Resolving collisions in Stokes suspensions with an efficient and stable potential-free constrained optimization algorithm

A common challenge in simulating dense suspension of rigid particles in Stokes flow is the numerical instability that arises due to particle collisions. To overcome this problem, often a strong repulsive potential between particles is prescribed. This in turn leads to numerical stiffness and dramatic reduction in stable time-step sizes. In this work, we eliminate such stiffness by introducing contact constraints explicitly and solving the hydrodynamic equations in tandem with a linear complementarity problem with inequality constraints. The Newton's third law of the collision force is explicitly guaranteed to allow consistent calculation of collision stresses. Efficient parallelization for shared-memory and distributed-memory architectures is also implemented. This method can be coupled to any Stokes hydrodynamics solver for particles with various shapes and allows us to simulate $10^4 \sim 10^7$ spheres on a laptop, depending on the cost of the Stokes hydrodynamics solver. We demonstrate its performance on a range of applications from active matter to multi-physics problems.