A new boundary meshfree method with distributed sources

Abstract

A new boundary meshfree method, to be called the boundary distributed source (BDS) method, is presented in this paper that is truly meshfree and easy to implement. The method is based on the same concept in the well-known method of fundamental solutions (MFS). However, in the BDS method the source points and collocation points coincide and both are placed on the boundary of the problem domain directly, unlike the traditional MFS that requires a fictitious boundary for placing the source points. To remove the singularities of the fundamental solutions, the concentrated point sources can be replaced by distributed sources over areas (for 2D problems) or volumes (for 3D problems) covering the source points. For Dirichlet boundary conditions, all the coefficients (either diagonal or off-diagonal) in the systems of equations can be determined analytically, leading to very simple implementation for this method. Methods to determine the diagonal coefficients for Neumann boundary conditions are discussed. Examples for 2D potential problems are presented to demonstrate the feasibility and accuracy of this new meshfree boundary-node method.

Introduction

The method of fundamental solutions (MFS) has been studied for many years along with the boundary element and other boundary methods [1]. The MFS uses only the fundamental solution, which is the response due to a concentrated point source, in the construction of the solution of a problem without using any integrals. It is a natural boundary meshfree method and offers several advantages as compared with the BEM. First, meshing a boundary with only nodes is certainly much easier than with elements. Second, singular integrals are avoided in the MFS (although singularities of the kernel still play an important role). Third, programming with the conventional MFS is significantly simplified as compared with the BEM. All these advantages with the MFS have attracted continued interests from researchers. Comprehensive reviews on the MFS for various applications can be found in Refs. [2], [3], [4]. Some work on the MFS can be found in Refs. [5], [6], [7], [8], [9], [10] for potential and elastostatic problems.

In the traditional MFS, a fictitious boundary slightly outside the problem domain is required in order to place the source points and avoid the singularity of fundamental solutions. The determination of the distance between the real boundary and the fictitious boundary is based on experience and therefore troublesome. In recent years, various efforts have been made aiming to remove this barrier in the MFS, so that the source points can be placed on the real boundary directly. Young et al. [11], [13] and Chen et al. [12] proposed to place the source points on the boundary in the MFS and novel ways to determine the diagonal coefficients directly for simple geometries or using the results from the BEM based on the fact that the MFS and the indirect boundary integral formulation are similar in nature. In their approach, information of the neighboring points before and after each source point is needed in general in order to form line segments for integrating the kernels to obtain the diagonal coefficients. This is essentially the same information of the element connectivity in a BEM mesh. Šarler [14] proposed a similar modified MFS, where the diagonal terms are determined by the integration of the fundamental solution on line segments formed by using neighboring points, and the use of a constant solution to determine the diagonal coefficients from the derivatives of the fundamental solution. This approach is very stable, as is also shown in this study, but it amounts to solving the problems twice and is therefore not amenable to the fast multipole method using iterative solvers for the MFS [10], [15]. Chen and Wang [16] recently proposed a similar method for determining the diagonal coefficients in the modified MFS by applying a known solution inside the domain, so that the diagonal coefficients from both the fundamental solution and its derivative can be determined indirectly, without using any element or integration concept. Again, this approach is appealing, stable, and accurate but is costly for solving large-scale problems due to the need to solve the problem twice.

In the spirit of pursuing truly meshfree boundary methods, we present in this paper a new boundary meshfree approach based on the modified MFS that has no fictitious boundaries and singularities. In this new approach, to be called boundary distributed source (BDS) method, the concentrated point sources are replaced with area-distributed sources covering the source points for 2D problems. These area-distributed sources are analytical integration of the original singular fundamental solution and its derivative so that they preserve the advantage of diagonal dominance for the system of equations, while they have no troublesome singularity issues. This BDS approach also does not require the information about the neighboring points for each source point, thus is a truly meshfree boundary method. Implementation of the method is easy and extension to 3D is straightforward where volume-distributed sources covering the source points can be applied. Although there are remaining issues with this BDS method, it is a very promising boundary meshfree method because it can be accelerated readily with the fast multipole method [10], [15] or other fast solution methods in solving large-scale engineering problems.