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

摘要

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 is the number of common vertices between the two triangles, in many cases we can reduce the dimension of the integral from to , obtaining a closed-form analytical result for (the common-triangle case).