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Saddle Connectors - An Approach to Visualizing the Topological Skeleton of Complex 3D Vector Fields
... Having described the extraction of 4D critical points, and having classified their properties in terms of inflow, outflow, and rotation, we now can address their representation by means of glyphs. Our approach is inspired by the glyphs for 3D vector field topology by Theisel et al. [TWHS03]. Similar to the 3D approach, we build our glyphs in the respective space, i.e., our glyphs are fourdimensional. ...
... There are two main difficulties with the visualization of 3D manifolds in 4D topology: (I) visual clutter due to their volumetric appearance, and (II) (self-)intersection with manifolds due to projection; although manifolds cannot intersect in 4D space because they consist of streamlines there, their projection in 3D can intersect and tends to do so, leading to 3D images that are hard to interpret and explore. This is problematic because topological structure is often conveyed by configurations where stable and unstable manifolds meet, e.g., at critical points and saddle connectors [TWHS03]. ...
... We thus complement it by computing for each vertex of each manifold the shortest 4D distance to all vertices of all other manifolds.We take the logarithm of this distance field and map values close to one to high contribution in the green channel and high opacity (see, e.g., Figure 4(d)). One can see that this technique successfully reveals regions where manifolds meet in 4D; in particular it reveals saddle connectors [TWHS03] in 4D. We leave the extraction of saddle connectors in 4D, however, as future work. ...
In this paper, we present an approach to the topological analysis of four-dimensional vector fields. In analogy to traditional 2D and 3D vector field topology, we provide a classification and visual representation of critical points, together with a technique for extracting their invariant manifolds. For effective exploration of the resulting four-dimensional structures, we present a 4D camera that provides concise representation by exploiting projection degeneracies, and a 4D clipping approach that avoids self-intersection in the 3D projection. We exemplify the properties and the utility of our approach using specific synthetic cases.
... For unsteady flow fields, it is also important to discover the temporal development of critical points detected at each time step. Previous approaches were developed to capture these patterns [1] and reveal their connections [2]. However, they usually serve specific purposes, provide very limited interaction support, and fail to meet various exploration needs. ...
... The shape of template can change depending on the degree of a critical point transitioning from one type into another. Theisel et al. [2] presented saddle connectors that visualize the topological skeleton of vector fields. Iconic representations are used for critical points, and the specific streamlines connecting different critical points are displayed. ...
... Support for unsteady flow fields: revealing the evolution of flows and features in unsteady flow fields. We compare our approach against four previous approaches: namely, FlowString [10], [35] (FS), flow web [22] (FW), FlowGraph [23], [24] (FG), and saddle connectors [2] (SC). Table 2 shows a summary of this comparison. ...
Visual exploration of flow fields is important for studying dynamic systems. We introduce semantic flow graph (SFG), a novel graph representation and interaction framework that enables users to explore the relationships among key objects (i.e., field lines, features, and spatiotemporal regions) of both steady and unsteady flow fields. The objects and their relationships are organized as a heterogeneous graph. We assign each object a set of attributes, based on which a semantic abstraction of the heterogeneous graph is generated. This semantic abstraction is SFG. We design a suite of operations to explore the underlying flow fields based on this graph representation and abstraction mechanism. Users can flexibly reconfigure SFG to examine the relationships among groups of objects at different abstraction levels. Three linked views are developed to display SFG, its node split criteria and history, and the objects in the spatial volume. For simplicity, we introduce SFG construction and exploration for steady flow fields with critical points being the only features. Then we demonstrate that SFG can be naturally extended to deal with unsteady flow fields and multiple types of features. We experiment with multiple data sets and conduct an empirical expert evaluation to demonstrate the effectiveness of our approach.
... De Leeuw and van Wijk [3] propose a glyph that, among others, contains derived values from the Jacobian, where the eigenvalues of the Jacobian are not directly encoded, and thus lacking (c). Theisel et al. [19] propose a glyph that lacks uniqueness (c). Palke et al. [13] use glyphs for asymmetric tensors with complex eigenvalues. ...
... Existing techniques are evaluated with respect to satisfying conditions (a)-(e) from section 2. In addition, the column (f) indicates if the technique is general, i.e., not restricted to symmetric tensors. method / satisfies (a) (b) (c) (d) (e) (f) Kindlmann and Schultz [8,15] tensor decomposition Globus et al. [6] de Leeuw and van Wijk [3] Theisel et al. [19] Mohr's circle [2] Haber glyph [7] this paper ...
... We use a rather straightforward color scheme to encode one continuous value, in 2d this is γ. We do so to ensure comparability with similar approaches [15,19] that proposed similar colors. Other and in particular more perception-oriented color maps are possible. ...
Glyphs are a powerful tool for visualizing second-order tensors in a variety of scientic data as they allow to encode physical behavior in geometric properties. Most existing techniques focus on symmetric tensors and exclude non-symmetric tensors where the eigenvectors can be non-orthogonal or complex. We present a new construction of 2d and 3d tensor glyphs based on piecewise rational curves and surfaces with the following properties: invariance to (a) isometries and (b) scaling, (c) direct encoding of all real eigenvalues and eigenvectors, (d) one-to-one relation between the tensors and glyphs, (e) glyph continuity under changing the tensor. We apply the glyphs to visualize the Jacobian matrix fields of a number of 2d and 3d vector fields.
... with eigenvalues λ 1,2 = a ± i and λ 3 = −2b < 0. Therefore, X * 0 is a repelling focus saddle (see [Theisel et al., 2003]). ...
... The results are centralized in Fig. 4, where the iconic representations indicate the stability type (see [Theisel et al., 2003]). ...
