【作者】 刘文军；

【导师】 黄斌；

【作者基本信息】 武汉理工大学 ， 结构工程， 2005， 硕士

【摘要】 本文首先回顾随机有限元法的研究状况，指出传统的摄动随机有限元法能够考虑结构系统的质量、阻尼、刚度、激励以及响应量的不确定性，但这种随机扰动量常常假定为很小。然后重点介绍了一种新的递推随机有限元方法，该方法可以解决大变异随机变量的问题，离散材料物理量的随机场，采用非正交多项式表示结构的特征值，建立了和摄动法类似的一系列确定的递推方程，再通过确定性有限元方法求解这些递推方程，得到结构特征值的均值和均方差。最后利用随机有限元法对随机结构的孤立特征值和重特征值问题进行分析。为此，本文就以下几个方面开展了探索性的研究工作： 1、简要介绍随机场的离散，比较了中心离散法、局部平均法和插值法的优缺点，然后阐述了随机过程的谱分解理论和随机过程的MC模拟方法。 2、阐述矩阵摄动理论，运用该方法对确定性特征值问题加以分析。利用一阶和二阶矩阵摄动法处理了确定性结构的孤立特征值问题，然后通过矩阵技巧将矩阵摄动法加以推广，使其能解决重特征值问题和密集特征值问题。 3、考虑结构材料的随机性，应用摄动随机有限元法对结构孤立特征值和重特征值问题进行分析，推导出特征值和特征向量的统计特征的数值表达式。 4、利用递推随机有限元法处理随机结构的孤立特征值和重特征值问题。考虑弹性模量的随机性，建立了一系列确定的递推方程组；求解方程组可得到结构特征值的均值和均方差的统计特征的数值表达式。 5、通过两个典型算例比较摄动、递推随机有限元法和MC模拟法，结果表明：摄动随机有限元方法仅对小变异的孤立、重特征值问题才行之有效；MC模拟法虽然能解决大变异的问题，但耗时很大；与摄动随机有限元法相比，在较宽随机涨落范围内递推随机有限元法结果都能更好的逼近MC模拟解；相对MC模拟法而言，递推随机有限元法能节省了大量计算时间。 最后，在总结全文工作的基础上，提出了文章尚需深入研究的问题。更多还原

【Abstract】 The history and designing research for stochastic finite element method are reviewed firstly. Then the merits and defects of conventional methods are proposed. The conventional perturbation stochastic finite element method can consider uncertainty of mass, damp, hardness, stimulus etc. as variables or uncertainties, but these variables are usually supposed smallness. One Recursive stochastic finite element method will be introduced in this paper, the new stochastic finite element method that can solve problems including random variables of large fluctuation levels uses discrete random fields to express elastic module. A series of certain recursive equations like perturbation method will be developed. Solving these equations, we can get expectation and mean square covariance of random structural eigenvalues. The stochastic structure with repeated eigenvalues is taken as research objects in this paper, and structural free vibration characters and solutions are investigated by the stochastic finite element method. Hence exploring research works on 5 aspects have been done around the purpose.1. Summarizing random field discretization, the paper compares the merits and defects of three conventional discrete methods, then introduce theories of spectrum decomposition and Monte-Carlo simulating method about stochastic process.2. Introducing the development of perturbation method, especially the principle of matrix perturbation method, this paper researches changes of structural eigenvalues and eigenvectors with structural parameters’ variables, and analyzes matrix perturbation method of one order and two order in structures with isolated eigenvalues, then extend this method to structures with repeated eigenvalues.3. Considering structural materials’ randomness and using perturbation stochastic finite element method to research it, isolated and repeated eigenvalues in structures will be analyzed. Then the computational expressions of numerical characters about structural modes are developed.4. Utilizing the new recursive stochastic finite element method to analyze free vibration of random structures with isolated and repeated eigenvalues, thencombining theories about random fields, a series of certain recursive equations will be developed. By solving these equations, statistical expressions of expectation and mean square covariance about eigenvalues will be gotten.5. Through two typical numerical examples about repeated eigenvalues, perturbation and recurisive stochastic finite element methods are compared with Monte-Carlo simulation method. We will find it that: perturbation stochastic finite element method can solve merely problems of isolated and repeated eigenvalues in the condition of random variables of small fluctuation levels; Monte-Carlo simulation method can solve these problems in the all fluctuation levels, but it costs too much time; the results of recursive stochastic finite element method can approach better those of Monte-Carlo simulation method than that of perturbation stochastic element method, even in the condition of large fluctuation levels, and this news stochastic finite element method can save much of much calculational time.In the end, the content of this paper has been summarized and some problems, which need further study, have been proposed.更多还原