Semi-discrete Model Research Articles

Stationary solution of semi-discrete traffic flow model on a ring road with slopes

The traffic bottleneck effects are investigated on a ring road with up- and down-slopes to the semi-discrete model under the Lagrange coordinate. The kinetic wave theory is applied to discuss all types of stationary solutions for a small relaxation time. Moreover, the analytical formulas of the queuing lengths before bottlenecks and the critical densities of the saturated flux are derived. The stability of the stationary solution on every road section under different mass increments is discussed. The relation between the slope gradient and saturated flux platform is also obtained. The numerical simulation reproduces many complex nonlinear phenomena, such as the steady-state flow, oscillatory congested traffic and saturated flux platform. This study helps to explain the empirical features on inhomogeneous roads in the real-world traffic.

Semi-discrete Green's function for solution of anisotropic thermal/electrostatic Boussinesq and Mindlin problems: Application to two-dimensional material systems

Green's function (GF) for steady state Laplace/Poisson equation is derived for an anisotropic, finite two-dimensional (2D) composite material by solving a combined Boussinesq–Mindlin problem. A semi-discrete model of the material is developed in which only one Cartesian coordinate axis is discretized, while the other is treated as a continuous variable. The Fourier integral for the continuous coordinate is obtained analytically. Thus, a 2D problem needs only a 1D discretization. An approximate analytical estimate shows that the numerical convergence of our model is at least an order of magnitude better than fully discretized models. Numerical results are reported for the GF for a phosphorene composite containing an array of metallic inclusions. The GF is useful as a starting solution in boundary element calculations. It can be used for deriving the full solution of the Laplace/Poisson equation for an arbitrary distribution of sources and boundary values, used for modeling heat flow and electrostatic potential distribution in a 2D composite. These material systems are of strong topical interest because of their potential application in revolutionary new solid-state devices for energy conversion and quantum computing. This paper is another step towards developing GF based characterization techniques for modern 2D materials.

Two- and three-dimensional elastic networks with rigid junctions: modeling within the theory of micropolar shells and solids

For two- and three-dimensional elastic structures made of families of flexible elastic fibers undergoing finite deformations, we propose homogenized models within the micropolar elasticity. Here we restrict ourselves to networks with rigid connections between fibers. In other words, we assume that the fibers keep their orthogonality during deformation. Starting from a fiber as the basic structured element modeled by the Cosserat curve beam model, we get 2D and 3D semi-discrete models. These models consist of systems of ordinary differential equations describing the statics of a collection of fibers with certain geometrical constraints. Using a specific homogenization technique, we introduce two- and three-dimensional equivalent continuum models which correspond to the six-parameter shell model and the micropolar continuum, respectively. We call two models equivalent if their approximations coincide with each other up to certain accuracy. The two- and three-dimensional constitutive equations of the networks are derived and discussed within the micropolar continua theory.

Time-asymptotic convergence rates towards discrete steady states of a nonlocal selection-mutation model

This paper is concerned with large time behavior of solutions to a semi-discrete model involving nonlinear competition that describes the evolution of a trait-structured population. Under some threshold assumptions, the steady solution is shown unique and strictly positive, and also globally stable. The exponential convergence rate to the steady state is also established. These results are consistent with the results in [P.-E. Jabin, H. L. Liu. Nonlinearity 30 (2017) 4220–4238] for the continuous model.

Darboux transformation and soliton solutions of the semi-discrete massive Thirring model

A one-fold Darboux transformation between solutions of the semi-discrete massive Thirring model is derived using the Lax pair and the dressing method. This transformation is used to find the exact expressions for soliton solutions on zero and nonzero backgrounds. It is shown that the discrete solitons have the same properties as their continuous counterparts.

Stochastic Dynamics of Discrete-Time Fuzzy Random BAM Neural Networks with Time Delays

By using the semidiscrete method of differential equations, a new version of discrete analogue of stochastic fuzzy BAM neural networks was formulated, which gives a more accurate characterization for continuous-time stochastic neural networks than that by the Euler scheme. Firstly, the existence of the 2p-th mean almost periodic sequence solution of the discrete-time stochastic fuzzy BAM neural networks is investigated with the help of Minkowski inequality, Hölder inequality, and Krasnoselskii’s fixed point theorem. Secondly, the 2p-th moment global exponential stability of the discrete-time stochastic fuzzy BAM neural networks is also studied by using some analytical skills in stochastic theory. Finally, two examples with computer simulations are given to demonstrate that our results are feasible. The main results obtained in this paper are completely new, and the methods used in this paper provide a possible technique to study 2p-th mean almost periodic sequence solution and 2p-th moment global exponential stability of semidiscrete stochastic fuzzy models.

