Nonlocal Model Research Articles

Threshold dynamics of a time-periodic nonlocal dispersal SIS epidemic model with Neumann boundary conditions

In this paper, we study a time-periodic nonlocal dispersal susceptible-infected-susceptible epidemic model with Neumann boundary conditions, where the total population number is constant. We first investigate limiting profile of the spectral bound for a time-periodic nonlocal dispersal operator, and then obtain asymptotic behavior of the basic reproduction ratio of the model as the dispersal rates go to zero and infinity, respectively. Next, we establish the existence, uniqueness and stability of steady states of the model in terms of the basic reproduction ratio. Finally, we discuss the impacts of small and large diffusion rates of the susceptible and infectious populations on the persistence and extinction of the disease.

Nonlocal geometric fracture theory: A nonlocal‐local asymptotically compatible framework for continuous–discontinuous deformation of solid materials

SummaryOne of the fundamental but unsolved problems in fracture mechanics of solid materials is how to describe the continuous and discontinuous deformation of continuum in a unified manner. This paper aims to develop a novel nonlocal geometric fracture theory attempting to address this essential issue. Using the concepts of nonlocal calculus, a new nonlocal equation for the deformation of continuum is obtained by minimizing a nonlocal Lagrangian. This is done to ensure that the proposed nonlocal model can be used to describe the continuous and discontinuous deformation of a continuum in a unified geometric manner, as well as to be asymptotically compatible with the classical continuum theory. Inspired by the phase field fracture theory, a nonlocal geometric crack surface functional is proposed to approximate the Griffith fracture energy in continuum. Note, however, that a distinguished feature of the proposed functional is that it enables strong discontinuities appear in the crack phase fields, which is difficult for the phase field fracture theory. After obtaining the nonlocal fracture energy, a nonlocal elastic energy is phenomenologically decomposed into volumetric and deviatoric parts, and then a system of nonlocal geometric fracture equations is derived to describe the damage nucleation, cracks initiation and propagation, which covers the whole physical process of fracture in solid materials. A staggered discontinuous Galerkin method is developed to reach an accurate and stable numerical solution. Numerical examples demonstrate the proposed theory can well capture the transition from the continuous to discontinuous deformation of the continuum and accurately describe the topological evolution of cracks in a unified framework.

Nonlocal strain gradient model for thermal buckling analysis of functionally graded nanobeams

Nonlocal strain gradient model for thermal buckling analysis of functionally graded nanobeams

Vibration properties of an elastic gold nanosphere submerged in viscoelastic fluid

In this paper, we propose a novel, simple and accurate analytical study based on nonlocal elasticity theory to forecast small-scale effects on the radial vibration of anisotropic gold nanospheres submerged in viscoelastic fluid (VEF). Eringen’s model is used to determine the motion equation for anisotropic nanospheres, with the fluid assumed to be viscoelastic and compressible. The frequency equation is derived by imposing the fluid-nanosphere interface continuity conditions. A comparison with the literature results is conducted to demonstrate the validity and correctness of this analysis, which indicates a very good agreement. The importance of small-scale effects in the radial vibration, which need to be included in the nonlocal elasticity model of submerged nanospheres, is eventually revealed by numerical examples. It is discovered that the nanosphere size, nonlocal parameter, and glycerol–water mixture have a significant impact on the vibration behaviors. Our results show that the small scale is crucial for the radial vibration of gold nanoparticles when the gold nanosphere is smaller than [Formula: see text]. Thus, the resulting frequency equation is very useful to interpret experimental measurements of the vibration characteristics of submerged gold nanospheres in VEF.

“Implicit” vs “Explicit” gradient plasticity models: Do they always remove mesh dependence in softening materials?

