A Vertex Function--Based Radial--Angular Quadrature for the Numerical Evaluation of Surface Test Integrals in the Method of Moments

Abstract

A novel integration scheme is proposed for the accurate numerical evaluation of test (reaction) integrals needed for solving complex direct or inverse electromagnetic problems using surface integral equation (SIE) formulations and the method of moments. Significant effort has already been devoted to improving the numerical evaluation of source integrals yielding potentials (or their derivatives), especially for triangular elements. However, numerical techniques for accurately evaluating the subsequent test integrals have been largely neglected, with simple numerical integration schemes being used that either ignore or are developed with incomplete knowledge of the detailed behavior of potentials (or their derivatives) near edges and vertices. Consequently, simple numerical quadrature schemes are found to be either of limited accuracy or slowly convergent with respect to increasing the sampling for self, edge- or vertex-adjacent source, and test triangle pairs. Here, we describe a simple model derived from static potential integrals that properly describes and bounds the potentials and their derivatives near vertices. From it, we are able to construct appropriate, separable radial-angular quadrature schemes that are both exponentially convergent and applicable to all potential forms of interest arising from both EFIE and MFIE operators. Numerical results are presented that demonstrate the wide-ranging applicability and improved convergence rates of the proposed scheme, and these are compared to some previously reported testing schemes. The method’s sensitivity to test triangle shape and the ratio choice of angular-to-radial sampling rates is also briefly explored.