The moving discontinuous Galerkin method with interface condition enforcement (MDG-ICE) is a high-order, r-adaptive method that treats the grid as a variable and weakly enforces the conservation law, constitutive law, and corresponding interface conditions in order to implicitly fit high-gradient flow features. In this paper, we develop an optimization solver based on the Levenberg-Marquardt algorithm that features an anisotropic, locally adaptive penalty method to enhance robustness and prevent cell degeneration in the computation of hypersonic, viscous flows. Specifically, we incorporate an anisotropic grid regularization based on the mesh-implied metric that inhibits grid motion in directions with small element length scales, an element shape regularization that inhibits nonlinear deformations of the high-order elements, and a penalty regularization that penalizes degenerate elements. Additionally, we introduce a procedure for locally scaling the regularization operators in an adaptive, elementwise manner in order to maintain grid validity. We apply the proposed MDG-ICE formulation to two- and three-dimensional test cases involving viscous shocks and/or boundary layers, including Mach 17.6 hypersonic viscous flow over a circular cylinder and Mach 5 hypersonic viscous flow over a sphere, which are very challenging test cases for conventional numerical schemes on simplicial grids. Even without artificial dissipation, the computed solutions are free from spurious oscillations and yield highly symmetric surface heat-flux profiles.

翻译：移动间断伽辽金方法结合界面条件强制（MDG-ICE）是一种高阶r自适应方法，该方法将网格视为变量，弱化守恒律、本构律及相应界面条件的约束，从而隐式拟合高梯度流动特征。本文基于Levenberg-Marquardt算法开发了一种优化求解器，其采用各向异性局部自适应惩罚方法提升鲁棒性，并防止高超声速黏性流动计算中的单元退化。具体而言，我们引入了基于网格隐含度量的各向异性网格正则化（抑制单元尺度较小方向上的网格运动）、单元形状正则化（抑制高阶单元的非线性变形）以及惩罚正则化（惩罚退化单元）。此外，我们提出了一种局部缩放正则化算子的自适应逐单元策略，以维持网格有效性。将所提出的MDG-ICE公式应用于包含黏性激波和/或边界层的二维及三维算例，包括马赫数17.6的高超声速黏性圆柱绕流和马赫数5的高超声速黏性球体绕流——这些算例对采用单纯形网格的传统数值格式极具挑战性。即使在无人工耗散的情况下，计算解仍无伪振荡，且产生高度对称的表面热流分布。