Visualization of Nonlinear Vector Field Topology III

Abstract: In this paper we present an approach to the analysis of the contribution of a small subregion in a dataset to the global flow. To this purpose, we subtract the potential flow that is induced by the boundary of the sub-domain from the original flow. Since the potential flow is free of both divergence and rotation, the localized flow field retains the original features. In contrast to similar approaches, by making explicit use of the boundary flow of the subre- gion, we manage to isolate the region-specific flow that contains exactly the local contribution of the considered subdomain to the global flow. In the remainder of the paper, we describe an implementation on unstructured grids in both two and three dimensions. We discuss the application of several widely used feature extraction methods on the localized flow, with an emphasis on topological schemes.

Abstract: It is a challenging task to visualize the behavior of time-dependent 3D vector fields. Most of the time an overview of unsteady fields is provided via animations, but, unfortunately, animations provide only transient impressions of momentary flow. In this paper we present two approaches to visualize time varying fields with fixed geometry. Path lines and streak lines represent such a steady visu- alization of unsteady vector fields, but because of occlusion and vi- sual clutter it is useless to draw them all over the spatial domain. A selection is needed. We show how bundles of streak lines and path lines, running at different times through one point in space, like through an eyelet, yield an insightful visualization of flow structure (“eyelet lines”). To provide a more intuitive and appealing visual- ization we also explain how to construct a surface from these lines. As second approach, we use a simple measurement of local changes of a field over time to determine regions with strong changes. We visualize these regions with isosurfaces to give an overview of the activity in the dataset. Finally we use the regions as a guide for placing eyelets.

Abstract: Predicates are functions that return Boolean values. They are an essential tool in computer science. A close look at flow feature definitions reveals that they can be seen as point predicates that tell if a specific feature exists at a certain point. Besides the information about features, scientists and engineers like to know the overall behavior of all streamlines in the flow, typically in the connection with the important features in their application domain. We call this a structure definition for the flow. A successful example for a structure definition is flow topology. In this paper, we present streamline predicates as functions that tell the user about the connection between streamlines and features selected by the user. This means answers to questions like: Which streamlines flow through a given vortex, separation bubble or shock wave? It can be shown that streamline predicates may refine flow topology so that it also reveals questions about vortices in 3D.

Abstract: This work is an attempt to test the concept of the hydrodynamic charge (analogous to the electric charge in electromagnetism) in the simple case of a coherent structure such as the Burgers vortex. We provide experimental measurements of both the so- called Lamb vector and its divergence (the charge) by two-dimensional particles images velocimetry. In addition, we perform a Helmholtz–Hodge decomposi- tion of the Lamb vector in order to explore its topological features. We compare the charge with the well-known Q-criterion in order to assess its interest in detecting and characterizing coherent structure. Usefulness of this concept in studies of vortex dynamics is demonstrated.

Abstract: Using topology for feature analysis in flow fields faces several problems. First of all, not all features can be detected using topology based methods. Second, while in flow feature analysis the user is interested in a quantification of feature parameters like position, size, shape, radial velocity and other parameters of feature models, many of these parameters can not be determined using topology based methods alone. Additionally, in some applications it is advantageous to regard the vector field as a superposition of several, possibly simple, features. As topology based methods are quite sensitive to superposition effects, their precision and usability is limited in these cases. In this paper, topology based analysis and visualization of flow fields is estimated and compared to other feature based approaches demonstrating these problems.

Abstract: We present, extend and apply a method to extract the contribution of a subregion of a data set to the global flow. To isolate this contribution we decompose the flow in the subregion into a potential flow that is induced by the original flow on the boundary and a localized flow. The localized flow is obtained by subtracting the potential flow from the original flow. Since the potential flow is free of both divergence and rotation the localized flow retains the original features and captures the region-specific flow that contains the local contribution of the considered sub- domain to the global flow. In the remainder of the paper, we describe an implementation on unstructured grids in both two and three dimensions for steady and unsteady flow fields. We discuss the application of some widely used feature extraction methods on the localized flow and describe applications like reverse-flow detection using the potential flow. Finally, we show that our algorithm is robust and scalable by applying it to various flow data sets and giving performance figures.

