Nonlocal Behavior Research Articles

Fractional-Order Shell Theory: Formulation and Application to the Analysis of Nonlocal Cylindrical Panels

AbstractWe present a theoretical and computational framework based on fractional calculus for the analysis of the nonlocal static response of cylindrical shell panels. The differ-integral nature of fractional derivatives allows an efficient and accurate methodology to account for the effect of long-range (nonlocal) interactions in curved structures. More specifically, the use of frame-invariant fractional-order kinematic relations enables a physically, mathematically, and thermodynamically consistent formulation to model the nonlocal elastic interactions. To evaluate the response of these nonlocal shells under practical scenarios involving generalized loads and boundary conditions, the fractional-finite element method (f-FEM) is extended to incorporate shell elements based on the first-order shear-deformable displacement theory. Finally, numerical studies are performed exploring both the linear and the geometrically nonlinear static response of nonlocal cylindrical shell panels. This study is intended to provide a general foundation to investigate the nonlocal behavior of curved structures by means of fractional-order models.

The analysis and computation on nonlocal thermoelastic problems of blend composites via enriched second-order multi-scale computational method

This paper proposes an innovative enriched second-order multi-scale (SOMS) computational method to simulate and analyze the nonlocal thermoelastic problems of blend composites with stress and heat flux gradient behaviors. The multiple periodical heterogeneities and periodic configurations of investigated blend composites in different substructures result in a huge computational cost for direct numerical simulations. The significant characteristics of this study are as follows. (1) The nonlocal properties of blend composites in constitutive equations are converted into the source terms of thermoelastic balance equations. The novel macro-micro coupled SOMS computational model for these transformed nonlocal multi-scale problems is derived on the basis of multi-scale asymptotic analysis. The nonlocal thermoelastic behaviors of blend composites can be merely uncovered in the enriched SOMS solutions. (2) The error analysis in the pointwise sense is presented to elucidate the importance and necessity of establishing the enriched SOMS solutions. Furthermore, an explicit error estimate for the SOMS approximate solutions is obtained in the integral sense for these nonlocal multi-scale problems. (3) A multi-scale numerical algorithm is presented to effectively simulate nonlocal thermoelastic problems of blend composites based on finite element method (FEM). Finally, the capability of the proposed enriched SOMS computational method is demonstrated by typical two-dimensional (2D) and three-dimensional (3D) blend composites, presenting not only the excellent numerical accuracy but also the less computational cost. This work proposes a unified multi-scale computational framework for enabling nonlocal thermoelastic behavior analysis of blend composites.

Multiscale nonlocal elasticity: A distributed order fractional formulation

This study presents a generalized multiscale nonlocal elasticity theory that leverages distributed order fractional calculus to accurately capture coexisting multiscale and nonlocal effects within a macroscopic continuum. The nonlocal multiscale behavior is captured via distributed order fractional constitutive relations derived from a nonlocal thermodynamic formulation. The governing equations of the inhomogeneous continuum are obtained via the Hamilton principle. As a generalization of the constant order fractional continuum theory, the distributed order theory can model complex media characterized by inhomogeneous nonlocality and multiscale effects. In order to understand the correspondence between microscopic effects and the properties of the continuum, an equivalent mass–spring lattice model is also developed by direct discretization of the distributed order elastic continuum. Detailed theoretical arguments are provided to show the equivalence between the discrete and the continuum distributed order models in terms of internal nonlocal forces, potential energy distribution, and boundary conditions. These theoretical arguments facilitate the physical interpretation of the role played by the distributed order framework within nonlocal elasticity theories. They also highlight the outstanding potential and opportunities offered by this methodology to account for multiscale nonlocal effects. The capabilities of the methodology are also illustrated via a numerical study that highlights the excellent agreement between the displacement profiles and the total potential energy predicted by the two models under various order distributions. Remarkably, multiscale effects such as displacement distortion, material softening, and energy concentration are well captured at the continuum level by the distributed order theory.

Self-testing quantum states via nonmaximal violation in Hardy's test of nonlocality

Self-testing protocols enable the certification of quantum devices without demanding full knowledge about their inner workings. A typical approach in designing such protocols is based on observing nonlocal correlations which exhibit maximum violation in a Bell test. We show that in the Bell experiment known as Hardy's test of nonlocality, not only does the maximally nonlocal correlation self-test a quantum state, rather a nonmaximal nonlocal behavior can serve the same purpose. We, in fact, completely characterize all such behaviors leading to a self-test of every pure two-qubit entangled state except for the maximally entangled ones. Apart from presenting an alternative self-testing protocol, our method provides a powerful tool towards characterizing the complex boundary of the set of quantum correlations.

