Topology-Based Methods in Visualization II

The depiction of a time-dependent flow in a way that effectively sup ports the structural analysis of its salient patterns is still a challenging problem for flow visualization research. While a variety of powerful approaches have been investigated for over a decade now, none of them so far has been able to yield repre sentations that effectively combine good visual quality and a physical interpretation that is both intuitive and reliable. Yet, with the huge amount of flow data generated by numerical computations of growing size and complexity, scientists and engineers are faced with a daunting analysis task in which the ability to identify, extract, and display the most meaningful information contained in the data is becoming absolutely indispensable.

This paper takes a look at the visualization side of vector field anal ysis based on Lagrangian coherent structures. The Lagrangian coherent structures are extracted as height ridges of finite-time Lyapunov exponent fields. The result ing visualizations are compared to those from traditional instantaneous vector field topology of steady and unsteady vector fields: they often provide more and better interpretable information. The examination is applied to 3D vector fields from a dynamical system and practical CFD simulations.

The study of flow separation from walls or solid objects is still an active research area in the fluid dynamics and flow visualization communities and many open questions remain. This paper aims at introducing a new method for the extraction of separation manifolds originating from separation lines. We address the problem from the flow visualization side by investigating features in flow cross sections around separation lines. We use the topological signature of the separation in these sections, in particular the presence of saddle points and their separatrices, as a guide to initiate the construction of the separation manifolds. Having this first part we use well known stream surface construction methods to propagate the surface further into the flow. Additionally, we discuss some lessons learned in the course of our experimentation with well known and new ideas for the extraction of separation lines.

The selection of appropriate level sets for the quantitative visualization of three dimensional imaging or simulation data is a problem that is both fundamental and essential. The selected level set needs to satisfy several topolog-ical and geometric constraints to be useful for subsequent quantitative processing and visualization. For an initial selection of an isosurface, guided by contour tree data structures, we detect the topological features by computing stable and unstable manifolds of the critical points of the distance function induced by the isosurface. We further enhance the description of these features by associating geometric attributes with them. We then rank the attributed features and provide a handle to them for curation of the topological anomalies.

Algorithms for computing contour trees for visualization commonly assume that the input is defined by barycentric interpolation on simplicial meshes or by trilinear interpolation on cubic meshes. In this paper, we describe a general framework for computing contour trees from a graph that captures all significant topological features. We show how to construct these graphs from any mesh-based interpolant by using cell-by-cell “widgets,” and also how to avoid constructing the entire graphs by making finite state machines that capture their traversals.

Our framework eases algorithm development and implementation, and can be used to establish relationships between interpolants. For example, we use it to demonstrate a formal equivalence between the topology defined by implicitly dis-ambiguated marching cubes cases and the topology induced by 8-/18- digital image connectivity.

We describe an approach to visually analyzing the dynamic behavior of 3D time-dependent flow fields by considering the behavior of the path lines. At selected positions in the 4D space-time domain, we compute a number of local and global properties of path lines describing relevant features of them. The resulting multivariate data set is analyzed by applying state-of-the-art information visualization approaches in the sense of a set of linked views (scatter plots, parallel coordinates, etc.) with interactive brushing and focus+context visualization. The selected path lines with certain properties are integrated and visualized as colored 3D curves. This approach allows an interactive exploration of intricate 4D flow structures. We apply our method to a number of flow data sets and describe how path line attributes are used for describing characteristic features of these flows.

Visualizing vector fields using streamlines or some derived applications is still one of the most popular flow visualization methods in use today. Besides the known trade-off between sufficient coverage in the field and cluttering of streamlines, the typical user question is: Where should I start my streamlines to see all important behavior?

In previous work, we define flow structures as an extension of flow topology that permits a partition of the whole flow tailored to the users needs. Based on the skeletal representation of the topology of flow structures, we propose a 3D streamline placement generating a minimal set of streamlines, that on the one hand exactly illustrates the desired property of the flow and on the other hand takes the topology of the specific flow structure into account. We present a heuristic and a deterministic approach and discuss their advantages and disadvantages.

The electric field around a molecule is generated by the charge dis tribution of its constituents: positively charged atomic nuclei, which are well approximated by point charges, and negatively charged electrons, whose proba bility density distribution can be computed from quantum mechanics (Atoms in Molecules: A Quantum Theory, Clarendon, Oxford, 1990). For the purposes of molecular mechanics or dynamics, the charge distribution is often approximated by a collection of point charges, with either a single partial charge at each atomic nucleus position, representing both the nucleus and the electrons near it, or as several different point charges per atom.

If one wants to study the global dynamics of a given system, key com ponents are the stable or unstable manifolds of invariant sets, such as equilibria and periodic orbits. Even in the simplest examples, these global manifolds must be approximated by means of numerical computations. We discuss an algorithm for computing global manifolds of vector fields that is decidedly geometric in nature. A two-dimensional manifold is built up as a collection of approximate geodesic level sets, i.e. topological smooth circles. Our method allows to visualize the resulting surface by making use of the geodesic parametrization.

