The validity of trilinear hexahedral (hex) mesh elements is a prerequisite for many applications of hex meshes, such as finite element analysis. A commonly used check for hex mesh validity evaluates mesh quality on the corners of the parameter domain of each hex, an insufficient condition that neglects invalidity elsewhere in the element, but is straightforward to compute. Hex mesh quality optimizations using this validity criterion suffer by being unable to detect invalidities in a hex mesh reliably, let alone fix them. We rectify these challenges by leveraging sum‐of‐squares relaxations to pinpoint invalidities in a hex mesh efficiently and robustly. Furthermore, we design a hex mesh repair algorithm that can certify validity of the entire hex mesh. We demonstrate our hex mesh repair algorithm on a dataset of meshes that include hexes with both corner and face‐interior invalidities and demonstrate that where naïve algorithms would fail to even detect invalidities, we are able to repair them. Our novel methodology also introduces the general machinery of sum‐of‐squares relaxation to geometry processing, where it has the potential to solve related problems.

中文翻译：

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通过平方和松弛修复六面体网格

三线六面体 (hex) 网格单元的有效性是许多六面体网格应用（例如有限元分析）的先决条件。一个常用的六边形网格有效性检查评估每个六边形参数域角落的网格质量，这是一种忽略元素中其他地方的无效性的不充分条件，但计算起来很简单。使用此有效性标准的六角网格质量优化无法可靠地检测六角网格中的无效性，更不用说修复它们了。我们通过利用平方和松弛来有效且稳健地查明十六进制网格中的无效性，从而纠正这些挑战。此外，我们设计了一种六角网格修复算法，可以证明整个六角网格的有效性。我们在网格数据集上演示了我们的六角网格修复算法，该数据集包括具有角和面内部无效性的六边形，并证明了在原始算法甚至无法检测到无效性的情况下，我们能够修复它们。我们的新方法还将平方和松弛的一般机制引入几何处理，它有可能解决相关问题。