We present an \(hr\)-adaptivity framework for optimization of high-order meshes. This work extends the r-adaptivity method by Dobrev et al. (Comput Fluids, 2020), where we utilized the Target-Matrix Optimization Paradigm (TMOP) to minimize a functional that depends on each element’s current and target geometric parameters: element aspect-ratio, size, skew, and rotation. Since fixed mesh topology limits the ability to achieve the target size and aspect-ratio at each position, in this paper, we augment the r-adaptivity framework with nonconforming adaptive mesh refinement to further reduce the error with respect to the target geometric parameters. The proposed formulation, referred to as \(hr\)-adaptivity, introduces TMOP-based quality estimators to satisfy the aspect-ratio target via anisotropic refinements and size target via isotropic refinements in each element of the mesh. The methodology presented is purely algebraic, extends to both simplices and hexahedra/quadrilaterals of any order, and supports nonconforming isotropic and anisotropic refinements in 2D and 3D. Using a problem with a known exact solution, we demonstrate the effectiveness of \(hr\)-adaptivity over both r- and \(h\)-adaptivity in obtaining similar accuracy in the solution with significantly fewer mesh nodes. We also present several examples that show that \(hr\)-adaptivity can help satisfy geometric targets even when \(r\)-adaptivity fails to do so, due to the topology of the initial mesh.

我们提出了一个\（hr \）-适应性框架，用于优化高阶网格。这项工作扩展了Dobrev等人的r适应性方法。（Comput Fluids，2020），我们利用目标矩阵优化范例（TMOP）来最小化依赖于每个元素当前和目标几何参数的功能：元素长宽比，尺寸，偏斜度和旋转度。由于固定网状拓扑限制，以实现在每个位置的目标尺寸和纵横比的能力，在本文中，我们增大了ř-自适应框架具有不合格的自适应网格细化功能，可以进一步减少有关目标几何参数的误差。所提出的公式称为\（hr \）-适应性，引入了基于TMOP的质量估计器，以在网格的每个元素中通过各向异性细化来满足长宽比目标，并通过各向同性细化来满足尺寸目标。所提供的方法是纯粹的代数方法，可扩展到任意阶的单纯形和六面体/四面体，并支持2D和3D中非一致性的各向同性和各向异性细化。通过使用已知精确解的问题，我们证明了\（hr \）-适应性对r-和\（h \）的有效性-适应性，以显着减少网格节点的方式在解决方案中获得相似的精度。我们还提供了一些示例，这些示例表明\（hr \）-适应性可以帮助满足几何目标，即使由于初始网格的拓扑结构而导致\（r \）-适应性未能做到这一点。