**1. Introduction**

The fractional calculus has recently been widely used to study the theory and applications of derivatives and integrals of arbitrary non-integer order. This branch of mathematical analysis has emerged in recent years as an effective and powerful tool for the mathematical modeling of various engineering, industrial, and materials science applications [1–3]. The fractional-order operators are useful in describing the memory and hereditary properties of various materials and processes, due to their nonlocal nature. It clearly reflects from the related literature produced by leading fractional calculus journals that the primary focus of the investigation had shifted from classical integer-order models to fractional order models [4, 5]. Fractional calculus has important applications in hereditary solid mechanics, fluid dynamics, viscoelasticity, heat conduction modeling and identification, biology, food engineering, econophysics, biophysics, biochemistry, robotics and control theory, signal and image processing, electronics, electric circuits, wave propagation, nanotechnology, etc. [6–8].

Numerous mathematicians have contributed to the history of fractional calculus, where Euler mentioned interpolating between integral orders of a derivative in 1730. Then, Laplace defined a fractional derivative by means of an integral in 1812.

Lacroix introduced the first fractional order derivative which appeared in a calculus in 1819, where he expressed the nth derivative of the function y <sup>¼</sup> xm as follows:

$$\frac{\mathbf{d}^{\mathfrak{n}}}{\mathbf{d}\mathbf{x}^{\mathfrak{n}}} = \frac{\Gamma(\mathfrak{m}+\mathfrak{1})}{\Gamma(\mathfrak{m}\cdot\mathfrak{n}+\mathfrak{1})} \mathbf{x}^{\mathfrak{m}\cdot\mathfrak{n}} \tag{1}$$

physics, nuclear reactors, and other industrial applications. Due to computational difficulties in solving nonlinear generalized magneto-thermoelastic problems in general analytically, many numerical techniques have been developed and implemented for solving such problems [9–17]. The boundary element method (BEM) [18–31] has been recognized as an attractive alternative numerical method to domain methods [32–36] like finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) in engineering applications. The superior feature of BEM over domain methods is that only the boundary of the domain needs to be discretized, which often leads to fewer elements and easier to use. This advantage of BEM over other domain methods has significant importance for modeling and optimization of thermoelastic problems which can be implemented using BEM with little cost and less input data. Nowadays, the BEM has emerged as an accurate and efficient computational technique for solving

*A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto…*

complicated inhomogeneous and non-linear problems in physical and engineering

A brief summary of the chapter is as follows: Section 1 introduces the background and provides the readers with the necessary information to books and articles for a better understanding of fractional order problems and their applications. Section 2 describes the physical modeling of fractional order problems in three-temperature nonlinear generalized magneto-thermoelastic ISMFGA structures. Section 3 outlines the BEM implementation for modeling of 3T fractional nonlinear generalized magneto-thermoelastic problems of multi-material ISMFGA structures subjected to moving heat source. Section 4 introduces an illustration of the mechanisms of solving design sensitivities and optimization problem of the current chapter. Section 5 presents the new numerical results that describe the effects of fractional order parameter on the problem's field variations and on the

Consider a multilayered structure with *n* functionally graded layers in the *xy*‐plane of a Cartesian coordinate. The *<sup>x</sup>*‐axis is the common normal to all layers as shown in **Figure 1**. The thickness of the layer is denoted by *h*. The considered multilayered structure has been placed in a primary magnetic field *H*<sup>0</sup> acting in the

In the present chapter, we introduce a practical engineering application of fractal analysis in the field of thermoelasticity, where the thermal field is described by time fractional three-temperature radiative heat conduction equations. Fractional order derivative considered in the current chapter has high ability to remove the difficulty of our numerical modeling. A new boundary element method for modeling and optimization of 3T fractional order nonlinear generalized thermoelastic multi-material initially stressed multilayered functionally graded anisotropic (ISMFGA) structures subjected to moving heat source is investigated. Numerical results show that the fractional order parameter has a significant effect on the sensitivities of displacements, total three-temperature, and thermal stresses. Numerical examples show that the fractional order parameter has a significant effect on the final topology of ISMFGA structures. Numerical results of the proposed model confirm the validity and accuracy of the proposed technique, and numerical examples results demonstrate the validity of the BESO multi-material

applications [37–69].

*DOI: http://dx.doi.org/10.5772/intechopen.92852*

topology optimization method.

final topology of multi-material ISMFGA structures.

**2. Formulation of the problem**

direction of the *<sup>y</sup>*‐axis.

**57**

Liouville assumed that dv dx<sup>v</sup> eax ð Þ¼ aveax for v<sup>&</sup>gt; 0 to obtain the following fractional order derivative:

$$\frac{\mathbf{d}^{\mathbf{v}}\mathbf{x}^{\mathbf{a}}}{d\mathbf{x}^{\mathbf{v}}} = \left(\cdot\mathbf{1}\right)^{\mathbf{v}} \frac{\Gamma(\mathbf{a}+\mathbf{v})}{\Gamma(\mathbf{a})} \mathbf{x}^{\mathbf{a}\cdot\mathbf{v}} \tag{2}$$

Laurent has been using the Cauchy's integral formula for complex valued analytical functions to define the integration of arbitrary order v >0 as follows:

$$\mathbf{^c}\_{\mathbf{c}} \mathbf{D\_x^v f}(\mathbf{x}) = \mathbf{^c}\_{\mathbf{c}} \mathbf{D\_x^{m \cdot \rho}} \mathbf{f}(\mathbf{x}) = \frac{\mathbf{d^m}}{\mathbf{d x}^m} \left[ \frac{1}{\Gamma(\rho)} \int\_{\mathbf{c}}^{\mathbf{x}} (\mathbf{x} - \mathbf{t})^{\rho - 1} \mathbf{f}(\mathbf{t}) \mathbf{d} \mathbf{t} \right], \mathbf{0} < \rho \le \mathbf{1} \tag{3}$$

where cD<sup>v</sup> <sup>x</sup> denotes differentiation of order v of the function f along the x‐axis. Cauchy introduced the following fractional order derivative:

$$\mathbf{f}\_{+}^{(a)} = \int \mathbf{f}(\mathbf{r}) \frac{(\mathbf{t} \cdot \mathbf{r})^{a \cdot 1}}{\Gamma(\cdot \mathbf{a})} d\mathbf{r} \tag{4}$$

Caputo introduced his fractional derivative of order α<0 to be defined as follows:

$$\mathbf{D}\_{\*}^{\mathfrak{a}}\mathbf{f}(\mathbf{t}) = \frac{\mathbf{1}}{\Gamma(\mathbf{m}\cdot\mathfrak{a})} \int\_{0}^{\mathfrak{t}} \frac{\mathbf{f}^{(\mathfrak{m})}(\tau)}{(\mathfrak{t}\cdot\mathfrak{r})^{\mathfrak{a}+1\cdot\mathfrak{m}}} d\tau, \mathbf{m}-\mathbf{1} < \mathfrak{a} < \mathbf{m}, \mathfrak{a} > \mathbf{0} \tag{5}$$

Recently, research on nonlinear generalized magneto-thermoelastic problems has received wide attention due to its practical applications in various fields such as geomechanics, geophysics, petroleum and mineral prospecting, earthquake engineering, astronautics, oceanology, aeronautics, materials science, fiber-optic communication, fluid mechanics, automobile industries, aircraft, space vehicles, plasma

#### *A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto… DOI: http://dx.doi.org/10.5772/intechopen.92852*

physics, nuclear reactors, and other industrial applications. Due to computational difficulties in solving nonlinear generalized magneto-thermoelastic problems in general analytically, many numerical techniques have been developed and implemented for solving such problems [9–17]. The boundary element method (BEM) [18–31] has been recognized as an attractive alternative numerical method to domain methods [32–36] like finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) in engineering applications. The superior feature of BEM over domain methods is that only the boundary of the domain needs to be discretized, which often leads to fewer elements and easier to use. This advantage of BEM over other domain methods has significant importance for modeling and optimization of thermoelastic problems which can be implemented using BEM with little cost and less input data. Nowadays, the BEM has emerged as an accurate and efficient computational technique for solving complicated inhomogeneous and non-linear problems in physical and engineering applications [37–69].

In the present chapter, we introduce a practical engineering application of fractal analysis in the field of thermoelasticity, where the thermal field is described by time fractional three-temperature radiative heat conduction equations. Fractional order derivative considered in the current chapter has high ability to remove the difficulty of our numerical modeling. A new boundary element method for modeling and optimization of 3T fractional order nonlinear generalized thermoelastic multi-material initially stressed multilayered functionally graded anisotropic (ISMFGA) structures subjected to moving heat source is investigated. Numerical results show that the fractional order parameter has a significant effect on the sensitivities of displacements, total three-temperature, and thermal stresses. Numerical examples show that the fractional order parameter has a significant effect on the final topology of ISMFGA structures. Numerical results of the proposed model confirm the validity and accuracy of the proposed technique, and numerical examples results demonstrate the validity of the BESO multi-material topology optimization method.

A brief summary of the chapter is as follows: Section 1 introduces the background and provides the readers with the necessary information to books and articles for a better understanding of fractional order problems and their applications. Section 2 describes the physical modeling of fractional order problems in three-temperature nonlinear generalized magneto-thermoelastic ISMFGA structures. Section 3 outlines the BEM implementation for modeling of 3T fractional nonlinear generalized magneto-thermoelastic problems of multi-material ISMFGA structures subjected to moving heat source. Section 4 introduces an illustration of the mechanisms of solving design sensitivities and optimization problem of the current chapter. Section 5 presents the new numerical results that describe the effects of fractional order parameter on the problem's field variations and on the final topology of multi-material ISMFGA structures.

### **2. Formulation of the problem**

Consider a multilayered structure with *n* functionally graded layers in the *xy*‐plane of a Cartesian coordinate. The *<sup>x</sup>*‐axis is the common normal to all layers as shown in **Figure 1**. The thickness of the layer is denoted by *h*. The considered multilayered structure has been placed in a primary magnetic field *H*<sup>0</sup> acting in the direction of the *<sup>y</sup>*‐axis.

of mathematical analysis has emerged in recent years as an effective and powerful tool for the mathematical modeling of various engineering, industrial, and materials science applications [1–3]. The fractional-order operators are useful in describing the memory and hereditary properties of various materials and processes, due to their nonlocal nature. It clearly reflects from the related literature produced by leading fractional calculus journals that the primary focus of the investigation had shifted from classical integer-order models to fractional order models [4, 5]. Fractional calculus has important applications in hereditary solid mechanics, fluid dynamics, viscoelasticity, heat conduction modeling and identification, biology, food engineering, econophysics, biophysics, biochemistry, robotics and control theory, signal and image processing, electronics, electric circuits, wave propagation,

Numerous mathematicians have contributed to the history of fractional calculus, where Euler mentioned interpolating between integral orders of a derivative in 1730. Then, Laplace defined a fractional derivative by means of an

Lacroix introduced the first fractional order derivative which appeared in a calculus in 1819, where he expressed the nth derivative of the function y <sup>¼</sup> xm as

dxn <sup>¼</sup> <sup>Γ</sup>ð Þ <sup>m</sup> <sup>þ</sup> <sup>1</sup>

dxv <sup>¼</sup> ð Þ ‐<sup>1</sup> <sup>v</sup> <sup>Γ</sup>ð Þ <sup>a</sup> <sup>þ</sup> <sup>v</sup>

lytical functions to define the integration of arbitrary order v >0 as follows:

1 Γ ρð Þ

dxm

Cauchy introduced the following fractional order derivative:

ðt 0 f ð Þ <sup>m</sup> ð Þ<sup>τ</sup>

f ð Þ α <sup>þ</sup> ¼ ð

<sup>Γ</sup>ð Þ <sup>m</sup>‐<sup>α</sup>

Laurent has been using the Cauchy's integral formula for complex valued ana-

ðx c

ð Þ <sup>x</sup> � <sup>t</sup> <sup>ρ</sup>�<sup>1</sup>

� �

<sup>x</sup> denotes differentiation of order v of the function f along the x‐axis.

<sup>f</sup>ð Þ<sup>τ</sup> ð Þ <sup>t</sup>‐<sup>τ</sup> <sup>α</sup>‐<sup>1</sup> <sup>Γ</sup>ð Þ ‐<sup>α</sup>

Caputo introduced his fractional derivative of order α<0 to be defined as follows:

Recently, research on nonlinear generalized magneto-thermoelastic problems has received wide attention due to its practical applications in various fields such as geomechanics, geophysics, petroleum and mineral prospecting, earthquake engineering, astronautics, oceanology, aeronautics, materials science, fiber-optic communication, fluid mechanics, automobile industries, aircraft, space vehicles, plasma

f tð Þdt

ð Þ <sup>t</sup>‐<sup>τ</sup> <sup>α</sup>þ1‐<sup>m</sup> <sup>d</sup>τ, m � <sup>1</sup><α<sup>&</sup>lt; m, <sup>α</sup> <sup>&</sup>gt;0 (5)

<sup>Γ</sup>ð Þ <sup>m</sup>‐<sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>x</sup><sup>m</sup>‐<sup>n</sup> (1)

<sup>Γ</sup>ð Þ<sup>a</sup> <sup>x</sup>‐a‐<sup>v</sup> (2)

, 0<ρ ≤1 (3)

dτ (4)

dx<sup>v</sup> eax ð Þ¼ aveax for v<sup>&</sup>gt; 0 to obtain the following

dn

dv x‐a

xf xð Þ¼ cD<sup>m</sup>‐<sup>ρ</sup> <sup>x</sup> f xð Þ¼ <sup>d</sup><sup>m</sup>

nanotechnology, etc. [6–8].

