Efficient numerical solution for fluid-structure interaction problems

Fluid-structure interaction (FSI) problems present a difficult challenge when modelling. They are time-dependent and highly non-linear, both in terms of the fluid and solid models. This project aims to explore numerical techniques to solve FSI problems within an arbitrary Lagrangian Eulerian (ALE) framework using the finite element method. By using a conformal meshing approach allows for the deformation of a single mesh throughout the domain, evolving with the fluid-structure interface.
The importance of using an appropriate pressure space approximation to accurately capture the discontinuous pressure at the fluid-structure interface will be demonstrated and described. The stable Taylor-Hood P2/P1 element pair, widely used for purely fluid cases, will be compared with the lower order but discontinuous pressure pairing P2/P0. These are compared to the P2/(P1+P0) pair, which enriches the Taylor-Hood discretisation with a piecewise discontinuous constant pressure, on each element. These finite element approximations are tested on two separate two- and three-dimensional test cases, where the extension of the three-dimensional approximation requires additional consideration to be stable.
For large three-dimensional cases, finding solutions of the resulting linear equation systems by direct solvers is severely limited by memory requirements, and thus efficient iterative methods are necessary. Efficient algorithms that employ block-preconditioned Krylov-subspace iterative methods are well known for the fluid flow problem. We consider the extension of these techniques to the discrete FSI problem, where the monolithic linear system contains contributions from both the solid and fluid models.
A block preconditioner, which uses algebraic multigrid approximation to diagonal velocity components, combined with an approximation to the action of the inverse of the Schur complement gives problem size-independent iteration counts when applied with GMRES.