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The State of the Art in Topology-Based Visualization of Unsteady Flow
Abstract
Vector fields are a common concept for the representation of many different kinds of flow phenomena in science and engineering. Methods based on vector field topology are known for their convenience for visualizing and analysing steady flows, but a counterpart for unsteady flows is still missing. However, a lot of good and relevant work aiming at such a solution is available. We give an overview of previous research leading towards topology-based and topology-inspired visualization of unsteady flow, pointing out the different approaches and methodologies involved as well as their relation to each other, taking classical (i.e. steady) vector field topology as our starting point. Particularly, we focus on Lagrangian methods, space–time domain approaches, local methods and stochastic and multifield approaches. Furthermore, we illustrate our review with practical examples for the different approaches.
... Second, our discussion of distributed-memory techniques does a new summarization of workloads and parallel characteristics (specifically Table 5), and also has been updated to include works appearing since their publication. There also have been many other excellent surveys involving flow visualization and particle advection: feature extraction and tracking [3], dense and texture-based techniques [4], topology-based flow techniques [5] and a subsequent survey focusing on topology for unsteady flow [6], integration-based, geometric flow visualization [7], and seed placement and streamline selection [8]. Our survey complements these existing surveys -while some of these works consider aspects of performance within their individual focal point, none of the surveys endeavor to provide a guide to particle advection performance. ...
... l ocat e i , j ,k + i nt er p i , j ,k + anal y ze i , j + t er m i , j (6) ...
... This section demonstrates cost estimation for a hypothetical workload based on Equation 6 and Table 7. The hypothetical workloads consists of a flow visualization algorithm advancing a million particles in a 3D uniform grid for a maximum of 1000 steps using a RK4 solver. ...
The performance of particle advection-based flow visualization techniques is complex, since computational work can vary based on many factors, including number of particles, duration, and mesh type. Further, while many approaches have been introduced to optimize performance, the efficacy of a given approach can be similarly complex. In this work, we seek to establish a guide for particle advection performance by conducting a comprehensive survey of the area. We begin by identifying the building blocks for particle advection and establishing a simple cost model incorporating these building blocks. We then survey existing optimizations for particle advection, using two high-level categories: algorithmic optimizations and hardware efficiency. The sub-categories of algorithmic optimizations include solvers, cell locators, I/O efficiency, and precomputation, while the sub-categories of hardware efficiency all involve parallelism: shared-memory, distributed-memory, and hybrid. Finally, we conclude the survey by identifying current gaps in particle advection performance, and in particular on achieving a workflow for predicting performance under various optimizations.
... Vector field analysis is critically important for many physics and environmental applications, such as combustion rate modeling, material sciences, climate research, or space science. In particular, vector field topology is one of the most popular visualization techniques for flow data [13], [15] because it breaks down even huge amounts of data into a compact, sparse, and easy to comprehend description with little information loss [4], [14], [19], [29]. ...
... The topological skeleton consisting of first order critical points and the invariant manifolds of saddles have been visualized for a long time [15]. Many extensions to 3D [12], higher order critical points [34], [36], [50], boundary switch points [10], [35], periodic orbits [51], [52], discrete topology [7], [20], [30], [40], and time-dependent flow [4], [29] have been suggested in the literature. Fig. 1: The different types of first order non-degenerate 2D critical points visualized with line integral convolution (LIC) [5] and arrow glyphs. ...
... For example, [1] developed and applied deep learning techniques in order to analyze the simulation while taking into account all the information produced by the simulation. Most existing feature extraction techniques, developed for CFD simulations focus on visualization and not machine learning [2], [3]. Deep learning techniques require exhaustive datasets with a very large number of data samples to produce reliable and generalizable performance. ...
... Most existing approaches for feature extraction on CFD simulations focus on visualization [2], [3]. Moreover, they focus specifically on the vector field and neglect other information, such as pressure and turbulent viscosity. ...
... For example, [1] developed and applied deep learning techniques in order to analyze the simulation while taking into account all the information produced by the simulation. Most existing feature extraction techniques, developed for CFD simulations focus on visualization and not machine learning [2], [3]. Deep learning techniques require exhaustive datasets with a very large number of data samples to produce reliable and generalizable performance. ...
... Most existing approaches for feature extraction on CFD simulations focus on visualization [2], [3]. Moreover, they focus specifically on the vector field and neglect other information, such as pressure and turbulent viscosity. ...
Computational Fluid Dynamics (CFD) simulations are a very important tool for many industrial applications, such as aerodynamic optimization of engineering designs like cars shapes, airplanes parts etc. The output of such simulations, in particular the calculated flow fields, are usually very complex and hard to interpret for realistic three-dimensional real-world applications, especially if time-dependent simulations are investigated. Automated data analysis methods are warranted but a non-trivial obstacle is given by the very large dimensionality of the data. A flow field typically consists of six measurement values for each point of the computational grid in 3D space and time (velocity vector values, turbulent kinetic energy, pressure and viscosity). In this paper we address the task of extracting meaningful results in an automated manner from such high dimensional data sets. We propose deep learning methods which are capable of processing such data and which can be trained to solve relevant tasks on simulation data, i.e. predicting drag and lift forces applied on an airfoil. We also propose an adaptation of the classical hand crafted features known from computer vision to address the same problem and compare a large variety of descriptors and detectors. Finally, we compile a large dataset of 2D simulations of the flow field around airfoils which contains 16000 flow fields with which we tested and compared approaches. Our results show that the deep learning-based methods, as well as hand crafted feature based approaches, are well-capable to accurately describe the content of the CFD simulation output on the proposed dataset.
... A large amount of work has been proposed to address the visualization and analysis of unsteady flow [13,14]. These techniques either extract the structural information or the local dynamics of the flow. ...
... These techniques either extract the structural information or the local dynamics of the flow. Structural analysis of unsteady flow aims to study the transport behavior of the flow and identify the boundaries of different regions, such that the particles within each region exhibit similar temporal transport behavior [13]. There are various methods to define and compute the structure of unsteady flow, including topological feature tracking [15][16][17] based on the bifurcation theory, and pathlinebased segmentation [18]. ...
Despite significant advances in the analysis and visualization of unsteady flow, the interpretation of it's behavior still remains a challenge. In this work, we focus on the linear correlation and non-linear dependency of different physical attributes of unsteady flows to aid their study from a new perspective. Specifically, we extend the existing spatial correlation quantification, i.e. the Local Correlation Coefficient (LCC), to the spatio-temporal domain to study the correlation of attribute-pairs from both the Eulerian and Lagrangian views. To study the dependency among attributes, which need not be linear, we extend and compute the mutual information (MI) among attributes over time. To help visualize and interpret the derived correlation and dependency among attributes associated with a particle, we encode the correlation and dependency values on individual pathlines. Finally, to utilize the correlation and MI computation results to identify regions with interesting flow behavior, we propose a segmentation strategy of the flow domain based on the ranking of the strength of the attributes relations. We have applied our correlation and dependency metrics to a number of 2D and 3D unsteady flows with varying spatio-temporal kernel sizes to demonstrate and assess their effectiveness.
... Vortex extraction and analysis continue to be active areas of research in fluid dynamics and scientific visualization [16,21,38,39]. Vortex definitions [16] commonly involve two key components: the vortex coreline, which represents the path around which fluid particles move [29], and a reference frame that reveals circular or spiral patterns when streamlines are projected onto a plane perpendicular to the coreline [40]. ...
Hairpin vortices are one of the most important vortical structures in turbulent flows. Extracting and characterizing hairpin vortices provides useful insight into many behaviors in turbulent flows. However, hairpin vortices have complex configurations and might be entangled with other vortices, making their extraction difficult. In this work, we introduce a framework to extract and separate hairpin vortices in shear driven turbulent flows for their study. Our method first extracts general vortical regions with a region-growing strategy based on certain vortex criteria (e.g., λ
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) and then separates those vortices with the help of progressive extraction of (λ
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) iso-surfaces in a top-down fashion. This leads to a hierarchical tree representing the spatial proximity and merging relation of vortices. After separating individual vortices, their shape and orientation information is extracted. Candidate hairpin vortices are identified based on their shape and orientation information as well as their physical characteristics. An interactive visualization system is developed to aid the exploration, classification, and analysis of hairpin vortices based on their geometric and physical attributes. We also present additional use cases of the proposed system for the analysis and study of general vortices in other types of flows.
... Vortex extraction and analysis continue to be active areas of research in fluid dynamics and scientific visualization [16,21,38,39]. Vortex definitions [16] commonly involve two key components: the vortex coreline, which represents the path around which fluid particles move [29], and a reference frame that reveals circular or spiral patterns when streamlines are projected onto a plane perpendicular to the coreline [40]. ...
