventional method to characterize the travelling and standing features in dynamical systems because the spatial variation of the original field are combined with the temporal variations. As such, conventional-EOF analysis detects only standing wave components, not travelling wave components. The key point to observe is that real-EOF analysis cannot deal with propagating features and it only uses spatial correlation of the data set, it is necessary to use both spatial and time information in order to identify such features (Esquivel, 2009). In this chapter, we extend the conventional empirical orthogonal function analysis to the study and detection of propagating features in nonlinear patters such as seismic wave propagation components that surge during an earthquake recorded from wide-area monitoring schemes such as GPS-based in multiple PMUs, most of the notation used in this text is standard, vectorial quantities are denoted by boldface letters and scalar quantities by italic letters; others symbols used in the text are too defined. Unlike the real case, complex EOF analysis allows compressing the data into the lowest possible number of spatial patters, each one composed of modes of variability, which may be either travelling or standing modes. The technique allows us to explicitly describe and localize standing and propagating oscillations to the leading seismic wave as a number of complex empirical modes.

In this section, we provide a spatio-temporal decomposition based in the use of time synchronized measured data recorded from multiple phasor measurement units (PMUs) in dynamical systems to cope with increasing complexity of information in the use of wide-area monitoring schemes. The methodology is proposed to identify and extract dynamically independent spatio-temporal patterns using a biorthogonal decomposition based in the complex EOF analysis and the separability of complex correlation functions considered from a statistical perspective (Aubry, et al., 1990; Dankowicz, et al., 1996; Spletzer, et al., 2010). This approach provides an efficient and accurate way to compute standing and propagating features of general nonstationary processes identifying important information for the analysis of dynamical phenomena such as seismic wave components recorded from earthquakes. Moreover, this may lead to greater understanding of the oscillatory activity in interconnected systems. The method allows the introduction of several measures that define moving features in space-time varying fields as: spatial amplitude and phase function, temporal amplitude and phase function, spatial and temporal energy, wave number, angular frequency and average phase speed (Barnett, 1983; Esquivel & Messina, 2008; Susanto, et al. 1997; Terradas, et al., 2004; Hannachi, et al., 2007). The method developed is general and could be applied without loss of generality to measured or simulated data. As an illustrative case, the method is applied to a synthetic example; additionally, data recorded from GPS-based multiple phasor measurements units from a real event of seismic wave components recorded during a submarine earthquake are used to study the practical applicability of the method to characterize spatio-temporal behaviour in wide-area systems.

#### **2.1 Theoretical development**

Empirical orthogonal function (EOF) analysis is a procedure for extracting a basis for a modal decomposition from an ensemble of signals in multidimensional measurements. A very appealing property of the basis is its optimality. Among all possible decompositions of a random field, the EOF analysis is the most efficient in the sense that for a given number of modes, the projection on the subspace used for modelling the random field will on average contain the most energy possible. Although EOF analysis has been regularly applied to nonlinear problems (Marrifield & Guza, 1990; Susanto, et al., 1997; Toh, 1987; Kaihatu, et al., 1997), it is essential to underline that it is a linear technique and that it is optimal only with respect to other linear representations. Empirical orthogonal function analysis, also known as proper orthogonal decomposition (POD) and Karhunen-Loève transform was introduced by (Kosambi, 1943). It is also worth pointing out that EOF analysis is closely related to principal component analysis (PCA) introduced by (Hotelling, 1933). For a detailed historical review of POD or PCA, the reader is referred to (Barnett, 1983; Hasselmann, 1988; Hostelling, 1933; Horel, 1984; Kosambi, 1943; Toh, 1987).

Let

376 Numerical Modelling

optimal persistent pattern (OPP) (DelSol, 2001), and the canonical correlation analysis (CCA) (Kwasniok, 2007) that capture various forms of spatio-temporal variability. Among other approaches, empirical orthogonal functions (EOFs) have been used since the mid-1970s for the identification of space-time dynamic systems. More recently, these techniques have gained wide popularity in applications related to wide-area data analysis and reduced-order modelling of various physical processes or models (Messina, et al. 2010; Spletzer, et al., 2010). Underlying issues of these techniques, such as the estimation and localization of propagating and standing features that may be associated with observed or measured data and their applications to space-time varying processes do not seem to be recognized or, at least, they have not been reported. This fact motivates the derivation of a model based on statistical techniques to identify the behaviour of multivariate processes such as the seismic wave propagation components that surge during an earthquake which involve variability over both space and time. These processes may contain moving patterns, travelling waves of different spatial scales and temporal frequencies that are proposed to identify in our study

**2. Theoretical fundamentals of empirical orthogonal functions** 

oscillations to the leading seismic wave as a number of complex empirical modes.

In this section, we provide a spatio-temporal decomposition based in the use of time synchronized measured data recorded from multiple phasor measurement units (PMUs) in dynamical systems to cope with increasing complexity of information in the use of wide-area monitoring schemes. The methodology is proposed to identify and extract dynamically independent spatio-temporal patterns using a biorthogonal decomposition based in the complex EOF analysis and the separability of complex correlation functions

The conventional analysis of empirical orthogonal functions is primarily a method of compressing of time and space variability of a data set into the lower possible number of spatial patterns. Each one of these patterns is composed of standing modes of variability and modulated by a time function. The conventional formulation of EOF analysis involves a set of optimal basis which is forced to approach the original field with modes at infinite frequency. In this section is shown that this requirement reduces the ability in the conventional method to characterize the travelling and standing features in dynamical systems because the spatial variation of the original field are combined with the temporal variations. As such, conventional-EOF analysis detects only standing wave components, not travelling wave components. The key point to observe is that real-EOF analysis cannot deal with propagating features and it only uses spatial correlation of the data set, it is necessary to use both spatial and time information in order to identify such features (Esquivel, 2009). In this chapter, we extend the conventional empirical orthogonal function analysis to the study and detection of propagating features in nonlinear patters such as seismic wave propagation components that surge during an earthquake recorded from wide-area monitoring schemes such as GPS-based in multiple PMUs, most of the notation used in this text is standard, vectorial quantities are denoted by boldface letters and scalar quantities by italic letters; others symbols used in the text are too defined. Unlike the real case, complex EOF analysis allows compressing the data into the lowest possible number of spatial patters, each one composed of modes of variability, which may be either travelling or standing modes. The technique allows us to explicitly describe and localize standing and propagating

using complex EOF analysis.

$$
\mu(\mathbf{x}, t) \tag{1}
$$

be a zero mean random field on a domain Ω. In practice, the field is sampled at a finite number of pints in time. Then, at time *tk*, the system displays a snapshot *u*(*x*,*tk*) which is a continuous function of *x* in Ω. The aim of the EOF analysis is to find the most persistent structure among the ensemble of *N* snapshots. More precisely, assume that **X**(*xj*,*tk*), *j*=1,...,*n* and *k*=1,...,*N* denotes a sequence of observations on some domain *x*єΩ where *x* is a vector of spatial variables, and *tk* is the time at which the observations are made. The method of EOF analysis, both spatial and time-dependent, is a specification of the general theory of expansion of random functions (random fields or random processes) in a series of some deterministic (nonrandom) functions with random uncorrelated coefficients (Feeny & Kappagantu, 1998). The essential idea of the proper orthogonal decomposition is to generate

Biorthogonal Decomposition for Wide-Area Wave Motion Monitoring Using Statistical Models 379

*x x x x dx dx x x dx* 1 1 <sup>1</sup> \* \*\* \*\*

such that, the inner product *x x* \* \* **C** 0 , with orthogonal eigenvectors (*x*), **ψ**(*x*),

From (4) it can be seen that if there exists an arbitrary variation (spatial), **ψ**\*(*x*)≠0, then the original field can be reconstructed using two optimal orthogonal basis given from (7). Based in this notion, an efficient technique to find the optimal basis using complex EOF analysis

Our proposed methodology based in EOF analysis and the Hilbert transform is developed to be applied for representations of complex data fields in a biorthogonal decomposition illustrating the phenomenon of spatial and temporal variability in interconnected systems. This method consists first in extend each real field data to the complex world using the Hilbert transform to provide the phase information; and second, the EOF analysis is developed to the complex data field for the detection and localization of propagation

Fig. 1. Global positioning system (GPS) and PMU data in terms of a time-space varying field.

Conventional EOF analysis of real-data fields is commonly carried out under the assumption that each field can be represented as a spatially fixed pattern of behaviour. This method, however, cannot be used for detection of propagating features because of the lack

*i j i j*

(8)

0, , 

*u u* (7)

 and, *T i j*

0 0 <sup>0</sup> ' ''

*i j i j*

0, , 

*T i j*

(CEOFs) is proposed (Esquivel, 2009).

features into dynamical systems.

**2.2 Complex data fields** 

i.e.,

an optimal basis, (*x*), for the representation of an ensemble of data collected from measurements or numerical simulations of a dynamic system as is shown in Fig. 1.

Given an ensemble of measured data, the data set can be written as the *N*×*n*-dimension matrix

$$\mathbf{X}(\mathbf{x}\_{j'}, t\_k) = \begin{bmatrix} \mu(\mathbf{x}\_1, t\_1) & \cdots & \mu(\mathbf{x}\_{n'}, t\_1) \\ \vdots & \ddots & \vdots \\ \mu(\mathbf{x}\_1, t\_N) & \cdots & \mu(\mathbf{x}\_{n'}, t\_N) \end{bmatrix} \tag{2}$$

where typically *n≠N*, so **X** is generally rectangular (Messina, et al., 2010). The technique yields an orthogonal basis for linear, infinite-dimensional Hilbert space *L*2([0,1]), that maximizes the averaged projection of the response matrix for the representation of the ensemble of data that is fully orthogonal, and it is assumed to be normalized, i.e.,

$$\max\_{\{\boldsymbol{\Psi}\_{\boldsymbol{\beta}}(\boldsymbol{\lambda}) \leftarrow \boldsymbol{\Lambda}^{2}(\{0,1\})\}} \frac{\left< \left(\mathcal{X}(\boldsymbol{x},t), \boldsymbol{\Phi}\_{\boldsymbol{\beta}}(\boldsymbol{x})\right)^{2} \right>}{\left\|\boldsymbol{\Phi}\_{\boldsymbol{\beta}}(\boldsymbol{x})\right\|^{2}} \text{ subject to } \left\|\boldsymbol{\Phi}\_{\boldsymbol{\beta}}(\boldsymbol{x})\right\|^{2} = 1\tag{3}$$

where . denotes the modulus, . is the *L*2-norm and, . implies the use of an average operation (Holmes, et al., 1996). The corresponding functional for the constrained variational problem is solved and reduced to:

$$\int\_{0}^{1} \left[ \int\_{0}^{1} \left\langle \mu(\mathbf{x}) \mu^{\*}(\mathbf{x}') \right\rangle \mathfrak{g}(\mathbf{x}') d\mathbf{x}' - \lambda \mathfrak{g}(\mathbf{x}) \right]^{\*} \mathfrak{g}^{\*}(\mathbf{x}) d\mathbf{x} = 0 \tag{4}$$

where the (\*) denotes the conjugate transpose (sometimes denoted as Hermitian, *H*), and the (') denotes transpose vector. Thus, if **ψ**\*(*x*)=0, the optimal basis are given by the eigenfunctions *j*(*x*) of the integral equation,

$$\int\_{0}^{1} \left< \mu(\mathbf{x}) \mu^\*(\mathbf{x}') \right> \mathfrak{sp}(\mathbf{x}') d\mathbf{x}' = \lambda \mathfrak{sp}(\mathbf{x}) \tag{5}$$

whose kernel is the averaged autocorrelation function *x x xx* \* *u u* **C** , ' . Under this assumption, the integral (5) can be written as

$$\mathbf{C}\mathfrak{op}(\boldsymbol{\mathfrak{x}}) = \mathbb{X}\mathfrak{op}(\boldsymbol{\mathfrak{x}}) \tag{6}$$

where the resulting autocorrelation matrix **C**, is real, symmetric, positive and semi-definite matrix. Therefore, the optimization problem can be recast as the problem of finding the largest eigenvectors, (*x*), of the equation (6), called empirical orthogonal functions (EOFs); its corresponding eigenvalues are real, nonnegative, and ordered so that 1 2 *j* ,, 0 . This method, also called conventional EOF analysis, cannot be used to detect propagation features due to the assumption that each field is represented as a spatial fixed pattern of behaviour and lack of phase information, becoming prohibitive to practical applications.

