Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 647038, 21 pages
Research Article

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

Department of Civil Engineering, University of Coimbra, Pólo II, Rua Luís Reis Santos, 3030-788 Coimbra, Portugal

Received 16 March 2012; Accepted 22 May 2012

Academic Editor: Alex Elias-Zuniga

Copyright © 2012 A. Tadeu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.