: Six timesteps of smoke emission from a fire inside a hall. The smoke propagation is simulated b... more : Six timesteps of smoke emission from a fire inside a hall. The smoke propagation is simulated by computational fluid dynamics and lit with an absorbtion and limited diffuse scattering-based lighting model. Illuminated exit signs shine through dense smoke, which is modeled by adjusting the transfer function in front of exit signs.
Lagrangian coherent structures play an important role in the analysis of unsteady vector fields b... more Lagrangian coherent structures play an important role in the analysis of unsteady vector fields because they represent the timedependent analog to vector field topology. Nowadays, they are often obtained as ridges in the finite-time Lyapunov exponent of the vector field. However, one drawback of this quantity is its very high computational cost because a trajectory needs to be computed for every sample in the space-time domain. One of the foci of this paper are Lagrangian coherent structures that are related to predefined regions such as boundaries, i.e. related to flow attachment and flow separation phenomena. It presents an efficient method for computing the finite-time Lyapunov exponent and its height ridges only in these regions, and in particular, grid advection for the efficient computation of time series of the finite-time Lyapunov exponent, exploiting temporal coherence.
The popularity of vector field topology in the visualization community is due mainly to the topol... more The popularity of vector field topology in the visualization community is due mainly to the topological skeleton which captures the essential information on a vector field in a set of lines or surfaces separating regions of different flow behavior. Unfortunately, vector field topology has no straightforward extension to unsteady flow, and the concept probably most closely related to the topological skeleton are the so-called Lagrangian coherent structures (LCS). LCS are material lines or material surfaces that separate regions of different flow behavior. Ideally, such structures are material lines (or surfaces) in an exact sense and at the same time maximally attracting or repelling, but practical realizations such as height ridges of the finite-time Lyapunov exponent (FTLE) fulfill these two requirements only in an approximate sense. In this paper, we quantify the deviation from exact material lines/surfaces for several FTLE-based concepts, and we propose a numerically simpler variants of FTLE ridges that has equal or better error characteristics than classical FTLE height ridges.
Figure 1: Cutaway image from a flow visualization. (a) Features of interest are colored orange. (... more Figure 1: Cutaway image from a flow visualization. (a) Features of interest are colored orange. (b) The algorithm determines the positions of the cutting objects in order to reveal as much of the features as possible. (c) The resulting cutaway.
Vorticity is the quantity used to describe the creation, transformation and extinction of vortice... more Vorticity is the quantity used to describe the creation, transformation and extinction of vortices. It is present not only in vortices but also in shear flow. Especially in ducted flows, most of the overall vorticity is usually contained in the boundary layer. When a vortex develops from the boundary layer, this can be described by transport of vorticity. For a better understanding of a flow it is therefore of interest to examine vorticity in all of its different roles. The goal of this application study was not primarily the visualization of vortices but of vorticity distribution and its role in vortex phenomena.
Livre: Topological methods in data analysis and visualization ii: theory, algorithms and applicat... more Livre: Topological methods in data analysis and visualization ii: theory, algorithms and applications (hardback) (series: mathematics and visualization) PEIKERT Ronald, HAUSER Helwig, CARR Hamish, FUCHS Raphael.
IEEE Transactions on Visualization and Computer Graphics, 2000
In this paper, we present a data-flow system which supports comparative analysis of time-dependen... more In this paper, we present a data-flow system which supports comparative analysis of time-dependent data and interactive simulation steering. The system creates data on-the-fly to allow for the exploration of different parameters and the investigation of multiple scenarios. Existing data-flow architectures provide no generic approach to handle modules that perform complex temporal processing such as particle tracing or statistical analysis over time. Moreover, there is no solution to create and manage module data, which is associated with alternative scenarios. Our solution is based on generic data-flow algorithms to automate this process, enabling elaborate data-flow procedures, such as simulation, temporal integration or data aggregation over many time steps in many worlds. To hide the complexity from the user, we extend the World Lines interaction techniques to control the novel data-flow architecture. The concept of multiple, special-purpose cursors is introduced to let users intuitively navigate through time and alternative scenarios. Users specify only what they want to see, the decision which data are required is handled automatically. The concepts are explained by taking the example of the simulation and analysis of material transport in levee-breach scenarios. To strengthen the general applicability, we demonstrate the investigation of vortices in an offline-simulated dam-break data set.
IEEE Transactions on Visualization and Computer Graphics, 2000
Feature-based flow visualization is naturally dependent on feature extraction. To extract flow fe... more Feature-based flow visualization is naturally dependent on feature extraction. To extract flow features, often higher-order properties of the flow data are used such as the Jacobian or curvature properties, implicitly describing the flow features in terms of their inherent flow characteristics (e.g., collinear flow and vorticity vectors). In this paper we present recent research which leads to the conclusion that feature extraction algorithms need to be extended to a time-dependent analysis framework (in terms of time derivatives) when dealing with unsteady flow data. Accordingly, we present extensions of the parallel vectors vortex criteria to the timedependent domain and show the improvements of the feature-based flow visualization results in comparison to the steady versions of this extraction algorithm both in the context of a high-resolution dataset, i.e. a simulation specifically designed to evaluate our new approach, as well as for a real-world dataset from a concrete application.
IEEE Transactions on Visualization and Computer Graphics, 2000
ABSTRACT Room air flow and air exchange are important aspects for the design of energy-efficient ... more ABSTRACT Room air flow and air exchange are important aspects for the design of energy-efficient buildings. As a result, simulations are increasingly used prior to construction to achieve an energy-efficient design. We present a visual analysis of air flow generated at building entrances, which uses a combination of revolving doors and air curtains. The resulting flow pattern is challenging because of two interacting flow patterns: On the one hand, the revolving door acts as a pump, on the other hand, the air curtain creates a layer of uniformly moving warm air between the interior of the building and the revolving door. Lagrangian coherent structures (LCS), which by definition are flow barriers, are the method of choice for visualizing the separation and recirculation behavior of warm and cold air flow. The extraction of LCS is based on the finite-time Lyapunov exponent (FTLE) and makes use of a ridge definition which is consistent with the concept of weak LCS. Both FTLE computation and ridge extraction are done in a robust and efficient way by making use of the fast Fourier transform for computing scale-space derivatives.
IEEE Transactions on Visualization and Computer Graphics, 2000
The perception of transparency and the underlying neural mechanisms have been subject to extensiv... more The perception of transparency and the underlying neural mechanisms have been subject to extensive research in the cognitive sciences. However, we have yet to develop visualization techniques that optimally convey the inner structure of complex transparent shapes. In this paper, we apply the findings of perception research to develop a novel illustrative rendering method that enhances surface transparency nonlocally. Rendering of transparent geometry is computationally expensive since many optimizations, such as visibility culling, are not applicable and fragments have to be sorted by depth for correct blending. In order to overcome these difficulties efficiently, we propose the illustration buffer. This novel data structure combines the ideas of the A and G-buffers to store a list of all surface layers for each pixel. A set of local and nonlocal operators is then used to process these depth-lists to generate the final image. Our technique is interactive on current graphics hardware and is only limited by the available graphics memory. Based on this framework, we present an efficient algorithm for a nonlocal transparency enhancement that creates expressive renderings of transparent surfaces. A controlled quantitative double blind user study shows that the presented approach improves the understanding of complex transparent surfaces significantly.
The fractal dimension of the Apollonian sphere packing has been computed numerically up to six tr... more The fractal dimension of the Apollonian sphere packing has been computed numerically up to six trusty decimal digits. Based on the 31 944 875 541 924 spheres of radius greater than 2 -19 contained in the Apollonian packing of the unit sphere, we obtained an estimate of 2.4739465, where the last digit is questionable. Two fundamentally different algorithms have been employed. Outlines of both algorithms are given.
Vector fields are a common concept for the representation of many different kinds of flow phenome... more 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.
In this paper we present an extended critical point concept which allows us to apply vector field... more 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.
: Scalar field on a space-time parameterization of a cylinder flow. All separating structures in ... more : Scalar field on a space-time parameterization of a cylinder flow. All separating structures in space-time are described by this field. The dark lines are 5 different path lines. (FTLE). Experience has shown that the resulting structures are often close to material structures (i.e., material lines or material surfaces). Moreover, it has been proven that for an integration time converging to infinity, they converge to exact material structures. However, due to the finite integration time in FTLE, they are generally not exact material structures. In this paper we introduce a modification of the FTLE method which is guaranteed to produce separating material structures as features of a scalar field. We achieve this by incorporating the complete available integration time both in forward and backward direction, and by choosing an appropriate definition for separating structures. We apply our method to two test data sets and show the differences to classical FTLE.
We present a dense visualization of vector fields on multi-layered surfaces. The method is based ... more We present a dense visualization of vector fields on multi-layered surfaces. The method is based on the illustration buffer, which provides a screen space representation of the surface, where each pixel stores a list of all surface layers. This representation is implemented on the GPU using shaders and leads to a fast, output sensitive technique. In our approach, we first use procedural noise to create an initial spot pattern on the surface that has both an almost constant screen space frequency and is view independent. Then, we perform anisotropic diffusion simultaneously on all surface layers using a discretization scheme that maintains second order convergence while only accessing the four neighboring pixels. Finally, we enhance this result with illustrative techniques and composite the final image. Our method works with time-evolving surfaces, time-dependent vector fields, and moving cameras. We apply our method to CFD data sets from engineering and astronomy as well as synthetic velocity fields.
