**3. A new mathematical modelling of nonlinear generalized micropolar thermoviscoelasticity problem**

With reference to a Cartesian coordinate system ð Þ *x*1, *x*2, *x*<sup>3</sup> , we consider an anisotropic micropolar thermoviscoelastic structure occupies the region R which bounded by a closed surface S, and Sið Þ i ¼ 1, 2, 3, 4 denotes subsets of S such that S1 þ S2 ¼ S3 þ S4 ¼ S*:*

#### **3.1 BEM simulation for temperature field**

The 2D-3T radiative heat conduction Eqs. (6)–(8) can be expressed as [53]

$$\nabla[\mathbb{K}\_{\alpha}\nabla\mathbb{T}\_{\alpha}(\mathbf{r},\mathbf{r})] + \overline{\mathbb{W}}(\mathbf{r},\mathbf{r}) = \mathbf{c}\_{\alpha}\rho\delta\_{1}\frac{\partial\mathbb{T}\_{\alpha}(\mathbf{r},\mathbf{r})}{\partial\mathbf{r}}\tag{9}$$

where

$$\overline{\boldsymbol{W}}(\mathbf{r},\mathbf{r}) = \begin{cases} -\rho \,\mathrm{W}\_{\mathrm{ei}}(\mathrm{T}\_{\mathrm{e}} - \mathrm{T}\_{\mathrm{i}}) - \rho \,\mathrm{W}\_{\mathrm{ep}}\left(\mathrm{T}\_{\mathrm{e}} - \mathrm{T}\_{\mathrm{p}}\right), \boldsymbol{\alpha} = \mathbf{e}, \boldsymbol{\delta}\_{\mathrm{1}} = \mathbf{1} \\\ \rho \,\mathrm{W}\_{\mathrm{ei}}(\mathrm{T}\_{\mathrm{e}} - \mathrm{T}\_{\mathrm{i}}), \boldsymbol{\alpha} = \mathbf{i}, \boldsymbol{\delta}\_{\mathrm{1}} = \mathbf{1} \\\ \rho \,\mathrm{W}\_{\mathrm{ep}}(\mathrm{T}\_{\mathrm{e}} - \mathrm{T}\_{\mathrm{p}}), \boldsymbol{\alpha} = \mathbf{p}, \boldsymbol{\delta}\_{\mathrm{1}} = \frac{\mathbf{4}}{\rho} \mathbf{T}\_{\mathrm{p}}^{3} \end{cases} \tag{10}$$

and

$$\mathbb{W}\_{\rm ei} = \rho \mathbb{A}\_{\rm ei} \mathbb{T}\_{\rm e}^{-2/3}, \mathbb{W}\_{\rm ep} = \rho \mathbb{A}\_{\rm ep} \mathbb{T}\_{\rm e}^{-1/2}, \mathbb{K}\_{\rm a} = \mathbb{A}\_{\rm a} \mathbb{T}\_{\rm a}^{5/2}, \mathfrak{a} = \mathfrak{e}, \mathrm{i}, \mathbb{K}\_{\rm p} = \mathbb{A}\_{\rm p} \mathbb{T}\_{\rm p}^{3+\mathbb{B}} \tag{11}$$

where parameters cα, αð Þ α ¼ e, i, p , , ei, ep are constant inside each subdomain, but they are discontinuous on the interfaces between subdomains. The total energy of unit mass can be described by

$$\mathbf{P} = \mathbf{P\_e} + \mathbf{P\_i} + \mathbf{P\_p}, \mathbf{P\_e} = \mathbf{c\_e} \mathbf{T\_e}, \mathbf{P\_i} = \mathbf{c\_i} \mathbf{T\_i}, \mathbf{P\_p} = \frac{1}{\rho} \mathbf{c\_p} \mathbf{T\_p^4} \tag{12}$$

Initial and boundary conditions can be written as

$$\mathbf{T}\_a(\mathbf{x}, \mathbf{y}, \mathbf{0}) = \mathbf{T}\_a^0(\mathbf{x}, \mathbf{y}) = \mathbf{g}\_1(\mathbf{x}, \mathbf{r}) \tag{13}$$

where

and

In which the entries of f�<sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.90824*

The diffusion matrix can be defined as

<sup>τ</sup>mþ1�τ<sup>m</sup> <sup>≤</sup> 1. The time derivative of (29) can be expressed as

� �Tmþ<sup>1</sup> <sup>α</sup> � <sup>θ</sup>GQ <sup>m</sup>þ<sup>1</sup> <sup>¼</sup> <sup>C</sup>

<sup>T</sup>\_ <sup>α</sup> <sup>¼</sup> dT<sup>α</sup> dθ

dθ

By substituting from Eqs. (29)–(31) into Eq. (25), we obtain

the previous time step solution as initial values for next step, we get

use of (23), we get [53]

vectors, respectively.

where 0 <sup>≤</sup><sup>θ</sup> <sup>¼</sup> <sup>τ</sup>�τ<sup>m</sup>

C *<sup>Δ</sup>*τ<sup>m</sup> <sup>þ</sup> <sup>θ</sup><sup>H</sup>

**191**

with

<sup>b</sup>q<sup>j</sup> ¼ �<sup>α</sup>

*Boundary Element Mathematical Modelling and Boundary Element Numerical Techniques…*

i¼1 f �1 ji

f g<sup>F</sup> ji <sup>¼</sup> <sup>f</sup> <sup>j</sup>

The boundary integral discretization scheme has been applied to (21) with the

where T<sup>α</sup> and Q are temperature, heat flux vectors and internal heat generation

<sup>C</sup> ¼ � <sup>H</sup> <sup>T</sup>b<sup>α</sup> � <sup>G</sup> <sup>Q</sup><sup>b</sup> h iF�<sup>1</sup>

ij <sup>¼</sup> <sup>T</sup>b<sup>j</sup>

ij <sup>¼</sup> <sup>b</sup>qj

Tb n o

Qb n o

<sup>T</sup><sup>α</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>θ</sup> <sup>T</sup><sup>m</sup>

For solving (25) numerically, we interpolate the functions T<sup>α</sup> and q as

<sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>T</sup><sup>m</sup>þ<sup>1</sup> <sup>α</sup> � <sup>T</sup><sup>m</sup>

α <sup>τ</sup><sup>m</sup>þ<sup>1</sup> � <sup>τ</sup><sup>m</sup> <sup>¼</sup> <sup>T</sup><sup>m</sup>þ<sup>1</sup> <sup>α</sup> � <sup>T</sup><sup>m</sup>

*<sup>Δ</sup>*τ<sup>m</sup> � ð Þ <sup>1</sup> � <sup>θ</sup> <sup>H</sup> � �Tm

Making use of initial conditions and boundary conditions at *Δ*τ<sup>m</sup> and considering

aj

ð Þ¼ <sup>τ</sup> <sup>X</sup> N

*∂*Tb j α ∂n

∂T ri ð Þ , τ ∂τ

ji are the coefficients of F�<sup>1</sup> which described in [34].

<sup>C</sup> <sup>T</sup>\_ <sup>α</sup> <sup>þ</sup> H T<sup>α</sup> <sup>¼</sup> G Q (25)

ð Þ ri (24)

D�<sup>1</sup> (26)

ð Þ xi (27)

ð Þ xi (28)

<sup>α</sup> <sup>þ</sup> <sup>θ</sup> <sup>T</sup><sup>m</sup>þ<sup>1</sup> <sup>α</sup> (29)

α

X ¼ (33)

*<sup>Δ</sup>*τ<sup>m</sup> (31)

<sup>α</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>θ</sup> GQ <sup>m</sup> (32)

<sup>q</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>θ</sup> <sup>q</sup><sup>m</sup> <sup>þ</sup> <sup>θ</sup> qmþ<sup>1</sup> (30)

(22)

(23)

$$\mathbb{K}\_{\mathfrak{a}} \frac{\partial \mathbf{T}\_{\mathfrak{a}}}{\partial \mathbf{n}} \Big|\_{\Gamma\_{1}} = \mathbf{0}, \mathfrak{a} = \mathbf{e}, \mathbf{i}, \operatorname{\mathbf{T}}\_{\mathbb{P}} \big|\_{\Gamma\_{1}} = \operatorname{\mathbf{g}}\_{2}(\mathbf{x}, \mathfrak{r}) \tag{14}$$

$$\mathbb{K}\_{\mathfrak{a}} \frac{\partial \mathbf{T}\_{\mathfrak{a}}}{\partial \mathbf{n}} \bigg|\_{\Gamma\_2} = \mathbf{0}, \mathfrak{a} = \mathbf{e}, \mathrm{i}, \mathrm{p} \tag{15}$$

we use the time-dependent fundamental solution which is a solution of the following differential equation

$$\mathrm{D}\mathbf{V}^{2}\mathrm{T}\_{a} + \frac{\partial\mathrm{T}\_{a}^{\*}}{\partial\mathbf{n}} = -\boldsymbol{\mathsf{S}}\left(\mathbf{r} - \mathbf{p}\_{i}\right)\boldsymbol{\mathsf{S}}\left(\mathbf{r} - \mathbf{r}\right), \mathrm{D} = \frac{\mathbb{K}\_{a}}{\rho\mathbf{c}}\tag{16}$$

In which the points pi are the singularities, where the temperatures are not defined there. Singular integrals are those whose kernels are not defined at the singularities on the integration domain R. They are defined by eliminating a small space including the singularity, and obtaining the limit when this small space tends to zero [40, 46].

The boundary integral equation corresponding to our considered heat conduction can be written as in Fahmy [46–48] as follows

$$\mathbf{CT}\_{a} = \frac{\mathbf{D}}{\mathbb{K}\_{a}} \int\_{\mathcal{O}}^{\pi} \int\_{\mathcal{S}} \left[ \mathbf{T}\_{a} \mathbf{q}^{\ast} - \mathbf{T}\_{a}^{\ast} \mathbf{q} \right] d\mathbf{S} \, d\mathbf{\tau} + \frac{\mathbf{D}}{\mathbb{K}\_{a}} \int\_{\mathcal{O}}^{\pi} \int\_{\mathcal{R}} \mathbf{b} \mathbf{T}\_{a}^{\ast} \, d\mathbf{R} \, d\mathbf{\tau} + \int\_{\mathcal{R}} \mathbf{T}\_{a}^{\dagger} \mathbf{T}\_{a}^{\ast} \big|\_{\mathbf{r} = 0} \tag{17}$$

which can be expressed in the following form [53].

$$\mathbf{CT}\_{a} = \int\_{\mathcal{S}} \left[ \mathbf{T}\_{a} \mathbf{q}^{\*} - \mathbf{T}\_{a}^{\*} \mathbf{q} \right] \, \mathrm{d}\mathbf{S} - \int\_{\mathcal{R}} \frac{\mathbb{K}\_{a}}{\mathbf{D}} \frac{\partial \mathbf{T}\_{a}^{\*}}{\partial \boldsymbol{\pi}} \mathbf{T}\_{a} \, \mathrm{d}\mathbf{R} \tag{18}$$

The time derivative of temperature T<sup>α</sup> can be approximated as

$$\frac{\partial \mathbf{T}\_{\alpha}}{\partial \mathbf{r}} \cong \sum\_{j=1}^{N} \mathbf{f}^{j}(\mathbf{r})^{j} \mathbf{a}^{j}(\mathbf{r}).\tag{19}$$

where f<sup>j</sup> ð Þ<sup>r</sup> and aj ð Þτ are known functions and unknown coefficients, respectively.

Also, we assume that <sup>T</sup><sup>b</sup> <sup>j</sup> <sup>α</sup> is a solution of

$$\nabla^2 \widehat{\mathbf{T}}\_a^\dagger = \mathbf{f}^\dagger \tag{20}$$

Thus, Eq. (18) results in the following boundary integral equation [53]

$$\mathbf{C} \cdot \mathbf{T}\_a = \int\_{\mathcal{S}} \left[ \mathbf{T}\_a \mathbf{q}^\* - \mathbf{T}\_a^\* \mathbf{q} \right] \, \mathrm{d}\mathbf{S} + \sum\_{j=1}^{N} \mathbf{a}^j(\tau) \mathbf{D}^{-1} \left( \mathbf{C} \hat{\mathbf{T}}\_a^\dagger - \int\_{\mathcal{S}} \left[ \mathbf{T}\_a^\dagger \mathbf{q}^\* - \hat{\mathbf{q}}^\dagger \mathbf{T}\_a^\* \right] \, \mathrm{d}\mathbf{S} \right) \tag{21}$$

*Boundary Element Mathematical Modelling and Boundary Element Numerical Techniques… DOI: http://dx.doi.org/10.5772/intechopen.90824*

where

The total energy of unit mass can be described by

Initial and boundary conditions can be written as

<sup>α</sup> *∂*T<sup>α</sup> ∂n � � � � Γ1

D∇<sup>2</sup> T<sup>α</sup> þ

tion can be written as in Fahmy [46–48] as follows

<sup>T</sup>αq<sup>∗</sup> � <sup>T</sup><sup>∗</sup>

CT<sup>α</sup> ¼

<sup>α</sup> <sup>q</sup> � �dS d<sup>τ</sup> <sup>þ</sup>

which can be expressed in the following form [53].

<sup>T</sup>αq<sup>∗</sup> � <sup>T</sup><sup>∗</sup> <sup>α</sup> <sup>q</sup> � � dS �

The time derivative of temperature T<sup>α</sup> can be approximated as

ffi <sup>X</sup> N

j¼1 f j ð Þr j aj

*∂*T<sup>α</sup> ∂τ

<sup>α</sup> is a solution of

N

j¼1 aj

∇2 Tb j <sup>α</sup> ¼ f

Thus, Eq. (18) results in the following boundary integral equation [53]

ð Þ<sup>τ</sup> <sup>D</sup>�<sup>1</sup> CT<sup>b</sup> <sup>j</sup>

<sup>α</sup> � ð S Tj

ð S

following differential equation

to zero [40, 46].

CT<sup>α</sup> <sup>¼</sup> <sup>D</sup> <sup>α</sup> ðτ O ð S

where f<sup>j</sup>

respectively.

C T<sup>α</sup> ¼

**190**

ð S

ð Þ<sup>r</sup> and aj

Also, we assume that <sup>T</sup><sup>b</sup> <sup>j</sup>

<sup>T</sup>αq<sup>∗</sup> � <sup>T</sup><sup>∗</sup>

<sup>α</sup> <sup>q</sup> � � dS <sup>þ</sup><sup>X</sup>

T<sup>α</sup> x, y, 0 � � <sup>¼</sup> <sup>T</sup><sup>0</sup>

> <sup>α</sup> *∂*T<sup>α</sup> ∂n � � � � Γ2

*∂*T<sup>∗</sup> α

<sup>P</sup> <sup>¼</sup> Pe <sup>þ</sup> Pi <sup>þ</sup> Pp, Pe <sup>¼</sup> ceTe, Pi <sup>¼</sup> ciTi, Pp <sup>¼</sup> <sup>1</sup>

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

<sup>α</sup> x, y

� �

� �δ τð Þ � <sup>r</sup> , D <sup>¼</sup> <sup>α</sup>

<sup>α</sup> dR dτ þ

*∂*T<sup>∗</sup> α ∂τ

¼ 0, α ¼ e, i, Tp

we use the time-dependent fundamental solution which is a solution of the

In which the points pi are the singularities, where the temperatures are not defined there. Singular integrals are those whose kernels are not defined at the singularities on the integration domain R. They are defined by eliminating a small space including the singularity, and obtaining the limit when this small space tends

The boundary integral equation corresponding to our considered heat conduc-

D <sup>α</sup> ðτ O ð R bT<sup>∗</sup>

> ð R <sup>α</sup> D

ð Þτ are known functions and unknown coefficients,

<sup>∂</sup><sup>n</sup> ¼ �<sup>δ</sup> <sup>r</sup> � pi

ρ cpT<sup>4</sup>

� � <sup>¼</sup> g1ð Þ x, <sup>τ</sup> (13)

¼ 0, α ¼ e, i, p (15)

ρc

ð R Ti αT<sup>∗</sup> α � �

<sup>Γ</sup><sup>1</sup> ¼ g2ð Þ x, τ (14)

<sup>p</sup> (12)

(16)

<sup>τ</sup>¼<sup>0</sup> (17)

T<sup>α</sup> dR (18)

ð Þτ *:* (19)

<sup>j</sup> (20)

<sup>α</sup>q<sup>∗</sup> � <sup>b</sup>q<sup>j</sup>

� �

h i

T∗ α

dS

(21)

$$
\widehat{\mathbf{q}}^{\mathbf{j}} = -\mathbb{K}\_{\mathbf{a}} \frac{\partial \widehat{\mathbf{T}}\_{\mathbf{a}}^{\mathbf{j}}}{\partial \mathbf{n}} \tag{22}
$$

and

$$\mathbf{a}^{\dagger}(\mathbf{r}) = \sum\_{\mathbf{i}=1}^{N} \mathbf{f}\_{\text{ji}}^{-1} \frac{\partial \mathbf{T}(\mathbf{r}\_{\text{i}}, \mathbf{r})}{\partial \mathbf{r}} \tag{23}$$

In which the entries of f�<sup>1</sup> ji are the coefficients of F�<sup>1</sup> which described in [34].

$$\{\mathbf{F}\}\_{\mathbf{j}\mathbf{i}} = \mathbf{f}^{\mathbf{j}}(\mathbf{r}\_{\mathbf{i}}) \tag{24}$$

The boundary integral discretization scheme has been applied to (21) with the use of (23), we get [53]

$$\mathbf{C}\,\dot{\mathbf{T}}\_a + \mathbf{H}\,\mathbf{T}\_a = \mathbf{G}\,\mathbf{Q} \tag{25}$$

where T<sup>α</sup> and Q are temperature, heat flux vectors and internal heat generation vectors, respectively.

The diffusion matrix can be defined as

$$\mathbf{C} = -\left[\mathbf{H}^{\widehat{\mathbf{T}}} \widehat{\mathbf{T}}\_a - \mathbf{G} \,\, \widehat{\mathbf{Q}}\right] \mathbf{F}^{-1} \mathbf{D}^{-1} \tag{26}$$

with

$$\left\{\widehat{\mathbf{T}}\right\}\_{\stackrel{\rightleftharpoons}{\doteq}} = \widehat{\mathbf{T}}^{\dagger}(\mathbf{x}\_{i})\tag{27}$$

$$\left\{\hat{\mathbb{Q}}\right\}\_{\mathfrak{j}} = \hat{\mathbb{q}}^{\downarrow}(\mathbf{x}\_{\mathfrak{i}}) \tag{28}$$

For solving (25) numerically, we interpolate the functions T<sup>α</sup> and q as

$$\mathbf{T}\_a = (\mathbf{1} - \boldsymbol{\Theta})\mathbf{T}\_a^{\mathbf{m}} + \boldsymbol{\Theta}\,\mathbf{T}\_a^{\mathbf{m}+1} \tag{29}$$

$$\mathbf{q} = (\mathbf{1} - \boldsymbol{\theta})\mathbf{q}^{\mathbf{m}} + \boldsymbol{\theta}\,\mathbf{q}^{\mathbf{m}+1} \tag{30}$$

where 0 <sup>≤</sup><sup>θ</sup> <sup>¼</sup> <sup>τ</sup>�τ<sup>m</sup> <sup>τ</sup>mþ1�τ<sup>m</sup> <sup>≤</sup> 1. The time derivative of (29) can be expressed as

$$\dot{\mathbf{T}}\_a = \frac{\mathbf{d}\mathbf{T}\_a}{\mathbf{d}\theta} \frac{\mathbf{d}\theta}{\mathbf{d}\tau} = \frac{\mathbf{T}\_a^{\mathrm{m}+1} - \mathbf{T}\_a^{\mathrm{m}}}{\mathbf{\tau}^{\mathrm{m}+1} - \mathbf{\tau}^{\mathrm{m}}} = \frac{\mathbf{T}\_a^{\mathrm{m}+1} - \mathbf{T}\_a^{\mathrm{m}}}{\Delta\mathbf{\tau}^{\mathrm{m}}} \tag{31}$$

By substituting from Eqs. (29)–(31) into Eq. (25), we obtain

$$\left(\frac{\mathbf{C}}{\Delta \mathbf{t}^{\mathbf{m}}} + \theta \mathbf{H}\right) \mathbf{T}\_{a}^{\mathbf{m}+1} - \theta \mathbf{G} \mathbf{Q}^{\mathbf{m}+1} = \left(\frac{\mathbf{C}}{\Delta \mathbf{t}^{\mathbf{m}}} - (\mathbf{1} - \theta) \mathbf{H}\right) \mathbf{T}\_{a}^{\mathbf{m}} + (\mathbf{1} - \theta) \mathbf{G} \mathbf{Q}^{\mathbf{m}} \tag{32}$$

Making use of initial conditions and boundary conditions at *Δ*τ<sup>m</sup> and considering the previous time step solution as initial values for next step, we get

$$\mathbf{a} \mathbf{X} = \mathbf{b} \tag{33}$$

The Adaptive Smoothing and Prolongation Algebraic Multigrid (aSP-AMG) method, which uses an adaptive Factorized Sparse Approximate Inverse (aFSAI) [67] preconditioner as high performance technique that has been implemented efficiently in Matlab (R2018a) for solving the resulting simultaneous linear algebraic systems (33).

## **3.2 BEM simulation for micropolar thermoviscoelastic fields**

According to the weighted residual method, we can write the differential Eqs. (1) and (2) in the following integral form

$$\int\_{\mathbb{R}} \left(\boldsymbol{\sigma}\_{\text{ij},\mathfrak{j}} + \mathbf{U}\_{\text{i}}\right) \mathbf{u}\_{\text{i}}^{\*} \, d\mathbf{R} = \mathbf{0} \tag{34}$$

According to Eringen [68], the elastic and couple stresses can be written in the

*Boundary Element Mathematical Modelling and Boundary Element Numerical Techniques…*

<sup>i</sup> dR þ

ð Þ ui � ui <sup>λ</sup> <sup>∗</sup>

By applying integration by parts again, the left-hand side of (46) can be written

S u∗ <sup>i</sup> λ<sup>i</sup> dS �

þ ð S μ∗ <sup>i</sup> ω<sup>i</sup> dS

The obtained weighting functions for Ui <sup>¼</sup> *<sup>Δ</sup>*<sup>n</sup> and Vi <sup>¼</sup> 0 along el were first

ij,j <sup>þ</sup> <sup>ε</sup>ijkσ<sup>∗</sup>

li el, λ <sup>∗</sup>

The obtained weighting functions for Ui <sup>¼</sup> 0 and Vi <sup>¼</sup> *<sup>Δ</sup>*<sup>n</sup> along el were next

The fundamental solutions that have been obtained analytically by Dragos [69]

<sup>i</sup> <sup>¼</sup> <sup>λ</sup> ∗ ∗

ð S u∗ li λ<sup>i</sup> dS þ

ð S u∗ ∗ li λ<sup>i</sup> dS þ

li el, μ<sup>∗</sup>

li el, λ <sup>∗</sup>

σ∗ ∗

lj,j <sup>þ</sup> <sup>ε</sup>ljkσ∗ ∗

<sup>i</sup> <sup>¼</sup> <sup>ω</sup>∗ ∗

Using the above two sets of weighting functions into (47) we have

ð S μ∗ li ωidS þ

ð S μ∗ ∗ li ωidS þ

<sup>i</sup> <sup>¼</sup> <sup>λ</sup> <sup>∗</sup>

li el, μ<sup>∗</sup>

According to Dragos [69], the fundamental solutions can be written as

σ∗

m<sup>∗</sup>

<sup>i</sup> <sup>¼</sup> <sup>ω</sup><sup>∗</sup>

li el, ω<sup>∗</sup>

m∗ ∗

li el, ω<sup>∗</sup>

S λ ∗ li uidS � ð R Viω<sup>∗</sup> <sup>i</sup> dR

<sup>i</sup> dS �

ð S4 μi ω∗ <sup>i</sup> dS �

ð S ω∗ <sup>i</sup> μ<sup>i</sup> dS þ

lj,j <sup>þ</sup> *<sup>Δ</sup>*nel <sup>¼</sup> <sup>0</sup> (48)

<sup>i</sup> <sup>¼</sup> <sup>μ</sup><sup>∗</sup>

ij,j ¼ 0 (51)

jk <sup>þ</sup> *<sup>Δ</sup>*nel <sup>¼</sup> <sup>0</sup> (52)

<sup>i</sup> <sup>¼</sup> <sup>μ</sup>∗ ∗

ð S ω∗

ð S ω∗ ∗

jk ¼ 0 (49)

ð S3 μiω<sup>∗</sup> <sup>i</sup> dS

ð S λ ∗ <sup>i</sup> ui dS

li el, (50)

li el, (53)

li μ<sup>i</sup> dS (54)

li μ<sup>i</sup> dS (55)

(46)

(47)

where ijkl ¼ klij and ijkl ¼ klij as shown in [68]. Hence, Eq. (44) can be re-expressed as [53]

> ð R Uiu<sup>∗</sup>

ð S1

ij,jωi,j dR þ

ij,j <sup>þ</sup> <sup>ε</sup>ijkσ<sup>∗</sup>

jk � �ω<sup>i</sup> dR ¼ �<sup>ð</sup>

ð S1 λiu<sup>∗</sup> <sup>i</sup> dS þ

<sup>i</sup> dS

σij ¼ ijklεkl, mij ¼ ijklωk,l (45)

following form

� ð R σ∗

as [53]

ð R σ∗

used as follows:

used as follows:

can be written as

Cn liω<sup>n</sup> <sup>i</sup> ¼ �<sup>ð</sup>

**193**

¼ �<sup>ð</sup> S2 λ<sup>i</sup> u<sup>∗</sup>

þ ð S3

ij,jui dR þ

ij εij dR �

ð R m<sup>∗</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.90824*

<sup>i</sup> dS �

ð Þ <sup>ω</sup><sup>i</sup> � <sup>ω</sup><sup>i</sup> <sup>μ</sup><sup>∗</sup>

ð R m<sup>∗</sup>

> u∗ <sup>i</sup> <sup>¼</sup> <sup>u</sup><sup>∗</sup>

u∗ <sup>i</sup> <sup>¼</sup> <sup>u</sup>∗ ∗

> S λ ∗ ∗ li uidS �

Cn liu<sup>n</sup> <sup>i</sup> ¼ �<sup>ð</sup>

$$\int\_{R} \left(\mathbf{m}\_{\text{ij,j}} + \varepsilon\_{\text{ijk}} \sigma\_{\text{jk}} + \mathbf{V}\_{\text{i}}\right) \mathbf{o}\_{\text{i}}^{\*} \, \mathrm{d}R = \mathbf{0} \tag{35}$$

where

$$\mathbf{U}\_{\mathbf{i}} = \rho \mathbf{F}\_{\mathbf{i}} - \rho \ddot{\mathbf{u}}\_{\mathbf{i}},\\\mathbf{V}\_{\mathbf{i}} = \rho (\mathbf{M}\_{\mathbf{i}} - \mathbf{J}\ddot{\mathbf{u}}\_{\mathbf{i}}) \tag{36}$$

The boundary conditions are

$$\mathbf{u}\_{\mathbf{i}} = \overline{\mathbf{u}}\_{\mathbf{i}} \text{ on } \mathbb{S}\_{\mathbf{i}} \tag{37}$$

$$
\lambda\_{\mathbf{i}} = \sigma\_{\mathbf{i}\dagger} \mathbf{n}\_{\mathbf{j}} = \overline{\lambda}\_{\mathbf{i}} \text{ on } \mathbf{S}\_2 \tag{38}
$$

$$\mathbf{a}\_{\mathbf{i}} = \overline{\mathbf{a}}\_{\mathbf{i}} \text{ on } \mathbf{S}\_{\mathbf{i}} \tag{39}$$

$$
\mu\_{\mathbf{i}} = \mathbf{m}\_{\mathbf{i}\dagger} \mathbf{n}\_{\mathbf{j}} = \overline{\mu}\_{\mathbf{i}} \text{ on } \mathbb{S}\_4 \tag{40}
$$

By integrating by parts the first term of Eqs. (34) and (35), we obtain

$$-\int\_{R} \sigma\_{\text{ij}} \mathbf{u}\_{\text{i},\text{j}}^{\*} \, \text{dR} + \int\_{R} \mathbf{U}\_{\text{i}} \mathbf{u}\_{\text{i}}^{\*} \, \text{dR} = -\int\_{S\_{2}} \lambda\_{\text{i}} \mathbf{u}\_{\text{i}}^{\*} \, \text{dS} \tag{41}$$

$$-\int\_{\mathcal{R}} \mathbf{m}\_{\text{ij}} \boldsymbol{\alpha}\_{\text{i,j}}^{\*} \, \text{d}\mathbf{R} + \int\_{\mathcal{R}} \boldsymbol{\varepsilon}\_{\text{j}\mathbf{k}} \boldsymbol{\sigma}\_{\text{jk}} \boldsymbol{\alpha}\_{\text{i}}^{\*} \, \text{d}\mathbf{R} + \int\_{\mathcal{R}} \mathbf{V}\_{\text{i}} \boldsymbol{\alpha}\_{\text{i}}^{\*} \, \text{d}\mathbf{R} = -\int\_{\mathcal{S}\_{\text{4}}} \boldsymbol{\mu}\_{\text{i}} \boldsymbol{\alpha}\_{\text{i}}^{\*} \, \text{d}\mathbf{S} \tag{42}$$

On the basis of Huang and Liang [64], we can write

$$\begin{aligned} & -\int\_{\mathcal{R}} \boldsymbol{\sigma}\_{\overline{\mathsf{u}}\boldsymbol{\mathsf{i}}} \mathbf{u}\_{\mathrm{i}}^{\*} \, \mathrm{d}\mathbf{R} + \int\_{\mathcal{R}} (\mathbf{m}\_{\overline{\mathsf{u}}\boldsymbol{\mathsf{i}}} + \boldsymbol{\varepsilon}\_{\overline{\mathsf{u}}\boldsymbol{\mathsf{k}}} \boldsymbol{\sigma}\_{\overline{\mathsf{k}}\boldsymbol{\mathsf{i}}}) \boldsymbol{\alpha}\_{\mathrm{i}}^{\*} \, \mathrm{d}\mathbf{R} + \int\_{\mathcal{R}} \mathbf{U}\_{\mathrm{i}} \mathbf{u}\_{\mathrm{i}}^{\*} \, \mathrm{d}\mathbf{R} + \int\_{\mathcal{R}} \mathbf{V}\_{\mathrm{i}} \boldsymbol{\alpha}\_{\mathrm{i}}^{\*} \, \mathrm{d}\mathbf{R} \\ & = \int\_{\mathcal{S}\_{\mathsf{I}}} (\lambda\_{\mathrm{i}} - \overline{\lambda}\_{\mathrm{i}}) \mathbf{u}\_{\mathrm{i}}^{\*} \, \mathrm{d}\mathbf{S} + \int\_{\mathcal{S}\_{\mathsf{I}}} (\overline{\mathbf{u}\_{\mathrm{i}}} - \mathbf{u}\_{\mathrm{i}}) \lambda\_{\mathrm{i}}^{\*} \, \mathrm{d}\mathbf{S} + \int\_{\mathcal{S}\_{\mathsf{I}}} (\mu\_{\mathrm{i}} - \overline{\mu}\_{\mathrm{i}}) \mathbf{o}\_{\mathrm{i}}^{\*} \, \mathrm{d}\mathbf{S} + \int\_{\mathcal{S}\_{\mathsf{J}}} (\overline{\mathbf{u}\_{\mathrm{i}}} - \mathbf{o}\_{\mathrm{i}}) \boldsymbol{\mu}\_{\mathrm{i}}^{\*} \, \mathrm{d}\mathbf{S}, \end{aligned} \tag{43}$$

By integrating by parts, the left-hand side of (43) can be written as

$$\begin{aligned} & -\int\_{\mathcal{R}} \boldsymbol{\sigma}\_{\text{ij}} \mathbf{e}\_{\text{ij}}^{\*} \, \mathrm{d}\mathbf{R} - \int\_{\mathcal{R}} \mathbf{m}\_{\text{ij}0} \mathbf{o}\_{\text{ij}}^{\*} \, \mathrm{d}\mathbf{R} + \int\_{\mathcal{R}} \mathbf{U}\_{\text{i}} \mathbf{u}\_{\text{i}}^{\*} \, \mathrm{d}\mathbf{R} + \int\_{\mathcal{R}} \mathbf{V}\_{\text{i}} \mathbf{o}\_{\text{i}}^{\*} \, \mathrm{d}\mathbf{R} \\ & = -\int\_{S\_{1}} \overline{\lambda}\_{\text{i}} \mathbf{u}\_{\text{i}}^{\*} \, \mathrm{d}\mathbf{S} - \int\_{S\_{1}} \lambda\_{\text{i}} \mathbf{u}\_{\text{i}}^{\*} \, \mathrm{d}\mathbf{S} + \int\_{S\_{1}} (\overline{\mathbf{u}\_{\text{i}}} - \mathbf{u}\_{\text{i}}) \lambda\_{\text{i}}^{\*} \, \mathrm{d}\mathbf{S} - \int\_{S\_{4}} \overline{\mu}\_{\text{i}} \mathbf{o}\_{\text{i}}^{\*} \, \mathrm{d}\mathbf{S} - \int\_{S\_{3}} \mu\_{\text{i}} \mathbf{o}\_{\text{i}}^{\*} \, \mathrm{d}\mathbf{S} \\ & + \int\_{S\_{\text{\mathcal{S}}}} (\overline{\mathbf{u}\_{\text{i}}} - \mathbf{u}\_{\text{i}}) \mu\_{\text{i}}^{\*} \, \mathrm{d}\mathbf{S} \end{aligned} \tag{44}$$

*Boundary Element Mathematical Modelling and Boundary Element Numerical Techniques… DOI: http://dx.doi.org/10.5772/intechopen.90824*

According to Eringen [68], the elastic and couple stresses can be written in the following form

$$
\sigma\_{\text{ij}} = \mathbb{A}\_{\text{ijkl}} \mathbf{e}\_{\text{kl}}, \mathbf{m}\_{\text{ij}} = \mathbb{B}\_{\text{ijkl}} \mathbf{o}\_{\text{k},\text{l}} \tag{45}
$$

where ijkl ¼ klij and ijkl ¼ klij as shown in [68]. Hence, Eq. (44) can be re-expressed as [53]

The Adaptive Smoothing and Prolongation Algebraic Multigrid (aSP-AMG) method, which uses an adaptive Factorized Sparse Approximate Inverse (aFSAI) [67] preconditioner as high performance technique that has been implemented efficiently in Matlab (R2018a) for solving the resulting simultaneous linear alge-

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

According to the weighted residual method, we can write the differential

<sup>i</sup> dR ¼ 0 (34)

Ui ¼ ρFi � ρu€i, Vi ¼ ρð Þ Mi � Jω€<sup>i</sup> (36)

ui ¼ ui on S1 (37) λ<sup>i</sup> ¼ σijnj ¼ λ<sup>i</sup> on S2 (38) ω<sup>i</sup> ¼ ω<sup>i</sup> on S3 (39) μ<sup>i</sup> ¼ mijnj ¼ μ<sup>i</sup> on S4 (40)

<sup>i</sup> dR ¼ 0 (35)

<sup>i</sup> dS (41)

<sup>i</sup> dS (42)

ð Þ <sup>ω</sup><sup>i</sup> � <sup>ω</sup><sup>i</sup> <sup>μ</sup><sup>∗</sup>

ð S3 μiω<sup>∗</sup> <sup>i</sup> dS

<sup>i</sup> dS

(43)

(44)

σij,j þ Ui � �u<sup>∗</sup>

mij,j þ εijkσjk þ Vi � �ω<sup>∗</sup>

By integrating by parts the first term of Eqs. (34) and (35), we obtain

<sup>i</sup> dR ¼ �

ð S2 λiu<sup>∗</sup>

<sup>i</sup> dR ¼ �

<sup>i</sup> dR þ

<sup>μ</sup><sup>i</sup> � <sup>μ</sup><sup>i</sup> ð Þω<sup>∗</sup>

<sup>i</sup> dS �

ð S4 μiω<sup>∗</sup> <sup>i</sup> dS �

ð S4 μiω<sup>∗</sup>

ð R Viω<sup>∗</sup> <sup>i</sup> dR

<sup>i</sup> dS þ

ð S3

ð R Uiu<sup>∗</sup>

<sup>i</sup> dR þ

<sup>i</sup> dR þ

By integrating by parts, the left-hand side of (43) can be written as

ð S1

<sup>i</sup> dS þ

<sup>i</sup> dR þ

ð Þ ui � ui <sup>λ</sup> <sup>∗</sup>

ð R Viω<sup>∗</sup> <sup>i</sup> dR

ð Þ ui � ui <sup>λ</sup> <sup>∗</sup>

ð R Uiu<sup>∗</sup> ð R Viω<sup>∗</sup>

> ð R Uiu<sup>∗</sup>

ð S4

i,j dR þ

εijkσjkω<sup>∗</sup>

On the basis of Huang and Liang [64], we can write

i,j dR þ

mij,j þ εijkσjk � �ω<sup>∗</sup>

> ð S1

**3.2 BEM simulation for micropolar thermoviscoelastic fields**

ð R

Eqs. (1) and (2) in the following integral form

The boundary conditions are

� ð R σiju<sup>∗</sup>

> ð R

i,j dR þ

<sup>i</sup> dS þ

ð R ð R

braic systems (33).

where

� ð R mijω<sup>∗</sup>

<sup>i</sup> dR þ

λ<sup>i</sup> � λ<sup>i</sup> � �u<sup>∗</sup>

ij dR �

ð R mij,jω<sup>∗</sup>

<sup>i</sup> dS �

ð Þ <sup>ω</sup><sup>i</sup> � <sup>ω</sup><sup>i</sup> <sup>μ</sup><sup>∗</sup>

ð S1 λiu<sup>∗</sup> <sup>i</sup> dS þ

<sup>i</sup> dS

� ð R σij,ju<sup>∗</sup>

¼ ð S2

� ð R σijε <sup>∗</sup>

**192**

¼ � ð S2 λ<sup>i</sup> u<sup>∗</sup>

þ ð S3

$$\begin{aligned} & -\int\_{\mathcal{R}} \boldsymbol{\sigma}\_{\text{ij}}^{\*} \boldsymbol{\varepsilon}\_{\text{ij}} \, \mathrm{d}\mathcal{R} - \int\_{\mathcal{R}} \mathbf{m}\_{\text{ij},j}^{\*} \boldsymbol{\alpha}\_{\text{i},j} \, \mathrm{d}\mathcal{R} + \int\_{\mathcal{R}} \mathbf{U}\_{\text{i}} \mathbf{u}\_{\text{i}}^{\*} \, \mathrm{d}\mathcal{R} + \int\_{\mathcal{R}} \mathbf{V}\_{\text{i}} \boldsymbol{\alpha}\_{\text{i}}^{\*} \, \mathrm{d}\mathcal{R} \\ & = -\int\_{\mathcal{S}\_{2}} \overline{\lambda}\_{\text{i}} \mathbf{u}\_{\text{i}}^{\*} \, \mathrm{d}\mathcal{S} - \int\_{\mathcal{S}\_{1}} \lambda\_{\text{i}} \mathbf{u}\_{\text{i}}^{\*} \, \mathrm{d}\mathcal{S} + \int\_{\mathcal{S}\_{1}} (\overline{\mathbf{u}}\_{\text{i}} - \mathbf{u}\_{\text{i}}) \lambda\_{\text{i}}^{\*} \, \mathrm{d}\mathcal{S} - \int\_{\mathcal{S}\_{4}} \overline{\mu}\_{\text{i}} \mathbf{o}\_{\text{i}}^{\*} \, \mathrm{d}\mathcal{S} - \int\_{\mathcal{S}\_{3}} \mu\_{\text{i}} \mathbf{o}\_{\text{i}}^{\*} \, \mathrm{d}\mathcal{S} \\ & + \int\_{\mathcal{S}\_{3}} (\overline{\mathbf{u}}\_{\text{i}} - \boldsymbol{\alpha}\_{\text{i}}) \boldsymbol{\mu}\_{\text{i}}^{\*} \, \mathrm{d}\mathcal{S} \end{aligned} \tag{46}$$

By applying integration by parts again, the left-hand side of (46) can be written as [53]

$$\begin{aligned} \int\_{\mathcal{R}} \boldsymbol{\sigma}\_{\text{ij},\text{j}}^{\*} \mathbf{u}\_{\text{i}} \, \text{d}\mathbf{R} + \int\_{\mathcal{R}} \left( \mathbf{m}\_{\text{ij},\text{j}}^{\*} + \boldsymbol{\varepsilon}\_{\text{ijk}} \boldsymbol{\sigma}\_{\text{jk}}^{\*} \right) \boldsymbol{\alpha}\_{\text{i}} \, \text{d}\mathbf{R} &= -\int\_{\mathcal{S}} \mathbf{u}\_{\text{i}}^{\*} \boldsymbol{\lambda}\_{\text{i}} \, \text{d}\mathbf{S} - \int\_{\mathcal{S}} \boldsymbol{\alpha}\_{\text{i}}^{\*} \, \boldsymbol{\mu}\_{\text{i}} \, \text{d}\mathbf{S} + \int\_{\mathcal{S}} \boldsymbol{\lambda}\_{\text{i}}^{\*} \, \text{u}\_{\text{i}} \, \text{d}\mathbf{S} \\ &+ \int\_{\mathcal{S}} \boldsymbol{\mu}\_{\text{i}}^{\*} \, \text{u}\_{\text{i}} \, \text{d}\mathbf{S} \end{aligned} \tag{47}$$

The obtained weighting functions for Ui <sup>¼</sup> *<sup>Δ</sup>*<sup>n</sup> and Vi <sup>¼</sup> 0 along el were first used as follows:

$$
\sigma\_{\mathbf{l}\mathbf{j},\mathbf{j}}^{\*} + \Delta^{\mathbf{n}} \mathbf{e}\_{\mathbf{l}} = \mathbf{0} \tag{48}
$$

$$\mathbf{m}^\*\_{\mathbf{i}\mathbf{j},\mathbf{j}} + \varepsilon\_{\mathbf{i}\mathbf{j}\mathbf{k}} \sigma^\*\_{\mathbf{j}\mathbf{k}} = \mathbf{0} \tag{49}$$

According to Dragos [69], the fundamental solutions can be written as

$$\mathbf{u}\_{\rm i}^{\*} = \mathbf{u}\_{\rm li}^{\*}\mathbf{e}\_{\rm l}, \mathbf{o}\_{\rm i}^{\*} = \mathbf{o}\_{\rm li}^{\*}\mathbf{e}\_{\rm l}, \lambda\_{\rm i}^{\*} = \lambda\_{\rm li}^{\*}\mathbf{e}\_{\rm l}, \mu\_{\rm i}^{\*} = \mu\_{\rm li}^{\*}\mathbf{e}\_{\rm l},\tag{50}$$

The obtained weighting functions for Ui <sup>¼</sup> 0 and Vi <sup>¼</sup> *<sup>Δ</sup>*<sup>n</sup> along el were next used as follows:

$$
\sigma\_{\text{ij},\text{j}}^{\*,\*} = \mathbf{0} \tag{51}
$$

$$
\mathbf{m}\_{\rm lj,j}^{\*,\*} + \varepsilon\_{\rm ljk} \sigma\_{\rm jk}^{\*,\*} + \Delta^{\rm n} \mathbf{e}\_{\rm l} = \mathbf{0} \tag{52}
$$

The fundamental solutions that have been obtained analytically by Dragos [69] can be written as

$$\mathbf{u}\_{\mathbf{i}}^{\*} = \mathbf{u}\_{\mathbf{i}\mathbf{i}}^{\*} \,^{\*} \mathbf{e}\_{\mathbf{l}},\\\boldsymbol{\alpha}\_{\mathbf{i}}^{\*} = \boldsymbol{\alpha}\_{\mathbf{li}}^{\*} \,^{\*} \mathbf{e}\_{\mathbf{l}},\\\boldsymbol{\lambda}\_{\mathbf{i}}^{\*} = \boldsymbol{\lambda}\_{\mathbf{li}}^{\*} \,^{\*} \mathbf{e}\_{\mathbf{l}},\\\boldsymbol{\mu}\_{\mathbf{i}}^{\*} = \boldsymbol{\mu}\_{\mathbf{li}}^{\*} \,^{\*} \mathbf{e}\_{\mathbf{l}},\tag{53}$$

Using the above two sets of weighting functions into (47) we have

$$\mathbf{C}\_{\mathrm{li}}^{\mathrm{n}}\mathbf{u}\_{\mathrm{i}}^{\mathrm{n}} = -\int\_{\mathrm{S}} \lambda\_{\mathrm{li}}^{\*}\,\mathbf{u}\_{\mathrm{i}}\,\mathrm{d}\mathbf{S} - \int\_{\mathrm{S}} \mu\_{\mathrm{li}}^{\*}\,\boldsymbol{\alpha}\_{\mathrm{i}}\,\mathrm{d}\mathbf{S} + \int\_{\mathrm{S}} \mathbf{u}\_{\mathrm{li}}^{\*}\,\lambda\_{\mathrm{i}}\,\mathrm{d}\mathbf{S} + \int\_{\mathrm{S}} \boldsymbol{\alpha}\_{\mathrm{li}}^{\*}\,\boldsymbol{\mu}\_{\mathrm{i}}\,\mathrm{d}\mathbf{S} \tag{54}$$

$$\mathbf{C}\_{\mathrm{li}}^{\mathrm{n}}\mathrm{o}\_{\mathrm{i}}^{\mathrm{n}} = -\int\_{\mathrm{S}} \lambda\_{\mathrm{li}}^{\ast \ast} \,^{\ast}\mathbf{u}\_{\mathrm{i}} \mathrm{d}\mathbf{S} - \int\_{\mathrm{S}} \mu\_{\mathrm{li}}^{\ast \ast} \,^{\ast}\mathrm{o}\_{\mathrm{i}} \mathrm{d}\mathbf{S} + \int\_{\mathrm{S}} \mathbf{u}\_{\mathrm{li}}^{\ast \ast} \,^{\ast}\lambda\_{\mathrm{i}} \mathrm{d}\mathbf{S} + \int\_{\mathrm{S}} \mathrm{o}\_{\mathrm{li}}^{\ast \ast} \,^{\ast}\mu\_{\mathrm{i}} \mathrm{d}\mathbf{S} \tag{55}$$

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

Thus, we can write

$$\mathbf{C}^{\mathbf{n}}\mathbf{q}^{\mathbf{n}} = -\int\_{\mathcal{S}} \mathbf{p}^{\*} \cdot \mathbf{q} d\mathbf{S} + \int\_{\mathcal{S}} \mathbf{q}^{\*} \cdot \mathbf{p} d\mathbf{S} \tag{56}$$

(aFSAI) as described in [67] for solving the resulting simultaneous linear algebraic

*Boundary Element Mathematical Modelling and Boundary Element Numerical Techniques…*

functions. Spline curves can be expressed in terms of k � order B-spline basis function. All B-spline basis functions are assumed to have their domain on [0,1]. B-splines basis functions are a more general type of curve than Bezier curves, where each control point Pi of i þ 1 control points Pð Þ 0, P1, P2, … , Pi is connected with a basis function Ni,k, the knots are the points that subdivide the domain [0,1] into knot spans. Also, each B-spline basis function is non-zero on the entire

vides the flexibility to design a large variety of shapes. The considered NURBS can be defined as follows

C tðÞ¼

P<sup>n</sup> P

Two criteria can be implemented during shape optimization of the solid

F ¼ <sup>1</sup> 2 ð

F ¼ <sup>ð</sup>

the functional (66). By minimizing the functional (66) σ*<sup>i</sup>*<sup>j</sup> are closer to σ0.

F ¼ <sup>ð</sup>

S

S

S

σij σ0 � �<sup>n</sup>

σij, σ<sup>0</sup> and n are equivalent boundary stresses, reference stress and natural number, respectively, where the greater value of n increases the speed of convergence of

In order to find the optimal boundary conditions for temperature the following

u u0 � �<sup>n</sup>

ð Þ λ � u dS (65)

dS (66)

dS (67)

n

where Ni,oð Þt and ϖ<sup>i</sup> are the B-spline basis functions of order o and the weights

The genetic algorithm greatly reduces computing time and computer memory of achieving an optimum solution, so, it can be used for solving multi-objective problems without needing to calculate the sensitivities. The profiles of the considered objects are represented based on the free form deformation (FFD) technique, where the FFD control points are considered as the genes and then the profiles of chromosomes are defined by the sequence of genes. The population is constructed by many individuals (chromosomes), where the fitness functions are evaluated by

B-spline basis functions are used as weights in the same manner as Bézier basis

The efficiency of our numerical modeling technique has been improved using a nonuniform rational B-spline curve (NURBS) to decrease the computation time and model optimized boundary where it reduces the number of control points and pro-

<sup>i</sup>¼<sup>0</sup>Ni,oð Þ<sup>t</sup> <sup>ϖ</sup>iPi

(64)

<sup>i</sup>¼<sup>0</sup>Ni,oð Þ<sup>t</sup> <sup>ϖ</sup><sup>i</sup>

system (63) in Matlab (R2018a).

*DOI: http://dx.doi.org/10.5772/intechopen.90824*

of control points Pi, respectively.

I. The minimum global compliance:

II. The minimum boundary equivalent stresses

using the BEM.

functional can be applied

**195**

bodies [70]

interval.

where

$$\begin{aligned} \mathbf{C}^{\mathfrak{u}} &= \begin{bmatrix} \mathbf{C}\_{11} & \mathbf{C}\_{12} & \mathbf{0} \\ \mathbf{C}\_{21} & \mathbf{C}\_{22} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \end{bmatrix}, \mathbf{q}^{\*} = \begin{bmatrix} u\_{11}^{\*} & u\_{12}^{\*} & \alpha\_{13}^{\*} \\ u\_{21}^{\*} & u\_{22}^{\*} & \alpha\_{23}^{\*} \\ u\_{31}^{\*} & u\_{32}^{\*} & \alpha\_{33}^{\*} \end{bmatrix}, \mathbf{p}^{\*} = \begin{bmatrix} \lambda\_{11}^{\*} & \lambda\_{12}^{\*} & \mu\_{13}^{\*} \\ \lambda\_{21}^{\*} & \lambda\_{22}^{\*} & \mu\_{23}^{\*} \\ \lambda\_{31}^{\*} & \lambda\_{32}^{\*} & \mu\_{33}^{\*} \end{bmatrix}, \\\ \mathbf{q} = \begin{bmatrix} u\_{1} \\ u\_{2} \\ \mu\_{3} \end{bmatrix}, \mathbf{p} = \begin{bmatrix} \lambda\_{1} \\ \lambda\_{2} \\ \mu\_{3} \end{bmatrix} \end{aligned}$$

In order to solve (56) numerically, we construct the following functions

$$\mathbf{q} = \boldsymbol{\Psi} \ \mathbf{q}^{\mathrm{j}}, \mathbf{p} = \boldsymbol{\Psi} \ \mathbf{p}^{\mathrm{j}} \tag{57}$$

substituting above functions into (56) and discretizing the boundary, we obtain

$$\mathbf{C}^{\mathbf{n}}\mathbf{q}^{\mathbf{n}} = \sum\_{j=1}^{N\_{\varepsilon}} \left[ -\int\_{\Gamma\_{j}} \mathbf{p}^{\*}\,\boldsymbol{\Psi}\,\mathrm{d}\Gamma \right] \mathbf{q}^{j} + \sum\_{j=1}^{N\_{\varepsilon}} \left[ \int\_{\Gamma\_{j}} \mathbf{q}^{\*}\,\boldsymbol{\Psi}\,\mathrm{d}\Gamma \right] \mathbf{p}^{j} \tag{58}$$

Equation after integration may be expressed as

$$\mathbf{C}^{\dot{\mathbf{q}}}\mathbf{q}^{\dot{\mathbf{i}}} = -\sum\_{\mathbf{j}=1}^{\mathbf{N}\_{\boldsymbol{\sigma}}} \hat{\mathbf{H}}^{\dot{\mathbf{j}}\mathbf{}}\mathbf{q}^{\mathbf{j}} + \sum\_{\mathbf{j}=1}^{\mathbf{N}\_{\boldsymbol{\sigma}}} \hat{\mathbf{G}}^{\dot{\mathbf{i}}\mathbf{j}}\mathbf{p}^{\mathbf{j}} \tag{59}$$

which can be expressed as

$$\sum\_{j=1}^{N\_o} \mathbb{H}^{\circ j} \mathbf{q}^{\circ} = \sum\_{j=1}^{N\_o} \widehat{\mathbb{G}}^{\circ j} \mathbf{p}^{\circ} \tag{60}$$

where

$$\mathbb{H}^{\mathsf{i}\mathsf{j}} = \begin{cases} \widehat{\mathbb{H}}^{\mathsf{i}\mathsf{j}} \text{ if } \mathsf{i} \neq \mathsf{j} \\ \widehat{\mathbb{H}}^{\mathsf{i}\mathsf{j}} + \mathbb{C}^{\mathsf{i}} \text{ if } \mathsf{i} = \mathsf{j} \end{cases} \tag{61}$$

Thus, we can write the following system of matrix equation as

$$\mathbb{HP} = \mathbb{GP} \tag{62}$$

Hence, we get the following system of linear algebraic equations

$$\mathbb{A}\,\mathbb{X}=\mathbb{B}\,\tag{63}$$
