2-D euler shape design on nonregular flows using adjoint rankine-hugoniot relations
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Optimal aerodynamic shape design aims to find the minimum of a functional that describes an aerodynamic property, by controlling the partial differential equation modeling the dynamics of the flow that surrounds an aircraft, by 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 apply methods inspired in control theory. One of the key ingredients relies on the usage of the adjoint methodology to simplify the computation of gradients. In this paper we restrict our attention to optimal shape design in two-dimensional systems governed by the steady Euler equations for flows whose steady-state solutions present discontinuities in the flow variables (an isolated shock wave). We first review some facts on control theory applied to optimal shape design and recall the 2-D Euler equations (including the Rankine-Hugoniot conditions). We then study the adjoint formulation, providing a detailed exposition of how the derivatives of functionals of aeronautical interest may be obtained when a discontinuity appears. Further on, adjoint equations will be discretized and analyzed and some novel numerical experiments with adjoint Rankine-Hugoniot relations will be shown. Finally, we expose some conclusions about the viability of a rigorous approach to the continuous Euler adjoint system with discontinuities in the flow variables.