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

TL;DR

This paper introduces a generalized Taylor-Duffy method for efficiently evaluating singular 6D integrals over tetrahedral domains, significantly improving accuracy and computational efficiency in volume integral equation solvers.

Contribution

It extends the Taylor-Duffy approach from triangle to tetrahedron integrals, enabling exact transformation to nonsingular integrals for VIE matrix element computation.

Findings

Achieves 12-digit accuracy in VIE matrix elements

Reduces computational cost by orders of magnitude

Applicable to various tetrahedral basis functions

Abstract

I present an accurate and efficient technique for numerical evaluation of singular 6-dimensional integrals over tetrahedon-product domains, with applications to calculation of Galerkin matrix elements for discretized volume-integral-equation (VIE) solvers using Schaubert-Wilton-Glisson (SWG) and other tetrahedral basis functions. My method extends the generalized Taylor-Duffy strategy---used to handle the singular \textit{triangle}-product integrals arising in discretized surface-integral-equation (SIE) formulations---to the tetrahedron-product case; it effects an exact transformation of a singular 6-dimensional integral to an nonsingular lower-dimensional integral that may be evaluated by simple numerical cubature The method is highly general and may---with the aid of automatic code generation facilitated by computer-algebra systems---be applied to a wide variety of singular integrals arising in various VIE formulations with various types of tetrahedral basis function, of which I present several examples. To demonstrate the accuracy and efficiency of my method, I apply it to the calculation of matrix elements for the volume electric-field integral equation (VEFIE) discretized with SWG basis functions, where the method yields 12-digit or higher accuracy with low computational cost---an improvement of many orders of magnitude compared to existing techniques.