Evaluation of 6-D Reaction Integrals via Double Application of the Divergence Theorem

This article addresses the problem of efficiently and accurately evaluating the singular or near-singular double volume (6-D) reaction integrals fundamental to the solution of volume integral equations (VIEs) using the method of moments (MoM). VIE techniques based on the MoM are used to obtain accurate solutions for electromagnetic (EM) problems of all types, and the accurate computation of the singular and near-singular reaction integrals that occur in the resulting system matrices is crucial for accurate EM modeling. Here we propose a scheme to treat, as a whole, the 6-D reaction integrals appearing in the MoM. The divergence theorem is directly applied twice in the physical space domain, not a transformed domain, thus eliminating restrictions to well-shaped elements. With appropriate integration reordering, the 6-D volume integrals are expressed as two radial integrals followed by two surface integrals over source and observation domain boundaries. We further smooth the integral by removing the static form of the kernel and evaluating it separately using a semi-analytical approach. The method is numerically validated for static and dynamic kernels arising in the electric field VIEs (i.e., for kernels having
$1/R$
singularities) and for linear basis functions.