A meshless BEM for solving transient non-homogeneous convection-diffusion problem with variable velocity and source term

Abstract

In this paper, a meshless BEM based on the radial integration method is developed to solve transient non-homogeneous convection-diffusion problem with spatially variable velocity and time-dependent source term. The Green function served as the fundamental solution is adopted to derive the boundary domain integral equation about the normalized field quantity. The two-point backward finite difference technique is utilized to discretize the time-dependent terms in the integral equation, which results in that the final integral equation formulation is only related with the normalized field quantity at the current time and has three domain integrals. Both two domain integrals regarding the normalized field quantity at the current and previous times are transformed into boundary integrals by using radial integration method and radial basis function approximation. For domain integral about the source term being known function of time and coordinate, radial basis functions approximation is still adopted to make the transformed boundary integral be evaluated only once, not at each time level. A pure boundary element algorithm with boundary-only discretization and internal points is established and the system of equations is assembled like the corresponding steady problem. Four numerical examples are given to demonstrate the accuracy and effectiveness of the present method.

Introduction

Numerical solution of transient convection-diffusion problem is required to analyze the time evolution of physical quantities of the fluid under consideration in many engineering applications. Different types of numerical methods have been developed to handle the challenging task due to the complexity of governing differential equation [1], [2], [3], [4], [5], [6]. The boundary element method (BEM), as a powerfully numerical solution technique with the semi-analytical feature, has been frequently utilized to solve the type of physical problem [7], [8], [9]. The significant advantage of BEM is to discretize only the boundary of computational domain, which would reduce the dimension of physical problem once. According to the time discretization for solving transient convection-diffusion problems in BEM, the solution methods can be mainly classified into the transform space approach [10], [11], [12] and the time domain approach [7], [8], [9], [13], [14], [15], [16]. The time-dependent derivative in governing equation can be eliminated in the transform space approach by using Laplace transform [10], [11] or Fourier transform [12], and the solution in time domain is reconstituted by using the inverse transform. For the alternative time domain approach, boundary integral equations are directly derived based on the governing equation by using the unsteady or steady fundamental solutions [14], [15], [16], and finite difference method (FDM) is often employed to obtain the time evolution of the field quantity. Due to the existence of convective term, time-dependent term, source term or variable physical parameters in governing equation, there are the difficulties to obtain the general fundamental solution of transient convection-diffusion problem, which can be used to derive the integral equation consisting of only the boundary integrals. Therefore, it is inevitable that fundamental solutions of the simplified problems are adopted to derive the integral equation formulation, and then the arising domain integrals would weaken the boundary-only discretization advantage of BEM [9], [14], [15], [16], [17], [18].

To overcome the deficiency of domain integrals in the integral equation, a simple and effective transformation technique, called radial integration method (RIM), was proposed to deal with the domain integrals by Gao [19] in 2002. This method can directly transform the domain integral with the known integrand into the equivalent boundary integral without the use of particular solution appearing in dual reciprocity method [9]. For the domain integrals including the unknown quantities, only the radial basis function approximation of unknown quantities is adopted to accomplish the RIM transformation of the kind of domain integrals [20], [21], [22]. Therefore, RIM can simultaneously convert the different types of domain integrals appearing in the integral equation in a unified way, which is very convenient for the unified programming. The RIM can also reduce the singularity of domain integrals. The combination of radial integration method and boundary element method is called as the radial integration boundary element method (RIBEM). After Gao [20] firstly applied RIBEM to analyze steady heat conduction problem with spatially variable thermal conductivity in 2006, various types of heat transfer problems have been solved effectively by using RIBEM, such as transient non-homogeneous heat conduction problem [21], [22], nonlinear heat conduction problem [23], [24], non-Fourier heat conduction problems [25], one-phase solidification problems [26], steady convection-conduction problem [27] and heat radiation problem [28]. Cui et al. [21], [29] developed a new polygonal boundary element algorithm to solve steady and unsteady heat conduction problems with variable thermal conductivity by using the RIM twice to transform 3D domain integrals into 1D line integrals. Peng et al. [30] adopted the three-step variable condensation technique to form multi-domain RIBEM for solving transient heat conduction problem with non-homogeneous multi-media. Yao and Yu [31], [32] had also carried out the RIBEM analysis of transient heat conduction problem by using the precise time integration method and precise integration method to discretize the time-derivative term. AL-Bayati and Wroble [33] applied the RIBEM to solve 2D non-homogeneous convection diffusion reaction problem with spatially variable source term. Peng et al. [34] analyzed the two-dimensional transient convection-diffusion problem with the constant parameters by using RIBEM. As a very powerful solution technique, there are still other heat transfer problems to be analyzed by using the RIBEM, e.g., transient non-homogeneous convection-diffusion problem with spatially variable velocity and time-dependent source term to be considered in the paper.