... Also, by assuming that between two saddles there exists a connection, for example, between the saddles X * 3,4 and X * 1,2 , this must be one of the trajectories contained in W u (X * 3,4 ) ∩ W s (X * 1,2 ). These kind of intersections can also be numerically determined [Theisel et al., 2003]. ...
Recently, we looked more closely into the Rabinovich–Fabrikant system, after a decade of study [Danca & Chen, 2004], discovering some new characteristics such as cycling chaos, transient chaos, chaotic hidden attractors and a new kind of saddle-like attractor. In addition to extensive and accurate numerical analysis, on the assumptive existence of heteroclinic orbits, we provide a few of their approximations.
... Motivated by the need for a more generic approach to characterize streaming flows, we turn to dynamical systems theory, as previously proposed for 2-D settings in Bhosale et al. (2020). This approach offers a sparse yet complete representation of the underlying flow topology and its dynamics, and generalizes to three dimensions (Chong, Perry & Cantwell 1990;Theisel et al. 2003). We first identify the zero velocity, critical points of the streaming field and classify these points based on their local flow properties, characterized through the eigenvalues/eigenvectors of the Jacobian J u associated with the velocity field (Chong et al. 1990). ...
... This allows for the existence of in-plane nodes (real eigenvalues of equal sign -figure 1f ) and in-plane foci (complex-conjugate eigenvalues -figure 1i), both of which can be unstable/repelling or stable/attracting in nature, depending on the signs of the eigenvalues. Under the incompressibility constraint, admissible combinations of local in-plane flows result in node-saddle-saddle (NSS, repelling example in figure 1g) and focus-saddle-saddle (FSS, repelling example in figure 1j) critical points (Chong et al. 1990;Theisel et al. 2003). ...
Recent studies on viscous streaming flows in two dimensions have elucidated the impact of body curvature variations on resulting flow topology and dynamics, with opportunities for microfluidic applications. Following that, we present here a three-dimensional characterization of streaming flows as functions of changes in body geometry and topology, starting from the well-known case of a sphere to progressively arrive at toroidal shapes. We leverage direct numerical simulations and dynamical systems theory to systematically analyse the reorganization of streaming flows into a dynamically rich set of regimes, the origins of which are explained using bifurcation theory.
... These stream surfaces tend to hide each other as well as other topological features and make complex 3D topology visualizations hardly interpretable. Saddle connectors are one solution of the occlusion problem [TWHS03]. They represent the separation surfaces as a finite number of stream lines. ...
... A special case of saddle connections is the so-called periodic blue sky bifurcation [AS92] where two separatrices of the same saddle collapse. Saddle connections of 2D timedependent vector fields can be extracted by making an adaption of the saddle connectors approach [TWHS03] which are the intersection curves of the separation surfaces of a 3D vector field starting in the outflow and inflow planes of the saddle points. The basic idea is to numerically integrate two separation surfaces until an intersection is found. ...
... While this strategy may work well for 2D vector fields [4], [5], [6], it is not straightforward to extend it to 3D vector fields due to the increasing complexity of 3D topology. In addition, the full 3D vector field topology can potentially be expensive to extract due to the increased dimensionality of the separatrices as well as numerical instabilities [3], [7], [8], making it less practical for large-scale datasets. Furthermore, such simplification typically does not take into account the influence of flow magnitude, an important physical property of the flow. ...
... This increased complexity has made the extraction and visualization of 3D vector field topology challenging [20], [21]. Theisel et al. introduced the saddle-connector to reduce the occlusion issue in the visualization of 3D topology [7]. Weinkauf et al. [3], [8] introduced a technique to visualize high-order critical points, which is achieved via the F-classification of a derived auxiliary tangential vector field defined on a closed surface surrounding each critical point. ...
Vector field topology has been successfully applied to represent the structure of steady vector fields. Critical points, one of the essential components of vector field topology, play an important role in describing the complexity of the extracted structure. Simplifying vector fields via critical point cancellation has practical merit for interpreting the behaviors of complex vector fields such as turbulence. However, there is no effective technique that allows direct cancellation of critical points in 3D. This work fills this gap and introduces the first framework to directly cancel pairs or groups of 3D critical points in a hierarchical manner with a guaranteed minimum amount of perturbation based on their robustness, a quantitative measure of their stability. In addition, our framework does not require the extraction of the entire 3D topology, which contains non-trivial separation structures, and thus is computationally effective. Furthermore, our algorithm can remove critical points in any subregion of the domain whose degree is zero and handle complex boundary configurations, making it capable of addressing challenging scenarios that may not be resolved otherwise. We apply our method to synthetic and simulation datasets to demonstrate its effectiveness.
... Lö elmann et al. additionally use short streamlines (streamlets) to depict the local ow behavior in the direct vicinity of critical points [111] and along selected streamlines [112]. Theisel et al. reduce clutter by visualizing saddle connectors obtained as the intersection curves from two 2D separatrices of an attracting and a repelling saddle [185]. As the connectors fail to separate the ow into di erent regions, the complete surfaces can still be shown by the system on demand. ...
... Topological techniques based on streamlines or stream surfaces fail to capture the true time-dependent separation behavior of the ow [175]. Theisel et al. presented an approach that determines the separation behavior of path lines locally [185], hence not allowing for an asymptotic analysis, like in the case of the steady vector eld topology. The path line oriented topology by Shi et al. [178] accounts for this issue, however, it restricts the analysis to time-periodic vector elds. ...
Computational Wuid dynamics (CFD) has become an important tool for predicting Fluid behavior in research and industry. Today, in the era of tera- and petascale computing, the complexity and the size of simulations have reached a state where an extremely large amount of data is generated that has to be stored and analyzed. An indispensable instrument for such analysis is provided by computational Wow visualization. It helps in gaining insight and understanding of the Wow and its underlying physics, which are subject to a complex spectrum of characteristic behavior, ranging from laminar to turbulent or even chaotic characteristics, all of these taking place on a wide range of length and time scales. The simulation side tries to address and control this vast complexity by developing new sophisticated models and adaptive discretization schemes, resulting in new types of data. Examples of such emerging simulations are generalized Vnite element methods or hp-adaptive discontinuous Galerkin schemes of high-order. This work addresses the direct visualization of the resulting higher-order Veld data, avoiding the traditional resampling approach to enable a more accurate visual analysis. The second major contribution of this thesis deals with the inherent complexity of Wuid dynamics. New feature-based and topology-based visualization algorithms for unsteady Wow are proposed to reduce the vast amounts of raw data to their essential structure. For the direct visualization pixel-accurate techniques are presented for 2D Veld data from generalized Vnite element simulations, which consist of a piecewise polynomial part of high order enriched with problem-dependent ansatz functions. Secondly, a direct volume rendering system for hp-adaptive Vnite elements, which combine an adaptive grid discretization with piecewise polynomial higher-order approximations, is presented. The parallel GPU implementation runs on single workstations, as well as on clusters, enabling a real-time generation of high quality images, and interactive exploration of the volumetric polynomial solution. Methods for visual debugging of these complex simulations are also important and presented. Direct Wow visualization is complemented by new feature and topology-based methods. A promising approach for analyzing the structure of time-dependent vector Velds is provided by Vnite-time Lyapunov exponent (FTLE) Velds. In this work, interactive methods are presented that help in understanding the cause of FTLE structures, and novel approaches to FTLE computation are developed to account for the linearization error made by traditional methods. Building on this, it is investigated under which circumstances FTLE ridges represent Lagrangian coherent structures (LCS)the timedependent counterpart to separatrices of traditional steady vector Veld topology. As a major result, a novel time-dependent 3D vector Veld topology concept based on streak surfaces is proposed. Streak LCS oUer a higher quality than corresponding FTLE ridges, and animations of streak LCS can be computed at comparably low cost, alleviating the topological analysis of complex time-dependent Velds.
... In the middle of the pressure taps 5-6 zone, there is a positive divergent source, also known as a repelling focus. [43][44][45] The streamlines start from the source and interact with the vortex in front of the source. Above the pressure tap 2, there is a negative divergent sink, i.e., attracting focus, where the streamlines rotate around the sink and end at this point. ...
This study uses two wind tunnel testing approaches: a sole high-frequency pressure scanning (HPS) approach, and a synchronized particle image velocimetry (PIV) and multipoint pressure scanning (SPMPS) approach, to investigate aerodynamic characteristics of a rectangular cylinder. HPS and SPMPS share identical experimental settings and device arrangements, while the SPMPS sampling frequency is much lower than HPS because the PIV system has a low sampling frequency in this study. SPMPS can simultaneously capture the flow field and surface pressure information. PIV measurement provides instantaneous flow field information, helping to analyze flow characteristics, and surface pressure taps offer both high- and low-frequency surface pressure information. It was found that for the rectangular cylinder, the zones of high turbulent kinetic energy and turbulent shear stress are associated with the lower negative pressure coefficient. In addition, diverse coherent structures in the instantaneous flow field resemble different critical points such as the saddle point, the repelling focus, and the attracting focus, and these coherent structures are associated with drastic changes in the pressure distribution or extreme pressure values. In particular, there is a visible flow reattachment, and the lift coefficient is more sensitive to the pressure distribution around the trailing edge of the rectangular cylinder. The convective velocity of wall pressure fluctuations is calculated based on the spatial temporal correlation of HPS information, and the convective velocity on the upper surface of the cylinder of SR = 3.25 is around 3.1 m/s providing evidence that Taylor's hypothesis breaks down for wall pressure fluctuations.
... In the vicinity of any CP, the two eigenvectors that have the same sign of their real parts of eigenvalues span a 2D eigenspace tangent to the separatrix surface, and the remaining one eigenvector spans a bidirector line. Any pair of an attracting CP and a repelling CP can be connected by a separator line and forms the so-called saddle-connection [6], as shown in Figure 1 (b). In the present report, we use ∑ to represent the separatrix surface and to represent the bidirector line. ...
... In the work of Globus et al. [12], the condition of uniqueness was not satisfied, because only the real eigenvalues were represented in the form of an ellipsoid, and it did not hold for complex values. Similarly, Theisel et al. [13] showed glyphs that failed to satisfy the condition of uniqueness, as well as that of change continuity. Technical mechanics Machines 2022, 10, 89 2 of 15 approaches the visualization of tautness tensors in several ways. ...
The rendering of tensor glyphs is a progressive process of visualizing the vector space both in fluid dynamics and the latest medical scanning. Nowadays, the rendering accuracy is ensured by numerical methods based on interpolation of tensor functions. The tensor glyph functions to visualize significant properties of the vector space. Not all these properties are visualized at all times. The number of properties and their unambiguity depend on the method chosen. This work presents a direct analytical expression covering rank two tensors in a plane. Unlike the methods used so far, this method is accurate and unambiguous one for tensor visualization. The method was applied to the simplest tensor type, which presented an advantage for the method’s analytical approach. The analytical approach to the planar case is significant also because it provides instruction on how to expand analytical calculations to cover higher spatial dimensions. In this way, numerical methods for tensor rendering can be replaced with an accurate analytical method.
