Nonlocal Phase Research Articles

Nanobeams with Internal Discontinuities: A Local/Nonlocal Approach.

The aim of the present work is to extend the two-phase local/nonlocal stress-driven integral model (SDM) to the case of nanobeams with internal discontinuities: as a matter of fact, the original formulation avoids the presence of any discontinuities. Consequently, here, for the first time, the problem of an internal discontinuity is addressed by using a convex combination of both local and nonlocal phases of the model by introducing a mixture parameter. The novel formulation here proposed was validated by considering six case studies involving different uncracked nanobeams by varying the constrains and the loading configurations, and the effect of nonlocality on the displacement field is discussed. Moreover, a centrally-cracked nanobeam, subjected to concentrated forces at the crack half-length, was studied. The size-dependent Mode I fracture behaviour of the cracked nanobeam was analysed in terms of crack opening displacement, energy release rate, and stress intensity factor, showing the strong dependency of the above fracture properties on both dimensionless characteristic length and mixture parameter values.

Quasi-static and dynamic fracture modeling by the nonlocal operator method

In this paper, a phase field model is developed and applied to the simulation of quasi-static and dynamic fracture using the nonlocal operator method (NOM). The phase field’s nonlocal weak and associated strong forms are derived by a variational principle. The NOM requires only the definition of the energy. Its differential operators replace the shape functions in methods such as FEM which drastically simplifies the implementation. We present both a nonlocal implicit phase field model and a nonlocal explicit phase field model for fracture; the first approach is better suited for quasi-static fracture problems while the key application of the latter one is dynamic fracture. To demonstrate the performance of the underlying approach, several benchmark examples for quasi-static and dynamic fracture are solved.

Second order linear thermodynamically consistent approximations to nonlocal phase field porous media models

We present a weakly compressible, thermodynamically consistent, hydrodynamical model for the binary fluid flow in porous media using the one-fluid multiple component formulation, consisting of a force balance equation with a weak inertia, weakly perturbed continuity equation, and a volume-preserving, nonlocal Allen–Cahn equation for the phase field variable. This model is called the nonlocal Allen–Cahn-Extended-Darcy (NACED) model. In the incompressible and/or the inertialess limit, it reduces to two limiting models that we focus on in this study: the incompressible, nonlocal Allen–Cahn-Extended-Darcy (NACED) model and nonlocal Allen–Cahn–Darcy (NACD) model, respectively. The weakly compressible model provides a pathway for us to develop thermodynamically consistent numerical algorithms for the incompressible models. Guided by the energy variational formulation of the models, we use the energy quadratization method in time and finite difference method in space on staggered grids to derive second-order, linear, coupled and decoupled, energy dissipation rate preserving schemes for the incompressible models. The decoupled schemes for NACED model are obtained by exploiting the intrinsic relation between the incompressible and weakly compressible model. We then prove that the linear systems resulted from the linear schemes are uniquely solvable. Mesh refinement tests are conducted to confirm the convergence rates and benchmark examples of binary fluid flow motion in porous media with and without gravity are presented to showcase the schemes’ accuracy and usefulness.

Nonlocal topological insulators: Deterministic aperiodic arrays supporting localized topological states protected by nonlocal symmetries

The properties of topological systems are inherently tied to their dimensionality. Indeed, higher-dimensional periodic systems exhibit topological phases not shared by their lower-dimensional counterparts. On the other hand, aperiodic arrays in lower-dimensional systems (e.g., the Harper model) have been successfully employed to emulate higher-dimensional physics. This raises a general question on the possibility of extended topological classification in lower dimensions, and whether the topological invariants of higher-dimensional periodic systems may assume a different meaning in their lower-dimensional aperiodic counterparts. Here, we demonstrate that, indeed, for a topological system in higher dimensions one can construct a one-dimensional (1D) deterministic aperiodic counterpart which retains its spectrum and topological characteristics. We consider a four-dimensional (4D) quantized hexadecapole higher-order topological insulator (HOTI) which supports topological corner modes. We apply the Lanczos transformation and map it onto an equivalent deterministic aperiodic 1D array (DAA) emulating 4D HOTI in 1D. We observe topological zero-energy zero-dimensional (0D) states of the DAA-the direct counterparts of corner states in 4D HOTI and the hallmark of the multipole topological phase, which is meaningless in lower dimensions. To explain this paradox, we show that higher-dimension invariant, the multipole polarization, retains its quantization in the DAA, yet changes its meaning by becoming a nonlocal correlator in the 1D system. By introducing nonlocal topological phases of DAAs, our discovery opens a direction in topological physics. It also unveils opportunities to engineer topological states in aperiodic systems and paves the path to application of resonances associates with such states protected by nonlocal symmetries.

