On the fast-multipole implementation of the simplified hybrid boundary element method

Resumo

The present paper is part of a research line to implement, test and apply a novel numerical tool that can simulate on a personal computer and in just a few minutes a problem of potential or elasticity with up to tens of millions of degrees of freedom. We have already developed our own version of the fast-multipole method (FMM), which relies on a consistent construction of the collocation boundary element method (BEM), so that ultimately only polynomial terms are required to be integrated ”“ and in fact can be given as a table of pre-integrated values ”“ for generally curved segments related to a given field expansion pole and no matter how complicated the problem topology and the underlying fundamental solution. The simplified hybrid BEM has a variational basis and in principle leads to a computationally less intensive analysis of large-scale 2D and 3D problems of potential and elasticity ”“ particularly if implemented in an expedite version. One of the matrix-vector products of this formulation deals with an equilibrium transformation matrix that comes out to be the transpose of the double-layer potential matrix of the conventional BEM. This in principle requires a reverse strategy as compared to our first developed (and reverse) FMM. The effective application of these strategies to any high-order boundary element and any curved geometry needs to be adequately assessed for both numerical accuracy and computational effort. This is the subject of the present investigations, which are far from a closure. A few numerical examples are shown and some initial conclusions can already be drawn.