Finite element quasi-static solutions of rate-independent softening plasticity models are known to depend on the mesh size (i.e., are unreliable) due to loss of ellipticity of the governing partial differential equations, which allows for the development of non-smooth solutions, such as shear bands of zero thickness. A “regularization” scheme that has been proposed is the use of “non-local” (Bažant et al., 1984) or “strain-gradient” theories (Aifantis, 1984). We examine in detail the families of “explicit” and “implicit” gradient plasticity models and show that use of such theories does not always eliminate mesh dependence of finite element solutions in rate-independent softening materials. Depending on the value of the non-local hardening modulus, the mathematical problem may still lose ellipticity and allow for the development of solutions with discontinuous spatial velocity gradients, which are responsible for the mesh dependence of numerical solutions. As an example, a rate-independent “implicit” gradient version of the von Mises yield condition with the associated flow rule is implemented in ABAQUS, and the formation of shear bands in plane strain tension is analyzed numerically; it is shown that, unless the material parameters in the non-local model are such that ellipticity is retained, the finite element solutions depend on the mesh size. The example of isotropic porous metal plasticity with a hardening matrix material is considered (e.g., the Gurson (1977) model); it is shown that regularization is achieved if an implicit non-local model is used, in which the local porosity f is replaced by its non-local (neighborhood average) counterpart f̄ in the yield condition. In all non-local plasticity models, dimensional consistency requires the introduction of one or more “material lengths” ℓi, which multiply the additional terms with the higher-order spatial derivatives introduced by the non-local models. The boundary layers associated with the singular perturbation problem when the material lengths ℓi→0 are examined. A linear stability analysis is carried out to study the stability of homogeneous problems under small harmonic perturbations; it is shown that the stability range of the non-local models is always larger than that of the corresponding local models.

Local discontinuous Galerkin method for a nonlocal viscous water wave model

In this article, the fully discrete local discontinuous Galerkin (LDG) method is presented for numerically solving a class of nonlocal viscous water wave model, where the BDF2 with the L1 formula of the time Caputo fractional derivative is applied to deal with the time direction, and the LDG method is developed to approximate the space direction. The stability of the fully discrete LDG scheme is proven, and the a priori error result with O(τ32+hk+12) in L2-norm is derived in detail. Finally, numerical examples supporting theoretical error result are provided, and the decay rates under different coefficients are also discussed.

Amplitude-phase modulated composite waves in coupled nonlocal system with self-induced PT symmetric potential.

We consider a coupled nonlocal nonlinear Schrödinger equation (nNLSE) with self-induced parity-time (PT) symmetric potential and investigate abundant amplitude-phase modulated composite waves manifesting diverse evolution patterns. It is found that the coupled nonlocal model can be decoupled into nNLSEs with self-induced PT symmetric potential under certain constraints through a general linear transformation with amplitude and phase modulation. Based on the exact solutions of the nNLSEs with self-induced PT potential, we study various composite waves superposed by bright and/or dark soliton solutions, rogue waves, bright/dark soliton and periodic soliton, and present the abundant evolution patterns under amplitude-phase modulation. The results here only demonstrate the characteristics of limited superposed composite waves. In fact, there exist infinite possible evolution patterns of composite waves due to the arbitrary amplitude-phase modulation in coupled nonlocal nonlinear system with self-induced PT symmetric potential.

A nonlocal theory of heat transfer and micro-phase separation of nanostructured copolymers

Directed self-assembly of block copolymer (BCP) has shown promise as a nano-patterning technology for various applications such as microelectronics, semiconductor manufacturing and biotechnology. In this paper, we present a thermodynamically consistent nonlocal model of phase transformation and heat transfer for BCP via the continuum theory of mixtures. Our model is developed based on the mass and energy balances of each phase and a set of microforce balances to describe the dynamics of microphase separation during self-assembly and the size and memory effects of heat conduction in BCP thin films. We derive constitutive relations for coupling the phase field and temperature through upscaling the self-consistent field theory of polymers, and present the connections between the derived nonlocal heat conduction model and the phonon Boltzmann transport equation. We provide a finite element solution of the proposed model that includes unconditionally stable time-stepping schemes, and we perform various numerical experiments that showcase different features of the model. Furthermore, we conduct three-dimensional numerical experiments of laser zone annealing process to qualitatively investigate the effect of laser velocity on the self-assembly behavior of BCP thin films. Our results show that the proposed model can simulate essential features of thermally induced phase responses of BCPs, consistent with experimental observations and high-fidelity simulations.

