本文選題：結(jié)構(gòu)可靠性 + Copula函數(shù)��； 參考：《湖南大學(xué)》2015年碩士論文

【摘要】：載荷、材料屬性、結(jié)構(gòu)尺寸等的不確定性廣泛存在于工程結(jié)構(gòu)中,可靠性分析是處理這類問(wèn)題的一種有效方法�，F(xiàn)有可靠性方法大都假設(shè)各輸入變量相互獨(dú)立,并轉(zhuǎn)換到標(biāo)準(zhǔn)正態(tài)空間進(jìn)行求解。然而,在很多情況下,隨機(jī)變量間具有相關(guān)性,且變量間的相關(guān)性可能對(duì)可靠性分析結(jié)果產(chǎn)生較大影響。目前處理相關(guān)性的可靠性方法主要有Nataf變換和Rosenblatt變換。然而,Nataf變換僅考慮了變量間的線性相關(guān)性,只能在某些特定樣本分布的情況下較好地度量變量間相關(guān)性,對(duì)于很多樣本分布類型或者變量間的聯(lián)合分布函數(shù)不服從高斯分布時(shí),該方法可能存在較大誤差；Rosenblatt變換是一種精確的相關(guān)性處理方法,但是,Rosenblatt變換必須基于精確的聯(lián)合概率分布函數(shù),而實(shí)際應(yīng)用中多維變量的聯(lián)合概率分布函數(shù)通常是未知的,所以其實(shí)際應(yīng)用受到很大限制。因此,開(kāi)發(fā)一種能克服上述缺陷的新方法,對(duì)于復(fù)雜結(jié)構(gòu)的可靠性分析與設(shè)計(jì)具有重要意義。本文針對(duì)近年來(lái)可靠性分析領(lǐng)域發(fā)展出的一種處理相關(guān)性的新工具,即Copula函數(shù),開(kāi)展了一系列研究,其主要工作如下：(1)提出了一種基于Copula函數(shù)的證據(jù)理論相關(guān)性分析模型及結(jié)構(gòu)可靠性計(jì)算方法,可處理證據(jù)變量間具有相關(guān)性的可靠性分析問(wèn)題。該方法引入Copula函數(shù)描述證據(jù)變量間的相關(guān)性,計(jì)算證據(jù)變量樣本的權(quán)重獲得結(jié)構(gòu)輸入變量間的最優(yōu)Copula函數(shù)。通過(guò)最優(yōu)Copula函數(shù)對(duì)證據(jù)變量邊緣基本可信度分配函數(shù)差分獲得聯(lián)合可信度分配函數(shù),并對(duì)每個(gè)焦元進(jìn)行極值分析,計(jì)算可靠域內(nèi)焦元的累積聯(lián)合BPA值獲得結(jié)構(gòu)的可靠性區(qū)間。(2)提出了一種基于Vine Copula函數(shù)的結(jié)構(gòu)可靠性分析方法,為復(fù)雜多維相關(guān)性問(wèn)題的可靠性分析提供了有效手段。通過(guò)Vine Copula建立多維隨機(jī)變量間的聯(lián)合概率分布函數(shù),并構(gòu)建相應(yīng)的可靠性分析模型。針對(duì)該可靠性分析模型,提出了兩類求解算法,即基于蒙特卡羅模擬的求解算法(VC-MCS)和基于一次二階矩的求解算法(VC-FORM)。VC-MCS方法效率較低,但可為其他高效算法的開(kāi)發(fā)提供重要的參考解；VC-FORM方法效率較高,可用于實(shí)際工程問(wèn)題的求解。(3)將Vine Copula函數(shù)引入結(jié)構(gòu)體系可靠性分析中,構(gòu)建了基于Vine Copula函數(shù)的結(jié)構(gòu)體系可靠性分析方法。通過(guò)Vine Copula函數(shù)描述結(jié)構(gòu)中不同功能函數(shù)間的相關(guān)性,并通過(guò)邊緣失效概率及蒙特卡羅積分求解結(jié)構(gòu)體系失效概率。該方法將邊緣失效概率和體系失效概率分開(kāi)處理,并可描述不同失效模式之間的不同相關(guān)特性,具有較高精度。
[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.
【學(xué)位授予單位】：湖南大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：TB114.3

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