The Neumann Series As A New Challenge For The BEM - A Concept For Two And Three Dimensional Problems In Elasticity

Abstract

Iterative solvers have become very popular in structural analysis in the context of using parallel processing computers besides scalar machines. Symmetric and positive definite operators can be treated on the basis of conjugate gradient solvers. On the other hand, most of the numerical BEM schemes are realized by the collocation method using a Gaussian elimination technique. This leads to a remaining approximation error for the resolution of critical zones, even if the discrete structure is refined step by step. If the collocation method is replaced by the Neumann series, one has the chance to control the solution iteratively in dependence on a priori introduced control parameters. This approximation property is essential for the application in combined boundary and finite element algorithms, where it offers the possibility to control the properties of the coupling matrix with an arbitrary accuracy. Numerical examples for two- and three-dimensional structural analysis give evidence of the precise approximation properties of the proposed expansion of the boundary integral equation in a Neumann series.