... Governing the asymptotic motion are a number of topological elements, which were described by Helman and Hesselink [26], including critical points (sinks, sources, centers, saddles), boundary elements (attachment and detachment points),the manifolds that separate flow regions of homogeneous asymptotic behavior (separatrices), and periodic orbits [2]. The extension to the 3D case [27] gave rise to a broader variety of elements, such as bifurcation lines [39] (lines to which nearby streamlines are asymptotically drawn to or repelled away from at an exponential rate) or saddle connectors [52] (individual streamlines that connect saddles). Aside from characterizations as extremal lines [32] of vortex-related scalar fields [46,47], vortex corelines have also been expressed as lines along which the velocity vector aligns with the single real-valued eigenvector of the Jacobian matrix [51]. ...
... Intersections of streak manifolds play an important role in unsteady flow analysis [MW98], and they manifest themselves as false-positive HTs (FTLE ridge intersections). Analogously to 2D unsteady flows [HS20], where such connectors can be related to saddle connectors in the corresponding steady 3D space-time vector field [TWHS03], the corresponding configurations in 3D unsteady flow can be found in the steady 4D space-time vector field [HRS18]. As indicated by the authors, two 4D saddle-type critical points can posses surfaces of saddle connectors. ...
We present an approach to local extraction of 3D time‐dependent vector field topology. In this concept, Lagrangian coherent structures, which represent the separating manifolds in time‐dependent transport, correspond to generalized streak manifolds seeded along hyperbolic path surfaces (HPSs). Instead of expensive and numerically challenging direct computation of the HPSs by intersection of ridges in the forward and backward finite‐time Lyapunov exponent (FTLE) fields, our approach employs local extraction of respective candidates in the four‐dimensional space‐time domain. These candidates are subsequently refined toward the hyperbolic path surfaces, which provides unsteady equivalents of saddle‐type critical points, periodic orbits, and bifurcation lines from steady, traditional vector field topology. In contrast to FTLE‐based methods, we obtain an explicit geometric representation of the topological skeleton of the flow, which for steady flows coincides with the hyperbolic invariant manifolds of vector field topology. We evaluate our approach on analytical flows, as well as data from computational fluid dynamics, using the FTLE as a ground truth superset, i.e., we also show that FTLE ridges exhibit several types of false positives.
... In 3D steady vector fields, the separatrices of saddle-type critical points can intersect in streamlines, which form connections between two different (heteroclinic) or one (homoclinic) saddle-type critical point. These are also called saddle connectors [TWHS03]. Similar connections can be formed by hyperbolic trajectories in unsteady vector fields [MW98]. ...
This paper does two main contributions to 2D time‐dependent vector field topology. First, we present a technique for robust, accurate, and efficient extraction of distinguished hyperbolic trajectories (DHT), the generative structures of 2D time‐dependent vector field topology. It is based on refinement of initial candidate curves. In contrast to previous approaches, it is robust because the refinement converges for reasonably close initial candidates, it is accurate due to its adaptive scheme, and it is efficient due to its high convergence speed. Second, we provide a detailed evaluation and discussion of previous approaches for the extraction of DHTs and time‐dependent vector field topology in general. We demonstrate the utility of our approach using analytical flows, as well as data from computational fluid dynamics.
... Governing the asymptotic motion are a number of topological elements, which were described by Helman and Hesselink [26], including critical points (sinks, sources, centers, saddles), boundary elements (attachment and detachment points),the manifolds that separate flow regions of homogeneous asymptotic behavior (separatrices), and periodic orbits [2]. The extension to the 3D case [27] gave rise to a broader variety of elements, such as bifurcation lines [39] (lines to which nearby streamlines are asymptotically drawn to or repelled away from at an exponential rate) or saddle connectors [52] (individual streamlines that connect saddles). Aside from characterizations as extremal lines [32] of vortex-related scalar fields [46,47], vortex corelines have also been expressed as lines along which the velocity vector aligns with the single real-valued eigenvector of the Jacobian matrix [51]. ...
We extend the definition of the classic instantaneous vector field saddles, sinks, and sources to the finite-time setting by categorizing the domain based on the behavior of the flow map w.r.t. contraction or expansion. Since the intuitive Lagrangian approach turns out to be unusable in practice because it requires advection in unstable regions, we provide an alternative, sufficient criterion that can be computed in a robust way. We show that both definitions are objective, relate them to existing approaches, and show how the generalized critical points and their separatrices can be visualized.
... Theisel et al's approach to constructing saddle connectors in place of separating stream surfaces addresses the challenges of occlusion in 3D flow visualisation (38) . This was extended by Weinkauf et al (44) using separating stream surfaces originating from boundary switch curves. ...
This application paper describes a novel, cluster-based, semi-automatic, stream surface placement strategy for structured and unstructured computational fluid dynamics (CFD) data, tailored towards a specific application: The BLOODHOUND jet and rocket propelled land speed record vehicle. An existing automatic stream surface placement algorithm
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, is extensively modified to cater for large unstructured CFD simulation data. The existing algorithm uses hierarchical clustering of velocity and distance vectors to find potential stream surface seeding locations. This work replaces the hierarchical clustering algorithm, designed to work with small regular grids, with a K-means clustering approach suitable for large unstructured grids. Modifications are made to the seeding curve construction algorithm, improving the smoothness and distribution of the discretised curve in complex cases. A new distance function is described which allows the user to target particular characteristics of simulation data. The proposed algorithm reduces the required memory footprint and computational requirement compared to previous work
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. The performance and effectiveness of the proposed algorithm is demonstrated, and CFD domain expert evaluation is provided describing the value of this approach.
