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arXiv 2210.01255math.NAcs.NA
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任意周期性下Stokes流的快速Ewald求和

Fast Ewald summation for Stokes flow with arbitrary periodicity

Joar Bagge, Anna-Karin Tornberg

备注：
54 pages, 15 figures
发表信息：
J. Comput. Phys. 493 (2023) 112473
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AI总结：

本文将Spectral Ewald方法扩展到任意周期性边界条件，通过新的修正核和截断误差估计实现三维Stokes流势的快速谱精度计算，并给出参数自动选择策略。

摘要：

本文提出了一种快速且具有谱精度的Ewald求和方法，用于计算三维Stokes流的stokeslet、stresslet和rotlet势。该工作将先前为Stokes流开发的Spectral Ewald方法扩展到任意数量（三个、两个、一个或零个）空间方向上的周期性边界条件，并统一在一个框架中。周期性势被分解为短程部分和长程部分，其中长程部分在Fourier空间中使用快速Fourier变换处理。该方法的一个关键组成部分是用于处理奇异积分的修正核。我们推导了新的修正核，并为stokeslet和stresslet给出了新的改进截断误差估计。设计并测试了一种基于给定误差容限自动选择参数的程序。给出了双重周期和单重周期情形下用于验证的解析公式。我们表明，对于N个源点和目标点，该方法的计算时间按O(N log N)缩放，并研究了计算时间如何依赖于误差容限和窗函数，即用于将不规则点数据平滑扩展到均匀网格的函数。该方法在完全周期情形下最快，而自由空间情形下的运行时间约为其三倍。此外，当将方法应用于低纵横比原胞中的均匀源分布时，可达到最高效率。本文所呈现的工作使得利用边界积分和势方法等对具有任意周期性的三维Stokes流进行高效且精确的模拟成为可能。

英文摘要：

A fast and spectrally accurate Ewald summation method for the evaluation of stokeslet, stresslet and rotlet potentials of three-dimensional Stokes flow is presented. This work extends the previously developed Spectral Ewald method for Stokes flow to periodic boundary conditions in any number (three, two, one, or none) of the spatial directions, in a unified framework. The periodic potential is split into a short-range and a long-range part, where the latter is treated in Fourier space using the fast Fourier transform. A crucial component of the method is the modified kernels used to treat singular integration. We derive new modified kernels, and new improved truncation error estimates for the stokeslet and stresslet. An automated procedure for selecting parameters based on a given error tolerance is designed and tested. Analytical formulas for validation in the doubly and singly periodic cases are presented. We show that the computational time of the method scales like O(N log N) for N sources and targets, and investigate how the time depends on the error tolerance and window function, i.e. the function used to smoothly spread irregular point data to a uniform grid. The method is fastest in the fully periodic case, while the run time in the free-space case is around three times as large. Furthermore, the highest efficiency is reached when applying the method to a uniform source distribution in a primary cell with low aspect ratio. The work presented in this paper enables efficient and accurate simulations of three-dimensional Stokes flow with arbitrary periodicity using e.g. boundary integral and potential methods.

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