Three-dimensional Vector Fields Research Articles

Entropy for Three-dimensional Vector Fields

Entropy for Three-dimensional Vector Fields

Invariant manifolds in a reversible Hamiltonian system: The tentacle-like geometry

We study a one-parameter family of time-reversible Hamiltonian vector fields in R4, which has received great attention in the literature. On the one hand, it is due to the role it plays in the context of certain applications in the field of Physics or Engineering and, on the other hand, we especially highlight its relevance within the framework of generic unfoldings of the four-dimensional nilpotent singularity of codimension four.The system exhibits a bifocal equilibrium point for a range of parameter values. The associated two-dimensional invariant manifolds, stable and unstable, fold into the phase space in such a way that they produce intricate patterns. This entangled geometry has previously been called tentacular geometry.We consider a three-dimensional level set containing the bifocal equilibrium point to gain insight into the folding behavior of these invariant manifolds. Our method consists of describing the traces left by invariant manifolds when crossing an invariant cross section by the reversibility map. With this new approach, we provide a better understanding of how tentacular geometry evolves with respect to the parameter.Our techniques enables us to link the tentacular geometry on the cross section with the study of cocooning cascades of homoclinic tangencies. Indeed, we present a general theory to extend to R4 the phenomena related to cocoon bifurcation that classically develop within families of three-dimensional reversible vector fields. On the basis of these results, we conjecture the existence of heteroclinic cycles consisting of two orbits connecting the bifocus with a saddle node periodic orbit.

Free Surface Energy and Hansen Solubility Parameter Vector Field. Interface Thickness

In this paper, a three-dimensional vector field model is proposed, whose dimensions are the Hansen Solubility Parameters: dispersion parameter (δD), polarity parameter (δP), and hydrogen bonding parameter (δH). The vector space that defines the field has the peculiarity of having a dispersion vector with a magnitude of 2 as its base vector, while the polarity and hydrogen bonding vectors have a magnitude of 1. A substance is characterised as a position vector, and the interaction between two substances is determined by calculating the vector difference of both, known as the interaction vector. The interaction among substances may involve solubility, swelling, cracking, surface tension, interface tension, and any physical phenomena where the intermolecular energies of dispersion, polarity or hydrogen bonding come into play. This paper studies free surface energy (surface and interfacial tension). It has been found that free surface energy is directly proportional to the square of the magnitude of the interaction vector. The proportionality constant, τ, is expressed in length units, has a value of 0.025 nm, and does not depend on the chemical nature of the substance or state of matter (solid, liquid or gas). The constant value τ appears universal and aligns with the thickness of interfaces, thereby supporting Guggenheim’s hypothesis. This hypothesis asserts that interfaces possess actual thickness and are not merely mathematical surfaces, as originally postulated by Gibbs. Moreover, it also has been found that the interface thickness, τ, is approximately equal to half of the Bohr radius, a0, which is defined by universal constants. Because the solubility parameters of thousands of substances are known and can be easily determined from their molecular structure, a good approximation of the surface and interfacial tension of any given substance can now be calculated. It has also been found that the contact angles of sessile droplets in three-phased systems can be calculated from the interaction vectors of the implicated substances.

Three dimensional analysis of the exhalation flow in the proximity of the mouth

The human exhalation flow is characterized in this work from the three-dimensional velocimetry results obtained by using the stereo particle image velocimetry (SPIV) measurement technique on the flow emitted from a realistic airway model. For this purpose, the transient exhalation flow through the mouth of a person performing two different breaths corresponding to two metabolic rates, standing relaxed (SR) and walking active (WA), is emulated and studied. To reproduce the flow realistically, a detailed three-dimensional model obtained from computed tomography measurements on real subjects is used. To cope with the variability of the experimental data, a subsequent analysis of the results is performed using the TR-PIV (time resolved particle image velocimetry) technique. Exhalation produces a transient jet that becomes a puff when flow emission ends. Three-dimensional vector fields of the jet velocity are obtained in five equally spaced transverse planes up to a distance of ▪ from the mouth at equally spaced time instants ▪ which will be referred to as phases (φ), from the beginning to the end of exhalation. The time evolution during exhalation of the jet area of influence, the velocity field and the jet air entrainment have been characterized for each of the jet cross sections. The importance of the use of realistic airway models for the study of this type of flow and the influence of the metabolic rate on its development are also analyzed. The results obtained contribute to the characterization of the human exhalation as a pathway of the transmission of pathogens such as SARS-CoV-2 virus.