... Weinkauf et al. [32] extended this approach to three dimensions. Theisel et al. [29] introduced the concept of saddle connectors, which are the intersection of the separation surfaces emanating from the saddles points. Wischgoll et al. [34] extracted attracting and repelling periodic orbits in planar flows by searching for cell cycles. ...
Gauss' theorem, which relates the flow through a surface to the vector field inside the surface, is an important tool in Flow Visualization. We are exploit the fact that the theorem can be further refined on polygonal cells and construct a process that encodes the particle movement through the boundary facets of these cells using transition matrices. By pure power iteration of transition matrices, various topological features, such as separation and invariant sets, can be extracted without having to rely on the classical techniques, e.g., interpolation, differentiation and numerical streamline integration. We will apply our method to steady vector fields with a focus on three dimensions.
... We distinguish two abstraction types: Semantic abstractions A i simplify the information in d by showing varying amounts of the information present in d using different visual representations. For example, a fluid flow volume d ⊂ R 3 can be rendered as an entire flow volume using LIC [5, 22], as stream LIC structures for a set of given streamlines [14], and as flow topology [25] ; these are increasingly simplified semantic representations. Sampling abstractions reduce the amount of points produced by a given semantic abstraction A i using data sampling. ...
... As visualizations of the topology of three-dimensional flow fields typically involve a large number of separating surfaces they suffer from occlusion problems and accordingly tend to get visually cluttered. To deal with that problem, Theisel et al. introduced the concept of saddle connectors [206] and boundary switch connectors [219] as a method for the simplified visualization of the topological skeleton of complex three-dimensional vector fields. However, the visual complexity, even of these simplified visualizations, depends heavily on the number of critical points involved. ...
Modern numerical simulation and data acquisition techniques create a multitude of different data fields. The interactive visualization of these large, three-dimensional, and often also time-dependent scalar, vector, and tensor fields plays an integral part in analyzing and understanding this data. Although basic visualization techniques vary significantly depending on the type of the respective data fields, there is one key feature that is dominating in today's visualization research. Driven by the need for interactive data inspection and exploration and by the extraordinary rate of increase of the computational power provided by modern graphics processing units, the attempt for consequent application of graphics hardware in all stages of the visualization pipeline has become a central theme in order to cope with the challenges of data set sizes growing at an ever increasing pace and advancing demands on the accuracy and complexity of visualizations. Contemporary graphics processing units now have reached a level of programmability roughly resembling their CPU counterparts. However, there are still important differences that strongly influence the design and implementation of GPU-based visualization algorithms.
This thesis addresses the problem of how to efficiently exploit the programmability and parallel processing capabilities of modern graphics processors for interactive visualization of three-dimensional data fields of varying data complexity and abstraction level. In particular new methods and GPU-based solutions for high-quality volume ray casting, the reconstruction of polygonal isosurfaces, and the point-based visualization of symmetric, second-order tensor fields, such as obtained by diffusion tensor imaging or resulting from CFD simulations, by means of ellipsoidal glyphs are presented that by combining the mapping and rendering stage onto the GPU result in an improved visualization cycle. Furthermore, a new approach for the topological analysis of noisy vector fields is described.
Although this work is focused on a number of specific visualization problems, it also intends to identify general design principles for GPU-based visualization algorithms that may prove useful in the context of topics not covered by this thesis.
... More recently, vector field topology has been extended to uncertain [19,20] and time-dependent [28] vector fields. Other conceptual contributions to this field include saddle connectors [25] and connectrices [5]. Scheuermann et al. [24] include the domain boundary in topological analysis to avoid missed topological structures in 2D vector fields. ...
We present a technique to visualize the streamline-based mapping between the boundary of a simply-connected subregion of arbitrary 3D vector fields. While the streamlines are seeded on one part of the boundary, the remaining part serves as escape border. Hence, the seeding part of the boundary represents a map of streamline behavior, indicating if streamlines reach the escape border or not. Since the resulting maps typically exhibit a very fine and complex structure and are thus not amenable to direct sampling, our approach instead aims at topologically consistent extraction of their boundary. We show that isocline surfaces of the projected vector field provide a robust basis for stream-surface-based extraction of these boundaries. The utility of our technique is demonstrated in the context of transport processes using vector field data from different domains.
... As visualizations of the topology of three-dimensional flow fields typically involve a large number of separating surfaces they suffer from occlusion problems and accordingly tend to get visually cluttered. To deal with that problem, Theisel et al. introduced the concept of saddle connectors [206] and boundary switch connectors [219] as a method for the simplified visualization of the topological skeleton of complex three-dimensional vector fields. However, the visual complexity, even of these simplified visualizations, depends heavily on the number of critical points involved. ...
In view of the inconvenient display of children’s clothing, the poor experience of designers when modifying sample clothes, and the immature display of children’s clothing, which affects the rational judgment of fashion buyers, this paper puts forward the modular design method of children’s clothing virtual fitting. This research uses Lectra CAD software to make children’s clothing pattern, Clo3D for virtual modular design, and “Insanda children’s clothing virtual design and show” for virtual try-on and show. The simulation results show that the research can effectively improve the efficiency of designers and buyers, can save costs, and can be effectively applied in the commercial field. At the same time, it provides a practical basis for the development of children’s clothing virtual display technology and the formulation of standards related to children’s clothing in the future.
