Full numerical quadrature of weakly singular double surface integrals in Galerkin boundary element methods
D'ELÍA, JORGE; BATTAGLIA, LAURA; CARDONA, ALBERTO; STORTI, MARIO
International Journal for Numerical Methods in Biomedical Engineering
John Wiley & Sons, Ltd.
Año: 2011 vol. 27 p. 314 - 334
When a Galerkin discretization of a boundary integral equation with a weakly singular kernel is performed over triangles, a double surface integral must be evaluated for each pair of them. If these pairs are not contiguous or not coincident the kernel is regular and a Gauss-Legendre quadrature can be employed. When the pairs have a common edge or a common vertex, then edge and vertex weak singularities appear. If the pairs have both facets coincident the whole integration domain is weakly singular. D. J. Taylor (IEEE Trans. on Antennas and Propagation, 51(7):1630-1637 (2003)) proposed a systematic evaluation based on a reordering and partitioning of the integration domain, together with a use of the Duffy transformations in order to remove the singularities, in such a way that a Gauss-Legendre quadrature was performed on three coordinates with an analytic integration in the fourth coordinate. Since this scheme is a bit restrictive because it was designed for electromagnetic kernels, a full numerical quadrature is proposed in order to handle kernels with a weak singularity with a general framework. Numerical tests based on modifications of the one proposed by W. Wang and N. Atalla are included.