Fully numerical evaluation of singular integrals in Moment methods: A novel strategy for the coplanar element case

In the Method of Moments the evaluation of reaction integrals for the critical case when the source integral is (near-) singular usually employs the so-called “singularity subtraction” scheme for the source (inner) integral, followed by ordinary Gauss-Legendre quadrature for the resulting test (outer) integral. Recent literature has focused on the “singularity cancellation” scheme to evaluate the source, still followed by the Gauss-Legendre quadrature for the test integral. In both integration schemes, little attention has been given to the test integral; however it is fairly well-known that the source integral, though resulting in bounded potentials, has lower order singularities near the source domain boundaries that can render standard quadrature schemes inefficient. Hence treating the entire 4-D reaction integral as a whole can have significant advantages. Polimeridis and co-authors (A. G. Polimeridis, F. Vipiana, J. R. Mosig, D. R. Wilton, IEEE Trans. Ant. Prop., 61, 3112-3122, 2013) have proposed, when the source and observation triangles coincide or share a common edge or vertex, the “direct evaluation method” (DEM) that treats the entire 4-D integration through a combination of variable transformations together with appropriate integration re-orderings.