Streamlines are an extensively utilized flow visualization technique for understanding, verifying, and exploring computational fluid dynamics simulations. One of the major challenges associated with the technique is selecting which streamlines to display. Using a large number of streamlines results in dense, cluttered visualizations, often containing redundant information and occluding important regions, whereas using a small number of streamlines could result in missing key features of the flow. Many solutions to select a representative set of streamlines have been proposed by researchers over the past two decades. In this state‐of‐the‐art report, we analyze and classify seed placement and streamline selection (SPSS) techniques used by the scientific flow visualization community. At a high‐level, we classify techniques into automatic and manual techniques, and further divide automatic techniques into three strategies: density‐based, feature‐based, and similarity‐based. Our analysis evaluates the identified strategy groups with respect to focus on regions of interest, minimization of redundancy, and overall computational performance. Finally, we consider the application contexts and tasks for which SPSS techniques are currently applied and have potential applications in the future.
One of the barriers to visualization-enabled scientific discovery is the difficulty in clearly and quantitatively articulating the meaning of a visualization, particularly in the exploration of relationships between multiple variables in large-scale data sets. This issue becomes more complicated in the visualization of three-dimensional turbu- lence, since geometry, topology, and statistics play complicated, intertwined roles in the definitions of the features of interest, making them difficult or impossible to precisely describe.
This dissertation develops and evaluates a novel interactive multivariate volume visualization framework that allows features to be progressively isolated and defined using a combination of global and feature-local properties. I argue that a progressive and interactive multivariate feature-local approach is advantageous when investigating ill-defined features because it provides a physically meaningful, quantitatively rich en- vironment within which to examine the sensitivity of the structure properties to the identification parameters. The efficacy of this approach is demonstrated in the analysis of vortical structures in Taylor-Green turbulence. Through this analysis, two distinct structure populations have been discovered in these data: structures with minimal and maximal local absolute helicity distributions. These populations cannot be distinguished via global distributions; however, they were readily identified by this approach, since their feature-local statistics are distinctive.
In mathematical modeling, it is often required the analysis of the vector field topology in order to predict the evolution of the variables involved. When a dynamical system is multistable, the trajectories approach different stable states, depending on the initial conditions. The aim of this work is the detection of the invariant manifolds of the saddle points to analyze the boundaries of the basins of attraction. Once that a sufficient number of separatrix points is found, a moving least squares meshfree method is involved to reconstruct the separatrix manifolds. Numerical results are presented to assess the method referring to tri‐stable models with complex attractors such as limit cycles or limit tori.
This work presents a novel method of visually capturing the relative importance of different flow regions with respect to a quantity of interest. Existing flow visualization techniques are enhanced to convey flow importance. This is accomplished by filtering their output with additional information obtained from the adjoint counterpart of the physical flow field. The additional adjoint data is of equal resolution to the physical flow field. The adjoint flow data links physical fluid regions to user chosen quantities of interest. Thus, regions that are less relevant to the quantity of interest can be masked or deemphasized to reduce clutter and to highlight the behavior of the more important flow regions. The concept is demonstrated on a series of fluid simulations. The visualizations highlight the temporally changing importance of flow features, regions of influence in complex flows, and occlusion reduction in 3D flows. The method is also demonstrated by checking the impact of perturbations introduced in flow regions that were found to be both important and unimportant based on the adjoint data.
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Vortices are commonly understood as rotating motions in fluid flows. The analysis of vortices plays an important role in numerous scientific applications, such as in engineering, meteorology, oceanology, medicine and many more. The successful analysis consists of three steps: vortex definition, extraction and visualization. All three have a long history, and the early themes and topics from the 70s survived to this day, namely the identification of vortex cores, their extent and the choice of suitable reference frames. This paper provides an overview over the advances that have been made in the last forty years. We provide sufficient background on differential vector field calculus, extraction techniques like critical point search and the parallel vectors operator, and we introduce the notion of reference frame invariance. We explain the most important region-based and line-based methods, integration-based and geometry-based approaches, recent objective techniques, the selection of reference frames by means of flow decompositions, as well as a recent local optimization-based technique. We point out relationships between the various approaches, classify the literature and identify open problems and challenges for future work.
An escape map is the partial mapping from seed points to exit points of streamlines in a bounded domain. The escape map is piecewise continuous, and a topological segmentation of the domain boundary yields the regions and curve segments on which it is continuous. Escape maps have recently been introduced in the context of studying the connectivity of coronal holes. Computation of escape maps faces the problem of exponentially diverging streamlines, where standard adaptive streamsurface methods can fail. As a tool to detect such places and to guide escape map computation, a technique based on isoclines has recently been proposed (Machado et al., IEEE Trans. Vis. Comput. Graph. 20(12):2604–2613, 2014). We show in this paper that, in the case of a divergence-free vector field, boundary switch connectors can be used as a purely topological alternative, which to the best of our knowledge is the first practical application of boundary switch connectors. We provide a systematic approach to the topological segmentation of 3D domain boundaries for divergence-free vector fields. Finally, we explore an alternative approach based on streamtubes and targeted at robustness in escape map computation. Simulation results as well as a synthetic vector field are used for validation.
Morse decomposition has been shown a reliable way to compute and represent vector field topology. Its computation first converts the original vector field into a directed graph representation, so that flow recurrent dynamics (i. e., Morse sets) can be identified as some strongly connected components of the graph. In this paper, we present a framework that enables the user to efficiently compute Morse decompositions of 3D piecewise linear vector fields defined on regular grids. Specifically, we extend the 2D adaptive edge sampling technique to 3D for the outer approximation computation of the image of any 3D cell for the construction of the directed graph. To achieve finer decomposition, a hierarchical refinement framework is applied to procedurally increase the integration steps and subdivide the underlying grids that contain certain Morse sets. To improve the computational performance, we implement our Morse decomposition framework using CUDA. We have applied our framework to a number of analytic and real-world 3D steady vector fields to demonstrate its utility.