Hairpin vortices are one of the most important vortical structures in turbulent flows. Extracting and characterizing hairpin vortices provides useful insight into many behaviors in turbulent flows. However, hairpin vortices have complex configurations and might be entangled with other vortices, making their extraction difficult. In this work, we introduce a framework to extract and separate hairpin vortices in shear driven turbulent flows for their study. Our method first extracts general vortical regions with a region-growing strategy based on certain vortex criteria (e.g., $\lambda_2$) and then separates those vortices with the help of progressive extraction of ($\lambda_2$) iso-surfaces in a top-down fashion. This leads to a hierarchical tree representing the spatial proximity and merging relation of vortices. After separating individual vortices, their shape and orientation information is extracted. Candidate hairpin vortices are identified based on their shape and orientation information as well as their physical characteristics. An interactive visualization system is developed to aid the exploration, classification, and analysis of hairpin vortices based on their geometric and physical attributes. We also present additional use cases of the proposed system for the analysis and study of general vortices in other types of flows.
... Vector field topology. There are many approaches to extract and visualize topological characteristics of vector fields in steady and unsteady flow fields [209,301]. These methods have a great potential to reduce the complexity of the flow data. ...
Cerebral aneurysms are weak vessel areas that can bulge out and balloon, caused by a pathologically altered structure of the vascular wall. They bear the risk of rupture, leading to internal bleeding causing high risks of mortality. Although most aneurysms will never rupture, the potential risk of bleeding makes the detection and risk-assessment of aneurysms a critical issue. Imaging methods are used for the detection and localization of aneurysms. The decision as to whether or not aneurysms should be treated must be carefully considered, as there is a risk of fatal outcome during surgery. Their initiation and progression depend strongly on the interplay of vascular morphology and hemodynamics. Unfortunately, the processes causing aneurysm growth and rupture are not well understood. Blood flow simulations can obtain information about the patient-specific hemodynamics.
It is also the basis for the development for new, low-risk treatment options since treatment success depends on blood flow characteristics. In clinical routine, risk assessment and treatment planning are just based on morphological characteristics
of an aneurysm and its surrounding vasculature. However, this information allows no reliable evaluation of the aneurysm state. To improve decision-making, medical and biomedical researchers analyze simulated flow data, which are multi-attribute data with high spatial and temporal complexity. The data exploration is performed quantitatively and qualitatively, where the former focuses on the evaluation of specific scalar values such as pressure or wall thickness and the latter focuses
on the analysis of flow patterns such as vortices. Correlations between qualitative and quantitative characteristics can be revealed and formed into hypotheses that can lead to a better understanding of the internal aneurysm procedures. However, the visual exploration of flow data is a time-consuming process, which is affected by visual clutter and occlusions. The goal of our work is to develop computer-aided methods that support the quantitative and qualitative visual exploration of morphological and hemodynamic characteristics in cerebral aneurysm data sets. Since this is an interdisciplinary process involving both physicians and fluid mechanics experts, redundancy-free management of aneurysm data sets is required to enable efficient analysis of the information. We developed a consistent structure to document aneurysm data sets, where users can search for specific cohorts, and individual cases can be analyzed more detailed to assess the aneurysm state as well as to weigh different treatment scenarios. The prerequisite for the visual exploration is the extraction of the ostium, which is a curved surface that separates the parent vessel from an aneurysm. We provide an automatic determination of the ostium. Based on this several other morphological descriptors are computed automatically. Besides an analysis of morphological aspects,
the aneurysm data exploration comprises four more parts: a simultaneous evaluation of wall- and flow-related characteristics, a simultaneous analysis of multiple scalar information on the aneurysm wall, the analysis of mechanical wall processes as well as a qualitative characterization of the internal flow behavior. We provide methods for each of these parts: occlusion-free depictions of the vessel morphology and internal blood flow, interactive 2D and 3D visualizations to explore multi-attribute
correlations, comparative glyph-based visualizations to explore mechanical wall forces and automatic classification of qualitative flow patterns. Our methods were designed and evaluated in collaboration with domain experts who confirmed their usefulness and clinical necessity.
... An unsteady vector field v(x, t) describes the time-varying flow in a certain domain [5,6]. To understand the Lagrangian behavior of flow, the trajectories of mass-less particles are often analyzed. ...
The finite-time Lyapunov exponent (FTLE) is widely used for understanding the Lagrangian behavior of unsteady flow fields. The FTLE field contains many important fine-level structures (e.g., Lagrangian coherent structures). These structures are often thin in depth, requiring Monte Carlo rendering for unbiased visualization. However, Monte Carlo rendering requires hundreds of billions of samples for a high-resolution FTLE visualization, which may cost up to hundreds of hours for rendering a single frame on a multi-core CPU. In this paper, we propose a neural representation of the flow map and FTLE field to reduce the cost of expensive FTLE computation. We demonstrate that a simple multi-layer perceptron (MLP)-based network can accelerate the FTLE computation by up to hundreds of times, and speed up the rendering by tens of times, while producing satisfactory rendering results. We also study the impact of the network size, the amount of training, and the predicted property, which may serve as guidance for selecting appropriate network structures.
... Besides, specific mathematical techniques are required to develop algorithms to analyze and classify their topology. The readers interested in the general VFA are referred to surveys by Jiang et al. [223], Laramee et al. [224], Pobitzer et al. [225], and Edmunds et al. [226]. ...
Toolpath generation (TPG) for multi-axis milling machines using vector fields (VF) and vector field analysis (VFA) is becoming increasingly popular in the manufacturing industry. Therefore, the paper presents a survey of algorithms and methods of TPG based on the vector fields of preferred directions (VFPD) for five-axis CNC machining. Two hundred relevant citations in top manufacturing and optimization journals during 1995–2021 have been presented and discussed. Additional 79 references in Appendices are related to a classification of five-axis machines, the theory and recent advances in the area of the vector and tensor fields.
... By doing so, even complex flow structures can be described with a few graphical primitives (topological skeleton). Topological methods are an active topic in research and an overview is provided by Pobitzer et al. [31]. Extraction of topological structures in 2D and 3D vector fields is discussed by Weinkauf [32]. ...
... For a survey on the substantial history of structure extraction and visualization, see reviews on geometric-topological objects (McLoughlin et al., 2010;Pobitzer et al., 2011), on coherent set detection (Hadjighasem et al., 2017), on feature tracking in meteorological contexts (Clark et al., 2014;Cai and Dumais Jr, 2015), and on the related problem of image segmentation (El-Baz, Jiang, and Suri, 2017). Most approaches in data visualization refer to structures defined by single scalar or vector fields, e.g., level-sets or ridges of Lyaponov exponents or, more application-related, the centerlines of vortices or jet streams (Kern et al., 2018). ...
Our ability to grasp and understand complex phenomena is essentially based on recognizing structures and relating these to each other. For example, any meteorological description of a weather condition and explanation of its evolution recurs to meteorological structures, such as convection and circulation structures, cloud fields and rain fronts. All of these are spatiotemporal structures, defined by time-dependent patterns in the underlying fields. Typically, such a structure is defined by a verbal description that corresponds to the more or less uniform, often somewhat vague mental images of the experts. However, a precise, formal definition of the structures or, more generally, concepts is often desirable, e.g., to enable automated data analysis or the development of phenomenological models. Here, we present a systematic approach and an interactive tool to obtain formal definitions of spatiotemporal structures. The tool enables experts to evaluate and compare different structure definitions on the basis of data sets with time-dependent fields that contain the respective structure. Since structure definitions are typically parameterized, an essential part is to identify parameter ranges that lead to desired structures in all time steps. In addition, it is important to allow a quantitative assessment of the resulting structures simultaneously. We demonstrate the use of the tool by applying it to two meteorological examples: finding structure definitions for vortex cores and center lines of temporarily evolving tropical cyclones. Ideally, structure definitions should be objective and applicable to as many data sets as possible. However, finding such definitions, e.g., for the common atmospheric structures in meteorology, can only be a long-term goal. The proposed procedure, together with the presented tool, is just a first approach to facilitate this long and arduous way.
... Streamlines vs. Pathlines. More recent research concentrated on the definition and extraction of topological structures in time-dependent flows [40], in which we face two major challenges. First, aside from periodic flows, the temporal domain is usually bounded, which does not permit the observation of asymptotic motion. ...
... Other, less obvious types of separatrices exist (e.g., closed trajectories, boundary switches [5]), but these are not computed in the current version of the code (this may be added in future releases). A significant amount of research has been done on topology-based visualization techniques [9][10][11][12] building on the foundation of critical points and separatrices. ...
A myriad of physical phenomena, such as fluid flows, magnetic fields, and population dynamics are described by vector fields. More often than not, vector fields are complex and their analysis is challenging.
