Aerodynamic Shape Optimization with Hybridized Discontinuous Galerkin Schemes

We present a discrete adjoint approach to aerodynamic shape optimization (ASO) based on a hybridized discontinuous Galerkin (HDG) discretization. Our implementation is designed to tie in as seamlessly as possible into a solver architecture written for general balance laws, thus adding design capability to a tool with a wide range of applicability. Design variables are introduced on designated surfaces using the knots of a spline-based geometry representation, while gradients are computed from the adjoint solution using a difference approximation of residual perturbations. A suitable optimization algorithm, such as an in-house steepest descent or the Preconditioned Sequential Quadratic Programming (PSQP) approach from the pyOpt framework, is then employed to find an improved geometry. The resulting ASO module is currently set up for 2D test cases governed by balance laws, including linear scalar equations or nonlinear systems of equations. We present verification of the implementation, including drag or heat flux minimization in compressible flows, as well as inverse design.