Now, if we assume that **ψ**\*(*x*)≠0, then (4) can be rewritten as

$$\int\_{0}^{1} \int\_{0}^{1} \boldsymbol{\Phi}^\*(\mathbf{x}') \Big\langle \boldsymbol{\mu}(\mathbf{x}) \boldsymbol{\mu}^\*(\mathbf{x}') \Big\rangle \boldsymbol{\Psi}^\*(\mathbf{x}) d\mathbf{x}' d\mathbf{x} = \int\_{0}^{1} \boldsymbol{\Phi}^\*(\mathbf{x}) \boldsymbol{\lambda} \boldsymbol{\Psi}^\*(\mathbf{x}) d\mathbf{x} \tag{7}$$

such that, the inner product *x x* \* \* **C** 0 , with orthogonal eigenvectors (*x*), **ψ**(*x*), i.e.,

$$\boldsymbol{\mathfrak{op}}\_{i}^{T}\boldsymbol{\mathfrak{op}}\_{j} = \begin{cases} 0, & \text{ $i \neq j$ } \\ \boldsymbol{\delta}(\boldsymbol{\mathfrak{op}}), & \text{ $i = j$ } \end{cases} \text{ and, } \boldsymbol{\Psi}\_{i}^{T}\boldsymbol{\mathfrak{w}}\_{j} = \begin{cases} 0, & \text{ $i \neq j$ } \\ \boldsymbol{\delta}(\boldsymbol{\mathfrak{w}}), & \text{ $i = j$ } \end{cases} \tag{8}$$

From (4) it can be seen that if there exists an arbitrary variation (spatial), **ψ**\*(*x*)≠0, then the original field can be reconstructed using two optimal orthogonal basis given from (7). Based in this notion, an efficient technique to find the optimal basis using complex EOF analysis (CEOFs) is proposed (Esquivel, 2009).

Our proposed methodology based in EOF analysis and the Hilbert transform is developed to be applied for representations of complex data fields in a biorthogonal decomposition illustrating the phenomenon of spatial and temporal variability in interconnected systems. This method consists first in extend each real field data to the complex world using the Hilbert transform to provide the phase information; and second, the EOF analysis is developed to the complex data field for the detection and localization of propagation features into dynamical systems.

Fig. 1. Global positioning system (GPS) and PMU data in terms of a time-space varying field.

#### **2.2 Complex data fields**

378 Numerical Modelling

an optimal basis, (*x*), for the representation of an ensemble of data collected from

Given an ensemble of measured data, the data set can be written as the *N*×*n*-dimension

1

ensemble of data that is fully orthogonal, and it is assumed to be normalized, i.e.,

*xt x*


**X φ**

**φ**

0 0 ' ''

*j*

*j <sup>j</sup> x L*

where typically *n≠N*, so **X** is generally rectangular (Messina, et al., 2010). The technique yields an orthogonal basis for linear, infinite-dimensional Hilbert space *L*2([0,1]), that maximizes the averaged projection of the response matrix for the representation of the

2

max subject to ( ) 1

where . denotes the modulus, . is the *L*2-norm and, . implies the use of an average operation (Holmes, et al., 1996). The corresponding functional for the constrained

*x x x dx x x dx* \* 1 1 \* \*

where the (\*) denotes the conjugate transpose (sometimes denoted as Hermitian, *H*), and the (') denotes transpose vector. Thus, if **ψ**\*(*x*)=0, the optimal basis are given by the

> *x x x dx x* <sup>1</sup> \* <sup>0</sup> ' ''

whose kernel is the averaged autocorrelation function *x x xx* \* *u u* **C** , ' . Under this

**C***x x* 

where the resulting autocorrelation matrix **C**, is real, symmetric, positive and semi-definite matrix. Therefore, the optimization problem can be recast as the problem of finding the largest eigenvectors, (*x*), of the equation (6), called empirical orthogonal functions (EOFs); its corresponding eigenvalues are real, nonnegative, and ordered so that

detect propagation features due to the assumption that each field is represented as a spatial fixed pattern of behaviour and lack of phase information, becoming prohibitive to practical

,, 0 . This method, also called conventional EOF analysis, cannot be used to

*n*

 *x*

<sup>0</sup> *u u* (4)

*u u* (5)

**φ**

2

(6)

(2)

(3)

*N n N*

*ux t ux t*

1 1 1

( ,) ( ,)

 

(,) (,) 

*ux t ux t*

measurements or numerical simulations of a dynamic system as is shown in Fig. 1.

*j k*

*x t*

*x* <sup>2</sup>

<sup>2</sup> ( ) ([0,1])

( )

**X**

*j*

variational problem is solved and reduced to:

eigenfunctions *j*(*x*) of the integral equation,

assumption, the integral (5) can be written as

Now, if we assume that **ψ**\*(*x*)≠0, then (4) can be rewritten as

1 2 *j*

applications.

 

 

**φ**

(,)

matrix

Conventional EOF analysis of real-data fields is commonly carried out under the assumption that each field can be represented as a spatially fixed pattern of behaviour. This method, however, cannot be used for detection of propagating features because of the lack

Biorthogonal Decomposition for Wide-Area Wave Motion Monitoring Using Statistical Models 381

For discretely sampled data, the Hilbert transform can be derived in the time domain by

2 ( (2 1) ) ( ,) 2 1

*k*

*ut k ut k*

*l*

*u* (16)

2 1 [ ( 2 1) ( 2 1)] 2 1

In previous formulations, the Hilbert transform was estimated by truncating the series (15).

where *h* is a convolution filter with an unit amplitude response and 90º phase shift. In this research, it has been found that a simple filter that has the desired properties of approximate

*l l h l <sup>l</sup>*

<sup>2</sup> <sup>2</sup> sin ( 2) , 0 ( )

As *L*→∞, equation (16) yields an exact Hilbert transform. This represents a filtering operation upon *u*(*xj*,*t*) in which the amplitude of each Fourier spectral component remains unchanged while its phase is advantaged by *π*/2. In (Hannachi, et al., 2007) has been found

In what follows, we discuss the extension of the conventional EOF analysis using the above approach to compute standing and propagating features of general nonstationary processes where the eigenvectors of the covariance matrix are complex and it can be expressed

The method of complex EOF analysis is an optimal technique of biorthogonal decomposition to find a spatial and temporal basis that spans an ensemble of data collected from experiments or numerical simulations. The method essentially decomposes a fluctuating field into a weighted linear sum of spatial orthogonal modes and temporal

orthogonal modes such that the projection onto the first few modes is optimal.

 

0 , 0

*<sup>u</sup>* (14)

(15)

(17)

*k ut k x t*

*k*

 

where *τ* is the step size. When (13) is applied to a discrete time series *u*(*xj*,*t*), *k*=0,±1,...,

applying a rectangular rule to (13). It can be shown that

*H jk*

*u*

we get

*H j*

*k*

*ut k x t*

2 ( 2 1) (,) 2 1

*k*

This truncation was approximated using a convolution filter as

unit amplitude response and *π*/2 phase shift is given by

that 7≤*L*≤25 provides adequate values for the filter response.

**3. Complex empirical orthogonal function analysis** 

where -*L*≤*l*≤*L*. We omit the calculations.

alternatively as a magnitude and phase pair.

*k* <sup>0</sup>

*L H jk j k L* ( , ) ( , ) ( ), *x t ux t h L*

of phase information (Esquivel & Messina, 2008). To fully utilize the data, a technique is necessary unknowing the nonstationarity of the time-series data.

Let *u*(*xj*,*tk*) be a space-time varying scalar field representing a time series recorded from a wide-area distribution system, where *xj*, *j*=1,...,*n* is a set of spatial variables on a space Ω, and *tk*, *k*=1,...,*N* is the time at which the observations are made. Provide *u*(*x*,*t*) is simple and square integrable, it has a Fourier representation of the form

$$\mathfrak{u}(\mathbf{x}\_{j},t) = \sum\_{m=1}^{\infty} \left[ a\_{j(m)}(\alpha) \cos(m\alpha t) + b\_{j(m)}(\alpha) \sin(m\alpha t) \right] \tag{9}$$

where *α<sup>j</sup>*(*<sup>m</sup>*)(*ω*) and *bj*(*<sup>m</sup>*)(*ω*) are the Fourier coefficients defined as

$$\begin{aligned} a\_{j(m)} &= \frac{1}{\pi} \int\_{-\pi}^{\pi} u(\mathbf{x}\_j, t) \cos(m \alpha t) d\alpha \\ b\_{j(m)} &= \frac{1}{\pi} \int\_{-\pi}^{\pi} u(\mathbf{x}\_j, t) \sin(m \alpha t) d\alpha \end{aligned} \tag{10}$$

This allows the description of travelling waves propagating throughout the system. Equation (9) can be rewritten in the form

$$\mathfrak{w}\_c(\mathbf{x}\_{j'}, t) = \sum\_{m=1}^{\infty} c\_{j(m)}(\alpha) e^{-im\alpha t} \tag{11}$$

where *c<sup>j</sup>*(*<sup>m</sup>*)(*ω*)= *α<sup>j</sup>*(*<sup>m</sup>*)(*ω*)+*ibj*(*<sup>m</sup>*)(*ω*), *i* 1 is the unit complex number. Expanding (11) and collecting terms gives

$$\begin{aligned} \mathfrak{u}\_c(\mathbf{x}\_j, t) &= \sum\_{m=1}^{\infty} \left\{ \left[ a\_{j(m)}(\alpha) \cos(m\omega t) + b\_{j(m)}(\alpha) \sin(m\omega t) \right] \right\} \\ &+ i \sum\_{m=1}^{\infty} \left\{ \left[ b\_{j(m)}(\alpha) \cos(m\omega t) - a\_{j(m)}(\alpha) \sin(m\omega t) \right] \right\} \\ &= \mathfrak{u}(\mathbf{x}\_j, t) + i \mathfrak{u}\_H(\mathbf{x}\_j, t) \end{aligned} \tag{12}$$

where the real part of *uc*(*xj*,*t*) is given by (9) and the imaginary part is the Hilbert transform of *u*(*xj*,*t*). In formal terms, the Hilbert transform of a continuous time series *u*(*xj*,*t*) is defined by the convolution

$$
\mu\_H(\mathbf{x}\_j, t) = \frac{1}{\pi} \int\_{-\infty}^{\infty} \frac{\mu(y)}{t - y} dy \tag{13}
$$

where the integral is taken to mean the Cauchy principal value. The most well-known classical methods for computing the Hilbert transform are derived from the Fourier transform. However, this transform has a global character and hence, it is not suitable for the characterization of local signal parameters. Alternatives for local implementation of the Hilbert transformation which are based on local properties are developed and tested in this analysis (Hannchi, et al., 2007; Lezama, et al., 2009; Terradas, et al., 2004, Barnett, 1983).

