The subject of the lecture is the finite element solution of fluid-structure interaction problems. Our main attention will be paid to the simulation of flow induced airfoil vibrations. A solid airfoil with two degrees of freedom, which can rotate around the elastic axis and oscillate in the vertical direction, is considered. The numerical simulation consists of the finite element solution of the Navier-Stokes equations coupled with the system of ordinary differential equations describing the airfoil motion. The time dependent computational domain and a moving grid are taken into account with the aid of the Arbitrary Lagrangian- Eulerian (ALE) formulation of the Navier-Stokes equations. The time discretization is carried out by the backward difference formula. In space the conforming stabilized finite element method is used. As a result a sufficiently accurate and robust method is developed, which is applied to the case of flow induced airfoil vibrations with large amplitudes after loosing the aeroelastic stability. The developed method was tested on several problems analyzed also experimentally in a wind tunnel.
A class of cell-centered Finite Volume (F.V.) Schemes can be efficiently analyzed in the framework of Mixed Finite Element (M.F.E.) Theory. For the model problem, the F.V. equations in the scalar field can be obtained from the dual mixed system using Raviart-Thomas element and a well-chosen numerical integration formula ("mass lumping").
This techniques is a good way for exhibiting a priori estimates in natural norms for the F.V. Schemes, the proof of which using only classical results of M.F.E. theory. It has been possible for 2-d triangular cells (but still not for n-d simplicial ones!), and for n-d rectangular cells.
This techniques allows also to construct a posteriori estimators for F.V. Schemes, which is important for F.V. codes. The method is based on a generalization to the case with numerical integration of Arnold-Brezzi results which associate to the mixed system a resolving nonconforming F.E. problem. We emphasize that there are significant differences between triangles and rectangles in proving the results.
Most of the present results have been obtained in collaboration with A.Agouzal, J.Baranger, J.Olaiz and F.Oudin.
