Two-dimensional vector field topology and scalar fields in viscous flows: Reconstruction methods

This article is concerned with the problem of reconstructing the curl and divergence of two-dimensional (2D) vector fields from ultrasound time-of-flight measurements taking the refraction of the emitted ultrasound rays into account. Usually, in 2D vector field tomography the refraction of the ultrasound signal is neglected; i.e., one assumes that the signal propagates along straight lines. We address refractive effects, assuming that the ultrasound signal propagates along the geodesics of a Riemannian metric that is associated with the refractive index which is due to Fermat's principle. The investigated vector field, however, is still defined on a domain in $\mathbb{R}^2$ that is equipped with the Euclidean metric. We show a connection between the ray transforms and the Radon transform along geodesic curves which generalizes a well-known result from the Euclidean case. Relying on this relation, we develop an asymptotic reconstruction formula for computing the curl and divergence of the vector field using Fourier integral operators. This article concludes with detailed numerical tests proving on the one hand a good performance of our method and showing on the other hand an improvement compared with computations that assume a propagation along straight lines.

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

In this letter, we have considered the universe is filled with the mixture of tachyonic field and scalar or phantom field. If the tachyonic field interacts with scalar or phantom field, the interaction term decays with time and the energy for scalar field is transferred to tachyonic field or the energy for phantom field is transferred to tachyonic field. The tachyonic field and scalar field potentials always decrease, but phantom field potential always increases.

We describe a correspondence between domain wall solutions of Einstein gravity with a single scalar field and self-interaction potential. The correspondence we call conformal scale factor inversion (CSFI) is a map comprising the inversion of the scale factor in conformal coordinates, and a transformation of the field and its potential which preserves the form of the Einstein equations for static and isotropic domain walls. By construction, CSFI maps the asymptotic anti--de Sitter boundary to the vicinity of a naked singularity in a theory with a special Liouville (exponential) scalar field potential; it is also a map in the parameter space of exponential potentials. The correspondence can be extended to linear fluctuations, being akin to an S-duality, and can be interpreted in terms of ``SUSY quantum mechanics'' for the fluctuation modes. The holographic implementation of CSFI relates the UV and IR regimes of a pair of holographic renormalization group flows; in particular, it is a symmetry of the GPPZ flow.

This paper describes an approach for interactive visualization of mixed scalar and vector fields, in which vector icons are generated from pre-voxelized icon templates and volume-rendered together with the volumetric scalar data. This approach displays simultaneously the global structure of the scalar field and the detailed features of the vector field. Interactive visualization is achieved with incremental image update, by re-rendering only a small portion of the image wherever and whenever a change occurs. This technique supports a set of interactive visualization tools, including change of vector field visualization parameters, real-time animation of vector icons advected within the scalar field, a zooming lens, and a local probe. Subject Terms: data visualisation; interactive systems; graphical user interfaces; rendering (computer graphics); interactive visualization; scalar fields; vector fields; vector icons; pre-voxelized icon templates; volume-rendered; volumetric scalar data; incremental image update; real-time animation; zooming lens; local probe

In flow visualization, the streamline has always been used to display the vector fields, but only using streamlines cannot reveal the nature of flowing intuitively. The dynamic glyphs technology is an important method to show flowing 3D vector fields. However, existing methods of dynamic glyphs are hard to convey the global patterns and local entities at the same time. In this paper, we present a practical method for dynamic visualization of 3D vector fields which combine dynamic glyphs and streamlines, which are rendered respectively according to the information of 3D vector fields. Our method integrates virtues of these two usual techniques, and provides high performance visualization of flowing 3D vector fields directly. Experiments with practical datasets certify that our method could not only depict the local and the global patterns, but more importantly reveal the dynamic nature of flowing 3D vector data.

Recent advances in vector field topologymake it possible to compute its multi-scale graph representations for autonomous 2D vector fields in a robust and efficient manner. One of these representations is a Morse Connection Graph (MCG), a directed graph whose nodes correspond to Morse sets, generalizing stationary points and periodic trajectories, and arcs - to trajectories connecting them. While being useful for simple vector fields, the MCG can be hard to comprehend for topologically rich vector fields, containing a large number of features. This paper describes a visual representation of the MCG, inspired by previous work on graph visualization. Our approach aims to preserve the spatial relationships between the MCG arcs and nodes and highlight the coherent behavior of connecting trajectories. Using simulations of ocean flow, we show that it can provide useful information on the flow structure. This paper focuses specifically on MCGs computed for piecewise constant (PC) vector fields. In particular, we describe extensions of the PC framework that make it more flexible and better suited for analysis of data on complex shaped domains with a boundary. We also describe a topology simplification scheme that makes our MCG visualizations less ambiguous. Despite the focus on the PC framework, our approach could also be applied to graph representations or topological skeletons computed using different methods.

