Chapter Eight - The BEM for Potential Problems in Inhomogeneous Anisotropic Bodies

Abstract

In this chapter, the boundary element method (BEM) is developed for solving problems described by the general second order elliptic partial differential equation with variable coefficients. The BEM applies only if a reciprocal identity for the governing operator and its fundamental solution can be established. While for the problem at hand the reciprocal identity can be established, it is not possible to derive the fundamental solution for the general second order elliptic partial differential equation. This is overcome by developing BEM formulations, which use the simple known fundamental solution of the Laplace equation. Among them the dual reciprocity method (DRM) and the analog equation method (ΑΕΜ) are proven to be the most efficient methods. These two methods are presented in this chapter. First the DRM, emphasizing its capabilities and limitations, and then the AEM. The latter, alleviated from any restrictions, renders the BEM an efficient computational tool for solving all problems described by the complete second order differential equation and systems of them as well. This fact is demonstrated by solving several representative problems.