Nonlocal Variables Research Articles

Quantum Correction for Newton’s Law of Motion

A description of the motion in noninertial reference frames by means of the inclusion of high time derivatives is studied. Incompleteness of the description of physical reality is a problem of any theory, both in quantum mechanics and classical physics. The “stability principle” is put forward. We also provide macroscopic examples of noninertial mechanics and verify the use of high-order derivatives as nonlocal hidden variables on the basis of the equivalence principle when acceleration is equal to the gravitational field. Acceleration in this case is a function of high derivatives with respect to time. The definition of dark metrics for matter and energy is presented to replace the standard notions of dark matter and dark energy. In the Conclusion section, problem symmetry is noted for noninertial mechanics.

Causal interpretation of quantum mechanics: The Schrodinger equation as a condition of stability

This work presents a further development of causal interpretation of quantum mechanics. Existence of ‘non-local hidden variables’ like a physical field is taken as an assumption. Under this assumption, on the basis of the causal approach it is proven that Schrödinger equation is a necessary condition for stability of the motion of a particle. For the case of a hydrogen atom this approach gives a mathematical base with which to suggest that (1) the electron’s spin in an atom is precessing; (2) the energy of the precessional motion on Bohr orbits satisfies the Rydberg formula, and (3) structures are formed in the physical vacuum which stabilize the motion of an electron on Bohr orbits.

Gradient-enhanced continuum models of healing in damaged soft tissues.

Healing of soft biological tissue is the process of self-recovering or self-repairing the injured or damaged extracellular matrix (ECM). Healing is assumed to be stress-driven, with the objective of returning to a homeostatic stress metrics in the tissue after replacing the damaged ECM with new undamaged one. However, based on the existence of intrinsic length scales in soft tissues, it is thought that computational models of healing should be non-local. In the present study, we introduce for the first time two gradient-enhanced constitutive healing models for soft tissues including non-local variables. The first model combines a continuum damage model with a temporally homogenized growth model, where the growth direction is determined according to local principal stress directions. The second one is based on a gradient-enhanced healing model with continuously recoverable damage variable. Both models are implemented in the finite-element package Abaqus by means of a user subroutine UEL. Three two-dimensional situations simulating the healing process of soft tissues are modeled numerically with both models, and their application for simulation of balloon angioplasty is provided by illustrating the change of damage field and geometry in the media layer throughout the healing process.

Fully Local Amplification Factor Transport Equation for Stationary Crossflow Instabilities

The traditional linear stability analysis for transition prediction includes the following steps: solving velocity profiles using boundary-layer equations or a computational fluid dynamics solver, guessing and searching operations for the eigenvalues, computing the linear perturbation equations, and integrating the amplification factor . It is difficult to implement into modern computational fluid dynamics solvers with parallel computations because it requires many nonlocal variables and coordinate system transformations. In this Paper, an amplification factor transport equation, which can be coupled with a modern computational fluid dynamics parallel solver, is established based on the analysis results through linear stability theory for stationary crossflow instabilities. According to the analysis results of Falkner–Skan–Cooke similarity velocity profiles using linear stability theory analysis, the source term of the present transport equation is formulated locally to describe the growth of stationary crossflow waves in the laminar boundary layer. This new equation is added to the equation proposed by Coder and Maughmer for Tollmien–Schlichting instabilities. Finally, coupling of these two amplification factor transport equations and Menter’s shear stress transport turbulence model completes an effective transition turbulence model. Comparisons between the present predictions, wind tunnel experiments and the standard linear stability theory analysis on an NLF(2)-0415 infinite swept wing, Petzold’s sickle-shaped wing and 6:1 inclined prolate spheroid validate the reasonable establishment of the present amplification factor transport equation for stationary crossflow instabilities.

Measurements of Nonlocal Variables and Demonstration of the Failure of the Product Rule for a Pre- and Postselected Pair of Photons.

