A New BEM for Fractional Nonlinear Generalized Porothermoelastic Wave Propagation Problems

: 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.


Introduction
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][3][4][5][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, Several researchers have contributed to the background of fractional calculus [7][8][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], viscothermoelasticity [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][24][25][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][28][29] and finite volume method (FVM) [30][31][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 fractionalorder 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 temperaturedependent FGA structures. Numerical results demonstrate the validity, accuracy and efficiency of our proposed model and technique.

Formulation of the Problem
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] σ ij, j + ρF i = ρü i + φρ Fvi (1) where σ ij is the mechanical stress tensor, ρ is the bulk density, ρ F is the fluid density, F i is the bulk body forces, φ is the porosity, u i is the solid displacement and v i is the fluid-solid displacement.
where ζ is the variation of the fluid volume per unit reference volume, q is the instantaneous flux and C i is the source term.
The fractional nonlinear heat conduction equation can be expressed in non-dimensionless form as in which where ε ij = 1 2 u i, j + u j, i , e = ε ii and A = φ 1 + Q R . in which the heat source function h (X , T, t) can be written as where T is the temperature,λ is the thermal conductivity, C ijkl is the constant elastic moduli, A is the Biot's effective stress coefficient, p is the fluid pressure, β ij is the stress-temperature coefficients, k is the permeability, T 0 is the reference temperature, Å is a unified parameter that introduces all generalized thermoelasticity theories into a unified system of equations, Q and R are solid-fluid coupling parameters, τ 0 , τ 1 , and τ 2 are relaxation times, ρ 0 = ηφρ F and η is the shape factor. where On the basis of Eq. (7), the fractional heat conduction Eq. (3) can be expressed as where J = 1, 2, . . . , F and f = 0, 1, 2, . . ., F.

BEM Implementation for Temperature Field
By using the transformation of Kirchhoff = can be written as [35] 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 Now, by using the fundamental solution of (9), we can write the boundary integral equation corresponding to (13) as [36] By substituting of (P, 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 and Q are M dimension vectors, and H and G are M × M dimension matrices.
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, H, and G can be computed at all time steps. Also, F, F Nl , F, and F Nl can be computed at all time steps using CTM.

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 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 M equals the summation of boundary nodes M and internal points M .
Based on [40], the function p (x 1 , x 2 ) may be described as To build the RPIM shape functions, we applied the following Gaussian radial basis function where α i and b j are unknown coefficients which can be computed from the following system and the following m conditions By using Eqs. (27) and (28), we can express α i and b j as 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 p contains boundary and internal p values.

CTM Evaluation of the Domain Integrals with Regularized Kernels
We now consider the following domain integrals that appear in the integral Eq. (16) where Ei (x) = −0.57721566 + ∞ n=1 (−1) n−1 x n n.n! − ln (x) 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 a is an unknown matrix, while X and b are known matrices.

BEM Implementation for Displacement Field
Based on the weighted residual technique, we can write Eqs. (1) and (2) as follows where 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 C n = Now, we consider the following definitions Substituting above definitions into (47), we get which after integration can be written as where which can be expressed as follows where the vectors Q, P, i, and j are displacements, tractions, pore pressure, and pore pressure gradients, respectively.
Substituting the boundary conditions into (54), we obtain the following system of equations in which A represents unknown matrix, while X and B represent known matrices.
According to Breuer et al. [41], a robust and efficient partitioned semi-implicit predictorcorrector 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.

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 σ 11 , σ 12 , and σ 22 for different values (a = 0.4, 0.7 and 1.0) of the fractional order parameter (FOP). It can be seen from these figures that the FOP has a great influence on the nonlinear thermal stress waves of FGA porous structures.
According to the relationship of elastic constants for anisotropic, isotropic, and orthotropic materials [44]. We therefore considered these three materials in the current study.  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 σ 11 are plotted in Fig. 12 for the proposed BEM and compared with the FDM results obtained by Awrejcewicz et al. [46] and FEM results obtained by Shakeriaski et al. [47], it can be shown from Fig. 12 that the BEM outcomes are in very good agreement with the FDM and FEM outcomes. Thus, the validity, accuracy, and usefulness of the proposed BEM have been demonstrated.     Figure 12: Propagation of the nonlinear thermal stress σ 11 waves with time t for a special case and different methods

Conclusion
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.