An implicit stress update algorithm for the plastic nonlocal damage model of concrete

The numerical implementation of advanced plastic damage models has been challenging due to the diverse computational difficulties caused by material nonlinearity. In this study, firstly, a plastic damage model is developed by a nonsmoothed Mohr–Coulomb yield criterion. The nonorthogonal flow rule without plastic potential function is employed to describe the dilatancy behavior of concrete. The stiffness degradation is considered through two tensile and compressive damage variables. Subsequently, an implicit stress update algorithm incorporating several advanced numerical methods is proposed to tackle difficulties arising in the numerical calculation of the plastic damage model. In the model implementation, the Mohr–Coulomb yield function with corners is replaced by a single smooth expression to ensure the stability of the numerical calculation. A line search method is used to solve nonlinear systems of integral stress equations to obtain better convergence than Newton method. The complex step derivative approximation is used to evaluate the consistent tangent operator to provide quadratic convergence speed of the global equilibrium iterations, thus avoiding tedious analytical derivative operations. The integral nonlocal method is also used to regularize the finite element solution. At the material point scale, the ability of the proposed model to describe the complex mechanical behavior of concrete is verified by utilizing nine sets of monotonic and cyclic loading and unloading tests. Finally, the algorithm and model’s performance is well tested by the failure analysis of five numerical examples. The code of this work will be freely available at https://github.com/zhouxin615/PD_model.

A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity

The purpose of this paper is to investigate a space-time Sinc-collocation method for solving the fourth-order nonlocal heat model arising in viscoelasticity, which is a class of partial integro-differential equations (PIDEs) whose solution typically exhibits the weak singularities at the initial time. Most of existing approximate methods for solving such kind PIDEs are unbalanced, since most work have used a low order scheme for integrating the temporal variable and a high order numerical framework for discretization of space variables. The current paper contributes to a space-time spectral-order Sinc-collocation method which is balanced in both time and space variables. In order to deal with the initial singularity of solution in the time direction, the Sinc-collocation method with single exponential (SE) transformation is used for the first time, which is an effective method to deal with the singularity of the equation. Also, we fully analyse the error bounds of the method which show the exponential convergence rate in space and time. Meanwhile, we consider several test problems for examining the suggested scheme. The experiment results highlight a good precision of the new Sinc-colloction method and the correctness of the theoretical prediction for this kind nonlocal heat problems.

Dynamics of nonlocal and local SIR diffusive epidemic model with free boundaries

This paper is concerned with nonlocal and local diffusive SIR epidemic model with free boundaries including convolution, which is natural extension of reaction diffusion systems with free boundary problems and local diffusions. The existence of unique global solution for this model is considered. Dichotomy of the spreading and vanishing is established. A spreading barrier line is found to determine whether the spreading of disease will fail finally. The spreading of disease will fail when it cannot spread across the spreading barrier line l∗, while it will be successful when it transcends over this barrier line. The results show that if the basic reproduction number R0<1, the spreading of disease will fail eventually, and if R0>1+d2μ2+α, the spreading of disease will get success finally. We also find that the spreading coefficients play important role in the spreading achievement. When 1+βθμ1<R0<1+d2μ2+α, the spreading coefficient decides whether the spreading of disease will be successful. It is shown that the spreading will be successful when the spreading coefficient is relatively big, while the spreading will fail if the spreading coefficient is relatively small.

Nonlocal boundaries and paradoxes in thermoelastic vibrations of functionally graded Non-Uniform cantilever nanobeams and annular nanoplates

The paradoxical behaviour of free vibrations of cantilever nanostructures using Eringen's nonlocal elasticity model is always a debatable issue for researchers in the past few years. According to this, the frequency of cantilever nanobeams increases with the increase in the value of the nonlocal parameter while it decreases for other boundary conditions. In thermoelastic vibrations, the frequency of nanobeams increases and decreases with the increase in temperature at the surfaces for clamped-free and other boundary conditions, respectively. Therefore, investigation of these structures is of critical importance. In view of this, the authors have proposed the modified boundary conditions for the well-established differential model of Eringen’s nonlocal elasticity theory to deal with the free edges of the structural components. Using these conditions, the vibration analysis of nonuniform functionally graded nanobeams and axisymmetric annular nanoplates is presented for clamped-free boundary conditions on the basis of first-order shear deformation theory and physical neutral plane. The thickness of these structural elements (nanobeams and annular nanoplates) is assumed to be varying linearly and parabolically in the axial direction. The temperature-dependent mechanical properties of the material are obtained by the power law. The nonlocal governing equations and modified boundary conditions for the structural elements are discretized using the generalized differential quadrature method and solved by MATLAB to obtain the fundamental frequency. For the present modifications, the behaviour of frequency parameter for different values of nonlocal parameter, volume fraction index, temperature difference, thickness parameter, and taper parameter is discussed.

