The method of moments (MoM) is generally employed to solve the electric-field integral equation (EFIE), the magnetic-field integral equation (MFIE), and the combined-field integral equation (CFIE), upon discretizing surfaces using planar or curvilinear mesh elements. Through this method, the integration over source and test elements leads to the evaluation of four-dimensional integrals. However, the presence of a Green's function in these equations yields scalar and vector potential terms with singularities (in their higher-order derivatives) when the test and source elements share one or more edges or vertices and near-singularities when they are otherwise close.
Many approaches have been developed to address the singularity and near-singularity for the inner, source-element integral; fewer approaches have been developed to address the singularity in the outer, test-element integral. In this work, we introduce geometrically symmetric quadrature rules better suited for evaluating the logarithmic singularities in the test integral. Symmetric rules that can efficiently handle singularities are desirable because their mapping to the integration domain is straightforward and points are not heavily concentrated near some vertices. Asymmetric rules, on the other hand, generally employed to integrate singularities, require the determination of vertex mapping, and points may be concentrated nonuniformly at the vertices.
We demonstrate the effectiveness of these rules for several examples encountered in both the scalar and vector potentials of the electric-field integral equation (singular, near-singular, and far interactions), and we compare their performance to existing rules. 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.