Mean almost periodicity and moment exponential stability of semi-discrete random cellular neural networks with fuzzy operations.

By using the semi-discretization technique of differential equations, the discrete analogue of a kind of cellular neural networks with stochastic perturbations and fuzzy operations is formulated, which gives a more accurate characterization for continuous-time models than that by Euler scheme. Firstly, the existence of at least one p-th mean almost periodic sequence solution of the semi-discrete stochastic models with almost periodic coefficients is investigated by using Minkowski inequality, Hölder inequality and Krasnoselskii’s fixed point theorem. Secondly, the p-th moment global exponential stability of the semi-discrete stochastic models is also studied by using some analytical skills and the proof of contradiction. Finally, a problem of stochastic stabilization for discrete cellular neural networks is studied.

A computational approach towards modelling dislocation transmission across phase boundaries

ABSTRACTTo study the nanoscopic interaction between edge dislocations and a phase boundary within a two-phase microstructure the effect of the phase contrast on the internal stress field due to the dislocations needs to be taken into account. For this purpose a 2D semi-discrete model is proposed in this paper. It consists of two distinct phases, each with its specific material properties, separated by a fully coherent and non-damaging phase boundary. Each phase is modelled as a continuum enriched with a Peierls–Nabarro (PN) dislocation region, confining dislocation motion to a discrete plane, the glide plane. In this paper, a single glide plane perpendicular to and continuous across the phase boundary is considered. Along the glide plane bulk induced shear tractions are balanced by glide plane shear tractions based on the classical PN model. The model's ability to capture dislocation obstruction at phase boundaries, dislocation pile-ups and dislocation transmission is studied. Results show that the phase contrast in material properties (e.g. elastic stiffness, glide plane properties) alone creates a barrier to the motion of dislocations from a soft to a hard phase. The proposed model accounts for the interplay between dislocations, external boundaries and phase boundary and thus represents a suitable tool for studying edge dislocation–phase boundary interaction in two-phase microstructures.

Borodin\u2013P\xe9ch\xe9 Fluctuations of the Free Energy in Directed Random Polymer Models

We consider two directed polymer models in the Kardar–Parisi–Zhang (KPZ) universality class: the O’Connell–Yor semi-discrete directed polymer with boundary sources and the continuum directed random polymer with (m, n)-spiked boundary perturbations. The free energy of the continuum polymer is the Hopf–Cole solution of the KPZ equation with the corresponding (m, n)-spiked initial condition. This new initial condition is constructed using two semi-discrete polymer models with independent bulk randomness and coupled boundary sources. We prove that the limiting fluctuations of the free energies rescaled by the 1 / 3rd power of time in both polymer models converge to the Borodin–Péché-type deformations of the GUE Tracy–Widom distribution.

Unraveling the dislocation core structure at a van der Waals gap in bismuth telluride

Tetradymite-structured chalcogenides such as bismuth telluride (Bi2Te3) are of significant interest for thermoelectric energy conversion and as topological insulators. Dislocations play a critical role during synthesis and processing of such materials and can strongly affect their functional properties. The dislocations between quintuple layers present special interest since their core structure is controlled by the van der Waals interactions between the layers. In this work, using atomic-resolution electron microscopy, we resolve the basal dislocation core structure in Bi2Te3, quantifying the disregistry of the atomic planes across the core. We show that, despite the existence of a stable stacking fault in the basal plane gamma surface, the dislocation core spreading is mainly due to the weak bonding between the layers, which leads to a small energy penalty for layer sliding parallel to the van der Waals gap. Calculations within a semidiscrete variational Peierls-Nabarro model informed by first-principles calculations support our experimental findings.

$2p$ -th Moment Global Exponential Stability of a Unique $2P$ -th Mean Almost Periodic Oscillation for Semi-Discrete Takagi-Sugeno Fuzzy Stochastic Cellular Neural Networks With Time Delays

By using semi-discrete and Takagi-Sugeno fuzzy methods, a new version of discrete analogue of stochastic fuzzy cellular neural networks is formulated, which gives a more accurate characterization for continuous-time stochastic model than that by Euler scheme. Firstly, the 2p-th moment global exponential stability for the obtained semi-discrete stochastic Takagi-Sugeno fuzzy model is studied with the help of Minkowski inequality and Holder inequality. Secondly, the 2p-th mean almost periodic outputs of the model is investigated by using Krasnoselskii's fixed point theorem. Finally, illustrative examples and numerical simulations are given to demonstrate that our results are feasible.