For discretely sampled data, the Hilbert transform can be derived in the time domain by applying a rectangular rule to (13). It can be shown that

$$
\mu\_H(\mathbf{x}\_{j'}, t) \approx \frac{2}{\pi} \sum\_{k=-\infty}^{\infty} \frac{\mu(t + (2k + 1)\tau)}{2k + 1} \tag{14}
$$

where *τ* is the step size. When (13) is applied to a discrete time series *u*(*xj*,*t*), *k*=0,±1,..., we get

$$\begin{split} \mu\_{H}(\mathbf{x}\_{j}, t\_{k}) &= \frac{2}{\pi} \sum\_{k=-\infty}^{\infty} \frac{\mu(t+2k+1)}{2k+1} \\ &= \frac{2}{\pi} \sum\_{k\geq 0} \frac{1}{2k+1} [\mu(t+2k+1) - \mu(t-2k-1)] \end{split} \tag{15}$$

In previous formulations, the Hilbert transform was estimated by truncating the series (15). This truncation was approximated using a convolution filter as

$$\ln \mu\_H(\mathbf{x}\_j, t\_k) = \sum\_{\ell=-L}^{L} \mu(\mathbf{x}\_j, t\_k - \ell) h(\ell) \,, \quad L = \infty \tag{16}$$

where *h* is a convolution filter with an unit amplitude response and 90º phase shift. In this research, it has been found that a simple filter that has the desired properties of approximate unit amplitude response and *π*/2 phase shift is given by

$$h(l) = \begin{cases} \frac{2}{\pi l} \sin^2(\pi l/2), & l \neq 0\\ 0, & l = 0 \end{cases} \tag{17}$$

where -*L*≤*l*≤*L*. We omit the calculations.

380 Numerical Modelling

of phase information (Esquivel & Messina, 2008). To fully utilize the data, a technique is

Let *u*(*xj*,*tk*) be a space-time varying scalar field representing a time series recorded from a wide-area distribution system, where *xj*, *j*=1,...,*n* is a set of spatial variables on a space Ω, and *tk*, *k*=1,...,*N* is the time at which the observations are made. Provide *u*(*x*,*t*) is simple and

*xt a ω mωt b ω mωt* ( ) ( )

*a ux t mωt dω*

<sup>1</sup> ( , )cos

*b ux t mωt dω*

This allows the description of travelling waves propagating throughout the system.

*i b ω mωt a ω mωt*

*j m j m*

( ) ( )

[ ( )cos( ) ( )sin( )]

<sup>1</sup> ( , )sin

( , ) [ ( )cos( ) ( )sin( )]

*<sup>u</sup>* (9)

(10)

(12)

*imωt*

*<sup>u</sup>* (11)

*<sup>u</sup>* (13)

*j j m j m*

*j m j*

*c j j m m xt c ω e* ( ) 1 ( ,) ( )

*xt a ω mωt b ω mωt*

( , ) [ ( )cos( ) ( )sin( )]

( ) ( )

where the real part of *uc*(*xj*,*t*) is given by (9) and the imaginary part is the Hilbert transform of *u*(*xj*,*t*). In formal terms, the Hilbert transform of a continuous time series *u*(*xj*,*t*) is defined

> *u y x t dy t y* <sup>1</sup> ( ) ( ,)

where the integral is taken to mean the Cauchy principal value. The most well-known classical methods for computing the Hilbert transform are derived from the Fourier transform. However, this transform has a global character and hence, it is not suitable for the characterization of local signal parameters. Alternatives for local implementation of the Hilbert transformation which are based on local properties are developed and tested in this analysis (Hannchi, et al., 2007; Lezama, et al., 2009; Terradas, et al., 2004, Barnett, 1983).

where *c<sup>j</sup>*(*<sup>m</sup>*)(*ω*)= *α<sup>j</sup>*(*<sup>m</sup>*)(*ω*)+*ibj*(*<sup>m</sup>*)(*ω*), *i* 1 is the unit complex number. Expanding (11) and

*j m j*

*c j j m j m*

*j Hj*

*H j*

*xt i xt*

( ,) ( ,)

*m*

1

 

*m*

1

*u u*

necessary unknowing the nonstationarity of the time-series data.

square integrable, it has a Fourier representation of the form

*m*

where *α<sup>j</sup>*(*<sup>m</sup>*)(*ω*) and *bj*(*<sup>m</sup>*)(*ω*) are the Fourier coefficients defined as

1

( )

( )

Equation (9) can be rewritten in the form

*u*

collecting terms gives

by the convolution

As *L*→∞, equation (16) yields an exact Hilbert transform. This represents a filtering operation upon *u*(*xj*,*t*) in which the amplitude of each Fourier spectral component remains unchanged while its phase is advantaged by *π*/2. In (Hannachi, et al., 2007) has been found that 7≤*L*≤25 provides adequate values for the filter response.

In what follows, we discuss the extension of the conventional EOF analysis using the above approach to compute standing and propagating features of general nonstationary processes where the eigenvectors of the covariance matrix are complex and it can be expressed alternatively as a magnitude and phase pair.

#### **3. Complex empirical orthogonal function analysis**

The method of complex EOF analysis is an optimal technique of biorthogonal decomposition to find a spatial and temporal basis that spans an ensemble of data collected from experiments or numerical simulations. The method essentially decomposes a fluctuating field into a weighted linear sum of spatial orthogonal modes and temporal orthogonal modes such that the projection onto the first few modes is optimal.

Biorthogonal Decomposition for Wide-Area Wave Motion Monitoring Using Statistical Models 383

Once the spatial eigenvectors associated with real and imaginary part of (20) are computed, the original field can be approximated by a spatio-temporal model. Assuming that this model is composed of standing and travelling wave components, the space-time varying

where **X***swc* and **X***twc* denotes the standing and travelling wave components respectively. Therefore, the associated approximation for the complex data field (19) in terms of a

> *xt t x i t x* () () () () 1 1 ( ,) () ( ) () ( )

where the time-dependent complex coefficients associated with each eigenfuntion, **A***<sup>R</sup>*(*<sup>j</sup>*)(*t*) and **A***<sup>I</sup>*(*<sup>j</sup>*)(*t*) are obtained as the projection of the basis *<sup>R</sup>*(*<sup>j</sup>*)(*x*) and *<sup>I</sup>*(*<sup>j</sup>*)(*x*) respectively into

> *Rj c Rj Ij c Ij*

These complex coefficients are conveniently split into their amplitude and phase, therefore,

 **A X φ A X φ**

*t x t x* ( ) ( ) ( ) ( )

() ( ) () ( )

*p q it x it x <sup>π</sup>*

**θ Φ θ Φ X RS R S** (25)

*xt t xe t xe* () () () () ( ( ) ( )) ( () ( ) ) () () () ()

( ,) () ( ) () ( )

where **R**(*t*) and **S**(*x*) are the temporal and spatial amplitude functions associated with the wave decomposition respectively and, **θ**(*t*) and **Φ**(*x*) are the temporal and spatial phase

The succeeding sections describe the properties of these representations to assess and to extract swing oscillations patterns and modal characteristics directly from recorded data in wide-area dynamical systems. It is shown that the proposed method can be used to predict

This function shows the spatial distribution of variability associated with each eigenmode. The spatial amplitude functions in the proposed model (25) are defined as (Hannchi, et al.,

Now, four measurements that define moving features in *u*(x,*t*) can then be defined:

*p q c Rj Rj Ij Ij j j*

truncated sum of dominant modes (EOFs basis) *p* and *q*, is defined as

from the complex model (23), the ensemble of data can be expressed as

*c Rj Rj Ij Ij j j*

1 1

 

the correct spatial location in the modal distribution of seismic wave.

1. Spatial distribution of variability of each eigenmode

4. Variability of the phase of a particular oscillation

*swc twc* **XX X** ( ,) ( ,) ( ,) *xt xt xt* (22)

(24)

**X A <sup>φ</sup> <sup>A</sup> <sup>φ</sup>** (23)

*Rj Rj Ij Ij*

field can be written as

complex field **X***c* of the form

2. Relative phase fluctuation

3. Temporal variability in magnitude

**3.1 Spatial amplitude function, S(***x***)** 

function.

Drawing on the above approach, an efficient formulation to compute a complex expansion for the data set has been derived.

Assume that **X**(*x*,*t*) is augmented by their imaginary components to form a complex data matrix such as (Esquivel, 2009)

$$\mathbf{X}\_c(\mathbf{x}, t) = \mathbf{X}\_R(\mathbf{x}, t) + i\mathbf{X}\_I(\mathbf{x}, t) \tag{18}$$

where the subscripts *c*, *R* and *I* indicate the complex, real and imaginary vectors respectively. Implicit in the model is the assumption that **X***c* can be represented as

$$\mathbf{X}\_c = \left\| \mathbf{X}\_c \right\| \left[ \cos(\mathbf{\theta}\_{X\_c} t) + i \sin(\mathbf{\theta}\_{X\_c} t) \right] \tag{19}$$

where **Χ***c* and *Xc* **θ** are the magnitude and phase of **X***c*. Under this assumption, the complex autocorrelation matrix becomes,

$$\mathbf{C} = \frac{1}{N} \mathbf{X}\_c^H \mathbf{X}\_c \tag{20}$$

where it is straightforward to show that the autocorrelation matrix ,**C**, for the case complex data can be written in the form **C**=**C***R*+*i***C***I* which the real part is a symmetrical matrix, (i.e., *<sup>T</sup>* **C C** *R R* ) and the imaginary part is a skew-symmetric matrix (i.e., *<sup>T</sup>* **C C** *I I* ). If the size of **C***I* is odd, then the determinant of **C***I* will always be zero. Because the symmetrical matrix is a particular case of the Hermitian matrix, then all its eigenvectors are real. Furthermore, the eigenvalues of the skew-symmetric matrix are all imaginary pure and, it is a normal matrix; its eigenvectors are complex conjugate.

From (20), It can be easily verified that

$$\begin{split} \mathbf{X}\_c^H \mathbf{X}\_c &= \begin{bmatrix} \mathbf{X}\_c^T \end{bmatrix} \begin{bmatrix} \mathbf{X}\_c \end{bmatrix} [\cos(\boldsymbol{\theta}\_{\mathbf{X}\_c^T} t) - i \sin(\boldsymbol{\theta}\_{\mathbf{X}\_c^T} t)] [\cos(\boldsymbol{\theta}\_{X\_c} t) + i \sin(\boldsymbol{\theta}\_{X\_c} t)] \\ &= \begin{bmatrix} \mathbf{X}\_c^T \end{bmatrix} \begin{bmatrix} \mathbf{X}\_c \end{bmatrix} \{ [\cos(\boldsymbol{\theta}\_{\mathbf{X}\_c^T} t) \cos(\boldsymbol{\theta}\_{X\_c} t) + \sin(\boldsymbol{\theta}\_{\mathbf{X}\_c^T} t) \sin(\boldsymbol{\theta}\_{X\_c} t)] \\ &+ i [\cos(\boldsymbol{\theta}\_{\mathbf{X}\_c^T} t) \sin(\boldsymbol{\theta}\_{X\_c} t) - \sin(\boldsymbol{\theta}\_{\mathbf{X}\_c^T} t) \cos(\boldsymbol{\theta}\_{X\_c} t)] \} \end{split} \tag{21}$$

From the decomposition given in (21) can be seen that the imaginary part is zero when the time is in phase with the extremum of the cosine or sine, that is, the sum of the two components is zero; at this time instant both are symmetrical matrices (Feeny, 2008). The imaginary part of (21) measures the degree of asymmetry when the sum of both matrices is different from zero; this is used to define the existence of arbitrary variations into the space, **ψ**\*(*x*)≠0; this feature is used to define the existence of travelling wave components in the space-time varying fields and to determine leading seismic wave propagation components.

