Analytical Integrals of Fundamental Solution of Three-Dimensional Laplace Equation and Their Gradients.

Abstract

New integrals of fundamental solution of three-dimensional Laplace equation are derived by using Gauss' divergence theorem. These are useful for boundary element method. One of the integrals is single layer potential of a constant triangular element and others are single and double layer potential of a linear triangular element. There are two advantages of these integrals. First, coordinate transformation and subdivision of a triangular element are not necessary to evaluate these integrals. Second, it is possible to evaluate formulas of single and double layer potential effectively, because the formula of double layer potential is related to the formula of single layer potential. The validity of these integrals is confirmed by comparing with numerical integration of fundamental solution over a triangular element by using Lachat algorithm. The effective gradient formulas are derived by differentiation of these integrals analytically. Present integrals can be applied for not only collocation BEM but also Galerkin BEM and fast multipole BEM.