We define a regularized size-shape distortion (quality) measure for curved high-order elements on a Riemannian space. To this end, we measure the deviation of a given element, straight-sided or curved, from the stretching, alignment, and sizing determined by a target metric. The defined distortion (quality) is suitable to check the validity and the quality of straight-sided and curved elements on Riemannian spaces determined by constant and point-wise varying metrics. The examples illustrate that the distortion can be minimized to curve (deform) the elements of a given high-order (linear) mesh and try to match with curved (linear) elements the point-wise stretching, alignment, and sizing of a discrete target metric tensor. In addition, the resulting meshes simultaneously match the curved features of the target metric and boundary. Finally, to verify if the minimization of the metric-aware size-shape distortion leads to meshes approximating the target metric, we compute the Riemannian measures for the element edges, faces, and cells. The results show that, when compared to anisotropic straight-sided meshes, the Riemannian measures of the curved high-order mesh entities are closer to unit. Furthermore, the optimized meshes illustrate the potential of curved $r$-adaptation to improve the accuracy of a function representation.

翻译：我们在黎曼空间上定义了一种正则化的尺寸-形状畸变（质量）度量，用于弯曲高阶单元。为此，我们度量给定单元（直边或弯曲）与目标度量所确定的拉伸、对齐和尺寸之间的偏差。所定义的畸变（质量）适用于检验由恒定和逐点变化度量确定的黎曼空间上直边单元和弯曲单元的有效性和质量。实例表明，该畸变可被最小化，从而对给定高阶（线性）网格的单元进行弯曲（变形），并尝试使用弯曲（线性）单元匹配离散目标度量张量的逐点拉伸、对齐和尺寸。此外，所得网格同时匹配目标度量和边界的弯曲特征。最后，为验证度量感知的尺寸-形状畸变的最小化是否导致网格逼近目标度量，我们计算了单元边、面和体的黎曼度量。结果表明，与各向异性直边网格相比，弯曲高阶网格实体的黎曼度量更接近单位值。此外，优化后的网格展示了弯曲$r$-自适应在提高函数表示精度方面的潜力。