Browsing by Author "Förster, Christiane"

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Robust methods for fluid-structure interaction with stabilised finite elements
(2007) Förster, Christiane; Ramm, Ekkehard (Prof. Dr.-Ing. Dr.-Ing. E.h. Dr. h.c.)
Various multifield problems and among them fluid-structure interaction applications arise in nearly all fields of engineering. The present work contributes to the development of a stable and robust approach for the numerical simulation of fluid-structure interaction problems. In particular two-dimensional and three-dimensional elastic structues interacting with incompressible flow are considered. The structural field is governed by the nonlinear elastodynamic equations while the dynamics of the fluid field are described by the incompressible Navier-Stokes equations. Both fields are discretised by finite elements in space and finite difference methods in time. An iteratively staggered partitioned coupling procedure with relaxation is applied to obtain the overall coupled solution. This work focuses on methodological aspects and contributes to a deeper understanding of the theoretical foundations of the approach. This is necessary to ensure that the formulation is stable and offers reliable results for a wide range of parameters. In particular the flow solver formulated in an arbitrary Lagrangean Eulerian approach is considered. In addition to the classical conservation laws of mass, linear momentum and energy geometric conservation has to be considered. This is a consequence of the formulation of the flow equations with respect to a moving frame of reference. The relationship of these conservation laws and the stability of the numerical scheme is investigated and stability limits in terms of maximal time step sizes for different formulations are established. It is further shown how an unconditionally stable ALE formulation has to be constructed. Another key issue is the stabilised finite element method employed on the fluid domain. The derivation of the method from a virtual bubble approach is revisited while special attention is turned to the fact that the domain is moving. A version of the stabilisation is derived which is nearly unaffected by the motion of the frame of reference. Further the sensitivity of the stabilised formulation with respect to critical parameters such as very small time steps, steep gradients and distorted meshes is assessed. At least for higher order elements where full consistency of the formulation is assured very accurate results can be obtained on highly distorted meshes. As another main issue the coupling of fluid and structure within a partitioned scheme is considered. A first concern in this context is the exchange of proper coupling data at the interface which is crucial for the consistency of the overall scheme. Subsequently the so-called artificial added mass effect is analysed. This effect is responsible for an inherent instability of sequentially staggered coupling schemes applied to the coupling of lightweight structures and incompressible flow. It is essentially the influence of the incompressibilty which excludes the successful use of simple staggered schemes. The analysis derived in the course of this work reveals why the artificial added mass instability depends upon the mass ratio but further on the specific time discretisation used on the fluid and structural field. In particular it is shown why more accurate temporal discretisation results in an earlier onset of the instability. While the theoretical considerations are accompanied by small numerical examples highlighting particular aspects some larger applications of the method are finally presented.