Original articles
The localized method of fundamental solutions for 2D and 3D inhomogeneous problems

https://doi.org/10.1016/j.matcom.2022.04.024Get rights and content

Abstract

In this paper, the newly-developed localized method of fundamental solutions (LMFS) is extended to analyze multi-dimensional boundary value problems governed by inhomogeneous partial differential equations (PDEs). The LMFS can acquire highly accurate numerical results for the homogeneous PDEs with an incredible improvement of the computational speed. However, the LMFS cannot be directly used for inhomogeneous PDEs. Traditional two-steps scheme has difficulties in finding the particular solutions, and will lead to a loss of the accuracy and efficiency. In this paper, the recursive composite multiple reciprocity method (RC-MRM) is adopted to re-formulate the inhomogeneous PDEs to higher-order homogeneous PDEs with additional boundary conditions, which can be solved by the LMFS efficiently and accurately. The proposed combination of the RC-MRM and the LMFS can analyze inhomogeneous governing equation directly and avoid troublesome caused by the two-steps schemes. The details of the numerical discretization of the RC-MRM and the LMFS are elaborated. Some numerical examples are provided to demonstrate the accuracy and efficiency of the proposed scheme. Furthermore, some key factors of the LMFS are systematically investigated to show the merits of the proposed meshless scheme.

Introduction

During the past half century, many numerical methods have been developed to solve boundary values problems, since the computer technology is booming for several decades. Among them, some numerical methods employ the fundamental solutions of the governing equation for approximation of solutions, so only boundary discretization is required in these boundary-type methods, such as the boundary element method (BEM) [1], the method of fundamental solutions (MFS) [21], the boundary knot method (BKM) [3], etc. The BEM used the surface mesh and boundary integral equations for discretization as well as requires the singular and nearly-singular integrals, which are mathematical sophisticated [13]. In order to avoid mesh generation and singularity, the MFS locates the fictitious source points outside the computational domain and expresses the numerical solutions as a linear combination of fundamental solutions. However, it is non-trivial to determine the optimal distance between the fictitious source points and physical boundary, which will influence the performance of the MFS [6]. In addition, another boundary-type collocation method based on non-singular general solution was proposed by Chen and Tanaka, which is named as the BKM [8], [9]. By introducing the concept of the origin intensity factor, the singular boundary method (SBM) was proposed to eliminate the singularity of fundamental solutions [4], so the fictitious boundary of the MFS is removed. The above-mentioned boundary-type methods adopted only boundary nodes and source nodes for simulation, so they are very efficient and simple in comparing with mesh-based methods. Besides, these boundary-type methods can obtain extremely-accurate numerical results due to the use of fundamental solution, which already satisfies the governing equation of the boundary value problem. Although these boundary-type methods are very powerful in dealing with some boundary value problems and engineering applications [11], the resultant system of linear algebraic equations are usually dense and sometimes ill-conditioning. In order to stably and accurately solve large-scale engineering problems by these boundary-type methods, the troublesome problems of dense matrices or ill-conditioning matrices in these boundary-type methods should be alleviated.

In 2019, Fan et al. [14] proposed the localized method of fundamental solutions (LMFS) by combining the MFS and the concept of localization. The LMFS is evolved from the classical MFS, so it inherits the merits of the MFS. The LMFS used the fundamental solutions of the governing equation to derive the solution expression, so it can obtain extremely accurate results. In addition, the localization in the LMFS is very similar with the concept in the localized radial basis function collocation method (LRBFCM) [30], [31]. Although the LMFS uses both of interior nodes and boundary nodes at the same time, the localization in the LMFS will yield a sparse system of linear algebraic equations instead of dense matrix in the conventional MFS. Accordingly, due to the simplicity, accuracy and efficiency of the LMFS, the LMFS has been applied to some mathematical or physical problems, such as the three-dimensional interior acoustic fields [26], the anisotropic heat conduction [18], the elasticity problems [19], etc. Similar with the conventional MFS, it can be easily found that the LMFS is very powerful in dealing with boundary value problems with homogeneous governing equations [14], [18], [19], [26].

