We detail how to use Newton's method for distortion-based curved $r$-adaption to a discrete high-order metric field while matching a target geometry. Specifically, we combine two terms: a distortion measuring the deviation from the target metric; and a penalty term measuring the deviation from the target boundary. For this combination, we consider four ingredients. First, to represent the metric field, we detail a log-Euclidean high-order metric interpolation on a curved (straight-edged) mesh. Second, for this metric interpolation, we detail the first and second derivatives in physical coordinates. Third, to represent the domain boundaries, we propose an implicit representation for 2D and 3D NURBS models. Fourth, for this implicit representation, we obtain the first and second derivatives. The derivatives of the metric interpolation and the implicit representation allow minimizing the objective function with Newton's method. For this second-order minimization, the resulting meshes simultaneously match the curved features of the target metric and boundary. Matching the metric and the geometry using second-order optimization is an unprecedented capability in curved (straight-edged) $r$-adaption. This capability will be critical in global and cavity-based curved (straight-edged) high-order mesh adaption.

翻译：我们详细阐述了如何利用牛顿法实现基于畸变的曲边$r$-自适应，以匹配离散高阶度量场的同时契合目标几何。具体而言，我们组合了两项指标：度量偏离目标度量的畸变项，以及度量偏离目标边界的惩罚项。对于该组合，我们考虑了四个要素。首先，为表示度量场，我们在曲边（直边）网格上详述了对数-欧几里得高阶度量插值方法。其次，针对该度量插值，我们推导了其在物理坐标下的一阶和二阶导数。第三，为表示域边界，我们提出了二维和三维NURBS模型的隐式表示方法。第四，针对该隐式表示，我们获取了其一阶和二阶导数。度量插值与隐式表示的导数使得目标函数可通过牛顿法最小化。通过这种二阶优化，生成的网格能同时匹配目标度量的曲边特征与边界。采用二阶优化同时契合度量与几何的能力，在曲边（直边）$r$-自适应领域尚属首次。该能力对于全局及基于腔体的曲边（直边）高阶网格自适应至关重要。