Fast hybrid numerical-asymptotic boundary element methods for high frequency screen and aperture problems based on least-squares collocation
Fast hybrid numerical-asymptotic boundary element methods for high frequency screen and aperture problems based on least-squares collocation
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基于最小二乘配置的高频屏蔽和孔径问题的快速混合数值渐近边界元方法
DOI：
10.1007/s42985-020-00013-3
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Emile Parolin
中科院分区：
文献类型：
--
作者：
A. Gibbs;D. Hewett;D. Huybrechs;Emile Parolin
本文提出了一种数值-渐近(HNA)混合边界元方法(BEM)来计算二维屏蔽层和孔板的高频散射，其计算成本随着频率的增加而有界。我们的方法是Hewett等人的高阶hp-HNA近似空间的配置实现。(IMA J Numer Anal 35：1698-1728,2015)，其中研究了Galerkin实现。这种配置方案的一个优点是，边界元矩阵项中出现的一维高振荡奇异积分比Galerkin情形中出现的二维积分更容易计算，从而使计算速度大大加快。在这里，我们使用最陡下降的数值方法来计算所需的积分，其代价与频率无关，这涉及复杂的轮廓变形。从Galerkin到配置的变化不是平凡的，因为基于平方线性系统的朴素配置实现受到与海航基数值冗余相关的严重的数值不稳定性的影响，这会产生高度病态的边界元矩阵。在本文中，我们展示了如何通过过采样来消除这些不稳定性，并使用截断奇异值分解来求解所产生的加权最小二乘意义下的超定配置系统。在我们的数值实验的基础上，稳定该方法所需的过采样量是适度的(通常在25%左右就足够了)，并且与频率无关。作为我们方法的一个应用，我们给出了高频散射的数值结果，这些结果是通过对中-三分之一Cantor集的预分形近似得到的。
We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM) for high frequency scattering by two-dimensional screens and apertures, whose computational cost to achieve any prescribed accuracy remains bounded with increasing frequency. Our method is a collocation implementation of the high order hp HNA approximation space of Hewett et al. (IMA J Numer Anal 35:1698–1728, 2015), where a Galerkin implementation was studied. An advantage of the current collocation scheme is that the one-dimensional highly oscillatory singular integrals appearing in the BEM matrix entries are significantly easier to evaluate than the two-dimensional integrals appearing in the Galerkin case, which leads to much faster computation times. Here we compute the required integrals at frequency-independent cost using the numerical method of steepest descent, which involves complex contour deformation. The change from Galerkin to collocation is nontrivial because naive collocation implementations based on square linear systems suffer from severe numerical instabilities associated with the numerical redundancy of the HNA basis, which produces highly ill-conditioned BEM matrices. In this paper we show how these instabilities can be removed by oversampling, and solving the resulting overdetermined collocation system in a weighted least-squares sense using a truncated singular value decomposition. On the basis of our numerical experiments, the amount of oversampling required to stabilise the method is modest (around 25% typically suffices), and independent of frequency. As an application of our method we present numerical results for high frequency scattering by prefractal approximations to the middle-third Cantor set.
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