Adjoint-based $hp$-adaptivity on anisotropic meshes for high-order compressible flow simulations

High-order numerical methods such as Discontinuous Galerkin, Spectral Difference, and Flux Reconstruction etc, which use polynomials that are local to each mesh element to represent the solution field, are becoming increasingly popular in solving convection-dominated flows. This is due to their potential in giving accurate results more efficiently than lower order methods such as the classical Finite Volume methods. In most engineering applications, we are more interested in some specific scalar quantities rather than the full flow details. In the case of aerodynamic flow simulations, these quantities can be lift or drag coefficient. To get accurate values for such target functional quantities, adjoint-based error estimators, along with a high-order solver, have been found to be quite useful. They can identify the mesh elements that contribute the most to the error, and adapting these elements should result in a more accurate target functional. To adapt a mesh element, one can either do mesh refinement (h-adaptation) or polynomial space enrichment (p-adaptation) or both (hp-adaptation). Of these, hp-adaptation offers the most efficient way for adaptation, since one can locally choose between mesh refinement or polynomial space enrichment based on what is more efficient in resolving the local solution features. We present efficient adjoint-based hp-adaptation methodologies on isotropic and anisotropic meshes for the recently developed high order Hybridized Discontinuous Galerkin scheme for (nonlinear) convection-diffusion problems, including the compressible Euler and Navier-Stokes equations. hp-adaptation on isotropic meshes is based on the spatial error distribution for a given target functional given by the adjoint error estimator and the solution regularity given by a regularity indicator. For anisotropic meshes, we extend the refinement strategy based on an interpolation error estimate, due to Dolejsi, by incorporating an adjoint-based error estimate. Using the two error estimates we determine the size and the shape of the triangular mesh elements on the desired mesh to be used for the subsequent adaptation steps. This is done using the concept of mesh-metric duality, where the metric tensors can encode information about mesh elements, which can be passed to a metric-conforming mesh generator to generate the required anisotropic mesh. The effectiveness of the adaptation methodology is demonstrated using numerical results: for a scalar convection-diffusion case with a strong boundary layer; inviscid subsonic, transonic and supersonic flows and viscous subsonic flow around a NACA0012 airfoil