Control of Euler and Navier-Stokes equations: Applications to optimal shape design in aeronautics

Recently, shape design has been acquiring increasing relevance within the aeronautical community, both in the industrial sector as well as in research centers. This increasing interest stems from the speed and precision of computations which are being achieved with the use of CFD (Computational Fluid Dynamics) techniques.Optimal shape design aims at finding the minimum of a functional by controlling the PDE modelling the flow using surface (domain boundaries) deformation techniques. As a solution to the enormous computational resources required for classical shape optimization of functionals of aerodynamic interest, the best strategy is to use in a systematic way methods inspired in control theory. To do this one assumes that a given aerodynamic surface (typically a wing) is an element that produces lift or drag by controlling or modifying the flow. One of the key ingredients in the application of control theoretical methods relies on the use of the adjoint system techniques to simplify the computation of gradients. Some of the groundbreaking works in this field are due to J.-L. Lions [1], followed by the developments of O. Pironneau [2], who was a pioneer in the applications to CFD. A. Jameson [3] was the first to apply these techniques to the Euler and Navier-Stokes equations in the field of aeronautical applications. In the present paper we will restrict our attention to optimal shape design in systems governed by the Euler or Navier-Stokes equations. We first review some standard facts on control theory applied to optimal shape design, and recall the Euler and Navier-Stokes equations in aerodynamical problems. We then study the adjoint formulation, providing a detailed exposition of how the derivatives of functionals may be obtained. Finally, the application of the Level Set methodology to optimal shape design is reviewed.