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

Abstract

In this paper, we present two approaches for designing geometrically symmetric quadrature rules to address the logarithmic singularities arising in the method of moments from the Green's function in integrals over the test domain. 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.

Manufactured Solutions for the Method-of-Moments Implementation of the Electric-Field Integral Equation

“…However, as described in the introduction, the integral in E i MS (x) (15) is not only unable to be computed analytically, but the singularity in the Green's function (4) when R → 0 complicates its accurate approximation, potentially contaminating convergence studies.…”

“…However, the presence of a Green's function in these equations yields scalar and vector potential terms with singularities 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 [1][2][3][4][5][6][7][8][9][10], as well as for the outer, test-element integral [11][12][13][14][15].…”

“…Exact solutions exist for curved surfaces, such as the Mie solution to Maxwell's equations for a sphere, permitting Combination 7 to be verified. Combination 5 can also be verified; however, while Error Source 3 can be easily studied for nonsingular integrands, for singular and nearly singular integrands, Error Source 3 is most likely better studied using unit tests, due to the computationally expensive adaptive integration techniques required to compute accurate reference solutions [15,41].…”