Inversion of generalized Radon transforms acting on 3D vector and symmetric tensor fields

Currently, theory of the ray transforms of vector and tensor fields is well developed, but the generalized Radon transforms of such fields have not been fully studied. We consider the normal, longitudinal and mixed Radon transforms (with integration over planes) acting on three-dimensional vector and symmetric tensor fields. We prove that these operators are continuous. In case when values of all generalized Radon transforms are known, inversion formulas are derived for componentwise reconstruction of vector and symmetric m-tensor fields, . Novel detailed decompositions of 3D vector and symmetric 2-tensor fields as a sum of pairwise orthogonal terms are obtained. For construction of each term in the sum only one function is required. With usage of these decompositions we have described the kernels and images of the generalized Radon transforms. In addition, we have obtained inversion formulas for each of the generalized Radon transforms acting on 3D vector and symmetric 2-tensor fields and have formulated theorems similar to the projection theorem for the Radon transform. For the cases similar statements are formulated as hypotheses. In addition, we consider the weighted longitudinal Radon transforms of 3D vector fields. Formulas are obtained for reconstructing the potential part of a 3D vector field from the known values of the longitudinal Radon transforms and one weighted Radon transform. Finally, we discuss the problem of vector fields reconstruction in , .

Abstract 14470: Right Ventricular Flow Visualization and Characterization via High Speed Volumetric Echo-PIV

Introduction: Although 4D echocardiography is used for many clinical applications, the evolving vortex flow in the right ventricle (RV) is relatively unknown and cannot be characterized with conventional echocardiography. An obstacle is that acquiring high frame rate (>100fps) data in presence of contrast is challenging and has not yet reported. Hypothesis: High frame-rate 4D echocardiographic data in presence of contrast can elucidate RV vortex flow via volumetric Echo-PIV (V-Echo-PIV). Methods: We devised a new transthoracic echocardiography (TTE) protocol using a Philips EPIQ CV9 ultrasound system with a 3D matrix probe (X5-1c) to acquire high frame-rate 4D data for processing with V-Echo-PIV. Apical RV focused view was used for full-volume acquisition by limiting the sector width and imaging depth solely to the entire RV. Visualization of the RV flow and wall dynamics was achieved by IV administration of Optison™ and Normal Saline. VolDICOMs were processed with V-Echo-PIV to obtain RV blood flow velocity field during the full cardiac cycle. Results: We improve the frame rates' scan lines to +30Hz. A 100Hz is achieved by stitching 4 cycles. We show that 4D RV flow can be quantified and characterized at high temporal resolution using V-Echo-PIV. Fig. 1A shows cine blood velocity in a healthy subject with a 51bpm heart rate. For which, V-Echo-PIV acquired 117 three-dimensional vector fields in one cardiac cycle, given the setting of 100Hz acquisition rate. The figure features flow patterns from systole to diastole, from left to right. Fig. 1B depicts the iso-surface of transient tricuspid vortex during diastolic RV filling. Conclusions: Enabled by high speed acquition in presence of contrast, V-Echo-PIV can be used as a viable alternative to 4D Flow MRI to visualize and evaluate real-time intra-cardiac blood flow, as it requires only a very short acquisition time and can be used wherever an echocardiography machine with matrix probes is available.

Three-dimensional acoustic vector field model using Gaussian beam tracing.

A three-dimensional acoustic vector field model was developed using Gaussian beam tracing. The field at any location is computed by coherently summing all contributing beams, where the beam equation of particle velocity depends only on beam width and eikonal. A deep-sea long-range sound propagation experiment in the South China Sea was analyzed using vector hydrophone measurements of particle velocity and sound pressure. The results were compared with the model predictions, indicating that beam tracing is effective in predicting sound pressure and acoustic vector fields in deep water.

