The multiple-reciprocity method. A new approach for transforming BEM domain integrals to the boundary

Abstract

The application of the boundary element method for solving boundary value problems with body forces, time dependent effects or certain class of non-linearities generally leads to an integral equation which contains domain integrals (e.g. Ref. 1). Although these integrals do not introduce any new unknowns they detract from the elegance of the formulation and affect the efficiency of the method as integrations over the whole volume are required. Hence, a substantial amount of research has been carried out to find a general and efficient method of transforming volume integrals into equivalent boundary ones. The approaches which have so far been proposed can be divided into three groups. The first approach started in 1982 and is called the Dual Reciprocity Method, This approach was developed by Nardini and Brebbia 2'3 and afterwards extended for a variety of problems by Brebbia and WrobeP and by Loeffler and Mansur 5. It has been proved that the Dual Reciprocity Method (e.g. Refs 6, 7, 8) permits to solve a wide range of problems and is accurate. Another group of approaches is based on the expansion of the source term into the Fourier series. The method was proposed by Tang 9 to deal with potential and elasticity problems and has also been successfully applied for solving neutron diffusion problems by Itagaki and Brebbia 1 o. The third class of methods is the use of particular solutions which can convert domain integrals into boundary integrals. This technique offers some advantages versus the previous approaches in some specific applications as demonstrated by Azevedo and Brebbial 1. It has also been pointed out in Ref. 11 that the use of global particular solutions can be thought of as a special case of Dual Reciprocity. In this paper a new method of transforming domain integrals into boundary integrals is proposed. The Multiple Reciprocity Method (MRM) can be thought of as an extension of the idea of Dual Reciprocity. However, instead of approximating the source term by the set of coordinate functions, a sequence of functions related to the fundamental solution is introduced. These constitute a set of higher order fundamental solutions which permit the second Green's identity to be applied to each term of