Metric-Based HP-Adaptive Discontinuous Galerkin Simulations on Quasi-Structured Quad-Dominant Meshes

High-order discontinuous Galerkin (DG) methods, due to their element-wise approximation of the solution using polynomials, are suitable for hp-adaptation, which can result in an efficient distribution of degrees of freedom (DoF) throughout the computational domain. On adapted unstructured meshes, it is desirable to have "structured" like mesh elements (triangles or quads) aligned with the anisotropic flow features like boundary layer, or shocks, because of their favorable numerical properties. Thus, the current work focuses on metric-based hp-adaptation on quasi-structured quad-dominant meshes for DG methods. An optimum metric field that minimizes the Lq norm of the high-order interpolation error estimate of a given solution variable and a polynomial distribution that minimizes the local (DoF) density are used for hp-adaptation. From the metric field, a quasi-structured mesh is obtained using the metric-orthogonal approach, which uses the information from the metric eigenvectors, resulting in right triangles. These right triangles are then combined in an optimal manner to get the quasi-structured quad-dominant mesh. The optimum polynomial distribution evaluated on the original mesh is interpolated to the adapted quad-dominant mesh. Three test cases are considered: an analytical metric function corresponding to a quarter-circle-shaped anisotropic feature, a two-dimensional scalar advection-diffusion equation with a sharp boundary layer, and laminar flow over a two-dimensional cylinder. For the analytical metric function, the developed mesh generator is able to produce a mesh having aligned quads with the curved anisotropic feature. For the scalar case, we get optimal slopes for error convergence on both h- and hp-adapted quasi-structured quad-dominant meshes, and hp-meshes give much better convergence than purely h-adapted meshes. For the laminar cylinder case, the hp-adaptation accurately captures the flow separation region with anisotropic quasi-structured quad elements.