馬爾科夫鏈方法在可靠性分析中的應(yīng)用研究

1、西 北 工 業(yè) 大 學(xué)博 士 學(xué) 位 論 文學(xué)位研究生題目：馬爾科夫鏈方法在可靠性分析中的應(yīng)用研究作 者： 袁 修 開 學(xué)科專業(yè)： 飛行器設(shè)計 指導(dǎo)教師： 呂 震 宙 2021年11月Application Research of Markov Chain Simulation in Reliability AnalysisA Thesis Submitted for the Degree of Doctor of PhilosophyYuan XiukaiAcademic Advisor: Professor Lu ZhenzhouNorthwestern Polytechnical Uni

2、versityNovember 2021摘要馬爾科夫鏈方法由于具有自適應(yīng)性以及較高的模擬效率，在許多領(lǐng)域中得到了廣泛的應(yīng)用。本文將針對可靠性分析中遇到的各種問題，包括非正態(tài)變量問題，小失效概率問題，可靠性靈敏度問題，擴展可靠性問題以及動態(tài)可靠性問題等，探討了馬爾科夫鏈在可靠性分析中的應(yīng)用，開展了一系列的可靠性分析方法，主要內(nèi)容如下：(1)基于對失效域樣本的高效馬爾可夫鏈模擬方法和鞍點估計法，提出了兩種可快速分析小失效概率情況下含非正態(tài)變量的非線性極限狀態(tài)函數(shù)的可靠性方法。第一種方法采用高效的改良的馬爾科夫鏈方法來獲得近似設(shè)計點，然后在近似設(shè)計點處將極限狀態(tài)函數(shù)線性化，最后采用鞍點估計法求解失效

3、概率。所提方法相對于基于均值點處線性化的鞍點估計方法，僅需增加少量計算量即可以提高近似求解的精度。第一種方法那么在非正態(tài)空間中，將所求失效概率轉(zhuǎn)化為線性極限狀態(tài)函數(shù)的失效概率與一個特征比例因子的乘積。線性極限狀態(tài)函數(shù)是通過馬爾可夫鏈模擬非線性極限狀態(tài)函數(shù)失效域中的樣本而獲得的，它與非線性極限狀態(tài)函數(shù)具有近似相同的設(shè)計點。而概率論中的乘法定理是獲取特征比例因子的理論依據(jù)，它反映了非線性極限狀態(tài)函數(shù)失效概率與線性極限狀態(tài)函數(shù)失效概率的關(guān)系。線性極限狀態(tài)函數(shù)的失效概率可以由鞍點估計法求得，而特征比例因子可以由馬爾可夫鏈快速模擬線性與非線性失效域中的樣本而近似算得。定性分析和定量的算例比照分析說明：所

4、提算法具有較廣的適用范圍，并且它的實現(xiàn)過程較為簡單，計算精度和效率均較高。(2)針對實際工程中可靠性分析設(shè)計的極限狀態(tài)方程為隱式的情況，提出了兩種基于馬爾科夫鏈模擬的支持向量機可靠性分析方法。所提的第一種方法方法采用改良的馬爾科夫鏈來產(chǎn)生極限狀態(tài)重要區(qū)域上的樣本點，再采用支持向量機方法求得相應(yīng)的函數(shù)替代模型來進行可靠性分析。由于馬爾科夫鏈能夠自適應(yīng)的模擬極限狀態(tài)重要失效區(qū)域附近的樣本，并且由于采用馬爾科夫鏈備選樣本點而非狀態(tài)點作為訓(xùn)練樣本，因而所提方法能夠高效快速逼近對失效概率奉獻較大區(qū)域的極限狀態(tài)方程，并且充分利用了模擬過程產(chǎn)生的有用信息。所提方法還采用了一種漸變方差的模擬策略，改善了馬爾科

5、夫鏈模擬樣本的質(zhì)量。另外所提方法分別采用支持向量機分類方法和回歸方法來構(gòu)建函數(shù)替代模型，能夠?qū)崿F(xiàn)風(fēng)險最小化的極限狀態(tài)方程的替代，使得失效概率可以高效高精度地被逼近。最后給出了數(shù)值算例和工程算例，說明本文所提方法在計算效率和精度上具有較好的性能。所提出的第二種方法是基于篩選的馬爾科夫鏈模擬的支持向量機可靠性分析方法，該方法將馬爾科夫鏈與支持向量機特有的邊界性質(zhì)相結(jié)合來產(chǎn)生訓(xùn)練樣本，既充分利用了馬爾科夫鏈的自適應(yīng)性，使抽樣的樣本點落在重要區(qū)域，又利用了支持向量機分類面的邊界的漸近逼近性，使訓(xùn)練樣本以較高效率和精度來產(chǎn)生，這樣就保證了擬合的支持向量機在失效概率的近似上具有較高的精度。所提方法能夠很好