*Fractal Analysis - Selected Examples*

Liouville assumed that dv

Dα

<sup>∗</sup> f tðÞ¼ <sup>1</sup>

fractional order derivative:

cD<sup>v</sup>

where cD<sup>v</sup>

**56**

integral in 1812.

follows:

**Figure 1.** *Geometry of the considered problem.*

According to the three-temperature theory, the governing equations of nonlinear generalized magneto-thermoelasticity in an initially stressed multilayered functionally graded anisotropic (ISMFGA) structure for the *i*th layer can be written in the following form:

$$
\sigma\_{ab,b} + \tau\_{ab,b} - \Gamma\_{ab} = \rho^i (\mathfrak{x} + \mathfrak{1})^m \ddot{u}\_a^i \tag{6}
$$

The total energy of unit mass can be described by

*abfg* , and *β<sup>i</sup>*

moving heat source is assumed to have the following form:

where, *Q*<sup>0</sup> is the heat source strength and *δ* is the delta function.

*<sup>a</sup>* � *DaT<sup>i</sup>*

*j*¼0

*<sup>α</sup>* � *Pi <sup>∂</sup>u<sup>i</sup>*

where inertia term, temperature gradient, and initial stress terms are treated as

According to finite difference scheme of Caputo at times ð Þ *f* þ 1 Δ*t* and *f*Δ*τ*, we

Based on Eq. (17), the fractional order heat Eq. (10) can be replaced by the

� � <sup>þ</sup> *i f* ð Þ <sup>þ</sup><sup>1</sup>

*<sup>α</sup>*,*II* ð Þ� *<sup>r</sup> <sup>α</sup>*,*<sup>I</sup>*ð Þ *<sup>x</sup> <sup>T</sup>i f* ð Þ <sup>þ</sup><sup>1</sup>

*Wa*,*<sup>j</sup> <sup>T</sup>i f* ð Þ <sup>þ</sup>1�*<sup>j</sup> <sup>α</sup>* ð Þ� *<sup>r</sup> <sup>T</sup>i f* ð Þ �*<sup>j</sup> <sup>α</sup>* ð Þ*<sup>r</sup>*

In this section, we are interested in using a boundary element method for modeling the two-dimensional three-temperature radiation heat conduction equa-

� � � �

*b ∂xa*

*Wa*,*<sup>j</sup> <sup>T</sup>i f* ð Þ <sup>þ</sup>1�*<sup>j</sup> <sup>α</sup>* ð Þ� *<sup>r</sup> <sup>T</sup>i f* ð Þ �*<sup>j</sup> <sup>α</sup>* ð Þ*<sup>r</sup>*

*<sup>Γ</sup>*ð Þ <sup>2</sup> � *<sup>a</sup>* ,*Wa*,*<sup>j</sup>* <sup>¼</sup> *Wa*,0 ð Þ *<sup>j</sup>* <sup>þ</sup> <sup>1</sup> <sup>1</sup>�*<sup>a</sup>* � ð Þ *<sup>j</sup>* � <sup>1</sup> <sup>1</sup>�*<sup>a</sup>* � � (18)

*<sup>α</sup>*,*<sup>I</sup>* ð Þ¼ *<sup>r</sup> Wa*,0*Ti f* ð Þ

� � (17)

� *∂ui a ∂xb*

where *σab*, *τab*, and *ui*

*<sup>α</sup>*0, *T<sup>i</sup> <sup>α</sup>*, *C<sup>i</sup>*

, and *c<sup>i</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.92852*

**3. BEM numerical implementation**

*Lgbui*

By using Eqs. (7)–(9), we can write (6) as

*<sup>f</sup>* <sup>¼</sup> *<sup>ρ</sup><sup>i</sup> u*€*i*

tions coupled with electron, ion, and phonon temperatures.

*τTi f* ð Þ *<sup>α</sup>* <sup>≈</sup> <sup>X</sup> *k*

are constant *T<sup>i</sup>*

, ~ *h*, *Pi* , *ρ<sup>i</sup>*

the body forces.

obtain [1].

where

following system:

�*<sup>α</sup>*,*<sup>I</sup>*ð Þ *<sup>x</sup> <sup>T</sup>i f* ð Þ

**59**

*Wa*,0*Ti f* ð Þ <sup>þ</sup><sup>1</sup> *<sup>α</sup>* ð Þ� *<sup>r</sup> α*ð Þ *<sup>x</sup> <sup>T</sup>i f* ð Þ <sup>þ</sup><sup>1</sup>

*<sup>α</sup>*,*<sup>J</sup>* ð Þ� *<sup>r</sup>* <sup>X</sup>

*Da*

*<sup>τ</sup>Ti f* ð Þ <sup>þ</sup><sup>1</sup> *<sup>α</sup>* <sup>þ</sup> *<sup>D</sup><sup>a</sup>*

*Wa*,0 <sup>¼</sup> ð Þ <sup>Δ</sup>*<sup>τ</sup>* �*<sup>a</sup>*

*f*

*j*¼1

layer: *μ<sup>i</sup>*

*<sup>P</sup>* <sup>¼</sup> *Pe* <sup>þ</sup> *PI* <sup>þ</sup> *pp*, *Pe* <sup>¼</sup> *ceT<sup>i</sup>*

*<sup>e</sup>*, *PI* <sup>¼</sup> *cIT<sup>i</sup>*

stress tensor, and displacement vector in the *i*th layer, respectively, *cα*(*α = c, I, p*)

*A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto…*

temperature, constant elastic moduli, and stress-temperature coefficients in the *i*th

magnetic field, initial stress, density, isochore specific heat coefficients in the *i*th layer; *τ* is the time; *τ*<sup>0</sup> and *τ*<sup>1</sup> are the relaxation times; *i* ¼ 1, 2, … , *n* represents the parameters in multilayered structure; and *m* is a functionally graded parameter. Also, we considered in the current study that the medium is subjected to a moving heat source of constant strength moving along *<sup>x</sup>*‐axis with a constant velocity *<sup>v</sup>*. This

*<sup>I</sup>*, *Pp* <sup>¼</sup> <sup>1</sup> 4 *cpT*<sup>4</sup>*<sup>i</sup>*

*<sup>k</sup>* are mechanical stress tensor, Maxwell's electromagnetic

*<sup>s</sup><sup>α</sup>* are, respectively, magnetic permeability, perturbed

*ab* are, respectively, reference temperature,

*Q x*ð Þ¼ , *τ Q*0*δ*ð Þ *x* � *vτ* (15)

*<sup>p</sup>* (14)

¼ *f gb* (16)

*<sup>α</sup>* ð Þ� *<sup>r</sup> α*ð Þ *<sup>x</sup> <sup>T</sup>i f* ð Þ

*<sup>m</sup>* ð Þþ *<sup>x</sup>*, *<sup>τ</sup> i f* ð Þ

*<sup>α</sup>*,*II* ð Þ*r*

*<sup>m</sup>* ð Þ *x*, *τ*

(19)

$$
\sigma\_{ab} = (\boldsymbol{\pi} + \mathbf{1})^m \left[ C\_{ab\text{jg}}^i u\_{f\text{g}}^i - \beta\_{ab}^i \left( T\_a^i - T\_{a0}^i + \boldsymbol{\pi}\_1 \dot{T}\_a^1 \right) \right] \tag{7}
$$

$$\tau\_{ab} = \mu^i (\varkappa + 1)^m \left( \tilde{h}\_{\bar{a}} H\_b + \tilde{h}\_b H\_a - \delta\_{ba} \left( \tilde{h}\_f H\_f \right) \right) \tag{8}$$

$$
\Gamma\_{ab} = P^i(\mathfrak{x} + \mathbf{1})^m \left( \frac{\partial u^i\_a}{\partial \mathfrak{x}\_b} - \frac{\partial u^i\_b}{\partial \mathfrak{x}\_a} \right) \tag{9}
$$

According to Fahmy [10], the time fractional order two-dimensional threetemperature (2D-3 T) radiative heat conduction equations in nondimensionless form can be expressed as follows:

$$D^a\_\tau T^i\_a(r,\tau) = \xi \nabla \left[ \mathbb{K}^i\_a \nabla T^i\_a(r,\tau) \right] + \xi \overline{\mathbb{W}}(r,\tau), \qquad \xi = \frac{1}{c^i\_{sa} \rho^i \delta\_1} \tag{10}$$

where

$$\overline{\mathbb{W}}(r,\tau) = \begin{cases} -\rho^i \mathbb{W}\_{el}\left(T^i\_\varepsilon - T^i\_l\right) - \rho^i \mathbb{W}\_{\sigma}\left(T^i\_\varepsilon - T^i\_p\right) + \overline{\overline{\mathbb{W}}}, & a = e, \delta\_1 = 1 \\\rho^i \mathbb{W}\_{el}\left(T^i\_\varepsilon - T^i\_l\right) + \overline{\overline{\mathbb{W}}}, & a = I, \delta\_1 = 1 \\\rho^i \mathbb{W}\_{\sigma}\left(T^i\_\varepsilon - T^i\_p\right) + \overline{\overline{\mathbb{W}}}, & a = p, \delta\_1 = T^3\_p \end{cases} \tag{11}$$

in which

$$\overline{\overline{\mathcal{W}}}(\mathbf{r},\mathbf{r}) = -\mathbb{K}\_{a}^{i}\dot{T}\_{a,ab}^{i} + \rho\_{ab}^{i}T\_{a0}^{i}\tau\_{0}\ddot{u}\_{a,b}^{i} + \rho^{i}\mathbf{c}\_{sa}^{i}\tau\_{0}\ddot{T}\_{a}^{i} - \mathcal{Q}(\mathbf{x},\boldsymbol{\pi})\tag{12}$$

$$\mathbb{V}\_{el} = \rho^i \mathbb{A}\_{el} T\_\epsilon^{i\left(-\frac{2}{3}\right)},\\ \mathbb{W}\_{cr} = \rho^i \mathbb{A}\_{cr} T\_\epsilon^{i\left(-\frac{1}{2}\right)},\\ \mathbb{K}\_a = \mathbb{A}\_a T\_a^{i\left(\frac{5}{2}\right)},\\ a = e, I, \mathbb{K}\_p = \mathbb{A}\_p T\_p^{i\left(3+\mathbb{B}\right)} \tag{13}$$

*A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto… DOI: http://dx.doi.org/10.5772/intechopen.92852*

The total energy of unit mass can be described by

$$P = P\_e + P\_I + p\_p,\\ P\_e = c\_\epsilon T\_\epsilon^i,\\ P\_I = c\_I T\_I^i,\\ P\_p = \frac{1}{4} c\_p T\_p^{4i} \tag{14}$$

where *σab*, *τab*, and *ui <sup>k</sup>* are mechanical stress tensor, Maxwell's electromagnetic stress tensor, and displacement vector in the *i*th layer, respectively, *cα*(*α = c, I, p*) are constant *T<sup>i</sup> <sup>α</sup>*0, *T<sup>i</sup> <sup>α</sup>*, *C<sup>i</sup> abfg* , and *β<sup>i</sup> ab* are, respectively, reference temperature, temperature, constant elastic moduli, and stress-temperature coefficients in the *i*th layer: *μ<sup>i</sup>* , ~ *h*, *Pi* , *ρ<sup>i</sup>* , and *c<sup>i</sup> <sup>s</sup><sup>α</sup>* are, respectively, magnetic permeability, perturbed magnetic field, initial stress, density, isochore specific heat coefficients in the *i*th layer; *τ* is the time; *τ*<sup>0</sup> and *τ*<sup>1</sup> are the relaxation times; *i* ¼ 1, 2, … , *n* represents the parameters in multilayered structure; and *m* is a functionally graded parameter. Also, we considered in the current study that the medium is subjected to a moving heat source of constant strength moving along *<sup>x</sup>*‐axis with a constant velocity *<sup>v</sup>*. This moving heat source is assumed to have the following form:

$$Q(\mathbf{x}, \tau) = Q\_0 \delta(\mathbf{x} - \nu \tau) \tag{15}$$

where, *Q*<sup>0</sup> is the heat source strength and *δ* is the delta function.

## **3. BEM numerical implementation**

By using Eqs. (7)–(9), we can write (6) as

$$L\_{gb}u^i\_f = \rho^i \ddot{u}^i\_a - \left(D\_a T^i\_a - P^i \left(\frac{\partial u^i\_b}{\partial \mathbf{x}\_d} - \frac{\partial u^i\_a}{\partial \mathbf{x}\_b}\right)\right) = f\_{gb} \tag{16}$$

where inertia term, temperature gradient, and initial stress terms are treated as the body forces.

In this section, we are interested in using a boundary element method for modeling the two-dimensional three-temperature radiation heat conduction equations coupled with electron, ion, and phonon temperatures.

According to finite difference scheme of Caputo at times ð Þ *f* þ 1 Δ*t* and *f*Δ*τ*, we obtain [1].

$$D\_\mathbf{r}^a T\_a^{i(\ \ f+1)} + D\_\mathbf{r}^a T\_a^{i(\ f)} \approx \sum\_{j=0}^k \mathcal{W}\_{aj} \left( T\_a^{i(\ \ f+1-j)}(r) - T\_a^{i(\ \ f-j)}(r) \right) \tag{17}$$

where

According to the three-temperature theory, the governing equations of nonlinear generalized magneto-thermoelasticity in an initially stressed multilayered functionally graded anisotropic (ISMFGA) structure for the *i*th layer can be written

*<sup>σ</sup>ab*,*<sup>b</sup>* <sup>þ</sup> *<sup>τ</sup>ab*,*<sup>b</sup>* � <sup>Γ</sup>*ab* <sup>¼</sup> *<sup>ρ</sup><sup>i</sup>*

*abfgu<sup>i</sup>*

ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>* <sup>~</sup>

<sup>Γ</sup>*ab* <sup>¼</sup> *<sup>P</sup><sup>i</sup>*

*<sup>α</sup>* ∇*T<sup>i</sup>*

*<sup>e</sup>* � *<sup>T</sup><sup>i</sup> I* � � � *<sup>ρ</sup><sup>i</sup>*

*<sup>e</sup>* � *<sup>T</sup><sup>i</sup> I*

*<sup>e</sup>* � *<sup>T</sup><sup>i</sup> p* � �

*αT*\_ *i*

*<sup>α</sup>*,*ab* <sup>þ</sup> *<sup>β</sup><sup>i</sup>*

*erT<sup>i</sup>* �<sup>1</sup> ð Þ<sup>2</sup>

*abT<sup>i</sup> a*0*τ*0*u*€*<sup>i</sup>*

*<sup>e</sup>* , *<sup>α</sup>* <sup>¼</sup> *αT<sup>i</sup>* <sup>5</sup> ð Þ<sup>2</sup>

*<sup>f</sup>*,*<sup>g</sup>* � *<sup>β</sup><sup>i</sup>*

*haHb* <sup>þ</sup> <sup>~</sup>

ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>∂</sup>u<sup>i</sup>*

According to Fahmy [10], the time fractional order two-dimensional threetemperature (2D-3 T) radiative heat conduction equations in nondimensionless

*er T<sup>i</sup>*

*<sup>α</sup>*ð Þ *<sup>r</sup>*, *<sup>τ</sup>* � � <sup>þ</sup> *<sup>ξ</sup>*ð Þ *<sup>r</sup>*, *<sup>τ</sup>* , *<sup>ξ</sup>* <sup>¼</sup> <sup>1</sup>

*<sup>e</sup>* � *<sup>T</sup><sup>i</sup> p* � �

� � <sup>þ</sup> , *<sup>α</sup>* <sup>¼</sup> *<sup>I</sup>*, *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>

<sup>þ</sup> , *<sup>α</sup>* <sup>¼</sup> *<sup>p</sup>*, *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>T</sup>*<sup>3</sup>

*<sup>a</sup>*,*<sup>b</sup>* <sup>þ</sup> *<sup>ρ</sup><sup>i</sup> c i sατ*0*T*€*<sup>i</sup>*

*ab T<sup>i</sup>*

*a ∂xb*

h i � �

*<sup>σ</sup>ab* <sup>¼</sup> ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>C</sup><sup>i</sup>*

*<sup>τ</sup>ab* <sup>¼</sup> *<sup>μ</sup><sup>i</sup>*

form can be expressed as follows:

�*ρ<sup>i</sup>*

8 >>>><

>>>>:

*eIT<sup>i</sup>* �<sup>2</sup> ð Þ<sup>3</sup>

*ρi eI T<sup>i</sup>*

*ρi er T<sup>i</sup>*

*<sup>α</sup>*ð Þ¼ *<sup>r</sup>*, *<sup>τ</sup> <sup>ξ</sup>*<sup>∇</sup> *<sup>i</sup>*

*eI T<sup>i</sup>*

ð Þ¼� *<sup>r</sup>*, *<sup>τ</sup> <sup>i</sup>*

*<sup>e</sup>* , *er* <sup>¼</sup> *<sup>ρ</sup><sup>i</sup>*

*Da τTi*

where

ð Þ¼ *r*, *τ*

in which

*eI* <sup>¼</sup> *<sup>ρ</sup><sup>i</sup>*

**58**

ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *mu*€*<sup>i</sup>*

*<sup>α</sup>* � *<sup>T</sup><sup>i</sup>*

*hbHa* � *<sup>δ</sup>ba* <sup>~</sup>

� *∂ui b ∂xa* � �

� � � �

*<sup>α</sup>*<sup>0</sup> <sup>þ</sup> *<sup>τ</sup>*1*T*\_ <sup>1</sup> *α*

*h fH <sup>f</sup>*

*ci <sup>s</sup><sup>α</sup> ρ<sup>i</sup> δ*1

þ , *α* ¼ *e*, *δ*<sup>1</sup> ¼ 1

*<sup>α</sup>* , *<sup>α</sup>* <sup>¼</sup> *<sup>e</sup>*,*I*, *<sup>p</sup>* <sup>¼</sup> *pT<sup>i</sup>*ð Þ <sup>3</sup>þ

*<sup>a</sup>* (6)

(7)

(8)

(9)

(10)

(11)

*p*

*<sup>p</sup>* (13)

*<sup>α</sup>* � *Q x*ð Þ , *τ* (12)

in the following form:

*Geometry of the considered problem.*

*Fractal Analysis - Selected Examples*

**Figure 1.**

$$\mathcal{W}\_{a,0} = \frac{(\Delta \tau)^{-a}}{\Gamma(2-a)}, \mathcal{W}\_{a\dot{\jmath}} = \mathcal{W}\_{a,0} \left( (\dot{\jmath} + \mathbf{1})^{1-a} - (\dot{\jmath} - \mathbf{1})^{1-a} \right) \tag{18}$$

Based on Eq. (17), the fractional order heat Eq. (10) can be replaced by the following system:

$$\begin{split} & \mathbb{W}\_{a,0} \mathbf{T}\_a^{i(\ \ f+1)}(r) - \mathbb{K}\_a(\mathbf{x}) T\_{a,\mathbf{f}}^{i(\ \ f+1)}(r) - \mathbb{K}\_{a,\mathbf{f}}(\mathbf{x}) T\_{a,\mathbf{f}}^{i(\ \ f+1)}(r) = \mathbb{W}\_{a,0} T\_a^{i(\ \ f)}(r) - \mathbb{K}\_a(\mathbf{x}) T\_{a,\mathbf{f}}^{i(\ \ f)}(r) \\ & - \mathbb{K}\_{a,\mathbf{f}}(\mathbf{x}) T\_{a,\mathbf{f}}^{i(\ \ f)}(r) - \sum\_{j=1}^f \mathbb{W}\_{a,j} \left( T\_a^{i(\ \ f+1-j)}(r) - T\_a^{i(\ \ f-j)}(r) \right) + \overline{\mathbb{W}}\_m^{i(\ \ f+1)}(\mathbf{x}, \mathbf{r}) + \overline{\mathbb{W}}\_m^{i(\ \ f)}(\mathbf{x}, \mathbf{r}) \end{split} \tag{19}$$

where *j* ¼ 1, 2, … *:*, *F*, *f* ¼ 0, 1, 2, … , *F*.

Now, according to Fahmy [10], and applying the fundamental solution which satisfies (19), the boundary integral equations corresponding to (10) without heat sources can be expressed as

$$T\_a^i(\xi) = \int\_{\mathcal{S}} \left[ T\_a^i q^{i\*} - T\_a^{i\*} q^i \right] d\mathcal{C} - \int\_R f\_{ab} \, T\_a^{i\*} \, d\mathcal{R} \tag{20}$$

Thus, the governing equations can be written in operator form as follows:

$$L\_{\mathfrak{g}^b} u^i\_f = f\_{\mathfrak{g}^b},\tag{21}$$

The fundamental solution *T<sup>i</sup>* <sup>∗</sup> can be defined as

ð

*LabT<sup>i</sup> αT<sup>i</sup>* <sup>∗</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.92852*

*Ti <sup>α</sup>*ð Þ¼ *ξ*

By combining (30) and (35), we obtain

� *t i* ∗ *da* �*ui* <sup>∗</sup>

> *ui* <sup>∗</sup> *da* 0 <sup>0</sup> �*T<sup>i</sup>* <sup>∗</sup> *α*

*R*

where

sentation formula

*ui <sup>d</sup>*ð Þ*ξ Ti <sup>α</sup>*ð Þ*ξ*

" #

¼ ð

contracted notation form as

*C*

� ð

*R*

*U<sup>i</sup>* <sup>∗</sup> *DA* ¼

*T*~*i* ∗ *<sup>α</sup>DA* ¼

**61**

*LabT<sup>i</sup>* <sup>∗</sup>

*<sup>α</sup>* � *LabT<sup>i</sup>* <sup>∗</sup>

ð

*q<sup>i</sup>* <sup>∗</sup> *T<sup>i</sup>*

*da βabnb*

" # *ui*

*C*

<sup>0</sup> �*qi* <sup>∗</sup>

" # *<sup>f</sup> gb*

*Ui*

T*i <sup>α</sup><sup>A</sup>* <sup>¼</sup> *<sup>t</sup>*

*ui* <sup>∗</sup>

8 >>>>><

>>>>>:

8 >>>><

>>>>:

�*T<sup>i</sup>* <sup>∗</sup>

�*u*~*<sup>i</sup>* <sup>∗</sup>

*t i* ∗

*<sup>A</sup>* <sup>¼</sup> *<sup>u</sup><sup>i</sup>*

(

*Ti*

*i*

(

� �*dR* <sup>¼</sup>

By using WRM and integration by parts, we can write (23) as follows:

*A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto…*

ð

*q<sup>i</sup>* <sup>∗</sup> *T<sup>i</sup>*

ð

*f abT<sup>i</sup>* <sup>∗</sup>

" # *<sup>t</sup>*

�*<sup>f</sup> ab* " #*dR* (36)

*da βaf n <sup>f</sup>* (41)

*C*

*ui* <sup>∗</sup> *da* 0 <sup>0</sup> �*T<sup>i</sup>* <sup>∗</sup> *α*

*<sup>α</sup>* � *<sup>q</sup><sup>i</sup> T<sup>i</sup>* <sup>∗</sup> *α* � �*dC* (32)

*C*

*<sup>α</sup> T<sup>i</sup> α*

*<sup>q</sup><sup>i</sup>* ¼ �*<sup>i</sup>*

*qi* <sup>∗</sup> ¼ �*<sup>i</sup>*

*<sup>α</sup>* � *<sup>q</sup><sup>i</sup> T<sup>i</sup>* <sup>∗</sup> *α* � �*dC* �

*αTi*

By the use of sifting property, we obtain from (32) the thermal integral repre-

*a Ti α*

( ) " #

þ

*<sup>a</sup> a* ¼ *A* ¼ 1, 2, 3

*<sup>a</sup> a* ¼ *A* ¼ 1, 2, 3 *<sup>q</sup><sup>i</sup> <sup>A</sup>* <sup>¼</sup> <sup>4</sup>

*da d* ¼ *D* ¼ 1, 2, 3; *a* ¼ *A* ¼ 1, 2, 3

*da d* ¼ *D* ¼ 1, 2, 3; *a* ¼ *A* ¼ 1, 2, 3

*<sup>d</sup> d* ¼ *D* ¼ 1, 2, 3; *A* ¼ 4 0 *D* ¼ 4; *a* ¼ *A* ¼ 1, 2, 3

*<sup>α</sup> A* ¼ 4

0 *d* ¼ *D* ¼ 1, 2, 3; *A* ¼ 4 0 *D* ¼ 4; *a* ¼ *A* ¼ 1, 2, 3

*<sup>α</sup> D* ¼ 4; *A* ¼ 4

�*qi* <sup>∗</sup> *<sup>D</sup>* <sup>¼</sup> 4; *<sup>A</sup>* <sup>¼</sup> <sup>4</sup>

*u*~*<sup>i</sup>* <sup>∗</sup> *<sup>d</sup>* <sup>¼</sup> *ui* <sup>∗</sup>

" #

The nonlinear generalized magneto-thermoelastic vectors can be written in

*αT<sup>i</sup>* <sup>∗</sup>

*<sup>α</sup>* ¼ �*δ*ð Þ *x*, *ξ* (31)

*<sup>α</sup>*,*bna* (33)

*<sup>α</sup>*,*bna* (34)

*<sup>α</sup> dR* (35)

*dC*

(37)

(38)

(39)

(40)

*i a qi*

$$L\_{ab}T^i\_a = f\_{ab} \tag{22}$$

where the operators *Lgb*, *f gb*, *Lab*, and *f ab* are as follows:

$$L\_{\rm gb} D\_{\rm abf} \frac{\partial}{\partial \mathbf{x}\_b} + D\_{\rm af} + \Lambda D\_{\rm aff}, \quad L\_{\rm ab} = D\_\mathbf{r}^a \tag{23}$$

$$f\_{gb} = \rho^i \ddot{u}\_a^i - \left(D\_a T\_a^i - P^i \left(\frac{\partial u\_b^i}{\partial \mathbf{x}\_a} - \frac{\partial u\_a^i}{\partial \mathbf{x}\_b}\right)\right) \tag{24}$$

$$f\_{ab} = \frac{\mathbb{K}\_a}{D} T\_a^i \frac{\partial}{\partial \tau} \tag{25}$$

where

$$\begin{split} D\_{\mathrm{adj}} &= \mathsf{C}\_{\mathrm{adj}} \varepsilon, \quad \varepsilon = \frac{\partial}{\partial \mathsf{x}\_{\mathrm{g}}}, \quad D\_{\mathrm{af}} = \mu H\_{0}^{2} \left( \frac{\partial}{\partial \mathsf{x}\_{a}} + \delta\_{a1} \Lambda \right) \frac{\partial}{\partial \mathsf{x}\_{f}}, \\ D\_{a} &= -\beta\_{\mathrm{ab}}^{i} \left( \frac{\partial}{\partial \mathsf{x}\_{b}} + \delta\_{b1} \Lambda + \mathsf{r}\_{1} \left( \frac{\partial}{\partial \mathsf{x}\_{b}} + \Lambda \right) \frac{\partial}{\partial \mathsf{x}} \right), \quad \Lambda = \frac{m}{\varkappa + 1}. \end{split}$$