Similar with other boundary-type methods, the LMFS cannot directly deal with inhomogeneous problems. For inhomogeneous problems, when the above-mentioned boundary-type methods are considered, the numerical solution can be expressed as a linear combination of a homogeneous solution and a particular solution. Generally, dual reciprocity method (DRM) [24], [25] and Chebyshev collocation method [7] are adopted to approximate the particular solutions. The forcing term of the governing equation is interpolated by a number of the radial basis functions [2], the Chebyshev polynomial [7], or other base functions [12] at some internal points. Once the particular solution is obtained, we can use one of the boundary-type methods to solve the homogeneous solution, which satisfies the homogeneous equation and the modified boundary conditions. Since the particular solution and the homogeneous solution are sequentially acquired, these procedures in the boundary-type methods are known as the two-steps schemes. Besides, it is well-known that the accuracy of the inhomogeneous problem by using the Dual reciprocity method (DRM) mainly depends on the approximation of the particular solutions. In order to avoid the numerical error from the approximation of particular solution and simplify the numerical procedures, the recursive composite multiple reciprocity method (RC-MRM) [5], a method evaluated from MRM [23], [24], [29], is adopted in this paper to annihilate the inhomogeneous terms without residuals of approximations for particular solution. By using the RC-MRM, the inhomogeneous governing equation can be converted to a higher-order homogeneous partial differential equation (PDE), which can be directly solved by using only the LMFS in this paper. As a result, the accuracy of solving the inhomogeneous problem by using the RC-MRM will only be depended on the LMFS  [16], [17].

From some previous study of the LMFS [14], [18], [19], [26], it is evident that the LMFS can acquire extremely accurate solution and outperform other domain-type collocation methods when boundary value problems with homogeneous equations are considered. However, the extension and the performance of the LMFS for solving inhomogeneous PDEs are still unclear. In this paper, we make the first attempt to study the performance of the LMFS for inhomogeneous PDEs by using the RC-MRM. By considering the RC-MRM, the inhomogeneous problem of lower-order PDEs will be re-formulated to a higher-order homogeneous problem with additional boundary conditions. Then, the LMFS is adopted to accurately and efficiently solve the higher-order homogeneous governing equations. The proposed combination of the RC-MRM and LMFS can simply analyze the boundary value problem by solving one sparse system of linear algebraic equations in comparing with well-known two-steps schemes in conventional boundary-type methods. Besides, we used the unphysical nodes, which are out of computational domain, to handle the additional boundary conditions. This technique has been proved and applied for BVPs in the premise research [15], [22], [28]. Finally, numerical results and comparisons of some examples will be provided in this paper to emphasize the accuracy and the simplicity of the proposed numerical scheme. The influence of several factors will also be investigated in order to present the merits of the combination of the RC-MRM and the LMFS.

The organization of this paper can be given as follows: The motivation of this study and the discussions of relevant literatures are presented in the first Section. Section 2 introduces the PDEs and the boundary conditions of the considered boundary value problems. The numerical procedures of the LMFS and the RC-MRM are provided in Section 3, several numerical examples are provided to demonstrate the performance of the proposed meshless scheme in Section 4. Finally, some conclusions and remarks are drawn in Section 5.

Section snippets

Mathematical formulation of the boundary value problem

In this section, the considered boundary value problem, governed by inhomogeneous PDEs, is described. In this paper, both two-dimensional (2D) and three-dimensional (3D) problems are considered and can be efficiently solved by the same proposed numerical procedures. The governing equation can be demonstrated as follows, u(x)=f(x),xΩ,with boundary conditions u(x)=h(x),xΩD, u(x)n=g(x),xΩN,where () is a linear second-order partial differential operator.x=x,y and x=x,y,z represent the

The dual reciprocity method

The DRM is proposed to handle the inhomogeneous term of Eq. (1), the numerical solution should be expressed as the linear combination of a particular solution up(x) and a homogeneous solution uh(x), u(x)=up(x)+uh(x)The particular solution satisfies the inhomogeneous PDE, and does not necessary satisfy any boundary condition, up(x)=f(x)On the other hand, the homogeneous solution satisfies the corresponding homogeneous governing equation, uh(x)=0,and the modified boundary conditions, uh(x)=h(x

Numerical examples

In this section, four numerical examples are provided to validated the accuracy and effectiveness of the proposed numerical method for inhomogeneous PDEs. For each example, the RC-MRM will be implemented to converted the PDE. Once the governing equation is re-formulated to a higher-order homogeneous PDE, the LMFS is adopted to efficiently and accurately solve the resultant boundary value problem. In order to tackle the additional boundary conditions, generated due to increasing the degree of

Conclusions

In this paper, a meshless numerical scheme, which is the combination of the LMFS, the RC-MRM and the unphysical nodes, is proposed to deal with 2D and 3D boundary value problems, governing by inhomogeneous PDE accurately and efficiently. By using the RC-MRM, the inhomogeneous governing equation is converted to a homogeneous higher-order PDE. Since the degree of the governing equation has been increased, some additional boundary conditions are derived to ensure the uniqueness of solution. Then,

Acknowledgment

This work is supported by the National Natural Science Foundation of China (No: 12172159), National Natural Science Foundation of Jiangxi Provence (No: 20212BAB211022). The authors are very thankful for Jiangxi double thousand talents support (No: jxsq2018106027).

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