6、的解決原有傳統(tǒng)響應(yīng)面存在的兩個根本問題，并通過數(shù)值算例和工程算例證實了所提方法的精度、效率。(3)提出了一種基于馬爾可夫鏈模擬的可靠性靈敏度分析方法，該方法將原極限狀態(tài)靈敏度轉(zhuǎn)化成線性超平面靈敏度與兩個修正失效域靈敏度的代數(shù)和的形式，通過兩次馬爾可夫鏈模擬得到超平面的靈敏度解析解及修正失效域靈敏度的估計，進而得到原極限狀態(tài)的靈敏度解。文中給出了可靠性靈敏度估計值及估計值方差和變異系數(shù)的計算公式。所提方法采用快捷的馬爾可夫鏈模擬，因而效率遠比Monte Carlo模擬要高。將所提方法運用到粉末冶金渦輪盤的低周疲勞壽命可靠性分析上，得到影響渦輪盤的壽命可靠度的各參數(shù)的靈敏度，為工程設(shè)計提供了指導(dǎo)。

7、(4)可靠性分析中根本變量分布參數(shù)為區(qū)間均勻變量時，失效概率及可靠性靈敏度為分布參數(shù)的函數(shù)，基于條件概率馬爾科夫鏈模擬提出了一種失效概率函數(shù)及其靈敏度的求解方法，并提出了新的度量指標(biāo)，它為失效概率函數(shù)和可靠性靈敏度在分布參數(shù)空間上的統(tǒng)計特征值。所提方法的主要思路是首先采用貝葉斯公式將失效概率函數(shù)轉(zhuǎn)化成全局失效概率與參數(shù)后驗密度的表達式。再通過條件概率模擬方法來求解全局失效概率，還采用了三階最大熵法來得到失效域樣本的條件密度分布，最終得到所求的失效概率函數(shù)，而可靠性靈敏度函數(shù)為失效概率函數(shù)的導(dǎo)數(shù)。文中結(jié)合算例探討了所提方法的精度、效率和適用性。(5)針對臨界水平和變量分布參數(shù)同時變化的情況，極限

8、狀態(tài)函數(shù)的累積分布及其導(dǎo)數(shù)均為二維函數(shù)，其求解較困難。本文提出基于貝葉斯公式的累積分布及其導(dǎo)數(shù)的求解方法。所提方法首先采用貝葉斯公式將所求二維函數(shù)轉(zhuǎn)化成全局失效概率及后驗密度的表述式，再采用Monte Carlo方法及條件概率模擬方法來進行求解。所求方法與傳統(tǒng)的求解方法相比可以給出二維擴展累積分布函數(shù)的表達式，且僅需一次可靠性分析即可求解。(6)針對隨機鼓勵下隨機結(jié)構(gòu)系統(tǒng)的動力可靠性分析問題，探討了本文所提方法在動力學(xué)問題中的應(yīng)用。首先將隨機鼓勵隨機結(jié)構(gòu)系統(tǒng)的動力可靠性問題轉(zhuǎn)化成一般的可靠性問題，然后采用前文提出的方法來進行求解，得到了相應(yīng)的失效概率、失效概率函數(shù)及靈敏度函數(shù)等，論證了前文所提

9、方法的可行性，也為工程中動力可靠性分析設(shè)計提供了可選方法。關(guān)鍵詞：可靠性，靈敏度分析，馬爾科夫鏈，鞍點估計，支持向量機，失效概率函數(shù)，累積分布函數(shù)，最大熵法，蒙特卡洛，動態(tài)可靠性，隨機鼓勵A(yù)bstractFor the high adaptability and high simulation efficiency, Markov chain algorithm has wide application. The problems, including nonnormal variables, small failure probability, reliability sensitivity

10、, augment reliability and dynamical reliability, are frequently encountered in practical reliability analysis. For these problems, the solutions are investigated with Markov chain algorithm, and some novel reliability analysis methods are presented. The main contributions are listed as follows:(1)Ba

