Fast and accurate computation of hyper-singular integrals in Galerkin surface integral equation formulations via the direct evaluation method

Hypersingular 4-D integrals, arising in the Galerkin discretization of surface integral equation formulations, are computed by means of the direct evaluation method. The proposed scheme extends the basic idea of the singularity cancellation methods, usually employed for the regularization of the singular integral kernel, by utilizing a series of coordinate transformations combined with a reordering of the integrations. The overall algebraic manipulation results in smooth 2-D integrals that can be easily evaluated via standard quadrature rules. Finally, the reduction of the dimensionality of the original integrals together with the smooth behavior of the associated integrands lead up to unmatched accuracy and efficiency.