Generalized Taylor–Duffy Method for Efficient Evaluation of Galerkin Integrals in Boundary-Element Method Computations

We present a generic technique, automated by computer-algebra systems and available as open-source software, for efficient numerical evaluation of a large family of singular and nonsingular four-dimensional integrals over triangle-product domains, such as those arising in the boundary-element method (BEM) of computational electromagnetism. Previously, practical implementation of BEM solvers often required the aggregation of multiple disparate integral-evaluation schemes in order to treat all of the distinct types of integrals needed for a given BEM formulation; in contrast, our technique allows many different types of integrals to be handled by the same algorithm and the same code implementation. Our method is a significant generalization of the Taylor-Duffy approach, which was originally presented for just a single type of integrand; in addition to generalizing this technique to a broad class of integrands, we also achieve a significant improvement in its efficiency by showing how the dimension of the final numerical integral may often reduced by one. In particular, if n is the number of common vertices between the two triangles, in many cases we can reduce the dimension of the integral from 4-n to 3-n, obtaining a closed-form analytical result for n=3 (the common-triangle case).