11、sed on fast Markov chain simulation for the samples distributed in failure region and Saddlepoint Approximation (SA) technique, two efficient reliability analysis methods are presented for evaluating the small failure probability of non-linear limit state function (LSF) with non-normal variables. Th

12、e first method firstly utilizes the advanced Markov chain simulation to efficiently obtain the approximate design point, and then linearizes the LSF at the design point, at last uses the SA to compute the failure probability with the linear limit state function. Compared with the SA based on mean po

13、int, the proposed method improves the precision by just increasing a little computation. In the second presented method, the failure probability of the non-linear LSF is transformed into a product of the failure probability of a linear LSF and a feature ratio factor. The linear LSF is obtained with

14、the samples distributed in the failure region, which are generated by the fast Markov chain simulation. The maximum likelihood point in the failure region of the linear LSF is approximately the same as that of the non-linear LSF. The feature ratio factor, which can be evaluated on the basis of multi

15、plicative rule of probability, exposes the relation between the failure probability of the non-linear LSF and that of the linear LSF. The failure probability of the linear LSF can be calculated by SA technique, and the feature ratio factor can be fast computed by the samples distributed in the failu

16、re regions of the non-linear LSF and those of the linear LSF. Qualitative analysis and quantitative computation demonstrate that the presented method has wide application, and it can be easily implemented and possesses high precision and high efficiency.(2)For the implicit LSF usually encountered in

17、 engineering reliability analysis and design, two support vector machine (SVM) reliability analysis methods are proposed based on fast Markov chain simulation. In the proposed method, Markov chain is used to simulate the samples in the important region defined by the LSF, and the SVM is employed to

18、obtain the solver surrogate by use of these samples. Since Markov chain can adaptively simulate the samples of the important region, and the candidate state but not Markov state is used as the training samples, the proposed method can well approximate the limit state function in the zone surrounding

19、 the design points, and can make full use of the information provided by Markov chain simulation. In addition, gradual change on variance in simulation process is adopted to improve the quality of the Markov chain samples. Moreover, the proposed method uses the SVM regression method and classificati

20、on method to construct the solver surrogate, which can automatically apply the Structural Risk Minimization (SRM) inductive principle in approximating the limit state equation, and thus approximate the failure probability with high precision. Finally numerical and engineering examples illustrate tha

21、t the proposed method owns good performance in calculating efficiency and precision. In the second presented method, an advanced SVM reliability analysis method is proposed on filtered Markov chain simulation. This method combines the Markov chain simulation and the margin property of the SVM to gen

22、erate training samples, it utilizes the adaptability of the Markov chain which makes the samples fall in the important region of the LSF, and utilizes asymptotic margin property of the SVM classification method which makes the samples generate in high precision. The proposed method solves the two pr

23、oblems which are encountered in traditional response surface method. The numerical and engineering examples demonstrate the precision and efficiency of the proposed method. (3)A decomposed arithmetic of reliability sensitivity analysis is proposed on the basis of Markov chain simulation. The propose

24、d method decomposes the reliability sensitivity of the actual limit state into algebra sum of reliability sensitivities of a linear failure region and modified failure regions, which can be rapidly solved by two Markov chain simulations. The variance of the reliability sensitivity estimation is anal

25、yzed as well. Markov chain can simulate the samples of the specified failure region and own higher efficiency than Monte Carlo method, thus the proposed method based on it also has high efficiency; this conclusion is demonstrated by the numerical examples and the fatigue life reliability sensitivity

26、 results of the powder metallurgy turbine disk.(4)In case that the distribution parameters of the basic variables are uniformly distributed interval variables in reliability analysis, the failure probability and the reliability sensitivity are the functions of the distribution parameters. The condit

27、ional probability Markov chain simulation method is proposed to obtain the failure probability function, reliability sensitivity function and the new reliability measures, which are the statistics characteristic values of the failure probability function and the reliability sensitivity function in t

28、he space of the distribution parameters. The key idea of the proposed method is that the failure probability function is transformed into the expression of the global failure probability and the posterior density of the distribution parameter, then the global failure probability is computed by condi

29、tional probability simulation method based on Markov chain algorithm, and the posterior density is obtained by the third order maximum entropy method based on failure samples, finally the failure probability function is obtained and the reliability sensitivity function is the derivative of the failu

