Solving Monolithic Fluid-Structure Interaction Problems in Arbitrary Lagrangian Eulerian Coordinates with the deal.II Library

Abstract

We describe a setting of a nonlinear fluid-structure interaction problem and the corresponding solution process in the finite element software package deal.II. The fluid equations are transformed via the ALE mapping (Arbitrary Lagrangian Eulerian framework) to a reference configuration and these are coupled with the structure equations by a monolithic solution algorithm. To construct the ALE mapping, we use a biharmonic equation. Finite differences are used for temporal discretization. The derivation is realized in a general manner that serves for different time stepping schemes. Spatial discretization is based on a Galerkin finite element scheme. The nonlinear system is solved by a Newton method. Using this approach, the Jacobian matrix is constructed by exact computation of the directional derivatives.
The implementation using the software library package deal.II serves for the computation of different fluid-structure configurations. Specifically, our geometry data are taken from the fluid-structure benchmark configuration that was proposed in 2006 in the DFG project Fluid-Structure Interaction I: Modelling, Simulation, Optimisation. Our results show that this implementation using deal.II is able to produce comparable findings.