Efficient analytical integration of symmetric Galerkin boundary integrals over curved elements: Thermal conduction formulation

Abstract

Substantial improvements are reported in the computational efficiency of Galerkin boundary element analysis (BEA) employing curved continuous boundary elements. A direct analytical treatment of the singular double integrations involved in Galerkin BEA, adapting a limit to the boundary concept used successfully in collocation BEA, is used to obviate significant computation in the determination of the Galerkin coefficient matrices. Symbolic manipulation has been strategically employed to aid in the analytical evaluation of the singular contributions to these double integrals. The analytical regularization procedure separates the potentially singular Galerkin integrands into an essentially singular but simple part, plus a regular remainder that can be integrated numerically. The finite contribution from the simplified singular term is then computed analytically. It is shown that the key to containing the explosive growth in the length of the formulae associated with such a hybrid analytical/numerical integration scheme is the strategic timing of when to take the limit to the boundary. This regularization also isolates the contribution from the curvature of the boundary element, thus facilitating enhanced computational efficiency in problems with many straight elements. Example problems are presented to quantify the performance of this approach. It is concluded that with these techniques, Galerkin symmetric BEA can be more efficient than its collocation-based counterpart.