Space‐Time Bifurcation Lines for Extraction of 2D Lagrangian Coherent Structures

Self Cite

This paper does two main contributions to 2D time‐dependent vector field topology. First, we present a technique for robust, accurate, and efficient extraction of distinguished hyperbolic trajectories (DHT), the generative structures of 2D time‐dependent vector field topology. It is based on refinement of initial candidate curves. In contrast to previous approaches, it is robust because the refinement converges for reasonably close initial candidates, it is accurate due to its adaptive scheme, and it is efficient due to its high convergence speed. Second, we provide a detailed evaluation and discussion of previous approaches for the extraction of DHTs and time‐dependent vector field topology in general. We demonstrate the utility of our approach using analytical flows, as well as data from computational fluid dynamics.

Section: Methodsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Extraction of Distinguished Hyperbolic Trajectories for 2D Time‐Dependent Vector Field Topology

Hofmann

Sadlo

2020

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“…[MSE13] used the PV operator to extract bifurcation lines in 3D steady vector fields, and applied their method to the space-time domain of 2D flows for extracting hyperbolic trajectories [MBES16].…”

Section: Related Workmentioning

confidence: 99%

The Dependent Vectors Operator

Hofmann

Sadlo

2019

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In this paper, we generalize the parallel vectors operator due to Peikert and Roth to arbitrary dimension, i.e., to four‐dimensional fields and beyond. Whereas the original operator tested for parallelism of two (derived) 2D or 3D vector fields, we reformulate the concept in terms of linear dependency of sets of vector fields, and propose a generic technique to extract and filter the solution manifolds. We exemplify our approach for vortex cores, bifurcations, and ridges as well as valleys in higher dimensions.

“…The derivative of the tensor field ∇T is a third-order tensor, for which the definition of eigenvalues and eigenvectors is non-trivial [ZPM07]. This means that we also find structures similar to hyperbolic trajectories in vector fields [MSE13,MBES16]. Further research is necessary in order to be able to distinguish these different features.…”

Section: Performance and Parameter Studymentioning

confidence: 99%

“…Locations where the flow is parallel to an eigenvector of a Jacobian with three real eigenvalues also form stable lines. These hyperbolic trajectories are the centers of simultaneous converging and diverging behavior of the vector field and can be used to extract Lagrangian coherent structures [MSE13,MBES16]. Roth and Peikert [RP98] showed that vortex core lines can equivalently be described as locations where the curvature of streamlines locally vanishes.…”

Section: Introductionmentioning

confidence: 99%

Core Lines in 3D Second‐Order Tensor Fields

Oster

Rössl

Theisel

2018

Vortices are important features in vector fields that show a swirling behavior around a common core. The concept of a vortex core line describes the center of this swirling behavior. In this work, we examine the extension of this concept to 3D second-order tensor fields. Here, a behavior similar to vortices in vector fields can be observed for trajectories of the eigenvectors. Vortex core lines in vector fields were defined by Sujudi and Haimes to be the locations where stream lines are parallel to an eigenvector of the Jacobian. We show that a similar criterion applied to the eigenvector trajectories of a tensor field yields structurally stable lines that we call tensor core lines. We provide a formal definition of these structures and examine their mathematical properties. We also present a numerical algorithm for extracting tensor core lines in piecewise linear tensor fields. We find all intersections of tensor core lines with the faces of a dataset using a simple and robust root finding algorithm. Applying this algorithm to tensor fields obtained from structural mechanics simulations shows that it is able to effectively detect and visualize regions of rotational or hyperbolic behavior of eigenvector trajectories.