scholarly journals Solutions of a non‐local aggregation equation: Universal bounds, concavity changes, and efficient numerical solutions

2020 ◽  
Vol 43 (8) ◽  
pp. 5398-5429
Author(s):  
Klemens Fellner ◽  
Barry D. Hughes
Keyword(s):  
Numerical Solutions ◽  
Aggregation Equation ◽  
Local Aggregation ◽  
Non Local ◽  
Universal Bounds

scholarly journals Non-Linear Langevin and Fractional Fokker–Planck Equations for Anomalous Diffusion by Lévy Stable Processes

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 760 ◽  
Author(s):  
Johan Anderson ◽  
Sara Moradi ◽  
Tariq Rafiq
Keyword(s):  
Anomalous Diffusion ◽  
Transport Coefficient ◽  
Numerical Solutions ◽  
Distribution Functions ◽  
Space Distribution ◽  
Non Linear ◽  
Non Local ◽  
Local Transport ◽  
Fokker Planck Equations ◽  
Fokker Planck

The numerical solutions to a non-linear Fractional Fokker–Planck (FFP) equation are studied estimating the generalized diffusion coefficients. The aim is to model anomalous diffusion using an FFP description with fractional velocity derivatives and Langevin dynamics where Lévy fluctuations are introduced to model the effect of non-local transport due to fractional diffusion in velocity space. Distribution functions are found using numerical means for varying degrees of fractionality of the stable Lévy distribution as solutions to the FFP equation. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy and modified transport coefficient. The transport coefficient significantly increases with decreasing fractality which is corroborated by analysis of experimental data.

Material stability and instability in non-local continuum models for dense granular materials

2019 ◽  
Vol 871 ◽  
pp. 799-830 ◽  
Author(s):  
Shihong Li ◽  
David L. Henann
Keyword(s):  
Granular Materials ◽  
Recent Work ◽  
Strain Localization ◽  
Numerical Solutions ◽  
Higher Order ◽  
Continuum Models ◽  
Independent Manner ◽  
Stability And Instability ◽  
Non Local ◽  
Flow Gradients

A class of common and successful continuum models for steady, dense granular flows is based on the$\unicode[STIX]{x1D707}(I)$model for viscoplastic grain-inertial rheology. Recent work has shown that under certain conditions,$\unicode[STIX]{x1D707}(I)$-based models display a linear instability in which short-wavelength perturbations grow at an unbounded rate – i.e. a Hadamard instability. This observation indicates that$\unicode[STIX]{x1D707}(I)$models will predict strain localization arising due to material instability in dense granular materials; however, it also raises concerns regarding the robustness of numerical solutions obtained using these models. Several approaches to regularizing this instability have been suggested in the literature. Among these, it has been shown that the inclusion of higher-order velocity gradients into the constitutive equations can suppress the Hadamard instability, while not precluding the modelling of strain localization into diffuse shear bands. In our recent work (Henann & Kamrin,Proc. Natl Acad. Sci. USA, vol. 110, 2013, pp. 6730–6735), we have proposed a non-local model – called the non-local granular fluidity (NGF) model – which also involves higher-order flow gradients and has been shown to quantitatively describe a wide variety of steady, dense flows. In this work, we show that the NGF model also successfully regularizes the Hadamard instability of the$\unicode[STIX]{x1D707}(I)$model. We further apply the NGF model to the problem of strain localization in quasi-static plane-strain compression using nonlinear finite-element simulations in order to demonstrate that the model is capable of describing diffuse strain localization in a mesh-independent manner. Finally, we consider the linear stability of an alternative gradient–viscoplastic model (Bouzidet al.,Phys. Rev. Lett., vol. 111, 2013, 238301) and show that the inclusion of higher-order gradients does not guarantee the suppression of the Hadamard instability.

scholarly journals On non-local energy transfer via zonal flow in the Dimits shift

2017 ◽  
Vol 83 (5) ◽  
Author(s):  
Denis A. St-Onge
Keyword(s):  
Energy Transfer ◽  
Numerical Solutions ◽  
Zonal Flow ◽  
Two Dimensional ◽  
Transfer Energy ◽  
Wave Interactions ◽  
Local Mechanism ◽  
Non Local ◽  
Quasilinear Approximation ◽  
Mode Truncation