Strain-Gradient Plasticity as the $\Gamma$-Limit of a Nonlinear Dislocation Energy with Mixed Growth

In this paper, we derive by means of $\Gamma$-convergence a macroscopic strain-gradient plasticity theory from a semidiscrete model for edge dislocations in an infinite cylindrical crystal. We consider an energy with subquadratic growth close to the dislocations. This allows us to treat the stored elastic energy without the need to introduce an ad hoc cut-off radius. As the main tool to prove a complementing compactness statement, we present a generalized version of the geometric rigidity result for fields with nonvanishing $\operatorname{curl}$. One of the main ingredients for the proof is a fine decomposition result for $L^1$-functions whose divergence is in certain critical Sobolev spaces.

Wrinkling in engineering fabrics: a comparison between two different comprehensive modelling approaches.

We consider two ‘comprehensive’ modelling approaches for engineering fabrics. We distinguish the two approaches using the terms ‘semi-discrete’ and ‘continuum’, reflecting their natures. We demonstrate a fitting procedure, used to identify the constitutive parameters of the continuum model from predictions of the semi-discrete model, the parameters of which are in turn fitted to experimental data. We, then, check the effectiveness of the continuum model by verifying the correspondence between semi-discrete and continuum model predictions using test cases not previously used in the identification process. Predictions of both modelling approaches are compared against full-field experimental kinematic data, obtained using stereoscopic digital image correlation techniques, and also with measured force data. Being a reduced order model and being implemented in an implicit rather than an explicit finite-element code, the continuum model requires significantly less computational power than the semi-discrete model and could therefore be used to more efficiently explore the mechanical response of engineering fabrics.

Comparing modeling predictions of aluminum edge dislocations: semidiscrete variational Peierls-Nabarro versus atomistics.

Multiple computational methods for modeling dislocations are implemented within a high-throughput calculation framework allowing for rigorous investigations comparing the methodologies. Focusing on aluminum edge dislocations, twenty-one classical aluminum interatomic potentials are used to directly model dislocation core structures using molecular dynamics, as well as provide input data for solving the semidiscrete variational Peierls-Nabarro dislocation model. The predicted dislocation core spreading obtained from both computational methods show similar trends across the potentials. Additionally, tests are done to rigorously determine if a recent correction to the Peierls-Nabarro model results in better agreement with the atomistic calculations.

Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using a finite-difference method

We consider a finite-difference semi-discrete scheme for the approximation of boundary controls for the one-dimensional wave equation. The high frequency numerical spurious oscillations lead to a loss of the uniform (with respect to the mesh size) controllability property of the semi-discrete model in the natural setting. We prove that, by filtering the high frequencies of the initial data in an optimal range, we restore the uniform controllability property. Moreover, we obtain a relation between the range of filtration and the minimal time of control needed to ensure the uniform controllability. The proof is based on the moment method.

Mathematical aspects in the meteorological interpretation of satellite hyperspectral infrared measurements part II: estimates of the cloud absorption vertical profile of Hurricane Ioke on 28 August 2006

ABSTRACTThis study processes and analyses hyperspectral measurements from the Atmospheric Infrared Sounder on 28 August 2006 covering day and night scenes of Hurricane Ioke with a temporal spacing of 12 hours. A semi-discrete multilevel cloud model is used to describe the perturbation of the outgoing thermal radiation caused by cloudiness in the field of view (FOV) of a satellite instrument. Cloud spectral effects in the model are represented by an effective cloud absorption vertical profile (CAVP). The CAVP estimate is considered an indicator of the presence of clouds at specific atmospheric layers. The CAVP estimates are compared with lidar measurements. Results indicate a realistic characterization of cloud top and cloud vertical scale. The spatial distribution of the CAVP estimates is used to describe the spatial structure of the hurricane and to monitor its changes over time.Comparison of daytime and night-time CAVP estimates shows that the effective hurricane radius is drastically reduced from about 578 km during the day to about 463 km at night at an atmospheric layer above 212 hPa and from about 604 km to about 499 km at atmospheric layer above 300 hPa. In addition, there is indication of a significant presence of ice crystals in the upper troposphere 10–14 km at night over the large areas adjacent to the hurricane. These crystal clouds have the potential to affect the hurricane energy budget by reducing the night-time cooling rate and trapping heat and moisture in the lower atmospheric layers.

