New formulations for evaluating hypersingular and strongly singular integrals in electromagnetic integral equations

Abstract

Electromagnetic (EM) integral equations include the singular integral kernels related to the Green's function. For surface integral equations (SIEs), there are two kinds of kernels, i.e. the L operator and K operator. The L operator is the dyadic Green's function which includes a double gradient operation on the scalar Green's function that results in 1/R3 hypersingular integrals (HSIs), where R is the distance between a source point and an observation point or field point. However, the HSIs could be reduced to 1/R weakly singular integrals (WSIs) in the method of moments (MoM) solutions if divergence conforming basis function like the Rao-Wilton-Glisson (RWG) basis function is used as an expansion and testing function. Without the help of these basis functions, we must carefully handle the HSIs and this happens in the implementation of Nystr¨om method (NM) or boundary element method (BEM). The K operator includes a single gradient operation on the scalar Green's function, yielding 1/R2 strongly singular integrals (SSIs) in the matrix elements. The SSIs also exist in the L operator in the MoM when the RWG-like basis functions cannot be used as a testing function. The accurate and efficient evaluation for the HSIs and SSIs is essential in solving the SIEs because they have a significant impact on the numerical solutions.