线性哈密顿系统的本征摄动法

利用哈密顿算子辛自共轭的特点讨论了保守哈密顿体系的摄动问题,给出了哈密顿矩阵的本征值与本征向量的二阶摄动分析方法。即当系统在哈密顿框架下进行较小修改时,不重复求解大型哈密顿矩阵的本征问题,只需在原系统的模态参数基础上进行模态分析即可,这种矩阵摄动法给出了修改后矩阵的二阶本征值和本征向量,为一般线性保守体系的本征摄动求解提出了一个新方法。     

Stokes 流问题中的辛本征解方法

通过引入哈密顿体系，将二维Stokes流问题归结为哈密顿体系下的本 征值和本征解问题. 利用辛本征解空间的完备性，建立一套封闭的求解问题方法. 研究结果 表明零本征值本征解描述了基本的流动，而非零本征值本征解则显示着端部效应影响特点. 数值算例给出了辛本征值和本征解的一些规律和具体例子. 这些数值例子说明了端部非规则 流动的衰减规律. 为研究其它问题提供了一条路径.     

辛体系下曲线形水中悬浮隧道对波浪激励的动力响应研究

在哈密顿体系下建立曲线形水中悬浮隧道的运动方程,即哈密顿正则方程.用哈密顿混合能变分原理处理曲线形水中悬浮隧道的两端边界条件以及径向布设的张力腿弹性支承,建立满足边界条件的线性方程组,采用分离变量法和本征向量展开法求解哈密顿正则方程.为曲线形水中悬浮隧道的动力响应分析提供一类方法.最后结合案例,运用辛方法分析波浪作用下曲线形水中悬浮隧道的弯矩和扭矩,结果表明,可以忽略扭矩对弯矩的影响.     

圆柱型正交各向异性弹性楔的佯谬解

圆柱型正交各向异性弹性楔体顶端受有集中力偶的经典解,当顶角满足一定关系时,其应力成为无穷大,这是个佯谬．该文在哈密顿体系下将该问题进行重新求解,即利用极坐标各向异性弹性力学哈密顿体系．在原变量和其对偶变量组成的辛几何空间求解特殊本征值的约当型本征解,从而直接给出该佯谬问题的解析解．结果再次表明经典力学中的弹性楔佯谬解对应的是哈密顿体系下辛几何的约当型解．     

空腔内粘性流问题与哈密顿体系方法

以双板驱动空腔粘性流问题为研究对象,根据其特点建立了哈密顿体系下的对偶正则方程,将问题归结为辛体系下的本征值问题.利用辛本征解空间的完备性、正交性和展开理论,形成一套封闭的求解问题方法.算例的数值结果揭示了一些空腔流动的特点.同时这种方法也为研究其他问题提供了一条思路.     

哈密顿体系与弹性楔体问题

将哈密体系引入到级坐标下的弹性力学楔体问题,利用该体系辛空间的性质,将问题化为本征值和本征向量求解上,得到了完备的解空间,从而改变了弹性力学传统的拉格朗日体系以应力函数为特征的半逆法的讨论去解决该类问题的思路,给出了一条求解该类问题的直接法。     

粘弹性厚壁筒问题的辛本征解方法

将哈密顿体系引进到粘弹性力学厚壁筒问题中,在辛体系下重新描述了基本问题,即建立了正则方程组。借助于积分变换,得到了拉伸、扭转和弯曲等问题的解以及有边界局部效应的解。将原问题归结为辛几何空间中的零本征值本征解和非零本征值本征解问题,从而建立了一种有效的分析问题方法和数值方法。为解决同类问题提供了一条可行的路径。     

环扇形薄板弯曲问题的环向辛对偶求解方法

根据平面弹性与薄板弯曲问题的相似性原理,极坐标系板弯曲的弯矩函数被引入作为原变量,并通过恰当的辛内积定义建立了环扇形薄板弯曲问题的一个辛几何空间.然后应用类Hellinger-Reissner变分原理,导出了辛几何窄问的对偶方程,从而将环扇形薄板弯曲问题导入到辛对偶求解体系.于是,分离变量和本征展开的有效数学物理方法得以实施,给出环扇形薄板弯曲问题的一个分析求解方法.具体讨论了两弧边简支和两弧边一边固支一边自由薄板的本征问题,分别导出它们对应的本征值超越方程和本征向量,并给出原问题本征展开形式的通解.最后,给出了两个算例的分析解并与已有文献或数值方法的解进行了对比,结果表明该方法有很好的收敛性和精度.     

固支矩形Mindlin板弯曲问题的辛求解体系

本文利用二类变量广义变分原理推出了Mindlin板弯曲问题的Hamilton体系，利用辛几何方法对全状态向量进行分离变量，得到相应的横向本征问题，在求出其本征值后，按本征函数展开法导出了原问题的辛本征通解。给出了一个承受集中载荷的四边固支矩形薄板的算例，按本文求解体系得到的解与经典解吻合较好。本文直接从Mindlin板弯曲问题出发，在其Hamilton体系内使用辛几何方法给出了的一套新的求解体系，突破了传统解法的局限性，具有一般性及较高的理论推广价值。     

陀螺系统与反对称矩阵辛本征解的计算

用状态向量法,引出陀螺线性系统的广义本征问题,证明了本征向量之间的加权共轭辛正交关系,以及用本征向量对任意状态向量的展开定理。运用反对称矩阵胞块组成的LDL~T分解,将本征方程导向辛本征问题的标准型。这套方法适用于陀螺系统K阵不正定的情形。对于辛本征问题用SH变换将矩阵化为半边三对角线胞块阵或三对角线胞块阵,然后再求解其全部本征解。为陀螺系统的模态分析打下了基础。     

OBJECTIVITY REQUIREMENT FOR FLUID DYNAMICS

IntroductionAsextensionoftheNewtonsecondlaw ,thefundamentaltheoryofclassicalfluiddynamicsconsidersthefluidelementwithdensityρasfluidparticle,anddescribestherelationbetweenthevarietyanddistributionofvelocityVoverspace_timevolumeelementdv ∧dt,namely ,themomentum_impulseequationsbyρdV dt=ρf - P+ ·σ(V , V ,…) ,(1 )wheref,P ,σareexternal (body)force ,pressureandviscous (area)force ,respectively .Therepresentationsofmotionandforceusuallyneedspecificframeofreference.Forinstance,inthehomog…     

Domain Decomposition Methods for Sensitivity Analysis of a Nonlinear Aeroelasticity Problem

Abstract

We consider the nonlinear aeroelasticity problem of the interaction between a viscous, incompressible fluid and Lin elastic solid undergoing large displacement. The non-linearities of the problem formulation include the solid and fluid governing equations. as well as thc dependence of the How geometry on the solid deformation. The resulting coupling is thus two-way. We develop domain-decomposition methods for solution and sensitivity analysis of the coupled problem. The domain decomposition is in the form of a block-Gauss-Seidel-like prcconditioncr that decomposes ihc coupled-domain problem into distinct nonovcrlapping fluid and solid subdotnain problems. The preconditioner thus enables exploitation or single-domain algorithms for solid and fluid mechanics discretization and solution. On the other hand, two-way fluid-solid coupling is retained within the residuals, which is essential for correct sensitivities. Sensitivities of field quantities can be found with little additional work beyond that required for solving the coupled fluid-solid system. The methodology developed here is illustrated by the solution of a problem of viscous incompressible flow about an infinite clastic cylinder. Sensitivities of the resulting velocity and displacement fields with respect to elastic modulus and fluid viscosity are computed.     

Stationary cylindrical vortex in a viscous electrically conducting fluid

An exact solution of the magnetohydrodynamic equations is constructed which describes steady vortex flow in a stationary cylinder on the axis of which a conductor carrying a known current is located. The solution is obtained under the assumption that the fluid is viscous and has finite electrical conductivity and that the magnetic field has only the axial and azimuthal components in a cylindrical coordinate system. It is found that the action of the Lorentz force is compensated by changing the pressure. Fluid flow occurs from the periphery to the axis of the cylinder under a pressure gradient, with flow rotation and swirling. The fluid flow causes a concentration of the magnetic lines near the axis of the cylinder, providing an exponential decrease in the magnetic field strength with distance from the axis. This flow can be considered as a model of a local increase in the magnetic field strength due to the transfer of its force lines by the flow of the electrically conducting fluid.     

Self-similar “diffusion” of momentum in an ideal fluid

It is proposed to consider plane or axisymmetric incompressible flows when at a certain point in space a finite source of momentum is instantaneously created. This type of flow is characterized by the continuous setting in motion of new volumes of fluid with a simultaneous decrease in velocity. It is usual to associate this diffusional process with viscosity [1]. Here it is shown, that such processes can be described within the framework of an ideal fluid. The main concern is to prove the existence of four-parameter plane ideal-fluid flow. The method of constructing the solution is based on the conformal transformation of the dimensionless variable, so that from the relatively simple self-similar solution the unknown flow can be obtained. It is shown that this method can be applied to other problems. The results obtained are compared with the well-known [2] solution of the linearized dipole diffusion problem for the plane motion of a viscous fluid and with certain generalizations of that solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 73–83, July–August, 1990.The author is grateful to A. A. Barmin and A. G. Kulikovskii for useful discussions.     

Some properties of multi-fluid stokes flows

The solution of Stokes' equations for a rotating axisymmetric body which possesses reflection symmetry about a planar interface between two infinite immiscible quiescent viscous fluids is shown to be independent of the viscosities of the fluids and identical with the solution when the fluids have the same viscosity. The result is generalized to a rotating axisymmetric system of bodies which possesses reflection symmetry about each interface of a plane stratified system of fluids. An analogous result for two-fluid systems with a nonplanar static interface is also derived. The effect on torque reduction produced by the presence of a second fluid layer adjacent to a rotating axisymmetric body is considered and explicit calculations are given for the case of a sphere. A proof of uniqueness for unbounded multi-fluid Stokes' flow is given and the asymptotic far field structure of the velocity field is determined for axisymmetric flow caused by the rotation of axisymmetric bodies.     

Development of a doubly periodic system of cavities in viscous bodies

The plane unknown-boundary problem of the development of a doubly periodic system of cavities in viscous media in the presence of finite strains is considered. Under conditions of timedependent slow Newtonian viscous fluid flow the solution of the doubly periodic problem of the development of a system of identical cavities whose centers are arranged in square and triangular grids is obtained.     

Hall effects on hydromagnetic flow on an oscillating porous plate

In this paper, an analysis is made on the unsteady flow of an incompressible electrically conducting viscous fluid bounded by an infinite porous flat plate. The plate executes harmonic oscillations at a frequency n in its own plane. A uniform magnetic field Ho is imposed perpendicular to the direction of the flow. It is found that the solution also exists for blowing at the plate. The temperature distribution is also obtained by taking viscous and Joule dissipation into account. The mean wall temperature θo（O） decreases with the increase in the Hall parameter m. It is found that no temperature distribution exists for the blowing at the plate.     

基于格子-波尔兹曼法的流体能耗求解

格子-波尔兹曼法是近年来新兴的一种计算流体力学数值方法。随着这种方法的不断发展,人们将它用于流体的仿真、优化等不同场合。与此同时,一些与流场流速和压强相关的物理量(如能耗)的求解也成为关注的焦点。本文介绍了能耗这一流体宏观量的格子-波尔兹曼法求解及其实现。与传统的有限差分法不同,本文在求解有关的速度梯度时使用了格子-波尔兹曼-矩法,这种方法不但能够避免有限差分法在边界处失效的缺点,而且计算简单,算法局部性好,适合大规模并行计算。本文在分析其数值解精度的基础上,使用这种方法进行了以能耗极小为目标的直通道内椭圆挡块的参数优化。这些分析和算例分别定量和定性地说明了本文算法的准确性。     

The development of a Cartesian cut cell method for incompressible viscous flows

This paper describes the extension of the Cartesian cut cell method to applications involving unsteady incompressible viscous fluid flow. The underlying scheme is based on the solution of the full Navier–Stokes equations for a variable density fluid system using the artificial compressibility technique together with a Jameson‐type dual time iteration. The computational domain encompasses two fluid regions and the interface between them is treated as a contact discontinuity in the density field, thereby eliminating the need for special free surface tracking procedures. The Cartesian cut cell technique is used for fitting the complex geometry of solid boundaries across a stationary background Cartesian grid which is located inside the computational domain. A time accurate solution is achieved by using an implicit dual‐time iteration technique based on a slope‐limited, high‐order, Godunov‐type scheme for the inviscid fluxes, while the viscous fluxes are estimated using central differencing. Validation of the new technique is by modelling the unsteady Couette flow and the Rayleigh–Taylor instability problems. Finally, a test case for wave run‐up and overtopping over an impermeable sea dike is performed. Copyright © 2006 John Wiley & Sons, Ltd.     

On the uniqueness and stability of solutions to some continuum mechanics problems

One-dimensional shear flows of a perfect fluid, a perfect rigid-plastic body, a Newtonian viscous fluid, and a viscoplastic medium in a plane layer and the compression problem for a Euler beam are considered as examples to discuss a relation between the nonuniqueness of a solution to the basic continuum mechanics problem and the instability of the unique solution to this problem. When the basic solution is not unique, the question is raised on the mechanical interpretation of the linearized stability problem for this solution.