Semi-discrete Model Research Articles

Finite element model for NCF composite reinforcement preforming: Importance of inter-ply sliding

A semi-discrete model is proposed for the simulation of non-crimp fabric reinforcement preforming. This approach is based on experimental observations of in-plane shear deformations of the reinforcement, out-of-plane bending and inter-ply sliding during the non-crimp fabric preforming. The experiment highlights strong sliding between the two plies of the non-crimp fabric during forming. The two plies of fibers composing the reinforcement are modeled by two layers of shell elements; the stitches are modeled by bar elements. The relationship between bar and shell elements is managed with a specific contact and anisotropic friction law associated to sliding threshold. This meso-macroscopic approach allows the simulation of industrial part forming in a reasonable computational time. The model shows a good prediction of deformations during the forming on a hemispherical shape. In particular sliding between the two fiber layers of the NCF obtained by the simulation is in good agreement with that measured during the preforming experiment.

A Review of some Theoretical Models for Point Defect Calculations

A Brief Sketch of Different Models for the Calculation of Defect Parameters in Metals and Alloys, Comparison of Data and Limitations Has Been Reviewed here; Especially Relaxations due to a Vacancy Type of Point Defect, its Formation, Migration, Activation Energies and Related other Parameters Based upon the Present Experimental Status. the Models Reviewed Are the Bond Model, Continuum Model, Semi-Discrete Model, Jellium Model, Thermodynamic Model, Lattice Statics Model, Atomistic Continuum Model and Pseudopotential Model. the Main Thrust Concerns the Last Model. the Taylor, Vashishta and Singwi, Harrison, Kleinmann and King and Kutler Form of Exchange and Correlation Function Are Almost Similar, Give Moderate Results and May Be Trusted for Better Results.

Semi-discrete events and models in categorical language

Semi-discrete events and models in categorical language

Turing instability for a semi-discrete Gierer–Meinhardt system

In this paper, we investigate the spatial patterns of a Gierer–Meinhardt system where the space is discrete in two dimensions with the periodic boundary condition and time is continuous, in contrast to the continuum models. The conditions of Turing instability are obtained by linear analysis and a series of numerical simulations are performed. In the instability region, we have shown that this system can produce a number of different patterns such as stripes and snowflake pattern, other than ubiquitous fix-spotted patterns. As mentioned, the results are substantiated only by means of snapshots of the apatial grid. However, we also give some analysis by using the time series at three random grids and of the average value of states, that is, the stable state patterns can be observed. On the other hand, the effects of varying parameters on pattern formation are also discussed. Moreover, simulations for fixed parameters and special initial conditions indicate that the initial conditions can alter the structure of patterns. The patterns can form as a consequence of cellular interaction. So patterns arising from a semi-discrete model can present simulations on a geometrically accurate representation in biology. As a result, our work is interesting and important in ecology.

A semi-discrete model and its approach to a solution for a wide moving jam in traffic flow

This paper investigates the analytical and numerical solutions to wide moving jams in traffic flow. Under the framework of the Lagrange coordinates, a semi-discrete model and a continuum model correlate with each other, in which the former model approaches the latter as the increment ΔM in the former model vanishes. This implies that the solution to a wide moving jam in the latter model, which can be analytically derived using the known theory, can be conceivably taken as an approximation to that of the former model. These results were verified through numerical simulations. Because a detailed understanding of the traffic phase “wide moving jam” is very important for the further development of Kerner’s three-phase traffic theory, this study helps to explain the empirical features of traffic breakdown and resulting congested traffic patterns that are observed in real traffic.

Time and Laplace‐domain methods for MTL transient and sensitivity analysis

PurposeThe purpose of this paper is to present the development and comparison of selected time‐domain and Laplace‐domain methods for the simulation of waves propagating along multiconductor transmission lines (MTLs), both uniform and nonuniform, and sensitivities with respect to distributed and lumped parameters of MTL systems.Design/methodology/approachA methodology is based on discrete, semidiscrete and continuous MTL models formulated and solved both in the time and Laplace domains, latter combined with a numerical inverse Laplace transform (NILT).FindingsThe most accurate method is that based on the MTL Laplace‐domain continuous model, processed via the MTL chain matrix and connected with an NILT. This method concurrently shows minimal RAM requirements, and in case of uniform MTLs, it runs fastest. For nonuniform MTLs, however, the implicit Wendroff formula is fastest, as long as the RAM is available.Research limitations/implicationsThe research is limited to linear MTLs only and the methods suppose terminating circuits based on their generalized Thévenin equivalents. They can be, however, generalized for more complex systems via more sophisticated boundary conditions treatment. The time‐domain methods have further potential to be generalized towards nonlinear MTLs.Practical implicationsThe methods considered can contribute to solving signal integrity issues in high‐speed electronic systems, the Matlab routines developed can serve in education process as well.Originality/valueThe implicit Wendroff formula has been adapted to enable simulation of voltage and/or current distributions and their sensitivities along the nonuniform MTLs' wires. Besides, semidiscrete and continuous nonuniform MTL models have been elaborated to enable sensitivities determination, both in the time and Laplace domains, latter connected with the NILT technique based on fast Fourier transform/inverse fast Fourier transform and quotient‐difference algorithms.

Turing instability for a two dimensional semi-discrete Oregonator model

In this paper, a semi-discrete (time continuous but two-dimensional spatially discrete) Oregonator model has been given in the microscopic domain, and Turing instability theory analysis is discusse...

EXISTENCE OF SOLUTIONS TO A REGULARIZED MODEL OF DYNAMIC FRACTURE

Existence and convergence results are proved for a regularized model of dynamic brittle fracture based on the Ambrosio–Tortorelli approximation. We show that the sequence of solutions to the time-discrete elastodynamics, proposed by Bourdin, Larsen & Richardson as a semidiscrete numerical model for dynamic fracture, converges, as the time-step approaches zero, to a solution of the natural time-continuous elastodynamics model, and that this solution satisfies an energy balance. We emphasize that these models do not specify crack paths a priori, but predict them, including such complicated behavior as kinking, crack branching, and so forth, in any spatial dimension.

A note on semi-discrete modelling in the life sciences

Semi-discrete models are a particular class of hybrid dynamical systems that undergo continuous dynamics most of the time but repeatedly experience discrete changes at some given moments. In the life sciences, since the first semi-discrete model was derived to describe population dynamics by Beverton & Holt (Beverton & Holt 1957 In Fisheries investigations, series 2, vol. 19), a large body of literature has been concerned with such modelling approaches. The aim of the present contribution is twofold. On the one hand, it provides a comprehensive introduction to semi-discrete modelling through two illustrative examples: the classical work by Beverton and Holt is recalled and an original example on immigration in a population model affected by a strong Allee effect is worked out. On the other hand, a short overview of the different applications of semi-discrete models in the life sciences is proposed.

Car-following and the macroscopic Aw-Rascle traffic flow model

We consider a semi-discrete car-following model and the macroscopic Aw-Rascle model for traffic flow given in Lagrangian form. The solution of the car-following model converges to a weak entropy solution of the system of hyperbolic balance laws with Cauchy initial data. For the homogeneous system, we allow vacuum in the initial data. By using properties of the semi-discrete model, we show that this solution of the hyperbolic system is stable in the $L^1$-norm.

Partial homogenization of discrete models

This article is devoted to the approximation of discrete models by semi-discrete (and semi-continuous) models. The partial homogenization method is applied to this end. The interface conditions between the discrete part and the continuous part are discussed. The error estimates for the solutions of discrete and semi-discrete models are proved.

Modeling and Computation for the High-Speed Rotating Flexible Structure

This paper is presented to improve the modeling accuracy and the computational stability for a high-speed rotating flexible structure. The differential governing equations are derived based on the first-order approximation coupling (FOAC) model theory in the framework of the generalized Hamiltonian principle. The semi-discrete model is obtained by the finite element method, and a new shape function based on FOAC is established for the piezoelectric layers. To increase the efficiency, accuracy, and stability of computation, first, the second-order half-implicit symplectic Runge–Kutta method is presented to keep the computational stability of the numerical simulation in a long period of time. Then, the idea of a precise integration method is introduced into the symplectic geometric algorithm. An improved symplectic precise integration method is developed to increase accuracy and efficiency. Several numerical examples are adopted to show the promise of the modeling and the computational method.

BETWEEN DISCRETE AND CONTINUOUS: CONSUMER–RESOURCE DYNAMICS WITH SYNCHRONIZED REPRODUCTION

In many consumer-resource systems the consumer population has synchronized reproduction at regular intervals (e.g., years) but consumes the resource and dies continuously, while the resource population grows continuously or has overlapping generations that are short relative to the time between consumer reproductive events. Such systems require "semi-discrete" models that have both discrete and continuous components. This paper defines and analyzes a canonical, semi-discrete model for a widespread class of consumer-resource interactions in which the consumer is a discrete breeder and the resource reproduction can be described continuously. The model is the analog of the Nicholson-Bailey and Lotka-Volterra models for discrete and continuous systems, respectively. It thereby develops the basis for understanding more realistic, and hence more complex, semi-discrete models. The model can display stable equilibria, consumer-resource cycles, and single-species-like overcompensation cycles. Cycles are induced by high maximum fecundity in the consumer. If the resource grows rapidly and the consumer has high maximum fecundity, the model reduces to a single-species discrete-time model of the consumer, which can exhibit overcompensation cycles. By contrast, such cycles in discrete consumer-resource models typically occur only in the resource once the consumer is extinct. Also unlike a common class of discrete models that do not display consumer-resource cycles with periods below four years, semi-discrete models can exhibit consumer-resource cycles with periods as short as two years.

