Analysis of numerical integration in the evaluation of hyper-singular integrals in Galerkin surface integral equation formulations via the direct evaluation method

In this paper, some aspects of the direct evaluation method for the computation of hyper-singular 4-D integrals, arising in the Galerkin discretization of surface integral formulations, are analyzed. In its original version, utilizing a series of coordinate transformations combined with a re-ordering of the integrations, the integrand not only was regularized but also it was casted in a convenient form in order to perform two dimensions of the integrals analytically. Here, we are going to test and compare the behavior when only one or none of the dimensions is held analytically. Furthermore, the importance of the re-ordering will be outlined, even when no analytical integration is performed.