Nonlocal discrete time crystals in periodically driven surface codes

Discrete time crystals (DTCs) are nonequilibrium phases of matter characterized by robust subharmonic order parameter dynamics. We report a type of DTC in a periodically driven surface code, the subharmonic signature of which is only observable with respect to some nonlocal order parameter. By modifying the commonly used metrics for characterizing ordinary DTCs, i.e., the spectral functions, spin-glass order parameters, and two-point correlators, we further numerically demonstrate the rich phases of the proposed system. Specifically, depending on the system parameters and boundary conditions, we find that the system may fall into either a ``surface code'' phase, trivial paramagnet phase, nonlocal period-doubling DTC phase, or nonlocal period-quadrupling DTC phase. Our work thus demonstrates the prospect of exploring topological codes for discovering novel phases of matter.

Wave propagation analysis of the higher-order nanopanel with laminated composite core and magneto-electro-elastic face sheets via nonlocal strain gradient theory

In the current report, wave propagation analysis of a sandwich doubly curved nanopanel considering a laminated composite core and a couple of face sheets of magneto-electro-elastic (MEE) in the framework of nonlocal strain gradient theory (NSGT) is investigated. The higher-order-shear-deformable (HSD) model is employed to obtain the formulation of strain–stress equations. Maxwell equations are employed for modeling the behaviors of MEE materials. By considering Hamiltonian, the current model’s governing equations have been extracted. Afterward, in order to analyze the influence of wavenumber, inserted ampere, inserted voltage, the laminated structure’s geometry, nano-sized mechanics impact on the smart sandwich nano-scaled panel’s phase velocity, a parametric study has been considered. It is also observed that by raising the electric potential, the number of critical waves could be likely to be increased. A useful suggestion of this study is that the influence of the magnetic field on the phase velocity is more than the influence of the electric field of the MEE panel on the wave propagation of the smart panel. Besides, the nonlocal parameter and phase velocity have an exponential negative relation, while the length scale parameter and phase velocity have a linear direct relation.

Non-local phase field revisited

The phase field model is revisited as an approximation of a full non-local approach. We focus on van der Waals fluids, where the non-local interaction potential consists of short-range repulsion and long-range attraction, characterized by distances d (i.e. a typical molecular size d ≈ 10−9 m) and ℓ ≫ d. The first non-local correction to the thermodynamic limit introduces a density square gradient free energy, expressed in terms of a characteristic length, a. Imposing that the line integral of the non-local free energy functional across an interfacial region must equal the surface tension, we find that a is up to two orders of magnitude larger than the molecular size, in agreement with the local equilibrium assumption. Instead, by finding a directly from the interaction potential, when the attractive force follows a power-law, as in the Lennard-Jones potential, then a ≅ d, which contradicts that a ≈ 10−7 m as from surface tension measurements. Conversely, when the attractive force follows a Debye-like exponentially decaying potential of range ℓ, then a ≈ ℓ ≫ d, in agreement with surface tension measurements and the mean field theory assumption. The other point that we have addressed is determining when the mean field model can be applied. By directly simulating the phase separation of a far-from-critical van der Waals fluid mixture, we find that the growth laws of the mean nuclei size are not modified when higher-order gradient terms are retained. An exponentially decaying attractive potential must be used since the higher-order gradient terms diverge when power-law potentials are considered, confirming that a Lennard-Jones interaction potential is not compatible.

Dynamics of Stress-Driven Two-Phase Elastic Beams.

The dynamic behaviour of micro- and nano-beams is investigated by the nonlocal continuum mechanics, a computationally convenient approach with respect to atomistic strategies. Specifically, size effects are modelled by expressing elastic curvatures in terms of the integral mixture of stress-driven local and nonlocal phases, which leads to well-posed structural problems. Relevant nonlocal equations of the motion of slender beams are formulated and integrated by an analytical approach. The presented strategy is applied to simple case-problems of nanotechnological interest. Validation of the proposed nonlocal methodology is provided by comparing natural frequencies with the ones obtained by the classical strain gradient model of elasticity. The obtained outcomes can be useful for the design and optimisation of micro- and nano-electro-mechanical systems (M/NEMS).

A Non-local Phase Field Model of Bohm\u2019s Quantum Potential

Assuming that the free energy of a gas depends non-locally on the logarithm of its mass density, the body force in the resulting equation of motion consists of the sum of density gradient terms. Truncating this series after the second term, Bohm’s quantum potential and the Madelung equation are identically obtained, showing explicitly that some of the hypotheses that led to the formulation of quantum mechanics admit a classical interpretation based on non-locality.

