Detection and visualization of closed streamlines in planar flows

Abstract

The analysis and visualization of flows is a central problem in visualization. Topology based methods have gained increasing interest in recent years. This article describes a method for the detection of closed streamlines in flows. It is based on a special treatment of cases where a streamline reenters a cell to prevent infinite cycling during streamline calculation. The algorithm checks for possible exits of a loop of crossed edges and detects structurally stable closed streamlines. These global features are not detected by conventional topology and feature detection algorithms.

TADD: A Computational Framework for Data Analysis Using Discrete Morse Theory

“…The respective eigenvectors can be used to compute the separatrices as the solution of an autonomous ODE. For the numerical treatment of these problems and the extraction of the periodic orbits, we refer to [18,20,5].…”

“…The respective eigenvectors can be used to compute the separatrices as the solution of an autonomous ODE. For the numerical treatment of these problems and the extraction of the periodic orbits, we refer to [18,20,5].One of the biggest challenges that such numerical algorithms face is the discrete nature of the extremal structure which necessitates a lot of binary decisions. For example, the type of a critical point depends on the sign of the eigenvalues.…”

Topology tracking for the visualization of time-dependent two-dimensional flows

“…This work is part of the Visual Math Project that explains the basic mathematical notions by means of discerning sketches. Two of the authors [12] presented an algorithm that computes streamlines in the 2D steady case while detecting if they run into a closed streamline. This can also be used in time slices of a time-dependent dataset.…”

“…Consequently, this is a streamline c a , so that there is a t 0 ∈ R with c a (t + nt 0 ) = c a (t) ∀ n ∈ N. From a topological point of view, closed streamlines behave in the same way as sources or sinks. To detect these closed streamlines we use the algorithm proposed by two of the authors [12]. Interpolating linearly on the given grid we get a continuous vector eld.…”

“…Further, special streamlines are extracted that sketch the structure of the ow. This method has been widely applied in the last decade even if the visualization of closed orbits is a recent advance [12]. Now in the case of unsteady vector elds, the original method must still be extended to offer insight into the structural evolution of the ow over time.…”