Vector field topology is a powerful analysis technique that consists in identifying the most essential structure of a vector field. Its topological features include critical points and separatrices, which segment the domain into regions of coherent flow behavior, provide a sparse and semantically meaningful representation of the underlying data.
However, a broad adoption of this formidable technique has been hampered by the lack of open source software implementing it. The Visualization Toolkit (VTK) now contains the filter vtkVectorFieldTopology that extracts the topological skeleton of 2D and 3D vector fields. This paper describes our implementation and demonstrates its broad applicability with two real-world examples from hydrology and space physics.
... The three-body problem defines a vector field and the topological structure of the CR3BP that we aim to visualize in a Poincaré map is the discrete signature of a three-dimensional vector field topology. There exists an abundant literature in the visualization community focused on the use of topological methods to represent vector fields [12,20,10]. In the discrete realm, Löffelmann et al. introduced multiple methods for the visualization of maps [17,15,14,16], thereby emphasizing the continuous structures that control the dynamics. ...
Mission designers must study many dynamical models to plan a low-cost spacecraft trajectory that satisfies mission constraints. They routinely use Poincaré maps to search for a suitable path through the interconnected web of periodic orbits and invariant manifolds found in multi-body gravitational systems. This paper is concerned with the extraction and interactive visual exploration of this structural landscape to assist spacecraft trajectory planning. We propose algorithmic solutions that address the specific challenges posed by the characterization of the topology in astrodynamics problems and allow for an effective visual analysis of the resulting information. This visualization framework is applied to the circular restricted three-body problem (CR3BP), where it reveals novel periodic orbits with their relevant invariant manifolds in a suitable format for interactive transfer selection. Representative design problems illustrate how spacecraft path planners can leverage our topology visualization to fully exploit the natural dynamics pathways for energy-efficient trajectory designs.
... Vortex Extraction and Boundary Identification. Vortex extraction remained an active field of research in fluid dynamics and scientific visualization despite decades of work [12,22,34,36]. Extraction methods are generally characterized into region-based techniques that calculate scalar fields, such as λ 2 [24] or the Q-criterion [23], and line-based techniques that compute the vortex coreline around which the particles rotate [4,32,39,42]. ...
Feature extraction is an integral component of scientific visualization, and specifically in situations in which features are difficult to formalize, deep learning has great potential to aid in data analysis. In this paper, we develop a deep neural network that is capable of finding vortex boundaries. For training data generation, we employ a parametric flow model that generates thousands of vector field patches with known ground truth. Compared to previous methods, our approach does not require the manual setting of a threshold in order to generate the training data or to extract the vortices. After supervised learning, we apply the method to numerical fluid flow simulations, demonstrating its applicability in practice. Our results show that the vortices extracted using the proposed method can capture more accurate behavior of the vortices in the flow.
... The three-body problem defines a vector field and the topological structure of the CR3BP that we aim to visualize in a Poincaré map is the discrete signature of a three-dimensional vector field topology. There exists an abundant literature in the visualization community focused on the use of topological methods to represent vector fields [12,20,10]. In the discrete realm, Löffelmann et al. introduced multiple methods for the visualization of maps [17,15,14,16], thereby emphasizing the continuous structures that control the dynamics. ...
Mission designers must study many dynamical models to plan a low-cost spacecraft trajectory that satisfies mission constraints. They routinely use Poincar\'e maps to search for a suitable path through the interconnected web of periodic orbits and invariant manifolds found in multi-body gravitational systems. This paper is concerned with the extraction and interactive visual exploration of this structural landscape to assist spacecraft trajectory planning. We propose algorithmic solutions that address the specific challenges posed by the characterization of the topology in astrodynamics problems and allow for an effective visual analysis of the resulting information. This visualization framework is applied to the circular restricted three-body problem (CR3BP), where it reveals novel periodic orbits with their relevant invariant manifolds in a suitable format for interactive transfer selection. Representative design problems illustrate how spacecraft path planners can leverage our topology visualization to fully exploit the natural dynamics pathways for energy-efficient trajectory designs.
... The input vector field and/or the potential of the irrotational component can be analysed by classifying its singularities and streamlines, which are smoothed in order to preserve only the persistent ones [48] in case of noise. For more details on these topics, we refer the reader to survey papers on topology-based visualisation [49], [50], [51] and on vortex extraction [33]. ...
The analysis of vector fields is crucial for the understanding of several physical phenomena, such as natural events (e.g., analysis of waves), diffusive processes, electric and electromagnetic fields. While previous work has been focused mainly on the analysis of 2D or 3D vector fields on volumes or surfaces, we address the meshless analysis of a vector field defined on an arbitrary domain, without assumptions on its dimension and discretisation. The meshless approximation of the Helmholtz-Hodge decomposition of a vector field is achieved by expressing the potential of its components as a linear combination of radial basis functions and by computing the corresponding conservative, irrotational, and harmonic components as solution to a least-squares or to a differential problem. To this end, we identify the conditions on the kernel of the radial basis functions that guarantee the existence of their derivatives. Finally, we demonstrate our approach on 2D and 3D vector fields measured by sensors or generated through simulation.
... The input vector field and/or the potential of the irrotational component can be analysed by classifying its singularities and streamlines, which are smoothed in order to preserve only the persistent ones [48] in case of noise. For more details on these topics, we refer the reader to survey papers on topology-based visualisation [49], [50], [51] and on vortex extraction [33]. ...
The analysis of vector fields is crucial for the understanding of several physical phenomena, such as natural events (e.g., analysis of waves), diffusive processes, electric and electromagnetic fields. While previous work has been focused mainly on the analysis of 2D or 3D vector fields on volumes or surfaces, we address the meshless analysis of a vector field defined on an arbitrary domain, without assumptions on its dimension and discretisation. The meshless approximation of the Helmholtz-Hodge decomposition of a vector field is achieved by expressing the potential of its components as a linear combination of radial basis functions and by computing the corresponding conservative, irrotational, and harmonic components as solution to a least-squares or to a differential problem. To this end, we identify the conditions on the kernel of the radial basis functions that guarantee the existence of their derivatives. Finally, we demonstrate our approach on 2D and 3D vector fields measured by sensors or generated through simulation.
... Switching to the unsteady case has pushed the limits of studying such geometric flow features [38], leading to successes on tracking topological features [17] as well as defining pathline [45,49] and streakline [50] topology. Recently, an alternative characterization of unsteady flow topology has focused on studying separation through the Finite-Time Lyapunov Exponent (FTLE) [2,14,21]. ...
We introduce a new technique to visualize complex flowing phenomena by using concepts from shape analysis. Our approach uses techniques that examine the intrinsic geometry of manifolds through their heat kernel, to obtain representations of such manifolds that are isometry-invariant and multi-scale. These representations permit us to compute heat kernel signatures of each point on that manifold, and we can use these signatures as features for classification and segmentation that identify points that have similar structural properties. Our approach adapts heat kernel signatures to unsteady flows by formulating a notion of shape where pathlines are observations of a manifold living in a high-dimensional space. We use this space to compute and visualize heat kernel signatures associated with each pathline. Besides being able to capture the structural features of a pathline, heat kernel signatures allow the comparison of pathlines from different flow datasets through a shape matching pipeline. We demonstrate the analytic power of heat kernel signatures by comparing both (1) different timesteps from the same unsteady flow as well as (2) flow datasets taken from ensemble simulations with varying simulation parameters. Our analysis only requires the pathlines themselves, and thus it does not utilize the underlying vector field directly. We make minimal assumptions on the pathlines: while we assume they are sampled from a continuous, unsteady flow, our computations can tolerate pathlines that have varying density and potential unknown boundaries. We evaluate our approach through visualizations of a variety of two-dimensional unsteady flows.
... Streamlines vs. Pathlines. More recent research concentrated on the definition and extraction of topological structures in time-dependent flows [40], in which we face two major challenges. First, aside from periodic flows, the temporal domain is usually bounded, which does not permit the observation of asymptotic motion. ...
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.
... It is also a means for flow field decomposition, simplification (Tricoche et al. 2000), and design (Theisel 2002). Currently, the main challenge lies in generalizing flow topology to time-varying data (Pobitzer et al. 2011;Bujack et al. 2019). ...
Flow plays a major role in environmental sciences, because many of the Earth’s physical and biological processes involve movement. Yet, there are major differences between theoretically available and practically applied visualization techniques to represent flow. This paper surveys various techniques in computational and environmental flow visualization. Techniques from the computational flow visualization community are classified into geometric, texture-based, topology-based, and feature-based approaches. Environmental flow applications are categorized into four application domains (atmospheric science, ecology, geosciences, and urban environments). Computational and environmental visualization approaches are compared to exhibit gaps and suggest solutions on how to bridge the gap. Outcomes from this literature review will inform the development of strategic initiatives for both future flow visualization research and flow visualization in the environmental sciences.
