非概率可靠性理論及相關(guān)算法研究
發(fā)布時間：2018-11-14 18:18
【摘要】：工程結(jié)構(gòu)的可靠性是學術(shù)界和工程界長期關(guān)注和研究的問題。傳統(tǒng)可靠性存在一定局限性,需要足夠數(shù)量的統(tǒng)計樣本,用以建立參數(shù)分布密度函數(shù)或者隸屬度函數(shù)。然而,實際工程中,獲得的樣本數(shù)據(jù)總是有限的,或者當所需數(shù)據(jù)量較大時,試驗費用較高,經(jīng)濟上不可接受；再者,對于處于設計中的結(jié)構(gòu),其參數(shù)的概率分布無從得知,采用某些假定的分布并不能完全保證與實際情況相符。因此,在樣本數(shù)量有限的情況下,無法確定參數(shù)的概率分布或隸屬度函數(shù),但其界限易于確定,這樣將參數(shù)表示成非概率凸集合顯得更為合理。非概率可靠性理論正是建立在非概率凸集合理論之上。 自上世紀90年代,Ben-Haim和Elishakoff提出非概率可靠性概念,理論上正趨于完善,但也存在一些認識上的錯誤,在算法上,仍缺乏正確有效的算法。本文對區(qū)間和超橢球凸集模型非概率可靠性指標的算法進行了研究,對具有顯式極限狀態(tài)方程的可靠性分析采用優(yōu)化算法中的梯度投影法,或者蒙特卡羅法,對隱式或者復雜極限狀態(tài)方程的可靠指標求解,采用蒙特卡羅法,響應面法和支持向量機。不確定參數(shù)變量之間的獨立性與相關(guān)性,決定非概率可靠性模型的建立,影響可靠指標的求解,本文也對非概率可靠性中的獨立性與相關(guān)性進行了討論。本文的研究內(nèi)容和研究成果如下： (1)對非概率可靠性與傳統(tǒng)可靠性進行比較研究,分析不同參數(shù)凸集合對可靠性的影響,給出原因。 (2)基于約束優(yōu)化理論中的可行方向法,將可行方向法中的梯度投影法,運用于非概率可靠性指標的求解,針對非概率可靠指標的特征,在求解過程中,提出空間搜索算法,以達到算法的收斂性。 (3)蒙特卡羅法是一種直接的數(shù)值模擬技術(shù),雖然計算量巨大,但計算結(jié)果可信度高,本文針對非概率可靠指標的不同形式,建立針對非概率可靠指標的蒙特卡羅法,為非概率可靠指標的其它算法提供檢驗方法。 (4)響應面法是針對具有復雜或隱式極限狀態(tài)方程的結(jié)構(gòu)可靠性分析方法,本文依據(jù)非概率可靠指標的特性,結(jié)合梯度投影算法,建立針對非概率可靠性的線性響應面法、線性加權(quán)響應法和二階響應面法。 (5)支持向量機是一種建立在小樣本的統(tǒng)計學習理論基礎(chǔ)上的機器學習方法,本文分別利用支持向量回歸機和支持向量分類機,建立適用于隱式極限狀態(tài)方程的非概率可靠性問題,解決了蒙特卡羅法和響應面法的維數(shù)災難,計算成本大、分析效率低下等問題。 (6)不確定參數(shù)的獨立性與相關(guān)性是客觀存在的,不因參數(shù)樣本的多寡而改變；本文對非概率凸集合的獨立性和相關(guān)性進行初步探討,用凸集合表述不確定參數(shù)的獨立性,在凸集合內(nèi)部表述不確定參數(shù)之間的相關(guān)性,并依此進行非概率可靠性分析。
[Abstract]:The reliability of engineering structures has long been concerned and studied by academic and engineering circles. There are some limitations in traditional reliability and a sufficient number of statistical samples are needed to establish parameter distribution density function or membership function. However, in practical engineering, the sample data obtained are always limited, or when the amount of data required is large, the test cost is higher and the economic is unacceptable. Furthermore, the probability distribution of the parameters is unknown for the structure under design, and the distribution using some assumptions can not be completely consistent with the actual situation. Therefore, in the case of limited number of samples, it is impossible to determine the probability distribution or membership function of parameters, but its bounds are easy to determine, so it is more reasonable to express the parameters as non-probabilistic convex sets. The theory of non-probabilistic reliability is based on the theory of non-probabilistic convex set. Since the 1990s, Ben-Haim and Elishakoff put forward the concept of non-probabilistic reliability, which is becoming more and more perfect in theory, but there are still some errors in understanding, and there is still a lack of correct and effective algorithm in the algorithm. In this paper, the algorithm of non-probabilistic reliability index for interval and hyperellipsoidal convex set model is studied. The gradient projection method or Monte Carlo method is used to analyze the reliability of the model with explicit limit state equation. Monte Carlo method, response surface method and support vector machine are used to solve the implicit or complex limit state equations. The independence and correlation among uncertain parameter variables determine the establishment of non-probabilistic reliability model and affect the solution of reliability index. The independence and correlation of non-probabilistic reliability are also discussed in this paper. The contents and results of this paper are as follows: (1) compare the non-probabilistic reliability with the traditional reliability, analyze the influence of different parameter convex sets on the reliability, and give the reasons. (2) based on the feasible direction method in the constrained optimization theory, the gradient projection method of the feasible direction method is applied to solve the non-probabilistic reliability index. According to the characteristics of the non-probabilistic reliability index, a spatial search algorithm is proposed. In order to achieve the convergence of the algorithm. (3) Monte-Carlo method is a direct numerical simulation technique. Although the calculation amount is huge, the calculation result is reliable. In this paper, the Monte Carlo method for non-probabilistic reliability index is established for different forms of non-probabilistic reliability index. It provides a test method for other algorithms of non-probabilistic reliability index. (4) response surface method (RSM) is a structural reliability analysis method with complex or implicit limit state equations. According to the characteristics of non-probabilistic reliability index and gradient projection algorithm, a linear response surface method for non-probabilistic reliability is established in this paper. Linear weighted response method and second order response surface method. (5) support vector machine (SVM) is a kind of machine learning method based on statistical learning theory of small sample. In this paper, support vector regression machine (SVM) and support vector classifier (SVM) are used respectively. The non-probabilistic reliability problem suitable for implicit limit state equations is established, which solves the problems of dimension disaster of Monte Carlo method and response surface method, high calculation cost and low analysis efficiency. (6) the independence and correlation of uncertain parameters exist objectively and do not change with the number of parameter samples; In this paper, the independence and correlation of non-probabilistic convex sets are preliminarily discussed. The independence of uncertain parameters is expressed by convex sets, and the correlation among uncertain parameters is expressed within convex sets, and the non-probabilistic reliability analysis is carried out accordingly.