From the decomposition for the complex autocorrelation matrix (20), the optimal basis for the proposed spatio-temporal decomposition is defined by the eigenfunctions *R*(*x*) and *I*(*x*) for the real and imaginary part respectively. A test to split the spatial-temporal covariance functions is given by (Wallaschek, 1988; Fuentes, 2006).

Once the spatial eigenvectors associated with real and imaginary part of (20) are computed, the original field can be approximated by a spatio-temporal model. Assuming that this model is composed of standing and travelling wave components, the space-time varying field can be written as

$$\mathbf{X}(\mathbf{x},t) = \mathbf{X}\_{\text{succ}}(\mathbf{x},t) + \mathbf{X}\_{\text{truc}}(\mathbf{x},t) \tag{22}$$

where **X***swc* and **X***twc* denotes the standing and travelling wave components respectively. Therefore, the associated approximation for the complex data field (19) in terms of a truncated sum of dominant modes (EOFs basis) *p* and *q*, is defined as

$$\mathbf{X}\_c(\mathbf{x}, t) = \sum\_{j=1}^p \mathbf{A}\_{R(j)}(t) \boldsymbol{\upmu}\_{R(j)}^H(\mathbf{x}) + i \sum\_{j=1}^q \mathbf{A}\_{I(j)}(t) \boldsymbol{\upmu}\_{I(j)}^H(\mathbf{x}) \tag{23}$$

where the time-dependent complex coefficients associated with each eigenfuntion, **A***<sup>R</sup>*(*<sup>j</sup>*)(*t*) and **A***<sup>I</sup>*(*<sup>j</sup>*)(*t*) are obtained as the projection of the basis *<sup>R</sup>*(*<sup>j</sup>*)(*x*) and *<sup>I</sup>*(*<sup>j</sup>*)(*x*) respectively into complex field **X***c* of the form

$$\begin{aligned} \mathbf{A}\_{R(j)}(t) &= \mathbf{X}\_c \boldsymbol{\upPhi}\_{R(j)}(\mathbf{x}) \\ \mathbf{A}\_{I(j)}(t) &= \mathbf{X}\_c \boldsymbol{\upPhi}\_{I(j)}(\mathbf{x}) \end{aligned} \tag{24}$$

These complex coefficients are conveniently split into their amplitude and phase, therefore, from the complex model (23), the ensemble of data can be expressed as

$$\mathbf{X}\_c(\mathbf{x},t) = \sum\_{j=1}^p \mathbf{R}\_{R(j)}(t)\mathbf{S}\_{R(j)}(\mathbf{x})e^{i(\boldsymbol{\theta}\_{R(j)}(t) + \boldsymbol{\Phi}\_{R(j)}(\mathbf{x}))} + \sum\_{j=1}^q \mathbf{R}\_{I(j)}(t)\mathbf{S}\_{I(j)}(\mathbf{x})e^{i(\boldsymbol{\theta}\_{I(j)}(t) + \boldsymbol{\Phi}\_{I(j)}(\mathbf{x}) + \boldsymbol{\pi})} \tag{25}$$

where **R**(*t*) and **S**(*x*) are the temporal and spatial amplitude functions associated with the wave decomposition respectively and, **θ**(*t*) and **Φ**(*x*) are the temporal and spatial phase function.

Now, four measurements that define moving features in *u*(x,*t*) can then be defined:


382 Numerical Modelling

Drawing on the above approach, an efficient formulation to compute a complex expansion

Assume that **X**(*x*,*t*) is augmented by their imaginary components to form a complex data

where the subscripts *c*, *R* and *I* indicate the complex, real and imaginary vectors

where **Χ***c* and *Xc* **θ** are the magnitude and phase of **X***c*. Under this assumption, the

where it is straightforward to show that the autocorrelation matrix ,**C**, for the case complex data can be written in the form **C**=**C***R*+*i***C***I* which the real part is a symmetrical matrix, (i.e., *<sup>T</sup>* **C C** *R R* ) and the imaginary part is a skew-symmetric matrix (i.e., *<sup>T</sup>* **C C** *I I* ). If the size of **C***I* is odd, then the determinant of **C***I* will always be zero. Because the symmetrical matrix is a particular case of the Hermitian matrix, then all its eigenvectors are real. Furthermore, the eigenvalues of the skew-symmetric matrix are all imaginary pure and, it is a normal

*T T c c c c*

From the decomposition given in (21) can be seen that the imaginary part is zero when the time is in phase with the extremum of the cosine or sine, that is, the sum of the two components is zero; at this time instant both are symmetrical matrices (Feeny, 2008). The imaginary part of (21) measures the degree of asymmetry when the sum of both matrices is different from zero; this is used to define the existence of arbitrary variations into the space, **ψ**\*(*x*)≠0; this feature is used to define the existence of travelling wave components in the space-time varying fields and to determine leading seismic wave propagation

From the decomposition for the complex autocorrelation matrix (20), the optimal basis for the proposed spatio-temporal decomposition is defined by the eigenfunctions *R*(*x*) and *I*(*x*) for the real and imaginary part respectively. A test to split the spatial-temporal

*i cos t t t cos t*

**Χ Χ**

covariance functions is given by (Wallaschek, 1988; Fuentes, 2006).

[ ( )sin( ) sin( ) ( )]}

*X X*

**Χ Χ**

*c c X X*

*cc c c X X*

**ΧΧ θ θ θ θ θ θ θθ**

**Χ Χ**

**ΧΧ Χ Χ θ θ θ θ**

*H c c N*

*T T c c c c*

*T T c c c c*

*cos t cos t t t*

{[ ( ) ( ) sin( )sin( )]

*cos t i t cos t i t*

[ ( ) sin( )][ ( ) sin( )]

respectively. Implicit in the model is the assumption that **X***c* can be represented as

*cR I* **XX X** ( ,) ( ,) ( ,) *xt xt i xt* (18)

*c c cc X X* **ΧΧ θ θ** [ ( ) ( )] *cos t isin t* (19)

<sup>1</sup> **C XX** (20)

(21)

for the data set has been derived.

matrix such as (Esquivel, 2009)

complex autocorrelation matrix becomes,

matrix; its eigenvectors are complex conjugate.

*H T*

*T*

From (20), It can be easily verified that

components.


The succeeding sections describe the properties of these representations to assess and to extract swing oscillations patterns and modal characteristics directly from recorded data in wide-area dynamical systems. It is shown that the proposed method can be used to predict the correct spatial location in the modal distribution of seismic wave.

#### **3.1 Spatial amplitude function, S(***x***)**

This function shows the spatial distribution of variability associated with each eigenmode. The spatial amplitude functions in the proposed model (25) are defined as (Hannchi, et al.,

Biorthogonal Decomposition for Wide-Area Wave Motion Monitoring Using Statistical Models 385

In this section, we turn our attention to the analysis of spatial and temporal behaviour of

In order to investigate travelling and standing features into a space-time varying field, the real physical field is reconstructed by taking the real part of the complex model given in

*Rj Rj Rj Ij Ij Ij Ij*

**X RS <sup>ω</sup> R S ω Κ** (30)

*xt t x t t x t x π* () () ( ) () () () ()

where **K**(*x*) is the wave number, and **ω***R*(*t*), **ω***I*(*t*) represent the angular frequency of the real and imaginary wave components, respectively. The wave number is only defined for travelling waves and its components in terms of the complex representation (25) are given by: **K**=*d*(**Φ**)/*dx*, with physical units of *rad.m*-1, and **ω**=*d*(**θ**)/*dt*, in *rad/s*. The relationship between complex modes and the wave motion is given from average phase speeds *c*R(*<sup>j</sup>*), *c*I(*j*)

From (30), it can be seen that the term associated with the *j*-th travelling wave component

*Ij Ij Ij Ij Ij Ij*

() () () () () ()

where we can see that the travelling wave components are also identified as the sum of two intermodulated standing wave components with negative sign. To obtain the decomposition of the original data field in its pure standing wave components, it is necessary to compute

> *p swc twc Rj Rj R j j xt xt xt t x t* 1 ( ,) ( ,) ( ,) cos

> > *xt t x t x π* () () () ()

where **X***swc* and **X***twc* represent the decomposition of the original field given by the pure standing and travelling wave components respectively. Furthermore, the damping factor of

From the modal decomposition given in (32-33), the statistical modes are also called orthogonal temporal and spatial modes respectively. Based in the proposed model, a

( , ) ( ) ( )cos( )

*twc Ij Ij Ij Ij*

*t x π*

**R S ω ω Κ**

cos( )

**X XX** *R S*

*q*

*j*

each mode is given by its amplitude.

1

practical criterion for choosing the relevant modes is given in the next section.

*Ij Ij Ij Ij*

() () () ()

cos( )

(31)

(32)

*- tx*

**R S ω Κ**

*t Κ x tx*


**X RS ω Κ** (33)

{sin( )sin( ) cos( )cos( )}

( , ) ( ) ( )cos( ) ( ) ( )cos( )

**4. Analysis of propagating features in space-time varying fields** 

(25), so, its wave form is given by (Esquivel & Messina, 2008; Esquivel, 2009)

*p q*

*j j*

*Ij Ij Ij Ij*

**R S ω Κ**

the difference with the pure travelling wave components as

() () () ()

1 1

propagating features in space-time varying fields.

**4.1 Space-time biorthogonal decomposition** 

obtained by using the relation *c*=**ω**/**K**, in *m*/*s*.

can be expressed as

with

2007; Marrifield & Guza, 1990; Susanto, et al., 1997; Terradas, et al., 2004; Toh, 1987; Barnett, 1983).,

$$\begin{aligned} \mathbf{S}\_{R(j)}(\mathbf{x}) &= \sqrt{\mathbf{q}\_{R(j)}^{\mathrm{H}}(\mathbf{x}) \mathbf{q}\_{R(j)}(\mathbf{x})} \\ \mathbf{S}\_{I(j)}(\mathbf{x}) &= \sqrt{\mathbf{q}\_{I(j)}^{\mathrm{H}}(\mathbf{x}) \mathbf{q}\_{I(j)}(\mathbf{x})} \end{aligned} \tag{26}$$

#### **3.2 Spatial phase function, (***x***)**

This function shows the relative phase fluctuation among various spatial locations where *u*(*x*,*t*) is defined, it is given by

$$\begin{aligned} \mathbf{OP}\_{R(j)}(\mathbf{x}) &= \tan^{-1} \left\{ \frac{\operatorname{im}[\boldsymbol{\Phi}\_{R(j)}(\mathbf{x})]}{\operatorname{rel}[\boldsymbol{\Phi}\_{R(j)}(\mathbf{x})]} \right\} \\ \boldsymbol{\Phi}\_{I(j)}(\mathbf{x}) &= \tan^{-1} \left\{ \frac{\operatorname{im}[\boldsymbol{\Phi}\_{I(j)}(\mathbf{x})]}{\operatorname{rel}[\boldsymbol{\Phi}\_{I(j)}(\mathbf{x})]} \right\} \end{aligned} \tag{27}$$

