Closed Form Integration of Singular and Hypersingular Integrals in 3D BEM Formulations for Heat Conduction

Abstract

The evaluation of the singular and hypersingular integrals that appear in three-dimensional boundary element formulations for heat diffusion, in the frequency domain, is presented in analytical form. This improves computational efficiency and accuracy. Numerical integrations using existing techniques based on standard Gaussian integration schemes that incorporate an enormous amount of sampling points are used to verify the solutions of singular integrals. For the hypersingular integrals the comparison is evaluated by making use of an analytical solution that is valid for circular domains, combined with a standard Gaussian integration scheme for the remaining boundary element domain. Closed form solutions for cylindrical inclusions with null temperatures and null heat fluxes prescribed on the boundary are then derived and used to validate the three-dimensional boundary element formulations.