Assessing requirements for modeling radiation in diffusion flames using an analytical, non-local model

An analytical approach for the calculation of radiative self-absorption in flames is presented, with the objective of providing an effective and computationally inexpensive tool for selecting radiation modeling methods for combustion applications. This one-dimensional non-local model approximates the flame structure as an infinitely long cylinder where all properties vary only along the radial direction. Flamelet solutions are then mapped to the radial direction and the radiative source terms, in particular the absorption, are computed analytically. The model is validated for a laminar jet flame and a turbulent pool fire, with results for the net radiative loss and the flame self-absorption showing good agreement to those of multidimensional, coupled simulations. To illustrate its applicability, the model is employed to study the characteristics of radiation in high-pressure combustion, in fire suppression and in hydrogen combustion. For the pressurized case, increased significance of radiation self-absorption is observed compared to an atmospheric flame, and accounting for the non-gray nature of radiation is found to be critical. In the fire suppression case, comparisons of the S-curves show that the choice of radiation model modifies the lower extinction limit, which becomes more important with reduced oxygen content. Lastly, for the hydrogen flame, the Planck-mean absorption coefficient is found to be more uniformly distributed in the mixture fraction space, and the optical thickness is smaller than a hydrocarbon flame at comparable conditions, indicating that radiation modeling requirements for that flame are likely less stringent. In general, the findings regarding radiation characteristics made on the basis of the non-local model are consistent with observations reported in the literature, but without the need for complex coupled combustion-radiation calculations. Therefore, the model provides a viable approach for selecting appropriate radiation models in combustion simulations.

Traveling wave solutions in a nonlocal dispersal SIR epidemic model with nonlocal time-delay and general nonlinear incidences

This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects. First, the minimal wave speed [Formula: see text] and the basic reproduction number [Formula: see text] are defined, which determine the existence of traveling wave solutions. Second, with the help of the upper and lower solutions, Schauder’s fixed point theorem, and limiting techniques, the traveling waves satisfying some asymptotic boundary conditions are discussed. Specifically, when [Formula: see text], for every speed [Formula: see text] there exists a traveling wave solution satisfying the boundary conditions, and there is no such traveling wave solution for any [Formula: see text] when [Formula: see text] or [Formula: see text] when [Formula: see text]. Finally, we analyze the effects of nonlocal time delay on the minimum wave speed.

Reverse-time type nonlocal Sasa–Satsuma equation and its soliton solutions

In this work, we study the Riemann–Hilbert problem and the soliton solutions for a nonlocal Sasa–Satsuma equation with reverse-time type, which is deduced from a reduction of the coupled Sasa–Satsuma system. Since the coupled Sasa–Satsuma system can describe the dynamic behaviors of two ultrashort pulse envelopes in birefringent fiber, our equation presented here has great physical applications. The classification of soliton solutions is studied in this nonlocal model by considering an inverse scattering transform to the Riemann–Hilbert problem. Simultaneously, we find that the symmetry relations of discrete data in the special nonlocal model are very complicated. Especially, the eigenvectors in the scattering data are determined by the number and location of eigenvalues. Furthermore, multi-soliton solutions are not a simple nonlinear superposition of multiple single-solitons. They exhibit some novel dynamics of solitons, including meandering and sudden position shifts. Also, they have the bound state of multi-soliton entanglement and its interaction with solitons.

HDR video synthesis by a nonlocal regularization variational model

High dynamic range (HDR) video synthesis is a very challenging task. Consecutive frames are acquired with alternate expositions, generally two or three different exposure times. Classical methods aim at registering neighboring frames and fuse them using image HDR techniques. However, the registration often fails to obtain accurate results and the fusion produces ghosting artifacts. Deep learning techniques have recently appeared imitating the structure of existing classical methods. The neural network is intended to estimate the registration function and choose the fusion weights. In this paper, we propose a new method for HDR video synthesis using a variational model. The proposed model uses a nonlocal regularization term to combine pixel information from neighboring frames. The obtained results are competitive with state-of-the-art. Moreover, the proposed method gives a more reliable and understandable solution than deep-learning based ones.