#### **3.3 Temporal amplitude function, R(***t***)**

This function gives a measure of the temporal variability in the magnitude of the modal structure in the original field. Similar to the description of the spatial amplitude function, the temporal amplitude function is defined as

$$\begin{aligned} \mathbf{R}\_{R(j)}(t) &= \sqrt{\mathbf{A}\_{R(j)}^{\mathrm{H}}(t)\mathbf{A}\_{R(j)}(t)}\\ \mathbf{R}\_{I(j)}(t) &= \sqrt{\mathbf{A}\_{I(j)}^{\mathrm{H}}(t)\mathbf{A}\_{I(j)}(t)} \end{aligned} \tag{28}$$

#### **3.4 Temporal phase function, θ(***t***)**

This function shows the temporal variation of the phase associated with the magnitude of the modal structure of *u*(*x*,*t*). It is given by

$$\begin{aligned} \boldsymbol{\Theta}\_{R(j)}(t) &= \tan^{-1} \left\{ \frac{\operatorname{im}[\mathbf{A}\_{R(j)}(t)]}{\operatorname{re}[\mathbf{A}\_{R(j)}(t)]} \right\} \\ \boldsymbol{\Theta}\_{I(j)}(t) &= \tan^{-1} \left\{ \frac{\operatorname{im}[\mathbf{A}\_{I(j)}(t)]}{\operatorname{re}[\mathbf{A}\_{I(j)}(t)]} \right\} \end{aligned} \tag{29}$$

Equations (26-29) provide a complete characterization of any propagating effects and periodicity in the original data field which might be obscured by standard cross-spectral analysis. These equations give a measure of the space-time distribution and can be used to identify the dominant modes and their phase relationships. Furthermore, for each dominant mode of interest, a mode shape can be computed by using the spatial part of (23). This method effectively decomposes the data into spatial and temporal modes.

#### **4. Analysis of propagating features in space-time varying fields**

In this section, we turn our attention to the analysis of spatial and temporal behaviour of propagating features in space-time varying fields.

#### **4.1 Space-time biorthogonal decomposition**

In order to investigate travelling and standing features into a space-time varying field, the real physical field is reconstructed by taking the real part of the complex model given in (25), so, its wave form is given by (Esquivel & Messina, 2008; Esquivel, 2009)

$$\mathbf{X}(\mathbf{x},t) = \sum\_{j=1}^{p} \mathbf{R}\_{R(j)}(t) \mathbf{S}\_{R(j)}(\mathbf{x}) \cos(\mathbf{o}\_{R(j)}t) + \sum\_{j=1}^{q} \mathbf{R}\_{I(j)}(t) \mathbf{S}\_{I(j)}(\mathbf{x}) \cos(\mathbf{o}\_{I(j)}t + \mathbf{K}\_{I(j)}\mathbf{x} + \pi) \tag{30}$$

where **K**(*x*) is the wave number, and **ω***R*(*t*), **ω***I*(*t*) represent the angular frequency of the real and imaginary wave components, respectively. The wave number is only defined for travelling waves and its components in terms of the complex representation (25) are given by: **K**=*d*(**Φ**)/*dx*, with physical units of *rad.m*-1, and **ω**=*d*(**θ**)/*dt*, in *rad/s*. The relationship between complex modes and the wave motion is given from average phase speeds *c*R(*<sup>j</sup>*), *c*I(*j*) obtained by using the relation *c*=**ω**/**K**, in *m*/*s*.

From (30), it can be seen that the term associated with the *j*-th travelling wave component can be expressed as

$$\begin{aligned} \mathbf{R}\_{I(j)} \mathbf{S}\_{I(j)} \cos(\mathfrak{o}\_{I(j)} t + \mathbf{K}\_{I(j)} \mathbf{x} + \pi) &= \\ \mathbf{R}\_{I(j)} \mathbf{S}\_{I(j)} [\sin(\mathfrak{o}\_{I(j)} t) \sin(\mathbf{K}\_{I(j)} \mathbf{x}) - \cos(\mathfrak{o}\_{I(j)} t) \cos(\mathbf{K}\_{I(j)} \mathbf{x})] &\approx \\ &= -\mathbf{R}\_{I(j)} \mathbf{S}\_{I(j)} \cos(\mathfrak{o}\_{I(j)} t + \mathbf{K}\_{I(j)} \mathbf{x}) \end{aligned} \tag{31}$$

where we can see that the travelling wave components are also identified as the sum of two intermodulated standing wave components with negative sign. To obtain the decomposition of the original data field in its pure standing wave components, it is necessary to compute the difference with the pure travelling wave components as

$$\mathbf{X}\_{\rm succ}(\mathbf{x},t) = \mathbf{X}(\mathbf{x},t) \cdot \mathbf{X}\_{\rm twc}(\mathbf{x},t) = \sum\_{j=1}^{p} \mathbf{R}\_{\aleph(j)}(t) \mathbf{S}\_{\aleph(j)}(\mathbf{x}) \cos \left(\mathbf{o}\_{\aleph(j)} t\right) \tag{32}$$

with

384 Numerical Modelling

2007; Marrifield & Guza, 1990; Susanto, et al., 1997; Terradas, et al., 2004; Toh, 1987;

*R j Rj Rj*

**S φ φ**

**S φ φ**

*R j*

( )

*x*

*x*

*I j*

*R j*

**θ**

**θ**

( )

*t*

*t*

method effectively decomposes the data into spatial and temporal modes.

*I j*

( )

( )

*x xx*

() () ()

(26)

(27)

(28)

(29)

*x xx* ( ) () ()

() () ()

*I j Ij Ij*

( ) () ()

This function shows the relative phase fluctuation among various spatial locations where

*R j*

*im x*

1 ( )

[ ( )] ( ) tan [ ()

**<sup>φ</sup> <sup>Φ</sup>**

**<sup>φ</sup> <sup>Φ</sup>**

[ ( )] ( ) tan [ ()

This function gives a measure of the temporal variability in the magnitude of the modal structure in the original field. Similar to the description of the spatial amplitude function,

*R j Rj Rj*

**R AA**

**R AA**

*t tt*

() () ()

*t tt* ( ) () ()

() () ()

*I j Ij Ij*

( ) () ()

This function shows the temporal variation of the phase associated with the magnitude of

*R j*

*im t*

**A**

**A A**

1 ( )

[ ( )] ( ) tan [ ()

[ ( )] ( ) tan [ ()

Equations (26-29) provide a complete characterization of any propagating effects and periodicity in the original data field which might be obscured by standard cross-spectral analysis. These equations give a measure of the space-time distribution and can be used to identify the dominant modes and their phase relationships. Furthermore, for each dominant mode of interest, a mode shape can be computed by using the spatial part of (23). This

1 ( )

*re t im t*

*R j I j*

( )

*I j*

( )

*re t*

**A**

1 ( )

*R j I j*

( )

*re x im x*

**φ**

*I j*

( )

*re x*

**φ**

Barnett, 1983).,

**3.2 Spatial phase function, (***x***)** 

**3.3 Temporal amplitude function, R(***t***)** 

**3.4 Temporal phase function, θ(***t***)** 

the modal structure of *u*(*x*,*t*). It is given by

the temporal amplitude function is defined as

*u*(*x*,*t*) is defined, it is given by

$$\mathbf{X}\_{\text{true}}(\mathbf{x},t) = \sum\_{j=1}^{q} \mathbf{R}\_{I(j)}(t)\mathbf{S}\_{I(j)}(\mathbf{x})\cos(\mathfrak{so}\_{I(j)}t + \mathbf{K}\_{I(j)}\mathbf{x} + \pi) \tag{33}$$

where **X***swc* and **X***twc* represent the decomposition of the original field given by the pure standing and travelling wave components respectively. Furthermore, the damping factor of each mode is given by its amplitude.

From the modal decomposition given in (32-33), the statistical modes are also called orthogonal temporal and spatial modes respectively. Based in the proposed model, a practical criterion for choosing the relevant modes is given in the next section.

Biorthogonal Decomposition for Wide-Area Wave Motion Monitoring Using Statistical Models 387

space-time varying field. In an effort to better understand the mechanism of wave propagating using complex EOF analysis, in the next section is presented a first example as illustrative case to determine the theoretical fundamentals from the complex EOF

**5. Motivating examples: Modeling of propagating wave using the covariance** 

As illustrative case, in this section we consider a first example of wave propagation to study the modelling of propagating wave using the complex EOF analysis (Marrifield, 1990).

For simplicity, we consider a nondispersive plane wave propagating at phase speed *c*, and

*j jj* ( ) [ ( )cos( ) ( )sin( )] *t kx t kx t*

 (37)

(38)

2 1

 

> 

2

 

1

(39)

(40)

(41)

 

*j jj j jj a kx kx b kx kx* ( ) ( )cos( ) ( )sin( ) ( ) ( )sin( ) ( )cos( )

*jj j* ( ) ( )cos( ) ( )sin( ) *t a tb t*

*if if*

2 2 2 1 2

To obtain phase information between stations, a complex representation of (39) is invoked. Its complex covariance matrix *jk j k <sup>t</sup> t t* \* **<sup>C</sup>** *u u* () () , where *<sup>t</sup>* denotes time averaging and

 

*a ib e a ib e*

[ ( ) ( )] \* [ ( ) ( )]

[ ( ) ( ) ( ) ( )] [ ( ) ( ) ( ) ( )]

*jk ik x*

**<sup>C</sup>** (42)

*jk jk jk jk*

*a a b b ia b b a*

2 2 ( ) cos( ) sin( )

, () () 0, ,

 

 

> 

 

*i t i t*

 

wavenumber as *k*= *ω*/*c*, past an array of sensors at positions *j* given by

 

 

 

*<sup>u</sup>*

the asterisk complex conjugation, is given as

**C**

where, for this example, *uj*(*t*) is a white, band-limited signal given by

 

[ ( ) ( )] \* [ ( ) ( )] ,

 

*jk j k j k kx kx i kx kx e*

*jj kk*

*a ib a ib*

which, simplifying and using condition given in (40), **C***jk* can be rewritten as

*jk j j k k*

*<sup>u</sup>*

Expanding (37) and using identities

we can rewritten (37) as

analysis.

**matrix** 

#### **4.2 Approximation order and energy distribution in the space-time varying modes**

The relationship between spatial and temporal behaviour in space-time varying fields can be obtained by noting that the spatio-temporal information can be mapping into a space and time grid, i.e., each component *u*(*x*,*t*) of the space-time varying field is represented by the field value at time *t* and spatial position *x*. Based in the proposed biorthogonal method, the analysis is used to determine the spatial and temporal energy distribution in the space-time varying field, a criterion for choosing the number of relevant modes from proposed model is given by the energy percentage contained in the *p* and *q* dominant modes of the form

$$\begin{aligned} \%E(p,q) &= \frac{\sum\_{j=1}^{p} \lambda\_{(j)\text{succ}} + \sum\_{j=1}^{q} \lambda\_{(j)\text{true}}}{\left\| \mathbf{X} \right\|\_{F}^{2}} \times 100 = 99\,\% \\ &= \text{subject to } \text{argmin}\{ E(p,q) : E(p,q) \ge E\_{0} \} \end{aligned} \tag{34}$$

where *<sup>F</sup> .* 2 denotes the Frobenius norm, *E*0 is an appropriate energy level, and *0 %E(p,q) 100* is the percentage of energy that is captured by the optimal basis. By neglecting modes corresponding to the small eigenvalues a reduced-order model can be constructed (Esquivel, 2009; Messina, et al. 2010).