The differential Eq. (21) can be solved using the weighted residual method (WRM) to obtain the following integral equation:

$$\int\_{R} (L\_{\rm gb} u^i\_f - f\_{\rm gb}) u^{i\*}\_{da} dR = 0 \tag{26}$$

Now, the fundamental solution *ui* <sup>∗</sup> *df* and traction vectors *t i* ∗ *da* and *t i <sup>a</sup>* can be written as follows:

$$L\_{gb}u^{i\*}\_{df} = -\delta\_{ad}\delta(\mathfrak{x}, \mathfrak{f})\tag{27}$$

$$t\_{da}^{i\*} = \mathbf{C}\_{ab \circ \mathbf{g}} u\_{df, \mathbf{g}}^{i\*} n\_b \tag{28}$$

$$t\_a^i = \frac{\overline{t}\_a^i}{\left(\varkappa + 1\right)^m} = \left(C\_{abf\mathfrak{g}} u\_{f\mathfrak{g}}^i - \beta\_{ab}^i \left(T\_a^i + \tau\_1 T\_a^i\right)\right) n\_b \tag{29}$$

Using integration by parts and sifting property of the Dirac distribution for (26), then using Eqs. (27) and (29), we can write the following elastic integral representation formula:

$$u\_d^i(\xi) = \int\_C (u\_{da}^{i\*} t\_a^i - t\_{da}^{i\*} u\_a^i + u\_{da}^{i\*} \beta\_{ab}^i T\_a^i n\_b) d\mathcal{C} - \int\_R f\_{gb} u\_{da}^{i\*} d\mathcal{R} \tag{30}$$

*A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto… DOI: http://dx.doi.org/10.5772/intechopen.92852*

The fundamental solution *T<sup>i</sup>* <sup>∗</sup> can be defined as

$$L\_{ab}T^{i\*}\_{a} = -\delta(\mathfrak{x}, \mathfrak{xi})\tag{31}$$

By using WRM and integration by parts, we can write (23) as follows:

$$\int\_{R} (L\_{ab}T\_a^i T\_a^{i\*} - L\_{ab}T\_a^{i\*}T\_a^i)dR = \int\_{C} (q^{i\*}T\_a^i - q^i T\_a^{i\*})dC \tag{32}$$

where

where *j* ¼ 1, 2, … *:*, *F*, *f* ¼ 0, 1, 2, … , *F*.

*Ti <sup>α</sup>*ð Þ¼ *ξ* ð *S Ti*

where the operators *Lgb*, *f gb*, *Lab*, and *f ab* are as follows:

*∂ ∂xb*

*u*€*i*

*<sup>a</sup>* � *DaT<sup>i</sup>*

*∂xg*

þ *δ<sup>b</sup>*1Λ þ *τ*<sup>1</sup>

*Lgbui*

*Lgbu<sup>i</sup>* <sup>∗</sup>

*t i* ∗

ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>* <sup>¼</sup> *Cabfgui*

*<sup>f</sup> ab* <sup>¼</sup> *<sup>α</sup> D Ti α ∂*

*LgbDabf*

*<sup>f</sup> gb* <sup>¼</sup> *<sup>ρ</sup><sup>i</sup>*

*Dabf* <sup>¼</sup> *Cabfgε*, *<sup>ε</sup>* <sup>¼</sup> *<sup>∂</sup>*

*∂ ∂xb*

ð

*R*

*i a*

ð

*ui* <sup>∗</sup> *da t i <sup>a</sup>* � *t i* ∗ *dau<sup>i</sup>*

*C*

*ab*

Now, the fundamental solution *ui* <sup>∗</sup>

*t i <sup>a</sup>* <sup>¼</sup> *<sup>t</sup>*

*ui <sup>d</sup>*ð Þ¼ *ξ*

(WRM) to obtain the following integral equation:

*Da* ¼ �*β<sup>i</sup>*

sources can be expressed as

*Fractal Analysis - Selected Examples*

where

as follows:

tation formula:

**60**

Now, according to Fahmy [10], and applying the fundamental solution which satisfies (19), the boundary integral equations corresponding to (10) without heat

> ð *R*

<sup>þ</sup> *Daf* <sup>þ</sup> <sup>Λ</sup>*Da*1*<sup>f</sup>* , *Lab* <sup>¼</sup> *<sup>D</sup><sup>a</sup>*

� � � �

*<sup>α</sup>* � *Pi <sup>∂</sup>ui*

, *Daf* <sup>¼</sup> *<sup>μ</sup>H*<sup>2</sup>

*∂ ∂xb* þ Λ � � *∂*

*ui* <sup>∗</sup>

*<sup>f</sup>*,*<sup>g</sup>* � *<sup>β</sup><sup>i</sup>*

Using integration by parts and sifting property of the Dirac distribution for (26), then using Eqs. (27) and (29), we can write the following elastic integral represen-

*ab T<sup>i</sup>*

*df* and traction vectors *t*

� �

The differential Eq. (21) can be solved using the weighted residual method

*<sup>f</sup>* � *f gb* � �

*da* <sup>¼</sup> *Cabfgui* <sup>∗</sup>

*<sup>a</sup>* <sup>þ</sup> *ui* <sup>∗</sup> *da β<sup>i</sup> abT<sup>i</sup> <sup>α</sup>nb*

� �*dC* �

*b ∂xa*

0

� *∂ui a ∂xb*

*∂ ∂xa*

*∂τ*

þ *δ<sup>a</sup>*1Λ � � *∂*

, <sup>Λ</sup> <sup>¼</sup> *<sup>m</sup>*

*dadR* ¼ 0 (26)

*df*, *gnb* (28)

*i* ∗ *da* and *t i*

*df* ¼ �*δadδ*ð Þ *x*, *ξ* (27)

*<sup>α</sup>* <sup>þ</sup> *<sup>τ</sup>*1*T<sup>i</sup> α* � � � � *nb* (29)

ð

*f gbu<sup>i</sup>* <sup>∗</sup>

*dadR* (30)

*R*

*f ab T<sup>i</sup>* <sup>∗</sup>

*<sup>f</sup>* ¼ *f gb*, (21)

*<sup>α</sup>* ¼ *f ab* (22)

*<sup>∂</sup><sup>τ</sup>* (25)

*∂x <sup>f</sup>* ,

*x* þ 1 *:*

*<sup>a</sup>* can be written

*<sup>α</sup> dR* (20)

*<sup>τ</sup>* (23)

(24)

*<sup>α</sup>q<sup>i</sup>* <sup>∗</sup> � *<sup>T</sup><sup>i</sup>* <sup>∗</sup> *<sup>α</sup> <sup>q</sup><sup>i</sup>* � �*dC* �

*Lgbui*

*LabT<sup>i</sup>*

Thus, the governing equations can be written in operator form as follows:

$$q^i = -\mathbb{K}\_a^i T\_{a,b}^i \mathfrak{n}\_a \tag{33}$$

$$q^{i\*} = -\mathbb{K}\_a^i T\_{a,b}^{i\*} \mathfrak{n}\_a \tag{34}$$

By the use of sifting property, we obtain from (32) the thermal integral representation formula

$$T\_a^i(\xi) = \int\_C (q^{i\*}T\_a^i - q^i T\_a^{i\*}) dC - \int\_C f\_{ab} T\_a^{i\*} dR \tag{35}$$

By combining (30) and (35), we obtain

$$
\begin{split}
\begin{bmatrix} u\_d^i(\xi) \\ T\_a^i(\xi) \end{bmatrix} &= \int\_C \left\{ -\begin{bmatrix} t\_{da}^{i\*} & -u\_{da}^{i\*}\beta\_{ab}u\_b \\ \mathbf{0} & -q^{i\*} \end{bmatrix} \begin{bmatrix} u\_a^i \\ T\_a^i \end{bmatrix} + \begin{bmatrix} u\_{da}^{i\*} & \mathbf{0} \\ \mathbf{0} & -T\_a^{i\*} \end{bmatrix} \begin{bmatrix} t\_a^i \\ q^i \end{bmatrix} \right\} d\mathcal{C} \\ &- \int\_{\mathbb{R}} \begin{bmatrix} u\_{da}^{i\*} & \mathbf{0} \\ \mathbf{0} & -T\_a^{i\*} \end{bmatrix} \begin{bmatrix} f\_{gb} \\ -f\_{ab} \end{bmatrix} dR \end{split} \tag{36}
$$

The nonlinear generalized magneto-thermoelastic vectors can be written in contracted notation form as

$$U\_A^i = \begin{cases} u\_a^i & a = A = 1,2,3 \\ T\_a^i & A = 4 \end{cases} \tag{37}$$

$$\mathbf{T}\_{aA}^{i} = \begin{cases} t\_a^i & a=A=1,2,3\\ q^i & A=4 \end{cases} \tag{38}$$

$$U\_{DA}^{i\*} = \begin{cases} u\_{da}^{i\*} & d=D=1,2,3; a=A=1,2,3\\ 0 & d=D=1,2,3; A=4\\ 0 & D=4; a=A=1,2,3\\ -T\_a^{i\*} & D=4; A=4 \end{cases} \tag{39}$$

$$\begin{aligned} -T\_a^{i\*} & \quad D = 4; A = 4\\ \bar{T}\_{aDA}^{i\*} &= \begin{cases} t\_{da}^{i\*} & d = D = 1, 2, 3; a = A = 1, 2, 3\\ -\bar{u}\_d^{i\*} & d = D = 1, 2, 3; A = 4\\ \mathbf{0} & D = 4; a = A = 1, 2, 3 \end{cases} \\ \begin{aligned} -q^{i\*} & D = 4; A = 4\\ -q^{i\*} & D = 4; A = 4 \end{aligned} \end{aligned} \tag{40}$$

$$
\tilde{\boldsymbol{u}}\_d^{i\*} = \boldsymbol{u}\_{da}^{i\*} \boldsymbol{\beta}\_{\boldsymbol{af}} \boldsymbol{n}\_f \tag{41}
$$

By using the above vectors, we can express (36) as

$$\boldsymbol{U}\_{D}^{i}(\xi) = \int\_{C} \left( \boldsymbol{U}\_{DA}^{i\*} \boldsymbol{\mathsf{T}}\_{aA}^{i} - \boldsymbol{\tilde{T}}\_{aDA}^{i} \boldsymbol{U}\_{A}^{i} \right) d\boldsymbol{C} - \int\_{R} \boldsymbol{U}\_{DA}^{i\*} \boldsymbol{S}\_{A} d\boldsymbol{R} \tag{42}$$

The source vector *SA* can be divided as

$$\mathbf{S}\_{A} = \mathbf{S}\_{A}^{0} + \mathbf{S}\_{A}^{T} + \mathbf{S}\_{A}^{u} + \mathbf{S}\_{A}^{\hat{T}} + \mathbf{S}\_{A}^{\hat{T}} + \mathbf{S}\_{A}^{\hat{u}} \tag{43}$$

By implementing the WRM to the following equations

ð

*ui* <sup>∗</sup> *da t iq ae* � *t i* ∗ *dauiq ae* � �*dC* �

*qi* <sup>∗</sup> *Tiq*

*C*

ð

*C*

*U<sup>i</sup>* <sup>∗</sup> *DATiq*

*<sup>α</sup><sup>A</sup>* � *T ^ i* ∗ *αDAU<sup>i</sup> A*

*Uiq DE*ð Þþ *ξ*

� �*dC*

ð

*C*

� �*dC*

� ð

*Ui <sup>F</sup>* <sup>≈</sup> <sup>X</sup> *N*

*C*

*T<sup>i</sup>* <sup>∗</sup> *αDA*,*l Uiq*

According to the procedure of Fahmy [44], we can write (58) in the following

The generalized displacements and velocities are approximated in terms of

*FD* and unknown coefficients *γ*

*q*¼1 *f q FD*ð Þ *x γ q*

ð

*C*

*uiq de*ð Þ¼ *ξ*

*DOI: http://dx.doi.org/10.5772/intechopen.92852*

*Tiq <sup>α</sup>* ð Þ¼ *ξ*

*Uiq DE*ð Þ¼ *ξ*

coupled thermoelasticity formula:

ð

*U<sup>i</sup>* <sup>∗</sup> *DAT<sup>i</sup>*

0

B@

*C*

¼ �<sup>ð</sup>

*C*

þ<sup>X</sup> *E*

*q*¼1

þ<sup>X</sup> *E*

*q*¼1

*U<sup>i</sup>* <sup>∗</sup> *DA*,*l Ti <sup>α</sup><sup>A</sup>* � *T ^ i* ∗ *αDA*,*l Ui A*

0

B@

*∂Uiq DE*ð Þ*ξ ∂ξl*

*q*

(Fahmy [46]):

single equation:

*Ui <sup>D</sup>*ð Þ¼ *ξ*

respect to *ξ<sup>l</sup>* as follows:

known tensor functions *f*

*∂U<sup>i</sup> <sup>D</sup>*ð Þ*ξ ∂ξl*

form:

**63**

*Lgbuiq fe* ¼ *f q*

*A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto…*

*LabTiq <sup>α</sup>* ¼ *f q*

Then the elastic and thermal representation formulae are given as follows

*<sup>α</sup>* � *<sup>q</sup>iqTi* <sup>∗</sup> *α* � �*dC* �

The representation formulae (55) and (56) can be combined into the following

*αDAUiq AE* � �*dC* �

*<sup>α</sup>AE* � *<sup>T</sup><sup>i</sup>* <sup>∗</sup>

By substituting from Eq. (57) into Eq. (52), we obtain the following BEM

*T<sup>i</sup>* <sup>∗</sup> *αDAUiq*

In order to compute the displacement sensitivity, Eq. (58) is differentiated with

