Adaptive Aerodynamic Design Optimization for Navier-Stokes Using Shape Derivatives with Discontinuous Galerkin Methods

We state and analyze one-shot optimization methods in a function space setting for optimal control problems, for which the state equation is given in terms of a fixed-point equation. Further, we concentrate on the application of a design optimization problem incorporating the solution of the compressible Navier-Stokes equations using a discontinuous Galerkin method. For the given primal fixed-point solver an appropriate adjoint solver is constructed. For the following design update we compute the shape derivative analytically based on the weak formulation of the governing equations. The primal, adjoint and design updates are performed in a one-shot manner, i.e., the corresponding equations are not fully solved, instead only a few iteration steps are performed. Finally, we add an additional adaptive step. During the optimization routine we refine or coarsen the grid to obtain a better accuracy.