Original Vector Field Research Articles

An airflow velocity field reconstruction method with sparse or incomplete data using physics-informed neural network

Velocity field is critical for creating a healthy and comfortable indoor environment but it is difficult to be measured experimentally. Due to the limitations of measurement instruments and site conditions, the captured data are usually sparse, incomplete, and low-dimensional, which limits further research on indoor velocity field. This study proposes a practical reconstruction and super-resolution method based on physics-informed learning that seamlessly integrates imperfect measured data and physical laws. The ability to reconstruct the airflow velocity field with imperfect data as the training dataset was validated using a three-dimensional and three-component airflow field generated with an axial fan. This method successfully recovered the airflow field from sparse or insufficient data (even when the amount of data was only 5 % of the original vector fields) and incomplete data. This indicates that the proposed method can reconstruct and super-resolve the airflow velocity vector fields from sparse, incomplete, and low-dimensional data.

Computation of Normal Form and Unfolding of Codimension-3 Zero-Hopf–Hopf Bifurcation

The computation of the normal form as well as its unfolding is a key step to understand the topological structure of a bifurcation. Though a lot of results have been obtained, it still remains unsolved for higher co-dimensional bifurcations. The main purpose of this paper is devoted to the computation of a codimension-3 zero-Hopf–Hopf bifurcation, at which a zero as well as two pairs of pure imaginary eigenvalues can be found from the matrix evaluated at the equilibrium point. Different distributions of eigenvalues are considered, which may behave in a non-semisimple form for 1:1 internal resonance. Based on the combination of center manifold and normal form theory, all the coefficients of normal forms and nonlinear transformations are derived explicitly in terms of parameters of the original vector field, which are obtained via a recursive procedure. Accordingly, a user friendly computer program using a symbolic computation language Maple is developed to compute the coefficients up to an arbitrary order for a specific vector field with zero-Hopf–Hopf bifurcation. Furthermore, universal unfolding parameters are derived in terms of the perturbation of physical parameters, which can be employed to investigate the local behaviors in the neighborhood of the bifurcation point. Here, we emphasize that though different norm forms based on different choices may exist, their topological structures are the same, corresponding to qualitatively equivalent dynamics.

Normal Form and Unfolding of Vector Field with Codimension-3 Triple Hopf Bifurcation

The universal unfolding of a normal form can be employed to reveal the general behaviors of a specific local bifurcation, while the computation of the normal form for high codimensional bifurcation still remains unsolved. This paper focuses on a vector field with codimension-3 triple Hopf bifurcation. Besides 1:1 internal resonance for two frequencies in semi-simple form, two cases are considered, corresponding to internal resonance and noninternal resonance between the first two frequencies and the third frequency, respectively. Based on a combination of center manifold and normal theory, all the coefficients in the normal form and the nonlinear transformation are derived explicitly in terms of the coefficients of the original vector field. Upon the recursive procedure established, a user friendly computer program can be easily developed using a symbolic computation language Maple to compute the coefficients up to an arbitrary order for a specific vector field with triple Hopf bifurcation. Furthermore, universal unfolding of the normal form is obtained, which can be used to display the topological structure in the neighborhood of bifurcation point. It is pointed out that different choices of the remaining terms in the nonlinear transformation may lead to different expressions of the normal form and the unfolding, which are qualitatively equivalent to each other.

Singly TASE Operators for the Numerical Solution of Stiff Differential Equations by Explicit Runge–Kutta Schemes

In this paper new explicit integrators for numerical solution of stiff evolution equations are proposed. As shown by Bassenne, Fu and Mani in (J Comput Phys 424:109847, 2021), the action on the original vector field of the stiff equations of an appropriate time-accurate and highly-stable explicit (TASE) linear operator, allows us to use explicit Runge–Kutta (RK) schemes with these modified equations so that the resulting algorithm becomes stable for the original stiff equations. Here a new family of TASE operators is considered. The new operators, called Singly TASE, have the advantage over the TASE operators of Bassenne et al. that the action on the vector field depends on the powers of the inverse of only one matrix, which can be computationally more simple, without loosing stability properties. A complete study of the linear stability properties of k–stage, kth–order explicit RK schemes under the action of Singly TASE operators of the same order is carried out for k le 4. For orders two, three and four, particular schemes that are nearly strongly A–stable and therefore suitable for stiff problems are devised. Further, explicit RK schemes with orders three and four that can be implemented with only two storage locations under the action of Singly TASE operators of the same order are discussed. A particular implementation of the classical four–stage fourth–order RK scheme with two Singly TASE operators is presented. A set of numerical experiments has been conducted to demonstrate the performance of the new schemes by comparing with previous RKTASE and other established methods. The main conclusion is that the new integrators provide a very simple solver for stiff systems with good stability properties and avoids the difficulties of using implicit algorithms.

Well-posedness of the deterministic transport equation with singular velocity field perturbed along fractional Brownian paths

In this article we prove path-by-path uniqueness in the sense of Davie [25] and Shaposhnikov [46] for SDE's driven by a fractional Brownian motion with a sufficiently small Hurst parameter H∈(0,12), uniformly in the initial conditions, where the drift vector field is allowed to be merely bounded and measurable.Using this result, we construct weak unique regular solutions in Wlock,p([0,1]×Rd), p>d of the classical transport and continuity equations with singular velocity fields perturbed along fractional Brownian paths.The latter results provide a systematic way of producing examples of singular velocity fields, which cannot be treated by the regularity theory of DiPerna-Lions [28], Ambrosio [2] or Crippa-De Lellis [23].Our approach is based on a priori estimates at the level of flows generated by a sequence of mollified vector fields, converging to the original vector field, and which are uniform with respect to the mollification parameter. In addition, we use a compactness criterion based on Malliavin calculus from [24] as well as a supremum estimate in time of moments of the derivative of the flow of SDE solutions.

Modified vector field and nonlinear guidance law for low-cost UAV path following

Purpose The purpose of this paper is to present a high accuracy path following method for low-cost fixed-wing UAVs. Design/methodology/approach The original vector field (VF) algorithm is condensed. A spatial integration mechanism is added to the existing VF and nonlinear guidance law, aiming to decrease steady-state cross-track-error and cope with long-term disturbance. Findings Numerical simulations show the proposed method could diminish steady-state cross-track-error effectively. Test flights show the proposed method is applicable on low-cost fixed-wing UAVs. Practical implications The path following accuracy shown in simulations and test flights indicates the proposed method could be deployed in scenarios including inflight rendezvous, formation, trafficway take-off and landing. Originality/value This paper provides an improved high-accuracy path following method for low-cost fixed-wing UAVs.

Suboptimal control law for a near-space hypersonic vehicle based on Koopman operator and algebraic Riccati equation

This paper presents a novel suboptimal attitude tracking controller based on the algebraic Riccati equation for a near-space hypersonic vehicle (NSHV). Since the NSHV’s attitude dynamics is complexly nonlinear, it is hard to directly construct an appropriate algebraic Riccati equation. We design the construction based on the Chebyshev series and the Koopman operator theory, which includes three steps. First, the Chebyshev series are considered to transform the error dynamics of the NSHV’s attitude into a polynomial system. Second, the Koopman operator is used to obtain a series of high-dimensional linear dynamics to approximate each of the polynomial system’s vector fields. In this step, our contribution is to determine a well-posed linear dynamics with the minimal dimension to approximate the original nonlinear vector field, which helps to design the control law and analyze the control performance. Third, based on the high-dimensional dynamics, the NSHV’s attitude error dynamics is separated into the linear part and the nonlinear part, such that the algebraic Riccati equation can be constructed according to the linear part. Then, the suboptimal error feedback control law is derived from the algebraic Riccati equation. The closed-loop control system is proved to be locally exponentially stable. Finally, the numerical simulation demonstrates the effectiveness of the suboptimal control law.

Hierarchical Collaboration Approach for Tracking of Flying Target by UAVs

This paper investigates a hierarchical target tracking method based on the collaboration of unmanned aerial vehicles (UAVs) at different altitudes. To enhance the target tracking range, the high-altitude UAVs monitor the wide area, and transmit their surveillance information to low-altitude UAVs which directly detect and collect information about the target's movements. The contributions of this paper are threefold: First, to track the flying target in a dynamic environment, a modified Lyapunov guidance vector field (LGVF) method is used to plan velocities for UAVs, where a time-varying vertical component is incorporated into the traditional LGVF function to satisfy the constraints of cluster communication. Secondly, a three-dimensional local collision-free guidance vector field (TLCGVF) method is proposed for UAVs to plan collision-free paths on line. To simultaneously track the target and avoid obstacles, the vector field by LGVF is used as the original vector field of TLCGVF. Thirdly, the rolling optimization strategy is used to adjust the reactive parameters of TLCGVF to enhance the path quality. The simulation results confirm the feasibility of the above approach.

Flow Field Reduction Via Reconstructing Vector Data From 3-D Streamlines Using Deep Learning.

We present a new approach for streamline-based flow field representation and reduction. Our method can work in the in situ visualization setting by tracing streamlines from each time step of the simulation and storing compressed streamlines for post hoc analysis where users can afford longer reconstruction time for higher reconstruction quality using decompressed streamlines. At the heart of our approach is a deep learning method for vector field reconstruction that takes the streamlines traced from the original vector fields as input and applies a two-stage process to reconstruct high-quality vector fields. To demonstrate the effectiveness of our approach, we show qualitative and quantitative results with several data sets and compare our method against the de facto method of gradient vector flow in terms of speed and quality tradeoff.

Model order reduction for dynamical systems: A geometric approach

The aim of this paper is to ask the question as whether it is possible, for a given dynamical system defined by a vector field over a finite dimensional inner product space, to construct a reduced-order model over a finite dimensional manifold. In order to give a positive answer to this question, we prove that if the manifold under consideration is an immersed submanifold of the vector space, considered as ambient manifold, then it is possible to construct explicitly a reduced-order vector field over this submanifold. In particular, we found that the reduced-order vector field satisfies the variational principle of Dirac–Frenkel and that we can formulate the Proper Orthogonal Decomposition under this framework. Finally, we propose a local-point estimator of the time-dependent error between the original vector field and the reduced-order one.

Breakdown of a 2D Heteroclinic Connection in the Hopf-Zero Singularity (II): The Generic Case

In this paper we prove the breakdown of the two-dimensional stable and unstable manifolds associated to two saddle-focus points which appear in the unfoldings of the Hopf-zero singulariry. The method consists in obtaining an asymptotic formula for the difference between this manifolds which turns to be exponentially small respect to the unfolding parameter. The formula obtained is explicit but depends on the so-called Stokes constants, which arise in the study of original vector field and which corresponds to the so called inner equation in singular perturbation theory.

Crowd Behavior Analysis via Curl and Divergence of Motion Trajectories

In the field of crowd behavior analysis, existing methods mainly focus on using local representations inspired by models found in other disciplines (e.g., fluid dynamics and social dynamics) to describe motion patterns. However, less attention is paid to exploiting motion structures (e.g., visual information contained in trajectories) for behavior analysis. In this paper, we consider both local characteristics and global structures of a motion vector field, and propose the Curl and Divergence of motion Trajectories (CDT) descriptors to describe collective motion patterns. To this end, a trajectory-based motion coding algorithm is designed to extract the CDT descriptors. For each motion vector field we construct its conjugate field, in which each vector is perpendicular to the counterpart in the original vector field. The trajectories in the motion and corresponding conjugate fields indicate the tangential and radial motion structures, respectively. By integrating curl (and divergence, respectively) along the tangential paths (and the radial paths, respectively), the CDT descriptors are extracted. We show that the proposed motion descriptors are scale- and rotation-invariant for effective crowd behavior analysis. For concreteness, we apply the CDT descriptors to identify five typical crowd behaviors (lane, clockwise arch, counterclockwise arch, bottleneck and fountainhead) with a pipeline including motion decomposition. Extensive experimental results on two benchmark datasets demonstrate the effectiveness of the CDT descriptors for describing and classifying crowd behaviors.

Helmholtz–Hodge decomposition and the analysis of 2D vector field ensembles

The Helmholtz–Hodge decomposition is a very useful instrument for vector field interpretation because it traditionally distinguishes conservative (curl-free) components from mass-preserving (divergence-free) components. In this work, we propose to use the Helmholtz–Hodge decomposition as a basic tool for the analysis of an ensemble of 2D vector fields. Given a vector field ensemble E, our approach firstly generates the corresponding curl-free, divergence-free, and harmonic component ensembles by using the Natural Helmholtz–Hodge decomposition to each vector field in E. With these ensembles in hand, our proposal not only quantifies, via a statistical analysis, how much each component ensemble is point-wisely correlated to the original vector field ensemble, but also allows to investigate the uncertainty of curl-free, divergence-free, and harmonic components separately. We test the proposed method with a synthetic vector field ensemble and with a simulated vector field ensemble.

Submersions, Hamiltonian Systems, and Optimal Solutions to the Rolling Manifolds Problem

Given a submersion $\pi:Q \to M$ with an Ehresmann connection $\mathcal{H}$, we describe how to solve Hamiltonian systems on $M$ by lifting our problem to $Q$. Furthermore, we show that all solutions of these lifted Hamiltonian systems can be described using the original Hamiltonian vector field on $M$ along with a generalization of the magnetic force. This generalized force is described using the curvature of $\mathcal{H}$ along with a new form of parallel transport of covectors vanishing on $\mathcal{H}$. Using the Pontryagin maximum principle, we apply this theory to optimal control problems $M$ and $Q$ to get results on normal and abnormal extremals. We give a demonstration of our theory by considering the optimal control problem of one Riemannian manifold rolling on another without twisting or slipping along curves of minimal length.

Quantum field theory of interacting plasmon–photon–phonon system

This work is devoted to the construction of the quantum field theory of the interacting system of plasmons, photons and phonons on the basis of general fundamental principles of electrodynamics and quantum field theory of many-body systems. Since a plasmon is a quasiparticle appearing as a resonance in the collective oscillation of the interacting electron gas in solids, the starting point is the total action functional of the interacting system comprising electron gas, electromagnetic field and phonon fields. By means of the powerful functional integral technique, this original total action is transformed into that of the system of the quantum fields describing plasmons, transverse photons, acoustic as well as optic longitudinal and transverse phonons. The collective oscillations of the electron gas is characterized by a real scalar field φ(x) called the collective oscillation field. This field is split into the static background field φ0(x) and the fluctuation field ζ(x). The longitudinal phonon fields are also split into the background fields and dynamical fields while the transverse phonon fields themselves are dynamical fields without background fields. After the canonical quantization procedure, the background fields φ0(x), remain the classical fields, while the fluctuation fields ζ(x) and dynamical phonon fields become quantum fields. In quantum theory, a plasmon is the quantum of Hermitian scalar field σ(x) called the plasmon field, longitudinal phonons as complex spinless quasiparticles are the quanta of the effective longitudinal phonon Hermitian scalar fields while transverse phonons are the quanta of the original Hermitian transverse phonon vector fields By means of the functional integral technique the original action functional of the interacting system comprising electron gas, electromagnetic field and phonon fields is transformed into the total action functional of the resultant system comprising plasmon scalar quantum field σ(x), longitudinal phonon effective scalar quantum fields and transverse phonon vector quantum fields .

Membranes with a symmetry of cohomogeneity one

We study the dynamics of the Nambu-Goto membranes with cohomogeneity one symmetry, i.e., the membranes whose trajectories are foliated by homogeneous surfaces. It is shown that the equation of motion reduces to a geodesic equation on a certain manifold, which is constructed from the original spacetime and Killing vector fields thereon. A general method is presented for classifying the symmetry of cohomogeneity one membranes in a given spacetime. The classification is completely carried out in Minkowski spacetime. We analyze one of the obtained classes in depth and derive an exact solution.