The two-dimensional Terry–Horton equation is shown to exhibit the Dimits shift when suitably modified to capture both the nonlinear enhancement of zonal/drift-wave interactions and the existence of residual Rosenbluth–Hinton states. This phenomenon persists through numerous simplifications of the equation, including a quasilinear approximation as well as a four-mode truncation. It is shown that the use of an appropriate adiabatic electron response, for which the electrons are not affected by the flux-averaged potential, results in an $\boldsymbol{E}\times \boldsymbol{B}$ nonlinearity that can efficiently transfer energy non-locally to length scales of the order of the sound radius. The size of the shift for the nonlinear system is heuristically calculated and found to be in excellent agreement with numerical solutions. The existence of the Dimits shift for this system is then understood as an ability of the unstable primary modes to efficiently couple to stable modes at smaller scales, and the shift ends when these stable modes eventually destabilize as the density gradient is increased. This non-local mechanism of energy transfer is argued to be generically important even for more physically complete systems.

scholarly journals Numerical solutions of a 2D fluid problem coupled to a nonlinear non-local reaction-advection-diffusion problem for cell crawling migration in a discoidal domain

2018 ◽  
Vol 1 ◽  
pp. 122 ◽  
Author(s):  
Christelle Etchegaray ◽  
Nicolas Meunier
Keyword(s):  
Numerical Solutions ◽  
Local Reaction ◽  
Approximate Solutions ◽  
Diffusion Problem ◽  
Poisson Problem ◽  
Advection Diffusion ◽  
Cell Crawling ◽  
Non Local ◽  
Volume Method ◽  
Advection Velocity

In this work, we present a numerical scheme for the approximate solutions of a 2D crawling cell migration problem. The model, defined on a non-deformable discoidal domain, consists in a Darcy fluid problem coupled with a Poisson problem and a reaction-advection-diffusion problem. Moreover, the advection velocity depends on boundary values, making the problem nonlinear and non local. For a discoidal domain, numerical solutions can be obtained using the finite volume method on the polar formulation of the model. Simulations show that different migration behaviours can be captured.

scholarly journals Energy-based non-local plasticity models for deformation patterning, localization and fracture

2015 ◽  
Vol 471 (2180) ◽  
pp. 20150275 ◽  
Author(s):  
Giovanni Lancioni ◽  
Tuncay Yalçinkaya ◽  
Alan Cocks
Keyword(s):  
Numerical Solutions ◽  
Deformation Field ◽  
Energy Potential ◽  
Heterogeneous Distribution ◽  
One Dimensional ◽  
Plastic Energy ◽  
Local Plasticity ◽  
Rate Dependent ◽  
Shear Problem ◽  
Non Local

This paper analyses the effect of the form of the plastic energy potential on the (heterogeneous) distribution of the deformation field in a simple setting where the key physical aspects of the phenomenon could easily be extracted. This phenomenon is addressed through two different (rate-dependent and rate-independent) non-local plasticity models, by numerically solving two distinct one-dimensional problems, where the plastic energy potential has different non-convex contributions leading to patterning of the deformation field in a shear problem, and localization, resulting ultimately in fracture, in a tensile problem. Analytical and numerical solutions provided by the two models are analysed, and they are compared with experimental observations for certain cases.

scholarly journals Macroscopic model for unsteady flow in porous media

2019 ◽  
Vol 862 ◽  
pp. 283-311 ◽  
Author(s):  
Didier Lasseux ◽  
Francisco J. Valdés-Parada ◽  
Fabien Bellet
Keyword(s):  
Porous Media ◽  
Unsteady Flow ◽  
Numerical Solutions ◽  
Creeping Flow ◽  
Flow In Porous Media ◽  
Macroscopic Model ◽  
Initial Flow ◽  
Effective Coefficients ◽  
Wide Range ◽  
Non Local

The present article reports on a formal derivation of a macroscopic model for unsteady one-phase incompressible flow in rigid and periodic porous media using an upscaling technique. The derivation is carried out in the time domain in the general situation where inertia may have a significant impact. The resulting model is non-local in time and involves two effective coefficients in the macroscopic filtration law, namely a dynamic apparent permeability tensor,$\unicode[STIX]{x1D643}_{t}$, and a vector,$\unicode[STIX]{x1D736}$, accounting for the time-decaying influence of the flow initial condition. This model generalizes previous non-local macroscale models restricted to creeping flow conditions. Ancillary closure problems are provided, which allow the effective coefficients to be computed. Symmetry and positiveness analyses of$\unicode[STIX]{x1D643}_{t}$are carried out, showing that this tensor is symmetric only in the creeping regime. The effective coefficients are functions of time, geometry, macroscopic forcings and the initial flow condition. This is illustrated through numerical solutions of the closure problems. Predictions are made on a simple periodic structure for a wide range of Reynolds numbers smaller than the critical value characterizing the first Hopf bifurcation. Finally, the performance of the macroscopic model for a variety of macroscopic forcings and initial conditions is examined in several case studies. Validation through comparisons with direct numerical simulations is performed. It is shown that the purely heuristic classical model, widely used for unsteady flow, consisting of a Darcy-like model complemented with an accumulation term on the filtration velocity, is inappropriate.

