Computers, Materials & Continua DOI:10.32604/cmc.2021.015115 | |

Article |

A New BEM for Fractional Nonlinear Generalized Porothermoelastic Wave Propagation Problems

1Department of Mathematics, Jamoum University College, Umm Al-Qura University, Alshohdaa, 25371, Jamoum, Saudi Arabia

2Department of Basic Sciences, Faculty of Computers and Informatics, Suez Canal University, New Campus, Ismailia, 41522, Egypt

*Corresponding Author: Mohamed Abdelsabour Fahmy. Email: maselim@uqu.edu.sal

Received: 06 November 2020; Accepted: 24 January 2021

Abstract: The main purpose of the current article is to develop a novel boundary element model for solving fractional-order nonlinear generalized porothermoelastic wave propagation problems in the context of temperature-dependent functionally graded anisotropic (FGA) structures. The system of governing equations of the considered problem is extremely very difficult or impossible to solve analytically due to nonlinearity, fractional order diffusion and strongly anisotropic mechanical and physical properties of considered porous structures. Therefore, an efficient boundary element method (BEM) has been proposed to overcome this difficulty, where, the nonlinear terms were treated using the Kirchhoff transformation and the domain integrals were treated using the Cartesian transformation method (CTM). The generalized modified shift-splitting (GMSS) iteration method was used to solve the linear systems resulting from BEM, also, GMSS reduces the iterations number and CPU execution time of computations. The numerical findings show the effects of fractional order parameter, anisotropy and functionally graded material on the nonlinear porothermoelastic stress waves. The numerical outcomes are in very good agreement with those from existing literature and demonstrate the validity and reliability of the proposed methodology.

Keywords: Boundary element method; fractional-order; nonlinear generalized porothermoelasticity; wave propagation; functionally graded anisotropic structures; Cartesian transformation method

The fractional order calculus (FOC) is the branch of mathematical analysis dealing with non-integer order calculus and its applications. The essential viewpoints are sketched out for fractional calculus theory in [1] and for fractional calculus applications in [2–6]. FOC is nowadays extremely popular due to its applications in different fields such as diffusion equation, quantum mechanics, nanotechnology, solid mechanics, continuum mechanics, biochemistry, wave propagation theory, polymers, robotics and control theory, finance and control theory, electrochemistry, electrical engineering, fluid dynamics, signal and image processing, biophysics, electric circuits, viscoelasticity, electronics, field theory, group theory, etc.

Several researchers have contributed to the background of fractional calculus [7–9]. Recently, Yu et al. [10] introduced new definitions of fractional derivative in the context of thermoelasticity. Research on generalized thermo-elasticity theories [11] has attracted much attention from many scientists, among which are research in magneto-thermoelasticity [12], visco-thermoelasticity [13,14] and micropolar-thermoelasticity [15,16].

Because of computational complexity in solving complex fractional thermoelasticity problems not having any general analytical solution, computational techniques should be used to solve such problems. Among these computational techniques are the boundary element method (BEM) that has been used for magneto thermoviscoelasticity [17,18], computerized engineering models [19,20], and design sensitivity and optimization [21,22] and nonlinear problems [23–26]. The BEM presents an attractive alternative numerical method to the domain methods for the investigation of thermoelastic wave propagation problems, like finite element method (FEM) [27–29] and finite volume method (FVM) [30–32]. The main feature of BEM over the domain type methods is that it requires boundary-only discretization of the domain under consideration. This feature has significant importance for solving complex thermoelastic problems with fewer elements, and requires very little computational cost, much less preparation of input data, and therefore easier to use.

In the present paper, we introduce a new boundary element model for solving fractional-order nonlinear generalized porothermoelastic wave propagation problems. The nonlinear terms are treated using the Kirchhoff transformation. The domain integrals were treated using the Cartesian transformation method. In the proposed BEM technique, the temperature and displacement distributions were calculated using a partitioned semi-implicit predictor–corrector coupling algorithm. Then, we can obtain the propagation of porothermoelastic stress waves in temperature-dependent FGA structures. Numerical results demonstrate the validity, accuracy and efficiency of our proposed model and technique.

