Anna Broms, KTH Title: An accurate and efficient method of fundamental solutions for Stokesian particulate flows

We develop simulation techniques for systems of nano- to micro-scaled rigid particles immersed in a viscous fluid, ubiquitous in nature and industry. With negligible fluid inertia, the set of PDEs known as the Stokes equations can be used to model the hydrodynamics at a snapshot in time.  Hydrodynamic interactions are challenging: they are simultaneously long-ranged and expensive to resolve both in time and space for closely interacting particles. The latter is caused by near-singular lubrication forces resulting from bodies in relative motion.  With the aim of controlling the accuracy for dense suspensions with a computationally cheap method, we present a new technique that combines the method of fundamental solutions (MFS) with the method of images. For rigid spheres, we propose to represent the flow
using Stokeslet proxy sources on interior spheres, augmented by lines of image sources adapted to each near-contact to resolve lubrication. Source strengths are found by a least-squares solve at contact-adapted boundary collocation nodes.  A one-body preconditioning strategy allows acceleration with the fast multipole method, hence close to linear scaling in the number of particles. For large suspensions with moderate lubrication forces, MFS sources on inner proxy-surfaces give accuracy on par with a well-resolved boundary integral formulation, but at a significantly reduced computational cost. For instance, the solution for a suspension of 10000 nearby ellipsoids with 26 million degrees of freedom is determined to five digits on a single workstation in less than two hours.

We will also discuss some challenges in simulating particle dynamics, which can be either stochastic (Brownian) or deterministic, depending on the particle size. Incorporating Brownian fluctuations introduces further computational complexities, as the correlation of the fluctuating forces on the particles at each time step depends on the configuration of the entire system. Calculating and applying these forces therefore quickly becomes resource-intensive. Furthermore, to draw statistically meaningful conclusions, a large number of realizations of the particle system is required with associated repeated solutions to the Stokes equations.

Joint work with Alex H. Barnett and Anna-Karin Tornberg.