*AE* � *<sup>U</sup><sup>i</sup>* <sup>∗</sup>

*<sup>α</sup>AE* � *<sup>U</sup><sup>i</sup>* <sup>∗</sup>

*<sup>α</sup>AE* � �*dC*

*DA*,*l Tiq*

�*ζ<sup>U</sup>* � *<sup>η</sup>T<sup>α</sup>* <sup>¼</sup> *<sup>ζ</sup>U*� � *<sup>η</sup>*℘� � �*<sup>α</sup>* (60)

*q <sup>D</sup>* and ~*γ q D*:

*DATiq <sup>α</sup>AE* � �*dC*

1 CA*αq*

> 1 CA*αq*

*<sup>D</sup>* (61)

*ae* (53)

*pj* (54)

*aedR* (55)

*<sup>α</sup> dR* (56)

*AEdR* (57)

*<sup>E</sup>* (58)

*<sup>E</sup>* (59)

ð

*R ui* <sup>∗</sup> *da f q*

ð

*R f q T<sup>i</sup>* <sup>∗</sup>

ð

*R U<sup>i</sup>* <sup>∗</sup> *DA f iq*

where

$$S\_A^0 = \begin{cases} 0 & A = 1,2,3\\ Q\_0 \delta(\varkappa - v\tau) & A = 4 \end{cases} \tag{44}$$

$$\mathbf{S}\_A^T = \boldsymbol{\omega}\_{AF} \mathbf{U}\_F^i \text{ with } \boldsymbol{\alpha}\_{AF} = \begin{cases} -\mathbf{D}\_a & A = 1, 2, 3; F = 4\\ \xi \nabla \left[ \mathbb{K}\_a^i \nabla \right] & \text{otherwise} \end{cases} \tag{45}$$

$$S\_A^t = \mathcal{y}U\_F^i \text{ with } \mathcal{y} = \begin{cases} P^i \left( \frac{\partial}{\partial \mathbf{x}\_b} - \frac{\partial}{\partial \mathbf{x}\_a} \right) & A = 1,2,3; F = 1,2,3, \\ 0 & A = 4; F = 4 \end{cases} \tag{46}$$

$$\mathbf{S}\_A^T = \Gamma\_{AF} \dot{U}\_F^i \text{ with } \Gamma\_{AF} = \begin{cases} -\rho\_{ab}^i \tau\_1 \left(\frac{\partial}{\partial \mathbf{x}\_b} + \Lambda\right) \frac{\partial}{\partial \tau} & \text{\$A = 4\$; } F = 4\$\\ -\mathbb{K}\_a^i & \text{otherwise} \end{cases} \tag{47}$$

$$S\_A^{\bar{T}} = \delta\_{AF} \ddot{U}\_F^{\bar{t}} \text{ with } \delta\_{AF} = \begin{cases} 0 & A = 4; F = 4\\ \rho^i c\_{sa}^i \tau\_0 & \text{otherwise} \end{cases} \tag{48}$$

$$\mathbf{S}\_A^i = \pm \ddot{U}\_F^i \text{ with } \preceq = \begin{cases} \rho^i & A = 1, 2, 3; F = 1, 2, 3, \\ \boldsymbol{\beta}\_{ab}^i T\_{a0}^i \tau\_0 & A = 4; F = 4 \end{cases} \tag{49}$$

The representation formula (36) can also be written in matrix form as follows:

$$\begin{aligned} [\mathbf{S}\_{A}] &= -\begin{bmatrix} \mathbf{0} \\ \mathbf{Q}\_{0}\delta(\mathbf{x} - \nu\tau) \end{bmatrix} + \begin{bmatrix} -D\_{a}T^{i}\_{a} \\ \xi\nabla\left[\mathbb{K}^{i}\_{a}\nabla T^{i}\_{a}(r,\tau)\right] \end{bmatrix} + \begin{bmatrix} P^{i}\left(\boldsymbol{u}^{i}\_{b,a} - \boldsymbol{u}^{i}\_{a,b}\right) \\ \mathbf{0} \end{bmatrix} \\ &+ \begin{bmatrix} -\boldsymbol{\rho}^{i}\_{ab}\tau\_{1}\left(\frac{\partial}{\partial\mathbf{x}\_{b}} + \Lambda\right)\dot{\boldsymbol{T}}^{i}\_{a} \\ -\mathbb{K}^{i}\_{a}\dot{\boldsymbol{T}}^{i}\_{a} \end{bmatrix} + \rho^{i}\boldsymbol{\epsilon}^{i}\_{sa}\tau\_{0}\begin{bmatrix} \mathbf{0} \\ \ddot{\boldsymbol{T}}^{i}\_{a} \end{bmatrix} + \begin{bmatrix} \rho^{i}\ddot{\boldsymbol{u}}^{i}\_{a} \\ \boldsymbol{\rho}^{i}\_{ab}\boldsymbol{\tau}^{i}\_{a0}\tau\_{0}\ddot{\boldsymbol{u}}^{i}\_{f,\mathbf{g}} \end{bmatrix} \end{aligned} \tag{50}$$

In order to convert the domain integral in (42) into the boundary, we approximate the source vector *SA* by a series of known functions *f q AE* and unknown coefficients *α<sup>q</sup> <sup>E</sup>* as

$$S\_A \approx \sum\_{q=1}^{E} f\_{AE}^q a\_E^q \tag{51}$$

Thus, the representation formula (42) can be written as follows:

$$U\_D(\xi) = \int\_C \left( U\_{DA}^{i\*} T\_{aA}^i - \tilde{T}\_{aDA}^{i\*} U\_A^i \right) d\mathcal{C} - \sum\_{q=1}^N \int\_R U\_{DA}^{i\*} f\_{AE}^q d\mathcal{R} a\_E^q \tag{52}$$

*A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto… DOI: http://dx.doi.org/10.5772/intechopen.92852*

By implementing the WRM to the following equations

By using the above vectors, we can express (36) as

*U<sup>i</sup>* <sup>∗</sup> *DA*T*<sup>i</sup>*

*<sup>α</sup><sup>A</sup>* � *<sup>T</sup>*~*<sup>i</sup>*

*<sup>A</sup>* <sup>þ</sup> *<sup>S</sup><sup>u</sup>*

*<sup>A</sup>* <sup>¼</sup> <sup>0</sup> *<sup>A</sup>* <sup>¼</sup> 1, 2, 3 *Q*0*δ*ð Þ *x* � *vτ A* ¼ 4

*<sup>F</sup>* with *<sup>ω</sup>AF* <sup>¼</sup> �*Da <sup>A</sup>* <sup>¼</sup> 1, 2, 3; *<sup>F</sup>* <sup>¼</sup> <sup>4</sup> *ξ*∇ *<sup>i</sup>*

0 *A* ¼ 4; *F* ¼ 4

*<sup>F</sup>* with *<sup>δ</sup>AF* <sup>¼</sup> <sup>0</sup> *<sup>A</sup>* <sup>¼</sup> 4; *<sup>F</sup>* <sup>¼</sup> <sup>4</sup> *ρi ci*

*<sup>α</sup>*0*τ*<sup>0</sup> *A* ¼ 4; *F* ¼ 4

*∂ ∂xb* þ Λ � � *∂*

�

*<sup>F</sup>* with <sup>Ⅎ</sup> <sup>¼</sup> *<sup>ρ</sup><sup>i</sup> <sup>A</sup>* <sup>¼</sup> 1, 2, 3; *<sup>F</sup>* <sup>¼</sup> 1, 2, 3,

The representation formula (36) can also be written in matrix form as follows:

*α*∇*T<sup>i</sup> <sup>α</sup>*ð Þ *<sup>r</sup>*, *<sup>τ</sup>* � �

In order to convert the domain integral in (42) into the boundary, we approxi-

" #

*α*

� �

*αDAU<sup>i</sup> A*

*<sup>A</sup>* <sup>þ</sup> *<sup>S</sup>T*\_

*dC* � ð

*<sup>A</sup>* <sup>þ</sup> *<sup>S</sup>T*€

*<sup>α</sup>*<sup>∇</sup> � � otherwise

*<sup>α</sup>* otherwise

þ

þ

0 *T*€*i α* � �

*dC* �<sup>X</sup> *N*

*q*¼1

ð

*R U<sup>i</sup>* <sup>∗</sup> *DA f q AEdRα<sup>q</sup>*

*P<sup>i</sup> ui*

*βi abT<sup>i</sup> α*0*τ*0*u*€*<sup>i</sup> f*, *g*

*q*

*<sup>b</sup>*,*<sup>a</sup>* � *ui a*,*b* � �

" #

0

*ρi u*€*i a*

" #

*AE* and unknown

*<sup>E</sup>* (51)

*<sup>s</sup>ατ*<sup>0</sup> otherwise

*R U<sup>i</sup>* <sup>∗</sup>

*<sup>A</sup>* <sup>þ</sup> *<sup>S</sup>u*€

*A* ¼ 1, 2, 3; *F* ¼ 1, 2, 3,

*<sup>∂</sup><sup>τ</sup> <sup>A</sup>* <sup>¼</sup> 4; *<sup>F</sup>* <sup>¼</sup> <sup>4</sup>

*DASAdR* (42)

*<sup>A</sup>* (43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

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

ð

*C*

*SA* <sup>¼</sup> *<sup>S</sup>*<sup>0</sup>

*S*0

*<sup>F</sup>* with *<sup>ψ</sup>* <sup>¼</sup> *<sup>P</sup><sup>i</sup> <sup>∂</sup>*

8 < :

*<sup>F</sup>* with <sup>Γ</sup>*AF* <sup>¼</sup> �*β<sup>i</sup>*

*<sup>A</sup>* <sup>þ</sup> *<sup>S</sup><sup>T</sup>*

�

*∂xb*

8 ><

>:

*βi abT<sup>i</sup>*

<sup>þ</sup> �*DaT<sup>i</sup>*

3 7 <sup>5</sup> <sup>þ</sup> *<sup>ρ</sup><sup>i</sup> c i <sup>s</sup><sup>α</sup>τ*<sup>0</sup>

*SA* <sup>≈</sup> <sup>X</sup> *E*

Thus, the representation formula (42) can be written as follows:

*<sup>α</sup><sup>A</sup>* � *<sup>T</sup>*~*<sup>i</sup>* <sup>∗</sup>

� �

*q*¼1 *f q AEα<sup>q</sup>*

*αDAU<sup>i</sup> A*

*ξ*∇ *<sup>i</sup>*

*T*\_ *i α*

(

�*<sup>i</sup>*

� *∂ ∂xa* � �

*abτ*<sup>1</sup>

�

The source vector *SA* can be divided as

*Ui <sup>D</sup>*ð Þ¼ *ξ*

*Fractal Analysis - Selected Examples*

where

*ST*

*Su <sup>A</sup>* <sup>¼</sup> *<sup>ψ</sup>U<sup>i</sup>*

*ST*\_

*<sup>A</sup>* <sup>¼</sup> <sup>Γ</sup>*AFU*\_ *<sup>i</sup>*

*Su*€ *<sup>A</sup>* <sup>¼</sup> <sup>Ⅎ</sup>*U*€ *<sup>i</sup>*

½ �¼� *SA*

coefficients *α<sup>q</sup>*

**62**

*ST*€

0 *Q*0*δ*ð Þ *x* � *vτ* � �

*abτ*<sup>1</sup>

*∂ ∂xb* þ Λ � �

�*<sup>i</sup> αT*\_ *i α*

mate the source vector *SA* by a series of known functions *f*

<sup>þ</sup> �*β<sup>i</sup>*

*<sup>E</sup>* as

*UD*ð Þ¼ *ξ*

ð

*U<sup>i</sup>* <sup>∗</sup> *DAT<sup>i</sup>*

*C*

2 6 4 *<sup>A</sup>* <sup>¼</sup> *<sup>δ</sup>AFU*€ *<sup>i</sup>*

*<sup>A</sup>* <sup>¼</sup> *<sup>ω</sup>AFU<sup>i</sup>*

$$L\_{\rm gb}u^{\rm iq}\_{\rm fe} = f^q\_{\rm ae} \tag{53}$$

$$L\_{ab}T^{iq}\_a = f^q\_{pj} \tag{54}$$

Then the elastic and thermal representation formulae are given as follows (Fahmy [46]):

$$\boldsymbol{u}\_{d\epsilon}^{iq}(\xi) = \int\_{C} (\boldsymbol{u}\_{da}^{i\*} \boldsymbol{t}\_{a\epsilon}^{iq} - \boldsymbol{t}\_{da}^{i\*} \boldsymbol{u}\_{a\epsilon}^{iq}) d\mathbf{C} - \int\_{R} \boldsymbol{u}\_{da}^{i\*} \boldsymbol{f}\_{a\epsilon}^{q} d\mathbf{R} \tag{55}$$

$$T\_a^{iq}(\xi) = \int\_C (q^{i\ast} \, T\_a^{iq} - q^{iq} \, T\_a^{i\ast}) \, d\mathcal{C} - \int\_{\tilde{R}} f^q \, T\_a^{i\ast} \, dR \tag{56}$$

The representation formulae (55) and (56) can be combined into the following single equation:

$$dU\_{DE}^{iq}(\xi) = \int\_{C} \left( U\_{DA}^{i\*} T\_{aAE}^{iq} - T\_{aDA}^{i\*} U\_{AE}^{iq} \right) d\mathcal{C} - \int\_{R} U\_{DA}^{i\*} f\_{AE}^{iq} d\mathcal{R} \tag{57}$$

