Nonlocal Variables Research Articles

Computational integration and experimental validation of a two-length scale model for accurate prediction of steels’ plastic behavior during phase transformation

Transformation plasticity, essential during solid–solid phase transitions, significantly impacts industrial processes like welding and quenching. Accurately simulating these procedures necessitates understanding thermal, metallurgical, and mechanical effects. Leblond et al.’s model offers a foundation, but refinement for mixed isotropic/kinematic hardening is crucial. We enhance this model by introducing characteristic length scales through nonlocal variables, illuminating plastic deformation mechanisms in both phases. Our work includes numerical implementation within a finite element analysis framework and practical applications to phase transformation scenarios involving A.508 cl. 3 and A533 low-alloy steels. Results affirm model robustness and efficiency in predicting phase transformation phenomena, benefiting industrial applications.

Amplification factor transport equation modelling of Mack mode disturbances in hypersonic boundary layers

In the hypersonic boundary layer, the streamwise instability modes are mainly the Mack first mode and the Mack second mode. After the Mach number exceeds 4.0, the Mack second mode becomes the dominant instability mode for boundary layer transition. Currently, there is no complete analytical method to establish an amplification factor transport equation for the second mode. In order to improve the application of the amplification factor transport (AFT) model in hypersonic boundary layer transition prediction, this paper employs boundary layer similarity solutions to conduct stability analysis on the second mode and introduces a function to calculate the growth rate of disturbances. Additionally, a transport equation for the amplification factor of the second mode is formulated. For the non-local variables appearing in the equation, a local calculation method is provided. Combined with the intermittency factor transport equation, a transition prediction model for the second mode is formed. Hypersonic flat plate, wedge, and flared wedge are selected for model validation. The computed results show good agreement with standard stability analysis or experimental results, demonstrating the rationality of the transition model and its high prediction accuracy and reliability.

An efficient unconditional energy stable scheme for the simulation of droplet formation

We have developed an efficient and unconditionally energy-stable method for simulating droplet formation dynamics. Our approach involves a novel time-marching scheme based on the scalar auxiliary variable technique, specifically designed for solving the Cahn-Hilliard-Navier-Stokes phase field model with variable density and viscosity. We have successfully applied this method to simulate droplet formation in scenarios where a Newtonian fluid is injected through a vertical tube into another immiscible Newtonian fluid. To tackle the challenges posed by nonhomogeneous Dirichlet boundary conditions at the tube entrance, we have introduced additional nonlocal auxiliary variables and associated ordinary differential equations. These additions effectively eliminate the influence of boundary terms. Moreover, we have incorporated stabilization terms into the scheme to enhance its numerical effectiveness. Notably, our resulting scheme is fully decoupled, requiring the solution of only linear systems at each time step. We have also demonstrated the energy decaying property of the scheme, with suitable modifications. To assess the accuracy and stability of our algorithm, we have conducted extensive numerical simulations. Additionally, we have examined the dynamics of droplet formation and explored the impact of dimensionless parameters on the process. Overall, our work presents a refined method for simulating droplet formation dynamics, offering improved efficiency, energy stability, and accuracy.

The Soliton Solutions for Nonlocal Multi-Component Higher-Order Gerdjikov–Ivanov Equation via Riemann–Hilbert Problem

The Lax pairs of the higher-order Gerdjikov–Ivanov (HOGI) equation are extended to the multi-component formula. Then, we first derive four different types of nonlocal group reductions to this new system. To construct the solution of these four nonlocal equations, we utilize the Riemann–Hilbert method. Compared to the local HOGI equation, the solutions of nonlocal equations not only depend on the local spatial and time variables, but also the nonlocal variables. To exhibit the dynamic behavior, we consider the reverse-spacetime multi-component HOGI equation and its Riemann–Hilbert problem. When the Riemann–Hilbert problem is regular, the integral form solution can be given. Conversely, the exact solutions can be obtained explicitly. Finally, as concrete examples, the periodic solutions of the two-component nonlocal HOGI equation are given, which is different from the local equation.

Geometric interpretation and experimental test of Leggett inequalities with nonmaximally entangled states.

