Taylor–Duffy Method for Singular Tetrahedron-Product Integrals: Efficient Evaluation of Galerkin Integrals for VIE Solvers

This paper presents an accurate and efficient technique for the numerical evaluation of singular six-dimensional (6-D) integrals over tetrahedron-product domains, with applications to the calculation of Galerkin matrix elements for discretized volume-integral-equation (VIE) solvers using Schaubert–Wilton–Glisson (SWG) and other tetrahedral basis functions. The proposed method extends the generalized Taylor–Duffy strategy—used to handle the singular triangle -product integrals arising in discretized surface-integral-equation formulations—to the tetrahedron-product case; it effects an exact transformation of a singular 6-D integral into a nonsingular lower dimensional integral, which may be evaluated by simple numerical cubature. The method is very general and may be applied—with the aid of automatic code generation facilitated by computer-algebra systems— to a wide variety of singular integrals arising in various volume-integral-equation (VIE) formulations with various types of tetrahedral basis functions, several examples of which are presented in this paper. To demonstrate the accuracy and efficiency of the method, it has been applied in the calculation of matrix elements for the volume electric-field integral equation discretized with SWG basis functions, where the method yields 12-digit or higher accuracy with a low computational cost—an improvement of many orders of magnitude compared to existing techniques.