Generalized method of fundamental solutions (GMFS) for boundary value problems

Abstract

In order to cope with the instability of the method of fundamental solutions (MFS), which caused by source offset, or source location, or fictitious boundary, a generalized method of fundamental solutions (GMFS) is proposed. The crucial part of the GMFS is used a generalized fundamental solution approximation (GFSA), which adopts a bilinear combination of fundamental solutions to approximate, rather than the linear combination of the MFS. Then the numerical solution of the GMFS is decided by a group of offsets corresponding to an intervention-point diffuse (IPD), instead of the MFS' only offset of a single source. To demonstrate the effectiveness of the proposed approach, four numerical tests are given. The results have shown that the GMFS is more accurate, stable, and better convergence than the traditional MFS.

On a Semi-analytical Method for Solution of the 2d Laplace Equation in Arbitrary Domains

“…In this section we take a selection of test problems from the recent literature (see e.g. [24,25]) that involve solution of the BVP for the Laplace equation in domains with arbitrary shapes. The problems are chosen as they have readily available analytical solutions.…”

Semi-Infinite Structure Analysis with Bimodular Materials with Infinite Element

“…In 1995, Liu et al proposed a Reproducing Kernel Particle Method (RKPM) approximation [33][34][35]. Thereafter, several meshless methods were developed such as the Method of Fundamental Solution (MFS) [36][37][38], the local Radial Point Interpolation Method (RPIM) [39][40][41], the local Radial Basis Function (RBF) collocation method [42][43][44] and the Meshless Intervention-Point (MIP) method [45], etc. In 2014, Wen et al proposed the meshless FBM [46].…”