Well posedness for semidiscrete fractional Cauchy problems with finite delay

We address the study of well posedness on Lebesgue spaces of sequences for the following fractional semidiscrete model with finite delay (0.1) Δ α u ( n ) = T u ( n ) + β u ( n − τ ) + f ( n ) , n ∈ N , 0 < α ≤ 1 , β ∈ R , τ ∈ N 0 , where T is a bounded linear operator defined on a Banach space X (typically a space of functions like L p ( Ω ) , 1 < p < ∞ ) and Δ α corresponds to the time discretization of the continuous Riemann–Liouville fractional derivative by means of the Poisson distribution. We characterize the existence and uniqueness of solutions in vector-valued Lebesgue spaces of sequences of the model (0.1) in terms of boundedness of the operator-valued symbol ( ( z − 1 ) α z 1 − α I − β z − τ − T ) − 1 , | z | = 1 , z ≠ 1 , whenever 0 < α ≤ 1 and X satisfies a geometrical condition. For this purpose, we use methods from operator-valued Fourier multipliers and resolvent operator families associated to the homogeneous problem. We apply this result to show a practical and computational criterion in the context of Hilbert spaces.

First-principles investigation of strain effects on the stacking fault energies, dislocation core structure, and Peierls stress of magnesium and its alloys

Taking pure Mg, Mg-Al, and Mg-Zn as prototypes, the effects of strain on the stacking fault energies (SFEs), dislocation core structure, and Peierls stress were systematically investigated by means of density functional theory and the semidiscrete variational Peierls-Nabarro model. Our results suggest that volumetric strain may significantly influence the values of SFEs of both pure Mg and its alloys, which will eventually modify the dislocation core structure, Peierls stress, and preferred slip system, in agreement with recent experimental results. The so-called ``strain factor'' that was previously proposed for the solute strengthening could be justified as a major contribution to the strain effect on SFEs. Based on multivariate regression analysis, we proposed universal exponential relationships between the dislocation core structure, the Peierls stress, and the stable or unstable SFEs. Electronic structure calculations suggest that the variations of these critical parameters controlling strength and ductility under strain can be attributed to the strain-induced electronic polarization and redistribution of valence charge density at hollow sites. These findings provide a fundamental basis for tuning the strain effect to design novel Mg alloys with both high strength and ductility.

Integrable Semi-discrete Kundu–Eckhaus Equation: Darboux Transformation, Breather, Rogue Wave and Continuous Limit Theory

To get more insight into the relation between discrete model and continuous counterpart, a new integrable semi-discrete Kundu–Eckhaus equation is derived from the reduction in an extended Ablowitz–Ladik hierarchy. The integrability of the semi-discrete model is confirmed by showing the existence of Lax pair and infinite number of conservation laws. The dynamic characteristics of the breather and rational solutions have been analyzed in detail for our semi-discrete Kundu–Eckhaus equation to reveal some new interesting phenomena which was not found in continuous one. It is shown that the theory of the discrete system including Lax pair, Darboux transformation and explicit solutions systematically yields their continuous counterparts in the continuous limit.

Iterative Region-of-Interest Reconstruction from Limited Data Using Prior Information

In practice, computed tomography and computed laminography applications suffer from incomplete data. In particular, when inspecting large objects with extremely different diameters in longitudinal and transversal directions or when high resolution reconstructions are desired, the physical conditions of the scanning system lead to restricted data and truncated projections, also known as the interior or region-of-interest (ROI) problem. To recover the searched-for density function of the inspected object, we derive a semi-discrete model of the ROI problem that inherently allows the incorporation of geometrical prior information in an abstract Hilbert space setting for bounded linear operators. Assuming that the attenuation inside the object is approximately constant, as for fibre reinforced plastics parts or homogeneous objects where one is interested in locating defects like cracks or porosities, we apply the semi-discrete Landweber–Kaczmarz method to recover the inner structure of the object inside the ROI from the measured data resulting in a semi-discrete iteration method. Finally, numerical experiments for three-dimensional tomographic applications with both an inherent restricted source and ROI problem are provided to verify the proposed method for the ROI reconstruction.