Integration over simplexes for accurate domain and boundary integral evaluation in boundary element methods

Abstract

The accuracy of boundary element methods is heavily dependent on the accurate determination of singular domain and boundary integrals on element domains. Many methods have been developed for accurate integral determination but none that is universally applicable across the range of boundary element applications. This paper is concerned with a semi-analytical method that applies to simplexes of arbitrary dimension. The method involves the careful employment of multiple integration where the inner integral is performed along a radial direction. Evaluation of the radial-inner integral on a simplex of dimension n provides n+1 integrals on simplex domains of dimension n−1. It is shown in the paper that provided closed form solutions exist for the inner integrals the method can be repeated on these n+1 simplexes providing a recursive method of integration. Performing radial inner integration on a simplex is shown to be ideal for the correct treatment of the radial singularities present. If a closed form solution is not available then a radial function is added and subtracted to facilitate the use of numerical integration. In addition to singularity annihilation the form of this radial function is selected to facilitate the continued recursive application of radial integration. The method is tested on domain and boundary integrals present in thermoelastostatic, elastodynamic and on simple test problems for which known analytical solutions exist. The results obtained using the semi-analytical simplex approach are shown to be considerably more accurate than those obtained using standard quadrature methods.