By substituting from Eq. (57) into Eq. (52), we obtain the following BEM coupled thermoelasticity formula:

$$\begin{aligned} U\_D^i(\xi) &= \int\_C \left( U\_{DA}^{i\*} T\_{aA}^i - \bar{T}\_{aDA}^{i\*} U\_A^i \right) d\mathcal{C} \\ &+ \sum\_{q=1}^E \left( U\_{DE}^{iq}(\xi) + \int\_C \left( T\_{aDA}^{i\*} U\_{AE}^{iq} - U\_{DA}^{i\*} T\_{aAE}^{iq} \right) d\mathcal{C} \right) a\_E^q \end{aligned} \tag{58}$$

In order to compute the displacement sensitivity, Eq. (58) is differentiated with respect to *ξ<sup>l</sup>* as follows:

$$\begin{split} \frac{\partial \mathbf{U}\_{D}^{i}(\xi)}{\partial \xi\_{l}} &= -\int\_{C} \left( \mathbf{U}\_{DA,l}^{i\*} \mathbf{T}\_{aA}^{i} - \overset{\smile i\*}{\mathbf{T}\_{aDA,l}^{i\*}} \mathbf{U}\_{A}^{i} \right) d\mathbf{C} \\ &+ \sum\_{q=1}^{E} \left( \frac{\partial \mathbf{U}\_{DE}^{iq}(\xi)}{\partial \xi\_{l}} - \int\_{C} \left( \mathbf{T}\_{aDA,l}^{i\*} \mathbf{U}\_{aAE}^{iq} - \mathbf{U}\_{DA,l}^{i\*} \mathbf{T}\_{aAE}^{iq} \right) d\mathbf{C} \right) a\_{E}^{q} \end{split} \tag{59}$$

According to the procedure of Fahmy [44], we can write (58) in the following form:

$$
\check{\zeta}U - \eta T\_a = (\zeta \check{U} - \eta \check{\otimes}) \overline{\alpha} \tag{60}
$$

The generalized displacements and velocities are approximated in terms of known tensor functions *f q FD* and unknown coefficients *γ q <sup>D</sup>* and ~*γ q D*:

$$U\_F^i \approx \sum\_{q=1}^N f\_{FD}^q(\mathbf{x}) \chi\_D^q \tag{61}$$

where

$$f\_{FD}^q = \begin{cases} f\_{fd}^q & f = F = 1,2,3; d = D = 1,2,3 \\\\ f^q & F = 4; D = 4 \\\\ \mathbf{0} & \text{otherwise} \end{cases} \tag{62}$$

Now, the gradients of the generalized displacements and velocities can also be approximated in terms of the tensor function derivatives as

$$U\_{F, \mathfrak{g}}^i \approx \sum\_{q=1}^N f\_{FD, \mathfrak{g}}^q(\mathfrak{x}) \eta\_K^q \tag{63}$$

*M* z}|{ *U*€ *i* þ Γ z}|{ *U*\_ *i* þ *K* z}|{

�1 , *M*

*abT<sup>i</sup>*

ment, temperature, and external force vectors, and *V*, *M*

z}|{*<sup>i</sup>*

*<sup>n</sup>*þ<sup>1</sup> þ Γ z}|{ *U*\_ *i*

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> þ B

*S*

z}|{*<sup>i</sup>*

*<sup>α</sup>*0*τ*0,

*X* z}|{ *T*€*i <sup>α</sup>* þ *A* z}|{ *T*\_ *i <sup>α</sup>* þ *B* z}|{ *Ti <sup>α</sup>* ¼ z}|{ *U*€ *i* þ

*DOI: http://dx.doi.org/10.5772/intechopen.92852*

0�<sup>1</sup> <sup>þ</sup> *<sup>ψ</sup>* � �,

z}|{ <sup>¼</sup> *<sup>β</sup><sup>i</sup>*

, *T<sup>i</sup>* and

In many applications, the coupling term

*M* z}|{ *U*€ *i*

> z}|{ *T*\_ *i*

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*�

By integrating Eq. (73) and using Eq. (75), we get

*<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>U</sup>*€ *<sup>i</sup> n*

� �

*<sup>n</sup>* þ *M* z}|{�<sup>1</sup>

*<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>U</sup>*\_ *<sup>i</sup> n*

> *U*€ *i <sup>n</sup>* þ *M* z}|{�<sup>1</sup>

> > *<sup>n</sup>* þ *M* z}|{�<sup>1</sup>

� �

*<sup>n</sup>* þ Δ*τ*<sup>2</sup> 4

*<sup>n</sup>* þ Δ*τ* <sup>2</sup> *<sup>U</sup>*€ *<sup>i</sup>*

z}|{ � �.

Γ

<sup>2</sup> *M* z}|{�<sup>1</sup>

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> þ A

*<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>η</sup>Tip*

*<sup>n</sup>* þ Δ*τ* <sup>2</sup> *<sup>U</sup>*€ *<sup>i</sup>*

where *<sup>V</sup>* <sup>¼</sup> *<sup>η</sup>*℘� � *<sup>ζ</sup>U*� � �*<sup>J</sup>*

*<sup>α</sup>* <sup>∇</sup> � �,

, *U*\_ *<sup>i</sup>* , *U<sup>i</sup>*

algebraic equations (DAEs):

X z}|{ *T*€*i*

z}|{*ip*

*<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>U</sup>*\_ *<sup>i</sup>*

*<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>U</sup><sup>i</sup>*

*U*\_ *i*

**65**

where <sup>γ</sup> <sup>¼</sup> *<sup>I</sup>* <sup>þ</sup> <sup>Δ</sup>*<sup>τ</sup>*

<sup>¼</sup> *<sup>U</sup><sup>i</sup>*

From Eq. (77) we have

*<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> <sup>γ</sup>�<sup>1</sup> *<sup>U</sup>*\_ *<sup>i</sup>*

<sup>¼</sup> *<sup>U</sup>*\_ *<sup>i</sup> <sup>n</sup>* þ Δ*τ* <sup>2</sup> *<sup>U</sup>*€ *<sup>i</sup>*

> *<sup>n</sup>* þ Δ*τ* <sup>2</sup> *<sup>U</sup>*\_ *<sup>i</sup>*

*<sup>n</sup>* <sup>þ</sup> <sup>Δ</sup>*τU*\_ *<sup>i</sup>*

where

*U*\_ *i*

*Ui*

z}|{ ¼ ��*<sup>ζ</sup>* <sup>þ</sup> *<sup>V</sup>* <sup>B</sup>*TJ*

where *U*€ *<sup>i</sup>*

the displacement.

matrices.

*K*

*B* z}|{ <sup>¼</sup> *<sup>ξ</sup>*<sup>∇</sup> *<sup>i</sup>* *<sup>U</sup><sup>i</sup>* <sup>¼</sup> z}|{*<sup>i</sup>*

z}|{ <sup>¼</sup> *<sup>V</sup>*Γ*AF*,

are, respectively, acceleration, velocity, displace-

z}|{, <sup>Γ</sup>

*<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup>

z}|{ *U*€ *i*

z}|{*ip*

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> is the predicted temperature.

*<sup>n</sup>*þ<sup>1</sup> � *K* z}|{ *Ui n*þ1

> *<sup>n</sup>*þ<sup>1</sup> � *K* z}|{ *Ui n*þ1

*<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup>

*ci <sup>s</sup><sup>α</sup>τ*0, *A*

z}|{, *K* z}|{ , A z}|{

*<sup>n</sup>*þ<sup>1</sup> that appear in the heat con-

*<sup>n</sup>*þ<sup>1</sup> (75)

z}|{ (76)

(77)

(78)

z}|{ ¼ �*ρ<sup>i</sup>*

z}|{ <sup>¼</sup> *<sup>V</sup>* <sup>Ⅎ</sup> <sup>þ</sup> *<sup>δ</sup>AF* � �, <sup>Γ</sup>

*A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto…*

¼ �*ηT*� <sup>þ</sup> *<sup>V</sup>*�

are, respectively, volume, mass, damping, stiffness, capacity, and conductivity

duction equation is negligible. Therefore, it is easier to predict the temperature than

Hence Eqs. (73) and (74) lead to the following coupled system of differential-

*<sup>n</sup>*þ<sup>1</sup> þ *K* z}|{ *Ui*

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup>

z}|{ *Ti*

<sup>0</sup> and *Tip*

 z}|{*ip*

*<sup>n</sup>*þ<sup>1</sup> � Γ z}|{ *U*\_ *i*

� � � �

 z}|{*ip*

 z}|{*ip*

*<sup>n</sup>*þ<sup>1</sup> � Γ z}|{ *U*\_ *i*

*<sup>n</sup>*þ<sup>1</sup> � *K* z}|{ *Ui n*þ1

� � � � � � (79)

� � � �

*S* 0 , *X*

z}|{ ¼ �*Q*0*δ*ð Þ *<sup>x</sup>* � *<sup>v</sup><sup>τ</sup>* .

z}|{ *U*€ *i* (73)

*α*,

, and B z}|{

z}|{ (74)

z}|{ ¼ �*<sup>i</sup>*

By substituting (63) into Eq. (45), we get

$$\mathbf{S}\_A^T = \sum\_{q=1}^N \mathbf{S}\_{AF} f\_{FD, \mathbf{g}}^q \mathbf{y}\_D^q \tag{64}$$

By applying the point collocation procedure of Gaul et al. [43] to Eqs. (51) and (61), we obtain

$$
\check{S} = f \overline{a}, \quad U^i = f' \gamma,\tag{65}
$$

Similarly, applying the same point collocation procedure to Eqs. (64), (46), (47), (48), and (49) yields

$$
\check{\mathbf{S}}^{T\_a^i} = \mathcal{B}^T \boldsymbol{\chi} \tag{66}
$$

$$\mathbf{S}\_A^u = \boldsymbol{\psi} \boldsymbol{U}^i \tag{67}$$

$$
\check{\mathbf{S}}^{T^i\_{\
u}} = \overline{\Gamma}\_{AF} \dot{U}^i \tag{68}
$$

$$
\check{\mathbf{S}}^{T\_a^i} = \overline{\delta}\_{AF} \ddot{\mathbf{U}}^i \tag{69}
$$

$$
\dot{\mathbf{S}}^{\ddot{\imath}} = \overline{\mathbf{A}} \ddot{\mathbf{U}}^{\dot{\imath}} \tag{70}
$$

where *ψ*, Γ*AF*, *δAF*, and Ⅎ are assembled using the submatrices ½ � *ψ* , ½ � Γ*AF* , ½ � *δAF* , and ½ � Ⅎ , respectively.

Solving the system (65) for *α* and *γ* yields

$$\overline{a} = \boldsymbol{J}^{-1}\check{\mathbf{S}}, \quad \boldsymbol{\gamma} = \boldsymbol{J}'^{-1}\boldsymbol{U}^{\boldsymbol{i}} \tag{71}$$

Now, the coefficient *α* can be written in terms of the unknown displacements *U<sup>i</sup>* , velocities *U*\_ *<sup>i</sup>* , and accelerations *U*€ *<sup>i</sup>* as

$$\overline{\boldsymbol{a}} = \boldsymbol{J}^{-1} \left( \dot{\boldsymbol{S}}^{0} + \left( \mathcal{B}^{T} \boldsymbol{J}^{\prime -1} + \overline{\boldsymbol{\nu}} \right) \boldsymbol{U}^{i} + \overline{\boldsymbol{\Gamma}}\_{AF} \dot{\boldsymbol{U}}^{i} + \left( \overline{\boldsymbol{\Delta}} + \overline{\boldsymbol{\delta}}\_{AF} \right) \ddot{\boldsymbol{U}}^{i} \right) \tag{72}$$

An implicit-implicit staggered algorithm has been implemented for use with the BEM to solve the governing equations which can now be written in a suitable form after substitution of Eq. (72) into Eq. (60) as

*A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto… DOI: http://dx.doi.org/10.5772/intechopen.92852*

$$
\widehat{\mathbf{M}}^i \ddot{\mathbf{U}}^i + \widehat{\mathbf{\Gamma}}^i \dot{\mathbf{U}}^i + \widehat{\mathbf{K}}^i \mathbf{U}^i = \widehat{\mathbb{Q}}^i \tag{73}
$$

$$
\widehat{\phantom{X}\ddot{\phantom{x}}\ddot{T}^{i}\_{a}} + \widehat{\phantom{A}\ddot{\phantom{T}}\dot{T}^{i}\_{a}} + \widehat{\phantom{B}\dot{T}^{i}\_{a}} = \widehat{\phantom{\phantom{\varphi}\ddot{\phantom{\varphi}}\ddot{U}^{i}}} + \widehat{\phantom{\varphi}\ddot{U}^{i}} + \widehat{\phantom{\theta}\ddot{\phantom{\varphi}}\tag{74}}\tag{74}$$

$$\begin{aligned} \text{where } V &= \left(\eta\breve{\wp} - \zeta\breve{U}\right)\overline{J}^{-1}, \ \widetilde{M} = V\left(\overline{\mathfrak{J}} + \overline{\delta}\_{\text{AF}}\right), \ \overline{\Gamma} &= V\overline{\Gamma}\_{\text{AF}}, \\ \widetilde{K} &= -\breve{\zeta} + V\left(\overline{\mathfrak{J}^{T}}\overline{J}^{-1} + \overline{\eta}\right), \ \widetilde{\mathbb{Q}}^{i} &= -\eta\breve{T} + V\breve{\mathcal{S}}^{0}, \ \widetilde{X} &= -\rho^{i}\acute{c}\_{sa}^{i}\pi\_{0}, \ \widetilde{A} &= -\mathbb{K}\_{a}^{i}, \\ \widetilde{B} &= \xi\nabla\left[\overline{\mathbb{K}\_{a}^{i}}\,\nabla\right], \ \widetilde{\mathbb{Z}} &= \beta\_{ab}^{i}\overline{I}\_{a0}^{i}\pi\_{0}, \ \widetilde{\mathbb{R}} &= -Q\_{0}\delta(x - v\tau). \end{aligned}$$
 where  $\breve{U}$ ,  $\dot{U}$ ,  $\dot{U}$ ,  $T$  and  $\widetilde{\mathbb{Q}}^{i}$  are, respectively, acceleration, velocity, displacement, displacement, displacement, displacement, respectively, dimensionless, respectively, acceleration, respectively, and  $\tau$ ,  $\widetilde{M}$ ,  $\widetilde{\Gamma}$ ,  $\widetilde{K}$ ,  $\widetilde{A}$ , and  $\widetilde{B}$  are, respectively, volume, mass, damping, stiffness, capacity, and conductivity.

are, respectively, volume, mass, damping, stiffness, capacity, and conductivity matrices. z}|{

In many applications, the coupling term *U*€ *i <sup>n</sup>*þ<sup>1</sup> that appear in the heat conduction equation is negligible. Therefore, it is easier to predict the temperature than the displacement.

