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DOI: 10.32604/cmes.2021.012218

ARTICLE

A Novel BEM for Modeling and Simulation of 3T Nonlinear Generalized Anisotropic Micropolar-Thermoelasticity Theory with Memory Dependent Derivative

Mohamed Abdelsabour Fahmy1,2,*

1 Jamoum University College, Umm Al-Qura University, Makkah, Saudi Arabia
2Faculty of Computers and Informatics, Suez Canal University New Campus, Ismailia, 41522, Egypt
*Corresponding Author: Mohamed Abdelsabour Fahmy. Email: maselim@uqu.edu.sa; mohamed_fahmy@ci.suez.edu.eg
Received: 20 June 2020; Accepted: 10 September 2020

Abstract: The main aim of this paper is to propose a new memory dependent derivative (MDD) theory which called three-temperature nonlinear generalized anisotropic micropolar-thermoelasticity. The system of governing equations of the problems associated with the proposed theory is extremely difficult or impossible to solve analytically due to nonlinearity, MDD diffusion, multi-variable nature, multi-stage processing and anisotropic properties of the considered material. Therefore, we propose a novel boundary element method (BEM) formulation for modeling and simulation of such system. The computational performance of the proposed technique has been investigated. The numerical results illustrate the effects of time delays and kernel functions on the nonlinear three-temperature and nonlinear displacement components. The numerical results also demonstrate the validity, efficiency and accuracy of the proposed methodology. The findings and solutions of this study contribute to the further development of industrial applications and devices typically include micropolar-thermoelastic materials.

Keywords: Boundary element method; memory dependent derivative; three-temperature; nonlinear generalized anisotropic micropolar-thermoelasticity

1  Introduction

The study of thermoelastic models has recently gained growing attention due to its many applications in aerospace technologies, geophysics, aeronautics, astronautics, robotics, earthquake engineering, mining engineering, nuclear energy industry, military technologies, soil dynamics, high-energy particle accelerators and detectors, and other engineering and electronic industries [19].

The classical thermo-elasticity (CTE) theory of Duhamel [10] and Newman [11] has two deficiencies: the first deficiency is the heat conduction of CTE does not include any elastic term, whereas the second deficiency is that, the equation of heat conduction has infinite heat propagation velocities. In order to overcome the first deficiency, Biot [12] proposed classical coupled thermo-elasticity (CCTE). But CTE and CCTE have the second deficiency. Therefore, many generalized thermo-elasticity theories have been developed to overcome the second deficiency of CTE. Among these theories are extended thermo-elasticity (ETE) theory of Lord et al. [13], temperature-rate-dependent thermo-elasticity (TRDTE) theory of Green et al. [1416] namely I, II and III, respectively, (where, GN theory I is based on Fourier’s law of heat conduction and identical to CTE theory, GN theory II characterizes the thermoelasticity without energy dissipation (TEWOED), and GN theory III which characterizes the thermoelasticity with energy dissipation (TEWED)). Although most thermal phenomena are practically represented using the classical Fourier thermal conductivity equation [1722], there are a large number of applications that require the use of the nonlinear heat conduction equation, great attention has been paid to investigate of nonlinear generalized thermoelastic problems by using boundary element method [2326]. Fahmy [27] introduced the three-temperature theory in the context of nonlinear generalized thermoelasticity.

The fractional calculus is the mathematical branch that is used to study the theory and applications of derivatives and integrals of arbitrary non-integer order. Recently, this branch has emerged as an effective tool for modeling of various engineering and industrial applications [28,29]. Due to the nonlocal nature of fractional order operators, they are useful for describing the memory and hereditary properties of various materials and processes. Also, the fractional calculus has drawn wide attention from the researchers of various countries in recent years due to its applications in solid mechanics, fluid dynamics, quantum mechanics, viscoelasticity, heat conduction modeling and identification, biology, food engineering, econophysics, biophysics, biochemistry, electrochemistry, electrical engineering, finance and control theory, robotics and control theory, signal and image processing, electronics, electric circuits, wave propagation, nanotechnology, flabby, oscillation, stochastic diffusion theory and wave propagation, etc. [3032].

Several famous mathematicians have contributed to the development of fractional order calculus, where Euler mentioned interpolating between integral orders of a derivative in 1730. At that point, Laplace characterized a fractional derivative by implies of an integral in 1812.

