Generalized Taylor–Duffy Method for Efficient Evaluation of Galerkin Integrals in Boundary-Element Method Computations
Generalized Taylor–Duffy Method for Efficient Evaluation of Galerkin Integrals in Boundary-Element Method Computations
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DOI：
10.1109/tap.2014.2367492
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发表时间：
2013-12
影响因子：
5.7
通讯作者：
M. T. H. Reid;Jacob K. White;S. Johnson
中科院分区：
文献类型：
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作者：
M. T. H. Reid;Jacob K. White;S. Johnson
我们提出了一个通用的技术，自动化的计算机代数系统和开源软件，有效的数值评估的一个大家庭的奇异和非奇异的四维积分三角积域，如计算电磁学的边界元法（BEM）中产生的。在此之前，BEM求解器的实际实现通常需要聚合多个不同的积分评估方案，以处理给定BEM公式所需的所有不同类型的积分;相比之下，我们的技术允许许多不同类型的积分由相同的算法和相同的代码实现来处理。我们的方法是一个显着的推广泰勒-达菲的方法，这是最初提出的只是一个单一类型的被积函数;除了推广这种技术广泛的一类被积函数，我们还实现了显着的提高其效率，显示如何最终的数值积分的维数可能往往减少了一个。特别地，如果n是两个三角形之间的公共顶点的数目，在许多情况下，我们可以将积分的维数从4-n减少到3-n，得到n=3（公共三角形情况）的封闭形式的分析结果。
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).