Gerik Scheuermann

A topology simplification method for 2D vector fields

Topology analysis of plane, turbulent vector fields results in visual clutter caused by critical points indicating vortices of finer and finer scales. A simplification can be achieved by merging critical points within a prescribed radius into higher order critical points. After building clusters containing the singularities to merge, the method generates a piecewise linear representation of the vector field in each cluster containing only one (higher order) singularity. Any visualization method can be applied to the result after this process. Using different maximal distances for the critical points to be merged results in a hierarchy of simplified vector fields that can be used for analysis on different scales.

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.

Continuous topology simplification of planar vector fields

Vector fields can present complex structural behavior, especially in turbulent computational fluid dynamics. The topological analysis of these data sets reduces the information, but one is usually still left with too many details for interpretation. In this paper, we present a simplification approach that removes pairs of critical points from the data set, based on relevance measures. In contrast to earlier methods, no grid changes are necessary, since the whole method uses small local changes of the vector values defining the vector field. An interpretation in terms of bifurcations underlines the continuous, natural flavor of the algorithm.

Surface techniques for vortex visualization

This paper presents powerful surface based techniques for the analysis of complex flow fields resulting from CFD simulations. Emphasis is put on the examination of vortical structures. An improved method for stream surface computation that delivers accurate results in regions of intricate flow is presented, along with a novel method to determine boundary surfaces of vortex cores. A number of surface techniques are presented that aid in understanding the flow behavior displayed by these surfaces. Furthermore, a scheme for phenomenological extraction of vortex core lines using stream surfaces is discussed and its accuracy is compared to one of the most established standard techniques.

Topology Tracking for the Visualization of Time-Dependent Two-Dimensional Flows

The paper presents a topology-based visualization method for time-dependent two-dimensional vector uf0deelds. A time interpolation enables the accurate tracking of critical points and closed orbits as well as the detection and identiuf0decation of structural changes. This completely characterizes the topology of the unsteady uf0dfow. Bifurcation theory provides the theoretical framework. The results are conveyed by surfaces that separate subvolumes of uniform uf0dfow behavior in a three-dimensional space-time domain.

Topological segmentation in three-dimensional vector fields

We present a new method for topological segmentation in steady three-dimensional vector fields. Depending on desired properties, the algorithm replaces the original vector field by a derived segmented data set, which is utilized to produce separating surfaces in the vector field. We define the concept of a segmented data set, develop methods that produce the segmented data by sampling the vector field with streamlines, and describe algorithms that generate the separating surfaces. This method is applied to generate local separatrices in the field, defined by a movable boundary region placed in the field. The resulting partitions can be visualized using standard techniques for a visualization of a vector field at a higher level of abstraction.

Exploring scalar fields using critical isovalues

Isosurfaces are commonly used to visualize scalar fields. Critical isovalues indicate isosurface topology changes: the creation of new surface components, merging of surface components or the formation of holes in a surface component. Therefore, they highlight interesting isosurface behavior and are helpful in exploration of large trivariate data sets. We present a method that detects critical isovalues in a scalar field defined by piecewise trilinear interpolation over a rectilinear grid and describe how to use them when examining volume data. We further review varieties of the marching cubes (MC) algorithm, with the intention of preserving topology of the trilinear interpolant when extracting an isosurface. We combine and extend two approaches in such a way that it is possible to extract meaningful isosurfaces even when a critical value is chosen as the isovalue.

A tetrahedra-based stream surface algorithm

This paper presents a new algorithm for the calculation of stream surfaces for tetrahedral grids. It propagates the surface through the tetrahedra, one at a time, calculating the intersections with the tetrahedral faces. The method allows us to incorporate topological information from the cells, e.g. critical points. The calculations are based on barycentric coordinates, since this simplifies the theory and the algorithm. The stream surfaces are ruled surfaces inside each cell, and their construction starts with line segments on the faces. Our method supports the analysis of velocity fields resulting from computational fluid dynamics (CFD) simulations.

Topology-based visualization of time-dependent 2D vector fields

Topology-based methods have been successfully applied to the visualization of instantaneous planar vector fields. In this paper, we present the topology-based visualization of time-dependent 2D flows. Our method tracks critical points over time precisely. The detection and classification of bifurcations delivers the topological structure of time dependent vector fields. This offers a general framework for the qualitative analysis and visualization of parameterdependent 2D vector fields.

Visualization of higher order singularities in vector fields

Presents an algorithm for the visualization of vector field topology based on Clifford algebra. It allows the detection of higher-order singularities. This is accomplished by first analysing the possible critical points and then choosing a suitable polynomial approximation, because conventional methods based on piecewise linear or bilinear approximation do not allow higher-order critical points and destroy the topology in such cases. The algorithm is still very fast, because of using linear approximation outside the areas with several critical points.