Nonlocal Velocity Research Articles

Nonlinear Free Vibration of Postbuckled PFG Micro/nanotubes in Presence of Internal Resonances via Nonlocal Strain and Velocity Gradient Theory

Nonlinear Free Vibration of Postbuckled PFG Micro/nanotubes in Presence of Internal Resonances via Nonlocal Strain and Velocity Gradient Theory

Exponential Stability of the Numerical Solution of a Hyperbolic System with Nonlocal Characteristic Velocities

In this paper, we investigate the problem of the exponential stability of a stationary solution for a hyperbolic system with nonlocal characteristic velocities and measurement error. The formulation of the initial boundary value problem of boundary control for the specified hyperbolic system is given. A difference scheme is constructed for the numerical solution of the considered initial boundary value problem. The definition of the exponential stability of the numerical solution in ℓ2-norm with respect to a discrete perturbation of the equilibrium state of the initial boundary value difference problem is given. A discrete Lyapunov function for a numerical solution is constructed, and a theorem on the exponential stability of a stationary solution of the initial boundary value difference problem in ℓ2-norm with respect to a discrete perturbation is proved.

Asymptotic stability of rarefaction wave for compressible Euler system with velocity alignment

In this paper, we study the asymptotic stability of the rarefaction wave for the one-dimensional compressible Euler system with nonlocal velocity alignment. Namely, for the initial data approaching to rarefaction wave, we prove the corresponding solution converges toward the rarefaction wave. Moreover, we obtain this system has weak alignment behavior. We develop some promoted estimates for the smooth approximate rarefaction wave and new a priori estimates by Fourier analysis tools. Moreover, we introduce the weighted energy method and Besov spaces to obtain the key high-order derivative estimates, in which we overcome the difficulties caused by the nonlocal velocity alignment. It is worth mentioning that this is the first stability result of rarefaction wave for compressible Euler system with velocity alignment.

Nonlinear free vibration of pre-buckled PFG micro/nanotubes via nonlocal strain and velocity gradient theory

Nonlinear free vibration of pre-buckled PFG micro/nanotubes via nonlocal strain and velocity gradient theory

Conservation Laws with Nonlocal Velocity: The Singular Limit Problem

Conservation Laws with Nonlocal Velocity: The Singular Limit Problem

Updated Lagrangian nonlocal general particle dynamics for large deformation problems

The traditional general particle dynamics (GPD) has been successfully applied in modeling fracture propagation, frictional contact and stick–slip problems in rocks. In this study, the updated Lagrangian nonlocal general particle dynamics (UL-NGPD) method is proposed to solve large deformation problems. In the UL-NGPD, the continuity equation and momentum equation are reformulated in the integro-differential formulation by introducing nonlocal theories in the updated Lagrangian framework. The artificial viscosity and artificial force state are utilized to enhance the numerical stability and accuracy in modeling large deformation. The nonlocal velocity gradient in the deformed configuration is used to update the Cauchy stress rate from the Drucker-Prager elasto-plastic constitutive model with strain-softening behaviour for solids. The UL-NGPD paradigm is numerically implemented through an explicit Predictor-Corrector scheme for high-performance computing. The stability and accuracy of the UL-NGPD are verified by numerical tests on compression test, slope stability analysis and aluminum bar collapse experiment. Thereafter, simulations of granular soil collapse and retrogressive landslide in sensitive clays are presented to demonstrate its efficacy and robustness in modeling challenging geotechnical problems involving large deformation. The numerical results show that the proposed UL-NGPD is powerful and suitable for analyzing large deformation problems in geotechnical engineering.

Dynamic stability of porous functionally graded nanotubes via nonlocal strain and velocity gradient theory

Dynamic stability of porous functionally graded nanotubes via nonlocal strain and velocity gradient theory

Blowup of solutions for a transport equation with nonlocal velocity and damping

We study a transport equation with nonlocal velocity and power law type damping. We show that a certain class of initial data smooth solution blows up in finite time.

Lyapunov stabilization of a nonlocal LWR traffic flow model

AbstractUsing a nonlocal macroscopic LWR‐type traffic flow model, we present an approach to control the nonlocal velocity towards a given equilibrium velocity. Therefore, we present a Lyapunov function measuring the L2 distance between these velocities. We compute the explicit rate at which the system tends towards the stationary speed. The traffic is controlled by a leading vehicle. Numerical examples demonstrate the theoretical results and possible extensions of them.

Velocity dependent potential for alpha 40Ca-elastic scattering

Angular distribution of the differential cross section of α40Ca elastic scattering has been investigated in the anomalous large angle scattering (ALAS) region by a nonlocal velocity dependent optical potential with surface term. The geometrical potential parameters were fixed and the depths of the potential were determined by fitting elastic angular distribution over the energy range (23–53.9 MeV). The real and the imaginary parts of the optical potential used here are of Woods–Saxon form factors. An energy dependence of the potential depths has been investigated at fixed geometrical potential parameters.

STABILITY OF THE EQUILIBRIUM STATE FOR THE SCALAR CONSERVATION LAW WITH NONLOCAL CHARACTERISTIC VELOCITIES

In this paper, we study input-state stability (ISS) of equilibrium for a scalar conservation law with nonlocal velocity and measurement error occurring in a high-volume reproducible system . Using the corresponding Lyapunov function, we derive sufficient and necessary conditions on ISS. We propose a numerical discretization of the scalar conservation law with nonlocal velocity and measurement error. For the proposed numerical approximation, the appropriate discrete Lyapunov function is analyzed to provide the ISS of the discrete equilibrium. For the proposed numerical approximation, the appropriate discrete Lyapunov function is analyzed to provide the ISS of the discrete equilibrium. Finally, we show computational results to confirm the theoretical conclusions.

Updated Lagrangian unsaturated periporomechanics for extreme large deformation in unsaturated porous media

Unsaturated periporomechanics is a strong nonlocal poromechanics based on peridynamic state and effective force state concept. In the previous periporomechnics the total Lagrangian formulation is adopted for the solid skeleton of porous media. In this article as a new contribution we formulate and implement an updated Lagrangian unsaturated periporomechanics framework for modeling extreme large deformation in unsaturated soils under drained conditions. In this new framework the so-called bond-associated sub-horizon concept is utilized to enhance the stability and accuracy at extreme large deformation of the solid skeleton. The stabilized nonlocal velocity gradient in the deformed configuration is used to update the effective force state from a critical state based visco-plastic constitutive model for unsaturated soils. The updated Lagrangian periporomechanics paradigm is numerically implemented through an explicit Newmark scheme for high-performance computing. Numerical examples are presented to demonstrate the stability of the computational updated Lagrangian periporomechanics paradigm and its efficacy and robustness in modeling extreme large deformation in porous media under drained conditions.

Global existence, blow-up and stability for a stochastic transport equation with non-local velocity

In this paper we investigate a non-linear and non-local one dimensional transport equation under random perturbations on the real line. We first establish a local-in-time theory, i.e., existence, uniqueness and blow-up criterion for pathwise solutions in Sobolev spaces Hs with s>3. Thereafter, we give a picture of the long time behavior of the solutions based on the type of noise we consider. On one hand, we identify a family of noises such that blow-up can be prevented with probability 1, guaranteeing the existence and uniqueness of global solutions almost surely. On the other hand, in the particular linear noise case, we show that singularities occur in finite time with positive probability, and we derive lower bounds of these probabilities. To conclude, we introduce the notion of stability of exiting times and show that one cannot improve the stability of the exiting time and simultaneously improve the continuity of the dependence on initial data.

Controllability for a highly re-entrant manufacturing system with local and nonlocal velocity

In this paper, we study a scalar conservation law that models a high-volume, multistage continuous production through a re-entrant manufacturing system. The velocity has both local and nonlocal properties. We first prove the existence and uniqueness of the weak solution to the Cauchy problem with initial and boundary data in L∞ space. Based on this, a local exact controllability result is established, i.e., there exists a control that drives the weak solution from any given initial data to any desired final data around the origin in a certain time period.

Formation of singularities for multi-dimensional transport equations with nonlocal velocity

This paper is concerned with a class of multi-dimensional transport equations with nonlocal velocity. It is shown that the local smooth solution cannot exist globally in time via the De Giorgi iteration technique.

Defects in Liquid Crystal Flows

This paper concerns the dynamical properties of topological defects in two dimensional flows of liquid crystals modeled by the Ginzburg--Landau approximations. The fluid is transported by a nonlocal (an averaged) velocity and is coupled with effects of the elastic stress. The defects move along the trajectories of the flow associated with this averaged velocity, that is, $\frac{d}{dt}a_j(t)={u}(a_j(t), t).$

On a 1D model with nonlocal interactions and mass concentrations: an analytical-numerical approach* *Abreu was partially supported by CNPq 306385/2019-8, BR and Ferreira was partially supported by FAPESP and CNPq, BR.

We study global well-posedness and finite time blow-up of solutions for a nonlinear one-dimensional transport equation with nonlocal velocity , ν > 0, and measure initial data. Such model arises in fluid mechanics in vortex-sheet problems and its nonlocal feature comes from the presence of a singular integral operator (Hilbert transform ) in the velocity field. In the viscous case ν > 0, we analytically obtain an explicit condition on the size of the initial data for the global well-posedness in the framework of pseudomeasure spaces. In fact, we can give the condition depending on the initial-mass and analyze how the flow evolves from singular measures. Also, we numerically study blow-up of concentration type and global diffusion-smooth behavior of solutions. We obtain numerics that indicate the threshold value 2π for the initial-data mass that decides between blow-up or global smoothness of solutions. Such value is the same obtained for regular initial-data and by means of entropy methods. Thus, it seems to be intrinsic to the nonlocal PDE and independent of a particular framework, approach and initial-data regularity. The inviscid case ν = 0 is remarkable: simulations for model , evidence that the solution presents blow-up of concentration type with mass-preserving, while an attenuation effect is observed for the model with opposite sign in the nonlinearity , for any nontrivial (positive) measure as initial data.

Network models for nonlocal traffic flow

We present a network formulation for a traffic flow model with nonlocal velocity in the flux function. The modeling framework includes suitable coupling conditions at intersections to either ensure maximum flux or distribution parameters. In particular, we focus on 1-to-1, 2-to-1 and 1-to-2 junctions. Based on an upwind type numerical scheme, we prove the maximum principle and the existence of weak solutions on networks. We also investigate the limiting behavior of the proposed models when the nonlocal influence tends to infinity. Numerical examples show the difference between the proposed coupling conditions and a comparison to the Lighthill-Whitham-Richards network model.

Large friction-high force fields limit for the nonlinear Vlasov–Poisson–Fokker–Planck system

<p style='text-indent:20px;'>We provide a quantitative asymptotic analysis for the nonlinear Vlasov–Poisson–Fokker–Planck system with a large linear friction force and high force-fields. The limiting system is a diffusive model with nonlocal velocity fields often referred to as aggregation-diffusion equations. We show that a weak solution to the Vlasov–Poisson–Fokker–Planck system strongly converges to a strong solution to the diffusive model. Our proof relies on the modulated macroscopic kinetic energy estimate based on the weak-strong uniqueness principle together with a careful analysis of the Poisson equation.</p>

Chaotic vibrations of carbon nanotubes subjected to a traversing force considering nonlocal elasticity theory

The existence of chaos in the lateral vibration of the carbon nanotube (CNT) can contribute to source of instability and inaccuracy within the nano mechanical systems. So, chaotic vibrations of a simply supported CNT which is subjected to a traversing harmonic force are studied in this paper. The model of the system is formulated by using nonlocal Euler–Bernoulli beam theory. The equation of motion is solved using the Rung–Kutta method. The effects of the nonlocal parameter, velocity and amplitude of the traversing harmonic force on the nonlinear dynamic response of the system are analyzed by the bifurcation diagrams, phase plane portrait, power spectra analysis, Poincaré map and the maximum Lyapunov exponent. The results indicate that the nonlocal parameter, velocity and amplitude of the traversing harmonic force have considerable effects on the bifurcation behavior and can be used as effective control parameters for avoiding chaos.