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Linear stability analysis of strongly coupled fluid-structure problems with the arbitrary-Lagrangian-Eulerian method. (English) Zbl 1441.74067

Summary: The stability analysis of elastic structures strongly coupled to incompressible viscous flows is investigated in this paper, based on a linearization of the governing equations formulated with the Arbitrary-Lagrangian-Eulerian method. The exact linearized formulation, previously derived to solve the unsteady non-linear equations with implicit temporal schemes, is used here to determine the physical linear stability of steady states. Once discretized with a standard finite-element method based on Lagrange elements, the leading eigenvalues/eigenmodes of the linearized operator are computed for three configurations representative for classical fluid-structure interaction instabilities: the vortex-induced vibrations of an elastic plate clamped to the rear of a rigid cylinder, the flutter instability of a flag immersed in a channel flow and the vortex shedding behind a three-dimensional plate bent by the steady flow. The results are in good agreement with instability thresholds reported in the literature and obtained with time-marching simulations, at a much lower computational cost. To further decrease this computational cost, the equations governing the solid perturbations are projected onto a reduced basis of free-vibration modes. This projection allows to eliminate the extension perturbation, a non-physical variable introduced in the ALE formalism to propagate the infinitesimal displacement of the fluid – solid interface into the fluid domain.

MSC:

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
74H55 Stability of dynamical problems in solid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
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