Arbitrary Velocity Research Articles

Time-Optimal Interception of a Moving Target by a Dubins Car

The problem of intercepting a target moving along a prescribed trajectory by a Dubins caris stated and formalized as a time-optimal control problem with an arbitrary car velocity directionat the interception. The conditions available in the literature under which the optimal trajectoryis a geodesic line drawn from the initial position of the car to the interception point are refined.Algebraic equations for calculating the optimal interception time are obtained. The optimalcontrol is synthesized based on these equations. A software module is developed for constructingthe optimal car trajectories for various target trajectories.

Three-body capture, ejection, and the demographics of bound objects in binary systems

ABSTRACT We study the capture of light objects of arbitrary velocity by binary systems. Extending results for the capture of comets in the Solar system, we develop a simple geometric characterization of the capture cross-section, leading directly to the distribution of orbital parameters of captured objects. We use the same framework to study the lifetimes of these bound orbits prior to ejection, and find that a simplified version of the Öpik–Arnold approach readily yields a closed-form estimate for the ejection rate that agrees well with numerical experiments. Without any detailed-balance assumptions, our results make manifest the characteristics of close encounters leading to capture and ejection. As an application of our results, we demonstrate the estimation of the equilibrium population of captured dark matter particles in a binary system.

Combined Effects of Heat and Mass Transfer on MHD Free Convective Flow of Maxwell Fluid with Variable Temperature and Concentration

Heat and mass transfer combined effects on MHD natural convection for a viscoelastic fluid flow are investigated. The dynamics of the fluid are controlled by the motion of the plate with arbitrary velocity along with varying temperature and mass diffusion. The non-dimensional forms of the governing equations of the model are developed along with generalized boundary conditions and the resulting forms are solved by the classical integral (Laplace) transform technique/method and closed-form solutions are developed. Obtained generalized results are very important due to their vast applications in the field of engineering and applied sciences; few of them are highlighted here as limiting cases. Moreover, parametric analysis of system parameters P r , S , K c , G T , G c , M , S c , λ is done via graphical simulations.

Coordinated target tracking by multiple unmanned surface vehicles with communication delays based on a distributed event-triggered extended state observer

This paper is concerned with a coordinated target tracking problem for a network of unmanned surface vehicles (USVs) subject to communication delays and constrained actuation resources. The target to be tracked maneuvers with arbitrary velocity. Its position can be measured by a small fraction of USVs whereas its velocity is not known to follower USVs. A resource-aware event-triggered control design method is presented for coordinated target tracking. Specifically, a distributed event-triggered extended state observer (ESO) is designed to estimate the position and velocity of the target over a cooperative estimation network. Furthermore, the above result is extended to cooperative estimation subject to communication delays, and a time-delayed distributed event-triggered ESO is designed. In the control law design, an event-triggered anti-disturbance position tracking control law is proposed for each follower USV based on a reduced-order ESO. By the proposed resource-aware event-triggered control laws, cooperative target tracking can be achieved regardless of the model uncertainty and external disturbances. Moreover, both the execution rate of the actuators and communication burden of the network are reduced. The stability of the closed-loop system is proven through input-to-state stability and cascade stability analysis. Simulation results demonstrate the effectiveness of the proposed method for the coordinated target tracking of USVs.

Colorless movement of focal spot with arbitrary velocity based on mirrors with special shape

Focal spot with moving speed is a new concept proposed in recent years, which has many applications in physics of intense field. The current method is to control the moving speed of focal spot by chirped pulse technique and chromatic aberration focusing, which suffers from the change of frequency components with the movement of focal spot. In this paper, we propose a method to realize the colorless movement of focal spot at arbitrary speed by using two kinds of mirrors. One is an aspherical mirror with long focal depth and the other is a stepped mirror. For a specific moving speed, we derive the relationship between the surface of aspherical mirror and the wavefront of laser by the geometrical optics. The stepped mirror is used to form stepped wavefront to separate the relationship between the shape of wavefront and the direction of laser. Time-dependent simulations are carried out to verify the real focusing property after considering diffraction effects, and focal spots with moving speed of 0.5c, 2c and −c are obtained.

Survival probability of a run-and-tumble particle in the presence of a drift

We consider a one-dimensional run-and-tumble particle, or persistent random walk, in the presence of an absorbing boundary located at the origin. After each tumbling event, which occurs at a constant rate γ, the (new) velocity of the particle is drawn randomly from a distribution W(v). We study the survival probability S(x, t) of a particle starting from x ⩾ 0 up to time t and obtain an explicit expression for its double Laplace transform (with respect to both x and t) for an arbitrary velocity distribution W(v), not necessarily symmetric. This result is obtained as a consequence of Spitzer’s formula, which is well known in the theory of random walks and can be viewed as a generalization of the Sparre Andersen theorem. We then apply this general result to the specific case of a two-state particle with velocity ±v 0, the so-called persistent random walk (PRW), and in the presence of a constant drift μ and obtain an explicit expression for S(x, t), for which we present more detailed results. Depending on the drift μ, we find a rich variety of behaviors for S(x, t), leading to three distinct cases: (i) subcritical drift −v 0 < μ < v 0, (ii) supercritical drift μ < −v 0 and (iii) critical drift μ = −v 0. In these three cases, we obtain exact analytical expressions for the survival probability S(x, t) and establish connections with existing formulae in the mathematics literature. Finally, we discuss some applications of these results to record statistics and to the statistics of last-passage times.

On Problem of Internal Boundary Control for String Vibration Equation

The article deals with the vibration control problem described by one dimensional wave equation with integral type boundary condition. As usual, the initial and final moments of time for arbitrary displacements and velocities of the wave are specified by points on a string (Cauchy data). It is shown that the minimum time for the realizable control is uniquely determined by the condition of correct solvability to the Cauchy problem involving data lying on disconnected manifold. This suggests that the internal boundary conditions does not affect the minimum time value. Necessary and sufficient conditions for the existence of the desired internal-boundary controls that move the process from the state initially specified to a predetermined final one are obtained and written out. The controls are presented in explicit analytical form. Moreover, it is shown that for the inner-boundary controls expressions, one should use not the representation of the solution to the Cauchy problem in the sought-for domain, but the formula for the general solution of the string oscillation equation (d’Alembert’s formula).

Nernst–Planck–Navier–Stokes Systems far from Equilibrium

We consider ionic electrodiffusion in fluids, described by the Nernst-Planck-Navier-Stokes system. We prove that the system has global smooth solutions for arbitrary smooth data: arbitrary positive Dirichlet boundary conditions for the ionic concentrations, arbitrary Dirichlet boundary conditions for the potential, arbitrary positive initial concentrations, and arbitrary regular divergence-free initial velocities. The result holds for any positive diffusivities of ions, in bounded domains with smooth boundary in three space dimensions, in the case of two ionic species, coupled to Stokes equations for the fluid. The result also holds in the case of Navier-Stokes coupling, if the velocity is regular. The global smoothness of solutions is also true for arbitrarily many ionic species, if all their diffusivities are the same.

Helioseismic Modeling of Background Flows

Abstract We present a three-dimensional (3D) numerical solver of the linearized compressible Euler equations (Global Acoustic Linearized Euler), used to model acoustic oscillations throughout the solar interior. The governing equations are solved in conservation form on a fully global spherical mesh (0 ≤ ϕ ≤ 2π, 0 ≤ θ ≤ π, 0 ≤ r ≤ R ⊙) over a background state generated by the standard solar model S. We implement an efficient pseudospectral computational method to calculate the contribution of the compressible material derivative dyad to internal velocity perturbations, computing oscillations over arbitrary 3D background velocity fields. This model offers a foundation for a “forward-modeling” approach, using helioseismology techniques to explore various regimes of internal mass flows. We demonstrate the efficacy of the numerical method presented in this paper by reproducing observed solar power spectra, showing rotational splitting due to differential rotation, and applying local helioseismology techniques to measure travel times created by a simple model of single-cell meridional circulation.

Efficient Algorithms for Max-Weighted Point Sweep Coverage on Lines.

As an important application of wireless sensor networks (WSNs), deployment of mobile sensors to periodically monitor (sweep cover) a set of points of interest (PoIs) arises in various applications, such as environmental monitoring and data collection. For a set of PoIs in an Eulerian graph, the point sweep coverage problem of deploying the fewest sensors to periodically cover a set of PoIs is known to be Non-deterministic Polynomial Hard (NP-hard), even if all sensors have the same velocity. In this paper, we consider the problem of finding the set of PoIs on a line periodically covered by a given set of mobile sensors that has the maximum sum of weight. The problem is first proven NP-hard when sensors are with different velocities in this paper. Optimal and approximate solutions are also presented for sensors with the same and different velocities, respectively. For M sensors and N PoIs, the optimal algorithm for the case when sensors are with the same velocity runs in time; our polynomial-time approximation algorithm for the case when sensors have a constant number of velocities achieves approximation ratio ; for the general case of arbitrary velocities, and approximation algorithms are presented, respectively, where integer is the tradeoff factor between time complexity and approximation ratio.

The worm-LBM, an algorithm for a high number of propagation directions on a lattice Boltzmann grid: The case of phonon transport

The lattice Boltzmann method (LBM) is a numerical approach to tackle problems described by a Boltzmann type-equation, where time, space, and velocities are discretized to describe scattering and advection. Even though the LBM executes advection along a lattice direction without numerical error, its usage in the high Knudsen number regime (ballistic) has been hindered by the ray effect problem (for dimensions greater than 1D). This problem has its origin in the low number of available propagation directions on standard LBM lattices. Here, to overcome this limitation, we propose the worm-lattice Boltzmann method (worm-LBM), which allows a high number of lattice directions by alternating in time the basic directions described within the next neighbor schemes. Additionally, to overcome the velocity anisotropy issue, which otherwise clearly manifests itself in the ballistic regime (e.g. the 2 higher grid velocity of the D2Q8 scheme along the diagonal direction compared to the axial one), the time-adaptive scheme (TAS) is proposed. The TAS method makes use of pausing advection on the grid, allowing to impose not only isotropic propagation but also arbitrary direction-dependent grid velocity. Last but not least, we propose a grid-mean free path (grid-MFP) correction to correctly handle the aforementioned velocity issue in the diffusive limit, without affecting the ballistic one. We provide a detailed description of the TAS method and the worm-LBM algorithm, and verify their numerical accuracy by using several transient diffusive-ballistic phonon transport cases, including different initial and boundary conditions. We demonstrate the accuracy of the new worm-LBM to describe problems where a high angular resolution (i.e. a high number of propagation directions) is required, as the in-plane thermal transport problem under adiabatic-diffusive boundary conditions. In this particular case, we show that schemes with a low number of propagation directions (D2Q8) result in an overestimation of the analytical Fuchs-Sondheimer solution for intermediate and high Knudsen numbers, and that schemes with a higher number of propagation directions are required to correctly describe the problem. Overall, the new, very accurate, and efficient worm-LBM algorithm, free of numerical smearing and false scattering, has the potential to be at the forefront of the numerical solvers to tackle the advective part of different equations in a wide field of applications.

Spectrally recycling space-time wave packets

'Space-time' (ST) wave packets are propagation-invariant pulsed optical beams that travel rigidly in linear media without diffraction or dispersion at a potentially arbitrary group velocity. These unique characteristics are a result of spatio-temporal spectral correlations introduced into the field; specifically, each spatial frequency is associated with a single temporal frequency (or wavelength). Consequently, the spatial and temporal bandwidths of ST wave packets are correlated, so that exploiting an optical source with a large temporal bandwidth or achieving an ultralow group velocity necessitate an exorbitantly large numerical aperture. Here we show that `spectral recycling' can help overcome these challenges. 'Recycling' or `reusing' each spatial frequency by associating it with multiple distinct but widely separated temporal frequencies allows one to circumvent the proportionality between the spatial and temporal bandwidths of ST wave packets, which has been one of their permanent characteristics since their inception. We demonstrate experimentally that the propagation invariance, maximum propagation distance, and group velocity of ST wave packets are unaffected by spectral recycling. We also synthesize a ST wave packet with group velocity c/14.3 (c is the speed of light in vacuum) with a reasonable numerical aperture made possible by spectral recycling.

Solving the Coriolis Vibratory Gyroscope Motion Equations by Means of the Angular Rate B-Spline Approximation

The exact solution of the movement equation of the Coriolis vibratory gyroscope (CVG) with a linear law of variation of the angular rate of rotation of the base is given. The solution is expressed in terms of the Weber functions (the parabolic cylinder functions) and their asymptotic representations. On the basis of the obtained solution, an analytical solution to the equation of the ring dynamics in the case of piecewise linear approximation of an arbitrary angular velocity profile on a time grid is derived. The piecewise linear solution is compared with the more rough piecewise constant solution and the dependence of the error of such approximations on the sampling step in time is estimated numerically. The results obtained make it possible to significantly reduce the number of operations when it is necessary to study long-range dynamics of oscillations of the system, as well as quantitatively and qualitatively control the convergence of finite-difference schemes for solving the movement equations of the Coriolis vibratory gyroscope.

Faxén formulas for particles of arbitrary shape and material composition

Abstract

Space-time chaos in the nonlinear Schrödinger equation

The paper provides an analytical and numerical analysis of the transition to space-time chaos in the generalized nonlinear Schrödinger equation with complex, in the general case, parameters. Equation describes, in particular, the wave amplitude of a surface plasmon polariton propagating over the contact surface of a metal with a dielectric. It is proved that considered equation has an infinite number of different stable wave solutions running along the spatial axis with arbitrary velocities, as well as an infinite number of different modes of space-time chaos in full accordance with the universal Feigenbaum-Sharkovsky-Magnitskii bifurcation theory. In this case, the bifurcation parameter is the value of the speed of propagation of traveling waves along the spatial axis, which is clearly not included in the original equation.

ДИНАМИКА ДВУХМАССОВОГО МИКРОМЕХАНИЧЕСКОГО ГИРОСКОПА L-L-ТИПА В РЕЖИМЕ ВЫНУЖДЕННЫХ КОЛЕБАНИЙ

This paper is devoted to a study of dynamics of an L-L-type two-mass micromechanical gyroscope (abbreviated as MMG) under forced oscillations. The mechanical-theoretical model of MMG with a sensitive element in the form of a model with two active masses (“Inertial mass -Frame”) is obtained. Based on the model, the system of differential equations is written. A distinctive feature of this paper is that an arbitrary angular velocity of the base Ω is considered. A new method is used to find a solution to the system. This method is suitable for analyzing dualfrequency systems. Finally, the solution is determined, and the amplitude-frequency characteristics of the system are described. The obtained analytical expressions and graphs are analyzed, and the reasonable conclusions about system behavior under forced oscillations are made. The proposed method for analyzing complex systems is compared with other available methods.

Течії Стокса в тривимірних областях

This study consists of two parts. First we consider an analytical approach for solving the problem of steady Stokes flow in some 3D containers with arbitrary velocities prescribed over the surfaces. The approach is based on the superposition method. First we discuss the Stokes problem solution in a finite cylinder. This is the simplest problem because the flow domain is restricted with only two families of coordinate surfaces and the edge (rim) is a smooth line. Then we discuss the analytical solution of the Stokes problem in more complicated domains, such as a circular cone, a rectangular trihedral corner and a 3D rectangular cavity. The Moffatt eddies in such domains are described. In the second part of the study we consider the laminar mixing process in the Stokes flow in a 3D container. We show that in 3D flows a much richer variety of mixing regimes is observed than in 2D flow configurations. The mixing processes in a 3D flow, containing periodic lines, possess essentially two-dimensional characteristics. In the flows, where only isolated periodic points exist, the liquid elements stretch or compress in all three directions.

