Trefftz boundary elements - multi-region formulations

This paper is concerned with an e ective numerical implementation of the Tre tz boundary element method, for the analysis of two-dimensional potential problems, de ned in arbitrarily shaped domains. The domain is rst discretized into multiple subdomains or regions. Each region is treated as a single domain, either nite or in nite, for which a complete set of solutions of the problem is known in the form of an expansion with unknown coe cients. 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 integral equations must be used, along common region interfaces, in order to couple to each other the unknowns of the solution expansions relative to the neighbouring regions. These boundary integrals are obtained from weighted residuals of the coupling conditions which allow the implementation of any order of continuity of the potential eld, across the interface boundary, between neighbouring regions. The technique used in the formulation of the region-coupling conditions drives the performance of the Tre tz boundary element method. While both of the collocation and Galerkin techniques do not generate new unknowns in the problem, the technique of Galerkin presents an additional and unique feature: the size of the matrix of the nal algebraic system of equations which is always square and symmetric, does not depend on the number of boundary elements used in the discretization of both the actual and region-interface boundaries. This feature which is not shared by other numerical methods, allows the Galerkin technique of the Tre tz boundary element method to be e ectively applied to problems with multiple regions, as a simple, economic and accurate solution technique. A very di cult example is analysed with this procedure. The accuracy and e ciency of the implementations described herein make the Tre tz boundary element method ideal for the study of potential problems in general arbitrarily-shaped two-dimensional domains