30、re probability function. The accuracy, efficiency and applicability of the proposed method are demonstrated with several examples. (5)When the threshold value and the distribution parameter are both variables, the cumulative distribution function of the LSF and its derivatives are two-dimension func

31、tions, which are difficult to obtain in reliability analysis. The Bayes formula is adopted to transform the two-dimension function into the expression of global failure probability and the posterior distribution of the variables, and then Monte Carlo simulation and the conditional probability simula

32、tion method are adopted to analyze. Compared with the traditional analysis method, the proposed method can give the expression of the two-dimension augment cumulative distribution function, and only one reliability analysis is needed. (6)The methods proposed in this paper are applied to the dynamica

33、l reliability problem of random structure subjected to stochastic excitation. It firstly transforms the dynamical reliability problem into traditional one, and then uses the methods proposed in this paper to obtain the failure probability, the failure probability function and the reliability sensiti

34、vity function of the distribution parameters of the basic random variables. The feasibilities of the proposed methods, which can provide alternative solutions for the dynamical reliability, are demonstrated. Keyword: Reliability; sensitivity analysis; Markov chain; Saddlepoint Approximation (SA); su

35、pport vector machine (SVM); failure probability function; cumulative distribution function (CDF); maximum entropy method; Monte Carlo; dynamical reliability; stochastic excitation目錄 TOC o 1-3 h z u HYPERLINK l _Toc246387131 摘要 PAGEREF _Toc246387131 h V HYPERLINK l _Toc246387132 Abstract PAGEREF _Toc

36、246387132 h IX HYPERLINK l _Toc246387133 目錄 PAGEREF _Toc246387133 h XIII HYPERLINK l _Toc246387134 第一章 緒論 PAGEREF _Toc246387134 h 1 HYPERLINK l _Toc246387135 1.1.可靠性分析的近似解析法和數(shù)字模擬法 PAGEREF _Toc246387135 h 1 HYPERLINK l _Toc246387136 1.2.可靠性分析的函數(shù)替代方法 PAGEREF _Toc246387136 h 3 HYPERLINK l _Toc246387137

37、 1.3.可靠性優(yōu)化中失效概率函數(shù)的求解 PAGEREF _Toc246387137 h 4 HYPERLINK l _Toc246387138 1.4.可靠性靈敏度的分析求解 PAGEREF _Toc246387138 h 5 HYPERLINK l _Toc246387139 1.5.論文主要工作 PAGEREF _Toc246387139 h 6 HYPERLINK l _Toc246387140 第二章 基于馬爾科夫鏈模擬與鞍點估計的非正態(tài)變量可靠性分析方法 PAGEREF _Toc246387140 h 7 HYPERLINK l _Toc246387141 2.1.基于馬爾科夫鏈的

38、一次鞍點估計可靠性分析方法 PAGEREF _Toc246387141 h 8 HYPERLINK l _Toc246387142 2.1.1.根本思路 PAGEREF _Toc246387142 h 8 HYPERLINK l _Toc246387143 2.1.2.馬爾科夫鏈模擬 PAGEREF _Toc246387143 h 8 HYPERLINK l _Toc246387144 2.1.3.鞍點估計法 PAGEREF _Toc246387144 h 11 HYPERLINK l _Toc246387145 2.1.4.所提方法的根本步驟 PAGEREF _Toc246387145 h

39、15 HYPERLINK l _Toc246387146 2.1.5.算例 PAGEREF _Toc246387146 h 16 HYPERLINK l _Toc246387147 2.1.6.本節(jié)小結(jié) PAGEREF _Toc246387147 h 17 HYPERLINK l _Toc246387148 2.2.基于條件概率馬爾科夫鏈模擬的可靠性分析方法 PAGEREF _Toc246387148 h 18 HYPERLINK l _Toc246387149 2.2.1.條件概率方法的根本思路 PAGEREF _Toc246387149 h 18 HYPERLINK l _Toc24638

40、7150 2.2.2.線性極限狀態(tài)函數(shù)失效概率求解的鞍點估計法 PAGEREF _Toc246387150 h 20 HYPERLINK l _Toc246387151 2.2.3.特征因子的求解 PAGEREF _Toc246387151 h 20 HYPERLINK l _Toc246387152 2.2.4.失效概率的估計值和近似方差分析 PAGEREF _Toc246387152 h 21 HYPERLINK l _Toc246387153 2.2.5.基于條件概率馬爾科夫鏈模擬的可靠性分析方法步驟 PAGEREF _Toc246387153 h 24 HYPERLINK l _Toc

