An Optimization Framework for Anisotropic Simplex Mesh Adaptation: Application to Aerodynamic Flows

Abstract

We apply an optimization-based framework for anisotropic simplex mesh adaptation to high-order discontinuous Galerkin discretizations of two-dimensional, steady-state aerodynamic flows. The framework iterates toward a mesh that minimizes the output error for a given number of degrees of freedom by considering a continuous optimization problem of the Riemannian metric field. The adaptation procedure consists of three key steps: sampling of the anisotropic error behavior using element-wise local solves; synthesis of the local errors to construct a surrogate error model in the metric space; and optimization of the surrogate model to drive the mesh toward optimality. The anisotropic adaptation decisions are entirely driven by the behavior of the a posteriori error estimate without making a priori assumptions about the solution behavior. As a result, the method handles any discretization order, naturally incorporates both the primal and adjoint solution behaviors, and robustly treats irregular features. The numerical results demonstrate that the proposed method is at least as competitive as the previous method that relies on a priori assumption of the solution behavior, and, in many cases, outperforms the previous method by over an order of magnitude in terms of the output accuracy for a given number of degrees of freedom.