: Six timesteps of smoke emission from a fire inside a hall. The smoke propagation is simulated b... more : Six timesteps of smoke emission from a fire inside a hall. The smoke propagation is simulated by computational fluid dynamics and lit with an absorbtion and limited diffuse scattering-based lighting model. Illuminated exit signs shine through dense smoke, which is modeled by adjusting the transfer function in front of exit signs.
Lagrangian coherent structures play an important role in the analysis of unsteady vector fields b... more Lagrangian coherent structures play an important role in the analysis of unsteady vector fields because they represent the timedependent analog to vector field topology. Nowadays, they are often obtained as ridges in the finite-time Lyapunov exponent of the vector field. However, one drawback of this quantity is its very high computational cost because a trajectory needs to be computed for every sample in the space-time domain. One of the foci of this paper are Lagrangian coherent structures that are related to predefined regions such as boundaries, i.e. related to flow attachment and flow separation phenomena. It presents an efficient method for computing the finite-time Lyapunov exponent and its height ridges only in these regions, and in particular, grid advection for the efficient computation of time series of the finite-time Lyapunov exponent, exploiting temporal coherence.
The popularity of vector field topology in the visualization community is due mainly to the topol... more The popularity of vector field topology in the visualization community is due mainly to the topological skeleton which captures the essential information on a vector field in a set of lines or surfaces separating regions of different flow behavior. Unfortunately, vector field topology has no straightforward extension to unsteady flow, and the concept probably most closely related to the topological skeleton are the so-called Lagrangian coherent structures (LCS). LCS are material lines or material surfaces that separate regions of different flow behavior. Ideally, such structures are material lines (or surfaces) in an exact sense and at the same time maximally attracting or repelling, but practical realizations such as height ridges of the finite-time Lyapunov exponent (FTLE) fulfill these two requirements only in an approximate sense. In this paper, we quantify the deviation from exact material lines/surfaces for several FTLE-based concepts, and we propose a numerically simpler variants of FTLE ridges that has equal or better error characteristics than classical FTLE height ridges.
Figure 1: Cutaway image from a flow visualization. (a) Features of interest are colored orange. (... more Figure 1: Cutaway image from a flow visualization. (a) Features of interest are colored orange. (b) The algorithm determines the positions of the cutting objects in order to reveal as much of the features as possible. (c) The resulting cutaway.
Vorticity is the quantity used to describe the creation, transformation and extinction of vortice... more Vorticity is the quantity used to describe the creation, transformation and extinction of vortices. It is present not only in vortices but also in shear flow. Especially in ducted flows, most of the overall vorticity is usually contained in the boundary layer. When a vortex develops from the boundary layer, this can be described by transport of vorticity. For a better understanding of a flow it is therefore of interest to examine vorticity in all of its different roles. The goal of this application study was not primarily the visualization of vortices but of vorticity distribution and its role in vortex phenomena.
Livre: Topological methods in data analysis and visualization ii: theory, algorithms and applicat... more Livre: Topological methods in data analysis and visualization ii: theory, algorithms and applications (hardback) (series: mathematics and visualization) PEIKERT Ronald, HAUSER Helwig, CARR Hamish, FUCHS Raphael.
IEEE Transactions on Visualization and Computer Graphics, 2000
In this paper, we present a data-flow system which supports comparative analysis of time-dependen... more In this paper, we present a data-flow system which supports comparative analysis of time-dependent data and interactive simulation steering. The system creates data on-the-fly to allow for the exploration of different parameters and the investigation of multiple scenarios. Existing data-flow architectures provide no generic approach to handle modules that perform complex temporal processing such as particle tracing or statistical analysis over time. Moreover, there is no solution to create and manage module data, which is associated with alternative scenarios. Our solution is based on generic data-flow algorithms to automate this process, enabling elaborate data-flow procedures, such as simulation, temporal integration or data aggregation over many time steps in many worlds. To hide the complexity from the user, we extend the World Lines interaction techniques to control the novel data-flow architecture. The concept of multiple, special-purpose cursors is introduced to let users intuitively navigate through time and alternative scenarios. Users specify only what they want to see, the decision which data are required is handled automatically. The concepts are explained by taking the example of the simulation and analysis of material transport in levee-breach scenarios. To strengthen the general applicability, we demonstrate the investigation of vortices in an offline-simulated dam-break data set.
IEEE Transactions on Visualization and Computer Graphics, 2000
Feature-based flow visualization is naturally dependent on feature extraction. To extract flow fe... more Feature-based flow visualization is naturally dependent on feature extraction. To extract flow features, often higher-order properties of the flow data are used such as the Jacobian or curvature properties, implicitly describing the flow features in terms of their inherent flow characteristics (e.g., collinear flow and vorticity vectors). In this paper we present recent research which leads to the conclusion that feature extraction algorithms need to be extended to a time-dependent analysis framework (in terms of time derivatives) when dealing with unsteady flow data. Accordingly, we present extensions of the parallel vectors vortex criteria to the timedependent domain and show the improvements of the feature-based flow visualization results in comparison to the steady versions of this extraction algorithm both in the context of a high-resolution dataset, i.e. a simulation specifically designed to evaluate our new approach, as well as for a real-world dataset from a concrete application.
