A fully Eulerian approach for fluid-structure interaction problems

Dipl.Math. Stefan Frei

Dec. 11, 2014, 3:30 p.m. S2 219

Fluid-structure interaction problems play an important role in a variety of ap-
plications in biology, medicine or the engineering sciences. In many cases, the Arbitrary
Lagrangian Eulerian (ALE) method provides an elegant and robust technique to deal with
moving interfaces and subdomains. The ALE method shows problems for large deformations
or movements of the structure, however, as the underlying grid may degenerate.
In a Fully Eulerian ansatz, in contrast, domains and interface move over a fixed back-
ground mesh [1]. This ansatz is thus capable of dealing with arbitrary movements up to
contact of e.g. a structure with a wall. The difficulty in the Eulerian ansatz lies in the
correct treatment of cells that are split by the interface. Applying standard discretization
methods leads to a reduced order of convergence and stability issues.
In this talk, we present modified discretization schemes in both space and time in order
to avoid these issues. The proposed finite element discretization in space corresponds to a
fitted finite element method that uses a fixed patch mesh [2]. Instead of moving mesh nodes,
we resolve the interface locally by an adapted parametric approach. For time discretization,
we use a modified time-stepping scheme that is based on the fixed-mesh ALE method [3].
We show second-order convergence for both discretization in space and time and give a
bound on the condition of the system matrix. Finally, we illustrate the capability of our
approach by means of benchmark problems and prototypical applications [4].