The key procedure for transforming domain integrals to boundary integrals in RIM is the evaluation of the radial integral [19], which can be integrated analytically only for some simple integrands. Usually, the numerical evaluation of radial integral is carried out by using Gaussian quadrature. In the case, the radial integral needs to be computed at each Gaussian point of boundary element, and the evaluation of one radial integral requires many Gaussian points if the radial distance is relatively long. Thus, the numerical evaluation of radial integrals in the three-dimensional domain integrals of large-scale non-homogeneous and non-linear problems is highly time-consuming, which would reduce the performance and efficiency of RIBEM. Peng et al. [35] presented the radial integration BEM based on elemental node computation for solving non-homogeneous heat conduction problems. In the proposed method, the radial integral is required to be computed only at boundary nodes and the radial integral at Gaussian points of boundary elements is interpolated in terms of its nodal value through shape function, which can save relatively much computational time. The effective technique has also been successfully applied to solve the non-homogeneous elasticity [36], steady convection-conduction [27] and 2D unsteady convection-diffusion problems [34]. Yang el al. [37] presented the series of analytical expressions of the radial integrals associated with three types of thermal conductivities to improve the computational efficiency of RIBEM for solving non-homogeneous heat conduction problem. Based on Yang's work, Feng [38] considered some special circumstances and proposed a new type of analytical expressions of radial integrals in solving 3D transient heat conduction problem.

In this paper, the transient non-homogeneous convection-diffusion problem with spatially variable velocity and time-dependent source term is solved by using the element nodal computation-based RIBEM. The fundamental solution of Laplace equation is used to derive the normalized boundary integral equation, which results in the appearance of four domain integrals induced by variable diffusivity, source term, convective and time-dependent terms in governing equation. To avoid the simultaneous RBFs approximation of normalized field quantity and its time derivative in the RIM transformation of domain integrals, the two-point backward finite difference technique is used to discretize these unknown physical quantities. And then the final boundary integral equation formulation associated with the normalized field quantity at the current time is established. Although the domain integral induced by the source term can be directly transformed into the boundary integral by using RIM, the compactly supported fourth-order spline RBFs are still adopted to approximate the source term due to its time-dependence, which can result in the evaluation of the transformed boundary integral only once throughout the whole time iteration, not at each time step. Moreover, the domain integral associated with the known field quantity at the previous time has the same situation, so the RBFs approximation is also used to accomplish its RIM transformation. For the domain integral including the unknown normalized field quantity at the current time, the combination of the RIM and RBFs approximation must be adopted simultaneously to transform it into the series of equivalent boundary integrals. Finally, the integral equation formulation with only the boundary integrals is formed and the finial system of equations can be assembled like the corresponding steady problem. The BEM algorithm with the boundary-only discretization and some selected internal points instead of internal cells is developed to solve the transient non-homogeneous convection-diffusion problem with variable velocity and source term. Four numerical examples are given to demonstrate the accuracy and effectiveness of the present method.