Interactive visualization of mixed scalar and vector fields

The paper discusses CAVEvis and a related set of tools for the interactive visualization and exploration of large sets of time-varying scalar and vector fields using the CAVE virtual reality environment. Since visualization of large data sets can be very time-consuming in both computation and rendering time, the task is distributed over multiple machines, each of which is specialized for some aspect of the visualization process. All modules must run asynchronously to maintain the highest level of interactivity. A model of distributed visualization is introduced that addresses important issues related to the management of time-dependent data, module synchronization, and interactivity bottlenecks.

This paper discusses the reconstruction of the two-dimensional (2D) vector field topology (VFT) from a 2D scalar potential field and vice versa. The physical foundation of the proposed reconstruction method is the convection-type equation coupling a 2D vector field (e.g., skin friction) with a 2D potential field (e.g., surface pressure, temperature, or scalar concentration) in viscous flows. To reconstruct the VFT, a variational method is applied to this inverse problem, and then, an approximate method is proposed based on the linear superposition of some elemental potential field structures with simple analytical forms (source, vortex, saddle, etc.). As examples, the proposed method is applied to swept shock-wave/boundary-layer interaction and near-wall turbulence. Furthermore, in a reversed process to reconstruct a 2D potential field from a 2D vector field, a similar variational method is applied, and an approximate method with a constant source term in the convection-type relation is proposed, which is particularly applicable to reconstruction of a surface pressure field from global skin friction measurements in aerodynamics experiments. The significance of this work is that the complex VFT can be reconstructed based on a scalar potential field by using a semi-analytical approach. The proposed method can be used in fluid mechanics and other disciplines such as computer graphics and data visualization.

The analysis of data that consists of multiple vector fields can be greatly facilitated by the simultaneous visualization of the vector fields. An effective visualization must accurately reflect the key physical structures of the fields in a way that does not allow for an unintended bias toward one distribution. While there are several effective techniques to visualize a single vector field through equally spaced streamlines, applying these techniques to individual vector fields and combining them in a single image yields undesirable artifacts. In this paper, we present strategies for the effective visualization of two vector fields through the use of streamlines.

Feature-aware streamline generation of planar vector fields via topological methods

: Interactive visualization allows scientists and engineers to analyze massive data sets in new ways. They can prune useless data, identify important features, and see the science behind the data. We address computational tasks of image generation, dynamical visualization, and interactive manipulation and quantitative querying. The data sets comprise of scalar fields (volumetric histological scans of spinal cord injuries), vector fields (numerical simulation of turbulent flows) and tensor fields (Reynolds stresses produced by vortices in the flow). Image generation is the core of visualization. Dynamical visualization brings out time dependent features that are hard to recognize in static images. Interactive manipulation and querying allows users to focus and quantify key features in large data sets.

Streamline1 is one of the most widely used method in vector field visualization. However, with the expansion of data size, the existing methods cannot ensure the efficiency while maintaining high accuracy. In order to analyse the vector field efficiently, this paper proposes an algorithm for streamline visualization in unstructured vector field while keeping both efficiency and accuracy based on Kipfer's algorithm[1]. First, all grids are classified according to their Jordan canonical form. Then the streamline expressions in each grid cell are deduced for each kind of the grids, and we found that the expressions can be represented in a uniform way. Finally, the exit point of the streamline in the grid cell is calculated by solving the equation composed of the streamline expression and the equation of each face of the tetrahedron. The exit point can be obtained quickly and accurately by using Newton iterative method. The experimental results show that our algorithm is more efficient for streamline visualization with the same accuracy than Kipfer's algorithm.

Personal computer-based visualization of three-dimensional scalar and vector fields: An application to molecular graphics

A new method for the simplification and the visualization of vector fields is presented based on the notion of centroidal Voronoi tessellations (CVT's). A CVT is a special Voronoi tessellation for which the generators of the Voronoi regions in the tessellation are also the centers of mass (or means) with respect to a prescribed density. A distance function in both the spatial and vector spaces is introduced to measure the similarity of the spatially distributed vector fields. Based on such a distance, vector fields are naturally clustered and their simplified representations are obtained. Our method combines simple geometric intuitions with the rigorously established optimality properties of the CVTs. It is simple to describe, easy to understand and implement. Numerical examples are also provided to illustrate the effectiveness and competitiveness of the CVT-based vector simplification and visualization methodology.

The interactions between scalar and vector properties is vital in oceanic phenomena and energy/material circulation processes. This study integrates scalar and vector fields in ocean visualization to clarify their interplay and proposes a standardized approach for designing transfer functions that simplify ocean feature extraction. An interactive collaborative transfer function operational mode was introduced to visualise the scalar and vector fields, effectively capturing the dynamic characteristics of oceanic phenomena under mutual influence. Using the high-resolution GLORYS12V1 marine dataset, we visualized thermohaline fields and currents. Results show that the collaborative visualisation method and standardized transfer function effectively extract morphological outlines of ocean fronts and mesoscale eddies. Furthermore, the real-time collaborative transfer function reveals high spatial and directional consistency between ocean fronts (extracted via a 0.05 °C/km temperature gradient threshold) and Kuroshio axes (identified using a 0.1 m/s flow velocity threshold), which highlights the barrier effect of ocean fronts on meridional heat exchange. Additionally, we revealed the dynamic entrainment and transport capability of energy and material in mesoscale eddies. The above results demonstrate that the proposed synergistic transfer function in this study can effectively uncover the key role of scalar-vector field interactions in the evolution of oceanic phenomena.