We report the first implementation of the von Neumann instantaneous measurements of nonlocal variables, which becomes possible due to technological achievements in creating hyperentangled photons. Tests of reliability and of the nondemolition property of the measurements have been performed with high precision, showing the suitability of the scheme as a basic ingredient of numerous quantum information protocols. The method allows us to demonstrate for the first time with strong measurements a special feature of pre- and postselected quantum systems: the failure of the product rule. It has been verified experimentally that for a particular pre- and postselected pair of particles, a single measurement on particle A yields with certainty σ_{x}^{A}=-1, a single measurement on particle B yields with certainty σ_{y}^{B}=-1, and a single nonlocal measurement on particles A and B yields with certainty σ_{x}^{A}σ_{y}^{B}=-1.

Comprehensive Tap and Phase Analysis in a Framework External to the PF

This paper presents a novel and comprehensive method of analyzing transformer taps and phases in power flow (PF) studies. The method is based on constructing a criterion function, which encapsulates physical constraints for the analyzed taps and phases. The root of the criterion function corresponds to the correct solution. To find the root, the Newton method is used. The criterion function is external to the PF routine, and so, the root-finding (and with it, the tap and phase analysis) is done without altering the underlying PF routine. The method is developed for quadrature-boosting-transformers, as well as for regular phase shifting and tap changing transformers. A possibility of accounting for changing impedance is developed and incorporated in the procedure, using real-world phase-impedance relation data of a phase shifting transformer. Generalizing control modes to different types of variables and to non-local variables is illustrated. The method is tested on large benchmark systems: the IEEE118, IEEE300, and Pegase13 659 bus system. Tests show that (since the method is Newtonian) the convergence is normally achieved within 10 iterations.

A peridynamics model for strain localization analysis of geomaterials

SummaryGeomaterials such as sand and clay are highly heterogeneous multiphase materials. Nonlocality (or a characteristic length scale) in modeling geomaterials based on the continuum theory can be associated with several factors, for instance, the physical interactions of material points within finite distance, the homogenization or smoothing process of material heterogeneity, and the particle or problem size‐dependent mechanical behavior (eg, the thickness of shear bands) of geomaterials. In this article, we formulate a nonlocal elastoplastic constitutive model for geomaterials by adapting a local elastoplastic model for geomaterials at a constant suction through the constitutive correspondence principle of the state‐based peridynamics theory. We numerically implement this nonlocal constitutive model via the classical return‐mapping algorithm of computational plasticity. We first conduct a one‐dimensional compression test of a soil sample at a constant suction through the numerical model with three different values of the nonlocal variable (horizon) δ. We then present a strain localization analysis of a soil sample under the constant suction and plane strain conditions with different nonlocal variables. The numerical results show that the proposed nonlocal model can be used to simulate the inception and propagation of shear banding as well as to capture the thickness of shear bands in geomaterials at a constant suction.

Instability states and Ostrogradsky formalism

Contemporary physics, both Classical and Quantum, requires a notion of inertial reference frames. However, how to find a physical inertial frame in reality where there always exist random weak forces? We suggest a description of the motion in instability frames by means of inclusion of higher time derivatives. They may play a role of non-local hidden variables in a more general description complementing both classical and quantum mechanics.

Geomorphometry on the surface of a triaxial ellipsoid: towards the solution of the problem

ABSTRACTExisting algorithms of geomorphometry can be applied to digital elevation models (DEMs) given with plane square grids or spheroidal equal angular grids on the surface of an ellipsoid of revolution or a sphere. Computations on spheroidal equal angular grids are trivial for modelling of the Earth, Mars, the Moon, Venus, and Mercury. This is because: (a) forms of these celestial bodies can be described by an ellipsoid of revolution or a sphere and (b) for these surfaces, there are well-developed theory and algorithms to solve the inverse geodetic problem as well as to determine spheroidal trapezoidal areas. It is advisable to apply a triaxial ellipsoid for describing the forms of small moons and asteroids. However, there are no geomorphometric algorithms intended for such a surface. In this article, first, we formulate the problem of geomorphometric modelling on a triaxial ellipsoid surface. Then, we recall definitions and formulae for coordinate systems of a triaxial ellipsoid and their transformation. Next, we present analytical and computational solutions, which provide the basis for geomorphometric modelling on the surface of a triaxial ellipsoid. The Jacobi solution for the inverse geodetic problem has a fundamental mathematical character. The Bespalov solutions for determination of the length of meridian/parallel arcs and the spheroidal trapezoidal areas are computationally efficient. Finally, we describe easy-to-code algorithms for derivation of local and non-local morphometric variables from DEMs based on a spheroidal equal angular grid of a triaxial ellipsoid.