We note from (34) that *<sup>F</sup> <sup>E</sup>* <sup>2</sup> **<sup>X</sup>** ; so the spatial-temporal energy distribution can be computed by

$$\%E\_{\text{swc}} = \frac{\left\|\mathbf{X}\_{\text{suc}}\right\|\_{F}^{2}}{\left\|\mathbf{X}\right\|\_{F}^{2}} \times 100\tag{35}$$

which is associated with the temporal energy distribution, and

$$\% \acute{o} E\_{\text{rev}} = \frac{\left\| \mathbf{X}\_{\text{true}} \right\|\_{F}^{2}}{\left\| \mathbf{X} \right\|\_{F}^{2}} \times 100 \tag{36}$$

is associated with the spatial energy distribution. Figure 2 shows a conceptual representation of spatial and temporal variability illustrating the energy distribution in a

Fig. 2. Three-dimensional view of energy distribution of a time-space varying field.

space-time varying field. In an effort to better understand the mechanism of wave propagating using complex EOF analysis, in the next section is presented a first example as illustrative case to determine the theoretical fundamentals from the complex EOF analysis.

#### **5. Motivating examples: Modeling of propagating wave using the covariance matrix**

As illustrative case, in this section we consider a first example of wave propagation to study the modelling of propagating wave using the complex EOF analysis (Marrifield, 1990).

For simplicity, we consider a nondispersive plane wave propagating at phase speed *c*, and wavenumber as *k*= *ω*/*c*, past an array of sensors at positions *j* given by

$$\mu\_{\dot{j}}(t) = \sum\_{o} \left[ \alpha(o) \cos(k x\_{\dot{j}} - o t) + \beta(o) \sin(k x\_{\dot{j}} - o t) \right] \tag{37}$$

Expanding (37) and using identities

$$\begin{aligned} a\_j(o) &= \alpha(o)\cos(k\mathbf{x}\_j) + \beta(o)\sin(k\mathbf{x}\_j) \\ b\_j(o) &= \alpha(o)\sin(k\mathbf{x}\_j) - \beta(o)\cos(k\mathbf{x}\_j) \end{aligned} \tag{38}$$

we can rewritten (37) as

386 Numerical Modelling

**4.2 Approximation order and energy distribution in the space-time varying modes**  The relationship between spatial and temporal behaviour in space-time varying fields can be obtained by noting that the spatio-temporal information can be mapping into a space and time grid, i.e., each component *u*(*x*,*t*) of the space-time varying field is represented by the field value at time *t* and spatial position *x*. Based in the proposed biorthogonal method, the analysis is used to determine the spatial and temporal energy distribution in the space-time varying field, a criterion for choosing the number of relevant modes from proposed model is given by the energy percentage contained in the *p* and *q* dominant modes of the form

*p q*

*λ λ*

*j j*

 

*Epq*

constructed (Esquivel, 2009; Messina, et al. 2010).

which is associated with the temporal energy distribution, and

where *<sup>F</sup> .*

computed by

*j swc j twc*

( ) ( ) 1 1 2

%(,) 100 99%

*F*

*Epq Epq E*

subject toargmin{ ( , ) : ( , ) }

2 denotes the Frobenius norm, *E*0 is an appropriate energy level, and

*0 %E(p,q) 100* is the percentage of energy that is captured by the optimal basis. By neglecting modes corresponding to the small eigenvalues a reduced-order model can be

We note from (34) that *<sup>F</sup> <sup>E</sup>* <sup>2</sup> **<sup>X</sup>** ; so the spatial-temporal energy distribution can be

**X X**

**X X**

is associated with the spatial energy distribution. Figure 2 shows a conceptual representation of spatial and temporal variability illustrating the energy distribution in a

Fig. 2. Three-dimensional view of energy distribution of a time-space varying field.

*swc F F*

*twc F F*

2 <sup>2</sup> 100

2 <sup>2</sup> 100 0

**X** (34)

*%E = swc* (35)

*%E = twc* (36)

$$\mu\_j(t) = \sum\_{\alpha} \left( a\_j(\alpha) \cos(\alpha t) + b\_j(\alpha) \sin(\alpha t) \right) \tag{39}$$

where, for this example, *uj*(*t*) is a white, band-limited signal given by

$$\alpha^2(\alpha) + \beta^2(\alpha) = \begin{cases} \mathbf{A}^2, & \text{if} \quad \alpha\_1 \le \alpha \le \alpha\_2 \\ 0, & \text{if} \quad \alpha > \alpha\_{2'}, \alpha < \alpha\_1 \end{cases} \tag{40}$$

To obtain phase information between stations, a complex representation of (39) is invoked. Its complex covariance matrix *jk j k <sup>t</sup> t t* \* **<sup>C</sup>** *u u* () () , where *<sup>t</sup>* denotes time averaging and the asterisk complex conjugation, is given as

$$\begin{split} \mathbf{C}\_{jk} &= \sum\_{\alpha} \left\{ \left[ a\_j(\alpha) - ib\_j(\alpha) \right] e^{i\alpha t} \right\} \ast \sum\_{\alpha} \left\{ \left[ a\_k(\alpha) + ib\_k(\alpha) \right] e^{-i\alpha t} \right\} \\ &= \sum\_{\alpha} \left\{ \left[ a\_j(\alpha) - ib\_j(\alpha) \right] \ast \left[ a\_k(\alpha) + ib\_k(\alpha) \right] \right\}, \end{split} \tag{41}$$
 
$$\begin{split} \mathbf{C} &= \sum\_{\alpha} \left\{ \left[ a\_j(\alpha) a\_k(\alpha) + b\_j(\alpha) b\_k(\alpha) \right] + i \left[ a\_j(\alpha) b\_k(\alpha) - b\_j(\alpha) a\_k(\alpha) \right] \right\} \end{split} \tag{42}$$

which, simplifying and using condition given in (40), **C***jk* can be rewritten as

$$\mathbf{C}\_{jk} = \mathbf{A}^2 \sum\_{o} \left\{ \cos(k\mathbf{x}\_j - k\mathbf{x}\_k) - i \sin(k\mathbf{x}\_j - k\mathbf{x}\_k) \right\} = \mathbf{A}^2 \sum\_{o=o\_1}^{o\_2} e^{-ik(o)\Delta\mathbf{x}\_{jk}} \tag{42}$$

Biorthogonal Decomposition for Wide-Area Wave Motion Monitoring Using Statistical Models 389

*k x k x c c e*

*k x k x*

*ik x ik x e e and* 1,1 1,2

1,1 2,1 2,1 1,1

1,2 2,2 2,2 1,2

, with

1. As can be seen from (51) and (52), the method yields complex conjugate eigenvectors

2. The value *<sup>j</sup> kx* gives the mode shape; this can be used for detection of wave propagating into original field and can be useful to identify the dominant stations involved in the propagating wave of dynamical oscillations. We remark that the performance of the complex EOF analysis as measured by the percentage of variance given in (47) depends on the spread in wave number relative to the array size, as the parameter *k x*

decreases, more of the variance is contained in the lowest complex EOF modes. 3. A point of particular interest is that, as a standard technique for describing coherent variability in spatial data, a relatively wide number bandwidth [ 0(2 )] *k x*

The development given in this section indicates that modal spatial patterns from a time domain complex EOF analysis may be computed in a straightforward manner. In the next section, data obtained from GPS-based multiple phasor measurements units from a real event of seismic wave of an earthquake are used to study the practical applicability of the method to characterize spatio-temporal behaviour in wide-area systems. Additionally, we discuss the practical computation of mode shape identification in relation to the proposed decomposition from measurements data that can be used to identify coherence groups in

vast wide-area interconnected systems where the propagating wave are given.

, with

2,1 2,2 1 1 0 0 , 0 0 0 0 0 0 

*ik x ik x*

*e if e*

, ,

*ikx*

*ikx*

2

*e*

2

*e*

*ik x ik x*

*e e*

, if ,

2 2 2,2

2 2

*ik x*

*ikx*

1

2

*e*

1,1 2,1

1,2 2,2

and an average wave number, *k* .

from (47).

*e*

*ikx*

*ikx*

*e*

*e*

The following observations can be made from the analysis:

*ikx*

2

1

sin sin

sin sin

2 2

which can be simplified to

where

and

*ce c*

*ik x*

2 2 0

1,2

0

(49)

(50)

(51)

(52)

results

where *jk <sup>j</sup> <sup>k</sup> x xx* . Replacing the summatory of (42) with an integral, yields

$$\mathbf{C}\_{jk} = \frac{\mathbf{A}^2}{\Delta \alpha \rho} \int\_{\alpha\_1}^{\alpha\_2} e^{-ik(\alpha)\Delta x\_{jk}} d\alpha \quad \text{to} \quad \Delta \alpha = \frac{2\pi}{\mathbf{T}} \tag{43}$$

Integrating (43) by parts and after some algebra, we can show that

$$\mathbf{C}\_{jk} = \mathbf{A}^2 \mathbf{M} \sin c \left(\frac{\Delta k \Delta \mathbf{x}\_{jk}}{2}\right) e^{-i\overline{k} \Delta \mathbf{x}\_{jk}} \tag{44}$$

with

$$\begin{aligned} \Delta k &= k(\alpha\_2) - k(\alpha\_1) \\ \overline{k} &= \frac{k(\alpha\_2) + k(\alpha\_1)}{2} \\ \mathbf{M} &= \frac{\mathbf{T}(\alpha\_2 - \alpha\_1)}{2\pi} \end{aligned} \tag{45}$$

where *k* is the wave number bandwith, *x* is the array length and M is the frequency.

Equation (44) illustrates some important properties of **C**. General algebraic expression in order to computating the eigenvalues and eigenfunctions of **C** for an arbitrary number of sensors (*n*), are very difficult to determine and which are not purposed here. For the case of two sensors, the above model can be reduced to

$$\mathbf{C}\_{jk} = \begin{bmatrix} \mathbf{A}^2 \mathbf{M} & \mathbf{A}^2 \mathbf{M} \sin c \left( \frac{\Delta k (\mathbf{x}\_1 - \mathbf{x}\_2)}{2} \right) e^{-\overline{\mathbf{x}} \cdot (\mathbf{x}\_1 - \mathbf{x}\_2)} \\\\ \mathbf{A}^2 \mathbf{M} \sin c \left( \frac{\Delta k (\mathbf{x}\_2 - \mathbf{x}\_1)}{2} \right) e^{-\overline{\mathbf{x}} \cdot (\mathbf{x}\_2 - \mathbf{x}\_1)} & \mathbf{A}^2 \mathbf{M} \end{bmatrix}, j, k = 1, 2 \tag{46}$$

From the relation above is followed that the eigenvalues of **C** are given by det[ ] **C***<sup>j</sup>*,*<sup>k</sup>* , i.e., it is easy to see further that

$$\mathcal{A}\_{1,2} = \mathbf{A}^2 \mathbf{M} \pm \mathbf{A}^2 \mathbf{M} \left| \sin c \left( \frac{\Delta k \Delta \chi}{2} \right) \right| = \mathbf{A}^2 \mathbf{M} \left[ 1 \pm \left| \sin c \left( \frac{\Delta k \Delta \chi}{2} \right) \right| \right] \tag{47}$$

