多孔介质中两相可混溶驱动问题的特征-混合动态有限元方法(英文)

数学上,多孔介质中一种不可压流体对另一不可压流体的相溶驱动由两个耦合的非线性偏微分方程组成,其中一个是关于压力的椭圆方程,另一个是关于浓度的抛物方程。本文用特征有限元方法结合动态有限元空间来逼近浓度,而压力和达西速度则由混合元方法来同时逼近。通过采用负模估计,我们给出了收敛性分析与误差估计。     

非线性拟双曲方程的有限元配置法数值分析

基于有限元配置法,采用分片双三次Hermite插值多项式空间作为逼近函数空间,本文对粘性振动及神经传播过程中涉及的一类非线性拟双曲方程的初边值问题建立了二维半离散和全离散格式.并对两种格式证明了数值解的存在唯一性,应用微分方程先验估计的理论和技巧得到了L2模最佳阶误差估计.数值实验结果表明:所提方法在保证整体误差估计要求且不增加计算量的前提下,比传统有限元方法有更高的逼近精度,并扩展了配置法的应用范围.     

四阶拟线性椭圆方程的有限元误差估计

针对一类四阶拟线性椭圆方程,本文给出了它的协调有限元逼近。当网格参数h足够小时,得到了有限元逼近解与真解之间的误差估计,并且这些误差估计是最优的。最后,通过数值实验验证了理论分析的准确性。证明方法可以类似地应用到某些二阶拟线性椭圆方程的有限元逼近。     

考虑弹丸与身管轴向运动耦合的火炮系统时变动力学分析

将火炮发射系统简化为移动质量作用下的轴向运动悬臂梁，基于伯努利-欧拉梁理论和广义Hamilton原理，推导了考虑梁内张力的轴向运动梁振动方程。采用变量替换的方法把时变振动方程转换为固定区域的变系数微分方程。再用假设模态法和Galerkin法对控制微分方程进行空间离散求解。利用高精度时间有限元求解该时变方程，用两个算例验算振动方程的准确性。最后数值模拟移动弹丸对身管横向振动的影响。     

抛物型积分微分方程各向异性非协调有限元分析

各向异性有限元方法的显著的优点之一就是可以用较少的自由度得到与传统有限元正则剖分时同样的估计结果.然而,在这种情况下,Sobolev空间上的:Bramble-Hilbert引理在插值误差分析中不能直接应用,而且对于非协调元来说其传统边界估计技巧也不再适用.本文证明了一个非协调单元具有各向异性特征,并将它应用到研究抛物积分微分方程半离散格式下的Galerkin逼近.利用单元的特殊性,验证了Ritz-Volterra投影与有限元插值是相同的.在解适当光滑时,通过引入一些新的技巧,得到了与传统方法相同的收敛误差估计和超逼近性质.最后,通过构造适当的插值后处理算子,得到了各向异性网格下的整体超收敛结果.该文的结果对进一步探索和设计数值的自适应算法是有帮助的.     

Brinkman-Forchheimer方程的加罚有限元方法

Brinkman-Forchheimer方程(BF方程)是具有强非线性项并满足无散度条件的流动控制方程,其中无散度条件的精确满足对控制方程的数值求解极其重要.为了放松无散度条件的限制,本文采用了加罚方法.为了得到加罚问题解的适定性,首先,利用加罚关系将压力项消去,证明了速度所满足的具有单调性的非线性椭圆变分问题等价于对应能量泛函的极小化问题,从而得到了速度的存在唯一性.进一步,利用LBB条件证明了BF方程加罚问题压力的存在唯一性.其次,证明了BF方程加罚问题的Galerkin变分问题的解关于加罚参数收敛到BF方程的Galerkin变分问题的解.最后,给出了BF方程加罚问题Galerkin变分问题的有限维逼近问题及其解的存在唯一性,并且得出了采用协调有限元离散的误差估计.数值算例表明加罚方法是有效的.     

基于有限元思想的无限微热源冶炼炉温度机理研究

对无限微热源法制备β-SiC粉体的冶炼炉进行温度机理研究,根据传热形式、传热模型推导出无限微热源冶炼炉的导热微分方程,采用Galerkin法对方程进行变分并得出平面非稳态导热温度场分布方程,利用有限元思想对方程进行推导,再采用有限差分方法求解温度矩阵以得到炉内各个时刻温度的变化情况.     

线性弹性柔性壳非协调有限元计算模型

本文基于Ciarlet-Lods-Miara定义的柔性壳模型提出一种Galerkin非协调有限元离散格式.首先,对积分区域进行Delaunay三角剖分,并在三角网格上对位移前两个分量用一次Lagrange多项式逼近,对第三个分量(即法向位移)用非协调Morley元逼近.其次,讨论了构造的Galerkin非协调有限元离散格式解的存在性、唯一性和先验误差估计.最后对特殊边界条件下的锥壳采用该方法进行数值实验,计算出不同网格下锥壳的位移,并通过分析数值实验结果证明有限元离散格式的收敛性和有效性.     

圆柱体工件高压气淬过程中的数值模拟

利用导热微分方程和单值性条件推导了轴对称短圆柱体温度分布的通用数学微分方程,可计算工件上任意点的温度和任意时间的温度分布;采用伽辽金(Galerkin)加权余量法推导了二维轴对称圆柱体瞬态温度场的有限元计算的积分方程;采用大型有限元计算软件ANSYS对5Cr1MoV轴对称实心圆柱体工件淬火过程的温度变化及热应力/应变分布状况进行了模拟计算.     