... Comprehensive reviews of uncertainty visualization can be found in [13], [14], and reviews of flow visualization are in [15], [16], and [17]. ...
We present an efficient and scalable solution to estimate uncertain transport behaviors-stochastic flow maps (SFMs)-for visualizing and analyzing uncertain unsteady flows. Computing flow maps from uncertain flow fields is extremely expensive because it requires many Monte Carlo runs to trace densely seeded particles in the flow. We reduce the computational cost by decoupling the time dependencies in SFMs so that we can process shorter sub time intervals independently and then compose them together for longer time periods. Adaptive refinement is also used to reduce the number of runs for each location. We parallelize over tasks-packets of particles in our design-to achieve high efficiency in MPI/thread hybrid programming. Such a task model also enables CPU/GPU coprocessing. We show the scalability on two supercomputers, Mira (up to 256K Blue Gene/Q cores) and Titan (up to 128K Opteron cores and 8K GPUs), that can trace billions of particles in seconds.
... Topology, namely scalar and flow topology [60], [87] as well as skeletonization [98] are useful tools for geometric abstraction. Fig. 4 shows flow characteristics directly using the Line Integral Convolution visualization method and a corresponding flow topology which is a higher geometric abstraction depicting critical points and separatrices, partitioning the domain into conceptually different flow behavior. ...
We explore the concept of abstraction as it is used in visualization, with the ultimate goal of understanding and formally defining it. Researchers so far have used the concept of abstraction largely by intuition without a precise meaning. This lack of specificity left questions on the characteristics of abstraction, its variants, its control, or its ultimate potential for visualization and, in particular, illustrative visualization mostly unanswered. In this paper we thus provide a first formalization of the abstraction concept and discuss how this formalization affects the application of abstraction in a variety of visualization scenarios. Based on this discussion, we derive a number of open questions still waiting to be answered, thus formulating a research agenda for the use of abstraction for the visual representation and exploration of data. This paper, therefore, is intended to provide a contribution to the discussion of the theoretical foundations of our field, rather than attempting to provide a completed and final theory.
The computational work to perform particle advection‐based flow visualization techniques varies based on many factors, including number of particles, duration, and mesh type. In many cases, the total work is significant, and total execution time (“performance”) is a critical issue. This state‐of‐the‐art report considers existing optimizations for particle advection, using two high‐level categories: algorithmic optimizations and hardware efficiency. The sub‐categories for algorithmic optimizations include solvers, cell locators, I/O efficiency, and precomputation, while the sub‐categories for hardware efficiency all involve parallelism: shared‐memory, distributed‐memory, and hybrid. Finally, this STAR concludes by identifying current gaps in our understanding of particle advection performance and its optimizations.
Critical point tracking is a core topic in scientific visualization for understanding the dynamic behaviour of time‐varying vector field data. The topological notion of robustness has been introduced recently to quantify the structural stability of critical points, that is, the robustness of a critical point is the minimum amount of perturbation to the vector field necessary to cancel it. A theoretical basis has been established previously that relates critical point tracking with the notion of robustness, in particular, critical points could be tracked based on their closeness in stability, measured by robustness, instead of just distance proximity within the domain. However, in practice, the computation of classic robustness may produce artifacts when a critical point is close to the boundary of the domain; thus, we do not have a complete picture of the vector field behaviour within its local neighbourhood. To alleviate these issues, we introduce a multilevel robustness framework for the study of 2D time‐varying vector fields. We compute the robustness of critical points across varying neighbourhoods to capture the multiscale nature of the data and to mitigate the boundary effect suffered by the classic robustness computation. We demonstrate via experiments that such a new notion of robustness can be combined seamlessly with existing feature tracking algorithms to improve the visual interpretability of vector fields in terms of feature tracking, selection and comparison for large‐scale scientific simulations. We observe, for the first time, that the minimum multilevel robustness is highly correlated with physical quantities used by domain scientists in studying a real‐world tropical cyclone dataset. Such an observation helps to increase the physical interpretability of robustness.
Our ability to grasp and understand complex phenomena is essentially based on recognizing structures and relating these to each other. For example, any meteorological description of a weather condition and explanation of its evolution recurs to meteorological structures, such as convection and circulation structures, cloud fields and rain fronts. All of these are spatiotemporal structures, defined by time‐dependent patterns in the underlying fields. Typically, such a structure is defined by a verbal description that corresponds to the more or less uniform, often somewhat vague mental images of the experts. However, a precise, formal definition of the structures or, more generally, of the concepts is often desirable, e.g., to enable automated data analysis or the development of phenomenological models. Here, we present a systematic approach and an interactive tool to obtain formal definitions of spatiotemporal structures. The tool enables experts to evaluate and compare different structure definitions on the basis of data sets with time‐dependent fields that contain the respective structure. Since structure definitions are typically parameterized, an essential part is to identify parameter ranges that lead to desired structures in all time steps. In addition, it is important to allow a quantitative assessment of the resulting structures simultaneously. We demonstrate the use of the tool by applying it to two meteorological examples: finding structure definitions for vortex cores and center lines of temporarily evolving tropical cyclones.
Ideally, structure definitions should be objective and applicable to as many data sets as possible. However, finding such definitions, e.g., for the common atmospheric structures in meteorology, can only be a long‐term goal. The proposed procedure, together with the presented tool, is just a first systematic approach aiming at facilitating this long and arduous way.
This paper presents a framework for topological feature analysis in time-dependent climate ensembles. Important climate indices such as the El Nino Southern Oscillation Index (ENSO) or the North Atlantic Oscillation Index (NAO) are usually derived by evaluating scalar fields at fixed locations or regions where these extremal values occur most frequently today. However, under climate change, dynamic circulation changes are likely to cause shifts in the intensity, frequency and location of underlying physical phenomena. In case of the NAO for instance, climatologists are interested in the position and intensity of the Icelandic Low and the Azores High as their interplay strongly influences the European climate, especially during the winter season. To robustly extract and track such highly variable features without a-priori region information, we present a topology-based method for dynamic extraction of such uncertain critical points on a global scale. The system additionally integrates techniques to visualize the variability within the ensemble and correlations between features. To demonstrate the utility of our VTK+TTK-based software framework, we explore a 150-year climate projection consisting of 100 ensemble members and particularly concentrate on sea level pressure fields.
Large-scale structures have been observed in many shear flows which are the fluid generated between two surfaces moving with different velocity. A better understanding of the physics of the structures (especially large-scale structures) in shear flows will help explain a diverse range of physical phenomena and improve our capability of modeling more complex turbulence flows. Many efforts have been made in order to capture such structures; however, conventional methods have their limitations, such as arbitrariness in parameter choice or specificity to certain setups. To address this challenge, we propose to use Multi-Resolution Dynamic Mode Decomposition (mrDMD), for large-scale structure extraction in shear flows. In particular, we show that the slow-motion DMD modes are able to reveal large-scale structures in shear flows that also have slow dynamics. In most cases, we find that the slowest DMD mode and its reconstructed flow can sufficiently capture the large-scale dynamics in the shear flows, which leads to a parameter-free strategy for large-scale structure extraction. Effective visualization of the large-scale structures can then be produced with the aid of the slowest DMD mode. To speed up the computation of mrDMD, we provide a fast GPU-based implementation. We also apply our method to some non-shear flows that need not behave quasi-linearly to demonstrate the limitation of our strategy of using the slowest DMD mode. For non-shear flows, we show that multiple modes from different levels of mrDMD may be needed to sufficiently characterize the flow behavior.
The topological notion of robustness introduces mathematically rigorous approaches to interpret vector field data. Robustness quantifies the structural stability of critical points with respect to perturbations and has been shown to be useful for increasing the visual interpretability of vector fields. However, critical points, which are essential components of vector field topology, are defined with respect to a chosen frame of reference. The classical definition of robustness, therefore, depends also on the chosen frame of reference. We define a new Galilean invariant robustness framework that enables the simultaneous visualization of robust critical points across the dominating reference frames in different regions of the data. We also demonstrate a strong connection between such a robustness-based framework with the one recently proposed by Bujack et al., which is based on the determinant of the Jacobian. Our results include notable observations regarding the definition of stable features within the vector field data.
Most existing unsteady flow visualization techniques concentrate on the depiction of geometric patterns in flow, assuming the geometry information provides sufficient representation of the underlying physical characteristics, which is not always the case. To address this challenge, this work proposes to analyse the time‐dependent characteristics of the physical attributes measured along pathlines which can be represented as a series of time activity curves (TAC). We demonstrate that the temporal trends of these TACs can convey the relation between pathlines and certain well‐known flow features (e.g. vortices and shearing layers), which enables us to select pathlines that can effectively represent the physical characteristics of interest and their temporal behaviour in the unsteady flow. Inspired by this observation, a new TAC‐based unsteady flow visualization and analysis framework is proposed. The centre of this framework is a new similarity measure that compares the similarity of two TACs, from which a new spatio‐temporal, hierarchical clustering that classifies pathlines based on their physical attributes, and a TAC‐based pathline exploration and selection strategy are proposed. A visual analytic system incorporating the TAC‐based pathline clustering and exploration is developed, which also provides new visualizations to support the user exploration of unsteady flow using TACs. This visual analytic system is applied to a number of unsteady flow in 2D and 3D to demonstrate its utility. The new system successfully reveals the detailed structure of vortices, the relation between shear layer and vortex formation, and vortex breakdown, which are difficult to convey with conventional methods.
