Computation of Galerkin Double Surface Integrals in the 3-D Boundary Element Method

The Galerkin boundary element method (BEM), also known as the method of moments, is a powerful method for solving the Laplace equation in three dimensions. There are advantages to Galerkin formulations for integral equations, as they treat problems associated with kernel singularity, and lead to symmetric and better conditioned matrices. However, the Galerkin method requires the computation of double surface integral over pairs of triangles. There are many semianalytical methods to treat these integrals, which all have some issues and are discussed in this paper. Novel methods inspired by the treatment of these kernels in the fast multipole method are presented for computing all the integrals that arise in the Galerkin formulation to any accuracy. Integrals involving completely geometrically separated triangles are nonsingular, and are computed using a technique based on spherical harmonics and multipole expansions and translations, which require the integration of polynomial functions over the triangles. Integrals involving cases where the triangles have common vertices, edges, or are coincident are treated via scaling and symmetry arguments, combined with automatic recursive geometric decomposition of the integrals. The methods are validated, and example results are presented.