41、246387154 2.2.6.所提方法的進一步討論 PAGEREF _Toc246387154 h 24 HYPERLINK l _Toc246387155 2.2.7.算例 PAGEREF _Toc246387155 h 25 HYPERLINK l _Toc246387156 2.2.8.本節(jié)小結(jié) PAGEREF _Toc246387156 h 33 HYPERLINK l _Toc246387157 2.3.本章小結(jié) PAGEREF _Toc246387157 h 34 HYPERLINK l _Toc246387158 第三章 基于馬爾科夫鏈模擬的支持向量機方法 PAGEREF _T

42、oc246387158 h 35 HYPERLINK l _Toc246387159 3.1.可靠性分析的函數(shù)替代方法 PAGEREF _Toc246387159 h 35 HYPERLINK l _Toc246387160 3.1.1.響應(yīng)面方法 PAGEREF _Toc246387160 h 35 HYPERLINK l _Toc246387161 3.1.2.支持向量機方法 PAGEREF _Toc246387161 h 37 HYPERLINK l _Toc246387162 3.2.基于馬爾科夫鏈模擬的支持向量機可靠性分析方法 PAGEREF _Toc246387162 h 38 H

43、YPERLINK l _Toc246387163 3.2.1.改良的馬爾科夫鏈模擬產(chǎn)生訓(xùn)練樣本點 PAGEREF _Toc246387163 h 39 HYPERLINK l _Toc246387164 3.2.2.支持向量機回歸方法 PAGEREF _Toc246387164 h 40 HYPERLINK l _Toc246387165 3.2.3.支持向量機分類方法 PAGEREF _Toc246387165 h 42 HYPERLINK l _Toc246387166 3.2.4.基于馬爾科夫鏈模擬的支持向量機方法的步驟 PAGEREF _Toc246387166 h 45 HYPERL

44、INK l _Toc246387167 3.2.5.算例 PAGEREF _Toc246387167 h 46 HYPERLINK l _Toc246387168 3.2.6.本節(jié)小結(jié) PAGEREF _Toc246387168 h 55 HYPERLINK l _Toc246387169 3.3.基于篩選馬爾科夫鏈模擬的支持向量機分類可靠性分析方法 PAGEREF _Toc246387169 h 56 HYPERLINK l _Toc246387170 3.3.1.支持向量機分類方法的邊界特性 PAGEREF _Toc246387170 h 56 HYPERLINK l _Toc246387

45、171 3.3.2.基于支持向量機邊界的篩選馬爾科夫鏈模擬方法 PAGEREF _Toc246387171 h 57 HYPERLINK l _Toc246387172 3.3.3.基于篩選馬爾科夫鏈的支持向量機分類方法 PAGEREF _Toc246387172 h 58 HYPERLINK l _Toc246387173 3.3.4.算例 PAGEREF _Toc246387173 h 60 HYPERLINK l _Toc246387174 3.3.5.本節(jié)小結(jié) PAGEREF _Toc246387174 h 63 HYPERLINK l _Toc246387175 3.4.本章小結(jié) P

46、AGEREF _Toc246387175 h 64 HYPERLINK l _Toc246387176 第四章 可靠性靈敏度分析的分解算法 PAGEREF _Toc246387176 h 65 HYPERLINK l _Toc246387177 4.1.可靠性靈敏度及其抽樣模擬求解方法 PAGEREF _Toc246387177 h 65 HYPERLINK l _Toc246387178 4.2.可靠性靈敏度的分解算法 PAGEREF _Toc246387178 h 66 HYPERLINK l _Toc246387179 4.2.1.分解算法的根本思路 PAGEREF _Toc246387

47、179 h 66 HYPERLINK l _Toc246387180 4.2.2.馬爾科夫鏈模擬特定失效域上的樣本 PAGEREF _Toc246387180 h 67 HYPERLINK l _Toc246387181 4.2.3.線性超平面的可靠性靈敏度的求解 PAGEREF _Toc246387181 h 68 HYPERLINK l _Toc246387182 4.2.4.及的求解 PAGEREF _Toc246387182 h 69 HYPERLINK l _Toc246387183 4.2.5.可靠性靈敏度估計及其方差和變異系數(shù) PAGEREF _Toc246387183 h 70