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.
In this paper, we present an integrated visual analytics approach to support the parametrization and exploration of flow visualization based on the finite‐time Lyapunov exponent. Such visualization of time‐dependent flow faces various challenges, including the choice of appropriate advection times, temporal regions of interest, and spatial resolution. Our approach eases these challenges by providing the user with context by means of parametric aggregations, with support and guidance for a more directed exploration, and with a set of derived measures for better qualitative assessment. We demonstrate the utility of our approach with examples from computation fluid dynamics and time‐dependent dynamical systems.
We present a state‐of‐the‐art report on time‐dependent flow topology. We survey representative papers in visualization and provide a taxonomy of existing approaches that generalize flow topology from time‐independent to time‐dependent settings. The approaches are classified based upon four categories: tracking of steady topology, reference frame adaption, pathline classification or clustering, and generalization of critical points. Our unique contributions include introducing a set of desirable mathematical properties to interpret physical meaningfulness for time‐dependent flow visualization, inferring mathematical properties associated with selective research papers, and utilizing such properties for classification. The five most important properties identified in the existing literature include coincidence with the steady case, induction of a partition within the domain, Lagrangian invariance, objectivity, and Galilean invariance.
Extraction of multiscale features using scale-space is one of the fundamental approaches to analyze scalar fields. However, similar techniques for vector fields are much less common, even though it is well known that, for example, turbulent flows contain cascades of nested vortices at different scales. The challenge is that the ideas related to scale-space are based upon iteratively smoothing the data to extract features at progressively larger scale, making it difficult to extract overlapping features. Instead, we consider spatial regions of influence in vector fields as scale, and introduce a new approach for the multiscale analysis of vector fields. Rather than smoothing the flow, we use the natural Helmholtz-Hodge decomposition to split it into small-scale and large-scale components using progressively larger neighborhoods. Our approach creates a natural separation of features by extracting local flow behavior, for example, a small vortex, from large-scale effects, for example, a background flow. We demonstrate our technique on large-scale, turbulent flows, and show multiscale features that cannot be extracted using state of the art techniques.
The topological analysis of unsteady vector fields remains to this day one of the largest challenges in flow visualization. We build up on recent work on vortex extraction to define a time-dependent vector field topology for 2D and 3D flows. In our work, we split the vector field into two components: a vector field in which the flow becomes steady, and the remaining ambient flow that describes the motion of topological elements (such as sinks, sources and saddles) and feature curves (vortex corelines and bifurcation lines). To this end, we expand on recent local optimization approaches by modeling spatially-varying deformations through displacement transformations from continuum mechanics. We compare and discuss the relationships with existing local and integration-based topology extraction methods, showing for instance that separatrices seeded from saddles in the optimal frame align with the integrationbased streakline vector field topology. In contrast to the streakline-based approach, our method gives a complete picture of the topology for every time slice, including the steps near the temporal domain boundaries. With our work it now becomes possible to extract topological information even when only few time slices are available. We demonstrate the method in several analytical and numerically-simulated flows and discuss practical aspects, limitations and opportunities for future work.
Time-dependent fluid flows often contain numerous hyperbolic Lagrangian coherent structures, which act as transport barriers that guide the advection. The finite-time Lyapunov exponent is a commonly-used approximation to locate these repelling or attracting structures. Especially on large numerical simulations, the FTLE ridges can become arbitrarily sharp and very complex. Thus, the discrete sampling onto a grid for a subsequent direct volume rendering is likely to miss sharp ridges in the visualization. For this reason, an unbiased Monte Carlo-based rendering approach was recently proposed that treats the FTLE field as participating medium with single scattering. This method constructs a ground truth rendering without discretization, but it is prohibitively slow with render times in the order of days or weeks for a single image. In this paper, we accelerate the rendering process significantly, which allows us to compute video sequence of high-resolution FTLE animations in a much more reasonable time frame. For this, we follow two orthogonal approaches to improve on the rendering process: the volumetric light path integration in gradient domain and an acceleration of the transmittance estimation. We analyze the convergence and performance of the proposed method and demonstrate the approach by rendering complex FTLE fields in several 3D vector fields.
We introduce a visual analysis system with GPU acceleration techniques for large sets of trajectories from complex dynamical systems. The approach is based on an interactive Boolean combination of subsets into a Focus+Context phase‐space visualization. We achieve high performance through efficient bitwise algorithms utilizing runtime generated GPU shaders and kernels. This enables a higher level of interactivity for visualizing the large multivariate trajectory data. We explain how our design meets a set of carefully considered analysis requirements, provide performance results, and demonstrate utility through case studies with many‐particle simulation data from two application areas.
Based on an intuitive physical definition of what a finite-time saddle-like behavior is, we derive a mathematical definition. We show that this definition builds the link between two FTLE-based saddle generalizations, which is not only of theoretical interest but also provides a more robust extraction of finite-time saddles.
Many numerical methods that simulate groundwater flow, particularly the continuous Galerkin finite element method, do not produce velocity information directly. Many algorithms have been proposed to improve the accuracy of velocity fields computed from hydraulic potentials. The differences in the streamlines generated from velocity fields obtained using different algorithms are presented in this report. The superconvergence method employed by FEFLOW, a popular commercial code, and some dual-mesh methods proposed in recent years are selected for comparison. The applications to depict hydrogeologic conditions using streamlines are used, and errors in streamlines are shown to lead to notable errors in boundary conditions, the locations of material interfaces, fluxes and conductivities. Furthermore, the effects of the procedures used in these two types of methods, including velocity integration and local conservation, are analyzed. The method of interpolating velocities across edges using fluxes is shown to be able to eliminate errors associated with refraction points that are not located along material interfaces and streamline ends at no-flow boundaries. Local conservation is shown to be a crucial property of velocity fields and can result in more accurate streamline densities. A case study involving both three-dimensional and two-dimensional cross-sectional models of a coal mine in Inner Mongolia, China, are used to support the conclusions presented.
Simulations and measurements of blood and airflow inside the human circulatory and respiratory system play an increasingly important role in personalized medicine for prevention, diagnosis and treatment of diseases. This survey focuses on three main application areas. (1) Computational fluid dynamics (CFD) simulations of blood flow in cerebral aneurysms assist in predicting the outcome of this pathologic process and of therapeutic interventions. (2) CFD simulations of nasal airflow allow for investigating the effects of obstructions and deformities and provide therapy decision support. (3) 4D phase‐contrast (4D PC) magnetic resonance imaging of aortic haemodynamics supports the diagnosis of various vascular and valve pathologies as well as their treatment. An investigation of the complex and often dynamic simulation and measurement data requires the coupling of sophisticated visualization, interaction and data analysis techniques. In this paper, we survey the large body of work that has been conducted within this realm. We extend previous surveys by incorporating nasal airflow, addressing the joint investigation of blood flow and vessel wall properties and providing a more fine‐granular taxonomy of the existing techniques. From the survey, we extract major research trends and identify open problems and future challenges. The survey is intended for researchers interested in medical flow but also more general, in the combined visualization of physiology and anatomy, the extraction of features from flow field data and feature‐based visualization, the visual comparison of different simulation results and the interactive visual analysis of the flow field and derived characteristics.
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.
Integration-based geometric method is widely used in vector field visualization. To improve the efficiency of integration advection-based visualization, we propose a uniform integrated advection (UIA) algorithm on steady and unsteady vector field according to common piecewise linear field data set analysis. UIA employs cell gradient-based interpolation along spatial and temporal direction, and transforms multi-step advection into single-step advection in association with fourth-order Runge–Kutta advection process. UIA can significantly reduce computational load, and is applicable on arbitrary grid type with cell-center/cell-vertex data structure. The experiments are performed on steady/unsteady vector fields with two-dimensional cell-center unstructured grids and three-dimensional cell-vertex grids, and also on unsteady field from fluid dynamics numerical simulation. The result shows that the proposed algorithm can significantly improve advection efficiency and reduce visualization computational time compared with fourth-order Runge–Kutta.