IEEE Transactions on Visualization and Computer Graphics, 2000
ABSTRACT Room air flow and air exchange are important aspects for the design of energy-efficient ... more ABSTRACT Room air flow and air exchange are important aspects for the design of energy-efficient buildings. As a result, simulations are increasingly used prior to construction to achieve an energy-efficient design. We present a visual analysis of air flow generated at building entrances, which uses a combination of revolving doors and air curtains. The resulting flow pattern is challenging because of two interacting flow patterns: On the one hand, the revolving door acts as a pump, on the other hand, the air curtain creates a layer of uniformly moving warm air between the interior of the building and the revolving door. Lagrangian coherent structures (LCS), which by definition are flow barriers, are the method of choice for visualizing the separation and recirculation behavior of warm and cold air flow. The extraction of LCS is based on the finite-time Lyapunov exponent (FTLE) and makes use of a ridge definition which is consistent with the concept of weak LCS. Both FTLE computation and ridge extraction are done in a robust and efficient way by making use of the fast Fourier transform for computing scale-space derivatives.
IEEE Transactions on Visualization and Computer Graphics, 2000
The perception of transparency and the underlying neural mechanisms have been subject to extensiv... more The perception of transparency and the underlying neural mechanisms have been subject to extensive research in the cognitive sciences. However, we have yet to develop visualization techniques that optimally convey the inner structure of complex transparent shapes. In this paper, we apply the findings of perception research to develop a novel illustrative rendering method that enhances surface transparency nonlocally. Rendering of transparent geometry is computationally expensive since many optimizations, such as visibility culling, are not applicable and fragments have to be sorted by depth for correct blending. In order to overcome these difficulties efficiently, we propose the illustration buffer. This novel data structure combines the ideas of the A and G-buffers to store a list of all surface layers for each pixel. A set of local and nonlocal operators is then used to process these depth-lists to generate the final image. Our technique is interactive on current graphics hardware and is only limited by the available graphics memory. Based on this framework, we present an efficient algorithm for a nonlocal transparency enhancement that creates expressive renderings of transparent surfaces. A controlled quantitative double blind user study shows that the presented approach improves the understanding of complex transparent surfaces significantly.
The fractal dimension of the Apollonian sphere packing has been computed numerically up to six tr... more The fractal dimension of the Apollonian sphere packing has been computed numerically up to six trusty decimal digits. Based on the 31 944 875 541 924 spheres of radius greater than 2 -19 contained in the Apollonian packing of the unit sphere, we obtained an estimate of 2.4739465, where the last digit is questionable. Two fundamentally different algorithms have been employed. Outlines of both algorithms are given.
Vector fields are a common concept for the representation of many different kinds of flow phenome... more 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.
In this paper we present an extended critical point concept which allows us to apply vector field... more 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.
: Scalar field on a space-time parameterization of a cylinder flow. All separating structures in ... more : Scalar field on a space-time parameterization of a cylinder flow. All separating structures in space-time are described by this field. The dark lines are 5 different path lines. (FTLE). Experience has shown that the resulting structures are often close to material structures (i.e., material lines or material surfaces). Moreover, it has been proven that for an integration time converging to infinity, they converge to exact material structures. However, due to the finite integration time in FTLE, they are generally not exact material structures. In this paper we introduce a modification of the FTLE method which is guaranteed to produce separating material structures as features of a scalar field. We achieve this by incorporating the complete available integration time both in forward and backward direction, and by choosing an appropriate definition for separating structures. We apply our method to two test data sets and show the differences to classical FTLE.
We present a dense visualization of vector fields on multi-layered surfaces. The method is based ... more We present a dense visualization of vector fields on multi-layered surfaces. The method is based on the illustration buffer, which provides a screen space representation of the surface, where each pixel stores a list of all surface layers. This representation is implemented on the GPU using shaders and leads to a fast, output sensitive technique. In our approach, we first use procedural noise to create an initial spot pattern on the surface that has both an almost constant screen space frequency and is view independent. Then, we perform anisotropic diffusion simultaneously on all surface layers using a discretization scheme that maintains second order convergence while only accessing the four neighboring pixels. Finally, we enhance this result with illustrative techniques and composite the final image. Our method works with time-evolving surfaces, time-dependent vector fields, and moving cameras. We apply our method to CFD data sets from engineering and astronomy as well as synthetic velocity fields.
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Papers by R. Peikert