# zbMATH — the first resource for mathematics

Numerical simulation of laminar incompressible fluid-structure interaction for elastic material with point constraints. (English) Zbl 1374.74035
Rannacher, Rolf (ed.) et al., Advances in mathematical fluid mechanics. Dedicated to Giovanni Paolo Galdi on the occasion of his 60th birthday. Selected papers of the international conference on mathematical fluid mechanics, Estoril, Portugal, May 21–25, 2007. Berlin: Springer (ISBN 978-3-642-04067-2/hbk; 978-3-642-04068-9/ebook). 451-472 (2010).
Summary: We present numerical techniques for solving the problem of fluid structure interaction with a compressible elastic material in a laminar incompressible viscous flow via fully coupled monolithic Arbitrary Lagrangian-Eulerian (ALE) formulation. The mathematical description and the numerical schemes are designed in such a way that more complicated constitutive relations can be easily incorporated. The whole domain of interest is treated as one continuum and we utilize the well known $$Q_2P_1$$ finite element pair for discretization in space to gain high accuracy. We perform numerical comparisons for different time stepping schemes, including variants of the Fractional-Step-$$\theta$$-scheme, Backward Euler and Crank-Nicholson scheme for both solid and fluid parts. The resulting nonlinear discretized algebraic system is solved by a quasi-Newton method which approximates the Jacobian matrices by the divided differences approach and the resulting linear systems are solved by a geometric multigrid approach. In the numerical examples, a cylinder with attached flexible beam is allowed to freely rotate around its axis which requires a special numerical treatment. By identifying the center of the cylinder with one grid point of the computational mesh we prescribe a Dirichlet type boundary condition for the velocity and the displacement of the structure at this point, which allows free rotation around this point. We present numerical studies for different problem parameters on various mesh types and compare the results with experimental values from a corresponding benchmarking experiment.
For the entire collection see [Zbl 1185.76020].

##### MSC:
 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 76D05 Navier-Stokes equations for incompressible viscous fluids 76M20 Finite difference methods applied to problems in fluid mechanics 76M10 Finite element methods applied to problems in fluid mechanics 74S05 Finite element methods applied to problems in solid mechanics
SPLIB
Full Text: