Explicit energy calculation for a charged elliptical plate

Potential problems for thin elliptical plates are solved exactly with emphasis on computation of the electrostatic energy. Expansions in terms of Jacobi polynomials are used.     

拉伸偏心椭圆孔板的应力集中系数表达式

先用半解析半经验的方法推导出拉伸中心椭圆孔有限宽板应力集中系数的显式表达式．将其计算结果和Durelli的光弹性实验结果、Isida公式以及有限元分析结果比较可知,新推导公式的精度较高,且适用范围更广．再用类似的方法推导出拉伸偏心椭圆孔板应力集中系数的显式表达式．经与Isida的公式和有限元分析结果比较可知,该公式适用范围更广、精度更高．当偏心距在一定范围内,误差小于8%．根据应力集中系数与应力强度因子的关系,由已得到的应力集中系数得出拉伸中心裂纹有限宽板和拉伸偏心裂纹板的应力强度因子．经与已有公式以及有限元分析结果比较可知,该应力强度因子表达式也有足够的精度．     

Numerical simulation of sheath plasma by the particle method taking account of coulomb collisions

We study the problem of the behavior of a plasma bounded longitudinally by an absorbing sheath. This model contains charged particles (electrons and ions) moving subject to a self-consistent electrostatic field. New particle pairs are generated in the region of a distributed source. As a numerical model we used the electrostatic “particle-in-cell” method supplemented by the Emmert model for a bulk source and the algorithm of binary Coulomb collisions using the Monte Carlo method. We give a mathematical statement of the problem. The computations were carried out using the direct implicit method with the “explicit limit” time step. The results of numerical simulation of this system are given. We consider the formation and evoluiton of potential structures (multiple weak nonmonotonic double layers). Five figures. Bibliography: 35 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 75–89.     

Turbulent Energy Spectrum via an Interaction Potential

For a large system of identical particles interacting by means of a potential, we find that a strong large scale flow velocity can induce motions in the inertial range via the potential coupling. This forcing lies in special bundles in the Fourier space, which are formed by pairs of particles. These bundles are not present in the Boltzmann, Euler and Navier–Stokes equations, because they are destroyed by the Bogoliubov–Born–Green–Kirkwood–Yvon formalism. However, measurements of the flow can detect certain bulk effects shared across these bundles, such as the power scaling of the kinetic energy. We estimate the scaling effects produced by two types of potentials: the Thomas–Fermi interatomic potential (as well as its variations, such as the Ziegler–Biersack–Littmark potential), and the electrostatic potential. In the near-viscous inertial range, our estimates yield the inverse five-thirds power decay of the kinetic energy for both the Thomas–Fermi and electrostatic potentials. The electrostatic potential is also predicted to produce the inverse cubic power scaling of the kinetic energy at large inertial scales. Standard laboratory experiments confirm the scaling estimates for both the Thomas–Fermi and electrostatic potentials at near-viscous scales. Surprisingly, the observed kinetic energy spectrum in the Earth atmosphere at large scales behaves as if induced by the electrostatic potential. Given that the Earth atmosphere is not electrostatically neutral, we cautiously suggest a hypothesis that the atmospheric kinetic energy spectra in the inertial range are indeed driven by the large scale flow via the electrostatic potential coupling.

     

The interaction between a screw dislocation and a rigid line in a confocal elliptical inhomogeneity

The interaction between a screw dislocation and an elastic elliptical inhomogeneity which contains a confocal rigid line is investigated. The screw dislocation is located inside either the elliptical inhomogeneity or the infinite matrix. By using the complex potential method, explicit series solutions of complex potentials are obtained. The image force acting on the screw dislocation and the stress intensity factor at the tip of the rigid line are derived. As a result, the analysis and discussion show that the influence of the rigid line on the interaction effects between a screw dislocation and an elliptical inhomogeneity is significant. The rigid line enhances the repulsive force exerted on the dislocation produced by the stiff inhomogeneity and abates the attractive force produced by the soft inhomogeneity. For the soft inhomogeneity, there is an unstable equilibrium position when the dislocation is inside the matrix and there is a stable equilibrium position when the dislocation is inside the inhomogeneity. The stress intensity factor contour around the rigid line tip shows that when a dislocation with positive burgers vector is in the upper half-plane, stress intensity factor will be positive; while in the lower half-plane, stress intensity factor will be negative; and in the x-axis, it will be zero. The absolute value of the stress intensity factor will increase when the dislocation approaches the tip of the rigid line. The stress intensity factor at the rigid line tip is enhanced by a harder matrix and abated by a softer matrix.     

The Sample Covariance Is Not Efficient for Elliptical Distributions

It is well known that the sample covariance is not an efficient estimator of the covariance of a bivariate normal vector. We extend this result to elliptical distributions and we propose a simple explicit estimator, which is efficient in the normal case and which outperforms the sample covariance in general. Necessary and sufficient conditions are established under which this estimator is in general efficient for an elliptical distribution.     

半导体问题的交替方向有限元方法

该文用交替方向有限元方法求解半导体问题的Energy Trans port (ET)模型。对模型中椭圆型的电子位势方程采用交替方向迭代法，对流占优扩散的电子浓度和空穴浓度方程采用特征交替方向有限元方法，热传导方程利用Patch逼近采用交替方向有限元方法求解。利用微分方程的先验估计理论和技巧，分别得到了椭圆型方程和抛物型方程的最优H+1和L+2误差估计。     

一类具有Taylor阻尼的Klein-Gordon方程解的稳定性分析

研究了一类具有Taylor阻尼的Klein-Gordon方程解的稳定性.通过对松弛函数及初始值进行适当的限制,首先基于位势井理论,得到了整体解的存在性,然后构造新的能量函数,利用凸函数的性质及扰动能量方法,得到了显式的能量衰减估计.研究过程中弱化了对松弛函数的限制,同时得到了更广泛的能量衰减速率估计.     

具有随厚度变化膨胀特征应变的弹性层的分析

提出了含随厚度变化膨胀特征应变的弹性层在任意载荷作用下的一般分析步骤。该研究基于状态空间方法和一种渐近展开技巧。如果外载均匀,则展开式系数自前几项后为零,从而可得弹性层力学场量封闭形式的表达式。这一表达式仅依赖于中面的位移分量,后者由一组类似于经典板理论的二维微分方程控制。因此,获得二维方程的解便可立即给出弹性层的三维响应。作为例子,还详细分析了均匀分布载荷作用下的夹支椭圆层。     

多椭圆孔有限大复合材料层板的热弹性分析

本文利用各向异性体平面热传导，热弹性理论中的复势方法，以保角映射，Ｆａｂｅｒ级数展开以及最小二乘边界配置技术为工具，导出了内边界条件精确满足，外边界条件近似满足的多椭圆孔复合材料层板的热传导以及热弹性问题的级数解，详细探讨了层板大小，孔径，相对孔距，孔的设置方式，椭圆度以及层板的铺层比例诸参数的影响规律，得到了一些有益结论。     

On the Solution of Some Non-Local Problems

This paper deals with two types of non-local problems for the Poisson equation in the disc. The first of them deals with the situation when the function value on the circle is given as a combination of unknown function values in the disc. The other type deals with the situation when a combination of the value of the function and its derivative by radius on the circle are given as a combination of unknown function values in the disc. The existence and uniqueness of the classical solution of these problems is proved. The solutions are constructed in an explicit form.     

Linear and nonlinear properties of electron-acoustic solitary waves with non-thermal electrons

Propagation of linear and nonlinear electron-acoustic waves (EAWs) in an unmagnetized collisionless plasma consisting of a cold electron fluid, non-thermal hot electrons and stationary ions are investigated. The standard normal-mode analysis is used to study the stability condition of linear (EAWs) waves. For nonlinear (EAWs) waves, a reductive perturbation method was employed to obtain a Korteweg–de Vries (KdV) equation for the first-order potential. The effects of a non-thermal distribution of hot electrons on the amplitude, width and energy of electrostatic solitary structures are also discussed.     

夹杂与基体对界面层螺旋位错的干涉效应

研究圆形夹杂与基体对有限厚度界面层螺旋位错的干涉问题。结合复变函数的分区亚纯函数理论、施瓦兹对称原理与柯西型积分运算,发展了多连通域联结问题的一个有效分析方法,将3个区域应力函数的联结问题化归为界面层应力函数的函数方程,并求得了显式级数解。利用该结果,研究与讨论了界面层螺旋位错能与位错力。     

Nonlinear mobility continuity equations and generalized displacement convexity

We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a nonrigorous argument indicating that they are not displacement semiconvex.     

On numerical schemes for phase-field models for electrowetting with electrolyte solutions

We present an energy-stable, decoupled discrete scheme for a recent model (see [1]) supposed to describe electrokinetic phenomena in two-phase flow with general mass densities. This model couples momentum and Cahn–Hilliard type phase-field equations with Nernst–Planck equations for ion density evolution and an elliptic transmission problem for the electrostatic potential. The transport velocities in our scheme are based on the old velocity field updated via a discrete time integration of the force densities. This allows to split the equations into three blocks which can be treated sequentially: The phase-field equation, the equations for ion transport and electrostatic potential, and the Navier–Stokes type equations. By establishing a discrete counterpart of the continuous energy estimate, we are able to prove the stability of the scheme. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fully discrete energy stable scheme for a phase-field moving contact line model with variable densities and viscosities

In this study, we propose a fully discrete energy stable scheme for the phase-field moving contact line model with variable densities and viscosities. The mathematical model comprises a Cahn–Hilliard equation, Navier–Stokes equation, and the generalized Navier boundary condition for the moving contact line. A scalar auxiliary variable is employed to transform the governing system into an equivalent form, thereby allowing the double well potential to be treated semi-explicitly. A stabilization term is added to balance the explicit nonlinear term originating from the surface energy at the fluid–solid interface. A pressure stabilization method is used to decouple the velocity and pressure computations. Some subtle implicit–explicit treatments are employed to deal with convention and stress terms. We establish a rigorous proof of the energy stability for the proposed time-marching scheme. A finite difference method based on staggered grids is then used to spatially discretize the constructed time-marching scheme. We also prove that the fully discrete scheme satisfies the discrete energy dissipation law. Our numerical results demonstrate the accuracy and energy stability of the proposed scheme. Using our numerical scheme, we analyze the contact line dynamics based on a shear flow-driven droplet sliding case. Three-dimensional droplet spreading is also investigated based on a chemically patterned surface. Our numerical simulation accurately predicts the expected energy evolution and it successfully reproduces the expected phenomena where an oil droplet contracts inward on a hydrophobic zone and then spreads outward rapidly on a hydrophilic zone.     

The Stability Theory of Charge Distributions on Conductors

Stability of solution is considered for the charge-density problemof several rigid charged conductors, stability being with respectto the geometrical configuration of the system. Under the analyticalrestrictions usual in potential theory for the surfaces on whichthe charges reside, two theorems are proved establishing thestability: the first for when the potentials of the conductorsare fixed, and the second for when their total charges are fixed.Subsequently, it is shown how the rigidity condition can berelaxed, but a counterexample to stability is described fora situation where the surface analytical conditions are notfulfilled. Finally, an application of the theory is given, inwhich the second of the said theorems is used to close a gapin the theory by the energy method of the electrostatic forceon a conductor.     

An image system and surface singularity solutions for potential flow past an elliptical cylinder

The Milne-Thomson circle theorem is extended to give a simplegeneral expression for the image system in an elliptical cylinderintroduced into an otherwise specified unbounded potential flowwhich contains no singularities in the region to be occupiedby the ellipse. This image system is used to obtain an expressionfor the corresponding source-sink surface singularity distributionon the ellipse, thus providing new benchmark test cases forsource-sink solutions as obtained numerically by a panel method.Several typical examples are given to illustrate the generaltheoretical approach.     

Spatial Expectile Predictions for Elliptical Random Fields

In this work, we consider an elliptical random field. We propose some spatial expectile predictions at one site given observations of the field at some other locations. To this aim, we first give exact expressions for conditional expectiles, and discuss problems that occur for computing these values. A first affine expectile regression predictor is detailed, an explicit iterative algorithm is obtained, and its distribution is given. Direct simple expressions are derived for some particular elliptical random fields. The performance of this expectile regression is shown to be very poor for extremal expectile levels, so that a second predictor is proposed. We prove that this new extremal prediction is asymptotically equivalent to the true conditional expectile. We also provide some numerical illustrations, and conclude that Expectile Regression may perform poorly when one leaves the Gaussian random field setting.     

Numerical studies of time-independent and time-dependent scattering by several elliptical cylinders

A numerical solution to the problem of time-dependent scattering by an array of elliptical cylinders with parallel axes is presented. The solution is an exact one, based on the separation-of-variables technique in the elliptical coordinate system, the addition theorem for Mathieu functions, and numerical integration. Time-independent solutions are described by a system of linear equations of infinite order which are truncated for numerical computations. Time-dependent solutions are obtained by numerical integration involving a large number of these solutions. First results of a software package generating these solutions are presented: wave propagation around three impenetrable elliptical scatterers. As far as we know, this method described has never been used for time-dependent multiple scattering.