In this article, we give an introduction to the basic theory of dislocations and some dislocation models at different length scales. Dislocations are line defects in crystals. The continuum theory of dislocations works well at the length scale of several lattice constants away from the dislocations. In the region surrounding the dislocations (core region), the crystal lattice is heavily distorted, and atomistic models are used to describe the atomic arrangement and related properties. The Peierls-Nabarro models of dislocations incorporate the atomic features into the continuum theory, therefore provide an alternative way to understand the dislocation core properties. The numerical simulation of the collective motion and interactions of dislocations, known as dislocation dynamics, is becoming a more and more important tool for the investigation of the plastic behaviors of materials. Several simulation methods for dislocation dynamics are also reviewed in this article.

Are extensions to continuum formulations for solving fluid dynamic problems in the transition-to-rarefied regimes viable alternatives to particle methods? It is well known that for increasingly rarefied flow fields, the predictions from continuum formulation, such as the Navier-Stokes equations lose accuracy. These inaccuracies are attributed primarily to the linear approximations of the stress and heat flux terms in the Navier-Stokes equations. The inclusion of higher-order terms, such as Burnett or high-order moment equations, could improve the predictive capabilities of such continuum formulations, but there has been limited success in the shock structure calculations, especially for the high Mach number case. Here, after reformulating the viscosity and heat conduction coefficients appropriate for the rarefied flow regime, we will show that the Navier-Stokes-type continuum formulation may still be properly used. The equations with generalization of the dissipative coefficients based on the closed solution of the Bhatnagar-Gross-Krook (BGK) model of the Boltzmann equation, are solved using the gas-kinetic numerical scheme. This paper concentrates on the non-equilibrium shock structure calculations for both monatomic and diatomic gases. The Landau-Teller-Jeans relaxation model for the rotational energy is used to evaluate the quantitative difference between the translational and rotational temperatures inside the shock layer. Variations of shear stress, heat flux, temperatures, and densities in the internal structure of the shock waves are compared with, (a) existing theoretical solutions of the Boltzmann solution, (b) existing numerical predictions of the direct simulation Monte Carlo (DSMC) method, and (c) available experimental measurements. The present continuum formulation for calculating the shock structures for monatomic and diatomic gases in the Mach number range of 1.2 to 12.9 is found to be satisfactory.

In this article we propose a higher-order space-time conservative method for hyperbolic systems with stiff and non-stiff source terms as well as relaxation systems. We call the scheme a slope propagation (SP) method. It is an extension of our scheme derived for homogeneous hyperbolic systems [1]. In the present inhomogeneous systems the relaxation time may vary from order of one to a very small value. These small values make the relaxation term stronger and highly stiff. In such situations underresolved numerical schemes may produce spurious numerical results. However, our present scheme has the capability to correctly capture the behavior of the physical phenomena with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved. The scheme treats the space and time in a unified manner. The flow variables and their slopes are the basic unknowns in the scheme. The source term is treated by its volumetric integration over the space-time control volume and is a direct part of the overall space-time flux balance. We use two approaches for the slope calculations of the flow variables, the first one results directly from the flux balance over the control volumes, while in the second one we use a finite difference approach. The main features of the scheme are its simplicity, its Jacobian-free and Riemann solver-free recipe, as well as its efficiency and high order accuracy. In particular we show that the scheme has a discrete analog of the continuous asymptotic limit. We have implemented our scheme for various test models available in the literature such as the Broadwell model, the extended thermodynamics equations, the shallow water equations, traffic flow and the Euler equations with heat transfer. The numerical results validate the accuracy, versatility and robustness of the present scheme.

We propose an artificial boundary method for solving the deterministic Kardar-Parisi-Zhang equation in one-, two- and three- dimensional unbounded domains. The exact artificial boundary conditions are obtained on the artificial boundaries. Then the original problems are reduced to equivalent problems in bounded domains. A finite difference method is applied to solve the reduced problems, and some numerical examples are provided to show the effectiveness of the method.

In this work, we propose a kind of cellular automaton model to simulate the railway traffic which is based on NaSch traffic model. The signaling system adopted in this work is the three-aspect fixed autoblock system. We investigate the space-time diagram of traffic flow using our model. The velocities and headways of trains are discussed. The impact of the time interval of trains on train running is also analyzed. Numerical simulation results demonstrate that our cellular automaton model can successfully reproduce some actual railway traffic phenomena.

In our simplfied description 'wealth' is money (m). A kinetic theory of a gas like model of money is investigated where two agents interact (trade) selectively and exchange some amount of money between them so that the sum of their money is unchanged and thus the total money of all the agents remains conserved. The probability distributions of individual money (P(m) vs. m) is seen to be influenced by certain ways of selective interactions. The distributions shift away from Boltzmann-Gibbs like the exponential distribution, and in some cases distributions emerge with power law tails known as Pareto's law (P(m) ∝ m^−(1+α)). The power law is also observed in some other closely related conserved and discrete models. A discussion is provided with numerical support to obtain insight into the emergence of power laws in such models.

In this paper, we present a model for grown-in point defects inside indium antimonide crystals grown by the Czochralski (CZ) technique. Our model is similar to the ones used for silicon crystal, which includes the Fickian diffusion and a recombination mechanism. This type of models is used for the first time to analyze grown-in point defects in indium antimonide crystals. The temperature solution and the advance of the melt-crystal interface, which determines the time-dependent domain of the model, are based on a recently derived perturbation model. We propose a finite difference method which takes into account the moving interface. We study the effect of thermal flux on the point defect patterns during and at the end of the growth process. Our results show that the concentration of excessive point defects is positively correlated to the heat flux in the system.

The nearly analytic discrete method (NADM) is a perturbation method originally proposed by Yang et al. (2003) [26] for acoustic and elastic waves in elastic media. This method is based on a truncated Taylor series expansion and interpolation approximations and it can suppress effectively numerical dispersions caused by the discretizating the wave equations when too-coarse grids are used. In the present work, we apply the NADM to simulating acoustic and elastic wave propagations in 2D porous media. Our method enables wave propagation to be simulated in 2D porous isotropic and anisotropic media. Numerical experiments show that the error of the NADM for the porous case is less than those of the conventional finite-difference method (FDM) and the so-called Lax-Wendroff correction (LWC) schemes. The three-component seismic wave fields in the 2D porous isotropic medium are simulated and compared with those obtained by using the LWC method and exact solutions. Several characteristics of wave propagating in porous anisotropic media, computed by the NADM, are also reported in this study. Promising numerical results illustrate that the NADM provides a useful tool for large-scale porous problems and it can suppress effectively numerical dispersions.

The multi-domain hybrid Spectral-WENO method (Hybrid) is introduced for the numerical solution of two-dimensional nonlinear hyperbolic systems in a Cartesian physical domain which is partitioned into a grid of rectangular subdomains. The main idea of the Hybrid scheme is to conjugate the spectral and WENO methods for solving problems with shock or high gradients such that the scheme adapts its solver spatially and temporally depending on the smoothness of the solution in a given subdomain. Built as a multi-domain method, an adaptive algorithm is used to keep the solutions parts exhibiting high gradients and discontinuities always inside WENO subdomains while the smooth parts of the solution are kept inside spectral ones, avoiding oscillations related to the well-known Gibbs phenomenon and increasing the numerical efficiency of the overall scheme. A higher order version of the multi-resolution analysis proposed by Harten is used to determine the smoothness of the solution in each subdomain. We also discuss interface conditions for the two-dimensional problem and the switching procedure between WENO and spectral subdomains. The Hybrid method is applied to the two-dimensional Shock-Vortex Interaction and the Richtmyer-Meshkov Instability (RMI) problems.