A parallel fast multipole accelerated integral equation scheme for 3D Stokes equations

Abstract

SUMMARYIn this paper, we discuss a numerical scheme for the Stokes equations in three dimensions. It uses an integral equation formulation and is accelerated by the new version of fast multipole method first introduced by Greengard and Rokhlin in 1997 (Acta Numerica 1997; 6:229-269). The code is parallelized to solve problems of extremely large size. The resulting numerical solver can be applied to Stokes flows in complex geometry and also serves as a building block for solving the Navier-Stokes equations of low to moderate Reynold's numbers.

A fast multipole method for the three-dimensional Stokes equations

“…The cost of the method is, therefore, approximately four times that of a harmonic FMM. Recently, Wang et al [11] have presented an efficient, parallel implementation for Stokeslet and Stresslet summations along these lines. However, none of these schemes make direct use of existing ''black-box'' harmonic FMMs.…”

Algorithm for direct numerical simulation of emulsion flow through a granular material

“…Relevant here are previous works [11][12][13] on fast summation of Stokes interactions in 3D, with an asymptotically linear complexity. Sangani and Mo [11] developed the first hydrodynamical version of the traditional electrostatical FMM of Greengard and Rokhlin.…”

“…Ying et al [12] developed a kernel-independent version (applicable to Laplacian and Stokes interactions), which retains the logical scheme of FMM but is technically simpler and only slightly slower. Wang et al [13] recently developed a parallel hydrodynamical version of the new FMM [14]. Of these works, only the code of Sangani and Mo [11] implemented periodic boundaries (which is an additional burden in terms of efficiency) and is significantly oriented on dispersed media simulations.…”

The development of the pFFT accelerated BEM for 3-D acoustic scattering problems based on the Burton and Miller's integral formulation

“…To overcome these difficulties, several fast BEMs were developed in the past decades. The BEM accelerated by the fast multipole expansion method (FMM) [2][3][4][5][6] had been developed and applied to nearly all the fields that the conventional BEM can be employed. The BEM accelerated by the precorrected fast Fourier transform method (pFFT) [7] was widely applied as well [8][9][10][11][12].…”