A broadband fast multipole accelerated boundary element method for the three dimensional Helmholtz equation
A broadband fast multipole accelerated boundary element method for the three dimensional Helmholtz equation
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DOI：
10.1121/1.3021297
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发表时间：
2009-01-01
影响因子：
2.4
通讯作者：
Duraiswami, Ramani
中科院分区：
文献类型：
--
作者：
Gumerov, Nail A.;Duraiswami, Ramani
本文介绍了三维亥姆霍兹方程边界元快速多极子迭代解法的发展。对于低kD和高kD（其中k是波数，D是域大小）的问题，亥姆霍兹方程的FMM显著不同，对于大问题，方法必须在层次结构的级别之间切换。边界元法需要几次近似计算（数值求积、使用单元逼近边界形状），并且这些误差必须与FMM引入的近似和迭代解的收敛准则相平衡。这些不同的误差都必须以这样的方式来选择，一方面，不做过多的工作，另一方面，总计算所获得的误差是可接受的。详细的翻译运营商为低和高kD，选择的代表，和BEM正交计划，所有这些近似一致，进行了说明。一种新的预处理器使用低精度FMM加速求解器作为一个正确的预处理器也被描述。大边界值问题的0.0001小于或类似kD小于或类似于500的开发的求解器的结果，并表现出接近理论预期。
The development of a fast multipole method (FMM) accelerated iterative solution of the boundary element method (BEM) for the Helmholtz equations in three dimensions is described. The FMM for the Helmholtz equation is significantly different for problems with low and high kD (where k is the wavenumber and D the domain size), and for large problems the method must be switched between levels of the hierarchy. The BEM requires several approximate computations (numerical quadrature, approximations of the boundary shapes using elements), and these errors must be balanced against approximations introduced by the FMM and the convergence criterion for iterative solution. These different errors must all be chosen in a way that, on the one hand, excess work is not done and, on the other, that the error achieved by the overall computation is acceptable. Details of translation operators for low and high kD, choice of representations, and BEM quadrature schemes, all consistent with these approximations, are described. A novel preconditioner using a low accuracy FMM accelerated solver as a right preconditioner is also described. Results of the developed solvers for large boundary value problems with 0.0001 less than or similar to kD less than or similar to 500 are presented and shown to perform close to theoretical expectations.