Lacroix presented the first formula for the fractional order derivative appeared in 1819, where he introduced the imagesth derivative of the function images as follows

images

In 1967, the Italian mathematician Caputo presented his fractional derivative of order images as

images

Diethelm [33] has suggested the derivative of Caputo in the form below

images

where images is the images-th order derivative and images is an integer such that images

images

Wang et al. [34] have introduced MDD as follows

images

where the first order (images) of MDD for a differentiable function images can be expressed as

images

Based on several practical applications, the memory effect needs weight images for images, so the MDD magnitude images is usually smaller than images, where the kernel function (images for images) Can be randomly selected over a staggered interval images, the practical kernel functions are 1, images and images, imagesimages, 1, 2, etc. These functions are monotonically increasing with images for the past time images and images for the present time images. The main feature of MDD, that the real-time functional value depends also, on the past time images. So, images depends on the past time (nonlocal operator), while the integration doesn’t depend on the past time (local operator).

As a special case images we have

images

The above equation shows that the common derivative images is the limit of images as images. That is,

images

Due to the computational difficulties in solving nonlinear generalized anisotropic thermoelastic problems, the problems become too complicated with no general analytical solution. So, numerical solutions should be implemented instead of analytical solutions to obtain the approximate solutions for such problems, one of the best of these numerical methods is the boundary element method (BEM) [35,36], which also called boundary integral equation method. BEM has been extensively used for a large variety of engineering and industrial applications. In the BEM, only the boundary of the computational domain needs to be discretized, so, it has a major advantage over domain methods which requires the whole computational domain discretization such as the finite difference method (FDM) [3739] and finite element method (FEM) [4042]. This advantage of BEM over domain methods has significant importance for modeling of nonlinear generalized thermoelastic problems which can be implemented using BEM with little cost and less input data [4357]. Through this paper, we would like to guide the reader to this important paper of Cheng et al. [58] which narrates BEM history in a wonderful and interesting way. Sladek et al. [5961] and Huang et al. [62] developed the boundary element formulation for micropolar thermoelasticity.

Researchers in numerical methods were only aware of the importance of FEM which could solve complex engineering problems. But now after the huge achievements of BEM and their ability to solve inhomogeneous and non-linear problems involving infinite and semi-infinite domains very efficiently, they realized the power, ease and accuracy of BEM in solving their complex problems by using a lot of software like FastBEM and BEASY.

The main aim of this paper is to propose a new MDD theory, called three-temperature nonlinear generalized anisotropic micropolar-thermoelasticity and propose a novel BEM technique for solving problems associated with the proposed theory. The numerical findings are graphically represented to demonstrate the impacts of the time delays and kernel functions on the total nonlinear three-temperature and nonlinear displacement components and demonstrate the validity and exactness of the suggested technique.

A brief summary of this paper is as follows: Section 1 introduces the background and provides the readers with the necessary information to books and articles for a better understanding of thermoelasticity theories, memory dependent derivative history and their applications. Section 2 describes the physical modeling of memory dependent derivative problems of three-temperature nonlinear generalized anisotropic micropolar-thermoelasticity. Section 3 outlines the BEM implementation for obtaining the temperature field of the considered problem. Section 4 outlines the BEM implementation for obtaining the displacement field of the considered problem. Section 5 introduces the computational performance of the proposed technique. Section 6 presents the new numerical results that describe the effects of time delays and kernel functions on the total temperature and displacement components. Section 7 outlines the significant findings of this paper.

2  Formulation of the Problem

The geometry of the considered problem is shown in Fig. 1 for a structure which occupies the bounded region images that bounded by S, where Si images such that S1 +S2 = S3 +S4 = S.

images

Figure 1: Geometry of the considered problem

The memory dependent derivative governing equations for three-temperature nonlinear generalized anisotropic micropolar-thermoelasticity theory and its problems can be expressed as follows [27]

images

images

where

images

images

images

images

The two dimensions three temperature (2D-3T) radiative heat conduction equations can be expressed as

images

images

images

where images, mij, images, images, uk, images and images are the mechanical stress tensor, couple stress, strain tensor, micro-strain tensor, displacement vector, temperature and reference temperature, respectively, Cabfg (Cabfg = Cfgab = Cbafg) and images (images) are respectively, the constant elastic moduli and stress-temperature coefficients of the anisotropic medium, Fi, Mi and images are mass force, mass couple and micro-rotation, respectively, J is micro-inertia coefficient, images images are the thermal conductivity coefficients, e, i and p denote electron, ion and phonon, respectively, images is the second order tensor associated with the TEWED and TEWOED theories, images, images, images, images, images and Å are the electron-ion energy coefficient, electron-phonon energy coefficient, density, specific heat capacities, time and unified parameter which introduced to consolidate all theories into a unified equations system, respectively, images images and images are the relaxation times, m is a functionally graded parameter. Also, g1, g2, images and images are suitably prescribed functions, images are the tractions defined by images, images and images are the Kronecker delta functions.

3  BEM Solution for Temperature Field

The 2D-3T radiative heat conduction equations mentioned above (15)(17) coupled with electron, ion and phonon temperatures, can be expressed in the context of memory dependent derivative theory [27]

images

images

where

images

images

where

images

and

images

which can be written in memory dependent derivative form as follows

images

where

images

and

images

where images (i, j = 1, 2), images images and images are the Kronecker delta, delay times and kernel function, respectively.

images

Initial and boundary conditions of 3T field can be written as

images

images

images

By using the fundamental solutions images that satisfies the following differential equation

images

Now, by applying the technique of Fahmy [27] to (19) we can write

images

which, in the absence of heat sources, can be written as follows

images

In order to transform the domain integral of (33) into the boundary, the time derivative of temperature can be approximated as follows

images

where fj(r) are known functions and images are unknown coefficients.

We assume that images is a solution of

images

Then, Eq. (33) resulted in the following boundary integral equation

images

where

images

and

images

where images are the coefficients of images that described as [50]

images

By discretizing Eq. (36) and using Eq. (38), we get [46]

images

where images is the heat flux vector and images and images are matrices.

The diffusion matrix may be described as

images

where

images

images

To solve Eq. (41) numerically, the functions images and images have been interpolated as

images

images

where, images determines the practical time images of the current time step.

By differentiating Eq. (44) with respect to time, we get

images

By substituting from Eqs. (44)(46) into (40), we obtain

images

which can be written as follows

images

in which images represents unknown matrix while X and images represent known matrices. The above formula gives the temperature as a function of the displacement field.

4  BEM Solution for Displacement Fields

Use of the weighted residual method to the governing Eqs. (9) and (10) yields

images

images

where

images

images

in which ui and images are approximate solutions and images and images are weighting functions.

The boundary conditions are

images

images

images

images

By integrating the first term of Eqs. (49) and (50) by parts, we get

images

images

Based on Huang et al. [62], we can write the following boundary integral equation

images

By integrating the left-hand side of (59) by parts, we get

images

Based on Eringen [63], we can write

images

Thus, Eq. (60) can be reexpressed as

images

By integrating the left-hand side of (62) by parts again and neglecting body force Ui and body couple Vi, we get

images

The weighting functions for images and Vi = 0 along el direction can be obtained as:

images

images

Now, we consider the following analytic fundamental solution of Dragos [64]

images

The weighting functions for Ui = 0 and images along el direction can be expressed as:

images

images

The analytic fundamental solution of Dragos [64] can also be expressed as

images

By using the above weighting functions sets into (63) we have

images

images

Thus, we can write

images

where

images

In order to solve (72) numerically, we define the following functions

images

By substituting from (74) into (72), we get

images

which also can be written as

images

By applying the following definition

images

Thus, by using (77), we can write (76) as follows

images

Hence, the global matrix system can be expressed as

images

Now, by using the initial and boundary conditions, we can write (79) as follows

images

in which images represents unknown matrix while images and images represent known matrices. An explicit staggered predictor-corrector algorithm which is based on the generalized modified shift-splitting (GMSS) iteration method [65] is implemented in order to solve (48) and (80) for obtaining the nonlinear three-temperature and nonlinear displacement fields.

5  Computational Performance of the Problem

Nowadays, modern CPUs are very powerful, versatile and can perform very complex problems much faster than previous ones [65,66]. We used GMSS method for the iterative solution of the resulted linear systems of equations images, where A is nonsingular, dense and nonsymmetric. We demonstrated the efficiency of our implemented GMSS method which results in fast convergence to the actual solution and does not need to complicated calculations.

The main objective of this section is to implement an accurate and robust iteration technique for solving the dense nonsymmetric algebraic system of linear equations arising from the BEM. So, GMSS of Huang et al. [67] has been implemented for solving the resulting linear systems in order to reduce the number of iterations and the CPU time. The BEM discretization is employed 1280 quadrilateral elements, with 3964 degrees of freedom (DOF). The generalized modified shift-splitting (GMSS) iteration method of Huang et al. [67], Uzawa-HSS iteration method of Yang et al. [68] and regularized iteration method of Badahmane [69] were compared with each other in Tab. 1. From this table, one can see that GMSS efficiency is superior to other iteration methods.