It then follows that the eigenvectors of **C** defined as *j k*, [ ]0 **C** , are given by

$$\begin{bmatrix} \mathbf{A}^2 \mathbf{M} \text{sinc} \mathbf{c} \left(\frac{\Delta k \Delta \mathbf{x}}{2}\right) & -\mathbf{A}^2 \mathbf{M} \text{sinc} \mathbf{c} \left(\frac{\Delta k \Delta \mathbf{x}}{2}\right) e^{-\overline{k} \Delta \mathbf{x}} \\\\ -\mathbf{A}^2 \mathbf{M} \text{sinc} \mathbf{c} \left(\frac{\Delta k \Delta \mathbf{x}}{2}\right) e^{\overline{k} \Delta \mathbf{x}} & \mathbf{A}^2 \mathbf{M} \text{sinc} \mathbf{c} \left(\frac{\Delta k \Delta \mathbf{x}}{2}\right) \end{bmatrix} \begin{bmatrix} \mathbf{B}\_{1,1} \\\\ \mathbf{B}\_{2,1} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \tag{48}$$

and

$$
\begin{bmatrix}
\end{bmatrix}
\begin{bmatrix}
\mathbf{B}\_{1,2} \\
\mathbf{B}\_{2,2}
\end{bmatrix} = \begin{bmatrix}
0 \\
0
\end{bmatrix}
\tag{49}
$$

which can be simplified to

$$
\begin{bmatrix} \mathbf{1} & -e^{-\overline{k}\mathbf{A}\mathbf{x}} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{B}\_{1,1} \\ \mathbf{B}\_{2,1} \end{bmatrix} = \begin{bmatrix} \mathbf{0} \\ \mathbf{0} \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} \mathbf{1} & e^{-\overline{k}\mathbf{A}\mathbf{x}} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{B}\_{1,2} \\ \mathbf{B}\_{2,2} \end{bmatrix} = \begin{bmatrix} \mathbf{0} \\ \mathbf{0} \end{bmatrix} \tag{50}
$$

where

388 Numerical Modelling

<sup>2</sup> ( ) 2 , to

*k x c e* <sup>2</sup> sin 2 

> 2 1 2 1

 

*j k kx x*

<sup>2</sup> , , 1,2 ( ) sin

2 2 1 2 ( )

() () () () 2 ( ) 2

2 1

*kk k k k <sup>k</sup>*

 

 

where *k* is the wave number bandwith, *x* is the array length and M is the frequency.

 

From the relation above is followed that the eigenvalues of **C** are given by det[ ]

*c c* 2 2 <sup>2</sup> 1,2 sin 1 sin

> *k x k x c c e*

*k x k x*

2 2

2 2 2,1

**C** (46)

*ik x x*

2 2 2 1 ( )

It then follows that the eigenvectors of **C** defined as *j k*, [ ]0

2 2

*ik x*

sin sin

sin sin

*ce c*

2 1

Equation (44) illustrates some important properties of **C**. General algebraic expression in order to computating the eigenvalues and eigenfunctions of **C** for an arbitrary number of sensors (*n*), are very difficult to determine and which are not purposed here. For the case of

 

*jk jk ik x*

**<sup>C</sup>** (43)

**C** (44)

*kx x c e*

( ) sin

*k x k x*

2 2

2 2 0

(45)

*ik x x*

**C** , are given by

1,1

0

*ik x*

1 2

**C***<sup>j</sup>*,*<sup>k</sup>* ,

(47)

(48)

where *jk <sup>j</sup> <sup>k</sup> x xx* . Replacing the summatory of (42) with an integral, yields

*jk e d* 2

1

Integrating (43) by parts and after some algebra, we can show that

*jk*

two sensors, the above model can be reduced to

*c e*

2

with

*jk*

and

i.e., it is easy to see further that

*jk ik x*

$$\begin{aligned} \mathbf{B}\_{1,1} &= \mathbf{B}\_{2,1} e^{-\overline{K}\mathbf{A}\mathbf{x}}, \quad \text{if} \quad \mathbf{B}\_{2,1} = \mathbf{T}, \quad \mathbf{B}\_{1,1} = \mathbf{T} e^{-\overline{K}\mathbf{A}\mathbf{x}}\\ \begin{bmatrix} \mathbf{B}\_{1,1} \\ \mathbf{B}\_{2,1} \end{bmatrix} &= \begin{bmatrix} e^{-\overline{K}\mathbf{x}\_1} \\ e^{-\overline{K}\mathbf{x}\_2} \end{bmatrix} \mathbf{T}\_{\prime} \quad \text{with} \quad \mathbf{T} = e^{-\overline{K}\mathbf{x}\_2} \end{aligned} \tag{51}$$

and

$$\begin{aligned} \mathbf{B}\_{1,2} &= -\mathbf{B}\_{2,2} e^{-i\overline{\mathbf{K}}\mathbf{A}\mathbf{x}}, \quad \text{if} \quad \mathbf{B}\_{2,2} = -\mathbf{T}, \quad \mathbf{B}\_{1,2} = \mathbf{T} e^{-i\overline{\mathbf{K}}\mathbf{A}\mathbf{x}} \\ \begin{bmatrix} \mathbf{B}\_{1,2} \\ \mathbf{B}\_{2,2} \end{bmatrix} &= \begin{bmatrix} e^{-i\overline{\mathbf{K}}\mathbf{x}\_{1}} \\ -e^{-i\overline{\mathbf{K}}\mathbf{x}\_{2}} \end{bmatrix} \mathbf{T}\_{\prime} \quad \text{with} \quad \mathbf{T} = e^{-i\overline{\mathbf{K}}\mathbf{x}\_{2}} \end{aligned} \tag{52}$$

The following observations can be made from the analysis:


The development given in this section indicates that modal spatial patterns from a time domain complex EOF analysis may be computed in a straightforward manner. In the next section, data obtained from GPS-based multiple phasor measurements units from a real event of seismic wave of an earthquake are used to study the practical applicability of the method to characterize spatio-temporal behaviour in wide-area systems. Additionally, we discuss the practical computation of mode shape identification in relation to the proposed decomposition from measurements data that can be used to identify coherence groups in vast wide-area interconnected systems where the propagating wave are given.

Biorthogonal Decomposition for Wide-Area Wave Motion Monitoring Using Statistical Models 391

During the time interval 15:36:14.730-15:40:42 the earthquake experiment severe fluctuations where its seismic wave components such as frequency and amplitude were felt. Figures 4,5 and 6 give an extraction from PMUs measurements of this event showing the observed oscillations in the selected stations, where for simplicity, the seismic wave features are selected in longitude, latitude and altitude components. As a first step towards the development of the proposed methodology, the observed records are placed in a data matrix representing equally spaced measurements in sixteen different geographical locations. For our simulation, 45000 snapshots are available. Each time series is then augmented with an imaginary component by the Hilbert analysis to provide phase information and the corresponding birthogonal decomposition is applied to the dataset. System measurements in Figs. 4,5 and 6 demonstrate significant variability suggesting a nonstationary process in both space and time. Furthermore, in these figures are shown the associated mode to the travelling wave components based in the proposed method of birthogonal decomposition. The results clearly show the seismic wave decomposition, it is evident that the travelling wave mode in longitude and latitude is quite prominent at CICEJ and Oblatos stations, while that in the Ciudad Guzman and Santa Rosa stations are more stronger in altitude. A point of particular interest is the agreement between the results from the proposed model and the real behaviour of the space-time variability presented during the seismic wave. In (Ortiz, et al., 1998) was analyzed the tsunami data generated by the Colima-Jalisco earthquake, where the results of the tsunami arrival time are consistent with

Fig. 3. Schematic showing the location of stations of the PMUs.

the presented in this analysis.

### **6. Complex EOF analysis to wide-area system oscillatory dynamics**

This section examines the application of the proposed technique to assess oscillations patterns in dynamical systems. Attention is focused on the identification of critical modes and the associated areas involved in the oscillations. In order to test the ability of the method to analyze complex oscillations, we use data recorded from time-synchronized measurements. The data were obtained from the Geophysical Institute of the National Autonomous University of México. A brief description of the data is given below.

At local time 15:36:14.730, October 9, 1995 a submarine earthquake was occurred near the Mexican coast (Colima-Jalisco); this earthquake was recorded by sixteen stations of phasor measurement units (PMUs) over a 225 s window sampled with time interval of 0.005 s during its propagating that was felt over much of Jalisco and parts of Colima. We examined evidence of seismic wave arrival times of the earthquake in PMUs based in global positing system (GPS). For simplicity, in Table 1 is given the description of the locations of each station. This earthquake was located at a depth of 5km about (18.740ºN, 104.670ºW). Figure 3 shows with a geographical diagram the PMUs locations and the location of the event.


Table 1. Description and location of stations of phasor measurements units.

This section examines the application of the proposed technique to assess oscillations patterns in dynamical systems. Attention is focused on the identification of critical modes and the associated areas involved in the oscillations. In order to test the ability of the method to analyze complex oscillations, we use data recorded from time-synchronized measurements. The data were obtained from the Geophysical Institute of the National

At local time 15:36:14.730, October 9, 1995 a submarine earthquake was occurred near the Mexican coast (Colima-Jalisco); this earthquake was recorded by sixteen stations of phasor measurement units (PMUs) over a 225 s window sampled with time interval of 0.005 s during its propagating that was felt over much of Jalisco and parts of Colima. We examined evidence of seismic wave arrival times of the earthquake in PMUs based in global positing system (GPS). For simplicity, in Table 1 is given the description of the locations of each station. This earthquake was located at a depth of 5km about (18.740ºN, 104.670ºW). Figure 3 shows with a geographical diagram the PMUs locations and the location of the event.

PMUs Station Latitude N Longitude W Altitude (msnm) 1 Ciudad Guzman 19.6º 103.4º 1507 2 Santa Rosa corona centro 20.912º 103.708º 770 3 Santa Rosa margen izquierda 20.912º 103.708º 780 4 Ciudad Granja 20.672º 103.398º 1680 5 Jardines del sur 20.648º 103.366º 1583 6 Arcos 20.671º 103.362º 1585 7 Obras publicas Zapopan 20.699º 103.361º 1561 8 Miravalle 20.633º 103.342º 1610 9 Rotonda 20.673º 103.34º 1542 10 San Rafael 20.654º 103.311º 1560 11 Planetario 20.717º 103.308º 1543 12 Tonala 20.641º 103.279º 1660 13 CICEJ superficie 20.6º 103.2º 1575 14 CICEJ pozo 9m 20.6º 103.2º 1566 15 CICEJ pozo 35m 20.6º 103.2º 1540 16 Oblatos 20.6º 103.2º 1580

Earthquake 18.740º 104.670º 5km (depth)

Table 1. Description and location of stations of phasor measurements units.

Location

**6. Complex EOF analysis to wide-area system oscillatory dynamics** 

Autonomous University of México. A brief description of the data is given below.

Fig. 3. Schematic showing the location of stations of the PMUs.

During the time interval 15:36:14.730-15:40:42 the earthquake experiment severe fluctuations where its seismic wave components such as frequency and amplitude were felt. Figures 4,5 and 6 give an extraction from PMUs measurements of this event showing the observed oscillations in the selected stations, where for simplicity, the seismic wave features are selected in longitude, latitude and altitude components. As a first step towards the development of the proposed methodology, the observed records are placed in a data matrix representing equally spaced measurements in sixteen different geographical locations. For our simulation, 45000 snapshots are available. Each time series is then augmented with an imaginary component by the Hilbert analysis to provide phase information and the corresponding birthogonal decomposition is applied to the dataset. System measurements in Figs. 4,5 and 6 demonstrate significant variability suggesting a nonstationary process in both space and time. Furthermore, in these figures are shown the associated mode to the travelling wave components based in the proposed method of birthogonal decomposition. The results clearly show the seismic wave decomposition, it is evident that the travelling wave mode in longitude and latitude is quite prominent at CICEJ and Oblatos stations, while that in the Ciudad Guzman and Santa Rosa stations are more stronger in altitude. A point of particular interest is the agreement between the results from the proposed model and the real behaviour of the space-time variability presented during the seismic wave. In (Ortiz, et al., 1998) was analyzed the tsunami data generated by the Colima-Jalisco earthquake, where the results of the tsunami arrival time are consistent with the presented in this analysis.