Leggett inequality states that nonlocal hidden-variable models might still be incompatible with the predictions of quantum physics. However, its theoretical and experimental demonstration is only in the scenario of 2-dimensional maximally entangled systems. An open question remains as to whether the Leggett inequality can be violated by nonmaximally entangled states. Here, we answer this question both in theory and experiment. Specifically, from the point of view of geometry, we theoretically map the problem of maximizing the correlation measure in the Leggett inequality to maximizing the sum of an ellipse's diameter and semi-diameter axes, accordingly, demonstrating that the violation of the Leggett inequality requires a more robust entanglement than that of Bell's theory. Experimentally, by leveraging the controllable photonic orbital angular momentum entanglement, we demonstrate the violation of Leggett-type inequalities by more than 8.7 and 4.5 standard deviations under concurrence C = 0.95 and 0.9, respectively. Our observations indicate that, the requirement for quantum correlation should be increased to exclude a particular class of non-local hidden variable theories that abide by Leggett's model, providing insights into the boundaries of quantum correlation and the limitations imposed by non-local hidden variables.

The Method of Everything vs. Experimenter Bias of Loophole-Free Bell Experiments.

Experimenter bias compromises the integrity and advancement of science, especially when awarded as such. For example, the 2022 Nobel Prize in Physics awarded for the loophole-free experiments that tested physicist John S. Bell's inequality theorem. These experiments employed the logic of conducting local experiments to obtain local evidence that contradicted local realistic theories of nature, thereby validating quantum mechanics as a fundamental non-local theory. However, there was one loophole that was wittingly not tested by the Nobel laureates. The notable exception was Bell's "super-deterministic" loophole, which was validated (2000) (2001) (2002) (2003) (2004) (2005) (2006) (2007) (2008) (2009) (2010) (2011) (2012) non-locally, thus compromising the subsequent Nobel Prize. More importantly, the discovery of two mutually exclusive and jointly exhaustive non-local hidden variables revealed why local scientific methods obtain false-positive and false-negative results. With knowledge of this fundamental omission, the inclusion of the non-local hidden variables in the local methods used in science can then advance it to be a complete study of nature.

Non-Local Vectorial Internal Variables and Generalized Guyer-Krumhansl Evolution Equations for the Heat Flux.

In this paper, we ask ourselves how non-local effects affect the description of thermodynamic systems with internal variables. Usually, one assumes that the internal variables are local, but that their evolution equations are non-local, i.e., for instance, that their evolution equations contain non-local differential terms (gradients, Laplacians) or integral terms with memory kernels. In contrast to this typical situation, which has led to substantial progress in several fields, we ask ourselves whether in some cases it would be convenient to start from non-local internal variables with non-local evolution equations. We examine this point by considering three main lengths: the observation scale R defining the elementary volumes used in the description of the system, the mean free path l of the microscopic elements of the fluid (particles, phonons, photons, and molecules), and the overall characteristic size L of the global system. We illustrate these ideas by considering three-dimensional rigid heat conductors within the regime of phonon hydrodynamics in the presence of thermal vortices. In particular, we obtain a generalization of the Guyer-Krumhansl equation, which may be of interest for heat transport in nanosystems or in systems with small-scale inhomogeneities.

Simulation of hydrogen embrittlement of steel using mixed nonlocal finite elements

A new simulation strategy to model hydrogen embrittlement based on a multi-field finite element using displacements, pressure, volume variation, nonlocal damage variables and lattice hydrogen concentration as unknowns is proposed. The material is described using a modified GTN model, which includes the description of hydrogen enhanced decohesion (HEDE). The finite element problem is solved using a fully implicit formulation. Numerical problems related to volumetric locking are solved using a mixed pressure/volume variation formulation. The use of the mixed formulation allows a straightforward evaluation of the pressure gradient, which drives hydrogen diffusion. Mesh size dependence is solved using an implicit gradient nonlocal formulation. Two variables are used to represent damage: the plastic volume variation and the accumulated plastic strain, controlling nucleation. The model is used to simulate an existing experimental database (Moro et al., 2010; Briottet et al., 2012) including tensile, fracture toughness and pressurized disk tests. The model, after adjusting the various coefficients, can represent the main experimental findings: the effect of deformation rate on the failure of tensile specimens, the transition from surface to internal fracture with increasing deformation rate, the sharp toughness drop under hydrogen (CT specimen), the fracture location and the effect of the pressurization rate in the case of the tests on disks.

