Ney A. Dumon
Dept. of Mechanical Engineering, Pontifícia Universidade Católica, Rio de Janeiro, Brasil

Conventional, Variational, Simplified and Meshless Boundary Element Methods

The hybrid stress boundary element method (HSBEM) was introduced in 1987 on the basis of the Hellinger-Reissner potential, as a generalization of Pian’s hybrid finite element method. This new two- field formulation makes use of fundamental solutions to interpolate the stress field in the domain of an elastic body, which ends up discretized as a superelement with arbitrary shape and arbitrary number of degrees of freedom located along the boundary. More recently, a variational counterpart – the hybrid displacement boundary element method (HDBEM) – was proposed on the basis of three field functions, with equivalent advantages and disadvantages. The present paper discusses these methods as well as the conventional, collocation boundary element method (CBEM), not only investigating the mechanical properties of the resulting matrix equations, but mainly redefining a series of concepts in both HDBEM and CBEM that hadn’t been properly considered by previous authors, particularly in which concerns multiply connected domains and body forces. A theoretical consequence of these developments is also presented – the simplified hybrid stress boundary element method (SHSBEM) – which has been successfully applied to a wide class of problems in the last years. Moreover, other simplifying formulations derived from a thorough theoretical assessment of these methods are presented for the first time: the simplified hybrid displacement boundary element method (SHDBEM – a counterpart of the SHSBEM) and a still challenging super simplified version (SSHBEM) that combines SHSBEM and SHDBEM and closes a circle of theoretical achievements. This SSHBEM may become the ideal meshless BEM for the time saving, although accurate, analysis of large problems related to advanced materials and transient effects.