Boundary element analysis of 3D linear potential problems combining fast multipole expansion and machine-precision numerical integration

Resumo

This paper is part of a research work to implement, test, and apply a novel numerical tool that can simu-
late 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. The authors have already developed their own version of the fast multipole method (FMM)
for two-dimensional problems, which relies on a consistent construction of the single-layer potential matrix of the
collocation boundary element method (BEM) so that ultimately only polynomial terms (as for the double-layer
potential matrix) are required to be integrated along generally curved segments related to a given field expansion
pole. The core of the present paper is the mathematical assessment of the double expansions needed in the 3D
FMM. The 3D implementation is combined with a particular formulation for linear triangle elements in which all
integrations for adjacent source point and boundary element are carried out analytically. As a result, numerical
approximations are due exclusively to the FMM series truncations. This allows isolating and testing truncation
errors incurred in the series expansions – and thus for the first time properly assessing the mathematical features
of the FMM, as illustrated by means of an example. The complete solution of a mixed boundary problem using a
GMRES solver, for instance, is just an additional task and, although already implemented, is not reported herein.