High-order meshes are crucial for achieving optimal convergence rates in curvilinear domains, preserving symmetry, and aligning with key flow features in moving mesh simulations, but their quality is challenging to control. In prior work, we have developed techniques based on Target-Matrix Optimization Paradigm (TMOP) to adapt a given high-order mesh to the geometry and solution of the partial differential equation (PDE). Here, we extend this framework to address two key gaps in the literature for high-order mesh r-adaptivity. First, we introduce tangential relaxation on curved surfaces using solely the discrete mesh representation, eliminating the need for access to underlying geometry (e.g., CAD model). Second, we ensure a continuously positive Jacobian determinant throughout the domain. This determinant positivity is essential for using the high-order mesh resulting from r-adaptivity with arbitrary quadrature schemes in simulations. The proposed approach is demonstrated to be robust using a variety of numerical experiments.

翻译：高阶网格对于在曲线域中实现最优收敛速率、保持对称性以及在动网格模拟中对齐关键流动特征至关重要，但其质量难以控制。在先前工作中，我们基于目标矩阵优化范式（TMOP）开发了将给定高阶网格适配于偏微分方程几何结构与解的技术。本文扩展该框架以解决高阶网格r-自适应研究中两个关键空白：首先，我们仅利用离散网格表示在曲面上实现切向松弛，无需访问底层几何结构（如CAD模型）；其次，我们确保整个计算域内雅可比行列式持续保持正值。该行列式正定性对于在模拟中将r-自适应生成的高阶网格与任意数值积分方案结合使用具有决定性意义。通过系列数值实验验证，所提方法展现出优异的鲁棒性。