(92b) Rapid and Accurate Methods for Modeling Hydrodynamic Forces in Brownian Dynamics Simulations

Complex fluids composed of colloids, surfactants or dissolved polymers, generically referred to as ``particles", are ubiquitous in nature and industry. The dynamics of these materials are controlled by solvent-mediated interactions, called hydrodynamic interactions (HIs), as well as the stochastic motion associated with Brownian diffusion. Constructing accurate computational models to simulate such systems is difficult because the structure and dynamics of colloidal materials are inherently multiscale. Practically, this means that very large numbers of discrete elements need to be simulated to capture the full range of dynamics in complex materials. Performing such large simulations is difficult because conventional simulation methods that incorporate HIs with Brownian motion exhibit a superlinear scaling of calculation time with the number of particles, which limits the size of simulations to at most a few thousand particles. For many self assembling materials -- colloidal, polymeric, micellar networks, for example -- millions of particles would be preferred. A further complication is that high accuracy hydrodynamic models, which are constructed mathematically via additional constraints on particle motion, require additional computational effort and further restrict accessible system sizes.

We will present a novel approach to performing Brownian dynamics simulations with hydrodynamic interactions and constraints that results in an algorithm with linear scaling in the number of particles modeled. This method, called the positively-split Ewald (PSE) algorithm, decomposes the operator describing the hydrodynamic interactions into a sparse local contribution, and a low-rank long wavelength contribution. Decomposing the interactions in this way allows the stochastic displacements to be efficiently sampled from the distribution defined by these two independent operators, leveraging their algebraic structures to accelerate the sampling to stochastic displacements. Hydrodynamic constraints, for instance those associated with lubrication forces, stresslets, and rigid particle assemblies, are readily incorporated in this algorithm and are used to build high accuracy simulations. The PSE approach enables a speedup of two orders of magnitudes over existing techniques. A high-performance implementation of this algorithm on graphics processing units (GPUs) enables simulations of millions of particles in reasonable wall clock times on consumer grade hardware. Applications of the method to dispersions of colloidal particles are discussed.