Direct Evaluation of Hypersingular Galerkin Surface Integrals

A direct algorithm for evaluating hypersingular integrals arising in a three-dimensional Galerkin boundary integral analysis is presented. The singular integrals are defined as limits to the boundary, and by integrating two of the four dimensions analytically, the coincident integral is shown to be divergent. However, the divergent terms can be explicitly calculated and shown to cancel with corresponding singularities in the adjacent edge integrals. A single analytic integration is employed for the edge and vertex singular integrals. This is sufficient to display the divergent term in the edge-adjacent integral and to show that the vertex integral is finite. By explicitly identifying the divergent quantities, we can compute the hypersingular integral without recourse to Stokes’s theorem or the Hadamard finite part. The algorithms are developed in the context of a linear element approximation for the Laplace equation but are expected to be generally applicable. As an example, the algorithms are applied to solve a thermal problem in an exponentially graded material.