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

For illustration of the theoretical results of our proposed technique from the preceding sections, two numerical examples are analyzed below. The first example is the cantilever beam with inferior corner load, the second is the Michell-type structure, where the material has the following physical data [58]:

The elasticity tensor

$$C\_{ijkl} = \begin{pmatrix} 60.23 & 18.67 & 18.96 & -7.69 & 15.60 & -25.28 \\ 18.67 & 21.26 & 9.36 & -3.74 & 4.21 & -8.47 \\ 18.96 & 9.36 & 47.04 & -8.82 & 15.28 & -8.31 \\ -7.69 & -3.74 & -8.82 & 10.18 & -9.54 & 5.69 \\ 15.60 & 4.21 & 15.28 & -9.54 & 21.19 & -8.54 \\ -25.28 & -8.47 & -8.31 & 5.69 & -8.54 & 20.75 \end{pmatrix} \tag{69}$$

p ¼ 25 MPa, and Δt ¼ 0*:*0006 s.

**Example 1.** Cantilever beam structure.

**Figure 1.** *Cantilever beam structure geometry.*

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

As a practical example, the proposed algorithm is applied on the cantilever beam with inferior corner load P ¼ 100 N*=*mm. The geometry of the cantilever beam is illustrated in **Figure 1**. The initial boundary mesh of the cantilever beam composed of 14 quadratic elements is also illustrated in **Figure 2**. The BEM grid is composed of 76 nodes along *x* direction and 51 nodes along *y* direction. These mesh parameters were obtained after convergence analysis. In the process of optimization, the cantilever beam structure optimization results are presented in **Figure 3** from initial to final structure for different iterations.

The present measured boundary element method (BEM) optimization results of the first example are compared in **Figure 4** with measured finite difference method (FDM) optimization results obtained by Itzá et al. [71] and measured finite element method (FEM) optimization results obtained using the software package COMSOL Multiphysics, version 5.4. It is clear from this figure that the BEM results obtained by the proposed technique are in excellent agreement with the FDM results [71] and FEM results of the COMSOL Multiphysics.

**Table 1** shows that our proposed BEM modeling of cantilever beam with inferior corner load drastically reduces the manpower needed for modeling and computer resources needed for the calculation in comparison with the calculated results based on the FDM and FEM.

**Example 2.** Michell-type structure.

where *u* and *u*<sup>0</sup> are boundary displacement and reference displacement, respectively. Minimization of the functional (67) reduces displacements on the selected

In order to identify unknown inner boundary, we use the following functional

points *k* and *l* respectively, u<sup>k</sup> and T<sup>l</sup> are computed displacements and temperatures in boundary points k and l respectively, δ and η are weight coefficients, and M and

For illustration of the theoretical results of our proposed technique from the preceding sections, two numerical examples are analyzed below. The first example is the cantilever beam with inferior corner load, the second is the Michell-type

> *:*23 18*:*67 18*:*96 �7*:*69 15*:*60 �25*:*28 *:*67 21*:*26 9*:*36 �3*:*74 4*:*21 �8*:*47 *:*96 9*:*36 47*:*04 �8*:*82 15*:*28 �8*:*31 �7*:*69 �3*:*74 �8*:*82 10*:*18 �9*:*54 5*:*69 *:*60 4*:*21 15*:*28 �9*:*54 21*:*19 �8*:*54 �25*:*28 �8*:*47 �8*:*31 5*:*69 �8*:*54 20*:*75

þ η X N

l¼1

are measured displacements and temperatures in boundary

<sup>T</sup><sup>l</sup> � <sup>T</sup>b<sup>l</sup> � �

(68)

1

CCCCCCCCCCCA

(69)

<sup>u</sup><sup>k</sup> � <sup>u</sup>b<sup>k</sup> � �

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

part of the boundary.

where <sup>u</sup>b<sup>k</sup> and <sup>T</sup>b<sup>l</sup>

N are numbers of sensors.

The elasticity tensor

0

BBBBBBBBBBB@

p ¼ 25 MPa, and Δt ¼ 0*:*0006 s. **Example 1.** Cantilever beam structure.

*Cijkl* ¼

**Figure 1.**

**196**

*Cantilever beam structure geometry.*

F ¼ δ

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

structure, where the material has the following physical data [58]:

X M

k¼1

As application example, we use a beam with a mid-span load Pð Þ ¼ 100 N*=*mm (Michell-type structure) as shown in **Figure 5**. The initial boundary mesh of the Michell-type structure composed of 40 quadratic discontinuous elements is also

**Figure 2.** *Initial boundary of the cantilever beam structure.*

**Figure 3.**

*Cantilever beam optimization process from initial to final structure for different iterations.*

illustrated in **Figure 6**. The BEM grid is composed of 76 nodes along *x* direction, and 51 nodes along *y* direction. This grid density was obtained after convergence

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

**Figure 7** shows the cantilever beam optimization process from initial to final

The optimization results of the second example obtained with the proposed BEM are compared in **Figure 8** with FDM optimization results [71] and FEM optimization results of COMSOL Multiphysics software, version 5.4. It is clear from this figure that our BEM results obtained by the proposed technique are in excellent

analysis.