几何非线性效应对受迫振动索理论解的影响评估

为研究单跨水平拉索背景下几何非线性的强弱对Galerkin离散法适用性产生的影响,首先引入两种不同的无量纲参数，将拉索面内非线性运动方程无量纲化，运用Galerkin离散法将偏微分运动方程转化为常微分方程，利用多尺度法进行摄动求解，得到幅频响应函数，并绘制幅频响应曲线；接着应用MATLAB得出系统时间历程曲线，最后应用ABAQUS软件进行有限元模拟，将有限元解与本文方法计算得到的数值解进行对比。在通过数值模拟得到系统时间历程曲线的同时，将两种无量纲方法也进行对比。结果表明：Galerkin离散法对于几何非线性较弱的系统适用性较高。无量纲化会导致系统非线性项系数变大。两种无量纲方法随着Irvine系数的增大，所得数值解更接近。研究成果可为索的非线性振动数值求解提供依据且有助于理论的完善。     

Fractional integro-differential analysis of heat and mass transfer

Application of the methods of fractional integro-differential analysis to an inhomogeneous canonical heat-conduction (diffusion) equation with inhomogeneous boundary conditions has enabled us for the first time to reduce the canonical heat-conduction equation to three equations of lower order that contain fractional-derivative operators. Examples and an analysis of those fundamental new possibilities that are opened up by such an approach to a wide class of problems of heat and mass exchange, combustion, self-propagating high-temperature synthesis, etc., have been given.__________Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 78, No. 1, pp. 35–47, January–February, 2005.     

Calculation of static magnetization for thin-walled constructionsby boundary element method

Two methods for salving two-dimensional (2-D) and three-dimensional (3-D) problems of magnetization for arbitrary-form thin-walled ferromagnetic constructions are considered. The first method is based on replacing the universal magnetic flux balance equation for a volume element by the equation for a median-surface boundary element (BE). The magnetization field is approximated by the field of point charges placed on the vertices of BEs. The second method uses the point dipoles in the centers of BEs     

Rosenau-Burgers 方程的修正局部 Crank-Nicolson 格式

本文给出了 Rosenau-Burgers 方程的两种修正局部 Crank-Nicolson 格式．首先,求解原有的偏微分方程对空间方向进行有限差分离散而得到的常微分方程．其次,利用矩阵分裂技术对这个方程的指数系数矩阵分别按行和元素进行逼近．最后,利用修正局部 Crank-Nicolson 方法得到了两种格式．讨论了格式的稳定性、收敛性和先验误差估计．数值实验结果表明了理论证明的正确性及格式的有效性．该格式具有结构简单、精度高的优点．     

On the accuracy of finite element solutions to the transient heat-conduction equation

This paper discusses three numerical integration schemes connected with a finite element solution to the transient heat-conduction equation. The main emphasis is on the achievable accuracy. A simple recurrence relation derived from the Galerkin process is suggested for the treatment of fast-varying boundary conditions as being significantly better than the usual Crank-Nicholson scheme in matters of short-time accuracy.     

Some analytical methods of solving inverse (coefficient) problems of heat-conduction theory

Some analytical methods are presented for the determination of thermophysical parameters without linearization of the heat-conduction equation. A qualitative study of the temperature fields is used.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 33, No. 6, pp. 1090–1096, December, 1977.     

薄壁杆件翘曲剪应力的边界元精确积分解法

用非连续边界元对薄壁杆件的约束扭转进行了分析,推导出了求解边界点二次翘曲函数值的边界积分方程,给出了边界积分方程数值求解时积分计算的精确表达式。数值算例表明:利用边界积分方程方法分析薄壁杆件的约束扭转问题时效率和精度高,同时采用精确积分可以有效的处理"边界层效应"问题。     

Self-modeling solutions of the heat-conduction equation

Self-modeling solutions of the heat-conduction equation are described. The temperature moment is assumed constant.     

A simple,stable, and accurate linear tetrahedral finite element for transient,nearly, and fully incompressible solid dynamics: a dynamic variational multiscale approach

We propose a new approach for the stabilization of linear tetrahedral finite elements in the case of nearly incompressible transient solid dynamics computations. Our method is based on a mixed formulation, in which the momentum equation is complemented by a rate equation for the evolution of the pressure field, approximated with piecewise linear, continuous finite element functions. The pressure equation is stabilized to prevent spurious pressure oscillations in computations. Incidentally, it is also shown that many stabilized methods previously developed for the static case do not generalize easily to transient dynamics. Extensive tests in the context of linear and nonlinear elasticity are used to corroborate the claim that the proposed method is robust, stable, and accurate. Copyright © 2015 John Wiley & Sons, Ltd.     

Error estimates of mixed methods for optimal control problems governed by parabolic equations

In this paper, we investigate the error estimates for the solutions of parabolic optimal control problem by mixed finite element methods. The state and co‐state are approximated by the lowest‐order Raviart–Thomas mixed finite element spaces, and the control is approximated by piecewise constant functions. The convergence for the states, co‐states and the control is demonstrated. Copyright © 2008 John Wiley & Sons, Ltd.     

Simple model for scanning calorimeter with pyroelectric heat sensor

The heat-conduction equation is solved for a simplified model of a scanning heat-conducting calorimeter with a working cell in the form of a spherical pyroelectric thermometer. The optimum material and dimensions for the pyroelectric sensors are determined by numerical analysis of the solution.