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Navier-Stokes Shape Design Using a Level Set Method

Optimal aerodynamic shape design aims at finding the minimum of a functional by controlling the PDE modeling the flow using surface deformation techniques. As a solution to the enormous computational resources required for classical shape optimization of functionals of aerodynamic interest, probably the best strategy is to use in a systematic way, methods inspired in control theory. One of the key ingredients relies on the use of the adjoint system techniques to simplify the computation of gradients. In the present paper we will restrict our attention to optimal shape design in systems governed by the Navier-Stokes equations. We first review some standard facts on control theory applied to optimal shape design, and recall the 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. Some relevant numerical issues will also be discussed. Finally, recent computational results applying a Level Set methodology are reviewed.