Table 1: Numerical results for the tested iteration methods

images

5.1 Uzawa-HSS Iteration Method

Now, the resulted linear system images in Eqs. (48) and (80) can be considered in the following form

images

where images is a non-Hermitian positive definite coefficient matrix, images is a full-column-rank matrix such that images, images is a known vector with images and images.

The iteration scheme of Uzawa method can be defined as

images

Due to the effectiveness of Uzawa method, several generalized techniques of Uzawa method, such as parameterized Uzawa methods, preconditioned Uzawa methods, inexact Uzawa methods and parameterized inexact Uzawa methods, have been developed to solve (81).

In order to solve the linear system images, where, A is Hermitian positive definite matrix. Yun [70] developed three Uzawa methods based on one-step successive over relaxation (SOR) iteration method due to its high efficiency to approximate images in each step of Uzawa method. Yang et al. [68] proposed the Uzawa-HSS iteration method based on one-step HSS iteration instead of one-step SOR. Bai et al. [71] proposed the Hermitian and skew-Hermitian splitting (HSS) iteration method to solve the non-Hermitian linear systems taking into consideration that images where images and images are the Hermitian and skew-Hermitian matrices of A which can be written as

images

In order to describe the Uzawa-HSS, we consider the iteration scheme of HSS iteration method which is used for solving linear equations system images as follows

images

which equals to

images

where

images

images

Now, we can define the Uzawa–HSS iteration scheme as follows:

First, compute images from the following iteration scheme

images

Second, compute images from the following iteration scheme

images

where images is a Hermitian positive definite preconditioning matrix.

For images and images, images until images and images converges, compute

images

where

images

images

5.2 Generalized Modified Shift-Splitting (GMSS) Iteration Method

Now, the resulted linear system (48) or (80) can be considered in the following form

images

where images and images, images.

According to Cao et al. [72] and Zhou et al. [73] and based on the well-known Hermitian and skew-Hermitian splitting (HSS) of the matrix A images, of Bai et al. [71], the matrix images can be written as

images

Now, the iteration scheme of the modified shift-splitting (MSS) can be described for solving linear equations system images, as

images

Based on the MSS iteration method, the generalized modified shift-splitting (GMSS) for the nonsymmetric matrix images is derived as follows

images

For images, images and images until images converges, compute

images

where images0, images0 is, another given positive constant, and I is a unit matrix.

The GMSS iteration method can be expressed as

images

where

images

and

images

According to Huang et al. [67], who proposed the GMSS, we can write

images

Let imagesand images, where, images, images and images, images

Now, the GMSS iteration method can be derived using the following algorithm:

For a given vector images, the vector images can be computed from the following steps

Step 1.    Compute images,

Step 2.    Solve images,

Step 3.    Compute images.

It can be seen from algorithm 1 that a linear system with the coefficient matrix images should be solved at each iteration, where the incomplete Cholesky factorization has been used as a preconditioner for Preconditioned Conjugate Gradient (PCG) Method for solving the sub-linear systems with the coefficient matrix images.

5.3 Regularized Iteration Method

Badahmane [69] proposed a regularized iteration method for solving the following system

images

where images and images, images.

According to Badahmane [69], the non-symmetric matrix images can be written as follows

images

For images, images until images converges, compute

images

The GMSS iteration method can be expressed as

images

where

images

where the regularized preconditioner of the matrix images is

images

From (104), the regularized iteration method computes the approximate solutions of (102) by

images

which equals to

images

where images, images and images

At each iteration step of regularized iteration (108), (109) should be solved using the following algorithm

1.    Solve images where images and images

2.    Compute images where images is a diagonal matrix, images, images, images and images is the exact solution.

6  Numerical Results and Discussion

The technique proposed in the current study may be applicable to a wide variety of three-temperature micropolar-thermoelastic problems relating to the suggested theory. During the simulation process the effects of time-delay and kernel function play a very important role. The proposed technique has been proven to be successful and efficient.

In the considered boundary element model, the boundary has been discretized using 42 linear boundary elements and 68 internal points as shown in Fig. 2. Also, the FDM and FEM discretization of the domain has been performed using 1896 second order quadrilateral elements and 5986 nodes.

images

Figure 2: Boundary element model of the considered problem

Fig. 3 shows the variations of the nonlinear three-temperature (images) along images-axis for different values of time-delay images and kernel function images. It can be seen from this figure that the time-delay has a significant effect on the nonlinear three-temperature distribution.

images

Figure 3: Variation of the images (images) along images-axis for different values of time-delay images and kernel function images.

Figs. 4 and 5 show the variation of the nonlinear displacements u1 and u2 along images-axis for different values of time-delay images and kernel function images. It is clear from these figures that the time delay greatly affects the displacement components.

images

Figure 4: Variation of the displacement images along images-axis for different values of time-delay images and kernel function images

images

Figure 5: Variation of the displacement images along x-axis for different values of time-delay images and kernel function images

Fig. 6 shows the variation of the nonlinear three-temperature along images-axis for different forms of kernel function and time-delay images. It is shown from this figure that the kernel function form has a significant influence on the nonlinear three-temperature distribution.

images

Figure 6: Variation of the temperature 3T (T0 = 0.1) along x-axis for different forms of kernel function and time-delay images

Figs. 7 and 8 show the variation of the nonlinear displacements u1 and u2 along images-axis for different forms of kernel function and time-delay images. It can be seen from these figures that the kernel function form has a significant influence on the nonlinear displacement components.

images

Figure 7: Variation of the displacement u1 along x-axis for different forms of kernel function and time-delay images

images

Figure 8: Variation of the displacement u2 along x-axis for different forms of kernel function and time-delay images

As there are no findings available for the problem under consideration. So, some literatures may be regarded as special cases from our general BEM problem. For comparison purposes with other approaches special cases addressed by other authors, we considered only one-dimensional problem. In the special case under consideration, the results are plotted in Figs. 911 to illustrate the total three-temperature and displacements distributions with the time images. The validity and exactness of our suggested technique have been demonstrated by a graphical comparison of the BEM special case results for the considered problem with those obtained using the FDM results of Pazera et al. [74] and FEM results of Xiong et al. [75] based on the substitution of three-temperature heat conduction with one-temperature heat conduction, it should be noted that the BEM results have been found to be in excellent agreement with the FDM and FEM results.

images

Figure 9: Variation of the nonlinear total temperature with time images

images

Figure 10: Variation of the nonlinear displacement images with time images

images

Figure 11: Variation of the nonlinear displacement images with time images

The performance of GMSS iteration method is compared against Uzawa-HSS iteration method and regularized iteration method. In actual computation, the parameters, images for Uzawa-HSS iteration method, images for GMSS iteration method and images for regularized iteration method have been chosen to be the experimentally found optimal ones that minimize the total number of iterative steps of these methods. Tab. 1 reports the iteration number (IT), CPU time, relative residual (RES) and error (ERR) of the tested iteration methods with respect to different values of time-step size images. From Tab. 1, it can be observed that the GMSS requires lowest IT and CPU times, which implies that the GMSS is superior to the other methods in terms of computing efficiency.

7  Conclusion

The main purpose of the current paper is to propose a new MDD theory called three-temperature nonlinear generalized anisotropic micropolar-thermoelasticity. This theory forms a new and good research point in thermoelasticity, and the scientific community will be interested in studying this research point in the following years due to its numerous low-temperature and high-temperature applications. The problems related to the proposed theory are very difficult to solve analytically. Therefore, we propose a new boundary element technique for solving such problems. For comparison purposes with other researchers in the literature, we only considered the one-dimensional one-temperature heat conduction model as a special case of our three-temperature heat conduction model. The numerical results confirm the validity and exactness of our suggested technique, where the BEM results are in excellent agreement with the results of FDM and FEM.

The GMSS iteration method has been implemented for solving the resulting linear systems in order to reduce the iterations number and CPU time. The implemented GMSS iteration method is quickly convergent without needing complicated calculations and. On the other hand, it is anticipated that the GMSS iteration method with the optimal parameters images would be much better and superior than Uzawa-HSS and regularized iteration methods for solving the resulting linear system from BEM. How to select the optimal parameters images for GMSS iteration method is a very practical and interesting problem that still needs further research and can be suggested as a future work through the current study.

The numerical results of our considered study can provide data references for mechanical engineers, computer engineers, geotechnical engineers, geothermal engineers, technologists, new materials designers, physicists, material science researchers and those who are interested in novel technologies in the area of three-temperature micropolar generalized thermoelastic materials. Application of three-temperature theories in advanced manufacturing technologies, with the development of soft machines and robotics in biomedical engineering and advanced manufacturing, thermoelastic response will be encountered more often where three-temperature radiative heat conduction will turn out to be the best choice for thermomechanical analysis in the design and analysis of micropolar generalized thermoelastic materials and structures.

Acknowledgement: The authors would like to thank the anonymous reviewers and the editor for their useful suggestions and comments which gave rise to the opportunity to revise and improve this paper.

Funding Statement: The author received no specific funding for this study.

Conflict of Interest: The author declares that they have no conflicts of interest to report regarding the present study.

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