Conservation Laws and Nonlocally Related Systems of Two-Dimensional Boundary Layer Models

Abstract Local conservation laws, potential systems, and nonlocal conservation laws are systematically computed for three-equilibrium two-component boundary layer models that describe different physical situations: a plate flow, a flow parallel to the axis of a circular cylinder, and a radial jet striking a planar wall. First, local conservation laws of each model are computed using the direct method. For each of the three boundary layer models, two local conservation laws are found. The corresponding potential variables are introduced, and nonlocally related potential systems and subsystems are formed. Then nonlocal conservation laws are sought, arising as local conservation laws of nonlocally related systems. For each of the three physical models, similar nonlocal conservation laws arise. Further nonlocal variables that lead to further potential systems are considered. Trees of nonlocally related systems are constructed; their structure coincides for all three models. The three boundary layer models considered in this work provide rich and interesting examples of the construction of trees of nonlocally related systems. In particular, the trees involve spectral potential systems depending on a parameter; these spectral potential systems lead to nonlocal conservation laws. Moreover, potential variables that are not locally related on solution sets of some potential systems become local functions of each other on solution sets of other systems. The point symmetry analysis shows that the plate and radial jet flow models possess infinite-dimensional Lie algebras of point symmetries, whereas the Lie algebra of point symmetries for the cylinder flow model is three-dimensional. The computation of nonlocal symmetries reveals none that arise for the original model equations, which is common for partial differential equations (PDE) systems without constitutive parameters or functions, but does reveal nonlocal symmetries for some nonlocally related PDE systems.

Semiclassical approach to dynamics of interacting fermions

Understanding the behaviour of interacting fermions is of fundamental interest in many fields ranging from condensed matter to high energy physics. Developing numerically efficient and accurate simulation methods is an indispensable part of this. Already in equilibrium, fermions are notoriously hard to handle due to the sign problem. Out of equilibrium, an important outstanding problem is the efficient numerical simulation of the dynamics of these systems. In this work we develop a new semiclassical phase-space approach (a.k.a. the truncated Wigner approximation) for simulating the dynamics of interacting fermions in arbitrary dimensions. As fermions are essentially non-classical objects, a phase-space is constructed out of all fermionic bilinears. Classical phase-space is thus comprised of highly non-local (hidden) variables representing these bilinears, and the cost of the method is that it scales quadratic rather than linear with system size. We demonstrate the strength of the method by comparing the results to the exact quantum dynamics of fermion expansion in the Hubbard model and quantum thermalization in the Sachdev–Ye–Kitaev (SYK) model for small systems, where the semiclassics nearly perfectly reproduces correct results. We furthermore analyse fermion expansion in a larger, intractable by exact methods, 2D Hubbard model, which is directly relevant to recent cold atom experiments.

A meshless approach to non-local damage modelling of concrete

A non-linear continuum damage mechanics model for concrete constructions is analysed using a radial point interpolation meshless method (RPIM). The fundamental mathematical relations and the material model are fully characterized. The 2D plane stress RPIM formulation is extended to a rate-independent standard (local) damage model considering both tension and compression static states. Additionally, in this work, the local damage formulation is modified considering a non-local constitutive damage criterion with regard to a Helmholtz free energy potential. Here, the internal variational fields, such as local and non-local damage variables, are determined by a return-mapping damage algorithm. Due to the non-linear nature of the phenomenon, a displacement controlled Newton-Raphson iterative approach is adopted to attain the non-linear damage solution. In the end, the performance of the proposed non-local damage model is evaluated using an experimental test of a notched-three-point bending beam available in the literature. The obtained solution shows that the meshless methods are capable to effectively analyse concrete structures assuming a non-linear non-local continuum damage model.

