Abstract Flow maps are a fundamental concept for the visualization and analysis of time‐dependent flows. A common approach is to apply sampling and reconstruction – which comes with a number of problems: flow maps are high‐dimensional functions with locally large gradients which makes approximations very challenging. In general, one must apply large sampling resolutions and more advanced methods to achieve sufficiently good results. Existing methods for efficient, high‐quality sampling and approximation restrict to subsets of flow maps, e.g., with fixed start and end times, or at least fixed differences. In this paper, we propose the first method to handle the entire flow map by decreasing the dimensionality of the space‐time‐time domain. Essentially, it is a framework which splits the high‐dimensional flow map into few lower‐dimensional functions. This enables to achieve much larger resolutions than traditional sampling with the same memory consumption. We use our method in established applications and show that it can significantly decrease errors in the reconstruction by using these larger resolutions.

Abstract Lagrangian coherent structures (LCS) is an important concept for the visualization of unsteady flows. They describe the boundaries of regions for which material transport stays mostly coherent over time which can help for a better understanding of dynamical systems. One of the most common techniques for their computation is the extraction of ridges from the finite‐time Lyapunov exponent (FTLE) field. FTLE ridges are challenging to extract, both in terms of accuracy and performance, because they expose strong gradients of the underlying field, tend to come close to each other and are dynamic with respect to different time parameters. We present a new method for extracting FTLE ridges for series of integration times which is able to show how coherent regions and their borders evolve over time. Our techniques mainly build on a particle system which is used for sampling the ridges uniformly. This system is highly optimized for the challenges of FTLE ridge extraction. Further, it is able to take advantage of the continuous evolvement of the ridges which makes their sampling for multiple integration times much faster. We test our method on multiple 3D datasets and compare it to the standard Marching Ridges technique. For the extraction examples our method is 13 to over 300 times faster, suggesting a significant advantage.

Abstract The model of a Double Gyre flow by Shadden et al. is a standard benchmark data set for the computation of hyperbolic Lagrangian Coherent Structures (LCS) in flow data. While structurally extremely simple, it generates hyperbolic LCS of arbitrary complexity. Unfortunately, the Double Gyre does not come with a well‐defined ground truth: the location of hyperbolic LCS boundaries can only be approximated by numerical methods that usually involve the gradient of the flow map. We present a new benchmark data set that is a small but carefully designed modification of the Double Gyre, which comes with ground truth closed‐form hyperbolic trajectories. This allows for computing hyperbolic LCS boundaries by a simple particle integration without the consideration of the flow map gradient. We use these hyperbolic LCS as a ground truth solution for testing an existing numerical approach for extracting hyperbolic trajectories. In addition, we are able to construct hyperbolic LCS curves that are significantly longer than in existing numerical methods.

We present a formal approach to the visual analysis of recirculation in flows by introducing recirculation surfaces for 3D unsteady flow fields. Recirculation surfaces are the loci where massless particle integration returns to its starting point after some variable, finite integration. We give a rigorous definition of recirculation surfaces as 2-manifolds embedded in 5D space and study their properties. Based on this we construct an algorithm for their extraction, which searches for intersections of a recirculation surface with lines defined in 3D. This reduces the problem to a repeated search for critical points in 3D vector fields. We provide a uniform sampling of the search space paired with a surface reconstruction and visualize results. This way, we present the first algorithm for a comprehensive feature extraction in the 5D flow map of a 3D flow. The problem of finding isolated closed orbits in steady vector fields occurs as a special case of recirculation surfaces. This includes isolated closed orbits with saddle behavior. We show recirculation surfaces for a number of artificial and real flow data sets.