【學位授予單位】：華中科技大學
【學位級別】：博士
【學位授予年份】：2013
【分類號】：TU311.2
本文編號：2331932
[Abstract]:The reliability of engineering structures has long been concerned and studied by academic and engineering circles. There are some limitations in traditional reliability and a sufficient number of statistical samples are needed to establish parameter distribution density function or membership function. However, in practical engineering, the sample data obtained are always limited, or when the amount of data required is large, the test cost is higher and the economic is unacceptable. Furthermore, the probability distribution of the parameters is unknown for the structure under design, and the distribution using some assumptions can not be completely consistent with the actual situation. Therefore, in the case of limited number of samples, it is impossible to determine the probability distribution or membership function of parameters, but its bounds are easy to determine, so it is more reasonable to express the parameters as non-probabilistic convex sets. The theory of non-probabilistic reliability is based on the theory of non-probabilistic convex set. Since the 1990s, Ben-Haim and Elishakoff put forward the concept of non-probabilistic reliability, which is becoming more and more perfect in theory, but there are still some errors in understanding, and there is still a lack of correct and effective algorithm in the algorithm. In this paper, the algorithm of non-probabilistic reliability index for interval and hyperellipsoidal convex set model is studied. The gradient projection method or Monte Carlo method is used to analyze the reliability of the model with explicit limit state equation. Monte Carlo method, response surface method and support vector machine are used to solve the implicit or complex limit state equations. The independence and correlation among uncertain parameter variables determine the establishment of non-probabilistic reliability model and affect the solution of reliability index. The independence and correlation of non-probabilistic reliability are also discussed in this paper. The contents and results of this paper are as follows: (1) compare the non-probabilistic reliability with the traditional reliability, analyze the influence of different parameter convex sets on the reliability, and give the reasons. (2) based on the feasible direction method in the constrained optimization theory, the gradient projection method of the feasible direction method is applied to solve the non-probabilistic reliability index. According to the characteristics of the non-probabilistic reliability index, a spatial search algorithm is proposed. In order to achieve the convergence of the algorithm. (3) Monte-Carlo method is a direct numerical simulation technique. Although the calculation amount is huge, the calculation result is reliable. In this paper, the Monte Carlo method for non-probabilistic reliability index is established for different forms of non-probabilistic reliability index. It provides a test method for other algorithms of non-probabilistic reliability index. (4) response surface method (RSM) is a structural reliability analysis method with complex or implicit limit state equations. According to the characteristics of non-probabilistic reliability index and gradient projection algorithm, a linear response surface method for non-probabilistic reliability is established in this paper. Linear weighted response method and second order response surface method. (5) support vector machine (SVM) is a kind of machine learning method based on statistical learning theory of small sample. In this paper, support vector regression machine (SVM) and support vector classifier (SVM) are used respectively. The non-probabilistic reliability problem suitable for implicit limit state equations is established, which solves the problems of dimension disaster of Monte Carlo method and response surface method, high calculation cost and low analysis efficiency. (6) the independence and correlation of uncertain parameters exist objectively and do not change with the number of parameter samples; In this paper, the independence and correlation of non-probabilistic convex sets are preliminarily discussed. The independence of uncertain parameters is expressed by convex sets, and the correlation among uncertain parameters is expressed within convex sets, and the non-probabilistic reliability analysis is carried out accordingly.
【學位授予單位】：華中科技大學
【學位級別】：博士
【學位授予年份】：2013
【分類號】：TU311.2
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