Step forward in epidemiological modeling: Introducing the rate indicator function to capture waves

In the last centuries, mathematical models have been used to depict the dynamic evolution of infectious diseases’ spread. The aim is to determine the total numbers of infected, recovered, and susceptible individuals; however, they only represent accumulated values due to the complexities of collected data. Having the graphical representations of these classes as accumulated values, one may not directly be able to predict waves to determine a day-to-day number of newly infected cases as some additional calculations are required. Collected data from real-world situations are represented as day-to-day new infections; they help determine in addition numbers of waves. However, current existing mathematical models cannot be used for wave prediction. While knowing the predicted numbers of waves, policymakers can take adequate measures to control the situation. To solve this problem, we questioned the fact that the rates of infection and recovery are constant and suggested an indicator rate function obtained using experimental data. To see the effect of this function, we considered a simple SIR model, which was modified by introducing rate indicator functions for infected and recovered classes. To include nonlocal behaviors in the mathematical model, different types of differential operators, including classical and fractional derivatives, were used. The models were solved numerically using well-known numerical schemes. The numerical solutions were plotted for different theoretical parameters; the results depict real-world behaviors. To test the efficiency of this new approach, we collected data from South Africa and compared them with our model. Not only our model could predict waves, but also it fits experimental data very accurately. This new approach will open new doors of investigation toward a revolution in epidemiological modeling.

Quantum violation of local causality in an urban network using hybrid photonic technologies

Quantum networks play a crucial role in distributed quantum information processing, enabling the establishment of entanglement and quantum communication among distant nodes. Fundamentally, networks with independent sources allow for new forms of nonlocality, beyond the paradigmatic Bell’s theorem. Here we implement the simplest of such networks—the bilocality scenario—in an urban network connecting different buildings with a fully scalable and hybrid approach. Two independent sources using different technologies—a quantum dot and a nonlinear crystal—are used to share a photonic entangled state among three nodes connected through a 270 m free-space channel and fiber links. By violating a suitable nonlinear Bell inequality, we demonstrate the nonlocal behavior of the correlations among the nodes of the network. Our results pave the way towards the realization of more complex networks and the implementation of quantum communication protocols in an urban environment, leveraging the capabilities of hybrid photonic technologies.

Computationally efficient higher-order three-scale method for nonlocal gradient elasticity problems of heterogeneous structures with multiple spatial scales

In this work, an innovative higher-order three-scale (HOTS) computational approach is developed, which can handle and simulate the nonlocal strain-stress gradient elasticity model of heterogeneous structures with multiple spatial scales. The significant characteristics of this study are: (i) the raw fourth-order nonlocal gradient elasticity equations with three-scale structural characteristics are decoupled as the new multi-scale second-order equations for reducing the continuity requirements for multi-scale computational method. (ii) a new micro-meso-macro coupled HOTS computational model for accurately analyzing the nonlocal gradient elasticity behaviors of heterogeneous structures with three-scale structural configurations is rigorously devised by virtue of multi-scale asymptotic analysis. (iii) the small-scale approximate performance of HOTS solutions is derived in the pointwise sense, that illustrates the crucial indispensability of establishing the HOTS solutions for capturing the micro-scale, meso-scale and macro-scale gradient elasticity behaviors of the heterogeneous structures accurately. (iv) the efficient three-scale numerical algorithm on the basis of finite element method (FEM) is presented at length. Finally, several numerical experiments are carried out to validate the applicability of the established HOTS computational methodology, presenting not only the excellent numerical accuracy, but also the advanced computational efficiency. This work provides a unified three-scale computational framework which enables the accurate analysis and efficient simulation of nonlocal strain-stress gradient elasticity problems of heterogeneous structures with multiple spatial scales, and provides a potential application for practical engineering computation.

