Optimal Parallel Preconditioners for Three-Dimensional Particulate Flows

Simulation of incompressible flows with hundreds of rigid particles in three-dimensional domains require efficient iterative algorithms and parallel preconditioners. In addition to the difficulty associated with solving Navier-Stokes equations for incompressible fluids, one must simulate the motion of rigid bodies under the action of hydrodynamic forces and gravity. A distributed Lagrange multiplier based fictitious domain approach is used along with operator splitting that decouples each time step into three subproblems – a time-dependent Stokes problem, a linearized advection-diffusion problem, and a linearly constrained quadratic minimization problem that enforces rigid body motion for the particles. The linear systems arising in each subproblem are solved using the preconditioned conjugate gradients method. In this paper, we present optimal preconditioners for these linear systems that assure convergence in a fixed number of steps. Furthermore, these preconditioners do not need to be constructed, stored, or factored, and can be applied in a matrix-free form. While optimality of the Stokes preconditioner as well as the linearized advection-diffusion preconditioner is derived from known results, we believe that an optimal matrix-free preconditioner for the linearly constrained quadratic minimization problem of the type presented here has been developed for the first time. The resulting algorithm is highly parallel and scalable, and has shown impressive speed improvement on parallel computers.