A non-local approach for Reissner–Mindlin shell elements in dynamic simulations: Application with a Gurson model

Numerical prediction of ductile tearing during a car crash is necessary to ensure the structural integrity of the vehicle. Recently, encouraging results were obtained by Davaze et al., (2021) with a non-local damage approach, compatible with dynamic explicit simulations and preserving parallel computation: an implicit gradient model enabled to reproduce experimental results with DP450 steel sheets with 3D solid elements. However, in the automotive industry, thin sheets of metals are mostly modeled with shell elements for computation time efficiency. Shell elements have a more complex formulation: their kinematic description is enriched accounting for displacements but also rotations, leading to two different strain contributions: membrane strain and bending strain. The transverse strain is computed independently in a post-processing step from local internal variables with a zero normal stress assumption. This formulation is known to lead to strain localization after the onset of necking, while this is not the case with 3D solid elements. The problem is even more challenging when considering shells in association with a softening material accounting for damage as this also leads to strong mesh dependency, as with 3D solid elements. This complex problem is solved using non-local formulations for the material behavior. The proposed solution accounts for both in-plane and in-thickness damage gradients. It consists of a two-step regularization procedure introducing nested wire elements in the shell thickness. Using non-local variables to evaluate the transverse strain enables to obtain a continuous spatial distribution for strains, in agreement with the 3D solid case. An updated Lagrangian formulation is used to account for large strains. The performance of the strategy is demonstrated in three test cases, with a comparison with an experimental result for one of them. The results obtained following the proposed strategy lead are equivalent to the ones obtained with the non-local method proposed by Davaze et al., (2021) and are thus consistent with the experimental data.

Determinism in hidden variable interpretation and Turings halting problem

Heisenberg deduced the famous uncertainty principle which shows that there exist several conjugated quantities that can never be measured precisely at the same time. Further, Copenhagen interpretation gives a new perspective about quantum mechanics, which claims that particles do not have properties like position or momentum until people measure it. Hence, the famous EPR paradox was proposed to question the realism and locality of Quantum mechanics, which leads to the Hidden Variable explanation. However, this theorem was proved wrong by John Bell in 1964 with Bell Inequality. In addition, Hidden Variable Interpretation was developed by De Broglie and Bohm, they came up with the Bohmian Mechanics, which can be considered as Non-local Hidden Variable theorem. This interpretation gives physical meaning to waves. Unlike Copenhagen Interpretation, this theorem claims that particles do have a determined position. Some people may argue that it can be proved contradicted by using the Turing method (self-reference). Because if the algorithm represented by physics law described Hidden Mechanics to determine particles state, then the future is determined and predictable. Therefore, another algorithm can be established based on that, which will lead to Liar Paradox. This article will briefly introduce the uncertainty principle and some interpretations about quantum mechanics. Furthermore, this article will combine some ideas in Alan Turings Halting Problem to the universe of non-local hidden variables as a thought experiment, which involve self-reference, to give an interesting result of locality and determinism.

Thermomechanical vibration response of nanoplates with magneto-electro-elastic face layers and functionally graded porous core using nonlocal strain gradient elasticity

The thermal vibration and buckling behavior of a functionally graded nanoplate are examined in this study. The nanoplate is made up of a silicon nitride/stainless steel core plate and two cobalt-ferrite/barium-titanate face plates. Four alternative porosity models were used to simulate the porosity of the nanoplate, and numerous variables that can affect the nanoplate’s behavior were taken into account. The study found that the thermomechanical behavior of nanoplates with magneto-electro-elastic face layers and a functionally graded porous core plate is affected by material gradation indices, porosity ratios, nonlocal variables, and different core plate material porosity models.

