Trefftz boundary elements-multi-region formulations

This paper is concerned with an effective numerical implementation of the T refftz boundary element method, for the analysis of two-dimensional potenti al problems, defined in arbitrarily shaped domains. The domain is first discretized into multiple subdomains or regions. Each r egion is treated as a single domain, either finite or infinite, for which a complete set of solutions of the problem is known in the form of an expans ion with unknown coefficients. Through the use of weighted residuals, this solution expansion is then forced to satisfy the boundary conditions of the actual domain of the problem, leading thus to a system of equations, from which the unknowns can be readily determined. When this basic procedure is adopted, in the analysis of multiple-region problems, proper boundary integ ral equations must be used, along common region interfaces, in order to cou ple to each other the unknowns of the solution expansions relative to the n eighbouring regions. These boundary integrals are obtained from weighted re siduals of the coupling conditions which allow the implementation of any or der of continuity of the potential field, across the interface boundary, be tween neighbouring regions. The technique used in the formulation of the region-coupling conditions dri ves the performance of the Trefftz boundary element method. While both of t he collocation and Galerkin techniques do not generate new unknowns in the problem, the technique of Galerkin presents an additional and unique featur e: the size of the matrix of the final algebraic system of equations which is always square and symmetric, does not depend on the number of boundary e lements used in the discretization of both the actual and regiori-interface boundaries. This feature which is not shared by other numerical methods, a llows the Galerkin technique of the Trefftz boundary element method to be e ffectively applied to problems with multiple regions, as a simple, economic and accurate solution technique. A very difficult example is analysed with this procedure. The accuracy and efficiency of the implementations described herein make the Trefftz boundar y element method ideal for the study of potential problems in general arbit rarily-shaped two-dimensional domains. Copyright (C) 1999 John Wiley & Sons , Ltd.