The Method of Fundamental Solutions for Two-Dimensional Elastostatic Problems with Stress Concentration and Highly Anisotropic Materials

Abstract

The method of fundamental solutions (MFS) is a boundary-type and truly meshfree method, which is recognized as an efficient numerical tool for solving boundary value problems. The geometrical shape, boundary conditions, and applied loads can be easily modeled in the MFS. This capability makes the MFS particularly suitable for shape optimization, moving load, and inverse problems. However, it is observed that the standard MFS lead to inaccurate solutions for some elastostatic problems with stress concentration and/or highly anisotropic materials. In this work, by a numerical study, the important parameters, which have significant influence on the accuracy of the MFS for the analysis of two-dimensional anisotropic elastostatic problems, are investigated. The studied parameters are the degree of anisotropy of the problem, the ratio of the number of collocation points to the number of source points, and the distance between main and pseudo boundaries. It is observed that as the anisotropy of the material increases, there will be more errors in the results. It is also observed that for simple problems, increasing the distance between main and pseudo boundaries enhances the accuracy of the results; however, it is not the case for complicated problems. Moreover, it is concluded that more collocation points than source points can significantly improve the accuracy of the results.

An inverse method of fundamental solutions for the identification of 2D elastic properties of anisotropic solids

“…In this section, the MFS for anisotropic elasticity in two-dimensional media is briefly reviewed. More details can be found in [30,31]. Consider a two-dimensional anisotropic domain  with the boundary   .…”