Accurate computation of Galerkin double surface integrals in the 3-D boundary element method

摘要

Many boundary element integral equation kernels are based on the Green'sfunctions of the Laplace and Helmholtz equations in three dimensions. Theseinclude, for example, the Laplace, Helmholtz, elasticity, Stokes, and Maxwell'sequations. Integral equation formulations lead to more compact, but denselinear systems. These dense systems are often solved iteratively via Krylovsubspace methods, which may be accelerated via the fast multipole method. Thereare advantages to Galerkin formulations for such integral equations, as theytreat problems associated with kernel singularity, and lead to symmetric andbetter conditioned matrices. However, the Galerkin method requires each entryin the system matrix to be created via the computation of a double surfaceintegral over one or more pairs of triangles. There are a number ofsemi-analytical methods to treat these integrals, which all have some issues,and are discussed in this paper. We present novel methods to compute all theintegrals that arise in Galerkin formulations involving kernels based on theLaplace and Helmholtz Green's functions to any specified accuracy. Integralsinvolving completely geometrically separated triangles are non-singular and arecomputed using a technique based on spherical harmonics and multipoleexpansions and translations, which results in the integration of polynomialfunctions over the triangles. Integrals involving cases where the triangleshave common vertices, edges, or are coincident are treated via scaling andsymmetry arguments, combined with automatic recursive geometric decompositionof the integrals. Example results are presented, and the developed software isavailable as open source.