Abstract Topological methods are important tools for data analysis, and recently receiving more and more attention in vector field visualization. In this paper, we give an introductory description to some important topological methods in vector field visualization. Besides traditional methods of vector field topology, space-time method and finite-time Lyapunov exponent, we also include in this survey Hodge decomposition, combinatorial vector field topology, Morse decomposition, and robustness, etc. In addition to familiar numerical techniques, more and more combinatorial tools emerge in vector field visualization. The numerical methods often rely on error-prone interpolations and interpolations, while combinatorial techniques produce robust but coarse features. In this survey, we clarify the relevant concepts and hope to guide future topological research in vector field visualization. Graphical abstract
Tensors model a wide range of physical phenomena. While symmetric tensors are sufficient for some applications (such as diffusion), asymmetric tensors are required, for example, to describe differential properties of fluid flow. Glyphs permit inspecting individual tensor values, but existing tensor glyphs are fully defined only for symmetric tensors. We propose a glyph to visualize asymmetric second-order two-dimensional tensors. The glyph includes visual encoding for physically significant attributes of the tensor, including rotation, anisotropic stretching, and isotropic dilation. Our glyph design conserves the symmetry and continuity properties of the underlying tensor, in that transformations of a tensor (such as rotation or negation) correspond to analogous transformations of the glyph. We show results with synthetic data from computational fluid dynamics.
This paper presents the state of the art in the area of topology-based visualization. It describes the process and results of an extensive annotation for generating a definition and terminology for the field. The terminology enabled a typology for topological models which is used to organize research results and the state of the art. Our report discusses relations among topological models and for each model describes research results for the computation, simplification, visualization, and application. The paper identifies themes common to subfields, current frontiers, and unexplored territory in this research area.
Vector Field Topology describes the asymptotic behavior of a flow in a vector field, i.e., the behavior for an integration time converging to infinity. For some applications, a segmentation of the flow in areas of similar behavior for a finite integration time is desired. We introduce an approach for a finite-time segmentation of a steady 2D vector field which avoids the systematic evaluation of the flow map in the whole flow domain. Instead, we consider the separatrices of the topological skeleton and provide them with additional information on how the separation evolves at each point with ongoing integration time. We analyze this behavior and its distribution along a separatrix, and we provide a visual encoding for it. The result is an augmented topological skeleton. We demonstrate the approach on several artificial and simulated vector fields.
Concepts from vector field topology have been successfully applied to a wide range of phenomena so far—typically to problems involving the transport of a quantity, such as in flow fields, or to problems concerning the instantaneous structure, such as in the case of electric fields. However, transport of quantities in time-dependent flows has so far been topologically analyzed in terms of advection only, restricting the approach to quantities that are solely governed by advection. Nevertheless, the majority of quantities transported in flows undergoes simultaneous diffusion, leading to advection-diffusion problems. By extending topology-based concepts with diffusion, we provide an approach for visualizing the mechanisms in advection-diffusion flow. This helps answering many typical questions in science and engineering that have so far not been amenable to adequate visualization. We exemplify the utility of our technique by applying it to simulation data of advection-diffusion problems from different fields.
A common problem of vector field topology algorithms is the large number of the resulting topological features. This chapter describes a method to simplify Morse decompositions by iteratively merging pairs of Morse sets that are adjacent in the Morse Connection Graph (MCG). When Morse sets A and B are merged, they are replaced by a single Morse set, that can be thought of as the union of A, B and all trajectories connecting A and B. Pairs of Morse sets to be merged can be picked based on a variety of criteria. For example, one can allow only pairs whose merger results in a topologically simple Morse set to be selected, and give preference to mergers leading to small Morse sets.
Graphics hardware has in recent years become increasingly programmable, and its programming APIs use the stream processor model to expose massive parallelization to the programmer. Unfortunately, the inherent restrictions of the stream processor model, used by the GPU in order to maintain high performance, often pose a problem in porting CPU algorithms for both video and volume processing to graphics hardware. Serial data dependencies which accelerate CPU processing are counterproductive for the data-parallel GPU.
This thesis demonstrates new ways for tackling well-known problems of large scale video/volume analysis. In some instances, we enable processing on the restricted hardware model by re-introducing algorithms from early computer graphics research. On other occasions, we use newly discovered, hierarchical data structures to circumvent the random-access read/fixed write restriction that had previously kept sophisticated analysis algorithms from running solely on graphics hardware. For 3D processing, we apply known game graphics concepts such as mip-maps, projective texturing, and dependent texture lookups to show how video/volume processing can benefit algorithmically from being implemented in a graphics API.
The novel GPU data structures provide drastically increased processing speed, and lift processing heavy operations to real-time performance levels, paving the way for new and interactive vision/graphics applications.
Graphs represent general node-link diagrams and have long been utilized in scientific visualization for data organization and management. However, using graphs as a visual representation and interface for navigating and exploring scientific data sets has a much shorter history, yet the amount of work along this direction is clearly on the rise in recent years. In this paper, we take a holistic perspective and survey graph-based representations and techniques for scientific visualization. Specifically, we classify these representations and techniques into four categories, namely partition-wise, relationship-wise, structure-wise and provenance-wise. We survey related publications in each category, explaining the roles of graphs in related work and highlighting their similarities and differences. At the end, we reexamine these related publications following the graph-based visualization pipeline. We also point out research trends and remaining challenges in graph-based representations and techniques for scientific visualization.
