Rapid sampling of stochastic displacements in Brownian dynamics simulations with stresslet constraints

Abstract

Brownian Dynamics simulations are an important tool for modeling the dynamics of soft matter. However, accurate and rapid computations of the hydrodynamic interactions between suspended, microscopic components in a soft material are a significant computational challenge. Here, we present a new method for Brownian dynamics simulations of suspended colloidal scale particles such as colloids, polymers, surfactants, and proteins subject to a particular and important class of hydrodynamic constraints. The total computational cost of the algorithm is practically linear with the number of particles modeled and can be further optimized when the characteristic mass fractal dimension of the suspended particles is known. Specifically, we consider the so-called "stresslet" constraint for which suspended particles resist local deformation. This acts to produce a symmetric force dipole in the fluid and imparts rigidity to the particles. The presented method is an extension of the recently reported positively split formulation for Ewald summation of the Rotne-Prager-Yamakawa mobility tensor to higher order terms in the hydrodynamic scattering series accounting for force dipoles [A. M. Fiore et al., J. Chem. Phys. 146(12), 124116 (2017)]. The hydrodynamic mobility tensor, which is proportional to the covariance of particle Brownian displacements, is constructed as an Ewald sum in a novel way which guarantees that the real-space and wave-space contributions to the sum are independently symmetric and positive-definite for all possible particle configurations. This property of the Ewald sum is leveraged to rapidly sample the Brownian displacements from a superposition of statistically independent processes with the wave-space and real-space contributions as respective covariances. The cost of computing the Brownian displacements in this way is comparable to the cost of computing the deterministic displacements. The addition of a stresslet constraint to the over-damped particle equations of motion leads to a stochastic differential algebraic equation (SDAE) of index 1, which is integrated forward in time using a mid-point integration scheme that implicitly produces stochastic displacements consistent with the fluctuation-dissipation theorem for the constrained system. Calculations for hard sphere dispersions are illustrated and used to explore the performance of the algorithm. An open source, high-performance implementation on graphics processing units capable of dynamic simulations of millions of particles and integrated with the software package HOOMD-blue is used for benchmarking and made freely available in the supplementary material.

Hydrodynamics control shear-induced pattern formation in attractive suspensions

“…Simulations. The positively split Ewald (PSE) (47,48) is used for discrete element simulations of spherical particles with low Reynolds number hydrodynamic interactions accounted for by the RPY approximation. Particles in suspension have a mean radius a and an imposed polydispersity of σ = 5%.…”

Fast Stokesian dynamics

“…work by Rotne & Prager (1969), Ladd (1990), Cichocki et al. (1994), Ichiki (2002), Kim & Karrila (2005) and Fiore & Swan (2018), among others). Although we will discuss the multipole expansion specifically, the framework developed herein admits use of other methods for computing the hydrodynamic mobility, e.g.…”

“…2017). We extended this method to the dipole level of the multipole expansion (Fiore & Swan 2018), which is equivalent to SD without lubrication. In this work, we develop a method incorporating the PSE sampling technique to achieve an algorithm for SD with Brownian fluctuations.…”

“…This extension includes reformulating Stokesian dynamics itself as a saddle point problem, for which a highly effective constraint preconditioner is developed and applied. In this formulation, the Brownian stress is decomposed into a far-field Brownian slip velocity, computed using the PSE approach (Fiore & Swan 2018), and a near-field Brownian force, computed using a fast preconditioned Krylov subspace approximation (Chow & Saad 2014). We show that this approach maintains the previously reported scaling, and discuss algorithmic changes and optimization for graphics processing unit (GPU) implementation.…”

Leveraging collective effects in externally driven colloidal suspensions: experiments and simulations

“…The corresponding cost depends on the fluid solver and on the discretization of the particles. FH has been successfully implemented into various numerical methods for particulate flows with linear scaling (O(N )) such as the Immersed Boundary Method [145,146,147,148,149], Distributed Lagrange Multipliers [150], Fluid Particle Dynamics [151], Finite Element Method [152,153], Force Coupling Method [154,155], Boundary Element Method [156] or Particle Mesh Ewald Method [157,158].…”