Fast Ewald summation for Stokesian particle suspensions

Abstract

We present a numerical method for suspensions of spheroids of arbitrary aspect ratio which sediment under gravity. The method is based on a periodized boundary integral formulation using the Stokes double layer potential. The resulting discrete system is solved iteratively using GMRES accelerated by the spectral Ewald (SE) method, which reduces the computational complexity to O(N log N ), where N is the number of points used to discretize the particle surfaces. We develop predictive error estimates, which can be used to optimize the choice of parameters in the Ewald summation. Numerical tests show that the method is well conditioned and provides good accuracy when validated against reference solutions.

Brownian dynamics of confined suspensions of active microrollers

“…The Galerkin approach ensures an SPD mobility matrix so Brownian motion can be added straightforwardly, however, the key difficulty lies in efficient implementation for systems of many particles. Another approach is to use boundary integral methods [55,56], which explicitly discretize the boundary and are thus more general. These methods do not ensure an SPD matrix, and are also still too expensive to use for suspensions of thousands of particles.…”

A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

“…1 It is natural to ask how the unit nullity of the BVP manifests itself in the solution space for the pair . 1 It is natural to ask how the unit nullity of the BVP manifests itself in the solution space for the pair .…”

“…The macroscopic response of a given microscopic periodic composite medium can often be summarized by an effective material property (e.g., a conductivity or permeability tensor), a fact placed on a rigorous footing by the field of homogenization (for a review see [14]). Application areas span all of the major elliptic PDEs, including the Laplace equation (thermal/electrical conductivity, electrostatics and magnetostatics of composites [12,27,35,38]); the Stokes equations (porous flow in periodic solids [18,26,50,75], sedimentation [1], mobility [69], transport by cilia carpets [16], vesicle dynamics in microfluidic flows [58]); elastostatics (microstructured periodic or random composites [29,36,61,64]); and the Helmholtz and Maxwell equations (phononic and photonic crystals, bandgap materials [42,65]). Application areas span all of the major elliptic PDEs, including the Laplace equation (thermal/electrical conductivity, electrostatics and magnetostatics of composites [12,27,35,38]); the Stokes equations (porous flow in periodic solids [18,26,50,75], sedimentation [1], mobility [69], transport by cilia carpets [16], vesicle dynamics in microfluidic flows [58]); elastostatics (microstructured periodic or random composites [29,36,61,64]); and the Helmholtz and Maxwell equations (phononic and photonic crystals, bandgap materials [42,65]).…”

“…Furthermore, to obtain accurate average material properties for random media or suspensions via the Monte Carlo method, thousands of simulation runs may be needed [38]. Integral equation methods [45,49,55,67] are natural, since the PDE has piecewise constant coefficients, and are very popular [1,12,26,27,36,50,64,69]. 2.8]), including particular solutions (starting in 1892 with Rayleigh's method for cylinders and spheres [70], and, more recently, in Stokes flow [71]), eigenfunction expansions [75], lattice Boltzmann and finite differencing [48], and finite element methods [29,61].…”

“…Particle-mesh Ewald (PME) methods. Recently this has been improved by Lindbo-Tornberg to achieve overall spectral accuracy in the Laplace [53] and Stokes [51] settings, with applications to 3D fluid suspensions [1]. Numerically, the spatial part is now local, while the spectral part may be applied by "smearing" onto a uniform grid, using a pair of FFTs, and (as in the nonuniform FFT [19]) correcting for the smearing.…”