Formal equivalence of direct and indirect boundary element methods

Abstract

Over the last few years the great potential of boundary element methods has become more widely recognised and it is now generally accepted that, not only can they be applied to all the classical field problems such as steady state and transient potential flow 1, elastostatics 2, elastodynamics 3 and elastoplasticity 4 but that in many cases they offer the most attractive and efficient solution technique. In addition, problems with boundaries at infinity, for which the domain solutions such as finite elements or finite differences are obviously unsuitable, present no difficulties when using boundary elements. Boundary elements can be formulated using two different approaches called the direct and indirect boundary element methods. The first of these takes the form of a statement which provides values of the solution variables at any internal field point in terms of the complete set of all the boundary data, both the essential and natural. The statement can be formulated rigorously using either an approach based on Green's theorem, or as a particular case of weighted residual methods 5. The indirect method has usually been justified on a less rigorous basis. In this case the field of interest is visualized as a region, delineated within an infinite, or semi-infinite body of the material, around which fictitious singular 'sources' are distributed at an initially unknown density such that the real, specified boundary conditions are established on the boundaries of the fictitious region. This note starts from the direct boundary element method statement for the solution of potential problems and shows how this can be transformed into the equivalent indirect statement, thereby establishing the latter rigorously. Similar arguments are then applied to a more general class of differential operators.