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In this paper, we revisit the Lagrangian accumulation process that aggregates the local attribute information along integral curves for vector field visualization. Similar to the previous work, we adopt the notation of the Lagrangian accumulation field or field for the representation of the accumulation results. In contrast to the previous work, we provide a more in-depth discussion on the properties of fields and the meaning of the patterns exhibiting in fields. In particular, we revisit the discontinuity in the fields and provide a thorough explanation of its relation to the flow structure and the additional information about the flow that it may reveal. addition, other remaining questions about the field, such as its sensitivity to the selection of integration time, are also addressed. Based on these new insights, we demonstrate a number of enhanced flow visualizations aided by the accumulation framework and the fields, including a new field guided ribbon placement, an field guided stream surface seeding and the visualization of particle-based flow data. To further demonstrate the generality of the accumulation framework, we extend it to the non-integral geometric curves (i.e., streak lines), which enables us to reveal information about the flow behavior other than those revealed by the integral curves. Finally, we introduce the Eulerian accumulation, which can reveal different flow behaviors from those revealed by Lagrangian accumulation. In summary, we believe that Lagrangian accumulation and the resulting fields offer valuable techniques to investigate flow behavior complementary to the current state-of-the-art techniques.
The aim of this article is to highlight the interest to apply Differential Geometry and Mechanics concepts to chaotic dynamical systems study. Thus, the local metric properties of curvature and torsion will directly provide the analytical expression of the slow manifold equation of slow-fast autonomous dynamical systems starting from kinematics variables (velocity, acceleration and over-acceleration or jerk).
The attractivity of the slow manifold will be characterized thanks to a criterion proposed by Henri Poincaré. Moreover, the specific use of acceleration will make it possible on the one hand to define slow and fast domains of the phase space and on the other hand, to provide an analytical equation of the slow manifold towards which all the trajectories converge. The attractive slow manifold constitutes a part of these dynamical systems attractor. So, in order to propose a description of the geometrical structure of attractor, a new manifold called singular manifold will be introduced. Various applications of this new approach to the models of Van der Pol, cubic-Chua, Lorenz, and Volterra–Gause are proposed.
The analysis and visualization of flows is a central problem in visualization. Topology based methods have gained increasing interest in recent years. This article describes a method for the detection of closed streamlines in 3D flows. It is based on a special treatment of cases where a streamline reenters a cell to prevent infinite cycling during streamline calculation. The algorithm checks for possible exits of a loop of crossed faces and detects structurally stable closed streamlines. These global features are not detected by conventional topology and feature detection algorithms.
Topology-based methods have been successfully applied to the vi- sualization of instantaneous planar vector elds. In this paper, we present the topology-based visualization of time-dependent 2D o ws. Our method tracks critical points over time precisely. The detection and classication of bifurcations delivers the topological structure of time dependent vector elds. This offers a general framework for the qualitative analysis and visualization of parameter- dependent 2D vector elds.
Finite-time Lyapunov stability analysis is reviewed and applied to a low-order spectral model of barotropic flow in a midlatitude channel. The tangent linear equations of the model are used to investigate the growth of small perturbations superposed on a reference solution for a prescribed time interval. Three types of reference solutions of the model, stationary, periodic, and chaotic, are investigated to demonstrate usefulness of the analysis in the study of the atmospheric predictability problem.The finite-time Lyapunov exponents, which give the growth rate of small perturbations, depend upon the reference solution as well as the prescribed time interval. The finite-time Lyapunov vector corresponding to the largest Lyapunov exponent gives the streamfunction field of the fastest growing perturbation for the time interval. In the case of the chaotic reference solution, the streamfunction field has large amplitudes in limited areas for a small time interval. The areas of the large perturbation growth have some relation to the reference streamfunction field.A possible application of the finite-time Lyapunov exponents and vectors to the atmospheric predictability problem is discussed. These quantities might be used as several forecast measures of the time-dependent predictability in numerical weather predictions.
An abstract is not available.
WEAVE (Workbench Environment for Analysis and Visual Explo-ration) is an environment for creating interactive visualization ap-plications. WEAVE differs from previous systems in that it pro-vides transparent linking between custom 3-D visualizations and multidimensional statistical representations, and provides interac-tive color brushing betwen all visualizations. In this paper, we demonstrate how WEAVE can be used to rapidly prototype a biomedical application, weaving together simu-lation data, measurement data, and 3-d anatomical data concerning the propagation of excitation in the heart. These linked statistical and custom three-dimensional visualizations of the heart can allow scientists to more effectively study the correspondance of structure and behavior.
Salient edges are perceptually prominent features of a surface. Most previous extraction schemes utilize the notion of ridges and valleys for their detection, thereby requiring curvature derivatives which are rather sensitive to noise. We introduce a novel method for salient edge extraction which does not depend on curvature derivatives. It is based on a topological analysis of the principal curvatures and salient edges of the surface are identified as parts of separatrices of the topological skeleton. Previous topological approaches obtain results including non-salient edges due to inherent properties of the underlying algorithms. We extend the profound theory by introducing the novel concept of separatrix persistence, which is a smooth measure along a separatrix and allows to keep its most salient parts only. We compare our results with other methods for salient edge extraction.
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.
This paper proposes aGalilean invariant generalization of critical points ofvector field topology for 2D time-dependent flows. The approach is based upon a Lagrangian consideration of fluid particle motion. It extracts long-living
features, likesaddles and centers, and filters out short-living local structures. This is well suited for analysis ofturbulent
flow, where standard snapshot topology yields an unmanageable large number of topological structures that are barely related
to the few main long-living features employed in conceptual fluid mechanics models. Results are shown for periodic and chaoticvortex
motion.
Enabling insight into large and complex datasets is a prevalent theme in visualization research for which different approaches are pursued.
Topology-based methods are built on the idea of abstracting characteristic structures such as the topological skeleton from the data and to construct the visualizations accordingly. There are currently new demands for and renewed interest in topology-based visualization solutions. This book presents 13 peer-reviewed papers as written results from the 2005 workshop “Topology-Based Methods in Visualization” that was initiated to enable additional stimulation in this field. It contains a longer chapter dedicated to a survey of the state-of-the-art, as well as a great deal of original work by leading experts that has not been published before, spanning both theory and applications. It captures key concepts and novel ideas and serves as an overview of current trends in topology-based visualization research.
This paper develops the theory and computation of Lagrangian Coherent Structures (LCS), which are defined as ridges of Finite-Time Lyapunov Exponent (FTLE) fields. These ridges can be seen as finite-time mixing templates. Such a framework is common in dynamical systems theory for autonomous and time-periodic systems, in which examples of LCS are stable and unstable manifolds of fixed points and periodic orbits. The concepts defined in this paper remain applicable to flows with arbitrary time dependence and, in particular, to flows that are only defined (computed or measured) over a finite interval of time. Previous work has demonstrated the usefulness of FTLE fields and the associated LCSs for revealing the Lagrangian behavior of systems with general time dependence. However, ridges of the FTLE field need not be exactly advected with
the flow. The main result of this paper is an estimate for the flux across an LCS, which shows that the flux is small, and in most cases negligible, for well-defined LCSs or those that rotate at a speed comparable to the local Eulerian velocity field, and are computed from FTLE fields with a sufficiently long integration time. Under these hypotheses, the structures represent nearly invariant manifolds even in systems with arbitrary time dependence.
The results are illustrated on three examples. The first is a simplified dynamical model of a double-gyre flow. The second is surface current data collected by high-frequency radar stations along the coast of Florida and the third is unsteady separation over an airfoil. In all cases, the existence of LCSs governs the transport and it is verified numerically that the flux of particles through these distinguished lines is indeed negligible.
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.
Textures are a well known graphic representation with many useful properties, and they have been well supported on graphics hardware since the mid-1990s. Textures execute a wide variety of operations in hardware, including filtering, compression, and blending and offer the potential to greatly accelerate many advanced visualization algorithms. The main function of textures is to encode detailed information without the need for large-scale polygonal models. This chapter discusses flow-texture algorithms that encode dense representations of time-dependent vector fields into textures and evolve those representations in time. These algorithms have the distinctive property that the update mechanism for each texel is identical. These algorithms are ideally suited to modern graphics hardware that relies on a single instruction multiple data architecture. Two algorithms can be viewed as precursors to flow textures: spot noise and moving textures. These two algorithms motivated the development of dense methods and their eventual mapping onto graphics hardware. A key component of flow-texture algorithms is a convolution operator that acts along some path in a noise texture. This approach to dense vector-field representation was first applied to steady flows and called line integral convolution.
We present an algorithm for compressing 2D vector fields that preserves topology. Our approach is to simplify the given data set using constrained clustering. We employ different types of global and local error metrics including the earth mover's distance metric to measure the degradation in topology as well as weighted magnitude and angular errors. As a result, we obtain precise error bounds in the compressed vector fields. Experiments with both analytic and simulated data sets are presented. Results indicate that one can obtain significant compression with low errors without losing topology information.
