基于Copula函数的结构可靠性分析

发布时间：2018-05-10 07:37

本文选题：结构可靠性 + Copula函数　；参考：《湖南大学》2015年硕士论文

【摘要】：载荷、材料属性、结构尺寸等的不确定性广泛存在于工程结构中,可靠性分析是处理这类问题的一种有效方法。现有可靠性方法大都假设各输入变量相互独立,并转换到标准正态空间进行求解。然而,在很多情况下,随机变量间具有相关性,且变量间的相关性可能对可靠性分析结果产生较大影响。目前处理相关性的可靠性方法主要有Nataf变换和Rosenblatt变换。然而,Nataf变换仅考虑了变量间的线性相关性,只能在某些特定样本分布的情况下较好地度量变量间相关性,对于很多样本分布类型或者变量间的联合分布函数不服从高斯分布时,该方法可能存在较大误差；Rosenblatt变换是一种精确的相关性处理方法,但是,Rosenblatt变换必须基于精确的联合概率分布函数,而实际应用中多维变量的联合概率分布函数通常是未知的,所以其实际应用受到很大限制。因此,开发一种能克服上述缺陷的新方法,对于复杂结构的可靠性分析与设计具有重要意义。本文针对近年来可靠性分析领域发展出的一种处理相关性的新工具,即Copula函数,开展了一系列研究,其主要工作如下：(1)提出了一种基于Copula函数的证据理论相关性分析模型及结构可靠性计算方法,可处理证据变量间具有相关性的可靠性分析问题。该方法引入Copula函数描述证据变量间的相关性,计算证据变量样本的权重获得结构输入变量间的最优Copula函数。通过最优Copula函数对证据变量边缘基本可信度分配函数差分获得联合可信度分配函数,并对每个焦元进行极值分析,计算可靠域内焦元的累积联合BPA值获得结构的可靠性区间。(2)提出了一种基于Vine Copula函数的结构可靠性分析方法,为复杂多维相关性问题的可靠性分析提供了有效手段。通过Vine Copula建立多维随机变量间的联合概率分布函数,并构建相应的可靠性分析模型。针对该可靠性分析模型,提出了两类求解算法,即基于蒙特卡罗模拟的求解算法(VC-MCS)和基于一次二阶矩的求解算法(VC-FORM)。VC-MCS方法效率较低,但可为其他高效算法的开发提供重要的参考解；VC-FORM方法效率较高,可用于实际工程问题的求解。(3)将Vine Copula函数引入结构体系可靠性分析中,构建了基于Vine Copula函数的结构体系可靠性分析方法。通过Vine Copula函数描述结构中不同功能函数间的相关性,并通过边缘失效概率及蒙特卡罗积分求解结构体系失效概率。该方法将边缘失效概率和体系失效概率分开处理,并可描述不同失效模式之间的不同相关特性,具有较高精度。
[Abstract]:The uncertainties of load, material properties and structural dimensions are widely used in engineering structures. Reliability analysis is an effective method to deal with this kind of problems. Most of the existing reliability methods assume that the input variables are independent of each other and convert to the standard normal space to solve the problem. However, in many cases, random variables have a correlation, and the correlation between variables may have a great impact on the reliability analysis results. At present, the reliability methods of dealing with correlation mainly include Nataf transform and Rosenblatt transform. However, the Nataf transform only considers the linear correlation between variables, and can only measure the correlation between variables in the case of certain sample distribution. When many sample distribution types or the joint distribution function between variables are not satisfied with the Gao Si distribution, The Rosenblatt transform is a kind of accurate correlation processing method, but Rosenblatt transform must be based on the exact joint probability distribution function, and the joint probability distribution function of multidimensional variables is usually unknown in practical application. Therefore, its practical application is greatly restricted. Therefore, it is of great significance to develop a new method to overcome the above defects for the reliability analysis and design of complex structures. In this paper, a series of researches have been carried out on the Copula function, a new tool developed in the field of reliability analysis in recent years. The main work is as follows: (1) A correlation analysis model of evidence theory based on Copula function and a structural reliability calculation method are proposed, which can deal with reliability analysis problems with correlation between evidence variables. In this method, the Copula function is introduced to describe the correlation between evidence variables, and the optimal Copula function between structural input variables is obtained by calculating the weights of the samples of the evidence variables. The joint confidence distribution function is obtained by the difference of the basic confidence distribution function on the edge of the evidence variable by the optimal Copula function, and the extreme value of each focal element is analyzed. A method of structural reliability analysis based on Vine Copula function is proposed, which provides an effective method for reliability analysis of complex multidimensional correlation problems. The joint probability distribution function among multidimensional random variables is established by Vine Copula, and the corresponding reliability analysis model is constructed. For the reliability analysis model, two kinds of algorithms are proposed, namely, the algorithm based on Monte Carlo simulation (VC-MCS) and the algorithm based on the first order second order moment (QORM). The efficiency of VC-FORMN. VC-MCS method is relatively low. However, the VC-FORM method can provide an important reference for the development of other efficient algorithms. The VC-FORM method can be used to solve practical engineering problems. The VC-FORM method can be used to solve practical engineering problems. The Vine Copula function is introduced into the reliability analysis of the structural system. The reliability analysis method of structure system based on Vine Copula function is constructed. The correlation between different function functions is described by Vine Copula function, and the failure probability of structural system is solved by edge failure probability and Monte Carlo integral. In this method, the edge failure probability and the system failure probability are treated separately, and the different correlation characteristics between different failure modes can be described. The method has high accuracy.
【学位授予单位】：湖南大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：TB114.3

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