Almost complete analytical integration in Galerkin BEM

TL;DR

This paper derives semi-analytical formulas for efficiently and accurately evaluating surface integrals in 3D Galerkin boundary element methods, reducing computational complexity while handling singularities.

Contribution

It introduces new analytical and semi-analytical formulas for surface integral evaluation in 3D BEM, simplifying computations and improving accuracy over existing methods.

Findings

Formulas for identical and shared-edge triangles are analytical.

Integrals over triangles with common vertices are reduced to 1D.

Integrals over disjoint triangles are reduced to 2D.

Abstract

In this work, semi-analytical formulae for the numerical evaluation of surface integrals occurring in Galerkin boundary element methods (BEM) in 3D are derived. The integrals appear as the entries of BEM matrices and are formed over pairs of surface triangles. Since the integrands become singular if the triangles have non-empty intersection, the transformation presented by Sauter and Schwab is used to remove the singularities. It is shown that the resulting integrals admit analytical formulae if the triangles are identical or share a common edge. Moreover, the four-dimensional integrals are reduced to one- or two-dimensional integrals for triangle pairs with common vertices or disjoint triangles respectively. The efficiency and accuracy of the formulae is demonstrated in numerical experiments.