An O(1) integration scheme for three-dimensional surface scattering problems
An O(1) integration scheme for three-dimensional surface scattering problems
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三维表面散射问题的 O(1) 积分方案
DOI：
10.1016/j.cam.2006.02.050
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发表时间：
2007
影响因子：
2.4
通讯作者：
C. Geuzaine
中科院分区：
文献类型：
--
作者：
O. Bruno;C. Geuzaine
我们提出了一种关于频率的O(1)-复杂度的精确方法（即，为了达到规定的误差容限，需要对任意高频率进行有界计算成本的方法），用于计算三维空间中表面声散射问题的边界积分公式中产生的奇异振荡积分。与我们最近介绍的适用于平面曲线散射的二维对应算法一样，本方法基于两个主要元素的组合：(1)在问题的边界积分公式中对未知密度进行高频ansatz，以及(2)扩展了固定相位方法的思想，以允许振荡函数的O(1)（高阶精确）积分。我们介绍的在当前三维环境中实现高效O(1)积分器的技术与先前二维算法中使用的技术有很大不同。特别地，在本文中，我们引入了一种有效的“规范”（混合解析-数值）算法，除了允许围绕奇异点和平稳相位点的振荡函数的积分外，还可以处理在二维散射表面中奇异点和一个或多个平稳点相互接近时产生的重大困难。我们包括数值结果，说明了积分算法在直径高达5000波长的声软球上的行为：在这种情况下，对于单个积分，算法在不到两秒的计算时间内产生三位数的精度。在初步的全散射模拟中，在一个1.5GHz AMD Athlon处理器上，对于直径为500波长的球体，在大约3小时的运行时间内获得了表面密度精度为两位数的解决方案。
We present an accurate method of O(1)-complexity with respect to frequency (i.e., a method that, to achieve a prescribed error tolerance, requires a bounded computational cost for arbitrarily high frequencies) for the computation of singular oscillatory integrals arising in the boundary integral formulation of problems of acoustic scattering by surfaces in three-dimensional space. Like the two-dimensional counterpart of this algorithm, which we introduced recently and which is applicable to scattering by curves in the plane, the present method is based on a combination of two main elements: (1) a high-frequency ansatz for the unknown density in a boundary integral formulation of the problem, and (2) an extension of the ideas of the method of stationary phase to allow for O(1) (high-order-accurate) integration of oscillatory functions. The techniques we introduce to implement an efficient O(1) integrator in the present three-dimensional context differ significantly from those used in the earlier two-dimensional algorithm. In particular, in the present text, we introduce an efficient “canonical” (hybrid analytic-numerical) algorithm which, in addition to allowing for integration of oscillatory functions around both singular points and points of stationary phase, can handle the significant difficulty that arises as singular points and one or more stationary points approach each other within the two-dimensional scattering surface. We include numerical results illustrating the behavior of the integration algorithm on sound-soft spheres with diameters of up to 5000 wavelengths: in such cases, for a single integral, the algorithm yields accuracies of the order of three digits in computational times of less than two seconds. In a preliminary full scattering simulation we present, a solution with two digits of accuracy in the surface density was obtained in about three hours running time, in a single 1.5GHz AMD Athlon processor, for a sphere of 500 wavelengths in diameter.