48、 HYPERLINK l _Toc246387184 4.3.算例 PAGEREF _Toc246387184 h 71 HYPERLINK l _Toc246387185 4.4.循環(huán)載荷作用下渦輪盤壽命可靠性靈敏度分析 PAGEREF _Toc246387185 h 76 HYPERLINK l _Toc246387186 4.5.本章小結(jié) PAGEREF _Toc246387186 h 78 HYPERLINK l _Toc246387187 第五章 失效概率函數(shù)與靈敏度函數(shù)的求解方法 PAGEREF _Toc246387187 h 79 HYPERLINK l _Toc24638718

49、8 5.1.失效概率函數(shù)及其特征指標(biāo)的求解方法 PAGEREF _Toc246387188 h 79 HYPERLINK l _Toc246387189 5.1.1.區(qū)間分布參數(shù)下的失效概率函數(shù)及其指標(biāo) PAGEREF _Toc246387189 h 80 HYPERLINK l _Toc246387190 5.1.2.失效概率函數(shù)及其指標(biāo)求解的Monte Carlo 直接法 PAGEREF _Toc246387190 h 81 HYPERLINK l _Toc246387191 5.1.3.失效概率函數(shù)及其指標(biāo)求解的FOSM法 PAGEREF _Toc246387191 h 82 HYPER

50、LINK l _Toc246387192 5.1.4.基于貝葉斯公式的Monte Carlo模擬求解方法 PAGEREF _Toc246387192 h 83 HYPERLINK l _Toc246387193 5.1.5.基于貝葉斯公式的條件概率馬爾科夫鏈模擬求解方法 PAGEREF _Toc246387193 h 87 HYPERLINK l _Toc246387194 5.1.6.算例 PAGEREF _Toc246387194 h 90 HYPERLINK l _Toc246387195 5.1.7.本節(jié)小結(jié) PAGEREF _Toc246387195 h 101 HYPERLINK

51、l _Toc246387196 5.2.可靠性靈敏度函數(shù)及其特征指標(biāo)的求解 PAGEREF _Toc246387196 h 103 HYPERLINK l _Toc246387197 5.2.1.區(qū)間分布參數(shù)下的全局靈敏度及其指標(biāo) PAGEREF _Toc246387197 h 103 HYPERLINK l _Toc246387198 5.2.2.全局靈敏度指標(biāo)求解的一次二階矩(FOSM)法 PAGEREF _Toc246387198 h 104 HYPERLINK l _Toc246387199 5.2.3.全局靈敏度指標(biāo)求解的Monte Carlo法 PAGEREF _Toc246387

52、199 h 105 HYPERLINK l _Toc246387200 5.2.4.全局靈敏度指標(biāo)求解的條件概率馬爾科夫鏈模擬方法 PAGEREF _Toc246387200 h 106 HYPERLINK l _Toc246387201 5.2.5.算例 PAGEREF _Toc246387201 h 108 HYPERLINK l _Toc246387202 5.2.6.本節(jié)小結(jié) PAGEREF _Toc246387202 h 115 HYPERLINK l _Toc246387203 5.3.本章小結(jié) PAGEREF _Toc246387203 h 116 HYPERLINK l _To

53、c246387204 第六章 極限狀態(tài)擴展累積分布函數(shù)及其導(dǎo)數(shù)的分析方法 PAGEREF _Toc246387204 h 117 HYPERLINK l _Toc246387205 6.1.擴展累積分布函數(shù)和靈敏度函數(shù) PAGEREF _Toc246387205 h 117 HYPERLINK l _Toc246387206 6.1.1.擴展累積分布函數(shù) PAGEREF _Toc246387206 h 117 HYPERLINK l _Toc246387207 6.1.2.擴展累積分布函數(shù)的導(dǎo)函數(shù) PAGEREF _Toc246387207 h 118 HYPERLINK l _Toc2463

54、87208 6.1.3.極限狀態(tài)函數(shù)的擴展概率密度對參數(shù)的靈敏度函數(shù) PAGEREF _Toc246387208 h 118 HYPERLINK l _Toc246387209 6.2.Monte Carlo 直接求解方法 PAGEREF _Toc246387209 h 118 HYPERLINK l _Toc246387210 6.3.正態(tài)變量線性極限狀態(tài)函數(shù)情況下的FOSM求解方法 PAGEREF _Toc246387210 h 119 HYPERLINK l _Toc246387211 6.4.基于貝葉斯公式的求解方法 PAGEREF _Toc246387211 h 120 HYPERL

