Preconditioning for near-contacts in large 2D Stokes flows: a locally compressed method of fundamental solutions
This work addresses the computational bottlenecks of poor iterative solver convergence and high discretization cost in simulating dense particle suspensions, which is important for applications in fluid dynamics and soft matter.
The paper introduces a two-body preconditioning strategy for the method of fundamental solutions to efficiently solve 2D Stokes flow problems involving large dense collections of nearly-touching rigid particles. The method achieves rapid GMRES convergence, solving a mobility problem with 10,000 disks at area fraction 0.65 and minimum separation 10^{-3} in 47 iterations with five-digit accuracy.
We tackle two key difficulties in the simulation of the viscous hydrodynamics of a large dense collection of rigid particles: (i) the poor convergence rate of an iterative solution of the discretized linear system as particle gaps shrink, and (ii) the large number of unknowns needed to accurately discretize the resulting lubrication-driven flows. Our focus is the 2D Stokes resistance and mobility boundary value problems for nearly-touching disks. To address both challenges, we introduce a general two-body preconditioning strategy, and implement it with the method of fundamental solutions. For each close particle pair, the hard-to-resolve interaction is represented in a basis precomputed by solving a local boundary value problem on a fine grid. In an iterative solve, the resulting flow field corrects that obtained from a coarse representation of all particles. The local fine-grid correction can even be compressed so that all particles except the pair itself are affected by an equivalent set of coarse sources. Numerical experiments demonstrate rapid GMRES convergence in challenging multi-particle settings, with iteration counts remaining low even in densely packed suspensions. For example, the mobility problem is solved for a random close packing with area fraction $φ= 0.65$, $P = 10000$ monodisperse disks, and minimum separation $10^{-3}$, in just 47 GMRES iterations, achieving five digits of accuracy with 72 vector unknowns per body.