随机结构有限元法及可靠性研究
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摘要
工程结构受生产加工及装配等工艺的影响，其材料性能、结构尺寸、边界条件等具有随机性，实际工作状态下的载荷与设计规定也有偏差。传统的确定性分析方法未能定量考虑这些随机因素的影响，经证明不够科学。因此，发展将确定性有限元法与随机分析理论结合的随机有限元法，分析在这些随机因素影响下复杂结构的响应，以失效概率评价结构的安全性就显得尤为重要。本文针对工程结构中的随机因素，开展了线性与非线性，静力和动力问题的随机有限元法，及以此为基础的结构可靠性研究。主要包括以下几方面的研究工作
1 随机有限元法研究
采用Cholesky分解变换或Gram-Schmidt特征正交化变换，将相关随机变量变换为独立的随机变量，再利用Monte Carlo抽样模拟，特别是采用Gram-Schmidt特征正交化变换法可大大缩减随机变量的数目，达到减少计算量的目的。
研究用谱描述的平稳与非平稳随机过程的三角级数模拟方法，对非平稳随机过程采用确定性函数与平稳随机过程乘积表示。
结合随机变量的Monte Carlo模拟法，随机场的局部平均法与随机过程的三角级数模拟法，用均值、方差、相关长度及功率谱等，可有效地描述随机变量、随机场、随机过程，及同时具有场与过程随机的变量。
回顾了线弹性、弹塑性及大变形几何非线性的静力、动力有限元法的基本方程及求解过程。在线性随机方程Neumann级数求解方法的基础上，结合非线性有限元分析方法，进一步研究了非线性随机方程的Neumann级数与增量逐步求解技术的结合的求解方法。该方法可用于求解非线性静力和动力随机问题，其简化形式也可分析线性静力和动力随机问题。证明了Neumann级数求解非线性随机问题的收敛性，同时给出了改进收敛性和计算效率的方法。算例表明了该方法具有较高精度和良好的应用潜力。
2结构可靠性研究
在最大熵法中需求解的随机变量的矩约束积分方程组具有高度非线性，通常的非线性方程组的求解方法不能收敛。为此，引入同伦算法，有效求解该积分方程组。通过理论分布的拟合实例表明最大熵法具有高精度，是结构可靠性分析中的一种良好的分析方法。
基于功能随机过程的极小化变换和首次超越概率，研究结构动力可靠性，
通过完全Moflte Carlo模拟比较表明，这两种方法具有较好的精度和较高的计算
效率。
研究了串联、并联及混合联接系统的可靠性分析方法。其中，在系统的动
力可靠性分析中引入极值变换法，可大大降低分布函数的维数；提出处理系统
可靠性中的高维累积分布函数的数值积分的Monte Carlo模拟法，使问题得到简
化，计算分析表明该方法十分有效。研究结构系统可靠性分析的截止枚举法，
通过截止枚举法可有效的识别对结构系统可靠性较大影响的主要失效模式，从
而简化系统可靠性计算。研究随机场的相关长度对系统可靠性分析的影响，指
出一般的结构可靠性分析问题在本质上是系统问题，只有在随机场相关长度相
对于结构尺寸很大和结构有应力集中等少数情况下，按点可靠性分析简化才有
较为符合实际的结果，而当前采用随机有限元法进行结构可靠性分析时往往忽
视了该问题。
3结构随机性与可靠性分析的程序研制
研制了结构随机性与可靠性分析的程序系统 SDA，该系统有 Von Mises、
Tresca、Mohr Coulomb和 D。ekerPrager等四种材料模型，可进行确定性、随
机性和可靠性分析。主要有以下几个方面的分析能力。
确定性分析。结构线弹性、弹塑性及大变形几何非线性的静力、动力响应
分析。
随机性分析和可靠性分析。随机分析方法：Neurnann随机有限元法和Monte
car。随机有限元法；可考虑的随机因素有：材料随机性，弹性模量、泊松比、
屈服应力、密度、阻尼、硬化系数、强度等随机场模型；载荷随机性方面，随
机场和（或）过程模型；几何随机性，坐标的扰动随机场模型。SDA可求解结
构线弹性、弹塑性及大变形几何非线性的静力、动力问题结构响应的均值、方
差及点可靠性和系统可靠性。
Owing to the effects of manufacturing and assembling, material properties > sizes and boundaries of engineering structures are usually stochastic, actually applied loads are also different with planned ones. In traditional determinate method these stochastic factors are neglected originally, which has been proved not to be scientific enough. Therefore, it is especially important to develop stochastic finite element method (SFEM). SFEM combines finite element method with random theories to analyze the random responses of complicated structures and judge the security of structures in terms of failure probability. In this paper stochastic finite element method and SFEM-based reliability evaluation are developed for linear, nonlinear, static and dynamic problems. The research work is mainly concerned with the following three aspects.
