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A set of fully numerical algorithms for evaluating the 4-D singular integrals arising from Galerkin surface integral equation methods over conforming quadrilateral meshes is presented. This paper is an extension of DIRECTFN, which was recently developed for the case of triangular patches, utilizing in the same fashion a series of coordinate transformations together with appropriate integration reorderings, and is applicable to coincident, edge adjacent, or vertex adjacent elements. The resulting formulas consist of sufficiently smooth kernels and exhibit several favorable characteristics when compared with the vast majority of the methods currently available. More specifically, they can be applied - without modifications - to the following challenging cases: 1) weakly and strongly singular kernels; 2) basis and testing functions of arbitrary order; 3) planar and curvilinear patches; 4) problem-specific Green functions (e.g., expressed in spectral integral form); and 5) spectral convergence to machine precision. Finally, we show that the overall performance of the fully numerical schemes can be further improved by a judicious choice of the integration order for each dimension.

On the Generalization of DIRECTFN for Singular Integrals Over Quadrilateral Patches