**4. A new boundary element technique for simulation and optimization of solid deformable bodies under different loads**

In order to solve (63), we apply adaptive smoothing and prolongation algebraic multigrid (aSP-AMG) based on adaptive Factorized Sparse Approximate Inverse

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

(aFSAI) as described in [67] for solving the resulting simultaneous linear algebraic system (63) in Matlab (R2018a).

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

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 provides the flexibility to design a large variety of shapes.

The considered NURBS can be defined as follows

$$\mathbf{C}(\mathbf{t}) = \frac{\sum\_{i=0}^{n} \mathbf{N}\_{i,o}(\mathbf{t}) \boldsymbol{\varpi}\_i \mathbf{P}\_i}{\sum\_{i=0}^{n} \mathbf{N}\_{i,o}(\mathbf{t}) \boldsymbol{\varpi}\_i} \tag{64}$$

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

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 using the BEM.

Two criteria can be implemented during shape optimization of the solid bodies [70]

I. The minimum global compliance:

$$\mathcal{F} = \frac{1}{2} \int\_{\mathbf{S}} (\lambda \cdot \mathbf{u}) \, \text{d}\mathbf{S} \tag{65}$$

II. The minimum boundary equivalent stresses

$$\mathcal{F} = \int\_{\mathcal{S}} \left( \frac{\sigma\_{\text{ij}}}{\sigma\_0} \right)^{\text{n}} \text{d}\mathbf{S} \tag{66}$$

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

In order to find the optimal boundary conditions for temperature the following functional can be applied

$$\mathcal{F} = \int\_{\mathbf{S}} \left(\frac{\mathbf{u}}{\mathbf{u}\_0}\right)^{\mathbf{n}} \,\mathrm{d}\mathbf{S} \tag{67}$$

Thus, we can write

*C*<sup>11</sup> *C*<sup>12</sup> 0 *C*<sup>21</sup> *C*<sup>22</sup> 0 0 00 3 7 <sup>5</sup>, <sup>∗</sup> <sup>¼</sup>

*λ*1 *λ*2 *μ*3

Cn<sup>n</sup> <sup>¼</sup> <sup>X</sup> Ne

which can be expressed as

where

**194**

j¼1 � ð Γj

Equation after integration may be expressed as

Ci

<sup>i</sup> ¼ �<sup>X</sup> Ne

> X Ne

> > j¼1

ij <sup>¼</sup> bij

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

Hence, we get the following system of linear algebraic equations

**of solid deformable bodies under different loads**

8 < :

bij

**4. A new boundary element technique for simulation and optimization**

In order to solve (63), we apply adaptive smoothing and prolongation algebraic multigrid (aSP-AMG) based on adaptive Factorized Sparse Approximate Inverse

3 7 5

2 6 4

where

*<sup>C</sup><sup>n</sup>* <sup>¼</sup>

¼

2 6 4

> *u*1 *u*2 *ω*3

3 7 <sup>5</sup>, <sup>¼</sup>

2 6 4 <sup>C</sup>n<sup>n</sup> ¼ �

ð S

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

*u*∗ <sup>11</sup> *u*<sup>∗</sup>

2 6 4

*u*∗ <sup>21</sup> *u*<sup>∗</sup>

*u*∗ ∗ <sup>31</sup> *u*∗ ∗

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

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

<sup>j</sup> <sup>þ</sup><sup>X</sup> Ne

<sup>j</sup> <sup>þ</sup><sup>X</sup> Ne

j¼1

bij

j¼1

ð Γj

bij

<sup>q</sup> <sup>¼</sup> <sup>ψ</sup> <sup>q</sup><sup>j</sup>

<sup>∗</sup>ψ dΓ " #

> j¼1 bij

ij<sup>j</sup> <sup>¼</sup> <sup>X</sup> Ne

j¼1

if i 6¼ j

<sup>þ</sup> <sup>C</sup><sup>i</sup> if i <sup>¼</sup> <sup>j</sup>

<sup>∗</sup> dS <sup>þ</sup>

<sup>12</sup> *ω*<sup>∗</sup> 13

<sup>22</sup> *ω*<sup>∗</sup> 23

<sup>32</sup> *ω*∗ ∗ 33

ð S

> 3 7 <sup>5</sup>, <sup>∗</sup> <sup>¼</sup>

<sup>∗</sup> dS (56)

<sup>12</sup> *μ*<sup>∗</sup> 13 3 7 5,

<sup>22</sup> *μ*<sup>∗</sup> 23

<sup>32</sup> *μ*∗ ∗ 33

<sup>j</sup> (58)

(61)

<sup>j</sup> (59)

<sup>j</sup> (60)

*λ* ∗ <sup>11</sup> *λ* <sup>∗</sup>

2 6 4

*λ* ∗ <sup>21</sup> *λ* <sup>∗</sup>

*λ* ∗ ∗ <sup>31</sup> *λ* ∗ ∗

, p <sup>¼</sup> <sup>ψ</sup> <sup>p</sup><sup>j</sup> (57)

<sup>∗</sup>ψ dΓ " #

¼ (62)

¼ (63)

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

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

$$\mathcal{F} = \delta \sum\_{k=1}^{M} \left( \mathbf{u}^{k} - \widehat{\mathbf{u}}^{k} \right) + \eta \sum\_{l=1}^{N} \left( \mathbf{T}^{l} - \widehat{\mathbf{T}}^{l} \right) \tag{68}$$

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

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

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

**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

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

final structure for different iterations.

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

FEM results of the COMSOL Multiphysics.

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

*Initial boundary of the cantilever beam structure.*

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

on the FDM and FEM.

**Figure 2.**

**Figure 3.**

**197**

where <sup>u</sup>b<sup>k</sup> and <sup>T</sup>b<sup>l</sup> are measured displacements and temperatures in boundary 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 N are numbers of sensors.