1. SFEM
At first, Cholesky factorization method and Gram-Schmidt orthogonalization method is used to covert these correlated random variables into independent ones respectively; then, Monte Carlo sampling is applied to simulate them. The two methods are convenient to generate correlated random variables. Especially, Gram-Schmidt orthogonalization method can reduce the number of simulated random variables and hence cut down the computational amount.
Trigonometric series is used to simulate stationary, non-stationary, Gaussian and non-Gaussin stochastic processes by spectral representation in which the non-stationary stochastic processes are expressed as the product of determinate functions and stationary stochastic processes.
Combined with Monte Carlo sampling technique, local average discretization of random field and trigonometric series representation of stochastic processes are used to describe random variables, random fields, random processes. These random fields or random processes vary with spatial position and temporal course in aspects of means, variances, correlated lengths and power spectral densities.
FEM governing equations analyzing linear elastic, elastoplastic, static, dynamic and geometric nonlinear problems are introduced. Then, based on the solving method of linear SFEM equations by Neumann series, the solving method of nonlinear SFEM equations by Neumann expansion combined with increment-solving technique step by step is presented. This method can be used to solve both linear and nonlinear static and dynamic problems. Astringency of Neumann series expansion to solve nonlinear problems is proved and a method to improve the convergence and enhance efficiency is presented. Examples show that the method has higher accuracy and application potential.
2. Structural reliability
It is necessary to solve the integral equations with moment constraints of random variables in maximum entropy method of reliability analysis. These equations are
severe nonlinear and their solutions cannot be obtained by common solving methods of nonlinear equations. Cognate method is introduced effectively to solve them. The solutions obtained by maximum entropy method are consistent with corresponding theoretic distribution, which shows maximum entropy method is a sort of good method for reliability analysis.
Based on the extreme transform method and first-cross reliability, dynamic reliability of structures is analyzed. Compared with Monte Carlo method, the two methods have good accuracy and efficiency.
System reliability analysis methods are investigated for structures in series, parallel, and series-parallel. Among them, the extreme value transform method is presented to cut down the dimension of distribution function. Monte Carlo method is proposed to solve the integral of the multi-dimension probabilistic density function, and related example shows that the method leads to good accuracy and efficiency. Truncated enumeration method (TEM) is presented to identify primary failure modes, which seriously affect system reliability, and to reduce computational amount. Affects of correlated lengths of random fields are investigated. It is emphasized that reliability of general structu
1 随机有限元法研究
采用Cholesky分解变换或Gram-Schmidt特征正交化变换，将相关随机变量变换为独立的随机变量，再利用Monte Carlo抽样模拟，特别是采用Gram-Schmidt特征正交化变换法可大大缩减随机变量的数目，达到减少计算量的目的。
研究用谱描述的平稳与非平稳随机过程的三角级数模拟方法，对非平稳随机过程采用确定性函数与平稳随机过程乘积表示。
结合随机变量的Monte Carlo模拟法，随机场的局部平均法与随机过程的三角级数模拟法，用均值、方差、相关长度及功率谱等，可有效地描述随机变量、随机场、随机过程，及同时具有场与过程随机的变量。
回顾了线弹性、弹塑性及大变形几何非线性的静力、动力有限元法的基本方程及求解过程。在线性随机方程Neumann级数求解方法的基础上，结合非线性有限元分析方法，进一步研究了非线性随机方程的Neumann级数与增量逐步求解技术的结合的求解方法。该方法可用于求解非线性静力和动力随机问题，其简化形式也可分析线性静力和动力随机问题。证明了Neumann级数求解非线性随机问题的收敛性，同时给出了改进收敛性和计算效率的方法。算例表明了该方法具有较高精度和良好的应用潜力。
2结构可靠性研究
在最大熵法中需求解的随机变量的矩约束积分方程组具有高度非线性，通常的非线性方程组的求解方法不能收敛。为此，引入同伦算法，有效求解该积分方程组。通过理论分布的拟合实例表明最大熵法具有高精度，是结构可靠性分析中的一种良好的分析方法。
基于功能随机过程的极小化变换和首次超越概率，研究结构动力可靠性，
通过完全Moflte Carlo模拟比较表明，这两种方法具有较好的精度和较高的计算
效率。
研究了串联、并联及混合联接系统的可靠性分析方法。其中，在系统的动
力可靠性分析中引入极值变换法，可大大降低分布函数的维数；提出处理系统
可靠性中的高维累积分布函数的数值积分的Monte Carlo模拟法，使问题得到简
化，计算分析表明该方法十分有效。研究结构系统可靠性分析的截止枚举法，
通过截止枚举法可有效的识别对结构系统可靠性较大影响的主要失效模式，从
而简化系统可靠性计算。研究随机场的相关长度对系统可靠性分析的影响，指
出一般的结构可靠性分析问题在本质上是系统问题，只有在随机场相关长度相
对于结构尺寸很大和结构有应力集中等少数情况下，按点可靠性分析简化才有
较为符合实际的结果，而当前采用随机有限元法进行结构可靠性分析时往往忽
视了该问题。
3结构随机性与可靠性分析的程序研制
研制了结构随机性与可靠性分析的程序系统 SDA，该系统有 Von Mises、
Tresca、Mohr Coulomb和 D。ekerPrager等四种材料模型，可进行确定性、随