Hence Eqs. (73) and (74) lead to the following coupled system of differentialalgebraic equations (DAEs):

$$
\widehat{\boldsymbol{M}}^{i}\widetilde{\boldsymbol{U}}\_{n+1}^{i} + \widehat{\boldsymbol{\Gamma}}^{i}\widetilde{\boldsymbol{U}}\_{n+1}^{i} + \widehat{\boldsymbol{K}}^{i}\widetilde{\boldsymbol{U}}\_{n+1}^{i} = \widehat{\mathbb{Q}}\_{n+1}^{ip} \tag{75}
$$

$$\widehat{\mathbf{X}\cdot\ddot{T}^{i}}\_{a(n+1)} + \widehat{\mathbf{A}\cdot\ddot{T}^{i}}\_{a(n+1)} + \widehat{\mathbf{B}\cdot T^{i}}\_{a(n+1)} = \widehat{\mathbf{Z}\cdot\ddot{U}^{i}}\_{n+1} + \widehat{\mathbf{R}} \tag{76}$$

where z}|{*ip <sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>η</sup>Tip <sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*� *S* <sup>0</sup> and *Tip <sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> is the predicted temperature. By integrating Eq. (73) and using Eq. (75), we get

$$\begin{aligned} \dot{U}\_{n+1}^{i} &= \dot{U}\_{n}^{i} + \frac{\Delta\tau}{2} \left( \ddot{U}\_{n+1}^{i} + \ddot{U}\_{n}^{i} \right) \\ &= \dot{U}\_{n}^{i} + \frac{\Delta\tau}{2} \left[ \ddot{U}\_{n}^{i} + \widetilde{\mathcal{M}}^{-1} \left( \widetilde{\mathbb{Q}}\_{n+1}^{ip} - \widetilde{\Gamma}^{\circ} \dot{U}\_{n+1}^{i} - \widetilde{\mathcal{K}}^{\circ} \dot{U}\_{n+1}^{i} \right) \right] \\\\ \dot{U}\_{n+1}^{i} &= \dot{U}\_{n}^{i} + \frac{\Delta\tau}{2} \left( \dot{U}\_{n+1}^{i} + \dot{U}\_{n}^{i} \right) \\ &\quad \vdots \quad \ddots \quad \vdots \quad \widehat{\Delta}^{2} \left[ \ddot{U}\_{n}^{i} + \widetilde{\mathcal{K}}^{\circ} \dot{U}\_{n+1}^{i} - \widetilde{\mathcal{K}}^{\circ} \dot{U}\_{n+1}^{i} \right] \end{aligned} \tag{77}$$

$$=\boldsymbol{U}\_{n}^{i}+\Delta\boldsymbol{\pi}\dot{\boldsymbol{U}}\_{n}^{i}+\frac{\Delta\boldsymbol{\pi}^{2}}{4}\left[\ddot{\boldsymbol{U}}\_{n}^{i}+\widehat{\boldsymbol{M}}^{\cdot -1}\left(\widehat{\boldsymbol{\mathbb{Q}}}\_{n+1}^{ip}-\widehat{\boldsymbol{\Gamma}\boldsymbol{\Gamma}}\dot{\boldsymbol{U}}\_{n+1}^{i}-\widehat{\boldsymbol{K}}^{\cdot}\boldsymbol{U}\_{n+1}^{i}\right)\right] \tag{78}$$

From Eq. (77) we have

$$\begin{aligned} \dot{U}\_{n+1}^{i} &= \overline{\gamma}^{-1} \Big[ \dot{U}\_{n}^{i} + \frac{\Delta \tau}{2} \Big[ \dot{U}\_{n}^{i} + \widetilde{\mathcal{M}}^{-1} \Big( \widetilde{\mathbb{Q}}^{ip}\_{n+1} - \widetilde{\mathcal{K}}^{i} \dot{U}\_{n+1}^{i} \Big) \Big] \Big] \end{aligned} \tag{79}$$
  $\text{where } \overline{\gamma} = \Big( I + \frac{\Delta \tau}{2} \widetilde{\mathcal{M}}^{-1} \widehat{\Gamma} \Big) . $ 

where

(61), we obtain

(47), (48), and (49) yields

and ½ � Ⅎ , respectively.

velocities *U*\_ *<sup>i</sup>*

**64**

Solving the system (65) for *α* and *γ* yields

, and accelerations *U*€ *<sup>i</sup>*

�<sup>1</sup> � *S* 0

after substitution of Eq. (72) into Eq. (60) as

*α* ¼ *J*

*f q FD* ¼

*Fractal Analysis - Selected Examples*

*f q*

8 >>><

>>>:

By substituting (63) into Eq. (45), we get

*f*

approximated in terms of the tensor function derivatives as

*Ui*

*ST <sup>A</sup>* <sup>¼</sup> <sup>X</sup> *N*

�

*<sup>F</sup>*,*<sup>g</sup>* <sup>≈</sup> <sup>X</sup> *N*

*q*¼1 *f q FD*,*<sup>g</sup>* ð Þ *x γ q*

*q*¼1

*SAF f q FD*,*<sup>g</sup> γ q*

By applying the point collocation procedure of Gaul et al. [43] to Eqs. (51) and

0

*<sup>S</sup>* <sup>¼</sup> *<sup>J</sup>α*, *<sup>U</sup><sup>i</sup>* <sup>¼</sup> *<sup>J</sup>*

Similarly, applying the same point collocation procedure to Eqs. (64), (46),

� *S Ti*

*Su*

� *S T*\_*ι*

� *S T*€*ι*

*α* ¼ *J* �<sup>1</sup>�

as

0�<sup>1</sup> <sup>þ</sup> *<sup>ψ</sup>* � �

þ B*TJ*

� *S u*€

where *ψ*, Γ*AF*, *δAF*, and Ⅎ are assembled using the submatrices ½ � *ψ* , ½ � Γ*AF* , ½ � *δAF* ,

*S*, *γ* ¼ *J*

Now, the coefficient *α* can be written in terms of the unknown displacements *U<sup>i</sup>*

0�<sup>1</sup>

*<sup>U</sup><sup>i</sup>* <sup>þ</sup> <sup>Γ</sup>*AFU*\_ *<sup>i</sup>*

� �*U*€ *<sup>i</sup>* � �

An implicit-implicit staggered algorithm has been implemented for use with the BEM to solve the governing equations which can now be written in a suitable form

*fd f* ¼ *F* ¼ 1, 2, 3; *d* ¼ *D* ¼ 1, 2, 3

(62)

*<sup>K</sup>* (63)

*<sup>D</sup>* (64)

*γ*, (65)

*<sup>α</sup>* ¼ B*<sup>T</sup><sup>γ</sup>* (66)

*<sup>A</sup>* <sup>¼</sup> *<sup>ψ</sup>U<sup>i</sup>* (67)

*<sup>α</sup>* <sup>¼</sup> <sup>Γ</sup>*AFU*\_ *<sup>i</sup>* (68)

*<sup>α</sup>* <sup>¼</sup> *<sup>δ</sup>AFU*€ *<sup>i</sup>* (69)

<sup>¼</sup> <sup>Ⅎ</sup>*U*€ *<sup>i</sup>* (70)

þ Ⅎ þ *δAF*

*U<sup>i</sup>* (71)

,

(72)

*<sup>q</sup> <sup>F</sup>* <sup>¼</sup> 4; *<sup>D</sup>* <sup>¼</sup> <sup>4</sup>

0 otherwise

Now, the gradients of the generalized displacements and velocities can also be

Substituting Eq. (79) into Eq. (78), we derive

$$\begin{split} \boldsymbol{U}\_{n+1}^{\boldsymbol{i}} &= \boldsymbol{U}\_{n}^{\boldsymbol{i}} + \Delta \boldsymbol{\tau} \boldsymbol{\bar{U}}\_{n}^{\boldsymbol{i}} \\ &+ \frac{\Delta \boldsymbol{\tau}^{2}}{4} \Big[ \boldsymbol{\bar{U}}\_{n}^{\boldsymbol{i}} + \widetilde{\boldsymbol{\mathcal{M}}}^{\boldsymbol{i}} \Big( \widehat{\boldsymbol{\mathcal{Q}}}\_{n+1}^{\boldsymbol{i}\boldsymbol{p}} - \widehat{\boldsymbol{\nabla}}^{\boldsymbol{i}} \widetilde{\boldsymbol{\nabla}}^{\boldsymbol{i}} \Big[ \boldsymbol{\bar{U}}\_{n}^{\boldsymbol{i}} + \frac{\Delta \boldsymbol{\tau}}{2} \Big[ \boldsymbol{\bar{U}}\_{n}^{\boldsymbol{i}} + \widetilde{\boldsymbol{\mathcal{M}}}^{\boldsymbol{i}} \Big( \widehat{\boldsymbol{\mathcal{Q}}}\_{n+1}^{\boldsymbol{i}\boldsymbol{p}} - \widehat{\boldsymbol{\mathcal{K}}}^{\boldsymbol{i}} \boldsymbol{U}\_{n+1}^{\boldsymbol{i}} \Big) \Big] \Big] - \widehat{\boldsymbol{\mathcal{K}}}^{\boldsymbol{i}} \boldsymbol{U}\_{n+1}^{\boldsymbol{i}} \Big] \Big] \end{split} \tag{80}$$

Now, a displacement predicted staggered procedure for the solution of (80) and

*A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto…*

*<sup>n</sup>*þ<sup>1</sup> and *<sup>U</sup>*€ *<sup>i</sup>*

 

 

> 

 

*<sup>f</sup>*ð Þ¼ *x*, *z*, 0 0 for ð Þ *x*, *z* ∈*R* ∪*C* (91)

*<sup>α</sup>*ð Þ¼ *x*, *z*, 0 0 for ð Þ *x*, *z* ∈*R* ∪*C* (94)

*<sup>f</sup>*ð Þ¼ *x*, *z*, *τ* Ψ *<sup>f</sup>*ð Þ *x*, *z*, *τ* for ð Þ *x*, *z* ∈*C*<sup>3</sup> (92)

*<sup>a</sup>*ð Þ¼ *x*, *z*, *τ* Φ*f*ð Þ *x*, *z*, *τ* for ð Þ *x*, *z* ∈*C*4, *τ* >0 (93)

*<sup>α</sup>*ð Þ¼ *x*, *z*, *τ f x*ð Þ , *z*, *τ* for ð Þ *x*, *z* ∈*C*1, *τ* >0 (95)

ð Þ¼ *x*, *z*, *τ h x*ð Þ , *z*, *τ* for ð Þ *x*, *z* ∈*C*2, *τ* >0 (96)

*<sup>n</sup>*þ<sup>1</sup> from Eqs. (77) and

*<sup>x</sup>*¼*h<sup>i</sup>* (87)

*<sup>x</sup>*¼*h<sup>i</sup>* (88)

*<sup>x</sup>*¼*h<sup>i</sup>* (89)

*<sup>x</sup>*¼*h<sup>i</sup>* (90)

*<sup>n</sup>*þ1, *<sup>U</sup>*€ *<sup>i</sup> <sup>n</sup>*þ1,

The first step is to predict the propagation of the displacement wave field:

temperature fields. The third step is to correct the displacement using the computed

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> from Eqs. (79), (81), (82), and (86), respectively. The continuity conditions for temperature, heat flux, displacement, and traction

*<sup>x</sup>*¼*h<sup>i</sup>* <sup>¼</sup> *<sup>T</sup>*ð Þ *<sup>i</sup>*þ<sup>1</sup> *<sup>α</sup>* ð Þ *<sup>x</sup>*, *<sup>z</sup>*, *<sup>τ</sup>*

*<sup>x</sup>*¼*h<sup>i</sup>* <sup>¼</sup> *<sup>q</sup>*ð Þ *<sup>i</sup>*þ<sup>1</sup> ð Þ *<sup>x</sup>*, *<sup>z</sup>*, *<sup>τ</sup>*

ð Þ *i*þ1 *<sup>a</sup>* ð Þ *x*, *z*, *τ*

*<sup>f</sup>* ð Þ *x*, *z*, *τ*

*<sup>x</sup>*¼*h<sup>i</sup>* <sup>¼</sup> *<sup>u</sup>*ð Þ *<sup>i</sup>*þ<sup>1</sup>

where *n* is the total number of layers, *ta* are the tractions which is defined by

where Ψ *<sup>f</sup>* , Φ *<sup>f</sup>* , *f*, and *h* are prescribed functions, *C* ¼ *C*<sup>1</sup> ∪*C*<sup>2</sup> ¼ *C*<sup>3</sup> ∪*C*4, and

According to Fahmy [58, 60], the design sensitivities of displacements components and total 3T can be performed by implicit differentiation of (75) and (76), respectively, which describe the structural response with respect to the design vari-

The bi-directional evolutionary structural optimization (BESO) is the evolution-

ary topology optimization method that allows modification of the structure by either adding or removing material to or from the structure design. This addition or removal depends on the sensitivity analysis. Sensitivity analysis is the estimation of the response of the structure to the modification of design variables and is depen-

(75), respectively, in Eq. (85) and solve the resulted equation for the three-

three-temperature fields for the Eq. (80). The fourth step is to compute *U*\_ *<sup>i</sup>*

that have been considered in the current chapter can be expressed as

 

 *<sup>x</sup>*¼*h<sup>i</sup>* <sup>¼</sup> *<sup>t</sup>*

The initial and boundary conditions of the present study are

*<sup>n</sup>*. The second step is to substitute for *<sup>U</sup>*\_ *<sup>i</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.92852*

*Ti*

*qi* ð Þ *x*, *z*, *τ* 

*ui*

*t i <sup>a</sup>*ð Þ *x*, *z*, *τ*

*<sup>f</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>z</sup>*, 0 *<sup>u</sup>*\_ *<sup>i</sup>*

*<sup>α</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>z</sup>*, 0 *<sup>T</sup><sup>i</sup>*

*ui*

*ta* ¼ *σabnb*, and *i* ¼ 1, 2, … , *n* � 1.