Nested invariant tori foliating a vector field and its curl: Toward MHD equilibria and steady Euler flows in toroidal domains without continuous Euclidean isometries

This paper studies the problem of finding a three-dimensional solenoidal vector field such that both the vector field and its curl are tangential to a given family of toroidal surfaces. We show that this question can be translated into the problem of determining a periodic solution with periodic derivatives of a two-dimensional linear elliptic second-order partial differential equation on each toroidal surface and prove the existence of smooth solutions. Examples of smooth solutions foliated by toroidal surfaces that are not invariant under continuous Euclidean isometries are also constructed explicitly, and they are identified as equilibria of anisotropic magnetohydrodynamics. The problem examined here represents a weaker version of a fundamental mathematical problem that arises in the context of magnetohydrodynamics and fluid mechanics concerning the existence of regular equilibrium magnetic fields and steady Euler flows in bounded domains without continuous Euclidean isometries. The existence of such configurations represents a key theoretical issue for the design of the confining magnetic field in nuclear fusion reactors known as stellarators.

An effective model for boundary vortices in thin-film micromagnetics

Ferromagnetic materials are governed by a variational principle which is nonlocal, nonconvex and multiscale. The main object is given by a unit-length three-dimensional vector field, the magnetization, that corresponds to the stable states of the micromagnetic energy. Our aim is to analyze a thin film regime that captures the asymptotic behavior of boundary vortices generated by the magnetization and their interaction energy. This study is based on the notion of “global Jacobian” detecting the topological defects that a priori could be located in the interior and at the boundary of the film. A major difficulty consists in estimating the nonlocal part of the micromagnetic energy in order to isolate the exact terms corresponding to the topological defects. We prove the concentration of the energy around boundary vortices via a [Formula: see text]-convergence expansion at the second order. The second-order term is the renormalized energy that represents the interaction between the boundary vortices and governs their optimal position. We compute the expression of the renormalized energy for which we prove the existence of minimizers having two boundary vortices of multiplicity [Formula: see text]. Compactness results are also shown for the magnetization and the corresponding global Jacobian.

Invariant manifold growth formula in cylindrical coordinates and its application for magnetically confined fusion

For three-dimensional vector fields, the governing formula of invariant manifolds grown from a hyperbolic cycle is given in cylindrical coordinates. The initial growth directions depend on the Jacobians of Poincaré map on that cycle, for which an evolution formula is deduced to reveal the relationship among Jacobians of different Poincaré sections. The evolution formula also applies to cycles in arbitrary finite n-dimensional autonomous continuous-time dynamical systems. Non-Möbiusian/Möbiusian saddle cycles and a dummy X-cycle are constructed analytically as demonstration. A real-world numeric example of analyzing a magnetic field timeslice on EAST is presented.

Physical measures for ash attractors

In this paper we will prove that any asymptotically sectional-hyperbolic attractor associated to a three-dimensional C2 vector field X supports an unique physical measure. Asymptotic sectional-hyperbolicity means that any point outside of the stable manifolds of the singularities has arbitrarily large hyperbolic times. Besides, we will describe its main properties and we will deduce some dynamical consequences of the existence of this measure.

Electric field-based quantization of the gauge invariant Proca theory

We consider the gauge invariant version of the Proca theory, where besides the real vector field there is also the real scalar field. We quantize the theory such that the commutator of the scalar field operator and the electric field operator is given by a predefined three-dimensional vector field, say $\cal E$ up to a global prefactor. This happens when the field operators of the gauge invariant Proca theory satisfy the proper gauge constraint. In particular, we show that $\cal E$ given by the classical Coulomb field leads to the Coulomb gauge constraint making the vector field operator divergenceless. We also show that physically unreadable gauge constraints can have a strikingly simple $\cal E$-representation in our formalism. This leads to the discussion of Debye, Yukawa, etc. gauges. In general terms, we explore the mapping between classical vector fields and gauge constraints imposed on the operators of the studied theory.

Anomalous Energy Flux in Critical L^p-Based Spaces

We construct a three-dimensional vector field that exhibits positive energy flux at every Littlewood–Paley shell and has the best possible regularity in L^p-based spaces, p le 3; in particular, it belongs to H^{(frac{5}{6})^-}.

