On the Generalization of DIRECTFN for Singular Integrals Over Quadrilateral Patches

Abstract

Abstract-A set of fully numerical algorithms for evaluating the four-dimensional singular integrals arising from Galerkin surface integral equation methods over conforming quadrilateral meshes is presented. This work is an extension of DIRECTFN, which was recently developed for the case of triangular patches, utilizing in a same fashion a series of coordinate transformations together with appropriate integration re-orderings. 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), 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.Index Terms-Galerkin inner product, method of moments (MoM), quadrilateral discretization, singular integrals, surface integral equations.

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“…The main idea for tackling these cases is to reduce the dimensionality of the volume-volume integrals to surface-surface step-by-step, as it has been shown in [33], [36]. As a result, the initial volume-volume integral boils down to a sum of 4 surfacesurface integrals, over the faces of the interacting voxels, which have smoother kernels and can be calculated by modern algorithms, originally developed for Galerkin SIE methods over quadrilateral patches [43], [45].…”

“…Moreover, the resulting formulation calls for the numerical integration of singular volume-volume Galerkin inner products, but as it has been shown in previous work [33]- [36], there exist readily available formulas that reduce both the dimensionality and singularity order of the kernels, allowing for the fast and precise numerical evaluation of the singular integrals by means of well-established sophisticated cubatures [37]- [43], originally developed for SIE formulations. Recent contributions have also expanded the analysis for arbitrary quadrilateral patches [44], [45].…”