A Metric for the Evaluation of Dense Vector Field Visualizations

In this work, we present an intuitive image-quality metric that is derived from the motivation of DVF visualization. It utilizes the features of the resulting image and effectively measures the similarity between the output of the visualization method and the input flow data. We use the angle between the gradient direction and the original vector field as a measure of such similarity and the gradient magnitude as an importance measure. Our metric enables the automatic evaluation of images for a given vector field and allows the comparison of different methods, parameters sets, and quality improvement strategies for a specific vector field. By integrating the metric into the image-computation process, our approach can be used to generate improved images by choosing the best parameter set. To verify the effectiveness of our method, we conducted an extensive user study that demonstrated the metric’s applicability to various situations. For instance, our approach elucidated the robustness of a DVF visualization in the presence of data-altering filters, such as resampling.

Pattern forced geophysical vector field segmentation based on Clifford FFT

Vector field segmentation is gaining increasing importance in geophysics research. Existing vector field segmentation methods usually can only handle the statistical characteristics of the original data. It is hard to integrate the patterns forced by certain geophysical phenomena. In this paper, a template matching method is firstly constructed on the foundation of the Clifford Fourier Transformation (CFT). The geometric meanings of both inner and outer components can provide more attractive information about the similarities between original vector field and template data. A composed similarity field is constructed based on the coefficients fields. After that, a modified spatial consistency preserving K-Means cluster algorithm is proposed. This algorithm is applied to the similarity fields to extract the template forced spatial distribution pattern. The complete algorithm for the overall processing is given and the experiments of ENSO forced global ocean surface wind segmentation are configured to test our method. The results suggest that the pattern forced segmentation can extract more latent information that cannot be directly measured from the original data. And the spatial distribution of ENSO influence on the surface wind field is clearly given in the segmentation result. All the above suggest that the method we proposed provides powerful and new thoughts and tools for geophysical vector field data analysis.

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

Methods for vector field visualization strive to have a sparse representation of the field, while encoding all of its important features. Streamline visualization is one of the most popular such methods. Traditionally, a set of streamline methods have focused on capturing the salient features of the vector field such as sources, sinks, and vortices. However, not all features are created equal, and some features of the vector field are more important than others, which could simply be mere noise. It is this problem of characterizing feature importance through streamline visualization that we try to address in this paper. Specifically, a given 2D vector field can be decomposed into a rotation-free (gradient) component, divergence-free (curl) component and a harmonic component by the so-called Hodge decomposition. Features in the original vector field, in some sense, correspond to features in the first two components. Furthermore, the gradient and curl components are each induced by a scalar field. By analyzing these two corresponding scalar fields using topological methods (in particular the contour tree and the persistent homology), we develop a simple and novel algorithm whose streamline density tends to reflect the topological importance of the features in the input vector field. Such a feature-aware streamline sketch is more informative, yet still simple both visually and in terms of its generation. It enhances our understanding of the underlying vector field, which is demonstrated here by several experimental results.

PIV measurement of turbulent flow within a street canyon: Detection of coherent motion

Turbulent flow inside a street canyon was investigated in a wind-channel. Velocity measurements were performed in vertical planes by means of particle image velocimetry (PIV) at the repetition rate 500Hz. Two geometries were used for comparison purposes: buildings with pitched roofs and with flat roofs. Both induce different dynamic regimes in the street, so that the generated turbulent flows are of different categories. Velocity data were analysed by proper orthogonal decomposition (POD). POD identifies the most dominant modes with high coherency in the flow and also evaluates the relative contributions of each mode to the overall kinetic energy of turbulence. Rigorous analysis of correctness of the physical interpretation of such decomposition was carried out. POD reconstruction of the original vector field was performed and the accuracy of the method was evaluated. Wavelet analysis was applied to the time-series of the POD expansion coefficients in order to reveal the dynamics of the modes. Vorticity, calculated from the original velocity data, was decomposed by POD as well. Finally, the correlation between the vorticity and other methods for vortex detection was assessed.