# On the Generalization of DIRECTFN for Singular Integrals Over Quadrilateral Patches

## TL;DR

This paper extends the DIRECTFN numerical algorithm to evaluate four-dimensional singular integrals over quadrilateral patches, enabling accurate and efficient solutions for surface integral equations in computational electromagnetics.

## Contribution

It introduces a fully numerical method for singular integrals over quadrilaterals, generalizing previous triangular patch approaches with improved applicability and spectral convergence.

## Key findings

- Applicable to weakly and strongly singular kernels
- Achieves spectral convergence to machine precision
- Performance improved by optimized integration order

## 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.