Investigation of Nonlinear Thermo-Elastic Behavior of Fluid Conveying Piezoelectric Microtube Reinforced by Functionally Distributed Carbon Nanotubes on Viscoelastic-Hetenyi Foundation

In this paper, nonlinear and nonlocal thermo-elastic behavior of a microtube reinforced by Functionally Distributed Carbon Nanotubes, with internal and external piezoelectric layers, in the presence of nonlinear viscoelastic-Hetenyi foundation, and axial fluid flow inside the microtube is studied. Nonlinear partial differential equations governing the system are derived using Reddy’s third-order shear deformations theory along with the Von-Karman theory including the effect of fluid viscosity. Then, the equations are converted to time-dependent ordinary nonlinear equations using the Galerkin method. Afterward, the governing equations of the microtube’s lateral displacements are solved using the multiple scales method. The analysis of the piezoelectric’s parametric resonance is performed by obtaining trivial and nontrivial stationary solutions and plotting characteristic curves of the frequency response and voltage response. At the end, the effect of different parameters including the flow velocity, excitation voltage, parameters of the foundation, viscosity parameter, thermal loading and nanotubes’ volume fraction index on the nonlinear behavior of the system, under parametric resonance condition, is investigated.

Thermal control of nucleation and propagation transition stresses in discrete lattices with non-local interactions and non-convex energy

Non-local and non-convex energies represent fundamental interacting effects regulating the complex behavior of many systems in biophysics and materials science. We study one-dimensional, prototypical schemes able to represent the behavior of several biomacromolecules and the phase transformation phenomena in solid mechanics. To elucidate the effects of thermal fluctuations on the non-convex non-local behavior of such systems, we consider three models of different complexity relying on thermodynamics and statistical mechanics: (i) an Ising-type scheme with an arbitrary temperature-dependent number of interfaces between different domains, (ii) a zipper model with a single interface between two evolving domains, and (iii) an approximation based on the stationary phase method. In all three cases, we study the system under both isometric condition (prescribed extension, matching with the Helmholtz ensemble of the statistical mechanics) and isotensional condition (applied force, matching with the Gibbs ensemble). Interestingly, in the Helmholtz ensemble the analysis shows the possibility of interpreting the experimentally observed thermal effects with the theoretical force–extension relation characterized by a temperature-dependent force plateau (Maxwell stress) and a force peak (nucleation stress). We obtain explicit relations for the configurational properties of the system as well (expected values of the phase fractions and number of interfaces). Moreover, we are able to prove the equivalence of the two thermodynamic ensembles in the thermodynamic limit. We finally discuss the comparison with data from the literature showing the efficiency of the proposed model in describing known experimental effects.

Non-Markovian random walks characterize network robustness to nonlocal cascades

Localized perturbations in a real-world network have the potential to trigger cascade failures at the whole system level, hindering its operations and functions. Standard approaches analytically tackling this problem are mostly based either on static descriptions, such as percolation, or on models where the failure evolves through first-neighbor connections, crucially failing to capture the nonlocal behavior typical of real cascades. We introduce a dynamical model that maps the failure propagation across the network to a self-avoiding random walk that, at each step, has a probability to perform nonlocal jumps toward operational systems' units. Despite the inherent non-Markovian nature of the process, we are able to characterize the critical behavior of the system out of equilibrium, as well as the stopping time distribution of the cascades. Our numerical experiments on synthetic and empirical biological and transportation networks are in excellent agreement with theoretical expectation, demonstrating the ability of our framework to quantify the vulnerability to nonlocal cascade failures of complex systems with interconnected structure.

A numerical approach to the fractional Laplacian operator with applications to quasi-geostrophic flows

The quasi-geostrophic models have been very successful for the study of oceanic and atmospheric dynamics in the mid-to-high latitude region of the earth where the Coriolis effect is significant. The governing equation of a quasi-geostrophic model is a transport equation with a fractional dissipation term. Although fractional operators have become a topic of great interest in the research community, the numerical discretization of such operators is very challenging due to their non-local behavior. In this work, we propose the numerical approximations of the bounded fractional Laplacian on a finite element discretization. In particular, we rely on the Riesz method and use a semi-analytical technique to approximate all the integrals involving the interaction between the inner local and the outer region. We test the implemented algorithm with numerical benchmarks, and we apply it to quasi-geostrophic flows. All the presented simulations are computationally expensive, due to the non-local behavior of the fractional Laplacian. For this reason, a parallel implementation of the numerical code has been developed.

Entanglement and nonlocality of four-qubit connected hypergraph states

We study entanglement and nonlocality of connected four-qubit hypergraph states. One obtains the Stochastic Local Operations and Classical Communication (SLOCC) classification from the known local unitary (LU)-orbits. We then consider Mermin’s polynomials and show that the nonlocal behavior of hypergraph states can be detected by a suitable choice of Mermin’s inequality. Finally, we implement some of the corresponding inequalities on the IBM Quantum Experience and report the detection of the nonlocality of a three-qubit hypergraph states.