Relationship Between the Nonlocal Effect and Lagging Behavior in Bioheat Transfer

Abstract Lots of generalized heat conduction models have been developed in recent decades, such as local thermal nonequilibrium model, phase lagging model, and nonlocal heat conduction model. But no attempt was made to prove which model is better (or worse) than others, or whether there is a certain relationship between these different models. With this inspiration, we establish the nonlocal bioheat transfer equations with lagging time, and the two and three-temperature bioheat transfer equations with considering all the carrier's heat conduction effect are also constructed. Comparing the two (or three)-temperature equation model with the nonlocal bioheat transfer models with lagging time, one may obtain: the lagging time of temperature gradient τtand the nonlocal characteristic length λq in the space derivative items of heat flux have the same effect on heat transfer; when the heat transport occur among N energy carriers with considering the conduction effects of all carries, the heat transfer processes are dependent upon the high-order effect of τqN-1, τtN-1 and λt(2N-1) in nonlocal dual phase lag bioheat transfer model. This phenomenon is very important for biological and medical systems where numerous carriers may exist on the cellular level.

Closed-form solution in bi-Helmholtz kernel based two-phase nonlocal integral models for functionally graded Timoshenko beams

Static bending behavior of functionally graded (FG) Timoshenko beams is studied via both strain- and stress-driven two-phase local/nonlocal mixed integral models based on the bi-Helmholtz kernel. The differential governing equations and boundary conditions are derived via the principle of minimum potential energy, the relations between strain and nonlocal stress are expressed as integral equations with the bi-Helmholtz kernel. The Laplace transform technique is utilized to obtain closed-form solutions for the first time. The numerical results are validated by reverting to the local case, and it is concluded that the present model shows well-posedness for exploring bounded beams in contrast to Eringen's pure nonlocal model: for all boundary and loading conditions, the increasing nonlocal length-scale parameter shows consistently softening or stiffening effect on the beam behaviour, according to the strain- or the stress-driven model, respectively. Moreover, the effects of shear-deformation and FG characteristics are also investigated. As the FG parameter increases, the non-dimensional deflection increases, while the softening effect caused by the strain-driven nonlocal phase seems to be more significant.

Nonlocal phase field approach for modeling damage in brittle materials

In this work, we present a nonlocal phase field model for damage in brittle materials. We define a non-conservative order parameter for representing the damage. The Helmholtz free energy functional of the Ginzburg-Landau type that incorporates a new degradation function for the elastic strain energy is considered. The hypothesis of strain equivalence and strain energy equivalence from continuum damage mechanics are adopted to predict the evolution of damage. A conjugate force to damage is derived for both these hypothesis to arrive at a damage criterion. The strong interrelation between phase field models and gradient damage models has been brought out. The solution of the nonlinear system of coupled equation is achieved by a change in the modified Arc-length method and considering a staggered approach. In this approach a linearization of the Crisfield Arc-length quadratic equation for calculation of load increment factor results in an unique root. Numerical examples are presented to demonstrate the usefulness of the proposed damage model.

Strong well-posedness and inverse identification problem of a non-local phase field tumour model with degenerate mobilities

We extend previous weak well-posedness results obtained in Frigeri et al. (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Vol. 22, Springer, Cham, pp. 217–254) concerning a non-local variant of a diffuse interface tumour model proposed by Hawkins-Daarud et al. (2012, Int. J. Numer. Method Biomed. Engng.28, 3–24). The model consists of a non-local Cahn–Hilliard equation with degenerate mobility and singular potential for the phase field variable, coupled to a reaction–diffusion equation for the concentration of a nutrient. We prove the existence of strong solutions to the model and establish some high-order continuous dependence estimates, even in the presence of concentration-dependent mobilities for the nutrient variable in two spatial dimensions. Then, we apply the new regularity results to study an inverse problem identifying the initial tumour distribution from measurements at the terminal time. Formulating the Tikhonov regularised inverse problem as a constrained minimisation problem, we establish the existence of minimisers and derive first-order necessary optimality conditions.

A Nonlocal Noise Reduction Method Based on Fringe Frequency Compensation for SAR Interferogram

Phase noise reduction is one of the key steps for synthetic aperture radar interferometry data processing. In this article, a novel phase filtering method is proposed. The main innovation and contribution of this research is to 1) incorporate local fringe frequency (LFF) compensation technique into the nonlocal phase filtering method to include more independent and identically distributed samples for filtering; 2) modify the nonlocal phase filter from three aspects: 1) executing nonlocal filtering in the complex domain of the residual phase to avoid gray jumps in phase, 2) adaptively calculating the smoothing parameter based on the LFF and the coherence coefficient, and 3) using the integral image in similarity calculation to improve the efficiency; 3) perform Goldstein filter in high coherence areas to reduce the computation expense. Experiments based on both simulated and real data have shown that the proposed method has achieved a better performance in terms of both noise reduction and edge preservation than some existing phase filtering methods.

