Hung Shih-Hsuan, Zhang Yue, Zhang Eugene
IEEE Trans Vis Comput Graph. 2024 Jan;30(1):1282-1291. doi: 10.1109/TVCG.2023.3326933. Epub 2023 Dec 25.
There have been recent advances in the analysis and visualization of 3D symmetric tensor fields, with a focus on the robust extraction of tensor field topology. However, topological features such as degenerate curves and neutral surfaces do not live in isolation. Instead, they intriguingly interact with each other. In this paper, we introduce the notion of topological graph for 3D symmetric tensor fields to facilitate global topological analysis of such fields. The nodes of the graph include degenerate curves and regions bounded by neutral surfaces in the domain. The edges in the graph denote the adjacency information between the regions and degenerate curves. In addition, we observe that a degenerate curve can be a loop and even a knot and that two degenerate curves (whether in the same region or not) can form a link. We provide a definition and theoretical analysis of individual degenerate curves in order to help understand why knots and links may occur. Moreover, we differentiate between wedges and trisectors, thus making the analysis more detailed about degenerate curves. We incorporate this information into the topological graph. Such a graph can not only reveal the global structure in a 3D symmetric tensor field but also allow two symmetric tensor fields to be compared. We demonstrate our approach by applying it to solid mechanics and material science data sets.
最近在三维对称张量场的分析和可视化方面取得了进展，重点是张量场拓扑结构的稳健提取。然而，诸如退化曲线和中性面等拓扑特征并非孤立存在。相反，它们之间存在着有趣的相互作用。在本文中，我们引入了三维对称张量场拓扑图的概念，以促进对此类场的全局拓扑分析。图的节点包括退化曲线和域中由中性面包围的区域。图中的边表示区域和退化曲线之间的邻接信息。此外，我们观察到一条退化曲线可以是一个环甚至是一个纽结，并且两条退化曲线（无论是否在同一区域）可以形成一个链环。我们给出了单个退化曲线的定义和理论分析，以帮助理解纽结和链环为何会出现。此外，我们区分了楔和三分线，从而使对退化曲线的分析更加详细。我们将这些信息纳入拓扑图。这样的图不仅可以揭示三维对称张量场中的全局结构，还可以对两个对称张量场进行比较。我们通过将其应用于固体力学和材料科学数据集来展示我们的方法。