Physics-based compact modeling of electro-thermal memristors: Negative differential resistance, local activity, and non-local dynamical bifurcations

Leon Chua's Local Activity theory quantitatively relates the compact model of an isolated nonlinear circuit element, such as a memristor, to its potential for desired dynamical behaviors when externally coupled to passive elements in a circuit. However, the theory's use has often been limited to potentially unphysical toy models and analyses of small-signal linear circuits containing pseudo-elements (resistors, capacitors, and inductors), which provide little insight into required physical, material, and device properties. Furthermore, the Local Activity concept relies on a local analysis and must be complemented by examining dynamical behavior far away from the steady-states of a circuit. In this work, we review and study a class of generic and extended one-dimensional electro-thermal memristors (i.e., temperature is the sole state variable), re-framing the analysis in terms of physically motivated definitions and visualizations to derive intuitive compact models and simulate their dynamical behavior in terms of experimentally measurable properties, such as electrical and thermal conductance and capacitance and their derivatives with respect to voltage and temperature. Within this unified framework, we connect steady-state phenomena, such as negative differential resistance, and dynamical behaviors, such as instability, oscillations, and bifurcations, through a set of dimensionless nonlinearity parameters. In particular, we reveal that the reactance associated with electro-thermal memristors is the result of a phase shift between oscillating current and voltage induced by the dynamical delay and coupling between the electrical and thermal variables. We thus, demonstrate both the utility and limitations of local analyses to understand non-local dynamical behavior. Critically for future experimentation, the analyses show that external coupling of a memristor to impedances within modern sourcing and measurement instruments can dominate the response of the total circuit, making it impossible to characterize the response of an uncoupled circuit element for which a compact model is desired. However, these effects can be minimized by proper understanding of the Local Activity theory to design and utilize purpose-built instruments.

On integral and differential formulations in nonlocal elasticity

The paper is concerned with comparative analysis of differential and integral formulations for boundary value problems in nonlocal elasticity. For the sake of simplicity, the focus is on an antiplane problem for a half-space with prescribed shear stress along the surface. In addition, 1D exponential kernel depending on the vertical coordinate is considered.First, a surface loading in the form of a travelling harmonic wave is studied. This provides a counter-example, revealing that within the framework of Eringen’s theory the solution to the differential model does not satisfy the equation of motion in nonlocal stresses underlying the related integral formulation.A more general differential setup, starting from singularly perturbed equations expressing the local stresses through the nonlocal ones, is also investigated. It is emphasised that the transformation of the original integral formulation to the differential one in question is only possible provided that two additional conditions on nonlocal stresses hold on the surface. As a result, the formulated problem subject to three boundary conditions appears to be ill-posed, in line with earlier observations for equilibrium of a nonlocal cantilever beam.Next, the asymptotic solution of the singularly perturbed problem, subject to a prescribed stress on the boundary, together with only one of the aforementioned extra conditions, is obtained at a small internal size. Such simplification may be justified when only one of the stresses demonstrates nonlocal behaviour; a similar assumption has been recently made within the so-called dilatational gradient elasticity. Three-term expansion is obtained, leading to a boundary value problem in local stresses over interior domain. The associated differential equations are identical to those proposed by Eringen, however, the derived effective boundary condition incorporates the effect of a nonlocal boundary layer which has previously been ignored. Moreover, the calculated nonlocal correction to the classical antiplane problem for an elastic half-space, coming from the boundary conditions is by order of magnitude greater than that appearing in the equations of motion. Finally, it is shown that the proposed effective condition supports an antiplane surface wave.

A continuous–discontinuous localizing gradient damage framework for failure analysis of quasi-brittle materials

This article presents a continuous–discontinuous approach that provides a discontinuous character to the quasi-brittle fracture process, modeled in a continuous setting through a micromorphic stress-based localizing gradient damage framework. The discontinuous framework is developed by enhancing the problem fields using discontinuous interpolations in the critically damaged regions exploiting local partition of unity via the eXtended Finite Element Method (XFEM). The proposed approach incorporates an improved spatial nonlocal diffusive behavior in the continuous bulk through an additional microforce balance equation to ensure an improved prediction of the initial phase of the loading process. Level sets are computed to track the discontinuity within the finite element mesh, predicted via evolving micromorphic equivalent strain. Here, the proposed continuous–discontinuous approach combines the advantages of the gradient damage method and the XFEM while exhibiting improved convergence using low-order finite elements during numerical simulations. The enhanced numerical kinematics of the proposed approach successfully eliminates damage spreading during the final failure stages and provides an accurate description of a true discontinuity without using cohesive zone modeling. Various representative structural examples of benchmark tests involving varying loading conditions are investigated to demonstrate the accuracy and robustness of the proposed approach to provide qualitative and quantitative accurate numerical predictions.

