A Time-Dependent Vector Field Topology Based on Streak Surfaces
A Time-Dependent Vector Field Topology Based on Streak Surfaces
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DOI：
10.1109/tvcg.2012.131
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发表时间：
2013-03-01
影响因子：
5.2
通讯作者：
Ertl, Thomas
中科院分区：
文献类型：
--
作者：
Ueffinger, Markus;Sadlo, Filip;Ertl, Thomas
最近的研究表明，直接适用于定常矢量场的二维矢量场拓扑概念如何通过用条纹代替流线的作用来推广到含时矢量场[1]。本文将这一概念推广到三维矢量场。在传统的三维矢量场中，拓扑分离是通过对与临界点对应的0维种子进行流线积分得到的。相反，在我们的新概念中，计算基于条纹的分离矩阵需要一维种子构造。类似于2D推广，我们证明了在正向和反向有限时间Lyapunov指数(FTLE)场中，沿着由余维1脊交曲线发出的不同路径表面播种的条纹曲面可以得到不变流形。这些路径曲面表示临界点的随时间变化的泛化，并在向量场的随时间变化的拓扑中传达进一步的结构。与基于FTLE脊线的传统方法相比，所得到的条纹流形在视觉质量和计算成本方面简化了拉格朗日相干结构(LCS)的分析，特别是在计算LCS的时间序列时。我们用合成算例和计算流体力学结果验证了新方法的有效性和实用性。
It was shown recently how the 2D vector field topology concept, directly applicable to stationary vector fields only, can be generalized to time-dependent vector fields by replacing the role of stream lines by streak lines [1]. The present paper extends this concept to 3D vector fields. In traditional 3D vector field topology separatrices can be obtained by integrating stream lines from 0D seeds corresponding to critical points. We show that in our new concept, in contrast, 1D seeding constructs are required for computing streak-based separatrices. In analogy to the 2D generalization we show that invariant manifolds can be obtained by seeding streak surfaces along distinguished path surfaces emanating from intersection curves between codimension-1 ridges in the forward and reverse finite-time Lyapunov exponent (FTLE) fields. These path surfaces represent a time-dependent generalization of critical points and convey further structure in time-dependent topology of vector fields. Compared to the traditional approach based on FTLE ridges, the resulting streak manifolds ease the analysis of Lagrangian coherent structures (LCS) with respect to visual quality and computational cost, especially when time series of LCS are computed. We exemplify validity and utility of the new approach using both synthetic examples and computational fluid dynamics results.