Studying free hight-frequency vibrations of an inhomogeneous nanorod based on the nonlocal theory of elasticity

Free high-frequency longitudinal vibrations of an inhomogeneous nanosized rod are studied on the basis of the nonlocal theory of elasticity. The upper part of spectrum with the wavelength comparable to the internal characteristic dimension of a nanorod is examined. An equations in the integral form with the Helmholtz kernel, incorporating both local and nonlocal phases, is used as the constitutive one. The original integro-differential equation is reduced to the forth-order differential equation with variable coefficients, the pair of additional boundary conditions being deduced. UsingWKB-method, a solution of the boundaryvalue problem is constructed in the form of the superposition of a main solution and edge effect integrals. As an alternative model, we consider the purely nonlocal (one-phase) differential model which allows estimating the upper part of spectrum of eigen-frequencies. Considering the nanorod with a variable cross-section area, we revealed a fair convergence of eigen-frequencies found in the framework of two models when the local fraction in the two-phase model vanishes.

Self-Improving inequalities for bounded weak solutions to nonlocal double phase equations

<p style='text-indent:20px;'>We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" <i>Analysis & PDE</i>, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript.</p>

Non-local spiral phase contrast imaging with thermal light (Invited)

Non-local spiral phase contrast imaging with thermal light (Invited)

A range-separated generalized Kohn-Sham method including a long-range nonlocal random phase approximation correlation potential.

Based on our recently published range-separated random phase approximation (RPA) functional [Kreppel et al., "Range-separated density-functional theory in combination with the random phase approximation: An accuracy benchmark," J. Chem. Theory Comput. 16, 2985-2994 (2020)], we introduce self-consistent minimization with respect to the one-particle density matrix. In contrast to the range-separated RPA methods presented so far, the new method includes a long-range nonlocal RPA correlation potential in the orbital optimization process, making it a full-featured variational generalized Kohn-Sham (GKS) method. The new method not only improves upon all other tested RPA schemes including the standard post-GKS range-separated RPA for the investigated test cases covering general main group thermochemistry, kinetics, and noncovalent interactions but also significantly outperforms the popular G0W0 method in estimating the ionization potentials and fundamental gaps considered in this work using the eigenvalue spectra obtained from the GKS Hamiltonian.

Applications of GNSS-RO to Numerical Weather Prediction and Tropical Cyclone Forecast

The global navigation satellite system (GNSS) radio occultation (RO) technique is an atmospheric sounding technique that originated in the 1990s. The data provided by this approach are playing a consistently significant role in atmospheric research and related applications. This paper mainly summarizes the applications of RO to numerical weather prediction (NWP) generally and specifically for tropical cyclone (TC) forecast and outlines the prospects of the RO technique. With advantages such as high precision and accuracy, high vertical resolution, full-time and all-weather, and global coverage, RO data have made a remarkable contribution to NWP and TC forecasts. While accounting for only 7% of the total observations in European Centre for Medium-Range Weather Forecasts’ (ECMWF’s) assimilation system, RO has the fourth-largest impact on NWP. The greater the amount of RO data, the better the forecast of NWP. In cases of TC forecasts, assimilating RO data from heights below 6 km and from the upper troposphere and lower stratosphere (UTLS) region contributes to the forecasting accuracy of the track and intensity of TCs in different stages. A statistical analysis showed that assimilating RO data can help restore the critical characteristics of TCs, such as the location and intensity of the eye, eyewall, and rain bands. Moreover, a non-local excess phase assimilation operator can be employed to optimize the assimilation results. With denser RO profiles expected in the future, the accuracy of TC forecast can be further improved. Finally, future trends in RO are discussed, including advanced features, such as polarimetric RO, and RO strategies to increase the number of soundings, such as the use of a cube satellite constellation.

Phase transitions and correlations in fracture processes where disorder and stress compete

We study the effect of the competition between disorder and stress enhancement in fracture processes using the local load sharing fiber bundle model, a model that hovers on the border between analytical tractability and numerical accessibility. We implement a disorder distribution with one adjustable parameter. The model undergoes a localization transition as a function of this parameter. We identify an order parameter for this transition and find that the system is in the localized phase over a finite range of values of the parameter bounded by a transition to the non-localized phase on both sides. The transition is first order at the lower transition and second order at the upper transition. The critical exponents characterizing the second order transition are close to those characterizing the percolation transition. We determine the spatiotemporal correlation function in the localized phase. It is characterized by two power laws as in invasion percolation. We find exponents that are consistent with the values found in that problem.