We present a robust method to generate high-quality high-order tetrahedral meshes with bounded approximation errors and low mesh complexity. The success of our method relies on two key components. The first one is three novel local operations that robustly modify the topology of the high-order tetrahedral mesh while avoiding invalid (flipped or degenerate) elements. In practice, our meshing algorithm follows the edge-based remeshing algorithm that iteratively conducts these local topological operations and a geometric optimization operation to improve mesh quality. The second is a new containment check procedure that robustly judges whether the approximation error between the input mesh and the high-order mesh exceeds the user-specified bound. If one operation causes the error-bounded constraint to be violated, we reject this operation to ensure a bounded approximation error. Besides, the number of tetrahedrons of the high-order mesh is reduced by progressively increasing the target edge length in the edge-based remeshing algorithm. A large number of experimental results have shown the capability and feasibility of our method. Compared to other state-of-the-art methods, our method achieves higher robustness and quality.

我们提出了一种鲁棒的方法来生成具有有限近似误差和低网格复杂度的高质量高阶四面体网格。我们方法的成功依赖于两个关键组件。第一个是三个新颖的局部操作，它们可以稳健地修改高阶四面体网格的拓扑结构，同时避免无效（翻转或退化）元素。在实践中，我们的网格划分算法遵循基于边缘的重新网格划分算法，该算法迭代地进行这些局部拓扑操作和几何优化操作以提高网格质量。第二个是新的包含检查程序，可以稳健地判断输入网格和高阶网格之间的近似误差是否超过用户指定的界限。如果一个操作导致错误有界约束被违反，我们拒绝此操作以确保有界近似误差。此外，在基于边缘的网格划分算法中，通过逐渐增加目标边缘长度来减少高阶网格的四面体数量。大量的实验结果表明了我们方法的能力和可行性。与其他最先进的方法相比，我们的方法实现了更高的鲁棒性和质量。