Standard-BEM solutions to two types of anisotropic-diffusion convection reaction equations with variable coefficients

Azis, Moh. Ivan

doi:10.1016/j.enganabound.2019.04.006

Cited by 35 publications

(12 citation statements)

References 15 publications

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“…For the PD formulation of elasticity, consider a body having a region Ω as shown in Figure 2. With reference to Figure 1, the EOM for particle i at time t as proposed in [2] is given as (1). e original formulation in this paper [2] was the bond-based PD theory, where internal forces in a body are modeled as a network of pairwise interactions.…”

Section: Review Of Peridynamic Theorymentioning

confidence: 99%

“…e governing mathematical equations of many physical phenomena in engineering and science are described by local diffusion equations. In solving these equations, researchers employed several numerical techniques in the past several years including finite element method (FEM), boundary element method (BEM) [1], finite difference method (FDM), and meshless methods. At the continuum level, diffusion processes are typically defined by local models via the famous Fourier's law of heat conduction and Fick's law of mass transport.…”

Section: Introductionmentioning

confidence: 99%

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A Review of Peridynamics (PD) Theory of Diffusion Based Problems

Zeleke

Ageze

2021

Journal of Engineering

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The study of heat conduction phenomena using peridynamic (PD) theory has a paramount significance on the development of computational heat transfer. This is because PD theory has got an interesting feature to deal with the inherent nonlocal nature of heat transfer processes. Since the revolutionary work on PD theory by Silling (2000), extensive investigations have been devoted to PD theory. This paper provides a survey on the recent developments of PD theory mainly focusing on diffusion based peridynamic (PD) formulation. Both the bond-based and state-based PD formulations are revisited, and numerical examples of two-dimensional problems are presented.

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Section: Review Of Peridynamic Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Review of Peridynamics (PD) Theory of Diffusion Based Problems

Zeleke

Ageze

2021

Journal of Engineering

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“…In dealing with the problem of numerically solving the convection–diffusion equation, several techniques have been applied such as finite differences, finite elements, boundary elements and meshless methods. Comprehensive studies and extensive lists of works using finite differences and finite elements can be found in references (Lewis et al , 2004; Nithiarasu et al , 2016; Yu and Heinrich, 1987) and Li (1990), while for boundary elements, the papers (Taigbenu and Liggett, 1986; Tanaka et al , 1987; Al-Bayati and Wrobel, 2018b; Al-Bayati and Wrobel, 2018a; Tanaka and Honma, 1989; Al-Bayati and Wrobel, 2019; Al-Bayati, 2018; Al-Bayati and Wrobel, 2020; Azis, 2019; Chen and Wang, 2016) and Peng et al (2020), refer to the majority of works done on the subject. The motivation to work on boundary elements resides in the fact that when applying the first two mentioned approaches, several numerical problems such as oscillations and smoothing of wave fronts have been reported.…”

Section: Introductionmentioning

confidence: 99%

Numerical modelling of convection-diffusion problems with first-order chemical reaction using the dual reciprocity boundary element method

Al-Bayati

Wrobel

2021

HFF

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Purpose The purpose of this paper is to describe an extension of the boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) formulations developed for one- and two-dimensional steady-state problems, to analyse transient convection–diffusion problems associated with first-order chemical reaction. Design/methodology/approach The mathematical modelling has used a dual reciprocity approximation to transform the domain integrals arising in the transient equation into equivalent boundary integrals. The integral representation formula for the corresponding problem is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. The finite difference method is used to simulate the time evolution procedure for solving the resulting system of equations. Three different radial basis functions have been successfully implemented to increase the accuracy of the solution and improving the rate of convergence. Findings The numerical results obtained demonstrate the excellent agreement with the analytical solutions to establish the validity of the proposed approach and to confirm its efficiency. Originality/value Finally, the proposed BEM and DRBEM numerical solutions have not displayed any artificial diffusion, oscillatory behaviour or damping of the wave front, as appears in other different numerical methods.

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“…Rap et al [23], Ravnik and Škerget [25,26], Li et al [18] and Pettres and Lacerda [22] considered the case of isotropic diffusion and variable coefficients (inhomogeneous media). Recently Azis and co-workers had been working on steady state problems of anisotropic inhomogeneous media for several types of governing equations, for examples [5,32] for the modified Helmholtz equation, [4,14,24,30,27,11,17] for the diffusion convection reaction equation, [29,8,13,16] for the Laplace type equation, [10,2,20,21,15] for the Helmholtz equation.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation for Unsteady Anisotropic-Diffusion Convection Equation of Spatially Variable Coefficients and Incompressible Flow

Azis¹

2021

Tamkang J. Math.

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The anisotropic-diffusion convection equation of spatiallyvariable coefficients which is relevant for functionally graded mediais discussed in this paper to find numerical solutions by using acombined Laplace transform and boundary element method. The variablecoefficients equation is transformed to a constant coefficients equation.The constant coefficients equation is then Laplace-transformed sothat the time variable vanishes. The Laplace-transformed equationis consequently written in a pure boundary integral equation whichinvolves a time-free fundamental solution. The boundary integral equationis therefore employed to find numerical solutions using a standardboundary element method. Finally the results obtained are inverselytransformed numerically using the Stehfest formula to get solutionsin the time variable. The combined Laplace transform and boundaryelement method is easy to be implemented, efficient and accurate forsolving unsteady problems of anisotropic functionally graded mediagoverned by the diffusion convection equation.

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Standard-BEM solutions to two types of anisotropic-diffusion convection reaction equations with variable coefficients

Cited by 35 publications

References 15 publications

A Review of Peridynamics (PD) Theory of Diffusion Based Problems

A Review of Peridynamics (PD) Theory of Diffusion Based Problems

Numerical modelling of convection-diffusion problems with first-order chemical reaction using the dual reciprocity boundary element method

Numerical Simulation for Unsteady Anisotropic-Diffusion Convection Equation of Spatially Variable Coefficients and Incompressible Flow

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