Vector field topology is a powerful and matured tool for the study of the asymptotic behavior of tracer particles in steady flows. Yet, it does not capture the behavior of finite-sized particles, because they develop inertia and do not move tangential to the flow. In this paper, we use the fact that the trajectories of inertial particles can be described as tangent curves of a higher dimensional vector field. Using this, we conduct a full classification of the first-order critical points of this higher dimensional flow, and devise a method to their efficient extraction. Further, we interactively visualize the asymptotic behavior of finite-sized particles by a glyph visualization that encodes the outcome of any initial condition of the governing ODE, i.e., for a varying initial position and/or initial velocity. With this, we present a first approach to extend traditional vector field topology to the inertial case.
Stretching and compression in tangent directions of Lagrangian coherent structures (LCS) are of particular interest in the vicinity of hyperbolic trajectories and play a key role in turbulence and mixing. Since integration of hyperbolic trajectories is difficult, we propose to visualize them in 2D time-dependent vector fields by space-time intersection curves of LCS. LCS are present as ridge lines in the 2D finite-time Lyapunov exponent (FTLE) field and as ridge surfaces in its 3D space-time domain. We extract these ridge surfaces from the forward and reverse FTLE field and intersect them. Due to their advection property, LCS become stream surfaces in 3D space-time. This allows us to use line integral convolution on the LCS to visualize their intrinsic dynamics, in particular around hyperbolic trajectories. To reduce occlusion, we constrain the LCS to space-time bands around their intersection curves. We evaluate our approach using synthetic, simulated, and measured vector fields.
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.
Morse decompositions have been proposed to compute and represent the topological structure of steady vector fields. Compared to the conventional differential topology, Morse decomposition and the resulting Morse Connection Graph (MCG) is numerically stable. However, the granularity of the original Morse decomposition is constrained by the resolution of the underlying spatial discretization, which typically results in non-smooth representation. In this work, an Image-Space Morse decomposition (ISMD) framework is proposed to address this issue. Compared to the original method, ISMD first projects the original vector field onto an image plane, then computes the Morse decomposition based on the projected field with pixels as the smallest elements. Thus, pixel-level accuracy can be achieved. This ISMD framework has been applied to a number of synthetic and real-world steady vector fields to demonstrate its utility. The performance of the ISMD is carefully studied and reported. Finally, with ISMD an ensemble Morse decomposition can be studied and visualized, which is shown useful for visualizing the stability of the Morse sets with respect to the error introduced in the numerical computation and the perturbation to the input vector fields.
Many concepts in computational flow visualization operate in the Lagrangian frame—they involve the integration of trajectories. A problem inherent to these approaches is the choice of an appropriate time length for the integration of these curves. While for some applications the choice of such a finite time length is straightforward, it represents in most other applications a parameter that needs to be explored and well-chosen. This becomes even more difficult in situations where different regions of the vector field require different time scopes. In this chapter, we introduce Lyapunov time for this purpose. Lyapunov time, originally defined for predictability purposes, represents the time over which a trajectory is predictable, i.e., not dominated by error. We employ this concept for steering the integration time in direct visualization by trajectories, and for derived representations such as line integral convolution and delocalized quantities. This not only provides significant visualizations related to time-dependent vector field topology, but at the same time incorporates uncertainty into trajectory-based visualization.
This study examines the ability to detect the dynamic interactions of vortical structures generated from a Helmholtz instability caused by separation over bluff bodies at large Reynolds number of approximately 104 based on a cross stream characteristic length of the geometry. Accordingly, two configurations, a square cylinder with normally incident flow and a thin airfoil with flow at an angle of attack of 200 are examined. Direct numerical simulation is used to obtain flow over the square cylinder. A time-resolved, three-component PIV data set is collected in a symmetry plane for the airfoil. Different approaches analyzing vector field and tensor field topologies are considered to identify vortical structures and local, swirl regions: (i) the Γ function that maps the degree of rotation rate (or pressure-gradients) to identify local swirl regions, (ii) Entity Connection Graph (ECG) that combines the Conley theory and Morse decomposition to identify vector field topology consisting of fixed points (sources, sinks, saddles, and periodic orbits), together with separatrices (links connecting them), and (iii) the λ2 method that examines the gradient fields of velocity to identify local regions of pressure minima. Both velocity and pressure-gradient fields are analyzed for the DNS data, whereas only velocity field is used for the experimental data set. The vector-field topology requires spatial integration of the velocity or pressure-gradient fields. The tensor field topology, on the other hand, is based on gradients of the velocity of pressure-gradient vectors. A detailed comparison of these techniques is performed by applying them to velocity or pressure-based data and using spatial filtering of the data sets to identify the multiscale features of the flow. It is shown that various techniques provide useful information about the flowfield at different scales that can be used for further analysis of many fluid engineering problems of practical interest. Copyright © 2008 by American Institute of Aeronautics and Astronautics, Inc.
The control of the flow over the flap of a three-element high-lift configuration is investigated numerically by solving the unsteady Reynolds-averaged Navier-Stokes equations (URANS). At a Reynolds number of Re = 1 · 106 the flow is perturbed by periodic blowing/suction through a slot near the flap leading edge. The main focus is on the mechanisms of separation control, for which flow field structures at two different excitation parameters are studied. The simulations are conducted using a swept wing of infinite span in order to study the impact of different excitation parameters. The method of feature-based extraction will be used to identify dominant large scale structures in the unforced and excited flow field. Copyright © 2008 by B. Günther, F. Thiele, T. Weinkauf, J. Sahner and H.-C. Hege.
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