For visualization purposes, the depiction of the topology results in synthetic representations that transcribe the fundamental characteristics of the data. Further, topology-based visualization results in a dramatic decrease in the amount of data required for interpretation, which makes it very appealing for the analysis of large-scale datasets. This chapter introduces to the mathematical foundations of the topological approach to flow visualization along with a survey of existing techniques in this domain. The focus is on methods directly related to the depiction and analysis of the flow topology. The chapter considers vector fields and introduces basic theoretical notions. Theoretical notions result from the qualitative theory of dynamical systems. Nonlinear and parameter-dependent topologies are discussed in the chapter, along with the fundamental concept of bifurcation. The chapter also treats tensor fields and considers the topology of the eigenvector fields of symmetric, second-order tensor fields.
A vortex is characterized by the swirling motion of fluid around a central region. This characterization stems from the visual perception of swirling phenomena that are pervasive throughout the natural world. However, translating this intuitive description of a vortex into a formal definition has been quite a challenge. Despite the lack of a formal definition, various detection algorithms have been implemented that can adequately identify vortices in most computational datasets. This chapter presents an overview of existing detection methods; in particular, the focus is on nine methods that are representative of the state of the art. The chapter begins by presenting three taxonomies for classifying these nine detection methods. It then describes each algorithm, along with pseudo-code where appropriate. Next, the chapter describes a recently developed verification algorithm for swirling flows. The chapter also discusses the different visualization techniques for vortices.
This paper examines whether hyperbolic Lagrangian structures-such as stable and unstable manifolds-found in model velocity data represent reliable predictions for mixing in the true fluid velocity field. The error between the model and the true velocity field may result from velocity interpolation, extrapolation, measurement imprecisions, or any other deterministic source. We find that even large velocity errors lead to reliable predictions on Lagrangian coherent structures, as long as the errors remain small in a special time-weighted norm. More specifically, we show how model predictions from the Okubo-Weiss criterion or from finite-time Lyapunov exponents can be validated. We also estimate how close the true Lagrangian coherent structures are to those predicted by models.
If one direction of (three-dimensional) space is singled out, it makes sense to formulate the description of embedded curves and surfaces in a frame that is adapted both to the embedded manifold and to the special direction, rather than a frame based upon the curvature landscape. Such a case occurs often in computer vision, where the image plane plays a role that differs essentially from the direction of view. The classical case is that of geomorphology, where the vertical is the singled out dimension. In computer vision the `ridges' and `(water-)courses' are recognized as important entities and attempts have been made to make the intuitive notions precise. These attempts repeat the unfortunate misunderstandings that marked the course of the late 19th century struggle to define the `Talweg' (equals `valley path' or `(water-)course'). We elucidate the problems and their solution via novel examples.
A new method is proposed to extract the axes of tubular vorticesin turbulence.Loci of sectional local minimum of the pressure associated with theadvection acceleration are traced numerically.It is applied to a homogeneous isotropic turbulence to demonstrate that swirling motions actually exist around these axes.The present method is shown to be superior to several typical onescommonly used in identifying tubular vortices.
Enabling insight into large and complex datasets is a prevalent theme in visualization research for which different approaches are pursued. Topology-based methods are built on the idea of abstracting characteristic structures such as the topological skeleton from the data and to construct the visualizations accordingly. There are currently new demands for and renewed interest in topology-based visualization solutions. This book presents 13 peer-reviewed papers as written results from the 2005 workshop Topology-Based Methods in Visualization that was initiated to enable additional stimulation in this field. It contains a longer chapter dedicated to a survey of the state-of-the-art, as well as a great deal of original work by leading experts that has not been published before, spanning both theory and applications. It captures key concepts and novel ideas and serves as an overview of current trends in topology-based visualization research.
An abstract is not available.
A set of differential equations for the eigenvalues and eigenvectors of
the stability matrix of a dynamical system, as well as for the Lyapunov
exponents and the corresponding eigenvectors, is derived. The rate of
convergence of the Lyapunov eigenvectors is shown to be exponential. The
eigenvectors of the stability matrix can be grouped into sets, each
spanning a subspace which converges at an exponential rate. It is
demonstrated that, generically, the real parts of the eigenvalues of the
stability matrix equal the corresponding Lyapunov exponents. This
statement has been tested numerically. The values of the Lyapunov
exponents, μi, are shown to be related to the
corresponding finite time values of the Lyapunov exponents (e.g. those
deduced from a finite time numerical simulation), μi(t),
by: μi(t) = μi + (bi +
ξi(t))/t. The bi's are constants and
ξi(t) are ``noise'' terms of zero mean. This observation
leads to a method of extrapolation, which has been used to predict
Lyapunov exponents from a finite amount of data. It is shown that the
use of the standard (numerical) methods to compute Lyapunov exponents
introduces an error ai/t in the value of μi(t),
where the ai's are constants. Thus the standard method has a
rate of convergence which is the same as that of the exact
μi(t)'s. Finally, we have shown how one can compute the
eigenvectors associated with each of the eigenvalues of the stability
matrix as well as the Lyapunov eigenvectors.
An effort is made to demonstrate the usefulness of critical-point
concepts in the understanding of flow patterns. Critical points are the
salient features of a flow pattern; given a distribution of such points
and their type, much of the remaining flow field and its geometry and
topology can be deduced, since there is only a limited number of ways
that the streamlines can be joined. Although a field may be unsteady,
the instantaneous streamline patterns furnish an idea of the transport
properties of an array of eddies in jets and wakes or in complicated
three-dimensional separation patterns. In addition, knowledge of a flow
field in one plane can often give clues as to the shape of possible
flows in other planes.
We present a visual analysis and exploration of fluid flow through a cooling jacket. Engineers invest a large amount of time and serious effort to optimize the flow through this engine component because of its important role in transferring heat away from the engine block. In this study we examine the design goals that engineers apply in order to construct an ideal-as-possible cooling jacket geometry and use a broad range of visualization tools in order to analyze, explore, and present the results. We systematically employ direct, geometric, and texture-based flow visualization techniques as well as automatic feature extraction and interactive feature-based methodology. And we discuss the relative advantages and disadvantages of these approaches as well as the challenges, both technical and perceptual with this application. The result is a feature-rich state-of-the-art flow visualization analysis applied to an important and complex data set from real-world computational fluid dynamics simulations.
We use direct Lyapunov exponents (DLE) to identify Lagrangian coherent structures in two different three-dimensional flows, including a single isolated hairpin vortex, and a fully developed turbulent flow. These results are compared with commonly used Eulerian criteria for coherent vortices. We find that despite additional computational cost, the DLE method has several advantages over Eulerian methods, including greater detail and the ability to define structure boundaries without relying on a preselected threshold. As a further advantage, the DLE method requires no velocity derivatives, which are often too noisy to be useful in the study of a turbulent flow. We study the evolution of a single hairpin vortex into a packet of similar structures, and show that the birth of a secondary vortex corresponds to a loss of hyperbolicity of the Lagrangian coherent structures.
The vortex patterns which occur in coflowing jets and wakes at moderate Reynolds numbers (of order 500) are examined in detail. Flow visualization is used in conjunction with a flying hot-wire system which allows instantaneous velocity vector fields to be rapidly measured and related to the smoke patterns. The structures were made perfectly periodic in time by artificial stimulation. The experiments were therefore completely deterministic. This newly developed data-acquisition technique does not require the use of Taylor's hypothesis for inferring patterns from a fixed streamwise position. It therefore allows the vector fields of rapidly evolving patterns to be produced. It also allows the phenomenon of three-dimensional vortex pairing to be studied. The classification of patterns and conjectured topologies made by Perry & Lim (1978a,
b) and the interpretations of Perry, Lim & Chong (1980) and Perry & Watmuff (1981) are examined. In the light of more-detailed measurements, it is found that some of these interpretations require modification.
The most widely used definitions of a vortex are not objective: they identify different structures as vortices in frames that rotate relative to each other. Yet a frame-independent vortex definition is essential for rotating flows and for flows with interacting vortices. Here we define a vortex as a set of fluid trajectories along which the strain acceleration tensor is indefinite over directions of zero strain. Physically, this objective criterion identifies vortices as material tubes in which material elements do not align with directions suggested by the strain eigenvectors. We show using examples how this vortex criterion outperforms earlier frame-dependent criteria. As a side result, we also obtain an objective criterion for hyperbolic Lagrangian structures.
Preface. 1. Introduction. 2. Mathematical Preliminaries. 3. Ridges in Euclidean Geometry. 4. Ridges in Riemannian Geometry. 5. Ridges of Functions Defined on Manifolds. 6. Applications to Image and Data Analysis. 7. Implementation Issues. Bibliography. Index.