Wave propagation in a non-local magneto-thermoelastic medium permeated by heat source

In this paper, we consider the new concept of non-local heat conduction equation to generalized magneto-thermoelastic problem of two dimensional isotropic and homogeneous half-space in presence of heat-flux at the boundary surface. By using the harmonic plane waves, the governing equations are transformed to the vector matrix differential equation which is then solved by eigenvalue method. The analytical closed form solutions for displacement component, temperature distribution and stress components have been made and comparisons are also illustrated graphically with the theory of non-local dual-phase-lag (NLDPL) and non-local Lord-Shulman (NLLS) theory for different values of physical parameters. The significant effects of non-local variables as well as phase lagging parameters on displacements, temperature distribution and stress components are studied by means of graphically and concluding remarks are drawn.

Environmental Variables in Predictive Soil Mapping

In the well-known conceptual model SCORPAN, given soil property is considered as dependent on the following environmental factors: soil, climate, organisms, topography, time and space. Predictive mapping of soils in digital soil mapping is based on similar ideas, but environmental factors may include not only factors of soils formation, but also, for example, remote sensing data, and have found a wide distribution not only in soil science, but also in ecology, agriculture and geomorphology. This paper provides a review of environmental factors that are used in predictive mapping with a special attention to situations when wide sets of environmental factors may be used and a part of them is not quantitative, such as vegetation types. Most developed are systems of quantitative variables for topography and climate description, so that a special attention is paid to them. Land surface description is performed using both local and non-local variables that need integration. In climate description variables are essential that estimate dry or wet terrain features, such as moisture ratio or water deficit. They need evaluation of potential evapotranspiration that is not measured by meteo-stations, but may be calculated. Possibilities of accounting these and other environmental factors including non-quantitative ones in quantitative statistical models of predictive mapping are described together with principles of their construction, verification, comparison, choice of appropriate models. Examples of soil predictive mapping applications are given for various scales, their specifics for different scales is outlined. Some aspects of remote sensing data usage are discussed.

A decentralized feedback approach for flow control in highway traffic networks

Finite-time optimal feedback control for highway traffic networks under information constraints is studied. The primary objective is to design a fully decentralized state-feedback controller with ramp metering and variable speed limit control actions, with no information exchange between local controllers, that ensures feasibility and improves a global performance index. The design of a decentralized feedback controller with a one-hop information structure is examined, wherein the optimum outflow rate from each segment of the network depends only on the state of that segment and the state of all other segments directly connected to the downstream junction of that segment. It is demonstrated that when cost/constraints can be modeled or approximated by piecewise-affine functions, the control law has a closed-form state-feedback realization. The decentralization is based on the truncation of non-local variables in local optimization problems. The resulting decentralized control scheme has a simple closed-form representation and is scalable to arbitrary large networks; moreover, we demonstrate that, under certain conditions, with respect to certain meaningful performance indexes, the performance loss due to decentralization is zero; namely, the centralized optimal controller has a decentralized realization with a one-hop information structure.

An efficient time-dependent auxiliary variable approach for the three-phase conservative Allen–Cahn fluids

Based on a time-dependent auxiliary variable approach, we propose linear, totally decoupled, and energy dissipative methods for the three-phase conservative Allen–Cahn (CAC) fluid system. The three-phase CAC equation has been extensively applied in the simulation of multi-component fluid flows because of the following advantages: (i) Total mass is conserved, (ii) Topological change of the interface can be implicitly captured. Compared with the ternary Cahn–Hilliard (CH) model, the CAC-type model is simple to solve. When we solve the CAC model by using the classical scalar auxiliary variable (SAV) approach, extra computational time is needed because we must decouple the local and non-local variables. The variant of SAV approach considered in the present study not only leads to linear and energy stable schemes, but also achieves highly efficient computation. Linear and decoupled equations need to be updated at each time step. We adopt the linear multigrid algorithm to speed up the convergence. Extensive numerical experiments with and without fluid flows are conducted to validate the temporal accuracy, mass conservation, and energy law.

Ductile fracture of high entropy alloys: From the design of an experimental campaign to the development of a micromechanics-based modeling framework

Cantor-type high entropy alloys form a new family of metallic alloys characterized by a combination of high strength and high fracture toughness. An experimental study on the CoCrNi alloy is first performed to determine the damage and fracture mechanisms under various stress states. A micromechanics-based ductile fracture model is identified and validated using these experimental data. The model corresponds to a hyperelastic finite strain multi-yield surface constitutive description coupled with multiple nonlocal variables. The yield surfaces consist of three distinct nonlocal solutions corresponding to three different modes of void expansion within an elastoplastic matrix: a void growth mode governed by a Gurson-based yield surface corrected for shear effects, an internal necking-driven coalescence mode governed by an extension of the Thomason yield surface based on the maximum principal stress, and a shear-driven coalescence mode governed by the maximum shear stress. This advanced formulation embedded in a large strain finite element setup captures the effects not only of the stress triaxiality but also of the Lode variable. In particular, the analysis shows that a failure model accounting for these two invariants of the stress tensor captures the fracture in high-entropy alloys over a wide range of conditions.

