# Toward a Lagrangian Vector Field Topology

## Abstract

In this paper we present an extended critical point concept which allows us to apply vector field topology in the case of unsteady flow. We propose a measure for unsteadiness which describes the rate of change of the velocities ina fluid element over time. This measure allows us to select particles for which topological properties remain intact inside a finite spatio-temporal neighborhood. One benefit of this approach is that the classification of critical points based on the eigenvalues of the Jacobian remains meaningful. In the steady case the proposed criterion reduces to the classical definition of critical points. As a first step we show that finding an optimal Galilean frame of reference can be obtained implicitly by analyzing the acceleration field. In a second step we show that this can be extended by switching to the Lagrangian frame of reference. This way the criterion can detect critical points moving along intricate trajectories. We analyze the behavior of the proposed criterion based on two analytical vector fields for which a correct solution is defined by their inherent symmetries and present results for numerical vector fields.

R. Fuchs, J. Kemmler, B. Schindler, J. Waser, F. Sadlo, H. Hauser, and R. Peikert, "Toward a Lagrangian Vector Field Topology," Computer Graphics Forum, vol. 29, iss. 3, p. 1163–1172, 2010.

[BibTeX]

In this paper we present an extended critical point concept which allows us to apply vector field topology in the case of unsteady flow. We propose a measure for unsteadiness which describes the rate of change of the velocities ina fluid element over time. This measure allows us to select particles for which topological properties remain intact inside a finite spatio-temporal neighborhood. One benefit of this approach is that the classification of critical points based on the eigenvalues of the Jacobian remains meaningful. In the steady case the proposed criterion reduces to the classical definition of critical points. As a first step we show that finding an optimal Galilean frame of reference can be obtained implicitly by analyzing the acceleration field. In a second step we show that this can be extended by switching to the Lagrangian frame of reference. This way the criterion can detect critical points moving along intricate trajectories. We analyze the behavior of the proposed criterion based on two analytical vector fields for which a correct solution is defined by their inherent symmetries and present results for numerical vector fields.

```
@ARTICLE {fuchs10lagrangian,
author = "Raphael Fuchs and Jan Kemmler and Benjamin Schindler and Jrgen Waser and Filip Sadlo and Helwig Hauser and Ronald Peikert",
title = "Toward a Lagrangian Vector Field Topology",
journal = "Computer Graphics Forum",
year = "2010",
volume = "29",
number = "3",
pages = "1163--1172",
month = "june",
abstract = "In this paper we present an extended critical point concept which allows us to apply vector field topology in the case of unsteady flow. We propose a measure for unsteadiness which describes the rate of change of the velocities ina fluid element over time. This measure allows us to select particles for which topological properties remain intact inside a finite spatio-temporal neighborhood. One benefit of this approach is that the classification of critical points based on the eigenvalues of the Jacobian remains meaningful. In the steady case the proposed criterion reduces to the classical definition of critical points. As a first step we show that finding an optimal Galilean frame of reference can be obtained implicitly by analyzing the acceleration field. In a second step we show that this can be extended by switching to the Lagrangian frame of reference. This way the criterion can detect critical points moving along intricate trajectories. We analyze the behavior of the proposed criterion based on two analytical vector fields for which a correct solution is defined by their inherent symmetries and present results for numerical vector fields.",
images = "images/fuchs10lagrangian2.jpg, images/fuchs10lagrangian.jpg",
thumbnails = "images/fuchs10lagrangian2_thumb.jpg, images/fuchs10lagrangian_thumb.jpg",
event = "EuroVis 2010",
location = "Bordeaux, France",
url = "//dx.doi.org/10.1111/j.1467-8659.2009.01686.x",
project = "semseg"
}
```