A Precise Time-Step Integration Method for Transient Analysis of Lossy Nonuniform Transmission Lines

This paper presents a novel time-domain integration method for transient analysis of nonuniform multiconductor transmission lines (MTLs). It can solve the time response of various kinds of transmission lines with arbitrary coupling status. The spatial discretization in this method is the same as the finite-difference time-domain (FDTD) algorithm. However, in order to eliminate the Courant-Friedrich-Levy condition constraint, a precise time-step integration method is utilized in time-domain calculation. It gives an analytical solution in the time domain for the spatial discretized Telegrapher's equations with linear boundary conditions. Large time steps can be adopted in the integration process to achieve accurate results efficiently. In the analysis of transmission lines with frequency-dependent parameters, a passive equivalent model is introduced, which leads to the similar semidiscrete model as that for the frequency-independent case. In addition, a rigorous proof of the passivity of the model is provided. Numerical examples are presented to demonstrate the accuracy and stability of the proposed method.

Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square

This paper studies the numerical approximation of the boundary control for the wave equation in a square domain. It is known that the discrete and semi-discrete models obtained by discretizing the wave equation with the usual finite-difference or finite-element methods do not provide convergent sequences of approximations to the boundary control of the continuous wave equation as the mesh size goes to zero. Here, we introduce and analyse a new semi-discrete model based on the space discretization of the wave equation using a mixed finite-element method with two different basis functions for the position and velocity. The main theoretical result is a uniform observability inequality which allows us to construct a sequence of approximations converging to the minimal L2-norm control of the continuous wave equation. We also introduce a fully discrete system, obtained from our semi-discrete scheme, for which we conjecture that it provides a convergent sequence of discrete approximations as both h and Δt, the time discretization parameter, go to zero. We illustrate this fact with several numerical experiments.

Characteristic Lie algebra and classification of semidiscrete models

We study characteristic Lie algebras of semi-discrete chains and attempt to use this notion to classify Darboux-integrable chains.

Inverse problems of the mathematical theory of tides: tide potential problem for a semi-discrete model

We study an inverse problem of the mathematical tide theory: a tidal potential problem for a semi-discrete model. The closure equation is constructed using observation data on the sea-level function on a part of the boundary and in the interior of the domain. Problems of existence and uniqueness of the solution are considered. An iteration algorithm for solving the problem is formulated.

Numerical solution to the optimal feedback control of continuous casting process

Using a semi-discrete model that describes the heat transfer of a continuous casting process of steel, this paper is addressed to an optimal control problem of the continuous casting process in the secondary cooling zone with water spray control. The approach is based on the Hamilton---Jacobi---Bellman equation satisfied by the value function. It is shown that the value function is the viscosity solution of the Hamilton---Jacobi---Bellman equation. The optimal feedback control is found numerically by solving the associated Hamilton---Jacobi---Bellman equation through a designed finite difference scheme. The validity of the optimality of the obtained control is experimented numerically through comparisons with different admissible controls. Detailed study of a low-carbon billet caster is presented.

Staircasing in semidiscrete stabilised inverse linear diffusion algorithms

We consider a semidiscrete model problem for the approximation of stabilised inverse linear diffusion processes. The work is motivated by an important observation on fully discrete schemes concerning the so-called staircasing phenomenon: when sharpening monotone data profiles, fully discrete methods generally introduce stepfunction-type solutions reminiscent of staircases. In this work, we show by an analysis of dynamical systems in corresponding semidiscrete formulations that already the semidiscrete numerical model contains the relevant information on the occurrence of staircasing. Numerical experiments confirm and complement the theoretical results.

Binary to unary transition problem in nucleation theory

The limiting behavior in binary nucleation theory when one component vanishes is examined. Failure of the traditional continuum theory to predict the transition from binary to unary nucleation is due to breakdown of the continuum approximation. A semidiscrete model is constructed for the transition regime. For binary systems of arbitrary composition, a general formula that combines the semi-discrete and continuum nucleation rates is proposed. Comparison with the exact solution calculated using a fully discrete matrix method shows good agreement.