Vector Field Topology Research Articles

Visualizing local vector field topology

Author(s): Scheuermann, Gerik; Hamann, Bernd; Joy, Ken; Kollmann, Wolfgang | Abstract: The topology of vector fields offers a well known way to show a condensed view of the stream line behavior of a vector field. The global structure of a field can be shown without time-consuming user interaction. With regard to large data visualization, one encounters a major drawback: the necessity to analyze a whole data set, even when interested in only a small region. We show that one can localize the topology concept by including the boundary in the topology analysis. The idea is demonstrated for a turbulent swirling jet simulation example. Our concept works for all planar, piecewise analytic vector fields on bounded domains.

Feature extraction of separation and attachment lines

Separation and attachment lines are topologically significant curves that exist on 2D surfaces in 3D vector fields. Two algorithms are presented, one point-based and one element-based, that extract separation and attachment lines using eigenvalue analysis of a locally linear function. Unlike prior techniques based on piecewise numerical integration, these algorithms use robust analytical tests that can be applied independently to any point in a vector field. The feature extraction is fully automatic and suited to the analysis of large-scale numerical simulations. The strengths and weaknesses of the two algorithms are evaluated using analytic vector fields and also results from computational fluid dynamics (CFD) simulations. We show that both algorithms detect open separation lines-a type of separation that is not captured by conventional vector field topology algorithms.

Visualizing nonlinear vector field topology

We present our results on the visualization of nonlinear vector field topology. The underlying mathematics is done in Clifford algebra, a system describing geometry by extending the usual vector space by a multiplication of vectors. We started with the observation that all known algorithms for vector field topology are based on piecewise linear or bilinear approximation, and that these methods destroy the local topology if nonlinear behavior is present. Our algorithm looks for such situations, chooses an appropriate polynomial approximation in these areas, and, finally, visualizes the topology. This overcomes the problem, and the algorithm is still very fast because we are using linear approximation outside these small but important areas. The paper contains a detailed description of the algorithm and a basic introduction to Clifford algebra.

The topology of symmetric, second-order 3D tensor fields

The authors study the topology of symmetric, second-order tensor fields. The results of the study can be readily extended to include general tensor fields through linear combination of symmetric tensor fields and vector fields. The goal is to represent their complex structure by a simple set of carefully chosen points, lines, and surfaces analogous to approaches in vector field topology. They extract topological skeletons of the eigenvector fields and use them for a compact, comprehensive description of the tensor field. Their approach is based on the premise: analyze, then visualize. The basic constituents of tensor topology are the degenerate points, or points where eigenvalues are equal to each other. Degenerate points play a similar role as critical points in vector fields. In tensor fields they identify two kinds of elementary degenerate points, which they call wedge points and trisector points. They can combine to form more familiar singularities-such as saddles, nodes, centers, or foci. However, these are generally unstable structures in tensor fields. Based on the notions developed for 2D tensor fields, they extend the theory to include 3D degenerate points. Examples are given on the use of tensor field topology for the interpretation of physical systems.

Topology of a holomorphic vector field around an isolated singularity

Topology of a holomorphic vector field around an isolated singularity

Visualizing vector field topology in fluid flows

Methods for automating the analysis and display of vector field topology in general, and flow topology in particular, are described. By using techniques to extract and visualize topological information, it is possible to combine the simplicity of schematic depictions with the quantitative accuracy of curves and surfaces computed directly from the data. Two-dimensional vector field topology is discussed, covering critical points and time-dependent flows, to provide a basis for the examination of topology in three-dimensional separated flows. Surface topology and separation structures in three-dimensional flows are then addressed. The construction of representations of tangent surfaces that are accurate, as well as efficient to compute and display, is examined, covering tessellation, clipping, and refinement. Locating, characterizing, and displaying three-dimensional critical points are considered. >

Interactive analysis of the topology of 4D vector fields

Interactive visualization methods are now evolving in response to a need to provide more immediate access to particular features of interest to analysts at particular points in the space and time of their data. This paper focuses on feature extraction methods relevant to the analysis of vector fields. In vector fields, “critical points” are those points at which the vector magnitude passes through zero. The word “topology” is used to describe the interconnection patterns between critical points. Topology is central to the understanding of vector fields. It provides very succinct and precise summary information, and can be used to subdivide large fields into well-defined subregions. In this paper, methods for interactively creating maps of vector-field topology are described. The advantages offered by interactive methods in comparison with automatic methods are also discussed.

Representation and display of vector field topology in fluid flow data sets

The visualization of physical processes in general and of vector fields in particular is discussed. An approach to visualizing flow topology that is based on the physics and mathematics underlying the physical phenomenon is presented. It involves determining critical points in the flow where the velocity vector vanishes. The critical points, connected by principal lines or planes, determine the topology of the flow. The complexity of the data is reduced without sacrificing the quantitative nature of the data set. By reducing the original vector field to a set of critical points and their connections, a representation of the topology of a two-dimensional vector field is much smaller than the original data set but retains with full precision the information pertinent to the flow topology is obtained. This representation can be displayed as a set of points and tangent curves or as a graph. Analysis (including algorithms), display, interaction, and implementation aspects are discussed. >

Origin of nonunitarity in quantum gravity.

We study the mathematical structure of finite-dimensional parametrized systems. We find that the systems must possess a so-called global time, if they are to be reducible. We list all conditions on the topology of the Hamiltonian vector field, which follow from the existence of a global time. The topological obstructions to the existence of a global time are analogous to the Gribov effect. We consider the canonical quantization and show that each known method of constructing a unitary quantum theory is based on the use of a global time. Studying cosmological models we show how choices of wrong candidates for time, as well as extensions of the configuration spaces, lead to violations of unitarity. We give simple examples of configuration spaces in which there is no global time. The interpretation of these results is that we are quantizing in wrong coordinates. Indeed, there is a class of global times for each parametrized system. These times cannot be functions of the internal and external three-geometry and of the instantaneous states of all fields only. We give an example of the transformation from geometrodynamical variables to variables containing a global time; its inverse is a sort of covering map and the topology of the Hamiltonian vector field is changed.