Complete semi-analytical treatment of weakly singular integrals on planar triangles via the direct evaluation method

摘要

A complete semi-analytical treatment of the four-dimensional (4-D) weakly singular integrals over coincident, edge adjacent and vertex adjacent triangles, arising in the Galerkin discretization of mixed potential integral equation formulations, is presented. The overall analysis is based on the direct evaluation method, utilizing a series of coordinate transformations, together with a re-ordering of the integrations, in order to reduce the dimensionality of the original 4-D weakly singular integrals into, respectively, 1-D, 2-D and 3-D numerical integrations of smooth functions. The analytically obtained final formulas can be computed by using typical library routines for Gauss quadrature readily available in the literature. A comparison of the proposed method with singularity subtraction, singularity cancellation and fully numerical methods, often used to tackle the multi-dimensional singular integrals evaluation problem, is provided through several numerical examples, which clearly highlights the superior accuracy and efficiency of the direct evaluation scheme.