*ui*

*t i*

*Ti*

*Ti*

*qi*

�

**4. Design sensitivity and optimization**

dent on the calculation of derivatives [70–80].

ables, and then we can compute thermal stresses sensitivities.

*C*<sup>1</sup> ∩*C*<sup>2</sup> ¼ *C*<sup>3</sup> ∩*C*<sup>4</sup> ¼ 0.

**67**

*<sup>α</sup>*ð Þ *x*, *z*, *τ* 

*<sup>f</sup>*ð Þ *x*, *z*, *τ*

(85) is as follows:

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> , and *<sup>T</sup>*€*<sup>i</sup>*

*Uip <sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>U</sup><sup>i</sup>*

*T*\_ *i*

Substituting *U*\_ *<sup>i</sup> <sup>n</sup>*þ<sup>1</sup> from Eq. (79) into Eq. (75), we obtain

$$\ddot{U}\_{n+1}^{j} = \widehat{\mathbf{M}}^{-1} \left[ \widehat{\mathbf{Q}}\_{n+1}^{jp} - \widehat{\mathbf{T}} \left[ \overline{\mathbf{y}}^{-1} \left[ \dot{U}\_{n}^{j} + \frac{\Delta \mathbf{r}}{2} \left[ \ddot{U}\_{n}^{j} + \widehat{\mathbf{M}}^{-1} \left( \widehat{\mathbf{Q}}\_{n+1}^{jp} - \widehat{\mathbf{K}}^{j} \boldsymbol{U}\_{n+1}^{j} \right) \right] \right] \right] - \widehat{\mathbf{K}}^{r} \mathbf{U}\_{n+1}^{i} \right] \tag{81}$$

By integrating the heat Eq. (74) and using Eq. (76), we obtain

$$\begin{split} \dot{T}\_{a(n+1)}^{j} &= \dot{T}\_{n}^{j} + \frac{\Delta \tau}{2} \left( \ddot{T}\_{a(n+1)}^{j} + \dddot{T}\_{am}^{j} \right) \\ &= \dot{T}\_{an}^{j} + \frac{\Delta \tau}{2} \left( \widehat{\mathbf{X}}^{-1} \left[ \widehat{\mathbf{Z}}^{-1} \overleftarrow{\dot{U}}\_{n+1}^{j} + \widehat{\mathbf{R}}^{-1} - \widehat{\mathbf{A}}^{-1} \overleftarrow{T}\_{a(n+1)}^{j} - \widehat{\mathbf{B}}^{-1} \overleftarrow{T}\_{a(n+1)}^{i} \right] + \ddot{T}\_{an}^{i} \right) \end{split} \tag{82}$$

$$\begin{split} T\_{a(n+1)}^{i} &= T\_{an}^{i} + \frac{\Delta \mathbf{r}}{2} \Big( \dot{T}\_{a(n+1)}^{i} + \dot{T}\_{an}^{i} \Big) \\ &= T\_{an}^{i} + \Delta \dot{\mathbf{r}}\_{an}^{i} + \frac{\Delta \mathbf{r}^{2}}{4} \Big( \ddot{T}\_{an}^{i} + \widetilde{\mathbf{X}}^{-1} \Big[ \widetilde{\mathbf{Z}}^{i} \dddot{U}\_{n+1}^{i} + \widetilde{\mathbf{R}}^{-1} \widetilde{\mathbf{A}}^{i} \ddot{T}\_{a(n+1)}^{i} - \widetilde{\mathbf{B}}^{-1} \widetilde{T}\_{a(n+1)}^{i} \Big] \Big), \end{split} \tag{83}$$

From Eq. (82) we get

$$\begin{aligned} \dot{T}\_{a(n+1)}^{j} &= \gamma^{-1} \left[ \dot{T}\_{an}^{i} + \frac{\Delta \tau}{2} \left( \widetilde{\mathbf{X}^{-1}} \left[ \widetilde{\mathbf{U}^{i}} \ddot{U}\_{n+1}^{i} + \widetilde{\mathbf{R}^{-1}} \mathbf{S}^{i} \widetilde{\mathbf{Z}^{i}}\_{a(n+1)} \right] + \ddot{T}\_{an}^{i} \right) \right] \quad \text{(84)}\\ &\text{where } \mathbf{\gamma} = \left( I + \frac{1}{2} \widetilde{\mathbf{A} \cdot \Delta \tau} \widetilde{\mathbf{X}^{-1}} \right). \end{aligned} $$
  $\text{Substituting Eq. (84) into Eq. (83), we obtain}$ 

$$\begin{split} T^{i}\_{a(n+1)} &= T^{i}\_{an} + \Delta \tau \dot{T}^{i}\_{an} + \frac{\Delta \tau^{2}}{4} \Big( \ddot{T}^{i}\_{an} + \widehat{\phantom{\textbf{X}}^{-1}} \Big[ \widehat{\phantom{\tau \textbf{U}}}^{i} \ddot{U}^{i}\_{n+1} + \widehat{\phantom{\mathsf{R}} \textbf{-}} \widehat{\phantom{\mathsf{A}}} \Big( \eta^{-1} \Big[ \dot{T}^{i}\_{an} \\ &+ \frac{\Delta \tau}{2} \Big( \widehat{\phantom{\mathsf{X}}}^{-1} \Big[ \widehat{\phantom{\tau \textbf{U}}}^{i} \ddot{U}^{i}\_{n+1} + \widehat{\phantom{\mathsf{R}} \textbf{-}} \widehat{\phantom{\tau \textbf{U}}}^{i} \ddot{T}^{i}\_{a(n+1)} \Big] + \ddot{T}^{i}\_{an} \Big) \Big] \Big) - \widehat{\phantom{\tau \textbf{S}}}^{i} \dot{T}^{i}\_{a(n+1)} \Big] \Big) \end{split} \tag{85}$$

Substituting *T*\_ *<sup>i</sup> <sup>n</sup>*þ<sup>1</sup> from Eq. (84) into Eq. (76), we get

$$\begin{split} \ddot{T}\_{a(n+1)}^{i} &= \widehat{\mathbf{X}^{i}} \left[ \widehat{\mathbf{Z}^{i}} \ddot{\mathbf{U}}\_{n+1}^{i} + \widehat{\mathbf{R}^{i}} - \widehat{\mathbf{A}^{i}} \left( \boldsymbol{\gamma}^{-1} \Big[ \dot{T}\_{m}^{i} + \frac{\Delta \tau}{2} \Big( \widehat{\mathbf{X}^{i}} \ddot{\mathbf{U}}\_{n+1}^{i} + \widetilde{\mathbf{R}^{i}} \Big] \right) \right] \\ &- \widehat{\mathbf{B}^{i}} \boldsymbol{T}\_{a(n+1)}^{i} \Big] + \ddot{T}\_{a(n+1)}^{i} \Big] \Big) \hat{\mathbf{J}}^{i} \mathbf{B}^{i} \mathbf{T}\_{a(n+1)}^{i} \Big] \end{split} \tag{86}$$

*A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto… DOI: http://dx.doi.org/10.5772/intechopen.92852*

Now, a displacement predicted staggered procedure for the solution of (80) and (85) is as follows:

The first step is to predict the propagation of the displacement wave field: *Uip <sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>U</sup><sup>i</sup> <sup>n</sup>*. The second step is to substitute for *<sup>U</sup>*\_ *<sup>i</sup> <sup>n</sup>*þ<sup>1</sup> and *<sup>U</sup>*€ *<sup>i</sup> <sup>n</sup>*þ<sup>1</sup> from Eqs. (77) and (75), respectively, in Eq. (85) and solve the resulted equation for the threetemperature fields. The third step is to correct the displacement using the computed three-temperature fields for the Eq. (80). The fourth step is to compute *U*\_ *<sup>i</sup> <sup>n</sup>*þ1, *<sup>U</sup>*€ *<sup>i</sup> <sup>n</sup>*þ1, *T*\_ *i <sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> , and *<sup>T</sup>*€*<sup>i</sup> <sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> from Eqs. (79), (81), (82), and (86), respectively.

The continuity conditions for temperature, heat flux, displacement, and traction that have been considered in the current chapter can be expressed as

$$\left.T\_a^i(\mathbf{x}, \mathbf{z}, \mathbf{z})\right|\_{\mathbf{x}=\mathbf{h}^i} = \left.T\_a^{(i+1)}(\mathbf{x}, \mathbf{z}, \mathbf{z})\right|\_{\mathbf{x}=\mathbf{h}^i} \tag{87}$$

$$\left.q^{i}(\mathbf{x},\mathbf{z},\mathbf{r})\right|\_{\mathbf{x}=\mathbf{h}^{i}}=q^{(i+1)}(\mathbf{x},\mathbf{z},\mathbf{r})\big|\_{\mathbf{x}=\mathbf{h}^{i}}\tag{88}$$

$$\left.u\_f^i(\mathbf{x}, z, \mathbf{z})\right|\_{\mathbf{x}=h^i} = \left.u\_f^{(i+1)}(\mathbf{x}, z, \mathbf{z})\right|\_{\mathbf{x}=h^i} \tag{89}$$

$$\left. \overline{\mathfrak{F}}\_a^i(\infty, z, \mathfrak{r}) \right|\_{\mathfrak{x} = h^i} = \mathfrak{F}\_a^{(i+1)}(\infty, z, \mathfrak{r}) \Big|\_{\mathfrak{x} = h^i} \tag{90}$$

where *n* is the total number of layers, *ta* are the tractions which is defined by *ta* ¼ *σabnb*, and *i* ¼ 1, 2, … , *n* � 1.

The initial and boundary conditions of the present study are

$$u^i\_f(\mathbf{x}, z, \mathbf{0}) = \dot{u}^i\_f(\mathbf{x}, z, \mathbf{0}) = \mathbf{0} \quad \text{for} \quad (\mathbf{x}, z) \in \mathbb{R} \cup \mathbb{C} \tag{91}$$

$$u^i\_f(\mathbf{x}, \mathbf{z}, \tau) = \Psi\_f(\mathbf{x}, \mathbf{z}, \tau) \quad \text{for} \quad (\mathbf{x}, \mathbf{z}) \in \mathbf{C}\_3 \tag{92}$$

$$\vec{\mathcal{T}}\_a(\mathbf{x}, z, \tau) = \Phi\_f(\mathbf{x}, z, \tau) \quad \text{for} \quad (\mathbf{x}, z) \in \mathbb{C}\_4, \tau > 0 \tag{93}$$

$$T\_a^i(\mathbf{x}, z, \mathbf{0}) = T\_a^i(\mathbf{x}, z, \mathbf{0}) = \mathbf{0} \quad \text{for} \quad (\mathbf{x}, z) \in \mathcal{R} \cup \mathcal{C} \tag{94}$$

$$T\_a^\dagger(\mathbf{x}, z, \mathbf{z}) = \overline{f}(\mathbf{x}, z, \mathbf{z}) \quad \text{for} \quad (\mathbf{x}, z) \in \mathbf{C}\_1, \ \mathbf{z} > \mathbf{0} \tag{95}$$

$$q^i(\mathbf{x}, \mathbf{z}, \tau) = \overline{h}(\mathbf{x}, \mathbf{z}, \tau) \quad \text{for} \quad (\mathbf{x}, \mathbf{z}) \in \mathcal{C}\_2, \ \tau > 0 \tag{96}$$

where Ψ *<sup>f</sup>* , Φ *<sup>f</sup>* , *f*, and *h* are prescribed functions, *C* ¼ *C*<sup>1</sup> ∪*C*<sup>2</sup> ¼ *C*<sup>3</sup> ∪*C*4, and *C*<sup>1</sup> ∩*C*<sup>2</sup> ¼ *C*<sup>3</sup> ∩*C*<sup>4</sup> ¼ 0. �
