**4. Design sensitivity and optimization**

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 variables, and then we can compute thermal stresses sensitivities.

The bi-directional evolutionary structural optimization (BESO) is the evolutionary 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 dependent on the calculation of derivatives [70–80].

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

*<sup>n</sup>*þ<sup>1</sup> � Γ

<sup>γ</sup>�<sup>1</sup> *<sup>U</sup>*\_ *<sup>i</sup> <sup>n</sup>* þ Δ*τ* <sup>2</sup> *<sup>U</sup>*€ *<sup>i</sup>*

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

X z}|{�<sup>1</sup>

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>T</sup>*\_ *<sup>i</sup>*

*<sup>α</sup><sup>n</sup>* þ Δ*τ*<sup>2</sup> 4

*<sup>α</sup><sup>n</sup>* þ Δ*τ* 2

*<sup>α</sup><sup>n</sup>* <sup>þ</sup> <sup>Δ</sup>*τT*\_ *<sup>i</sup>*

<sup>2</sup> A z}|{

*<sup>α</sup><sup>n</sup>* þ

X z}|{�<sup>1</sup>

> z}|{ *U*€ *i*

� �

� �

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

*αn*

 z}|{ *U*€ *i*

*αn*

*T*€*i <sup>α</sup><sup>n</sup>* þ X z}|{�<sup>1</sup>

X z}|{�<sup>1</sup>

Δ*τ* X � � z}|{�<sup>1</sup>

> Δ*τ*<sup>2</sup> 4

 z}|{ *U*€ *i*

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

*T*€*i <sup>α</sup><sup>n</sup>* þ X z}|{�<sup>1</sup>

Substituting Eq. (84) into Eq. (83), we obtain

 z}|{ *U*€ *i*

.

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

*<sup>n</sup>*þ<sup>1</sup> from Eq. (84) into Eq. (76), we get

���

z}|{ � <sup>A</sup> z}|{

z}|{ � <sup>B</sup> z}|{ *Ti α*ð Þ *n*þ1

� � � � �

� B z}|{ *Ti α*ð Þ *n*þ1 �

� ���

� �

z}|{γ�<sup>1</sup> *<sup>U</sup>*\_ *<sup>i</sup>*

*n* þ Δ*τ* <sup>2</sup> *<sup>U</sup>*€ *<sup>i</sup> <sup>n</sup>* þ *M* z}|{�<sup>1</sup>

*<sup>n</sup>*þ<sup>1</sup> from Eq. (79) into Eq. (75), we obtain

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

 z}|{ *U*€ *i*

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

 z}|{ *U*€ *i*

<sup>γ</sup>�<sup>1</sup> *<sup>T</sup>*\_ *<sup>i</sup>*

*<sup>α</sup><sup>n</sup>* þ Δ*τ* 2

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

� h � �

z}|{ � <sup>A</sup> z}|{

<sup>þ</sup> *<sup>T</sup>*€*<sup>i</sup> αn*

> X z}|{�<sup>1</sup>

� � � �

z}|{ � <sup>A</sup> z}|{ *T*\_ *i*

� �

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

z}|{ � <sup>B</sup> z}|{ *Ti α*ð Þ *n*þ1

� �

z}|{ � <sup>A</sup> z}|{ *T*\_ *i*

� � � �

� �

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

 z}|{*ip*

� � � � � �

 z}|{*ip*

� � � � � � � �

� �

� � ��

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

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

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

z}|{ *Ti α*ð Þ *n*þ1

> *<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> � B z}|{ *Ti α*ð Þ *n*þ1

> > <sup>þ</sup> *<sup>T</sup>*€*<sup>i</sup> αn*

> > > <sup>γ</sup>�<sup>1</sup> *<sup>T</sup>*\_ *<sup>i</sup> αn*

� B z}|{ *Ti α*ð Þ *n*þ1

 z}|{ *U*€ *i n*þ1

� *K* z}|{*U<sup>i</sup> n*þ1

� *K* z}|{ *Ui n*þ1

(80)

(81)

<sup>þ</sup> *<sup>T</sup>*€*<sup>i</sup> αn*

(82)

(83)

(84)

��

(85)

þ z}|{

(86)

 z}|{*ip*

*<sup>n</sup>*þ<sup>1</sup> � Γ z}|{

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

*U*€ *i*

*T*\_ *i*

*Ti*

*T*\_ *i*

*Ti*

*T*€*i*

**66**

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

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>T</sup>*\_ *<sup>i</sup>*

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

<sup>¼</sup> *<sup>T</sup><sup>i</sup>*

<sup>¼</sup> *<sup>T</sup>*\_ *<sup>i</sup> <sup>α</sup><sup>n</sup>* þ Δ*τ* 2

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

*<sup>α</sup><sup>n</sup>* <sup>þ</sup> <sup>Δ</sup>*τT*\_ *<sup>i</sup>*

From Eq. (82) we get

*<sup>α</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> <sup>γ</sup>�<sup>1</sup> *<sup>T</sup>*\_ *<sup>i</sup>*

where <sup>γ</sup> <sup>¼</sup> *<sup>I</sup>* <sup>þ</sup> <sup>1</sup>

þ Δ*τ* 2

Substituting *T*\_ *<sup>i</sup>*

z}|{�<sup>1</sup>

� B z}|{ *Ti α*ð Þ *n*þ1 � <sup>þ</sup> *<sup>T</sup>*€*<sup>i</sup> α*ð Þ *n*þ1

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

þ Δ*τ*<sup>2</sup> 4 *U*€ *i <sup>n</sup>* þ *M* z}|{�<sup>1</sup>

*<sup>n</sup>*þ<sup>1</sup> ¼ *M* z}|{�<sup>1</sup>

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

*Fractal Analysis - Selected Examples*

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

 z}|{*ip*

*<sup>n</sup>* þ Δ*τ* 2 *T*€*i*

The homogenized vector of thermal expansion coefficients *α<sup>H</sup>* can be written in terms of the homogenized elasticity matrix *D<sup>H</sup>* and the homogenized vector of stress-temperature coefficients *β<sup>H</sup>* as follows:

$$a^H = \left(D^H\right)^{-1} \beta^H \tag{97}$$

The thermal conductivity tensor is

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

the total three-temperature sensitivity.

displacement sensitivities.

the thermal stress sensitivities.

**Figure 2.**

**69**

*Variation of the total 3T sensitivity along* x*-axis.*

*kpj* ¼

displacement components, and thermal stress components.

2 6 4

Mass density *<sup>ρ</sup>* <sup>¼</sup> 7820kg*=*m<sup>3</sup> and heat capacity *<sup>c</sup>* = 461 J/kg K.

5*:*20 0 0 7*:*6 0 0 0 38*:*3

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

The proposed technique that has been utilized in the present chapter can be applicable to a wide range of three-temperature nonlinear generalized thermoelastic problems of ISMFGA structures. The main aim of this chapter was to assess the impact of fractional order parameter on the sensitivities of total three-temperature,

**Figure 2** shows the variation of the total temperature sensitivity along the *<sup>x</sup>*‐axis. It was shown from this figure that the fraction order parameter has great effects on

**Figures 3** and **4** show the variation of the displacement components *u*<sup>1</sup> and *u*<sup>2</sup> along the *<sup>x</sup>*‐axis for different values of fractional order parameter. It was noticed from these figures that the fractional order parameter has great effects on the

**Figures 5–7** show the variation of the thermal stress components *σ*11, *σ*12, and *σ*22, respectively, along the *<sup>x</sup>*‐axis for different values of fractional order parameter. It was noted from these figures that the fractional order parameter has great influences on

Since there are no available results for the three-temperature thermoelastic problems, except for Fahmy's research [10–14]. For comparison purposes with the special cases of other methods treated by other authors, we only considered one-dimensional numerical results of the considered problem. In the special case under consideration,

3 7 5

W*=*Km (103)

For the material design, the derivative of the homogenized vector of thermal expansion coefficients can be written as

$$\frac{\partial \alpha^{H}}{\partial X\_{kl}^{m}} = \left(D^{H}\right)^{-1} \left(\frac{\partial \beta^{H}}{\partial X\_{kl}^{m}} - \frac{\partial D^{H}}{\partial X\_{kl}^{m}} \alpha^{H}\right) \tag{98}$$

where *<sup>∂</sup>DH ∂X<sup>m</sup> kl* and *<sup>∂</sup>β<sup>H</sup> ∂X<sup>m</sup> kl* for any *l*th material phase can be calculated using the adjoint variable method [73] as

$$\frac{\partial D^H}{\partial X\_{kl}^m} = \frac{1}{|\Omega|} \int\_Y \left( I - B^m U^m \right)^T \frac{\partial D^m}{\partial X\_{kl}^m} (I - B^m U^m) dy \tag{99}$$

and

$$\frac{\partial \rho^{H}}{\partial \mathbf{X}\_{kl}^{m}} = \frac{1}{|Y|} \int\_{Y} \left( I - \mathbf{B}^{m} \mathbf{U}^{m} \right)^{T} \frac{\partial \mathbf{D}^{m}}{\partial \mathbf{X}\_{kl}^{m}} \left( \boldsymbol{\alpha}^{m} - \mathbf{B}^{m} \boldsymbol{\varphi}^{m} \right) d\mathbf{y} + \frac{1}{|\boldsymbol{\Omega}|} \int\_{Y} \left( I - \mathbf{B}^{m} \mathbf{U}^{m} \right)^{T} \frac{\partial \boldsymbol{\alpha}^{m}}{\partial \mathbf{X}\_{kl}^{m}} d\mathbf{y} \tag{100}$$

where j j Ω is the volume of the base cell.

#### **5. Numerical examples, results, and discussion**

In order to show the numerical results of this study, we consider a monoclinic graphite-epoxy as an anisotropic thermoelastic material which has the following physical constants [57].

The elasticity tensor is expressed as

$$\mathbf{C}\_{p\bar{j}kl} = \begin{bmatrix} 430.1 & 130.4 & 18.2 & 0 & 0 & 201.3 \\\\ 130.4 & 116.7 & 21.0 & 0 & 0 & 70.1 \\\\ 18.2 & 21.0 & 73.6 & 0 & 0 & 2.4 \\\\ 0 & 0 & 0 & 19.8 & -8.0 & 0 \\\\ 0 & 0 & 0 & -8.0 & 29.1 & 0 \\\\ 201.3 & 70.1 & 2.4 & 0 & 0 & 147.3 \end{bmatrix} \text{GPa} \tag{101}$$

The mechanical temperature coefficient is

$$
\boldsymbol{\beta}\_{pj} = \begin{bmatrix} 1.01 & 2.00 & 0 \\ 2.00 & 1.48 & 0 \\ 0 & 0 & 7.52 \end{bmatrix} \cdot 10^6 \frac{\text{N}}{\text{Km}^2} \tag{102}
$$

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

The thermal conductivity tensor is

The homogenized vector of thermal expansion coefficients *α<sup>H</sup>* can be written in

*β<sup>H</sup>* (97)

*<sup>I</sup>* � *BmUm* ð Þ*dy* (99)

*<sup>I</sup>* � *<sup>B</sup>mUm* ð Þ*<sup>T</sup> <sup>∂</sup>α<sup>m</sup>*

*∂X<sup>m</sup> kl dy*

GPa (101)

Km<sup>2</sup> (102)

(100)

(98)

terms of the homogenized elasticity matrix *D<sup>H</sup>* and the homogenized vector of

*<sup>α</sup><sup>H</sup>* <sup>¼</sup> *DH* � ��<sup>1</sup>

<sup>¼</sup> *DH* � ��<sup>1</sup> *<sup>∂</sup>β<sup>H</sup>*

For the material design, the derivative of the homogenized vector of thermal

*∂X<sup>m</sup> kl*

*<sup>I</sup>* � *BmUm* ð Þ*<sup>T</sup> <sup>∂</sup>Dm*

*<sup>α</sup><sup>m</sup>* � *Bmφ<sup>m</sup>* ð Þ*dy* <sup>þ</sup>

In order to show the numerical results of this study, we consider a monoclinic graphite-epoxy as an anisotropic thermoelastic material which has the following

> 430*:*1 130*:*4 18*:*2 0 0 201*:*3 130*:*4 116*:*7 21*:*0 0 0 70*:*1 18*:*2 21*:*0 73*:*60 0 2*:*4

> 0 0 0 19*:*8 �8*:*0 0 0 00 �8*:*0 29*:*1 0 201*:*3 70*:*1 2*:*4 0 0 147*:*3

> > 1*:*01 2*:*00 0 2*:*00 1*:*48 0

3 7 7 <sup>5</sup> � <sup>10</sup><sup>6</sup> <sup>N</sup>

0 07*:*52

� *<sup>∂</sup>D<sup>H</sup> ∂X<sup>m</sup> kl αH*

for any *l*th material phase can be calculated using the adjoint

1 j j Ω ð *Y*

� �

*∂X<sup>m</sup> kl*

stress-temperature coefficients *β<sup>H</sup>* as follows:

*Fractal Analysis - Selected Examples*

expansion coefficients can be written as

and *<sup>∂</sup>β<sup>H</sup> ∂X<sup>m</sup> kl*

> *∂D<sup>H</sup> ∂X<sup>m</sup> kl* ¼ 1 j j Ω ð *Y*

*<sup>I</sup>* � *BmUm* ð Þ*<sup>T</sup> <sup>∂</sup>D<sup>m</sup>*

**5. Numerical examples, results, and discussion**

where j j Ω is the volume of the base cell.

The elasticity tensor is expressed as

The mechanical temperature coefficient is

*βpj* ¼

*∂X<sup>m</sup> kl*

where *<sup>∂</sup>DH ∂X<sup>m</sup> kl*

and

*∂β<sup>H</sup> ∂X<sup>m</sup> kl* ¼ 1 j j *Y* ð *Y*

variable method [73] as

physical constants [57].

*Cpjkl* ¼

**68**

*∂α<sup>H</sup> ∂X<sup>m</sup> kl*

$$k\_{pj} = \begin{bmatrix} 5.2 & 0 & 0 \\ 0 & 7.6 & 0 \\ 0 & 0 & 38.3 \end{bmatrix} \text{W/Km} \tag{103}$$

Mass density *<sup>ρ</sup>* <sup>¼</sup> 7820kg*=*m<sup>3</sup> and heat capacity *<sup>c</sup>* = 461 J/kg K.

The proposed technique that has been utilized in the present chapter can be applicable to a wide range of three-temperature nonlinear generalized thermoelastic problems of ISMFGA structures. The main aim of this chapter was to assess the impact of fractional order parameter on the sensitivities of total three-temperature, displacement components, and thermal stress components.

**Figure 2** shows the variation of the total temperature sensitivity along the *<sup>x</sup>*‐axis. It was shown from this figure that the fraction order parameter has great effects on the total three-temperature sensitivity.

**Figures 3** and **4** show the variation of the displacement components *u*<sup>1</sup> and *u*<sup>2</sup> along the *<sup>x</sup>*‐axis for different values of fractional order parameter. It was noticed from these figures that the fractional order parameter has great effects on the displacement sensitivities.

**Figures 5–7** show the variation of the thermal stress components *σ*11, *σ*12, and *σ*22, respectively, along the *<sup>x</sup>*‐axis for different values of fractional order parameter. It was noted from these figures that the fractional order parameter has great influences on the thermal stress sensitivities.

Since there are no available results for the three-temperature thermoelastic problems, except for Fahmy's research [10–14]. For comparison purposes with the special cases of other methods treated by other authors, we only considered one-dimensional numerical results of the considered problem. In the special case under consideration,

**Figure 2.** *Variation of the total 3T sensitivity along* x*-axis.*

**Figure 3.** *Variation of the displacement* u*<sup>1</sup> sensitivity along* x*-axis.*

**Figure 4.** *Variation of the displacement* u*<sup>2</sup> sensitivity along* x*-axis.*

the displacement *u*<sup>1</sup> and thermal stress *σ*<sup>11</sup> results are plotted in **Figures 8** and **9**. The validity and accuracy of our proposed BEM technique were demonstrated by comparing our BEM results with the FEM results of Xiong and Tian [81], it can be noticed that the BEM results are found to agree very well with the FEM results.

Example 2. MBB beam.

**Figure 5.**

**Figure 6.**

**71**

*Variation of the thermal stress* σ*<sup>11</sup> sensitivity along* x*-axis.*

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

material ISMFGA structure.

Example 3. Roller-supported beam.

*Variation of the thermal stress* σ*<sup>12</sup> sensitivity along* x*-axis.*

It is known that extraordinary thermo-mechanical properties can be accomplished by combining more than two materials phases with conventional materials [75]. For this reason, it is essential that the topology optimization strategy permits more than two materials phases within the design domain. In this example, we consider a MBB beam shown in **Figure 12**, where the BESO final topology of MBB beam has been shown in **Figure 13a** for *α* ¼ 0*:*5 and shown in **Figure 13b** for *α* ¼ 1*:*0 to show the effect of fractional order parameter on the final topology of the multi-

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

Example 1. Short cantilever beam.

The mean compliance has been minimized, to obtain the maximum stiffness, when the structure is subjected to moving heat source. In this example, we consider a short cantilever beam shown in **Figure 10**, where the BESO final topology of considered short cantilever beam shown in **Figure 11a** for *α* ¼ 0*:*5 and shown in **Figure 11b** for *α* ¼ 1*:*0. It is noticed from this figure that the fractional order parameter has a significant effect on the final topology of the multi-material ISMFGA structure.

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

**Figure 5.** *Variation of the thermal stress* σ*<sup>11</sup> sensitivity along* x*-axis.*

**Figure 6.** *Variation of the thermal stress* σ*<sup>12</sup> sensitivity along* x*-axis.*

Example 2. MBB beam.

It is known that extraordinary thermo-mechanical properties can be accomplished by combining more than two materials phases with conventional materials [75]. For this reason, it is essential that the topology optimization strategy permits more than two materials phases within the design domain. In this example, we consider a MBB beam shown in **Figure 12**, where the BESO final topology of MBB beam has been shown in **Figure 13a** for *α* ¼ 0*:*5 and shown in **Figure 13b** for *α* ¼ 1*:*0 to show the effect of fractional order parameter on the final topology of the multimaterial ISMFGA structure.

Example 3. Roller-supported beam.

the displacement *u*<sup>1</sup> and thermal stress *σ*<sup>11</sup> results are plotted in **Figures 8** and **9**. The validity and accuracy of our proposed BEM technique were demonstrated by comparing our BEM results with the FEM results of Xiong and Tian [81], it can be noticed

The mean compliance has been minimized, to obtain the maximum stiffness, when the structure is subjected to moving heat source. In this example, we consider a short cantilever beam shown in **Figure 10**, where the BESO final topology of considered short cantilever beam shown in **Figure 11a** for *α* ¼ 0*:*5 and shown in **Figure 11b** for *α* ¼ 1*:*0. It is noticed from this figure that the fractional order parameter has a significant effect on the final topology of the multi-material

that the BEM results are found to agree very well with the FEM results.

Example 1. Short cantilever beam.

*Variation of the displacement* u*<sup>2</sup> sensitivity along* x*-axis.*

*Variation of the displacement* u*<sup>1</sup> sensitivity along* x*-axis.*

*Fractal Analysis - Selected Examples*

ISMFGA structure.

**70**

**Figure 3.**

**Figure 4.**

**Figure 7.** *Variation of the thermal stress* σ*<sup>22</sup> sensitivity along* x*-axis.*

**Figure 10.**

**Figure 11.**

**Figure 12.**

**73**

*Design domain of a MBB beam.*

**Figure 9.**

*Design domain of a short cantilever beam.*

*The final topology of a short cantilever beam: (a)* α *= 0.5 and (b)* α *= 1.0.*

*Variation of the thermal stress* σ*<sup>11</sup> waves along* x*-axis.*

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

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

**Figure 8.** *Variation of the displacement* u*<sup>1</sup> sensitivity along* x*-axis.*

In this example, we consider a roller-supported beam shown in **Figure 14**, where the BESO final topology of a roller-supported beam shown in **Figure 15a** for *α* ¼ 0*:*5 and shown in **Figure 15b** for *α* ¼ 1*:*0.

Example 4. Cantilever beam (validation example).

In order to demonstrate the validity of our implemented BESO topology optimization technique, we consider isotropic case of a cantilever beam shown in **Figure 16** as a special case of our anisotropic study to interpolate the elasticity matrix and the stress-temperature coefficients using the design variables *XM*, then we compare our *A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto… DOI: http://dx.doi.org/10.5772/intechopen.92852*

**Figure 9.** *Variation of the thermal stress* σ*<sup>11</sup> waves along* x*-axis.*

**Figure 10.** *Design domain of a short cantilever beam.*

#### **Figure 11.**

In this example, we consider a roller-supported beam shown in **Figure 14**, where the BESO final topology of a roller-supported beam shown in **Figure 15a** for *α* ¼ 0*:*5

In order to demonstrate the validity of our implemented BESO topology optimization technique, we consider isotropic case of a cantilever beam shown in **Figure 16** as a special case of our anisotropic study to interpolate the elasticity matrix and the stress-temperature coefficients using the design variables *XM*, then we compare our

and shown in **Figure 15b** for *α* ¼ 1*:*0.

*Variation of the displacement* u*<sup>1</sup> sensitivity along* x*-axis.*

*Variation of the thermal stress* σ*<sup>22</sup> sensitivity along* x*-axis.*

*Fractal Analysis - Selected Examples*

**Figure 8.**

**72**

**Figure 7.**

Example 4. Cantilever beam (validation example).

*The final topology of a short cantilever beam: (a)* α *= 0.5 and (b)* α *= 1.0.*

**Figure 12.** *Design domain of a MBB beam.*

### *Fractal Analysis - Selected Examples*

**Figure 13.**

*The final topology of MBB beam: (a)* α *= 0.5 and (b)* α *= 1.0.*

Find *XM*

**Figure 17.**

That minimize *<sup>C</sup><sup>M</sup>* <sup>¼</sup> <sup>1</sup>

, � <sup>P</sup>*<sup>N</sup>*

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

Subject to *VM*<sup>∗</sup>

displacement vector; *V<sup>M</sup>*<sup>∗</sup>

**Tables 1**–**4**, respectively.

That minimize *<sup>C</sup><sup>M</sup>* <sup>¼</sup> <sup>1</sup>

, *<sup>j</sup>* � <sup>P</sup>*<sup>N</sup>*

Subject to *<sup>V</sup><sup>M</sup>*<sup>∗</sup>

design domain; *f*

stated as. Find *X<sup>M</sup>*

*V<sup>M</sup>*

*ARM*

*rM*

**Table 1.**

**75**

<sup>2</sup> *<sup>P</sup><sup>M</sup>* � �*<sup>T</sup>*

<sup>2</sup> *<sup>P</sup><sup>M</sup>* � �*<sup>T</sup>*

*<sup>i</sup>*¼<sup>1</sup>*V<sup>M</sup> <sup>i</sup> X<sup>M</sup>*

*BESO parameters for minimization of a short cantilever beam.*

*<sup>i</sup>*¼1*V<sup>M</sup> <sup>i</sup> X<sup>M</sup> <sup>i</sup>* ¼ 0

*The final topology of a cantilever beam: (a) MMA and (b) BESO.*

*uM* <sup>¼</sup> <sup>1</sup>

*X<sup>M</sup>*

on the structure, which is the sum of mechanical and thermal loads; *uM* is the

*<sup>M</sup>*,*mec* is the mechanical load vector; and *f*

*uM* <sup>¼</sup> <sup>1</sup>

*ij* � <sup>P</sup> *<sup>j</sup>*�<sup>1</sup>

*X<sup>M</sup>*

<sup>2</sup> *f*

*<sup>i</sup>*¼<sup>1</sup> *<sup>V</sup><sup>M</sup>*<sup>∗</sup> ,

*<sup>K</sup>MuM* <sup>¼</sup> *<sup>P</sup><sup>M</sup>*

*<sup>i</sup>* ¼ *xmin*V1; *j* ¼ 1, 2

**Variable name Variable description Variable value**

*<sup>f</sup>* Final volume fraction 0.5 *ERM* Evolutionary ratio 1%

*max* Volume addition ratio 5%

*min* Filter ratio 3 mm *τ* Convergence tolerance 0.1% *N* Convergence parameter 5

*<sup>M</sup>*,*ter* <sup>þ</sup> *<sup>f</sup> <sup>M</sup>*,*mec* � �*<sup>T</sup>*

*<sup>i</sup>* ¼ 0; *j* ¼ 1, 2

vector. Also, the BESO parameters considered in Examples 1–4 can be seen in

of elements; *K<sup>M</sup>* is the global stiffness matrix; *xmin* is a small value (e.g., 0.0001), which it guarantee that none of the elements will be removed completely from

<sup>2</sup> *f*

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

*<sup>K</sup>MuM* <sup>¼</sup> *<sup>P</sup><sup>M</sup>*

*<sup>i</sup>* ¼ *xmin*V1

where *X<sup>M</sup>* is the design variable; *C<sup>M</sup>* is the mean compliance; *P* is the total load

The BESO topology optimization problem implemented in Examples 2 and 3, to find the distribution of the two materials in the design domain, which minimize the compliance of the structure, subject to a volume constraint in both phases can be

*<sup>M</sup>*,*ter* <sup>þ</sup> *<sup>f</sup> <sup>M</sup>*,*mec* � �*<sup>T</sup>*

*uM*

*uM*

*<sup>M</sup>*,*ter* is the thermal load

, is the volume of the solid material; *N* is the total number

#### **Figure 14.**

*Design domain of a roller-supported beam.*

**Figure 15.** *The final topology of a multi-material roller-supported beam: (a)* α *= 0.5 and (b)* α *= 1.0.*

BESO final topology shown in **Figure 17a** with the material interpolation scheme of the solid isotropic material with penalization (SIMP) shown in **Figure 17b**.

The BESO topology optimization problem implemented in Examples 1 and 4, to find the distribution of the *M* material phases, with the volume constraint can be stated as

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

**Figure 17.** *The final topology of a cantilever beam: (a) MMA and (b) BESO.*

Find *XM* That minimize *<sup>C</sup><sup>M</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> *<sup>P</sup><sup>M</sup>* � �*<sup>T</sup> uM* <sup>¼</sup> <sup>1</sup> <sup>2</sup> *f <sup>M</sup>*,*ter* <sup>þ</sup> *<sup>f</sup> <sup>M</sup>*,*mec* � �*<sup>T</sup> uM* Subject to *VM*<sup>∗</sup> , � <sup>P</sup>*<sup>N</sup> <sup>i</sup>*¼1*V<sup>M</sup> <sup>i</sup> X<sup>M</sup> <sup>i</sup>* ¼ 0 *<sup>K</sup>MuM* <sup>¼</sup> *<sup>P</sup><sup>M</sup> X<sup>M</sup> <sup>i</sup>* ¼ *xmin*V1

where *X<sup>M</sup>* is the design variable; *C<sup>M</sup>* is the mean compliance; *P* is the total load on the structure, which is the sum of mechanical and thermal loads; *uM* is the displacement vector; *V<sup>M</sup>*<sup>∗</sup> , is the volume of the solid material; *N* is the total number of elements; *K<sup>M</sup>* is the global stiffness matrix; *xmin* is a small value (e.g., 0.0001), which it guarantee that none of the elements will be removed completely from design domain; *f <sup>M</sup>*,*mec* is the mechanical load vector; and *f <sup>M</sup>*,*ter* is the thermal load vector. Also, the BESO parameters considered in Examples 1–4 can be seen in **Tables 1**–**4**, respectively.

The BESO topology optimization problem implemented in Examples 2 and 3, to find the distribution of the two materials in the design domain, which minimize the compliance of the structure, subject to a volume constraint in both phases can be stated as.

Find *X<sup>M</sup>*

That minimize *<sup>C</sup><sup>M</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> *<sup>P</sup><sup>M</sup>* � �*<sup>T</sup> uM* <sup>¼</sup> <sup>1</sup> <sup>2</sup> *f <sup>M</sup>*,*ter* <sup>þ</sup> *<sup>f</sup> <sup>M</sup>*,*mec* � �*<sup>T</sup> uM* Subject to *<sup>V</sup><sup>M</sup>*<sup>∗</sup> , *<sup>j</sup>* � <sup>P</sup>*<sup>N</sup> <sup>i</sup>*¼<sup>1</sup>*V<sup>M</sup> <sup>i</sup> X<sup>M</sup> ij* � <sup>P</sup> *<sup>j</sup>*�<sup>1</sup> *<sup>i</sup>*¼<sup>1</sup> *<sup>V</sup><sup>M</sup>*<sup>∗</sup> , *<sup>i</sup>* ¼ 0; *j* ¼ 1, 2 *<sup>K</sup>MuM* <sup>¼</sup> *<sup>P</sup><sup>M</sup> X<sup>M</sup> <sup>i</sup>* ¼ *xmin*V1; *j* ¼ 1, 2


**Table 1.** *BESO parameters for minimization of a short cantilever beam.*

BESO final topology shown in **Figure 17a** with the material interpolation scheme of

The BESO topology optimization problem implemented in Examples 1 and 4, to find the distribution of the *M* material phases, with the volume constraint can be

the solid isotropic material with penalization (SIMP) shown in **Figure 17b**.

*The final topology of a multi-material roller-supported beam: (a)* α *= 0.5 and (b)* α *= 1.0.*

stated as

**74**

**Figure 16.**

*Design domain of a cantilever beam.*

**Figure 13.**

**Figure 14.**

**Figure 15.**

*Design domain of a roller-supported beam.*

*Fractal Analysis - Selected Examples*

*The final topology of MBB beam: (a)* α *= 0.5 and (b)* α *= 1.0.*


where *<sup>V</sup>M*<sup>∗</sup>

**6. Conclusion**

,

which is made of *j*th material.

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

tional tool for material design.

**Author details**

**77**

Mohamed Abdelsabour Fahmy

provided the original work is properly cited.

*<sup>j</sup>* is the volume of *j*th material phase and *i* and *j* denote the element *i*th

The main purpose of this chapter is to describe a new boundary element formulation for modeling and optimization of 3T time fractional order nonlinear generalized thermoelastic multi-material ISMFGA structures subjected to moving heat source, where we used the three-temperature nonlinear radiative heat conduction

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

Numerical results show the influence of fractional order parameter on the sensitivities of the study's fields. The validity of the present method is examined and demonstrated by comparing the obtained outcomes with those known in the literature. Because there are no available data to confirm the validity and accuracy of our proposed technique, we replace the three-temperature radiative heat conduction with one-temperature heat conduction as a special case from our current general study of three-temperature nonlinear generalized thermoelasticity. In the considered special case of 3T time fractional order nonlinear generalized thermoelastic multi-material ISMFGA structures, the BEM results have been compared graphically with the FEM results; it can be noticed that the BEM results are in excellent agreement with the FEM results. These results thus demonstrate the validity and accuracy of our proposed technique. Numerical examples are solved using the multi-material topology optimization algorithm based on the bi-evolutionary structural optimization method (BESO). Numerical results of these examples show that

equations combined with electron, ion, and phonon temperatures.

the fractional order parameter affects the final result of optimization. The implemented optimization algorithm has proven to be an appropriate computa-

Faculty of Computers and Informatics, Suez Canal University, Ismailia, Egypt

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: mohamed\_fahmy@ci.suez.edu.eg

chemical processes such as bond breaking and bond forming.

Nowadays, the knowledge of 3T fractional order optimization of multi-material ISMFGA structures, can be utilized by mechanical engineers for designing heat exchangers, semiconductor nano materials, thermoelastic actuators, shape memory actuators, bimetallic valves and boiler tubes. As well as for chemists to observe the

#### **Table 2.**

*Multi-material BESO parameters for minimization of a MBB beam.*


#### **Table 3.**

*Multi-material BESO parameters for minimization of a roller-supported beam.*


**Table 4.** *BESO parameters for minimization of a cantilever beam.* *A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto… DOI: http://dx.doi.org/10.5772/intechopen.92852*

where *<sup>V</sup>M*<sup>∗</sup> , *<sup>j</sup>* is the volume of *j*th material phase and *i* and *j* denote the element *i*th which is made of *j*th material.
