Evaluation of the 4-D singular and near singular potential integrals via the Stokes' theorem

A closed-form solution for singular and near-singular double surface integrals arising in boundary integral equations of electrostatics is given for the case of arbitrary coplanar polygonal surfaces. To date, a limited number of closed-form solutions for these integrals were published only for the case of coincident triangular surfaces. The second result of the paper is the evaluation method itself, which is extendable to non-coplanar and non-polygonal surfaces. The main idea of the evaluation method is to construct exact differential forms to perform integration via the Stokes' theorem. Free of coordinates, the differential forms approach has several important advantages over the traditional algebraic singularity subtraction and singularity cancellation methods. Numerical tests on triangles show that the proposed method maintains accuracy even as the aspect ratio of a triangle tends to infinity while the existing methods fail in that limit. Thus, the results of this paper are expected to greatly improve the accuracy and efficiency of the computational electrostatics codes. We also show that the proposed method is extendable to other kernels, e.g., Helmholtz, and therefore has the potential to greatly speed up the matrix build of the computational electromagnetics and acoustics codes.

Automated summary

This paper presents a closed-form solution for singular and near-singular double surface integrals on arbitrary coplanar polygonal surfaces, as well as an evaluation method extendable to non-coplanar and non-polygonal surfaces. The evaluation method constructs exact differential forms for integration via the Stokes' theorem, providing advantages over traditional singularity subtraction methods. Numerical tests show that the proposed method maintains accuracy for triangles with aspect ratios tending to infinity, unlike existing methods.