Shape sensitivity analysis for the compressible Navier–Stokes equations via discontinuous Galerkin methods

This paper describes the formulation of adjoint-based sensitivity analysis and optimization techniques for high-order discontinuous Galerkin discretizations applied to viscous compressible flow. The flow is modeled by the compressible Navier–Stokes equations and the discretization of the viscous flux terms is based on an explicit symmetric interior penalty method. The discrete adjoint equation arising from the sensitivity derivative calculation is formulated consistently with the analysis problem, including the treatment of boundary conditions. In this regard, the influence on the sensitivity derivatives resulting from the deformation of curved boundary elements must be taken into account. Several numerical examples are used to examine the order of accuracy (up to p=4p=4) achieved by the current DG discretizations, to verify the derived adjoint sensitivity formulations, and to demonstrate the effectiveness of the discrete adjoint algorithm in steady and unsteady design optimization for both two- and three-dimensional viscous design problems.

► Adjoint sensitivity formulation is derived for discontinuous Galerkin (DG) methods. ► Shape optimization problems are studied for steady and unsteady viscous flows. ► Adjoint-based sensitivity derivatives agree well with the finite-difference gradients. ► Optimal error convergence rates are obtained for various orders of DG schemes.