Decoupled finite element scheme of the variable-density and viscosity phase-field model of a two-phase incompressible fluid flow system using the volume-conserved Allen–Cahn dynamics

We aim to develop a fully-discrete version numerical scheme in this article for solving a variable density and viscosity phase-field model involving a nonlinear coupling between the conserved Allen–Cahn equation and the incompressible fluid dynamics. The scheme uses the finite element method for spatial discretization and the linearly stabilized-explicit method for time discretization. The fully-decoupled structure is achieved by applying the “zero-energy-contribution” feature satisfied by coupled nonlinear terms that include the advection and the surface tension, where two nonlocal auxiliary variables are used. These nonlocal auxiliary variables, combined with the operator Strang-splitting method, play as the keys to obtaining an efficient numerical algorithm. At each time step, only a series of full decoupling elliptic equations need to be solved. We rigorously demonstrate the solvability and unconditional energy stability of the scheme, and verify its effectiveness through carrying out various numerical examples including 3D droplets rising simulations.

A gradient enhanced constitutive framework for the investigation of ductile damage localization within semicrystalline polymers

The paper presents a gradient enhanced model dealing with nonlocal phenomena driven by the ductile damage in semicrystalline polymers that exhibit rate-dependent and rate-independent mechanisms. The study aims at capturing the material localization at high damage levels. To this end, a viscoelastic-viscoplastic constitutive model is formulated considering a gradient enhanced thermodynamic potential, function of local and nonlocal state variables. The model is based on two different options for the nonlocal state variable: the first option considers nonlocal damage scalar, while the second considers nonlocal hardening state variable. An appropriate user defined material subroutine is developed so as to define and to update the stress, the state variables, and the associated tangent moduli towards finite element structural computations. The analogy between the steady-state heat equation and the nonlocal gradient enhanced relation enables coupling the displacement and nonlocal fields within a finite element package code (i.e., ABAQUS). Structural analyses for polyamide 66 (PA66) material are presented to assess the efficiency of the developed nonlocal model. This research work demonstrates that the model is able to capture efficiently the ductile damage localization and simulate the related fields when using the nonlocal hardening or damage state variable. This study for the first time combines viscoelasticity, viscoplasticity, and damage with nonlocal approaches, and can be considered as an initial step towards developing a constitutive formulation towards multiscale analyses for polymer-based composites.

Thermodynamic framework for variance-based non-local constitutive models

The present work deals with the development of a framework dedicated to the construction of constitutive models with non-local internal variables. Such internal variables allow considering the impact of microstructural changes on the current state of a material point. Non-locality is introduced by considering, not only the spatial average, but also the spatial variance of an internal variable in constitutive relations. The proposed framework relies on continuum thermodynamics to construct the set of constitutive equations. Such a framework allows including some information regarding the spatial distribution of internal variables when constructing non-local models for thermomechanical applications. In contrast with gradient-type models, this strategy does not require additional equilibrium equations and boundary conditions. For the purpose of illustration, some numerical examples are presented. According to the numerical results, the proposed framework can be used to circumvent the difficulties associated with excessive spatial localization or to consider size effects.

Goldstone States as Non-Local Hidden Variables

We consider the theory of spinor fields in polar form, where the spinorial true degrees of freedom are isolated from their Goldstone states, and we show that these carry information about the frames which is not related to gravitation, so that their propagation is not restricted to be either causal or local: we use them to build a model of entangled spins where a singlet possesses a uniform rotation that can be made to collapse for both states simultaneously regardless their spatial distance. Models of entangled polarizations with similar properties are also sketched. An analogy with the double-slit experiment is also presented. General comments on features of Goldstone states are given.