Truncation of fractional derivative for online system identification

Fractional derivatives are non local operators that has compacity property in terms of parameter number for modeling diffusive phenomenon with very few parameters. One of its main properties is its non-local behavior, as it can be exploited to model long-memory phenomena such as heat transfers. However, such non-locality implies a constant knowledge of the full past of the function to be differentiated. In the context of real-time system identification, this may limit the experiences as calculations become slower as time progresses. This study deals with the relationship between frequency content of a signal and its truncation error in order to obtain real-time exploitable algorithms.

Hopf bifurcation analysis in a delayed diffusive predator-prey system with nonlocal competition and schooling behavior

<abstract><p>We consider a delayed diffusive predator-prey system with nonlocal competition in prey and schooling behavior in predator. We mainly study the local stability and Hopf bifurcation at the positive equilibrium by using time delay as the parameter. We also analyze the property of Hopf bifurcation by center manifold theorem and normal form method. Through the numerical simulation, we obtain that time delay can affect the stability of the positive equilibrium and induce spatial inhomogeneous periodic oscillations of prey and predator's population densities. In addition, we observe that the increase of space area will not be conducive to the stability of the positive equilibrium $ (u_*, v_*) $, and may induce the inhomogeneous periodic oscillations of prey and predator's population densities under some values of the parameters.</p></abstract>

Numerical analysis of some partial differential equations with fractal-fractional derivative

<abstract> <p>In this study, we expanded the partial differential equation framework to which fractal-fractional differentiation can be applied. For this, we employed the generalized Mittag-Leffler function, and the fractal-fractional derivatives based on the power-law kernel. A general partial differential equation with the fractal-fractional derivative, the power law kernel and the generalized Mittag-Leffler function was thoroughly examined. There is almost no numerical scheme for solving partial differential equations with fractal-fractional derivatives, as less investigation has been done in this direction in the last decades. In this work, therefore, we shall attempt to provide a numerical method that might be used to solve these equations in each circumstance. The heat equation was taken into consideration for the application and numerically solved using a few simulations for various values of fractional and fractal orders. It is observed that, when the fractal order is 1, one obtains fractional partial differential equations which have been known to replicate nonlocal behaviors. Meanwhile, if the fractional order is 1, one obtains fractal-partial differential equations. Thus, when the fractional order and fractal dimension are different from zero, nonlocal processes with similar features are developed.</p> </abstract>

Some new generalizations for exponentially (s, m)-preinvex functions considering generalized fractional integral operators

The generalized fractional integral has been one of the most useful operators for modelling non-local behaviors by fractional differential equations. It is considered, for several integral inequalities by introducing the concept of exponentially (s, m)-preinvexity. These variants derived via an extended Mittag-Leffler function based on boundedness, continuity and Hermite-Hadamard type inequalities. The consequences associated with fractional integral operators are more general and also present the results for convexity theory. Moreover, we point out that the variants are useful in solving the problems of science, engineering and technology where the Mittag-Leffler function occurs naturally.

Time fractional telegraph equation and its solution by Laplace transform method

In this paper, one-dimensional time fractional telegraph equation is deduced using fractional Ohm’s law in transmission lines. The nonlocal behavior, arising from the time fractality, is discussed via fractional calculus. Here, the time fractional differential equation related to electrical GLRC circuit is proposed in terms of the Liouville–Caputo fractional derivative. To keep the dimensionality of the physical parameters [Formula: see text] the new parameter [Formula: see text] is introduced. This parameter ([Formula: see text]) characterizes the existence of fractional structure in the system. The analytical solution of the proposed equation is obtained by using Laplace transform method in terms of Mittag-Leffler function which describes physical system with memory. The order of the time derivative being considered is [Formula: see text]. The result of an example shows that the behavior of voltage depends on the value of the fractional order of the proposed equation.