We propose a new class of vortex definitions for flows that are induced by rotating mechanical parts, such as stirring devices, helicopters, hydrocyclones, centrifugal pumps, or ventilators. Instead of a Galilean invariance, we enforce a rotation invariance, i.e., the invariance of a vortex under a uniform-speed rotation of the underlying coordinate system around a fixed axis. We provide a general approach to transform a Galilean invariant vortex concept to a rotation invariant one by simply adding a closed form matrix to the Jacobian. In particular, we present rotation invariant versions of the well-known Sujudi-Haimes, Lambda-2, and Q vortex criteria. We apply them to a number of artificial and real rotating flows, showing that for these cases rotation invariant vortices give better results than their Galilean invariant counterparts.

Abstract In flow visualization, integral surfaces rapidly tend to expand, fold and produce vast amounts of occlusion. While silhouette enhancements and local transparency mappings proved useful for semi‐transparent depictions, they still introduce visual clutter when surfaces grow more complex. An effective visualization of the flow requires a balance between the presentation of interesting surface parts and the avoidance of occlusions that hinder the view. In this paper, we extend the concept of opacity optimization to surfaces to obtain a global approach to the occlusion problem. Starting with a partition of the surfaces into patches, we compute per‐patch opacity as minimizer of a bounded‐variable least‐squares problem. For the final rendering, opacity is interpolated on the surfaces. The resulting visualization technique is interactive, frame‐coherent, view‐dependent and driven by domain knowledge.

Abstract Stream surfaces are a well‐studied and widely used tool for the visualization of 3D flow fields. Usually, stream surface seeding is carried out manually in time‐consuming trial and error procedures. Only recently automatic selection methods were proposed. Local methods support the selection of a set of stream surfaces, but, contrary to global selection methods, they evaluate only thequalityof the seeding lines but not the quality of the whole stream surfaces. Global methods, on the other hand, only support the selection of asingleoptimal stream surface until now. However, for certain flow fields a single stream surface is not sufficient to represent all flow features. In our work, we overcome this limitation by introducing a global selection technique for asetof stream surfaces. All selected surfaces optimize global stream surface quality measures and are guaranteed to be mutually distant, such that they can convey different flow features. Our approach is an efficient extension of the most recent global selection method for single stream surfaces. We illustrate its effectiveness on a number of analytical and simulated flow fields and analyze the quality of the results in a user study.

This paper applies Poisson-based methods to assist in interactive exploration of steady flow fields. Using data-driven deformations we obtain flow-orthogonal and flow-tangential surfaces by a flux-based optimization. Surfaces are positioned interactively and deformed in real-time according to local flow. The deformed surfaces are particularly useful for defining seed structures. We show how the same gradient-based computational framework can be applied to obtain parametrizations of flow-aligned surfaces. This way it is easy to define nontrivial seed structures for integration-based flow visualization methods. Additionally, the flow-aligned parametrizations are employed for view-independent surface-based LIC visualizations. We apply our method to a number of data sets to show the effectiveness of our deformations and parametrization-based seed extraction methods for interactive flow exploration.

Abstract Stream surfaces are well‐known and widely‐used structures for 3D flow visualization. A single surface can be sufficient to represent important global flow characteristics. Unfortunately, due to the huge space of possible stream surfaces, finding the globally most representative stream surface turns out to be a hard task that is usually performed by time‐consuming manual trial and error exploration using slight modifications of seed geometries. To assist users we propose a new stream surface selection method that acts as an automatic preprocessing step before data analysis. We measure stream surface relevance by a novel surface‐based quality measure that prefers surfaces where the flow is aligned with principal curvature directions. The problem of seed structure selection can then be reduced to the computation of simple minimal paths in a weighted graph spanning the domain. We apply a simulated annealing‐based optimization method to find smooth seed curves of globally near‐optimal stream surfaces. We illustrate the effectiveness of our method on a series of synthetic and real‐world data sets.

We define APAP surfaces, surfaces that are as perpendicular as possible to steady 3D vector fields, and present a method to construct discrete representations of them. Since, in general, a perfectly perpendicular surface to a vector field does not exist, we propose and minimize an error metric to enforce perpendicularity as much as possible. Our algorithm constructs an APAP surface by deforming a seed surface anchored in a domain point. In the discrete setting this minimization results in iteratively solving linear least-squares problems and integrating a locally scaled version of the vector field. The definition of the error metric and its numerical minimization guarantee that the minimum zero is attained for the perfectly perpendicular surface if it exists. Otherwise, the minimization converges to the same local minimum independent of the seed configuration, and the resulting surface is - in a least-squares sense - as perpendicular as possible to the flow. We apply these APAP surfaces as an interactive flow visualization tool which we demonstrate on a number of synthetic and real flow data sets.

Abstract The generation of discrete stream surfaces is an important and challenging task in scientific visualization, which can be considered a particular instance of geometric modeling. The quality of numerically integrated stream surfaces depends on a number of parameters that can be controlledlocally, such as time step or distance of adjacent vertices on the front line. In addition there is a parameter that cannot be controlled locally: stream surface meshes tend to show high quality, well‐shaped elements only if the current front line is “globally” approximately perpendicular to the flow direction. We analyze the impact of this geometric property and present a novel solution – a stream surface integrator that forces the front line to be perpendicular to the flow and that generates quad‐dominant meshes with well‐shaped and well‐aligned elements. It is based on the integration of a scaled version of the flow field, and requires repeated minimization of an error functional along the current front line. We show that this leads to computing the 1‐dimensional kernel of a bidiagonal matrix: a linear problem that can be solved efficiently. We compare our method with existing stream surface integrators and apply it to a number of synthetic and real world data sets.