The [formula omitted]-ALE-SPH model: An arbitrary Lagrangian-Eulerian framework for the [formula omitted]-SPH model with particle shifting technique

The behaviour of a weakly-compressible SPH scheme obtained by rewriting the Navier-Stokes equations in an arbitrary Lagrangian-Eulerian (ALE) format is studied. Differently from previous works on ALE, which generally adopt conservative variables (i.e. mass and momentum) and rely on the use of Riemann solvers inside the spatial operators, the proposed model is expressed in terms of primitive variables (i.e. density and velocity) and is written by using the standard differential formulations of the weakly-compressible SPH schemes. Similarly to ALE-SPH models, the arbitrary velocity field is obtained by modifying the pure Lagrangian velocity of the material point through a velocity δu→ given by a Particle Shifting Technique (PST). We show that the above-mentioned ALE-SPH equations are, however, unstable when they are integrated in time. The instability appears in the form of large volume variations in those fluid regions characterised by high velocity strain rates. Nonetheless, the scheme can be stabilised if appropriate diffusion terms are included in both the equations of density and mass. This latter scheme, hereinafter called δ-ALE-SPH scheme, is validated against reference benchmark test-cases: the viscous flow around an inclined elliptical cylinder, the lid-driven cavity and a dam-break flow impacting a vertical wall.

Stationary solution for quasi-homogeneous small-scale magnetic field advected by non-Gaussian turbulent flow

We consider fluctuations of a magnetic field excited by an external force and advected by isotropic turbulent flow. It appears that non-Gaussian velocity gradient statistics and a finite region of pumping force provide the existence of a stationary solution. The mean-square magnetic field is calculated for arbitrary velocity gradient statistics. An estimate for possible feedback of the magnetic field on velocity shows that, for a wide range of parameters, stationarity without feedback would take place even in the case of intensive pumping of the magnetic field.

Optimal tuning of engineering wake models through lidar measurements

Abstract. Engineering wake models provide the invaluable advantage to predict wind turbine wakes, power capture, and, in turn, annual energy production for an entire wind farm with very low computational costs compared to higher-fidelity numerical tools. However, wake and power predictions obtained with engineering wake models can be insufficiently accurate for wind farm optimization problems due to the ad hoc tuning of the model parameters, which are typically strongly dependent on the characteristics of the site and power plant under investigation. In this paper, lidar measurements collected for individual turbine wakes evolving over a flat terrain are leveraged to perform optimal tuning of the parameters of four widely used engineering wake models. The average wake velocity fields, used as a reference for the optimization problem, are obtained through a cluster analysis of lidar measurements performed under a broad range of turbine operative conditions, namely rotor thrust coefficients, and incoming wind characteristics, namely turbulence intensity at hub height. The sensitivity analysis of the optimally tuned model parameters and the respective physical interpretation are presented. The performance of the optimally tuned engineering wake models is discussed, while the results suggest that the optimally tuned Bastankhah and Ainslie wake models provide very good predictions of wind turbine wakes. Specifically, the Bastankhah wake model should be tuned only for the far-wake region, namely where the wake velocity field can be well approximated with a Gaussian profile in the radial direction. In contrast, the Ainslie model provides the advantage of using as input an arbitrary near-wake velocity profile, which can be obtained through other wake models, higher-fidelity tools, or experimental data. The good prediction capabilities of the Ainslie model indicate that the mixing-length model is a simple yet efficient turbulence closure to capture effects of incoming wind and wake-generated turbulence on the wake downstream evolution and predictions of turbine power yield.