The idea of independently moving, interacting graphical objects is introduced as a method for the visualization of continuous fields. Bird-oid objects or boids are discussed. These boids derive from: (1) icons which are geometric objects whose shape and appearance are related to the field variables, (2) three-dimensional cursors by which a user interactively picks a point in space, (3) particle traces, which are numerically integrated trajectories in space, (4) moving frames of vectors along space curves, and (5) actors, which are programming objects that can create and destroy instances of themselves, act according to internal logic, and communicate with each other and with a user. A software prototype in the C++ language has been developed which demonstrates some of the capabilities of these objects for the visualization of scalar, vector, and tensor fields defined over finite elements or finite volumes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We present the design of a scalable parallel pathline construction method for visualizing large time-varying 3D vector fields. A 4D (i.e., time and the 3D spatial domain) representation of the vector field is introduced to make a time-accurate depiction of the flow field. This representation also allows us to obtain pathlines through streamline tracing in the 4D space. Furthermore, a hierarchical representation of the 4D vector field, constructed by clustering the 4D field, makes possible interactive visualization of the flow field at different levels of abstraction. Based on this hierarchical representation, a data partitioning scheme is designed to achieve high parallel efficiency. We demonstrate the performance of parallel pathline visualization using data sets obtained from terascale flow simulations. This new capability will enable scientists to study their time-varying vector fields at the resolution and interactivity previously unavailable to them.

One major discipline of Scientific Visualization is the display of vector field data. We address the problem of visualizing vector fields containing critical points of higher order Poincare-Index. We use polynomial pieces so that the degree is high enough to get the right Poincare-Index at the location of the critical points. This is an essential improvement because nearly all visualization of vector fields up to now used piecewise linear interpolation and therefore can not show this kind of critical points. A description of vector fields in Clifford algebra opens the way to a better understanding of some plane polynomial vector fields. This is used to choose the polynomial approximations needed for the visualization.

We discuss new models of an “affine” theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein’s proposal to specify the space-time geometry by the use of the Hamilton principle. More specifically, the connection coefficients are determined using a “geometric” Lagrangian that is an arbitrary function of the generalized (nonsymmetric) Ricci curvature tensor (and, possibly, of other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein gravity with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) vector field (vecton), and massive (or tachyonic) scalar fields. These fields couple only to gravity and can generate dark matter and/or inflation. The new field masses (real or imaginary) have a geometric origin and must appear in any concrete model. The concrete choice of the geometric Lagrangian determines further details of the theory, for example, the nature of the vector and scalar fields that can describe massive particles, tachyons, or even “phantoms.” In “natural” geometric theories, which are discussed here, dark energy must also arise. We mainly focus on intricate relations between geometry and dynamics while only very briefly considering approximate cosmological models inspired by the geometric approach.

A classical relativistic approach based in the Vlasov equation is applied to the study of infinite nuclear matter. Hadronic matter couples to massive scalar and vector fields. The small-amplitude oscillations around a stationary state are studied. Orthogonality and completeness relations and the energy-weighted sum rule are obtained for the longitudinal modes. The appearance of the zero-sound mode and the distribution of the strength among the different modes are discussed. It is seen that the scalar field only couples to the low-energy excitations at high densities and for wavelengths not too long. The vector field couples strongly to the low-energy excitations. The long-wavelength limit and high-density limit are studied.

The extended scalar-tensor and vector-tensor theories admit black hole solutions with the nontrivial profiles of the scalar and vector fields, respectively. The disformal transformation maps a solution in a class of the scalar-tensor or vector-tensor theories to that in another class, and hence it can be a useful tool to construct a new nontrivial solution from the known one. First, we investigate how the stationary and axisymmetric solutions in the vector-tensor theories without and with the $U(1)$ gauge symmetry are disformally transformed. We start from a stationary and axisymmetric solution satisfying the circularity conditions, and show that in both the cases the metric of the disformed solution in general does not satisfy the circularity conditions. Using the fact that a solution in a class of the vector-tensor theories with the vanishing field strength is mapped to that in a class of the shift-symmetric scalar-tensor theories, we derive the disformed stationary and axisymmetric solutions in a class of these theories, and show that the metric of the disformed solutions does not satisfy the circularity conditions if the scalar field depends on the time or azimuthal coordinate. We also confirm that in the scalar-tensor theories without the shift symmetry, the disformed stationary and axisymmetric solutions satisfy the circularity conditions. Second, we investigate the disformal transformations of the stationary and axisymmetric black hole solutions in the generalized Proca theory with the nonminimal coupling to the Einstein tensor, the shift-symmetric scalar-tensor theory with the nonminimal derivative coupling to the Einstein tensor, the Einstein-Maxwell theory, and the Einstein-conformally coupled scalar field theory. We show that the disformal transformations modify the causal properties of the spacetime.