Adaptive finite element approximation of fluid-structure interaction based on ALE and Eulerian variational formulations



FLUSTRUK Benchmark


ALE framework, snapshot (click for movie, 22MB)


Eulerian framework, snapshot (click for movie, 34MB)
Computational fluid dynamics and computational structure mechanics are two major areas of numerical simulation of physical systems. With the introduction of high performance computing it has become possible to tackle systems with a coupling of fluid and structure dynamics. General examples of such fluid-structure interaction (FSI) problems are flow transporting elastic particles (particulate flow), flow around elastic structures (airplanes, submarines) and flow in elastic structures (haemodynamics, transport of fluids in closed containers). In all these settings the dilemma in modeling the coupled dynamics is that the fluid model is normally based on an Eulerian perspective in contrast to the usual Lagrangian approach for the solid model. This makes the setup of a common variational description difficult. However, such a variational formulation of FSI is needed as the basis of a consistent approach to residual-based a posteriori error estimation and mesh adaptation as well as to the solution of optimal control problems by the Euler-Lagrange method. This is the subject of the this project.

The work and results of this project were financed by the Deutsche Forschungsgemeinschaft (DFG) through the Project P2 of the joint Research Unit 493 'Fluid-Structure Interaction Modelling, Simulation, Optimisation'. One of the goals is the developement of a general FSI benchmark 'FLUSTRUK-A' (PDF) and the quantitative comparison of simulation results. The unit consists of nine groups, which have implemented a wide array of approaches to simulating FSI problems. The focus of the simulations always being the benchmark. The results of the first phase of the joint research group are availible in Fluid-Structure Interaction Modelling, Simulation, Optimisation.

Combining the Eulerian and the Lagrangian setting for describing FSI involves conceptional difficulties. On the one hand the fluid domain itself is time-dependent and depends on the deformation of the structure domain. On the other hand, for the structure the fluid boundary values (velocity and the normal stress) are needed. In both cases values from the one problem are used for the other, which is costly and can lead to a drastic loss of accuracy. A common approach to dealing with this problem is to separate the two models, solve each separately, and so converge iteratively to a solution, which satisfies both together with the interface conditions Solving the separated problems serially multiple times is referred to as a 'partitioned approach'.

A basic partitioned approach does not contain a variational equation for the fluid-structure interface. To achieve this, usually an auxiliary unknown coordinate transformation function is introduced for the fluid domain. With its help the fluid problem is rewritten as one on the transformed domain, which is fixed in time. Then, all computations are done on the fixed reference domain and as part of the computation the auxiliary transformation function has to be determined at each time step. This fully coupled approach to solving the whole problem on an (optionally arbitrary) reference domain is generally referred to as the 'arbitrary Lagrangian-Eulerian' (ALE) method.

In this project, we follow an additional (to the authors knowledge new) way of posing the fluid as well as the structure problem in a fully Eulerian framework. In the Eulerian setting a phase variable is employed on the fixed mesh to distinguish between the different phases, liquid and solid. This approach to identifying the fluid-structure interface is generally referred to as 'interface capturing', a method commonly used in the simulation of multiphase flows. Examples for the use of such a phase variable are the Volume of Fluid (VoF) method and the Level Set (LS) method. In the classical LS approach the distance function has to continually be reinitialized, due to the smearing effect by the convection velocity in the fluid domain. This makes the use of the LS method delicate for modeling FSI problems particularly in the presence of cornered structures. To cope with this difficulty, we introduce a variant, the 'Initial Position set method' (IP set), of the LS method that makes reinitialization unnecessary and which can easily cope with cornered structures.

Based on the Eulerian variational formulation of the FSI system, we use the 'dual weighted residual' (DWR) method to derive 'goal-oriented' a posteriori error estimates. The evaluation of these error estimates requires the approximate solution of a linear dual variational problem. The resulting a posteriori error indicators are then used for automatic local mesh adaption.

The methods have been (successfully) tested for both the ALE as well as the Eulerian framework for a wide range of tests, stationary and instationary.

driven cavity with elastic base problem


ALE framework, adaptively refined mesh
Eulerian framework, adaptively refined mesh