Robust semi-analytical integration of singular impedance matrix elements in surface integral equation solvers

The direct evaluation method tailored to the 4-D singular integrals arising in the second-kind Fredholm surface integral equation formulations is presented. The proposed semi-analytical integration scheme extends the existing singularity cancellation methods, often used to tackle singular integrals, by utilizing a series of coordinate transformations combined with the re-ordering of the integrations. The overall algebraic manipulation results in a reduction of the dimensionality and the final 2-D kernels comprise sufficiently smooth behavior, allowing a straightforward treatment by standard quadrature rules. Numerical results demonstrate that one of the main error sources of the second-kind integral equations is undoubtedly cured.