The geometry of the considered problem is depicted in Fig. 1. The governing equations for fractional-order nonlinear generalized porothermoelastic wave propagation problems in the context of FGA structures can be written as [33]

where

where

The fractional nonlinear heat conduction equation can be expressed in non-dimensionless form as

in which

where

in which the heat source function

where T is the temperature,

According to finite difference scheme of Caputo at times

where

On the basis of Eq. (7), the fractional heat conduction Eq. (3) can be expressed as

where

3 BEM Implementation for Temperature Field

By using the transformation of Kirchhoff

The decomposition of the right-hand side of (10) into linear and nonlinear sections, yields

The nonlinear section can be written as

Based on [24], we can write (11) into the following form

where

Now, by using the fundamental solution of (9), we can write the boundary integral equation corresponding to (13) as [36]

By substituting of

where

Now, the domain integrals in Eq. (16) can be computed using CTM. Thus, the unknown boundary values can be calculated from the following system

where

Thus, the unknown internal values can be calculated from the following system

If we have assumed that the time step size is constant, then, H, G,

3.1 CTM Evaluation of the Domain Integrals with Irregularly Spaced Data Kernels

Now, we are considering the following regular domain integral [37,38]

Based on Khosravifard et al. [39], we can write the domain integral (19) as follows

where

By applying the composite Gaussian quadrature method to (19), we obtain

which can be written as

By implementing the radial point interpolation method (RPIM) [40], we can write

where

Based on [40], the function

To build the RPIM shape functions, we applied the following Gaussian radial basis function

where

and the following

By using Eqs. (27) and (28), we can express

Thus, based on [40], and using (29), we can write Eq. (25) in the following form

Thus, we have

which can be written as

where

3.2 CTM Evaluation of the Domain Integrals with Regularized Kernels

We now consider the following domain integrals that appear in the integral Eq. (16)

where

According to [25], the weakly singular in (33) can be regularized to obtain

where

and

Also, the domain integral in (34) can be regularized to obtain

where

and

Hence, from (18) we get

where

4 BEM Implementation for Displacement Field

Based on the weighted residual technique, we can write Eqs. (1) and (2) as follows

where

in which

On using integration by parts for the first term of Eqs. (42) and (43), we get

Based on Fahmy [24], elastic stress can be expressed as

which can be expressed as

where

Now, we consider the following definitions

Substituting above definitions into (47), we get

which after integration can be written as

where

Now, we can write (50) as

which can be expressed as follows

where the vectors

Substituting the boundary conditions into (54), we obtain the following system of equations

in which

According to Breuer et al. [41], a robust and efficient partitioned semi-implicit predictor–corrector coupling algorithm was implemented with GMSS [42] for solving the resulting linear Eqs. (41) and (54) arising from the boundary element discretization, where poro-thermo-elastic coupling is considered instead of fluid-structure-interaction coupling.

5 Numerical Results and Discussion

The proposed BEM technique which is based on the coupling algorithm [41], should be applied to a wide variety of fractional-order nonlinear porothermoelastic wave propagation problems.

In the present paper, we considered the temperature-dependent properties of anisotropic porous copper material, where the specific heat and density are tabulated in Tab. 1 [43].

The thermal conductivity is given by

The domain boundary of the current problem has been discretized into 42 boundary elements and 68 internal points as depicted in Fig. 2.

Figs. 3–5 illustrate the propagation of nonlinear thermal stress waves

According to the relationship of elastic constants for anisotropic, isotropic, and orthotropic materials [44]. We therefore considered these three materials in the current study.

Figs. 6–8 show the propagation of nonlinear thermal stress waves

Figs. 9–11 display the propagation of nonlinear thermal stress waves

The effectiveness of our proposed approach has been established through the use of the GMSS which doesn’t need the entire matrix to be stored in the memory and converges quickly without the need for complicated calculations. During our treatment of the considered problem, we implemented GMSS, Uzawa-HSS, and regularized iteration methods [45]. Tab. 2 displays the number of iterations (IT), processor time (CPU), relative residual (RES), and error (ERR) of the considered methods computed for different fractional order values. It can be noted from Tab. 2 that the GMSS needs the lowest IT and CPU times, which means that GMSS method has better performance than Uzawa-HSS and regularized methods.

For comparison purposes with other methods, we only considered the one-dimensional special case. Therefore, the time distribution results of the nonlinear thermal stress

The main objective of the current paper is to develop a new boundary element model for solving fractional-order nonlinear generalized porothermoelastic wave propagation problems in FGA structures, which are difficult or impossible to solve analytically. Therefore, an efficient numerical procedure based on BEM has been proposed to overcome this challenge. The Kirchhoff transformation is first used to treat the nonlinear terms. Then, the Cartesian transformation method (CTM) has been applied to transform the domain integration into boundary integration, As a result, the computational complexity of integration and CPU computing time are significantly reduced. The memory requirements and Processing time are also reduced by applying the GMSS method which does not need that the entire matrix is stored in the memory, and it is rapidly converging without the need for complicated calculations. The numerical outcomes are presented graphically to show the effects of fractional parameter, anisotropy, and functionally graded material on the nonlinear thermal stress waves. The numerical outcomes also show very good agreement with the earlier work in the literature as a special case. These outcomes also confirm the validity, accuracy, and effectiveness of the proposed methodology.

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

Conflicts of Interest: The author declares that he has no conflicts of interest to report regarding the present study.

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