Fast algorithms for large dense matrices with applications to biofluids

Abstract

Numerical simulation of biofluids entails solving equations of fluid-structure interactions. At zero Reynolds number, solvers such as the Method of Regularized Stokeslets (MRS) give rise to large and dense matrices in practical applications where the number of structures immersed in the fluid is large. Building on previous work for an unbounded fluid domain, we first extend the Kernel-Independent Fast Multipole Method (KIFMM) for MRS to compute the matrix-vector products for the fluid flow induced by point forces above a stationary wall. In this case, the use of a regularized image system introduces additional terms to the solution which cause the matrix-vector multiplication to be quite challenging. In addition, we study the case where a linear system needs to be solved for the unknown forces that structures with known velocities exert on the fluid. Our main contribution is proposing several preconditioning techniques for the matrices associated with a few variants of MRS, including the case where a force-free, torque-free condition is imposed. They take advantage of the data-sparsity of FMM matrices as well as properties of Krylov subspaces. Our approach is memory efficient, capable of handling non-uniformly distributed structures and applicable to all FMM matrices. It enables efficient computation of the flow field surrounding a large group of dynamic micro-structures; in particular, we study the effects of fluid mixing caused by the periodic beating of a dense carpet of lung cilia.