针对高阶网格的靶矩阵优化范式
The Target-Matrix Optimization Paradigm for High-Order Meshes
AI总结：
本文提出基于Knupp扩展的靶矩阵优化范式，用于控制和提升高阶有限元网格质量，通过局部精确控制与全局优化相结合，解决通用等参数元映射中的网格质量度量、采样点选择及优化问题，展示了新算法在高阶ALE应用中的优势。
摘要：
我们描述了一个框架，用于基于Knupp扩展的靶矩阵优化范式（TMOP）来控制和改进高阶有限元网格的质量。该方法允许高阶应用对局部网格质量进行非常精确的控制，同时仍能提升网格的全局质量。我们处理了将各种TMOP组件适应到通用等参数元映射设置中的问题，包括二维和三维中的网格质量度量、采样点的选择以及由此产生的网格优化问题的求解。我们还探讨了其他实际考虑因素，如切向松弛和限制与原网格的偏差。新高阶TMOP算法的好处通过多个测试问题和来自高阶任意欧拉-拉格朗日（ALE）应用的例子来展示。我们的实现以开源库的形式免费提供。
英文摘要：
We describe a framework for controlling and improving the quality of high-order finite element meshes based on extensions of the Target-Matrix Optimization Paradigm (TMOP) of Knupp. This approach allows high-order applications to have a very precise control over local mesh quality, while still improving the mesh globally. We address the adaption of various TMOP components to the settings of general isoparametric element mappings, including the mesh quality metric in 2D and 3D, the selection of sample points and the solution of the resulting mesh optimization problem. We also investigate additional practical concerns, such as tangential relaxation and restricting the deviation from the original mesh. The benefits of the new high-order TMOP algorithms are illustrated on a number of test problems and examples from a high-order arbitrary Eulerian-Lagrangian (ALE) application. Our implementation is freely available in an open-source library form.
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