A Novel Approach for Evaluating Hypersingular and Strongly Singular Surface Integrals in Electromagnetics

Solving electromagnetic (EM) problems by integral equation methods requires an accurate and efficient treatment for the singular integral kernels related to the Green's function. For surface integral equations (SIEs), there are L and K operators which include hypersingular integrals (HSIs) and strongly singular integrals (SSIs), respectively. The HSIs are generated from the double gradient of the Green's function while the SSIs come from the single gradient of the Green's function. Although the HSIs could be reduced to weakly singular integrals (WSIs) in the method of moments (MoM) implementation with divergence conforming basis function such as the Rao-Wilton-Glisson (RWG) basis function, they do appear in Nyström method (NM) or boundary element method (BEM) and one has to tackle them. The SSIs always exist in the K operator and could also exist in the L operator when the testing function is not the RWG-like basis function. The treatment for the HSIs and SSIs is essential because they have a significant influence on the numerical solutions. There have been many publications dealing with the singular integrals, but they mainly focus on the WISs or SSIs, and the HSIs were seldom addressed. In this work, we develop a novel approach for evaluating those HSIs and SSIs based on the Stokes' theorem. The derived formulas are much simpler and more friendly in implementation since no polar coordinates or extra coordinate transformation are involved. Numerical experiments are presented to demonstrate the effectiveness of the approach.