**Figure 5.**

**Figure 6.**

**Figure 7.**

**199**

*Michel-type structure geometry.*

*Initial boundary of the Michel-type structure.*

*Michell-type structure optimization process from initial to final structure for different iterations.*

structure for different iterations.

agreement with the FDM and FEM results.

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

**Figure 4.**

*Final cantilever beam structure for BEM, FDM and FEM.*


**Table 1.** *Comparison of computer resources needed for FDM, FEM and BEM modeling of cantilever beam structure.* *Boundary Element Mathematical Modelling and Boundary Element Numerical Techniques… DOI: http://dx.doi.org/10.5772/intechopen.90824*

illustrated in **Figure 6**. The BEM grid is composed of 76 nodes along *x* direction, and 51 nodes along *y* direction. This grid density was obtained after convergence analysis.

**Figure 7** shows the cantilever beam optimization process from initial to final structure for different iterations.

The optimization results of the second example obtained with the proposed BEM are compared in **Figure 8** with FDM optimization results [71] and FEM optimization results of COMSOL Multiphysics software, version 5.4. It is clear from this figure that our BEM results obtained by the proposed technique are in excellent agreement with the FDM and FEM results.

**Figure 5.** *Michel-type structure geometry.*

**Figure 4.**

**Table 1.**

**198**

*Final cantilever beam structure for BEM, FDM and FEM.*

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

*Comparison of computer resources needed for FDM, FEM and BEM modeling of cantilever beam structure.*

**Figure 6.** *Initial boundary of the Michel-type structure.*

**Figure 7.**

*Michell-type structure optimization process from initial to final structure for different iterations.*

**Table 2** shows that our proposed BEM modeling of Michell-type structure dramatically reduces the computer resources necessary to calculate our proposed modeling in comparison with the calculated results based on the FDM and FEM.

**6. Conclusion**

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

efficiency of our proposed technique.

*α*, *α*, *α*̿, *α*ˇ micro-polar constants

*δ*, *η* weight coefficients *εij* strain tensor *εijk* alternate tensor *ϵij* micro-strain tensor

*λ* tractions

*τ* time

**201**

ℵ ¼ 1 þ *ϑ*<sup>0</sup>

*ρ* material density *σij* force stress tensor *σ*<sup>0</sup> reference stress

*τ*0, *τ*1, *τ*<sup>2</sup> relaxation times *ω<sup>i</sup>* micro-rotation vector

*c* specific heat capacity *Cijkl* constant elastic moduli *e* ¼ *εkk* ¼ *ϵkk* dilatation *lij* piezoelectric tensor *Fi* mass force vector *J* micro-inertia coefficient current density vector

*βij* stress–temperature coefficients *δij* Kronecker delta ð Þ *i*, *j* ¼ 1, 2

*ϑ*<sup>0</sup> viscoelastic relaxation time *ϖ* weights of control points

> *∂ ∂τ*

*b* internal heat generation vector

viscoelastic constant

**Nomenclature**

In the present paper, we propose a new theory called nonlinear micropolar thermoviscoelasticity involving three temperatures. A new mathematical modeling of nonlinear generalized micropolar thermoviscoelasticity problem. A new boundary element technique for simulation and optimization problems of mechanics of solid deformable bodies is implemented based on genetic algorithm (GA), free form deformation (FFD) method and nonuniform rational B-spline curve (NURBS) as the global optimization technique for solving complex simulation and optimization problems associated with the proposed theory. FFD is an efficient and versatile parameterization technique for treating shape optimization problems with complex shapes. It is implemented for simulation and optimization of the shape. In the formulation of the considered problem, solutions are obtained for specific arbitrary parameters which are the control points positions in the considered problem, the profiles of the considered objects are determined by FFD method, where the FFD control points positions are treated as genes, and then the chromosomes profiles are defined with the genes sequence. The population is founded by a number of individuals (chromosomes), where the objective functions of individuals are determined by the boundary element method (BEM). Due to the large amount of computer resources required by the FDM and FEM, our proposed BEM model can be applied to a wide range of simulation and optimization problems related with our proposed theory. The numerical results demonstrate the validity, accuracy and

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

**Figure 8.**

*Final Michell-type structure for BEM, FDM and FEM.*


**Table 2.** *Comparison of computer resources needed for FDM, FEM and BEM modeling of Michell-type structure.* *Boundary Element Mathematical Modelling and Boundary Element Numerical Techniques… DOI: http://dx.doi.org/10.5772/intechopen.90824*