Mathematical modeling of gravitating configurations of physical fields is one of the priority directions of the modern theory of gravity. Most of the exact solutions constructed within the framework of the general relativity are static or stationary configurations. This is due to the objective complexity of solving the Einstein equations under the assumption of nonstationarity. We present an approach to constructing nonstationary configurations of a spherically symmetric nonlinear real scalar field and the electromagnetic field, which are assumed both to be minimally coupled to gravity. It is based on the isolation of one invariant equation written in terms of the characteristic function and scalar field potential. Using the proposed method, an exact nonstationary solution with a nontrivial topology of space-time will be constructed.

Numerical study on the model of HTS bearing by coupling the magnetic scalar potential and magnetic field intensity

Detecting singular patterns in 2D vector fields using weighted Laurent polynomial

While 2D and 3D vector fields are ubiquitous in computational sciences, their use in graphics is often limited to regular grids, where computations are easily handled through finite-difference methods. In this paper, we propose a set of simple and accurate tools for the analysis of 3D discrete vector fields on arbitrary tetrahedral grids. We introduce a variational, multiscale decomposition of vector fields into three intuitive components: a divergence-free part, a curl-free part, and a harmonic part. We show how our discrete approach matches its well-known smooth analog, called the Helmotz-Hodge decomposition, and that the resulting computational tools have very intuitive geometric interpretation. We demonstrate the versatility of these tools in a series of applications, ranging from data visualization to fluid and deformable object simulation.

In this paper we propose a control law aimed at tracing level curves (isolines) in a scalar potential field. An exploration agent governed by such a law can map simply connected regions in space where the potential field exceeds a predefined threshold. The distinguishing feature of our control is that it does not rely on higher order characteristics of the field such as the gradient at a point or the curvature of the isolines, in contrast with these parameters appearing as inputs in other proposed control laws [1], [2]. Furthermore, we establish relationships between the performance of our control law and the geometry of the isoline and show results from implementing the algorithm in both a simulated environment and a testbed.

We investigate self-similar scalar field solutions to the Einstein equations in whole cylinder symmetry. Imposing self-similarity on the spacetime gives rise to a set of single variable functions describing the metric. Furthermore, it is shown that the scalar field is dependent on a single unknown function of the same variable and that the scalar field potential has exponential form. The Einstein equations then take the form of a set of ODEs. Self-similarity also gives rise to a singularity at the scaling origin. We extend the work of Condron and Nolan (2014 Class. Quantum Grav. 31 015015), which determined the global structure of all solutions with a regular axis in the causal past of the singularity. We identified a class of solutions that evolves through the past null cone of the singularity. We give the global structure of these solutions and show that the singularity is censored in all cases.

A novel approach is introduced to define a quantitative measure of closeness between vector fields. The usefulness of this measurement can be seen when comparing computational and experimental flow fields under the same conditions. Furthermore, its applicability can be extended to more cumbersome tasks, such as navigating through a large database, searching for similar topologies. This new measure relies on the use of critical points, which are a key feature in vector field topology. In order to characterize critical points, /spl alpha/ and /spl beta/ parameters are introduced. They are used to form a closed set of eight unique patterns for simple critical points. These patterns are also basic building blocks for higher-order nonlinear vector fields. In order to study and compare a given set of vector fields, a measure of distance between different patterns of critical points is introduced. The basic patterns of critical points are mapped onto a unit circle in /spl alpha/-/spl beta/ space. The concept of the "Earth mover's distance" is used to compute the closeness between various pairs of vector fields, and a nearest-neighbor query is thus produced to illustrate the relationship between the given set of vector fields. This approach quantitatively measures the similarity and dissimilarity between vector fields. It is ideal for data compression of a large flow field, since only the number and types of critical points along with their corresponding /spl alpha/ and /spl beta/ parameters are necessary to reconstruct the whole field. It can also be used to better quantify the changes in time-varying data sets.

This paper reports on a 3D temporal numerical simulation of the interaction between a jet flow and a trailing vortex. The simulation was performed from an experimental database of the vortex wake generated by a rectangular wing NACA0012 equipped with two jets beneath the airfoil. In this work, we focus on the 3D vorticity and scalar (initially included in the jet) fields in the vortex wake. The turbulence induced by the jet flow held interacts with the vortex, resulting in the large-scale structure generation outside the vortex core whereas the region very close to the core keeps its laminar state. The scalar field is subject to distortion and stretching inside the azimuthal vorticity tubes. The scalar concentration provides a reliable signature of where the helical instability occurs outside the vortex core. Later, the counter rotating vortices undergo a connection and consequently lose coherence in a very short time. The resulting scalar field concentrates then in small-scale eddies.