A gradient-based non-local GTN model: Explicit finite element simulation of ductile damage and fracture

Non-local models have over the years been established as an effective approach to solve the pathological mesh dependency problem observed in finite element simulation of strain-softening materials. This paper presents the formulation, implementation, and application of a gradient-based non-local Gurson–Tvergaard–Needleman (GTN) model for explicit finite element analysis. The porosity is taken as the non-local variable where the increment in porosity is averaged over the volume using an implicit gradient model. The gradient model is implemented in Abaqus/Explicit by utilising the coupled thermal–mechanical solver, which proves to be both a simple and computationally efficient approach. Due to the use of an explicit integration scheme, a transient term is introduced to the partial differential equation of the gradient formulation. The non-local GTN model is compared to the local counterpart for increasing mesh refinements using a plane strain shear band specimen, a plane strain tension specimen, and a plane strain compact tension specimen. The proposed approach can remedy the pathological mesh dependency problem. For the plane strain tension specimen, it is shown that the non-local GTN model will preserve the fracture mode (slant versus cup–cup fracture mode) when the mesh is refined. The non-local GTN model is also able to predict the same fracture mode as observed in ductile tearing experiments. However, non-local averaging can over-smooth the fields and exclude the slant fracture mode from occurring if the material length Lc is too large.

Theoretical thermal damping vibration analysis of functionally graded viscoelastic Timoshenko microbeam with integral nonlocal strain gradient model

In this work, nonlocal strain gradient integral model is applied to investigate the free damping vibration analysis of functionally graded (FG) viscoelastic Timoshenko microbeam with immovable boundary conditions in thermal environment. The microbeam is assumed to be viscoelastic and modeled by Kelvin–Voigt model. The differential governing equations and corresponding boundary conditions are derived with the Hamilton’s principle. Combining nonlocal strain gradient integral model and Kelvin–Voigt viscoelastic model, the integral constitutive equations of nonlocal stress with thermal effect are derived and then converted into differential form plus constitutive constraints. The size-dependent axial force due to thermal expansion is explicitly derived through the immovable boundary conditions. The bending deflection and moment, cross-sectional rotation as well as shear force are explicitly derived with Laplace transformation for linear thermo-elastic vibration. Taking into account the boundary conditions as well as constitutive constraints, one gets a nonlinear equation with complex coefficients, from which one can determine the complex characteristic frequency. A two-step numerical method is proposed to solve the elastic vibration frequency and damping ratio. The influence of length scale parameters, viscos coefficient, FG index, thermal effect, vibration order and ratio between beam length and thickness on the vibration frequencies and damping ratio is investigated numerically for FG viscoelastic Timoshenko microbeams under different immovable boundary conditions.

Nanostructure modeling for thermal properties of zigzag carbon nanotubes

In the nanotube design application, the case of dynamic damage is key problem attracted many engineering researchers. The thermal effect on the axial buckling of zigzag double walled carbon nanotubes (DWCNTs) embedded in an elastic medium is investigated in this article based on the nonlocal Timoshenko beam model. The effects of the van der Waals (vdW) forces between inner and outer tubes and the surrounding elastic medium are taken into account based on a Winkler model. Moreover, a modified molecular mechanics model is developed to study the effect of thermal environment on mechanical properties of zigzag double carbon nanotubes, in which the covalent bonds are traded as a truss. It is found that the axial buckling properties of zigzag DWCNTs increase with the increase of the temperature at room or low temperature, while the axial buckling properties of zigzag DWCNTs decrease with the increase of the temperature at high temperature. Moreover, the buckling strain considering the thermal effect is larger than the buckling strain ignoring the effect of temperature change, and increases with the increase of temperature change, which is contrary to the case of high temperature.

A non-local GTN model with two length scales – Application to ductile failure in a wide range of stress triaxiality

A non-local extension of the shear-modified Gurson model for porous plasticity is proposed, which is capable of preventing spurious strain localization and suitable for predicting crack initiation and growth under general loading conditions. In the extended model, the progression of shear failure is separated from flat dimple rupture by the assumption that these failure mechanisms are characterized by a deviatoric and a dilatational material length parameter, respectively. The two length parameters are introduced through a delocalization process based on the non-local integral approach evaluated on the current configuration. A comprehensive numerical investigation is conducted to disclose the influence of these length parameters. For this purpose, five different specimen geometries are considered covering a wide range of stress triaxiality including pure shear. It is observed that separation of the length parameters is essential for predicting crack branching driven by shear localization, e.g., the flat dimple-to-shear failure transition of a crack that approaches and penetrates a free surface.