scholarly journals Axially and spherically symmetric solitons in warm plasma

2011 ◽  
Vol 77 (6) ◽  
pp. 749-764 ◽  
Author(s):  
MAXIM DVORNIKOV
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Solutions ◽  
Theoretical Description ◽  
Nonlinear Schrodinger Equation ◽  
Spherically Symmetric ◽  
Local Electron ◽  
Non Local ◽  
Warm Plasma

AbstractWe study the existence of stable axially and spherically symmetric plasma structures on the basis of the new nonlinear Schrödinger equation (NLSE) accounting for non-local electron nonlinearities. The numerical solutions of NLSE having the form of spatial solitions are obtained and their stability is analyzed. We discuss the possible application of the obtained results to the theoretical description of natural plasmoids in the atmosphere.

Linear stability of finite-amplitude capillary waves on water of infinite depth

2012 ◽  
Vol 696 ◽  
pp. 402-422 ◽  
Author(s):  
Roxana Tiron ◽  
Wooyoung Choi
Keyword(s):  
Linear Stability ◽  
Evolution Equations ◽  
Numerical Solutions ◽  
Normal Modes ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
One Dimensional ◽  
Highly Nonlinear ◽  
Non Local ◽  
Two Bubbles

AbstractWe study the linear stability of the exact deep-water capillary wave solution of Crapper (J. Fluid Mech., vol. 2, 1957, pp. 532–540) subject to two-dimensional perturbations (both subharmonic and superharmonic). By linearizing a set of exact one-dimensional non-local evolution equations, a stability analysis is performed with the aid of Floquet theory. To validate our results, the exact evolution equations are integrated numerically in time and the numerical solutions are compared with the time evolution of linear normal modes. For superharmonic perturbations, contrary to Hogan (J. Fluid Mech., vol. 190, 1988, pp. 165–177), who detected two bubbles of instability for intermediate amplitudes, our results indicate that Crapper’s capillary waves are linearly stable to superharmonic disturbances for all wave amplitudes. For subharmonic perturbations, it is found that Crapper’s capillary waves are unstable, and our results generalize to the highly nonlinear regime the analysis for small amplitudes presented by Chen & Saffman (Stud. Appl. Maths, vol. 72, 1985, pp. 125–147).

scholarly journals Non-local Aggregation for RGB-D Semantic Segmentation

2021 ◽  
pp. 1-1
Author(s):  
Guodong Zhang ◽  
Jing-Hao Xue ◽  
Pengwei Xie ◽  
Fan Yang ◽  
Guijin Wang
Keyword(s):  
Semantic Segmentation ◽  
Local Aggregation ◽  
Non Local

scholarly journals Are Current Discontinuities in Molecular Devices Experimentally Observable?

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 691
Author(s):  
F. Minotti ◽  
G. Modanese
Keyword(s):  
Field Effect ◽  
Maxwell Equations ◽  
Numerical Solutions ◽  
Molecular Devices ◽  
Maxwell Theory ◽  
Current Conservation ◽  
Dependent Part ◽  
Correct Definition ◽  
Non Local ◽  
Local Term

An ongoing debate in the first-principles description of conduction in molecular devices concerns the correct definition of current in the presence of non-local potentials. If the physical current density j=(−ieℏ/2m)(Ψ*∇Ψ−Ψ∇Ψ*) is not locally conserved but can be re-adjusted by a non-local term, which current should be regarded as real? Situations of this kind have been studied for example, for currents in saturated chains of alkanes, silanes and germanes, and in linear carbon wires. We prove that in any case the extended Maxwell equations by Aharonov-Bohm give the e.m. field generated by such currents without any ambiguity. In fact, the wave equations have the same source terms as in Maxwell theory, but the local non-conservation of charge leads to longitudinal radiative contributions of E, as well as to additional transverse radiative terms in both E and B. For an oscillating dipole we show that the radiated electrical field has a longitudinal component proportional to ωP^, where P^ is the anomalous moment ∫I^(x)xd3x and I^ is the space-dependent part of the anomaly I=∂tρ+∇·j. For example, if a fraction η of a charge q oscillating over a distance 2a lacks a corresponding current, the predicted maximum longitudinal field (along the oscillation axis) is EL,max=2ηω2qa/(c2r). In the case of a stationary current in a molecular device, a failure of local current conservation causes a “missing field” effect that can be experimentally observable, especially if its entity depends on the total current; in this case one should observe at a fixed position changes in the ratio B/i in dependence on i, in contrast with the standard Maxwell equations. The missing field effect is confirmed by numerical solutions of the extended equations, which also show the spatial distribution of the non-local term in the current.

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