As we show with the example of the Lorenz system, this is a big advantage when one wants to understand the geometry of complicated two-dimensional global man ifolds. More precisely, for the standard system parameters, the origin of the Lorenz system has a two-dimensional stable manifold — called the Lorenz manifold — and the other two equilibria each have a two-dimensional unstable manifold. The inter sections of these manifolds in the three-dimensional phase space form heteroclinic connections from the nontrivial equilibria to the origin. A parameter-dependent visualization of these manifolds clarifies the transition to chaos in the Lorenz system.

The vortex breakdown phenomenon occurring in a rotating flow within a closed cylinder is still a challenging research field. In particular the goal to describe all significant order parameters of vortex breakdown is not reached. For further insight the viscous and laminar Newtonian flow inside a cylinder with a rotating lid has been calculated by solving the full Navier Stokes equations. During the sim ulation the rotational speed of the lid has been increased, which causes a gradual transition of the internal flow field topology. Starting from a flow field without any reversed flow at the vortex axis the vortex breakdown phenomenon develops indi cated by one or more vortex breakdown bubbles. A phenomenological description of the vortex breakdown process is given by applying a topological analysis to the flow field, which illustrates the main flow structures, their behaviour and changes. By visualization of critical points, at which the velocity magnitude vanishes, the topological flow structure change of the velocity field becomes obvious. Additionally their associated separatrices are integrated into the field, which allows to illustrate the shape of the vortex breakdown bubbles. In particular the spherical shape of the first appearing breakdown bubble leads to the idea to introduce a streamfunction, which describes the spherical breakdown bubble approximately. Applying a Taylor expansion of the velocity field leads to an analytical description of the local stream line topology nearby one critical point of a breakdown bubble. The interpretation of the appendant differential equations allows a deeper insight into the dynamical behaviour of the breakdown phenomenon and its main enforcing parameters. The paper presents the results of a local streamline and vortex line topology analysis, especially the dynamical relation between the velocity and vorticity field in regard to the topological structure of the vortex breakdown phenomenon in the lid driven cylinder.

A pattern often found in regions of recirculating flow is the vortex ring. Smoke rings and vortex breakdown bubbles are two familiar instances of this pattern. A vortex ring requires at least two critical points, and in fact this minimum number is observed in many synthetic or real-world examples. Based on this observation, we propose a visualization technique utilizing a Poincaré section that contains the pair of critical points. The Poincaré section by itself can be taken as a visualization of the vortex ring, especially if streamlines are seeded on the stable and unstable manifolds of the critical points. The resulting image reveals the extent of the structure, and more interestingly, regions of chaos and islands of stability. As a next step, we describe for the case of incompressible flow an algorithm for finding invariant tori in an island of stability. The basic idea is to find invariant closed curves in the Poincaré plane, which are then taken as seed curves for stream surfaces. For visualization the two extremes of the set of nested tori are computed. This is on the inner side the periodic orbit toward which the tori converge, and on the outer side, a torus which marks the boundary between ordered and chaotic flow, a distinction which is of importance for the mixing properties of the flow. For the purpose of testing, we developed a simple analytical model of a perturbed vortex ring based on Hill's spherical vortex. Finally, we applied the proposed visualization methods to this synthetic vector field and to two hydromechanical simulation results.

The visualization community is currently witnessing strong advances in topology-based flow visualization research. Numerous algorithms have been pro posed since the introduction of this class of approaches in 1989. Yet despite the many advances in the field, topology-based flow visualization methods have, until now, failed to penetrate industry. Application domain experts are still, in general, not using topological analysis and visualization in daily practice. We present a range of state-of-the art topology-based flow visualization methods such as vortex core line extraction, singularity and separatrix extraction, and periodic orbit extraction techniques, and apply them to real-world data sets. Applications include the visual ization of engine simulation data such as in-cylinder flow, cooling jacket flow, as well as flow around a spinning missile. The novel application of periodic orbit extraction to the boundary surface of a cooling jacket is presented. Based on our experiences, we then describe what we believe needs to be done in order to bring topological flow visualization methods to industry-level software applications. We believe this discussion will inspire useful directions for future work.

Curve-skeletons of 3-D objects are medial axes shrunk to a single line. There are several applications for curve-skeletons. For example, animation of 3-D objects, such as an animal or a human, as well as planning of flight paths for virtual colonoscopy. Other applications are the extraction of center lines within blood vessels where center lines are used to quantitatively measure vessel length, vessel diameter, and angles between vessels. The described method computes curve-skeletons based on a vector field that is orthogonal to the object's boundary surface. A topological analysis of this field then yields the center lines of the curve-skeletons. In contrast to previous methods, the vector field does not need to be computed for every sampled point of the entire volume. Instead, the vector field is determined only on the sample points on the boundary surface of the objects. Since most of the computational time was spent on calculating the force field in previous methods, the proposed approach requires significantly less time compared to previous vector-based techniques while still achieving a better accuracy and robustness compared to methods based on Voronoi tessellations.