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.

An optimization-based framework for anisotropic simplex mesh adaptation

“…The trend is more pronounced for a nonlinear problem, in which local problems require much fewer Newton iterations than the global problem. (For a typical two-dimensional Reynolds-averaged Navier-Stokes flow over an isolated airfoil, the adaptation cost constitutes less than 10% of the flow solve [27]. )…”

Solutions of High-order Methods for Three-dimensional Compressible Viscous Flows

“…[1][2][3][4][5][6][7] More and more physically realistic problems can be tackled through the use of such methods to obtain high accuracy solution, while reducing the need for extremely high-resolution meshes that are often required by lower-order methods. Moreover, a great deal of effort has been devoted to the versatility, robustness and efficiency of high-order flow solvers, including adaptive mesh refinement techniques, [8][9][10][11][12] solution limiting and shock capturing methods, 9,[13][14][15][16] hybrid methodologies and multigrid solution strategies. 2,[17][18][19] To this end, this paper continues on the development of high-order discretization methods, consisting of discontinuous Galerkin (DG) [1][2][3]6,18,20 and streamline/upwind Petrov-Galerkin (SUPG) [21][22][23][24] discretizations, to further expand the capability of high-order schemes in solving a wide range of viscous flow problems for complex geometries and to compare the accuracy of the high-order DG and SUPG methods.…”

ElVis: A System for the Accurate and Interactive Visualization of High-Order Finite Element Solutions

“…It also strives to increase the level of solution automation in modern CFD by taking the engineer "out of the loop" through estimation and control of errors in outputs of interest (e.g., lift, drag) [7,36]. Solution automation is accomplished through an iterative mesh optimization process, which is driven by error estimates based on the adjoints of outputs of interest [35].…”

“…The flow conditions are M ∞ = 0.3, Re c = 4000, and α = 12.5. The solution was obtained through ProjectX, using an output-adaptive automated solution strategy [35].…”

“…We demonstrate ElVis' utility by using it to visualize finite element simulations produced by ProjectX, which is a general-purpose PDE solver with an emphasis on aerospace applications [7,21,6,36,35]. Specifically, we will consider the visualizations necessary during the debugging and verification processes of model generation.…”