NUMERICAL STABILIZATION OF A FLUID-STRUCTURE INTERACTION SYSTEM

Michel Fournié (1) , Moctar Ndiaye , Jean-Pierre Raymond

1 IMT - Institut de Mathématiques de Toulouse UMR5219

We study the numerical stabilization of a fluid-structure interaction system in a wind tunnel, around an unstable stationary solution. The goal is to find a feedback control, acting only in the structure equation, able to stabilize locally the full coupled nonlinear system. The Finite Element Approximation of this coupled system leads to a Differential Algebraic Equation (D.A.E.) for the fluid velocity, the structure displacement and its velocity, and a Lagrange multiplier taking into account together the incompressibility condition and the equality of the fluid velocity and the displacement velocity of the structure at the fluid-structure interface. We determine a feedback control law based on a spectral decomposition of the D.A.E. But the D.A.E. is not standard because the operator acting on the Lagrange multiplier in the differential equation is not the transpose of the operator involved in the algebraic constraints. We overcome these difficulties by proving new relationships between eigenvalue problems involving Lagrange multipliers and those without Lagrange multipliers. In numerical simulations, we show how the calculation of the degree of stabilizability of different invariant subspaces (generalized eigenspaces) may be helpful to determine an efficient control strategy. In particular, the determined feedback law is able to reject a perturbation (leading to a complete stabilization of the nonlinear fluid-structure system) whose magnitude is of order 15% of the inflow velocity.

Mots clés: Fluid-structure interaction; feedback control; stabilization; Navier-Stokes equations; beam equations; Finite element approximation

Domaines: Optimisation et contrôle [math.OC]

Michel Fournié, Moctar Ndiaye, Jean-Pierre Raymond. NUMERICAL STABILIZATION OF A FLUID-STRUCTURE INTERACTION SYSTEM. 2018. ⟨hal-01743781⟩