旋转叶轮内含激波跨声速三元流动的变分原理与广义变分原理(Ⅱ)

作为文献[1]的继续和发展,本文将为势流、Beltrami流及旋流建立流函数的变分原理族。主要特点是应用泛函变域变分工具成功地处理了一些未知界面和间断面(激波、自由尾涡面、上下游驻点流面). 首先,列出转轮内理想流体三元相对定常绝热流动(图1)的气动方程组:     

变分法初步

本文介绍变分法的基础知识，它包括：泛函、函数的变分和泛函的变分等概念；变分的计算方法和运算规则；以及泛函的极值条件－欧拉方程等．     

透平S_2流面气动半反命题有限元解

一、变分原理 吴仲华教授的三元流动理论已在国内外透平机械计算中获得了广泛的应用。但基于吴氏理论的变分有限元计算实属少见。近几年来刘高联基于两个流面理论对透平机械中多种气动命题找到了泛函表达式,并编制了一些有限元程序。但S_2流面的计算尚少。 基于S_2流面半反命题的气动基本方程组,引入流函数后可得下列主方程:     

一类非线性扰动Burgers方程的孤子变分迭代解法

研究了一类非线性扰动Burgers方程的求解问题. 利用变分迭代方法, 首先引入一个泛函, 然后计算它的变分, 最后构造方程的迭代关系式, 得到了相应方程的孤子解的近似展开式.     

含冷却水管大体积混凝土温度场计算的一种新方法

水管冷却是混凝土坝施工期的主要温度控制措施,提出一种新的水管冷却数值模拟理论和计算方法.该方法将水管置于常规混凝土单元内部,在单元中把混凝土与水管的接触面作为散热面纳入控制方程的边界条件,把混凝土通过冷却水管壁面耗散的能量叠加到常规泛函中,根据此复合泛函由变分原理建立含冷却水管混凝土的有限元支配方程.编制相应的三维计算程序,在程序中水管网格的拓扑信息由程序自动完成,冷却水管可以从混凝土单元任意位置穿过,因此不增加网格剖分的难度,算例表明该方法能准确模拟混凝土的冷却效应.     

非线性广义Landau-Ginzburg-Higgs方程孤子解的变分迭代解法

利用扰动方法研究了一类非线性广义Landau-Ginzburg-Higgs方程.首先引入一个泛函，计算它的变分，求得Lagrange乘子.然后构造了原方程解的迭代关系式，得到了近似解. 关键词： 孤子 扰动 变分迭代     

导热几何形状反演的变分原理及边界条件的确立

本文采用变域变分原理,建立了导热几何形状反演问题的变分原理,同时获得了该问题所需满足的边界条件和附加条件.该变分原理能将未知形状的几何变量及控制方程结合在一个变分泛函中,使得数学描述简洁、紧凑,且几何变量及控制方程的求解能耦合地进行.介绍了运用该变分原理并结合有限元方法进行数值计算的方法. 关键词： 几何形状反演 变分原理 有限元 导热     

由电磁测量信号估算平衡磁面

在Tokamak等离子体实验结果分析中实际的平衡磁面是极需要的。本文试探从电磁测量信号推得平衡磁面。 一、自由边界平衡程序 借助于自由边界平衡程序计算中等托卡马克装置HT-6M的平衡量。从MHD平衡方程出发,     

强非线性发展方程孤波近似解

研究了一个强非线性发展方程.利用变分原理，首先构造了相应的泛函.选取Lagrange乘子，再用广义变分迭代方法得到了孤波的任意次精度的近似解. 关键词： 发展方程 非线性 孤立子 近似方法     

求解结构动力响应的卷积型DQ半解析法

通过卷积将原始控制方程构造成包含初始条件的新的具有完整初值问题特征的控制方程.该方程既与Gurtin变分原理一样有合理的数学内涵,又避免了卷积型Gurtin变分原理泛函和计算的繁复.对新的控制方程在时间域取解析函数,在空间域采用离散的DQ法,经对梁的动力响应问题的计算表明,该方法是一种精度好效率高的求解动力响应问题的计算方法.     

The kinetic boundary layer for the Klein-Kramers equation; A new numerical approach

We explore a numerical technique for determining the structure of the kinetic boundary layer of the Klein-Kramers equation for noninteracting Brownian particles in a fluid near a wall that absorbs the Brownian particles. The equation is of interest in the theory of diffusion-controlled reactions and of the coagulation of colloidal suspensions. By numerical simulation of the Langevin equation equivalent to the Klein-Kramers equation we amass statistics of the velocities at the first return to the wall and of the return times for particles injected into the fluid at the wall with given velocities. The data can be used to construct the solutions of the standard problems at an absorbing wall, the Milne and the albedo problem. We confirm and extend earlier results by Burschka and Titulaer, obtained by a variational method vexed by the slow convergence of the underlying eigenfunction expansion. We briefly discuss some further boundary layer problems that can be attacked by exploiting the results reported here.     

反射率的数值反演

通过分析热流边界条件下的一维热传导过程,提出并实现了由后表面温度变化历史曲线反演前表面反射率变化历史曲线的数值方法。通过一维热传导的正向和反演计算过程,验证了反演程序的可靠性。对数值反演计算和实验测量两种方式得到的反射率曲线进行了对比,结果显示,两条反射率曲线在变化趋势上符合较好,初步验证了一维温度场中数值反演方法的可行性。     

Exact boundary condition for time-dependent wave equation based on boundary integral

An exact non-reflecting boundary conditions based on a boundary integral equation or a modified Kirchhoff-type formula is derived for exterior three-dimensional wave equations. The Kirchhoff-type non-reflecting boundary condition is originally proposed by L. Ting and M.J. Miksis [J. Acoust. Soc. Am. 80 (1986) 1825] and numerically tested by D. Givoli and D. Cohen [J. Comput. Phys. 117 (1995) 102] for a spherically symmetric problem. The computational advantage of Ting–Miksis boundary condition is that its temporal non-locality is limited to a fixed amount of past information. However, a long-time instability is exhibited in testing numerical solutions by using a standard non-dissipative finite-difference scheme. The main purpose of this work is to present a new exact boundary condition and to eliminate the long-time instability. The proposed exact boundary condition can be considered as a limit case of Ting–Miksis boundary condition when the two artificial boundaries used in their method approach each other. Our boundary condition is actually a boundary integral equation on a single artificial boundary for wave equations, which is to be solved in conjunction with the interior wave equation. The new boundary condition needs only one artificial boundary, which can be of any shape, i.e., sphere, cubic surface, etc. It keeps all merits of the original Kirchhoff boundary condition such as restricting the temporal non-locality, free of numerical evaluation of any special functions and so on. Numerical approximation to the artificial boundary condition on cubic surface is derived and three-dimensional numerical tests are carried out on the cubic computational domain.     

An efficient numerical method for the equations of steady and unsteady flows of homogeneous incompressible Newtonian fluid

In the present paper, we present some numerical methods to solve the equations of steady and unsteady flows, such as those in the microcirculatory bed and large blood vessels (arteries and veins), respectively. In the case of steady flows, the method does not need neither any boundary conditions on pressure nor any small parameter, and the main computation consists of solving some Poisson equations. In the case of unsteady flows, the scheme uses a consistent Neumann boundary condition for the pressure Poisson equation. At each time step, a Poisson and heat equation are solved for the pressure and each velocity component, respectively. The accuracy and efficiency of scheme are checked by a set of numerical tests.     

A variational treatment of the mixed boundary condition for the equilibrium diffusion equation

We formulate the asymptotic boundary layer analysis which leads to a mixed (Robbin) boundary condition for the equilibrium diffusion equation of radiative transfer. The information required from the nonlinear boundary layer equations to obtain the boundary condition is obtained by deriving and applying an appropriate variational principle. Based upon an examination of certain limiting cases and the good accuracy previously reported in the literature for similar variational treatments, the accuracy of this variational procedure is expected to be quite good over a wide range of conditions.     

定态对流扩散方程的渐近数值方法

本文提出了一种求解定态对流扩散方程的渐近数值方法,在边界层附近不必取很细的网格,对模型问题的数值计算表明,利用中等大小的步长就可得到边界层内的数值解。     

A numerical comparison between two unconstrained variational formulations

In an effort to relieve the often cumbersome burden of meeting the requirements of the end conditions and to unify the solution formulation for boundary- and initial-value problems, unconstrained variational statements have been introduced in conjunction with some approximate methods. In the case of a boundary value problem, it is shown in this paper that two different variational statements can be established: one is arrived at by the use of the Lagrange multipliers, the other by energy considerations. The numerical convergence of the solutions associated with finite element schemes involving use of one of these two different variational statements is compared with that of the other. In the case of an initial value problem, both formulations can again be established when the adjoint field variable and the adjoint variational statement are introduced. The numerical data presented here indicate that while both methods generate excellent convergent results for the boundary problem, the method of stiff springs yields results which show much better convergence for the initial value problem than those achieved by Lagrange multipliers.     

The density profile for the Klein-Kramers equation near an absorbing wall

We derive asymptotic series for the expansion coefficients of a function in terms of the Pagani functions, which occur in the boundary layer solutions of the Klein-Kramers equation. The results enable us to determine the density profile in the stationary solution of this equation near an absorbing wall from the numerically determined velocity distribution at the wall, with an accuracy of about 2%. We also obtain information about the analytic behavior of the density profile: this profile increases near the wall with the square root of the distance to the wall. Finally, the asymptotic analysis leads to an understanding of the slow convergence of variational approximations to the solution of the absorbing-wall problem and of the exponents that occur when one studies the variational approximations to various quantities of interest as functions of the number of terms in the variational ansatz. This is used to obtain a better variational estimate for the density at the wall.