Explaining quantum correlations through evolution of causal models

We propose a framework for the systematic and quantitative generalization of Bell's theorem using causal networks. We first consider the multi-objective optimization problem of matching observed data while minimizing the causal effect of nonlocal variables and prove an inequality for the optimal region that both strengthens and generalizes Bell's theorem. To solve the optimization problem (rather than simply bound it), we develop a novel genetic algorithm treating as individuals causal networks. By applying our algorithm to a photonic Bell experiment, we demonstrate the trade-off between the quantitative relaxation of one or more local causality assumptions and the ability of data to match quantum correlations.

Potential Systems and Nonlocal Conservation Laws of Prandtl Boundary Layer Equations on the Surface of a Sphere

Abstract: The potential systems and nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere have been investigated. The multiplier approach yields two local conservation laws for the Prandtl boundary layer equations on the surface of a sphere. Two potential variables ψ and ϕ are introduced corresponding to first and second conservation law. Moreover, another potential variable p is introduced by considering the linear combination of both conservation laws. Two level one potential systems involving a single nonlocal variable ψ or ϕ are constructed. One level two potential system involving both nonlocal variables ψ and ϕ is established. The nonlocal variable p is utilised to derive a spectral potential system. The nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere are derived by computing the local conservation laws of its potential systems. The nonlocal conservation laws are utilised to derive the further nonlocally related systems.

A computational framework for modelling damage-induced softening in fibre-reinforced materials – Application to balloon angioplasty

A computational framework for modelling damage-induced softening in fibre-reinforced materials is presented. The main aspect of this framework is the proposed non-local gradient-enhanced continuum damage formulation. At the material level, the elastic constitutive behaviour is defined by a hyperelastic functional including a volumetric and an isochoric contribution. The isochoric contribution is subdivided into three contributions associated to three different phases i=0,1,2. Phase 0 is represented by an incompressible neo-Hookean material, whereas phases 1 and 2 are represented by an exponential format that accounts for the stretching along two preferred anisotropy directions, i.e. two fibre families. Furthermore, a 1−di–type damage function, is introduced to reproduce the loss of stiffness in each phase i. Following the ideas discussed in (Dimitrijević and Hackl, 2008; Waffenschmidt et al. 2014) and references cited therein, the model is built around the enhancement of the local free energy function by means of terms that contain the referential gradients of the non-local damage variables ϕi. The inclusion of these terms ensures an implicit regularisation of the finite element implementation. A finite element implementation of the non-local gradient-enhanced continuum damage model is presented. To this end we develop an 8-noded Q1Q1P0 hexahedral element following a variational approach, in order to efficiently model the quasi-incompressible behaviour of the hyperelastic material. This element is implemented in Abaqus by means of a user subroutine UEL. Three boundary value problems are studied: an anisotropic plate with a hole, a balloon angioplasty and a full-3D artery-like tube. These computational experiments serve to illustrate the main capabilities of the proposed model.

Spheroidal equal angular DEMs: The specificity of morphometric treatment

AbstractDigital elevation models (DEMs) are commonly constructed using two main types of regular grids: plane square grids and spheroidal equal angular grids. Methods and algorithms intended for plane square‐gridded DEMs should not be directly applied to spheroidal equal angular DEMs. This is because these grids have fundamentally different geometry. However, some researchers continue to apply square‐grid algorithms to spheroidal equal angular DEMs. It seems appropriate to consider once again the specifity of morphometric treatment of spheroidal equal angular DEMs. This article, first, demonstrates possibilities of direct calculation of local, nonlocal, and combined morphometric variables from spheroidal equal angular DEMs exemplified by slope gradient, catchment area, and topographic index. Second, the article shows computational errors when algorithms for plane square‐gridded DEMs are unreasonably applied to spheroidal equal angular DEMs. The study is exemplified by two DEMs. A medium‐resolution DEM of a relatively small, high‐mountainous area (Mount Elbrus) was extracted from the SRTM1 DEM. A low‐resolution DEM of a vast region with the diverse topography (the central and western regions of Kenya) was extracted from the SRTM30_PLUS DEM. The results show that application of square‐grid methods to spheroidal equal angular DEMs leads to substantial computational errors in models of morphometric variables.