机性和可靠性分析。主要有以下几个方面的分析能力。
确定性分析。结构线弹性、弹塑性及大变形几何非线性的静力、动力响应
分析。
随机性分析和可靠性分析。随机分析方法：Neurnann随机有限元法和Monte
car。随机有限元法；可考虑的随机因素有：材料随机性，弹性模量、泊松比、
屈服应力、密度、阻尼、硬化系数、强度等随机场模型；载荷随机性方面，随
机场和（或）过程模型；几何随机性，坐标的扰动随机场模型。SDA可求解结
构线弹性、弹塑性及大变形几何非线性的静力、动力问题结构响应的均值、方
差及点可靠性和系统可靠性。
Owing to the effects of manufacturing and assembling, material properties > sizes and boundaries of engineering structures are usually stochastic, actually applied loads are also different with planned ones. In traditional determinate method these stochastic factors are neglected originally, which has been proved not to be scientific enough. Therefore, it is especially important to develop stochastic finite element method (SFEM). SFEM combines finite element method with random theories to analyze the random responses of complicated structures and judge the security of structures in terms of failure probability. In this paper stochastic finite element method and SFEM-based reliability evaluation are developed for linear, nonlinear, static and dynamic problems. The research work is mainly concerned with the following three aspects.
1. SFEM
At first, Cholesky factorization method and Gram-Schmidt orthogonalization method is used to covert these correlated random variables into independent ones respectively; then, Monte Carlo sampling is applied to simulate them. The two methods are convenient to generate correlated random variables. Especially, Gram-Schmidt orthogonalization method can reduce the number of simulated random variables and hence cut down the computational amount.
Trigonometric series is used to simulate stationary, non-stationary, Gaussian and non-Gaussin stochastic processes by spectral representation in which the non-stationary stochastic processes are expressed as the product of determinate functions and stationary stochastic processes.
Combined with Monte Carlo sampling technique, local average discretization of random field and trigonometric series representation of stochastic processes are used to describe random variables, random fields, random processes. These random fields or random processes vary with spatial position and temporal course in aspects of means, variances, correlated lengths and power spectral densities.
FEM governing equations analyzing linear elastic, elastoplastic, static, dynamic and geometric nonlinear problems are introduced. Then, based on the solving method of linear SFEM equations by Neumann series, the solving method of nonlinear SFEM equations by Neumann expansion combined with increment-solving technique step by step is presented. This method can be used to solve both linear and nonlinear static and dynamic problems. Astringency of Neumann series expansion to solve nonlinear problems is proved and a method to improve the convergence and enhance efficiency is presented. Examples show that the method has higher accuracy and application potential.
2. Structural reliability
It is necessary to solve the integral equations with moment constraints of random variables in maximum entropy method of reliability analysis. These equations are
severe nonlinear and their solutions cannot be obtained by common solving methods of nonlinear equations. Cognate method is introduced effectively to solve them. The solutions obtained by maximum entropy method are consistent with corresponding theoretic distribution, which shows maximum entropy method is a sort of good method for reliability analysis.
Based on the extreme transform method and first-cross reliability, dynamic reliability of structures is analyzed. Compared with Monte Carlo method, the two methods have good accuracy and efficiency.
System reliability analysis methods are investigated for structures in series, parallel, and series-parallel. Among them, the extreme value transform method is presented to cut down the dimension of distribution function. Monte Carlo method is proposed to solve the integral of the multi-dimension probabilistic density function, and related example shows that the method leads to good accuracy and efficiency. Truncated enumeration method (TEM) is presented to identify primary failure modes, which seriously affect system reliability, and to reduce computational amount. Affects of correlated lengths of random fields are investigated. It is emphasized that reliability of general structu
引文
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