A new Fast Multipole formulation for the elastodynamic half-space Green's tensor
A new Fast Multipole formulation for the elastodynamic half-space Green's tensor
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DOI：
10.1016/j.jcp.2013.11.010
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发表时间：
2014-02
期刊：
影响因子：
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通讯作者：
S. Chaillat;M. Bonnet
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文献类型：
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作者：
S. Chaillat;M. Bonnet
本文提出了一种基于半空间绿色张量的半无限介质的频域弹性动力快速多极边界元法（FM-BEM），避免了平面无摩擦面的离散化。半空间绿色格林张量常用于计算土-结构相互作用中的弹性波传播，并应用于地震学或土木工程。然而，与全空间绿色格林张量不同，弹性动力学半空间绿色格林张量不能用亥姆霍兹基本解的导数表示。因此，该张量的多极展开不能直接从已知的展开式中获得，而是在这里通过相对于平行于自由表面的空间坐标的部分傅立叶变换导出。所得到的配方严格要求傅立叶积分，其被积函数是奇异和振荡的有效的积分。在这些条件下，经典的高斯求积将表现不佳，失败或需要大量的点。相反，一个版本定制的Rokhlin和合著者提出的方法，它产生广义高斯求积规则的特定类型的积分，目前的需求已经实现。通过对单层弹性动力势的数值实验，证明了该方法的准确性和有效性，其自由度约为N= 6× 10 5。特别是，显着低于非多极版本的复杂性被证明是实现。
In this article, a version of the frequency-domain elastodynamic Fast Multipole-Boundary Element Method (FM-BEM) for semi-infinite media, based on the half-space Greenʼs tensor (and hence avoiding any discretization of the planar traction-free surface), is presented. The half-space Greenʼs tensor is often used (in non-multipole form until now) for computing elastic wave propagation in the context of soil–structure interaction, with applications to seismology or civil engineering. However, unlike the full-space Greenʼs tensor, the elastodynamic half-space Greenʼs tensor cannot be expressed using derivatives of the Helmholtz fundamental solution. As a result, multipole expansions of that tensor cannot be obtained directly from known expansions, and are instead derived here by means of a partial Fourier transform with respect to the spatial coordinates parallel to the free surface. The obtained formulation critically requires an efficient quadrature for the Fourier integral, whose integrand is both singular and oscillatory. Under these conditions, classical Gaussian quadratures would perform poorly, fail or require a large number of points. Instead, a version custom-tailored for the present needs of a methodology proposed by Rokhlin and coauthors, which generates generalized Gaussian quadrature rules for specific types of integrals, has been implemented. The accuracy and efficiency of the proposed formulation is demonstrated through numerical experiments on single-layer elastodynamic potentials involving up to about N= 6× 10 5 degrees of freedom. In particular, a complexity significantly lower than that of the non-multipole version is shown to be achieved.