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In this article we investigate computational methods for the control and stabilization of systems modeled by parabolic equations of the advection-reaction-difTusion type and by the Navier-Stokes equations for incompressible viscous fluids. For the first class of control problems wc shall discuss open loop control methods and closed loop methods a la Riccati. Concerning the second class of problems we shall consider the Dirichlet boundary control of low around cylinders and attempt to reduce the viscous drag. The results of numerical experiments will validate the computational methods described in this article.

CFD played a critical role as design tool in the 1970-80s in French laboratories and aeronautical industries with the rapid progress of computing facilities. Three short contributions by senior scientists and technologists describe how CFD developed rapidly in the French scientific and industrial community and was recognized as a mature tool in aerospace engineering for the aerodynamic design of civil and military aircraft and space vehicles like the European HERMES Space plane.

In this paper we address the problem of solving the compressible Navier Stokes equations. The development of space programs in Europe and the U.S. has generated renewed interest in aerodynamic ~ aerothermal design. Significant progress has been made recently in the Finite Element solutions of the Euler equations or their coupling with species convection equations for reactive flows and the difficulties associated mainly accurate and stable shock capturing capabilities are well known. When solving the Navier Stokes equations, additionnal features have to he computed: a precise representation of the houndary layer yielding pressure, skin friction and heat transfer coefficients is required along with recirculation, vortex, wake and shock interaction structures. This is critical for a number of aerospace applications dealing with strongly separated flows, including the simulation of flight at high angle of attack or the evaluation of the efficiency of body flaps. We will outline two Finite Element methodologies coupled with mesh adaptation that are being developed to accurately capture those features. One is based on a Galerkin Least-square formulation which brings good stability properties and makes this approach especially attractive for high Mach number calculations (space applications). The other one relies on accurate centered space discretbation and makes use of compatible finite element approximation and is advocated for aerodynamic shape design at subsonic and transonic regimes(aircraft application). For both methods, the grid quality control is still a major open problem even in the context of triangular Finite Element unstructured meshes; in order to capture the essential features of the flow, the mesh has to take into account not only the geometry hut also he fine enough in those regions where the solution needs accuracy. The mesh adaption used in this paper relies on refinement and node movement procedures provided by physical or geometrical criteria. Numerical flow simulation around reference simplified 2D geometries and 3D industrial computations originating from aircraft aerospace applications are presented for the development and validation of the above methodologies.

Solutions of elliptic boundary value problems are obtained with variational formulations by minimization of functionals. Conventional tools of the finite element strategy are emphasized; linear or quadratic approximations on triangular elements, curved iso-parametric elements, Ni shape funtions, Li homogeneous co-ordinates. High speed of computation of the final assembled matrices require the use of the Formal–FORMAC calculus. Illusrations of the finite element method are chosen in the class of the 2d–3d compressible subsonic irrotational flows in a nozzle and around one or several lifting bodies of arbitary shape.

chapitre 7; Level set methods have recently gained much popularity to capture discontinuities, including their possible propagation. In this contribution we present a finite element approach for solving the governing equations of level set methods. After a review of the governing equations, the initialisation of the level sets, the discretisation on a finite domain and the stabilisation of the resulting finite element method will be discussed. Special attention will be given to the proper treatment of the internal boundary condition, which is achieved by exploiting the partition-of-unity property of finite element shape functions.