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

## Abstract

The main objective of this chapter is to introduce a new theory called three-temperature nonlinear generalized micropolar thermoviscoelasticity. Because of strong nonlinearity of simulation and optimization problems associated with this theory, the numerical solutions for problems related with the proposed theory are always very difficult and require the development of new numerical techniques. So, we propose a new boundary element technique for simulation and optimization of such problems based on genetic algorithm (GA), free form deformation (FFD) method and nonuniform rational B-spline curve (NURBS) as the shape optimization technique. In the formulation of 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). The numerical results verify the validity and accuracy of our proposed technique.

### Keywords

- boundary element modeling
- simulation
- optimization
- three-temperature
- nonlinear generalized micropolar thermoviscoelasticity
- solid deformable bodies

## 1. Introduction

The classical thermo-elasticity (CTE) theory which was introduced by Duhamel [1] and Neumann [2] characterized the strain-temperature gradients equations in an elastic body, but it has two shortcomings contrary to physical observations: First, the heat conduction equation of this theory does not include any elastic terms. Second, the heat conduction equation is of a parabolic type predicting infinite speeds of thermal waves. Biot [3] developed the classical coupled thermo-elasticity (CCTE) theory to overcome the first shortcoming in CTE. However, both theories share the second shortcoming. So, several generalized thermoelasticity theories that predict finite speeds of propagation for heat waves have been developed such as extended thermo-elasticity (ETE) theory of Lord and Shulman [4], temperature-rate-dependent thermo-elasticity (TRDTE) theory of Green and Lindsay [5], three linear generalized thermoelasticity theories (type I, II and III) of Green and Naghdi [6, 7], low-temperature thermoelasticity (LTTE) model of Hetnarski and Ignaczak [8], the dual phase-lag (DPL) heat conduction equation of Tzou [9, 10] which has been developed taking into consideration the phonons-electrons interactions to obtain dual phase-lag thermoelasticity (DPLTE) [11, 12], and three-phase-lag thermoelasticity (TPLTE) model of Choudhuri [13] who takes into consideration the phase-lags of heat flux, temperature gradient and thermal displacement gradient. Chen and Gurtin [14], introduced the theory of two-temperatures (conductive temperature and thermodynamic temperature) heat conduction in the context of elastic bodies, then Youssef [15] extended this theory to generalized thermoelasticity. Fahmy [16] introduced three-temperature nonlinear generalized micropolar-magneto-thermoelasticity theory and developed a new boundary element technique for Modeling and Simulation of complex problems associated with this theory. Theory of micropolar elasticity [17, 18] has been developed for studying the mechanical behavior of polymers and elastomers and applied in many applications [19, 20, 21, 22, 23, 24]. Then, Eringen [25] and Nowacki [26] extended it to micropolar thermoelasticity, and then implemented in various applications [27, 28, 29]. Because of strong nonlinearity of three-temperatures radiative heat conduction equations, the numerical solution and simulation of such problems are always difficult and require the development of new numerical schemes [30, 31]. In comparison with other numerical methods [32, 33, 34], the boundary element method has been successfully applied and performed for solving various applications [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. The boundary element technique has been formulated in the context of micropolar thermoelasticity by Sladek and Sladek [61, 62, 63] and Huang and Liang [64]. Through the current paper, the term three-temperatures introduced for the first time in the field of nonlinear generalized micropolar thermoviscoelasticity. Recently, evolutionary algorithms [65, 66] have received much attention of researchers. The genetic algorithm (GA) can deal with the multi-objective and complex shapes problems. Also, it could reach an optimal solution with highly reduced computational cost.

The main aim of this article is to introduce a new theory called nonlinear generalized micropolar thermoviscoelasticity involving three temperatures. Because of strong nonlinearity, it is very difficult to solve the problems related to this theory analytically. Therefore, we propose a new boundary element model for simulation and optimization of three temperatures nonlinear generalized micropolar thermoviscoelastic problems associated with this theory. The genetic algorithm (GA) was implemented based on free form deformation (FFD) technique and nonuniform rational B-spline (NURBS) curve as an optimization technique for the considered problems. The numerical results demonstrate the validity and accuracy of our proposed model.

## 2. Formulation of the problem

The governing equations for three-temperature anisotropic generalized micropolar thermoviscoelasticity problems can be expressed as [58]

where

The two-dimension three-temperature (2D-3T) radiative heat conduction equations can be expressed as [53]

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

With reference to a Cartesian coordinate system

### 3.1 BEM simulation for temperature field

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

where

and

where parameters

The total energy of unit mass can be described by

Initial and boundary conditions can be written as

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

In which the points

The boundary integral equation corresponding to our considered heat conduction can be written as in Fahmy [46, 47, 48] as follows

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

The time derivative of temperature

where

Also, we assume that

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

where

and

In which the entries of

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

where

The diffusion matrix can be defined as

with

For solving (25) numerically, we interpolate the functions

where

The time derivative of (29) can be expressed as

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

Making use of initial conditions and boundary conditions at

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

where

The boundary conditions are

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

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

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

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

where

Hence, Eq. (44) can be re-expressed as [53]

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

The obtained weighting functions for

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

The obtained weighting functions for

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

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

Thus, we can write

where

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

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

Equation after integration may be expressed as

which can be expressed as

where

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

Hence, we get the following system of linear algebraic equations

## 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 (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

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

where

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:

II. The minimum boundary equivalent stresses

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

where

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

where

## 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

**Example 1.** Cantilever beam structure.

As a practical example, the proposed algorithm is applied on the cantilever beam with inferior corner load

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.

As application example, we use a beam with a mid-span load

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.

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

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 efficiency of our proposed technique.

α , α ¯ , α ̿ , α ˇ | micro-polar constants |

β ij | stress–temperature coefficients |

δ ij | Kronecker delta i j = 1 2 |

δ , η | weight coefficients |

ε ij | strain tensor |

ε ijk | alternate tensor |

ϵ ij | micro-strain tensor |

λ | tractions |

ϑ 0 | viscoelastic relaxation time |

ϖ | weights of control points |

ρ | material density |

σ ij | force stress tensor |

σ 0 | reference stress |

τ | time |

τ 0 , τ 1 , τ 2 | relaxation times |

ω i | micro-rotation vector |

ℵ | = 1 + ϑ 0 ∂ ∂ τ viscoelastic constant |

b | internal heat generation vector |

c | specific heat capacity |

C ijkl | constant elastic moduli |

e | = ε kk = ϵ kk dilatation |

e lij | piezoelectric tensor |

F i | mass force vector |

J | micro-inertia coefficient |

J | current density vector |

K e | electron conductive coefficients |

K i | ion conductive coefficients |

K r | phonon conductive coefficients |

k , l | boundary points |

M , N | sensors numbers |

M i | mass couple vector |

m ij | couple stress tensor |

P | total energy of unit mass |

P e | = c e T e Electron energy |

P i | = c i T i Ion energy |

P r | = 1 ρ c r T r 4 Phonon energy |

ℙ | values vector of tractions and couple stress |

p i | singular points |

Q | heat flux vectors |

ℚ | values vector of displacements and microrotations |

R | problem’s boundary |

S | problem’s domain |

T e | electron temperature |

T i | ion temperature |

T r | phonon temperature |

u | boundary displacement |

u i | displacement vector |

u 0 | reference displacement |

u k | computed displacements |

u ̂ k | measured displacements |

W ei | electron-ion energy coefficient |

W er | electron–phonon energy coefficient |