On the numerically exact integration of singular Galerkin impedance matrix elements in computational electromagnetics

Abstract

Surface integral equation (SIE) formulations have reached a workhorse status in computational electromagnetics over the last decades. The numerical solution of Fredholm first and second kind SIEs is typically carried out by means of Galerkin (or Petrov-Galerkin) method of moments discretization schemes. The accuracy and stability of those schemes are strongly dependent on the accurate and efficient computation of the associated impedance matrix elements. In the case of disjoint supports of basis and testing functions, the arising multidimensional integrals are regular, allowing a straightforward numerical integration. Hence, special emphasis is naturally laid upon the most challenging cases that appear when those supports are overlapping or share some common points, thus, giving rise to singular integrals. More specifically, the singular integrals that stem from MoM SIE formulations can be categorized into weakly singular (improper Riemann integrable or 1/R) and strongly singular (Cauchy or 1/R2), provided certain restrictions to both basis and testing functions. We will present our latest advances on the fast and accurate integration of the above mentioned 4-D singular integrals for both div-conforming and curl-conforming testing functions over triangular tessellations. The numerical experiments have been undertaken on Matlab and C++ platforms with double precision arithmetic, while the reference values obtained with high precision arithmetic exhibit smooth convergence beyond 16 significant digits. As it will be clearly demonstrated by the results, the proposed method leads to exponential convergence both for 1/R and 1/R2 singularities with the accuracy being limited only by the incidental presence of error propagation effects in the numerical integration of sufficiently smooth functions. In any case, the results converge to a minimum of 13 significant digits (for most of the cases close to machine precision) with unmatched efficiency, thus allowing a safe shift of future research studies on other aspects of surface integral equation formulations.