Biorthogonal Decomposition for Wide-Area Wave Motion Monitoring Using Statistical Models 393

From this figure is evident that the earthquake was feeling with fluctuating components

Spectral analysis results for the leading travelling wave shows that the main power is concentred in oscillations with frequencies about 4.8, 5.2 and 4.0 Hz, to the longitude, latitude and altitude components which are associated with the major time interval of the

Fig. 7. Spectrograms to the seismic wave from the longitude, latitude and altitude

finally the Ciudad Guzman, Los Arcos, CICEJ (1,6,13,14) stations in altitude.

One of the most attractive features of the proposed technique is its ability to detect changes in the mode shape properties of critical modes arising from systems. Changes in the mode shape may indicate changes in topology of the dynamic systems and may be useful for the design of special protection systems. This is a problem that has been recently addressed

Using the spatial phase and amplitude (the mode shape), the phase relationship between key system locations can be determined. In this analysis, we display the complex values as a vector with the length of its arrow proportional to eigenvector magnitude and direction equal to the eigenvector phase. Figure 8 shows the mode shape for the three travelling wave computed from the longitude, latitude and altitude components for the seismic wave, this information is useful to identify the dominant stations involved in the oscillations. Simulation results to the mode shape clearly show that the CICEJ (13,14,15) stations are more stronger evident at the longitude components; CICEJ (13,14) stations in latitude, and

components using the travelling wave modes.

using spectral correlation analysis (Wallaschek, 1988).

after 10 s since it was occurred.

seismic wave.

Fig. 4. Seismic fluctuating components in longitude and the leading mode showing spatiotemporal variability in the location of stations.

Fig. 5. Seismic fluctuating components in latitude and the leading mode showing spatiotemporal variability in the location of stations.

Fig. 6. Seismic fluctuating components in altitude and the leading mode showing spatiotemporal variability in the location of stations.

Additional insight into the frequency variability of the seismic oscillations can be obtained from the analysis of instantaneous frequency. Recognizing that the instantaneous frequency is the derivative of the temporal phase function given from the proposed model (30), the instantaneous frequency is estimated for each mode of concern. However, other approach can be used to characterize the spectral behaviour that requires other analytical formulations (Ortiz, et al., 2000).

The study focuses on the travelling wave mode which is the mode that captures most of the variability in the seismic wave. Figure 7 gives the spectrogram of the travelling wave modes associated to the longitude, latitude and altitude for the interval of interest in this study.

Fig. 4. Seismic fluctuating components in longitude and the leading mode showing spatio-

Fig. 5. Seismic fluctuating components in latitude and the leading mode showing spatio-

Fig. 6. Seismic fluctuating components in altitude and the leading mode showing spatio-

Additional insight into the frequency variability of the seismic oscillations can be obtained from the analysis of instantaneous frequency. Recognizing that the instantaneous frequency is the derivative of the temporal phase function given from the proposed model (30), the instantaneous frequency is estimated for each mode of concern. However, other approach can be used to characterize the spectral behaviour that requires other analytical formulations

The study focuses on the travelling wave mode which is the mode that captures most of the variability in the seismic wave. Figure 7 gives the spectrogram of the travelling wave modes associated to the longitude, latitude and altitude for the interval of interest in this study.

temporal variability in the location of stations.

temporal variability in the location of stations.

temporal variability in the location of stations.

(Ortiz, et al., 2000).

From this figure is evident that the earthquake was feeling with fluctuating components after 10 s since it was occurred.

Spectral analysis results for the leading travelling wave shows that the main power is concentred in oscillations with frequencies about 4.8, 5.2 and 4.0 Hz, to the longitude, latitude and altitude components which are associated with the major time interval of the seismic wave.

Fig. 7. Spectrograms to the seismic wave from the longitude, latitude and altitude components using the travelling wave modes.

One of the most attractive features of the proposed technique is its ability to detect changes in the mode shape properties of critical modes arising from systems. Changes in the mode shape may indicate changes in topology of the dynamic systems and may be useful for the design of special protection systems. This is a problem that has been recently addressed using spectral correlation analysis (Wallaschek, 1988).

Using the spatial phase and amplitude (the mode shape), the phase relationship between key system locations can be determined. In this analysis, we display the complex values as a vector with the length of its arrow proportional to eigenvector magnitude and direction equal to the eigenvector phase. Figure 8 shows the mode shape for the three travelling wave computed from the longitude, latitude and altitude components for the seismic wave, this information is useful to identify the dominant stations involved in the oscillations. Simulation results to the mode shape clearly show that the CICEJ (13,14,15) stations are more stronger evident at the longitude components; CICEJ (13,14) stations in latitude, and finally the Ciudad Guzman, Los Arcos, CICEJ (1,6,13,14) stations in altitude.

Biorthogonal Decomposition for Wide-Area Wave Motion Monitoring Using Statistical Models 395

technique is especially attractive, because, it does not require previous notion about the behaviour associated with abrupt changes in system topology or operating conditions.

Complex empirical orthogonal function analysis is shown to be a useful method to identify standing and travelling patterns in wide-area system measurements. In the use of information in interconnected systems, spatio-temporal analysis of wide-area timesynchronized measurements shows that transient oscillations may manifest highly complex phenomena, including nonstatonary behaviour. Numerical results show that the proposed method can provide accurate estimation of nonstationary effects, modal frequency, mode shapes, and time instants of intermittent transient responses. This information is important

The main contributions in this chapter are based in the estimation of propagating and standing features in space-time varying processes using statistical techniques to identify oscillatory activity in interconnected systems through the use of wide-area monitoring

The proposed technique is based on the complex correlation structure from space-time varying fields, which can treat both, spatial and temporal information; this provides a global picture on the system behaviour to characterize oscillatory dynamics. Its significant drawbacks are associated to treat with their space-time scales. These include geographical distribution and the time interval to the modal extraction using measured data. For some applications may be desirable to have these data at very high space-time resolution that allow the study of processes close to inertial frequency, then a technique of space-time interpolation can be used. Wide-area monitoring may prove invaluable in interconnected system dynamic studies by giving a quick assessment of the damping and frequency content of dominant system modes after a critical contingency. The alternative technique based on space-time dependent complex EOF analysis of measured data is proposed to resolve the localized nature of transient processes and to extract dominant temporal and spatial

This work was supported under a scholarship granted by CONACYT México no. 172551 to CINVESTAV Guadalajara. We thank to the Geophysical Institute of the National Autonomous University of México for their support and facility to use the data presented in

Achatz, U.; Schmitz, G. & Greisiger, K.-M. (1995). Principal Interaction Patterns in Baroclinic

Aubry, N.; Guyonnet, R. & Lima, R. (1990). Spatiotemporal Analysis of Complex Signals:

Wave Life Cycles, *Journal of the Atmospheric Sciences*, Vol.52, No.18, (September

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to determine strategies for wide-area monitoring and special protection systems.

schemes in interconnected systems.

information.

this work.

**9. References** 

1995), pp. 3201-3213, ISSN 15200469

1991), pp. 683-738, ISSN 00224715

**8. Acknowledgment** 

These results are in general agreement with the shown in Figs. 4,5 and 6 from the observed oscillations giving validity to the results. The new results provide clarification on the exact phase relationship between key stations as a function of space.

Fig. 8. Mode shape fluctuating in longitude, latitude and amplitude of the leading mode showing the phase relationship between stations.

#### **7. Conclusion**

Approaches for detection of propagation features in space-time varying system measurements through its travelling and standing components are proposed.

The conceptual framework developed provides bases for the analysis, detection, and simplification of seismic wave components through use of wide-area monitoring schemes such as global positioning systems (GPS) based in multiple phasor measurements units (PMUs) for interconnected systems, and enables the simultaneous study of synchronized measurements. The main advantage of the approach is its ability to compress the variability of large data sets into the fewest possible number of spatial and temporal modes. The technique is especially attractive, because, it does not require previous notion about the behaviour associated with abrupt changes in system topology or operating conditions.

Complex empirical orthogonal function analysis is shown to be a useful method to identify standing and travelling patterns in wide-area system measurements. In the use of information in interconnected systems, spatio-temporal analysis of wide-area timesynchronized measurements shows that transient oscillations may manifest highly complex phenomena, including nonstatonary behaviour. Numerical results show that the proposed method can provide accurate estimation of nonstationary effects, modal frequency, mode shapes, and time instants of intermittent transient responses. This information is important to determine strategies for wide-area monitoring and special protection systems.

The main contributions in this chapter are based in the estimation of propagating and standing features in space-time varying processes using statistical techniques to identify oscillatory activity in interconnected systems through the use of wide-area monitoring schemes in interconnected systems.

The proposed technique is based on the complex correlation structure from space-time varying fields, which can treat both, spatial and temporal information; this provides a global picture on the system behaviour to characterize oscillatory dynamics. Its significant drawbacks are associated to treat with their space-time scales. These include geographical distribution and the time interval to the modal extraction using measured data. For some applications may be desirable to have these data at very high space-time resolution that allow the study of processes close to inertial frequency, then a technique of space-time interpolation can be used. Wide-area monitoring may prove invaluable in interconnected system dynamic studies by giving a quick assessment of the damping and frequency content of dominant system modes after a critical contingency. The alternative technique based on space-time dependent complex EOF analysis of measured data is proposed to resolve the localized nature of transient processes and to extract dominant temporal and spatial information.

#### **8. Acknowledgment**

394 Numerical Modelling

These results are in general agreement with the shown in Figs. 4,5 and 6 from the observed oscillations giving validity to the results. The new results provide clarification on the exact

Fig. 8. Mode shape fluctuating in longitude, latitude and amplitude of the leading mode

Approaches for detection of propagation features in space-time varying system

The conceptual framework developed provides bases for the analysis, detection, and simplification of seismic wave components through use of wide-area monitoring schemes such as global positioning systems (GPS) based in multiple phasor measurements units (PMUs) for interconnected systems, and enables the simultaneous study of synchronized measurements. The main advantage of the approach is its ability to compress the variability of large data sets into the fewest possible number of spatial and temporal modes. The

measurements through its travelling and standing components are proposed.

showing the phase relationship between stations.

**7. Conclusion** 

phase relationship between key stations as a function of space.

This work was supported under a scholarship granted by CONACYT México no. 172551 to CINVESTAV Guadalajara. We thank to the Geophysical Institute of the National Autonomous University of México for their support and facility to use the data presented in this work.

#### **9. References**


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### *Edited by Peep Miidla*

This book demonstrates applications and case studies performed by experts for professionals and students in the field of technology, engineering, materials, decision making management and other industries in which mathematical modelling plays a role. Each chapter discusses an example and these are ranging from well-known standards to novelty applications. Models are developed and analysed in details, authors carefully consider the procedure for constructing a mathematical replacement of phenomenon under consideration. For most of the cases this leads to the partial differential equations, for the solution of which numerical methods are necessary to use. The term Model is mainly understood as an ensemble of equations which describe the variables and interrelations of a physical system or process. Developments in computer technology and related software have provided numerous tools of increasing power for specialists in mathematical modelling. One finds a variety of these used to obtain the numerical results of the book.

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