On One Normal Boundary Value Problem of the Field Theory

One normal problem of the field theory on a plane for a system of Poisson equations is studied. This problem is one of implementations of the general approach to formulation of boundary value problems of the field theory, which is based on studying the kernels of trace operators and kernels of trace functionals, which are taken as the main space of desired solutions. A peculiarity of this problem is that one of the boundary conditions imposed on the desired solution is nonlocal in nature. The considered boundary condition nonlocality is due to the fact that the base space in which the solution of the problem is sought is the kernel of the regular trace functional in the Sobolev space. The need to supplement the problem on the one-dimensional cokernel of the above-mentioned functional, which arises in this case, determines the second boundary condition, which is local (pointwise) in nature. That is why, in the framework of the duality of the Sobolev space and its conjugate space, the solution to the problem is the pair (u, с), where с is a constant that compensates for the one-dimensional “defect” of the problem. It should be noted that the theory of boundary value problems for vector fields on a plane differs essentially from the theory of boundary value problems for three-dimensional vector fields. One of the reasons for the difficulties in formulating three-dimensional boundary value problems is that a number of important surfaces cannot be made parallel to each other, which rules out the possibility to consider “tangential” boundary value problems, that is, problems with boundary tangential fields. In this respect, the theory of boundary value problems of plane vector fields differs from the three-dimensional theory for the better. On the other hand, the fact that a formula that would interrelating the Laplace operator with first-order field theory operations is not available on a plane is an obstacle to identifying new non-standard boundary value problems for plane fields. Revealing of this interrelation is a separate issue, the solution of which leads to the concept of the trace of first-order field theory operators as a singular functional above the space of traces of functions from the first-order Sobolev space. Thus, boundary value problems for plane fields are a separate section of the general theory of field boundary value problems. One of the non-standard problems in the theory of plane vector fields, which contains non-local boundary conditions, is considered. A theorem for existence and uniqueness of a weak solution of the problem posed is formulated and proved. The proof of the theorem includes a few steps: using the Galerkin method, the existence of a solution to the equation is established; the passage to the limit is made, and the vector function u(x) is found; the number α is determined, and the uniqueness of the function u and the number α is established. The main result of the work is that the problem posed is correct in nature.

Анализ геоинформационных технологий в целях изучения визуализации и отображения вертикальных и горизонтальных деформаций поверхности Земли

Information on the movements and stress-strain state of the Earth`s surface and crust is necessary to predict catastrophic geodynamic processes, such as earthquakes, volcanic eruptions, landslides, glacier collapses, karst sinkholes and subsidence of soils in mining areas. The authors summarize the existing experience of applying geoinformation technologies for modeling Earth’s surface deformations; reveal the shortcomings of current software solutions and the approach to their joint use for analyzing these processes. The relevance of using spatial data infrastructure as a source of data on Earth`s surface deformations is noted. Solutions for improvement of Earth`s surface deformations imaging with the help of the three-dimensional vector fields visualization method, presented by the authors in previous surveys, as well as the results of visualization of the Earth`s surface displacements model, demonstrating applicability and operability of the method developed by the authors, are offered. The requirements and the general functional layout of the thematic geographic information system intended for geoinformation modeling of the Earth`s surface deformations and administrative tasks solution are formed.

Fractal analysis of degenerate spiral trajectories of a class of ordinary differential equations

In this paper we initiate the study of the Minkowski dimension, also called the box dimension, of degenerate spiral trajectories of a class of ordinary differential equations. A class of singularities of focus type with two zero eigenvalues (nilpotent or more degenerate) has been studied. We find the box dimension of a polynomial degenerate focus of type (n,n) by exploiting the well-known fractal results for α-power spirals. In the general (m,n) case, we formulate a conjecture about the box dimension of a degenerate focus using numerical experiments. Further, we reduce the fractal analysis of planar nilpotent contact points to the study of the box dimension of a slow-fast spiral generated by their “entry-exit” function. There exists a bijective correspondence between the box dimension of the slow-fast spirals and the codimension of contact points. We also construct a three-dimensional vector field that contains a degenerate spiral, called an elliptical power spiral, as a trajectory.

Small Amplitude Periodic Orbits in Three-Dimensional Quadratic Vector Fields with a Zero-Hopf Singularity

We consider some families of three-dimensional quadratic vector fields having a fixed zero-Hopf equilibrium. We are interested in the bifurcation of periodic ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document}-orbits from the singularity, that is, those small amplitude orbits that make a fixed arbitrary number ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} of revolutions about a rotation axis and then returns to the initial point closing the orbit. When the parameters of the family are restricted to certain explicitly computable open semi-algebraic sets Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}, we characterize those parameters that give rise to the appearance of local two-dimensional periodic invariant manifolds through the singularity. Also we use a Bautin-type analysis to study the maximum number of small-amplitude ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document}-limit cycles that can be made to bifurcate from the equilibrium when the parameters of the family are restricted to Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}. We obtain global upper bounds on the number of bifurcated ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document}-limit cycles.