Development of a boundary layer parameters identification method for transition prediction with complex grids

Predicting boundary layer transition accurately is important to thermal protection and drag reduction of flight vehicles. Up to now, there has been many transition prediction methods. However, most of those methods need boundary layer parameters, which are difficult to obtain in massively parallel execution since some parameters are nonlocal variables, thus greatly limiting the application of those methods. A grid-reorder method is developed to obtain the boundary layer parameters, which is suitable for parallel computing in this paper. With the grid-reorder method the wall normal grid cells can be easily found, and two criteria are used to determine the boundary layer edge in the wall normal direction, then the boundary layer parameters such as boundary layer thickness, boundary layer momentum thickness, boundary layer edge velocity, cross-flow velocity, and so on, can be obtained accurately and efficiently. The method has been coupled to three transition prediction methods, the γ-Reθ model, the k-ω-γ model, and the transition correlations, to validate its effectiveness. For the γ-Reθ model, the cross-flow velocity is obtained with the grid-reorder method, then a cross-flow intermittency factor is developed and introduced into the model, and the inclined prolate spheroid case is used to test the performance of the model. For the k-ω-γ model, the grid-reorder method is applied to obtain the boundary layer edge velocity and the inflection point velocity which are of vital importance to form the second-mode timescale for hypersonic transition prediction. For the transition correlations, Reθ/ Me is obtained effectively with the grid-reorder method. The X-51 forebody is selected to test the effectiveness of Reθ/Me for complex geometries and the results show a good correspondence with the experiment results. The successful application in three transition prediction methods demonstrates that the grid-reorder method has an excellent performance in obtaining the boundary layer parameters and can broaden the application of the existing transition prediction method in engineering.

A Gradient Inelastic Flexibility-Based Frame Element Formulation

AbstractThis paper introduces a novel gradient inelastic beam theory and a corresponding flexibility-based (FB) frame element formulation to treat major weaknesses of existing FB formulations in the presence of softening section constitutive relations (or softening material response). These weaknesses include (1) strain localization and loss of objectivity and (2) instabilities and convergence failures of the numerical solution algorithms. These weaknesses are shown to emanate from the pathogenies of Navier’s beam theory in the presence of softening section constitutive relations. These pathogenies include (1) loss of solution uniqueness; and (2) loss of spatial continuity of the strain field, under a continuous force field. The proposed gradient inelastic beam theory is derived by enriching Navier’s beam theory with nonlocal section deformation variables through gradient-based equations. For the proposed theory, a FB formulation is developed. This FB formulation (1) results in a single set of algebraic e...

Causal non-locality can arise from constrained replication

The fundamental theories of physics are local theories, depending on local interactions of local variables. It is not clear if and how strictly local theories can produce non-local variables that have causal effectiveness. Yet, non-local effectiveness appears to exist, such as in the form of memory (non-locality through time) and causally effective spatial structures (non-locality through space). Here it is shown, by construction, how such non-locality can be produced from elementary components: non-isolated systems, multiplicative noise, self-replication, and elimination. A theory is derived that explains how causal non-locality can arise from strictly local interactions.

Gardner's deformation of the Krasil'shchik—Kersten system

The classical problem of construction of Gardner's deformations for infinite-dimensional completely integrable systems of evolutionary partial differential equations (PDE) amounts essentially to finding the recurrence relations between the integrals of motion. Using the correspondence between the zero-curvature representations and Gardner deformations for PDE, we construct a Gardner's deformation for the Krasil'shchik-Kersten system. For this, we introduce the new nonlocal variables in such a way that the rules to differentiate them are consistent by virtue of the equations at hand and second, the full system of Krasil'shchik-Kersten's equations and the new rules contains the Korteweg-de Vries equation and classical Gardner's deformation for it.