Mode surfaces are the generalization of degenerate curves and neutral surfaces, which constitute 3D symmetric tensor field topology. Efficient analysis and visualization of mode surfaces can provide additional insight into not only degenerate curves and neutral surfaces, but also how these features transition into each other. Moreover, the geometry and topology of mode surfaces can help domain scientists better understand the tensor fields in their applications. Existing mode surface extraction methods can miss features in the surfaces. Moreover, the mode surfaces extracted from neighboring cells have gaps, which make their subsequent analysis difficult. In this paper, we provide novel analysis on the topological structures of mode surfaces, including a common parameterization of all mode surfaces of a tensor field using 2D asymmetric tensors. This allows us to not only better understand the structures in mode surfaces and their interactions with degenerate curves and neutral surfaces, but also develop an efficient algorithm to seamlessly extract mode surfaces, including neutral surfaces. The seamless mode surfaces enable efficient analysis of their geometric structures, such as the principal curvature directions. We apply our analysis and visualization to a number of solid mechanics data sets.

中文翻译：

---

对称张量场的模态面：拓扑分析和无缝提取

模态曲面是简并曲线和中性曲面的推广，它们构成了 3D 对称张量场拓扑。模式曲面的有效分析和可视化不仅可以提供对退化曲线和中性曲面的进一步了解，还可以提供对这些特征如何相互过渡的深入了解。此外，模式表面的几何形状和拓扑结构可以帮助领域科学家更好地理解其应用中的张量场。现有模式表面提取方法可能会遗漏表面中的特征。此外，从相邻单元中提取的模式面存在间隙，这使得后续分析变得困难。在本文中，我们对模式表面的拓扑结构进行了新颖的分析，包括使用 2D 非对称张量对张量场的所有模式表面进行通用参数化。这使我们不仅可以更好地理解模式曲面中的结构及其与退化曲线和中性曲面的相互作用，而且还可以开发一种有效的算法来无缝提取模式曲面，包括中性曲面。无缝模式表面能够有效分析其几何结构，例如主曲率方向。我们将我们的分析和可视化应用于许多固体力学数据集。