Characterization and Integration of the Singular Test Integrals in the Method-of-Moments Implementation of the Electric-Field Integral Equation

TL;DR

This paper develops symmetric quadrature rules to accurately integrate singular test integrals in the method of moments for electric-field integral equations, improving convergence and accuracy over existing polynomial-based methods.

Contribution

It introduces geometrically symmetric quadrature rules specifically designed for singular integrals in the method of moments, enhancing numerical accuracy and efficiency.

Findings

Symmetric quadrature rules outperform polynomial schemes in convergence.

The new rules improve accuracy for singular, near-singular, and far interactions.

Effective in both scalar and vector potentials of electric-field integral equations.

Abstract

In this paper, we characterize the logarithmic singularities arising in the method of moments from the Green's function in integrals over the test domain, and we use two approaches for designing geometrically symmetric quadrature rules to integrate these singular integrands. These rules exhibit better convergence properties than quadrature rules for polynomials and, in general, lead to better accuracy with a lower number of quadrature points. We demonstrate their effectiveness for several examples encountered in both the scalar and vector potentials of the electric-field integral equation (singular, near-singular, and far interactions) as compared to the commonly employed polynomial scheme and the double Ma--Rokhlin--Wandzura (DMRW) rules, whose sample points are located asymmetrically within triangles.