Abstract: Streamline predicates are simply boolean functions on the set of all streamlines in a flow field. A characteristic set of a streamline predicate is the set of all streamlines fulfilling the predicate. If streamline predicates are defined based on asymptotic be- havior, the characteristic sets become α- or ω-basins. Using boolean algebra on the streamline predicates, we obtain the usual flow topology. We show that these con- siderations allow us to generalize flow topology to flow structure definitions. These flow structure definitions can be flexibly adapted to typical analysis tasks arising in flow studies and taylored to the users’ needs

Abstract: Visualizing flow structures according to the users’ inter- ests provides insight to scientists and engineers. In previ- ous work, a flow structure based on streamline predicates, that examine, whether a streamline has a given property, was defined. Evaluating all streamlines results in charac- teristic sets grouping all streamlines with similar behavior with respect to a given predicate. Since there are infinitely many streamlines, the algorithm chooses a finite subset for the computation of an approximated discrete version of the characteristic sets. However, even the construction of char- acteristic sets based on a finite set of streamlines tends to be computationally expensive. Based on a thorough analysis of all processing steps, we present and compare different acceleration approaches. The techniques are based on sim- plifications that result in characteristic set boundaries devi- ating from the correct but computational expensive bound- aries. We develop measures for objective comparison of the introduced errors. An adaptive refinement approach turns out to be the best compromise between computation time and quality.

Abstract: We present a method to extract and visualize vortices that originate from bounding walls of three-dimensional time- dependent flows. These vortices can be detected using their footprint on the boundary, which consists of critical points in the wall shear stress vector field. In order to follow these critical points and detect their transformations, affected regions of the surface are parameterized. Thus, an existing singularity tracking algorithm devised for planar settings can be applied. The trajectories of the singularities are used as a basis for seeding particles. This leads to a new type of streak line visualization, in which particles are released from a moving source. These generalized streak lines visualize the particles that are ejected from the wall. We demonstrate the usefulness of our method on several transient fluid flow datasets from computational fluid dynamics simulations.

Abstract: In most fluid dynamics applications, unsteady flow is a natural phenomenon and steady models are just sim- plifications of the real situation. Since computing power in- creases, the number and complexity of unsteady flow simu- lations grows, too. Besides time-dependent features, scien- tists and engineers are essentially looking for a description of the overall flow behavior, usually with respect to the re- quirements of their application domain. We call such a de- scription a flow structure, requirering a framework of defi- nitions for unsteady flow structure. In this paper, we present such a framework based on pathline predicates. Using the common computer science definition, a predicate is a Boolean function, and a pathline predicate is a Boolean function on pathlines that decides if a pathline has a property of interest to the user. We will show that any suitable set of pathline predicates can be interpreted as an unsteady flow structure definition. The visualization of the resulting unsteady flow structure provides a visual description of overall flow behav- ior with respect to the user’s interest. Furthermore this flow structure serves as a basis for pathline placements tailored to the requirements of the application.

Abstract: Flow visualization has been a very active subfield of scientific visualization in recent years. From the resulting large variety of methods this paper discusses partition-based techniques. The aim of these approaches is to partition the flow in areas of common structure. Based on this partitioning, subsequent visualization techniques can be ap- plied. A classification is suggested and advantages/disadvantages of the different tech- niques are discussed as well.

Abstract: In this paper, we present an overview of a number of existing flow visualization meth- ods, developed by the authors in the recent past, that are specifically aimed at integrat- ing and leveraging domain-specific knowledge into the visualization process. These methods transcend the traditional divide between interactive exploration and feature- based schemes and allow a visualization user to benefit from the abstraction properties of feature extraction and topological methods while retaining intuitive and interactive control over the visual analysis process, as we demonstrate on a number of examples.

Abstract: 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.

Abstract: 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.