Flow visualization has been a very attractive component of scientific visualization research for a long time. Usu-ally very large multivariate datasets require processing. These datasets often consist of a large number of sample locations and several time steps. The steadily increasing performance of computers has recently become a driv-ing factor for a reemergence in flow visualization research, especially in texture-based techniques. In this paper, dense, texture-based flow visualization techniques are discussed. This class of techniques attempts to provide a complete, dense representation of the flow field with high spatio-temporal coherency. An attempt of categorizing closely related solutions is incorporated and presented. Fundamentals are shortly addressed as well as advantages and disadvantages of the methods.
In this paper we explore a novel combined application of two of our existing visualisation techniques to thetracking of 3D vortex tubes in an unsteady flow. The applied techniques are the winding-angle vortex extractiontechnique based on streamline geometry, and the attribute-based feature tracking technique. We have applied theseto the well-known case of an unsteady 3D flow past a tapered cylinder.
First, 2D vortices are detected in a number of horizontal slices for each time step, by means of the winding-anglevortex extraction method. For each 2D vortex a number of attributes are calculated and stored. These vorticesare visualised by a special type of ellipse icons, showing the position, shape and rotational direction and speed ineach slice.
Next, for each time step, 3D vortex tubes are constructed from the 2D vortices by applying the feature trackingprocedure in a spatial dimension to connect the corresponding vortices in adjacent slices. The result is a graphattribute set with the 2D vortex attributes in the nodes and the spatial correspondences as edges.
Finally, the 3D vortex tubes are tracked in time using the same tracking procedure, for finding the correspondingtubes in successive time steps. The result is a description of the evolution of the 3D vortices. An interactive, time-dependentvisualisation is generated using the temporal correspondences of each vortex tube. This analysis revealsa number of interesting patterns.
ACM CSS: I.3.8 Computer Graphics—Applications
In this paper we present an extended critical point concept which allows us to apply vector field topology in the case of unsteady flow. We propose a measure for unsteadiness which describes the rate of change of the velocities in a fluid element over time. This measure allows us to select particles for which topological properties remain intact inside a finite spatio-temporal neighborhood. One benefit of this approach is that the classification of critical points based on the eigenvalues of the Jacobian remains meaningful. In the steady case the proposed criterion reduces to the classical definition of critical points. As a first step we show that finding an optimal Galilean frame of reference can be obtained implicitly by analyzing the acceleration field. In a second step we show that this can be extended by switching to the Lagrangian frame of reference. This way the criterion can detect critical points moving along intricate trajectories. We analyze the behavior of the proposed criterion based on two analytical vector fields for which a correct solution is defined by their inherent symmetries and present results for numerical vector fields.
Current unsteady multi-field simulation data-sets consist of millions of data-points. To efficiently reduce this enormous amount of information, local statistical complexity was recently introduced as a method that identifies distinctive structures using concepts from information theory. Due to high computational costs this method was so far limited to 2D data. In this paper we propose a new strategy for the computation that is substantially faster and allows for a more precise analysis. The bottleneck of the original method is the division of spatio-temporal configurations in the field (light-cones) into different classes of behavior. The new algorithm uses a density-driven Voronoi tessellation for this task that more accurately captures the distribution of configurations in the sparsely sampled high-dimensional space. The efficient computation is achieved using structures and algorithms from graph theory. The ability of the method to detect distinctive regions in 3D is illustrated using flow and weather simulations.
We present the first algorithm for constructing 3D vector fields based on their topological skeleton. The skeleton itself is modeled by interactively moving a number of control polygons. Then a piecewise linear vector field is automatically constructed which has the same topological skeleton as modeled before. This approach is based on a complete segmentation of the areas around critical points into sectors of different flow behavior. Based on this, we present the first approach to visualizing higher order critical points of 3D vector fields.
Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Line and Curve Generation I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism
We introduce a scheme of control polygons to design topological skeletons for vector fields of arbitrary topology. Based on this we construct piecewise linear vector fields of exactly the topology specified by the control polygons. This way a controlled construction of vector fields of any topology is possible. Finally we apply this method for topology-preserving compression of vector fields consisting of a simple topology.
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 chapter gives an overview on topological methods for vector field processing. After introducing topological features
for 2D and 3D vector fields, we discuss how to extract and use them as visualization tools for complex flow phenomena. We
do so both for static and dynamic fields. Finally, we introduce further applications of topological methods for compressing,
simplifying, comparing, and constructing vector fields.
Focus+context visualization is well-known from information visualization: cer- tain data subsets of special interest are shown in more detail (locally enlarged) whereas the rest of the data is provided as context (in reduced space) to sup- port user orientation and navigation. The key point of this work is a generalized denition of focus+context vi- sualization which extends its applicability also to scientic visualization. We show how dieren t graphics resources such as space, opacity, color, etc., can be used to visually discriminate between data subsets in focus and their respec- tive context. To furthermore demonstrate its general use, we discuss several quite dieren t examples of focus+context visualization with respect to our generalized denition. Finally, we also discuss the very important interaction aspect of focus+context visualization.
This paper is the result of research and contemplation on the actual usefulness of topology-based methods in real-world applications.
We recapitulate commonly used arguments in favor of topology-based approaches first to realign our expectations with respect
to the utilization of topology extraction in the context of concrete applications. To illustrate some of our considerations,
we take a closer look at one specific example, i.e., the visual analysis of flow through a cooling jacket and we report our
respective experiences. After discussing the topology-based analysis of the cooling jacket case, we contrast topology-based
flow visualization with an alternative approach, i.e., the interactive feature extraction for feature-based visualization.
Without generalizing just from the one concrete example scenario, we still are able to conclude with some broader experiences
which we have made in the past and which seem to align well with the opinion of others in our field.
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 behavior, 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 considerations
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
Flow visualization research has made rapid advances in recent years, especially in the area of topology-based flow visualization.
The ever increasing size of scientific data sets favors algorithms that are capable of extracting important subsets of the
data, leaving the scientist with a more manageable representation that may be visualized interactively. Extracting the topology
of a flow achieves the goal of obtaining a compact representation of a vector or tensor field while simultaneously retaining
its most important features. We present the state of the art in topology-based flow visualization techniques. We outline numerous
topology-based algorithms categorized according to the type and dimensionality of data on which they operate and according
to the goal-oriented nature of each method. Topology tracking algorithms are also discussed. The result serves as a useful
introduction and overview to research literature concerned with the study of topology-based flow visualization.
Topological concepts provide highly comprehensible representations of the main features of a flow with a limited number of
elements. This paper presents an automated classification method of instantaneous velocity fields based on the analysis of
their critical points distribution and feature flow fields. It uses the fact that topological changes of a velocity field
are continuous in time to extract large scale periodic phenomena from insufficiently time-resolved datasets. This method is
applied to two test-cases : an analytical flow field and PIV planes acquired downstream a wall-mounted cube.
Some applications of critical point theory are shown for the description and identification of eddying motions in turbulence and in vortex shedding. This includes both large scale and fine scale motions. Difficulties in the interpretations of flow topology are outlined with some examples.
We present a technique to visualize global uncertainty in stationary 3D vector fields by a topological approach. We start from an existing approach for 2D uncertain vector field topology and extend this into 3D space. For this a number of conceptional and technical challenges in performance and visual representation arise. In order to solve them, we develop an acceleration for finding sink and source distributions. Having these distributions we use overlaps of their corresponding volumes to find separating structures and saddles. As part of the approach, we introduce uncertain saddle and boundary switch connectors and provide algorithms to extract them. For the visual representation, we use multiple direct volume renderings. We test our method on a number of synthetic and real data sets.
Streak surfaces are among the most important features to support 3D unsteady flow exploration, but they are also among the computationally most demanding. Furthermore, to enable a feature driven analysis of the flow, one is mainly interested in streak surfaces that show separation profiles and thus detect unstable manifolds in the flow. The computation of such separation surfaces requires to place seeding structures at the separation locations and to let the structures move correspondingly to these locations in the unsteady flow. Since only little knowledge exists about the time evolution of separating streak surfaces, at this time, an automated exploration of 3D unsteady flows using such surfaces is not feasible. Therefore, in this paper we present an interactive approach for the visual analysis of separating streak surfaces. Our method draws upon recent work on the extraction of Lagrangian coherent structures (LCS) and the real-time visualization of streak surfaces on the GPU. We propose an interactive technique for computing ridges in the finite time Lyapunov exponent (FTLE) field at each time step, and we use these ridges as seeding structures to track streak surfaces in the time-varying flow. By showing separation surfaces in combination with particle trajectories, and by letting the user interactively change seeding parameters such as particle density and position, visually guided exploration of separation profiles in 3D is provided. To the best of our knowledge, this is the first time that the reconstruction and display of semantic separable surfaces in 3D unsteady flows can be performed interactively, giving rise to new possibilities for gaining insight into complex flow phenomena.