Dislocation characterization in crystals of potash alum grown by seeded solution growth under conditions of low supersaturation

An analysis has been carried out of the genesis and character of growth dislocations present in all growth sectors of single crystals of potash alum. The crystals, grown from seeded solutions by the temperature lowering method under conditions of low supersaturation, presented the well-developed forms: {111} dominant, {100} and {110}. Growth dislocations formed predominately during refacetting of the edges and corners of the seed, rounded during preparation and insertion into the supersaturated solution. From here they become refracted into the {111} sectors which proved to be the most defective. Smaller numbers of dislocations form at the {111}, {100} and {110} seed interfaces and propagate in these sectors. In crystals of inferior quality, a number of inclusions were found predominantly in the fast growing {100} sectors which become the source of additional dislocations. Dislocations present in the original seed did not propagate across the interface into the developing crystal. Dislocations of all characters were observed. The principal Burgers vectors were found to be left angle bracket100right-pointing angle bracket, left angle bracket110right-pointing angle bracket and left angle bracket111right-pointing angle bracket.

Plate-shaped transformation products in zirconium-base alloys

Plate-shaped products resulting from martensitic, diffusional, and mixed mode transformations in zirconium-base alloys are compared. in the present study. These alloys are particularly suitable for the comparison in view of the fact that the lattice correspondence between the parent beta (bcc) and the product alpha (hcp) or gamma-hydride (fct) phases are remarkably similar for different types of transformations. Crystallographic features such as orientation relations, habit planes, and interface structures associated with these transformations have been compared:, with a view toward examining whether the transformation mechanisms have characteristic imprints on these experimental observables. Martensites exhibiting dislocated lath, internally twinned plate, and self-accommodating three-plate cluster morphologies have been encountered in Zr-2.5Nb alloy. Habit planes corresponding to all these morphologies have been found to be consistent with the predictions based on the invariant plane strain (IFS) criterion. Different morphologies have been found to reflect the manner in which the neighboring martensite variants are assembled. Lattice-invariant shears (LISs) for all these cases have been identified to be either {10 (1) over bar 1}(alpha) ((1) over bar 123)(alpha) slip or twinning on (10 (1) over bar 1)(alpha) planes. Widmanstatten alpha precipitates, forming in a step-quenching treatment, have been shown to have a lath morphology, the alpha/beta interface being decorated with a periodic array of (c + a) dislocations at a spacing of 8 to 10 nm. The line vectors of these dislocations are nearly parallel to the invariant lines. The alpha precipitates, forming in the retained beta phase on aging, exhibit an internally twinned structure with a zigzag habit plane. Average habit planes for the morphologies have been found to lie near the {103}(beta) - {113}(beta) poles, which are close to the specific variant of the {112}(beta) plane, which transforms into a prismatic plane of the type {1 (1) over bar 00}(alpha). The crystallography of the formation of the gamma-hydride phase (fct) from both the alpha and beta phases is seen to match the IFS predictions. While the beta-gamma transformation can be treated approximately as a simple shear on the basal plane involving a change in the stacking sequence, the alpha-gamma transformation call be conceptually broken into a alpha --> beta transformation following the Burgers correspondence and the simple beta-gamma shear process. The active eutectoid decomposition in the Zr-Cu system, beta --> alpha + beta', has been described in terms of cooperative growth of the alpha phase from the beta phase through the Burgers correspondence and of the partially ordered beta' (structurally similar to the equilibrium Zr2Cu phase) through an ordering process. Similarities and differences in crystallographic features of these transformations have been discussed. and the importance of the invariant line vector in deciding the geometry of the corresponding habit planes has been pointed out.

Real-Space Manifestations of Bottlenecks in Turbulence Spectra

An energy-spectrum bottleneck, a bump in the turbulence spectrum between the inertial and dissipation ranges, is shown to occur in the nonturbulent, one-dimensional, hyperviscous Burgers equation and found to be the Fourier-space signature of oscillations in the real-space velocity, which are explained by boundary-layer-expansion techniques. Pseudospectral simulations are used to show that such oscillations occur in velocity correlation functions in one- and three-dimensional hyperviscous hydrodynamical equations that display genuine turbulence. DOI: 10.1103/PhysRevLett.110.064501

Structure-function hierarchies and von Karman-Howarth relations for turbulence in magnetohydrodynamical equations

We generalize the method of A. M. Polyakov, Phys. Rev. E 52, 6183 (1995)] for obtaining structure-function relations in turbulence in the stochastically forced Burgers equation, to develop structure-function hierarchies for turbulence in three models for magnetohydrodynamics (MHD). These are the Burgers analogs of MHD in one dimension Eur. Phys. J.B 9, 725 (1999)], and in three dimensions (3DMHD and 3D Hall MHD). Our study provides a convenient and unified scheme for the development of structure-function hierarchies for turbulence in a variety of coupled hydrodynamical equations. For turbulence in the three sets of MHD equations mentioned above, we obtain exact relations for third-order structure functions and their derivatives; these expressions are the analogs of the von Karman-Howarth relations for fluid turbulence. We compare our work with earlier studies of such relations in 3DMHD and 3D Hall MHD.

A stabilization approach for mesh-free simulations of systems developing shocks or extreme strain localizations

A new stabilization scheme, based on a stochastic representation of the discretized field variables, is proposed with a view to reduce or even eliminate unphysical oscillations in the mesh-free numerical simulations of systems developing shocks or exhibiting localized bands of extreme deformation in the response. The origin of the stabilization scheme may be traced to nonlinear stochastic filtering and, consistent with a class of such filters, gain-based additive correction terms are applied to the simulated solution of the system, herein achieved through the element-free Galerkin method, in order to impose a set of constraints that help arresting the spurious oscillations. The method is numerically illustrated through its Applications to inviscid Burgers' equations, wherein shocks may develop as a result of intersections of the characteristics, and to a gradient plasticity model whose response is often characterized by a developing shear band as the external load is gradually increased. The potential of the method in stabilized yet accurate numerical simulations of such systems involving extreme gradient variations in the response is thus brought forth. (C) 2014 Elsevier Ltd. All rights reserved.

对流扩散方程的变步长摄动有限差分格式

摄动有限差分(PFD)方法是构造高精度差分格式的一种新方法。变步长摄动有限差分方法是等步长摄动有限差分方法的发展和推广。对需要局部加密网格的计算问题，变步长PFD格式不需要对自变量进行数学变换，且和等步长PFD格式一样，具有如下的共同特点：从变步长一阶迎风格式出发，通过把非微商项(对流系数和源项)作变步长摄动展开，展开幂级数系数通过消去摄动格式修正微分方程的截断误差项求出，由此获得高精度变步长PFD格式。该格式在一、二和三维情况下分别仅使用三、五和七个基点，且具有迎风性。文中利用变步长PFD格式对对流扩散反应模型方程，变系数方程及Burgers方程等进行了数值模拟，并与一阶迎风和二阶中心格式及其问题的精确解作了比较。数值试验表明，与一阶迎风和二阶中心格式相比，变步长PFD格式具有精度高，稳定性与收敛性好的特点。变步长PFD格式与等步长PFD格式相比，变步长PFD解在薄边界层型区域的分辨率得到了明显的提高。

粘性流动有限差分计算的新策略

对粘性流动计算，提出有限离散单元流动的流体分析(理论)和耦合离散流体理论(CDFT)的差分格式。利用CDFT差分格式计算Burgers方程和计算激波边界层干扰流动的数值实验表明：对计算精度和计算效率的提高，CDFT格式比提高常用差分格式(即离散流体力学方程得到的格式)精度和改进常用格式形式等更有效，且运算量小。

Calculations of longitudinal and transverse velocity structure functions using a vortex model of isotropic turbulence

The longitudinal structure function (LSF) and the transverse structure function (TSF) in isotropic turbulence are calculated using a vortex model. The vortex model is composed of the Rankine and Burgers vortices which have the exponential distributions in the vortex Reynolds number and vortex radii. This model exhibits a power law in the inertial range and satisfies the minimal condition of isotropy that the second-order exponent of the LSF in the inertial range is equal to that of the TSF. Also observed are differences between longitudinal and transverse structure functions caused by intermittency. These differences are related to their scaling differences which have been previously observed in experiments and numerical simulations.

NS方程激波计算的摄动有限差分方法

摄动有限差分（PFD）方法从一阶迎风差分格式出发，将差分系数展开为网格步长的幂级数，通过提高修正微分方程的逼近精度来获得更高精度的差分格式。由于格式基于一阶迎风格式，因此具有迎风效应、网格节点少等特点。本文首先通过对Burgers方程的摄动差分格式的推导，将摄动有限差分格式引入时间相关法的计算，并构造了守恒形式的摄动有限差分格式，然后推广到一维Navier-Stokes方程组的计算。数值比较研究表明：本文构造的NS方程摄动有限差分格式具有比一阶迎风较高的精度和分辨率，而且保持了一阶迎风格式的无振荡性质。

约化摄动法和非线性波远场分析

<正> 一、引言 如所周知,摄动法(特别是奇异摄动法)在研究弱非线性波方面有着广泛的应用[1—5]。其中,近年来发展形成的约化摄动法已经成了分析各种非线性波远场的有力工具[6—8]。 约化摄动法的实质是,对于一般的描述非线性波的复杂方程组,通过适当的坐标变形和摄动展开,在一阶近似下,把方程组约化成较为简单可解的单个非线性方程(例如Burgers方程、Korteweg-de Vries方程、非线性Schrodinger方程等),从而可以分析远离波的相互作用区的远场。

On the unstable stacking criterion for ideal and cracked copper-crystals

The unstable stacking criteria for an ideal copper crystal under homogeneous shearing and for a cracked copper crystal under pure mode II loading are analysed. For the ideal crystal under homogeneous shearing, the unstable stacking energy gamma(us) defined by Rice in 1992 results from shear with no relaxation in the direction normal to the slip plane. For the relaxed shear configuration, the critical condition for unstable stacking does not correspond to the relative displacement Delta = b(p)/2, where b(p) is the Burgers vector magnitude of the Shockley partial dislocation, but to the maximum shear stress. Based on this result, the unstable stacking energy Gamma(us) is defined for the relaxed lattice. For the cracked crystal under pure mode II loading, the dislocation configuration corresponding to Delta = b(p)/2 is a stable state and no instability occurs during the process of dislocation nucleation. The instability takes place at approximately Delta = 3b(p)/4. An unstable stacking energy Pi(us) is defined which corresponds to the unstable stacking state at which the dislocation emission takes place. A molecular dynamics method is applied to study this in an atomistic model and the results verify the analysis above.

H4 exponential finite-difference scheme for convective diffusion-equations

Perturbations are applied to the convective coefficients and source term of a convection-diffusion equation so that second-order corrections may be applied to a second-order exponential scheme. The basic Structure of the equations in the resulting fourth-order scheme is identical to that for the second order. Furthermore, the calculations are quite simple as the second-order corrections may be obtained in a single pass using a second-order scheme. For one to three dimensions, the fourth-order exponential scheme is unconditionally stable. As examples, the method is applied to Burgers' and other fluid mechanics problems. Compared with schemes normally used, the accuracies are found to be good and the method is applicable to regions with large gradients.

Numerical experiments on one-dimensional model of turbulence

The initial-value problem of a forced Burgers equation is numerically solved by the Fourier expansion method. It is found that its solutions finally reach a steady state of 'laminar flow' which has no randomness and is stable to disturbances. Hence, strictly speaking, the so-called Burgers turbulence is not a turbulence. A new one-dimensional model is proposed to simulate the Navier-Stokes turbulence. A series of numerical experiments on this one-dimensional turbulence is made and is successful in obtaining Kolmogorov's (1941) k exp(-5/3) inertial-range spectrum. The (one-dimensional) Kolmogorov constant ranges from 0.5 to 0.65.

迎风紧致格式与热流计算

该文在朱庆勇、马延文(1998)一文的基础上给出了利用迎风紧致格式求解NS方程.它是UCGVC格式在粘性流计算中的推广.对于方程中的无粘部分利用Steger-Warming的通量分裂技术将流通向量分裂成两部分,在此基础上据风向构造逼近于无粘项的三阶迎风紧致格式.对方程中的粘性部分采用通常的二阶中心差分格式.该文利用对Burgers方程的特征分析,研究了为正确模拟边界层内的流动特征对网格雷诺数的限制条件.通过对超声速粘性球头绕流的数值模拟表明:迎风紧致格式能在较大的网格雷诺数的条件下计算热流值,结果是令人满意的.

I. Exact solution of some nonlinear evolution equations. II. The similarity solution for the Korteweg-De Vries equation and the related Painleve Transcendent

In Part I, a method for finding solutions of certain diffusive dispersive nonlinear evolution equations is introduced. The method consists of a straightforward iteration procedure, applied to the equation as it stands (in most cases), which can be carried out to all terms, followed by a summation of the resulting infinite series, sometimes directly and other times in terms of traces of inverses of operators in an appropriate space.

We first illustrate our method with Burgers' and Thomas' equations, and show how it quickly leads to the Cole-Hopft transformation, which is known to linearize these equations.

We also apply this method to the Korteweg and de Vries, nonlinear (cubic) Schrödinger, Sine-Gordon, modified KdV and Boussinesq equations. In all these cases the multisoliton solutions are easily obtained and new expressions for some of them follow. More generally we show that the Marcenko integral equations, together with the inverse problem that originates them, follow naturally from our expressions.

Only solutions that are small in some sense (i.e., they tend to zero as the independent variable goes to ∞) are covered by our methods. However, by the study of the effect of writing the initial iterate u_1 = u_(1)(x,t) as a sum u_1 = ^∼/u_1 + ^≈/u_1 when we know the solution which results if u_1 = ^∼/u_1, we are led to expressions that describe the interaction of two arbitrary solutions, only one of which is small. This should not be confused with Backlund transformations and is more in the direction of performing the inverse scattering over an arbitrary “base” solution. Thus we are able to write expressions for the interaction of a cnoidal wave with a multisoliton in the case of the KdV equation; these expressions are somewhat different from the ones obtained by Wahlquist (1976). Similarly, we find multi-dark-pulse solutions and solutions describing the interaction of envelope-solitons with a uniform wave train in the case of the Schrodinger equation.

Other equations tractable by our method are presented. These include the following equations: Self-induced transparency, reduced Maxwell-Bloch, and a two-dimensional nonlinear Schrodinger. Higher order and matrix-valued equations with nonscalar dispersion functions are also presented.

In Part II, the second Painleve transcendent is treated in conjunction with the similarity solutions of the Korteweg-de Vries equat ion and the modified Korteweg-de Vries equation.