We present an approach to local extraction of 3D time-dependent vector field topology. In this concept, Lagrangian coherent structures, which represent the separating manifolds in time-dependent transport, correspond to generalized streak manifolds seeded along hyperbolic path surfaces (HPSs). Instead of expensive and numerically challenging direct computation of the HPSs by intersection of ridges in the forward and backward finite-time Lyapunov exponent (FTLE) fields, our approach employs local extraction of respective candidates in the four-dimensional space-time domain. These candidates are subsequently refined toward the hyperbolic path surfaces, which provides unsteady equivalents of saddle-type critical points, periodic orbits, and bifurcation lines from steady, traditional vector field topology. In contrast to FTLE-based methods, we obtain an explicit geometric representation of the topological skeleton of the flow, which for steady flows coincides with the hyperbolic invariant manifolds of vector field topology. We evaluate our approach on analytical flows, as well as data from computational fluid dynamics, using the FTLE as a ground truth superset, i.e., we also show that FTLE ridges exhibit several types of false positives.

中文翻译：

---

3D 瞬态矢量场拓扑的局部提取

我们提出了一种局部提取 3D 时间相关矢量场拓扑的方法。在这个概念中，代表时间相关传输中分离流形的拉格朗日相干结构对应于沿双曲路径表面 (HPS) 播种的广义条纹流形。我们的方法不是通过前向和后向有限时间李雅普诺夫指数 (FTLE) 场中脊的交叉来直接计算 HPS，而是在四维时空域中对各个候选进行局部提取。这些候选对象随后被改进为双曲线路径表面，它提供了稳定的传统矢量场拓扑结构中鞍型临界点、周期轨道和分岔线的不稳定等价物。与基于 FTLE 的方法相比，我们获得了流拓扑骨架的明确几何表示，对于稳定流，它与矢量场拓扑的双曲不变流形重合。我们使用 FTLE 作为地面实况超集评估我们对分析流的方法以及来自计算流体动力学的数据，即，我们还表明 FTLE 脊表现出几种类型的误报。