55、INK l _Toc246387212 6.4.1.求解擴展函數(shù)、及的貝葉斯公式 PAGEREF _Toc246387212 h 120 HYPERLINK l _Toc246387213 6.4.2.基于貝葉斯公式的Monte Carlo模擬求解方法 PAGEREF _Toc246387213 h 123 HYPERLINK l _Toc246387214 6.4.3.基于貝葉斯公式的條件概率模擬求解方法 PAGEREF _Toc246387214 h 124 HYPERLINK l _Toc246387215 6.5.算例 PAGEREF _Toc246387215 h 127 HYPER

56、LINK l _Toc246387216 6.6.本章小結(jié) PAGEREF _Toc246387216 h 142 HYPERLINK l _Toc246387217 第七章 隨機鼓勵下隨機結(jié)構(gòu)動力可靠性分析 PAGEREF _Toc246387217 h 143 HYPERLINK l _Toc246387218 7.1.隨機結(jié)構(gòu)系統(tǒng)的動力可靠性分析 PAGEREF _Toc246387218 h 143 HYPERLINK l _Toc246387219 7.1.1.平穩(wěn)隨機響應(yīng)分析的虛擬鼓勵法 PAGEREF _Toc246387219 h 143 HYPERLINK l _Toc246

57、387220 7.1.2.隨機結(jié)構(gòu)系統(tǒng)的動力可靠性分析 PAGEREF _Toc246387220 h 144 HYPERLINK l _Toc246387221 7.2.單自由度振子體系動力可靠性問題 PAGEREF _Toc246387221 h 145 HYPERLINK l _Toc246387222 7.3.十五桿結(jié)構(gòu)動力可靠性分析 PAGEREF _Toc246387222 h 148 HYPERLINK l _Toc246387223 7.4.地震隨機鼓勵下隨機鋼框架結(jié)構(gòu)的動力可靠性分析 PAGEREF _Toc246387223 h 151 HYPERLINK l _Toc24

58、6387224 7.5.本章小結(jié) PAGEREF _Toc246387224 h 153 HYPERLINK l _Toc246387225 第八章 結(jié)論與展望 PAGEREF _Toc246387225 h 155 HYPERLINK l _Toc246387226 參考文獻 PAGEREF _Toc246387226 h 159 HYPERLINK l _Toc246387227 發(fā)表論文及科研 PAGEREF _Toc246387227 h 165 HYPERLINK l _Toc246387228 獲獎情況 PAGEREF _Toc246387228 h 168 HYPERLINK l

59、 _Toc246387229 致謝 PAGEREF _Toc246387229 h 169 HYPERLINK l _Toc246387230 學(xué)位論文知識產(chǎn)權(quán)聲明書 PAGEREF _Toc246387230 h 170第一章 緒論在工程中，由于不確定因素的廣泛存在，可靠性被引入到產(chǎn)品的平安分析和設(shè)計中，并逐漸成為科學(xué)和工程中一個非常重要的概念。概率論和隨機過程理論的開展，為隨機可靠性理論體系的建立和開展奠定了堅實的根底。目前基于可靠性的分析設(shè)計方法已經(jīng)廣泛應(yīng)用于航空、航天、汽車、船舶和土木工程等領(lǐng)域。在可靠性理論日益廣泛應(yīng)用的同時，工程結(jié)構(gòu)機構(gòu)的復(fù)雜性，多樣性對可靠性分析設(shè)計方法也提出了更

60、高的要求。目前仍然存在一些難點問題，傳統(tǒng)的可靠性分析方法并沒有很好的解決。這些問題包括：1非正態(tài)變量問題2高維問題3隱式極限狀態(tài)問題4可靠性優(yōu)化設(shè)計。對于這幾方面的問題，傳統(tǒng)的可靠性分析方法還有待進一步的完善和開展?？煽啃苑治龅慕平馕龇ê蛿?shù)字模擬法可靠性分析方法簡單的分，一般可分為近似解析法和數(shù)字模擬法?？煽啃苑治鲋袀鹘y(tǒng)的近似解析方法具有計算快捷和計算效率高的優(yōu)勢，因此在工程上得到了廣泛的應(yīng)用。一次可靠度方法 REF _Ref242193636 r h * MERGEFORMAT 1- REF _Ref243016031 r h * MERGEFORMAT 9是工程常用的方法，它采用將極限狀