正交各向异性材料切口尖端热弹奇性特征分析

研究正交各向异性平面V形切口,计算其热弹奇性特征.通过引入切口尖端物理场的渐近级数展开式,将应力和热流平衡方程转化为关于奇性指数的特征常微分方程组,采用插值矩阵法求解,获取切口尖端的热流、应力奇性指数和对应的特征角函数.算例表明,该法精度高适应性强.     

双压电材料三维界面端部力电耦合场奇异性

基于切口根部物理场的幂级数渐近展开假设,从三维应力平衡方程和麦克斯韦方程组出发,导出关于双压电材料楔形界面切口端部奇性指数的特征微分方程组,并将切口的力电学边界条件表达为奇性指数和特征角函数的组合,从而将双压电材料楔形界面切口端部奇性指数的计算转化为相应边界条件下常微分方程组特征值的求解,运用插值矩阵法求解界面端部若干阶应力奇性指数和相应特征函数.计算结果与已有结果对比表明本文方法的有效性和具有较高的计算精度.     

电子-正电子散射与电子-电子散射的极化效应

计算了电子-正电子散射的16种极化散射截面.根据角分布的特征,它们可以分为三大类,其前向奇性分别是sin~(-4)(θ/2)、sin~(-2)(θ/2)和1.也计算了电子-电子散射的16种极化散射截面,它们既可以按前向奇性分为三大类,也可以按后向奇性分为三大类.三类奇性的特征与电子-正电子散射类似.对于三类奇性的出现给出了直观的物理说明.     

边界元法分析二维线弹性裂纹扩展

用边界元法研究裂纹扩展过程.首先将尖端区域Williams渐近展开的特征分析法与边界积分方程结合,解出切口尖端附近应力奇异性区域的各应力场渐近展开项系数,获得平面切口/裂纹结构完整的位移和应力场.再基于考虑非奇异应力项贡献的最大周向应力脆性断裂准则,运用边界元法分析边缘含裂纹半圆形弯曲试样在荷载作用下的启裂方向,对裂纹扩展过程给出自动跟踪方法,通过算例证明边界元法模拟裂纹扩展过程的正确性和有效性.     

圆介质参考波导特征方程的近似式及其应用

本文将作为参考波导的圆介质波导模式的特征方程进行了有效的近似简化,使之成为独立的四个简单特征方程组。利用这些方程组作为参考波导的特征方程来计算各种复杂波导的传输参数时,能够大大简化计算工作。     

扩展的双曲函数法和Zakharov方程组的新精确孤立波解

借助于符号计算软件Maple，利用扩展的双曲函数法求出了Zakharov方程组的精确孤立波解，包括钟状孤立波解、扭结状孤立波解、包络孤立波解、奇性孤立波解和一种新的形式的孤立波解.这种方法也适用于其他非线性波方程.     

基于广义交替数值通量的局部间断Galerkin方法求解二维波动方程

波的传播往往在复杂的地质结构中进行,如何有效地求解非均匀介质中的波动方程一直是研究的热点.本文将局部间断Galekin(local discontinuous Galerkin, LDG)方法引入到数值求解波动方程中.首先引入辅助变量,将二阶波动方程写成一阶偏微分方程组,然后对相应的线性化波动方程和伴随方程构造间断Galerkin格式;为了保证离散格式满足能量守恒,在单元边界上选取广义交替数值通量,理论证明该方法满足能量守恒性.在时间离散上,采用指数积分因子方法,为了提高计算效率,应用Krylov子空间方法近似指数矩阵与向量的乘积.数值实验中给出了带有精确解的算例,验证了LDG方法的数值精度和能量守恒性;此外,也考虑了非均匀介质和复杂计算区域的计算,结果表明LDG方法适合模拟具有复杂结构和多尺度结构介质中的传播.     

超导材料的混沌现象

讨论了一维稳态和发展型Ginzburg-Landau(缩写为G-L)超导方程组,从数学上论证了常稳态解的不稳定性及无穷维动力系统在行波解意义下的极限集之存在性.指出G-L超导方程组描述的超导材料具有作者意义下的无穷维动力系统的混沌现象,即超导材料在传导中可能出现奇特现象 关键词： 超导材料 G-L超导方程组 无穷维动力系统 混沌现象     

水平层状介质并矢Green函数的递推矩阵方法

采用递推矩阵方法计算任意数目水平层状介质的并矢Green函数.根据层界面处电场和磁场的连续性条件得到3个确定Sommerfeld积分待定系数的矩阵方程组,分别对应于垂向单位电偶极子产生的TM波、水平方向单位电偶极子产生的TE波和TM波,这些方程组均可通过递推方法快速求解.只需改变3个方程组中源项元素的位置,就可以方便地得到当源点和场点在任意层时的并矢Green函数.本文给出的并矢Green函数表达式形式简洁且不含指数增加项,计算时不会出现溢出现象.     

标量波传播问题的双互易精细积分法

为避免使用计算多种特征频率下的声场响应,采用双互易方法将边界积分方程中时间二次导数项的域积分转化为边界积分.首先,将计算场点配置在边界上并考虑边界条件,可以获得由内部节点上声压量线性表示的边界节点上的物理量;其次,将计算场点配置于域内离散节点上,将所得边界积分方程组中关于边界物理量用内部节点的声压量线性表示,获得关于声压量的二阶常微分方程组;第三,引入声压变化速度作为未知量,将二阶常微分方程组转化为一阶常微分方程组;最后,采用精细积分法精确求解常微分方程组.数值算例验证了双互易精细积分法的正确性和稳定性.     

The effect of the boundary conditions on in-plane and out-of-plane stress field in three dimensional plates weakened by free-clamped V-notches

Dealing with the material microstructure an analytical multiscale model has recently been developed by Sih. Physically, the different orders of the stress singularities are related to the different constraints associated with the defect thought as a microscopic V-notch at the tip of the main crack. Irregularities of the material microstructure tend to curl the crack tip being the clamped-free boundary conditions the more realistic and general representation of what occurs on the microscopic V-notch. As a result, mixed mode conditions are always present along the V-notch bisector line.It is known for a long time that at the antisymmetric (mode II) stress distribution ahead of the crack tip generates a coupled out-of-plane singular mode. Recent theoretical and numerical analyses have demonstrated that this out-of-plane mode due to three-dimensional effects occurs also in the case of large V-notches where the mode II stress field is no longer singular. In addition, when the notch opening angle is non-zero, the three-dimensional singular stress state is strongly influenced by the plate thickness.The aim of this paper is to investigate the effect of free-fixed boundary conditions along the notch edges in three dimensional plates weakened by pointed V-notches and quantify the intensity of the out-of-plane singularity occurring under this constraint configuration. For the sake of simplicity a macronotch is considered rather than a micronotch. A synthesis of the magnitude of the stress state through the plate thickness is carried out by using the mean value of the strain energy density over a given control volume embracing the notch tip. The capability of the strain energy density to capture all the combined effects due to the out-of-plane mode make it a powerful parameter at every scale levels.     

Spherical Gravitational Collapse: Tangential Pressure and Related Equations of State

We derive an equation for the acceleration of a fluid element in the spherical gravitational collapse of a bounded compact object made up of an imperfect fluid. We show that non-singular as well as singular solutions arise in the collapse of a fluid initially at rest and having only a tangential pressure. We obtain an exact solution of the Einstein equations, in the form of an infinite series, for collapse under tangential pressure with a linear equation of state. We show that if a singularity forms in the tangential pressure model, the conditions for the singularity to be naked are exactly the same as in the model of dust collapse.     

Nonsingular Collapse of a Perfect Fluid Sphere Within a Dilaton-Gravity Reformulation of the Oppenheimer Model

A generic four-dimensional dilaton gravity is considered as a basis for reformulating the paradigmatic Oppenheimer–Synder model of a gravitationally collapsing star modelled as a perfect fluid or dust sphere. Initially, the vacuum Einstein scalar-tensor equations are modified to Einstein–Langevin equations which incorporate a noise or micro-turbulence source term arising from Planck scale conformal, dilaton fluctuations which induce metric fluctuations. Coupling the energy-momentum tensor for pressureless dust or fluid to the Einstein–Langevin equations, a modification of the Oppenheimer–Snyder dust collapse model is derived. The Einstein–Langevin field equations for the collapse are of the form of a Langevin equation for a non-linear Brownian motion of a particle in a homogeneous noise bath. The smooth worldlines of collapsing matter become increasingly randomised Brownian motions as the star collapses, since the backreaction coupling to the fluctuations is non-linear; the input assumptions of the Hawking–Penrose singularity theorems are then violated. The solution of the Einstein–Langevin collapse equation can be found and is non-singular with the singularity being smeared out on the correlation length scale of the fluctuations, which is of the order of the Planck length. The standard singular Oppenheimer–Synder model is recovered in the limit of zero dilaton fluctuations.     

Elastic analysis of a mode Ⅱ crack in an icosahedral quasicrystal

Based on the displacement potential functions, the elastic analysis of a mode II crack in an icosahedral quasicrystal is performed by using the Fourier transform and dual integral equation theory. By the solution, the analytic expressions for the displacement field and stress field are obtained. The asymptotic behaviours of the phonon and phason stress fields around the crack tip indicate that the stresses near the crack tip exhibit a square root singularity. The most important physical quantities of fracture theory, crack stress intensity factor and energy release rate, are evaluated in an explicit version.     

Discrete phase space II. Relativistic lattice wave equations and divergence-free green's functions

Therelativistic lattice Klein-Gordon equation, Dirac equation, electromagnetic equations, and gauge field equations are presented as partialdifference equations. Various lattice Green's functions are constructed (except for non-abelian gauge fields). It is proved that many of the lattice Green's functions are non-singular or divergence-free. Moreover, it is conjectured that all lattice Green's functions are non-singular.     

三维预处理技术初探

本文应用高阶近似LU分解法对三维椭圆型方程的正规七点差分矩阵给出了预处理阵,并对典型问题给出了预处理共轭梯度法的计算结果。文中所采用的分阶方论以及阶矩阵等概念,适用于一般的非奇异对角优势稀疏矩阵。对不同网格点数的计算结果表明,三维预处理共轭梯度法仍具有超线性的收敛速率,在高阶情形下超线性的特征尤为突出,由于非零对角线数增长过快,高阶的三维预处理方法不宜采用,零阶方法和一阶方法是值得推荐的。