Background Oriented Schlieren Technique With Reconstruction Of Three-Dimensional Vector Fields For Pressure Measurement Of Laser Induced Underwater Shock Waves

We propose a novel reconstruction of the three-dimensional vector field (VR) of an axisymmetric target to overcome the problems that existing background-oriented schlieren (BOS) techniques based on reconstruction of three-dimensional scalar field (SR-BOS). This approach is based on an approximate relation between the projection of the axisymmetric vector field and the reconstructed vector field. The remarkable feature of the proposed technique is the reconstruction of an axisymmetric vector field with nonzero divergence, such as the field of a laser-induced underwater shock wave. Reconstruction of 3D vector fields, such as conventional vector tomography, is usually used in the field of electromagnetics with vector fields of zero divergence, and its application to fluid phenomena has been di fficult. We have successfully applied this technique to BOS (VR-BOS) and measured the pressure fields of underwater shock waves. Using the experimental data and previous research data, the superiority of VR-BOS was clarified by comparing the measurement results of SR-BOS and VR-BOS, for example, measurement accuracy, in better convergence, less dependence on the spatial resolution of the acquired images, and lower computational cost. In particular, the fact that the measurement accuracy of VR-BOS is almost independent of resolution is expected to expand the maximum pressure values that can be measured by the BOS technique. We also used synthetic data to investigate the effect of noise on VR-BOS. If the synthetic displacement contains noise, the noise is amplified in the vicinity of the axisymmetry axis in the reconstruction distribution calculated by VT. If the distance from the axisymmetry axis is more than 1/4 of the measurement range, the measurement accuracy of the reconstructed distribution can be sufficiently guaranteed. This proposed technique can be applied fluid dynamical fields, as well as other axisymmetric targets with vector reconstruction fields in other areas, such as electromagnetics and material mechanism.

Rod-packing arrangements of invariant tori in solenoidal vector fields with cubic symmetries

The arrangements of invariant tori that resemble rod packings with cubic symmetries are considered in three-dimensional solenoidal vector fields. To find them systematically, vector fields whose components are represented in the form of multiple Fourier series with finite terms are classified using magnetic groups. The maximal magnetic group compatible with each arrangement is specified on the assumption that the cores of the nested invariant tori are straight and located on the lines corresponding to the central axes of the rods packed. Desired rod-packing arrangements are demonstrated by selecting vector fields whose magnetic groups are the maximal ones and by drawing their integral curves that twine around invariant tori. In the demonstration of chiral arrangements, Beltrami flows (or force-free fields in plasma physics), which have the strongest chirality of all solenoidal vector fields satisfying the same vector Helmholtz equation, are used. As by-products, several chain-like arrangements of closed invariant tori were found. One of the chains consists of knotted invariant tori. In all vector fields (chiral or achiral) selected for the demonstration, the volume percentages of ordered regions formed by invariant tori in a unit cell were roughly measured with the aid of a supervised machine learning technique.

Resource-aware synchronized path following of multiple unmanned surface vehicles with experiments: A cooperative vector field approach

This paper presents a cooperative vector field approach for synchronized path following of multiple autonomous surface vehicles (USVs) with limited communication resources. First, a cooperative vector field is proposed by introducing a path coordination design into a three-dimensional vector field. Based on the cooperative vector field, concise guidance laws of surge speed and angular rate are derived for synchronized path following. In order to reduce communication times among USVs, an event-triggered cooperative vector field is next proposed. Specifically, a predictor is devised to predict the current path variables by using the last transmitting ones, and an event-triggered mechanism is developed based on the predicted path variables. Then, a self-triggered cooperative vector field is developed to avoid continuous listening. A self-triggered mechanism is designed such that each USV predicts its next triggering time and broadcasts it to its neighbors at the current triggering time. The stability of the three cooperative vector fields is analyzed, and Zeno behavior is excluded. Finally, both simulation and experiment results are provided to illustrate the efficacy of the proposed continuous, event-triggered, and self-triggered cooperative vector fields for resource-aware synchronized path following.