To achieve quick response times appropriate for interactive analysis we need a tradeoff between accuracy, flexibility and computational cost. In the context of our application we know that the engineers are interested in understanding the relationships between the
5.4. IMPLEMENTATION DETAILS 83
Figure 5.9: (a) A tradeoff between accuracy and interaction time: for interactive analysis we use particle paths computed offline for each volume cell. To reach interactive response rates, we restrict the system to one line segment per cell and particle and store the correct path length in an additional attribute. (b) When the number of pixels available is lower than the number of trajectories we combine groups of lines using a scan line algorithm.
This can result in a large number of segments for the combined line.
computational cells of the simulation grid. Therefore we seed one path- or streamline per cell to be able to compute a delocalized detector response for each cell. During interac-tion the main computainterac-tional cost lies in the line fusion approach and our prototype can compute the final texture for small regions of interest with up to 100K cells at interactive rates (se Figure 5.9). A more efficient implementation could be several magnitudes faster.
Trajectories are computed off line and stored in an additional data set, which takes several hours for the full two-stroke data set, but for a specific region of interest at a selected time step only small subset of these lines has to be computed.
Chapter Conclusions
The main drawback of the presented method is that the detected results are no longer objective in the sense that each engineer will come to exactly the same vortex detection results. The exact location of the vortex boundary is dependent on the specification of integration length parameters. These differences are typically small and as long as we do not have a general definition for what a vortex is, this fuzziness can be considered appropriate. The second disadvantage of the presented approach is the time consumed by integrating trajectories through the fluid. We used a rather inefficient implementation where the timings cannot be considered representative, and GPU-based implementations are reported to compute trajectories at nearly interactive rates [79]. Another drawback is that interaction is often a necessity. Using bad integration length parameters, the results are more blurred and worse than λ2 regions, even though the delocalized detectors have
In this chapter we have proposed delocalized vortex region detectors. With little inter-action to determine reasonable parameters, the delocalized vortex detectors improve the feature extraction process. We have also discussed how the ability to control the range of integration improves the expressiveness of the detectors over their local counterparts. The delocalized detectors are a combination of the Eulerian and the Lagrangian approach to vortex region extraction. The basic message here is that the Eulerian and the Lagrangian are not different alternatives to vortex extraction, opposite to each other, but that they can be combined to one technique sharing the benefits of both. The good local vortex de-tection performance of the Eulerian criteria and the global information of the Lagrangian view point combine to generate well separated and smooth detection results.
Chapter 6
Parallel Vectors Criteria for Unsteady Flow Vortices
”Often there’s no need to tear off an arm to remove a splinter. ” (Frank Herbert 1920 – 1986)
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 chapter we present recent research which leads to the (not really surprising) 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 two extensions of the parallel vectors based vortex extraction criteria to the time-dependent domain and show the improvements of feature-based flow visualization in comparison to the steady versions of this extraction algorithm both in the context of a high-resolution data set, i.e., a simulation specifically designed to evaluate our new approach, as well as for a real-world data set from a concrete application.
6.1 Motivation
We present a solution to the challenge of feature extraction when dealing with time-dependent simulation data from computational fluid dynamics. We aim at feature-based flow visualization with focus on vortices and their central locations. In an extension of the state of the art we present two new methods for the extraction of vortex core lines (aka.vortex axes1) in unsteady flowwhich are truthful to the time-dependent nature of the extracted features.
1 In other fields, e.g., in fluid mechanics,vortex coresare considered to be of regional type (and not of line type). We use the termvortex core line for line-type curve features which represent central locations in vortices.
the extraction criterion in a way that temporal derivatives are used for the local characteri-zation of vortices and not only the Jacobian of the flow. This is synonymous to considering pathlines for feature extraction from unsteady flow instead of streamlines. Even though we experienced in exchange with colleagues, reviewers, and others that this extension is easily and quickly considered to be logical and straight forward, the results improve more than expected.
Very often, flow phenomena such as gas flow during combustion or air flow around a vehicle are time-dependent in their nature and steady representations are just an ap-proximation. Data sets with time-independent flow are useful for domain experts as they provide information about general or large-scale characteristics of the flow, at a relatively low cost in terms of data set size, simulation time, as well as analysis time. However, we still observe a clear trend towards more unsteady flow data in scientific as well as in com-mercial applications mostly because of better results, especially when doing a more careful or detailed flow analysis, and also because of the availability of increased computing and storage resources.
Accordingly, we consider it important to explicitly demonstrate that feature extraction based on time, is not only logical to do, but indeed yields better results. In certain cases, we can even observe that the traditional, streamline-oriented approaches lead to displaced
“features”. Furthermore, we can find an improved agreement of the new approach with physical extraction schemes such as the low-pressure assumption in the midst of vortices (no need for a correction step). In Sections 6.2 and 6.4 we exemplify our point by means of selected cases both in analytic and computed form. The need for a new approach is demonstrated as well as the gain through improved results.
The algorithms, which we take as a basis for developing our new approach, are the proven method for extracting vortex core lines from steady flow data by Sujudi and Haimes [140] as well as the related, higher-order method by Roth and Peikert [122]. Both approaches were successfully applied in many cases, especially when dealing with time-independent data. As such, we consider them as a strong starting point for approaching the case of unsteady flow data. To do so, we adopt the principle of the parallel vectors operator [113] for extracting the vortex core lines in conjunction with modified extraction criteria that are based on temporal derivatives.
Weinkauf et al. [157] approach the question of vortex core line extraction in a similar fashion. For finding ”swirling particle cores” they analyze the real eigenvector of the