非概率可靠性理论及相关算法研究

发布时间：2018-11-14 18:18

【摘要】：工程结构的可靠性是学术界和工程界长期关注和研究的问题。传统可靠性存在一定局限性,需要足够数量的统计样本,用以建立参数分布密度函数或者隶属度函数。然而,实际工程中,获得的样本数据总是有限的,或者当所需数据量较大时,试验费用较高,经济上不可接受；再者,对于处于设计中的结构,其参数的概率分布无从得知,采用某些假定的分布并不能完全保证与实际情况相符。因此,在样本数量有限的情况下,无法确定参数的概率分布或隶属度函数,但其界限易于确定,这样将参数表示成非概率凸集合显得更为合理。非概率可靠性理论正是建立在非概率凸集合理论之上。自上世纪90年代,Ben-Haim和Elishakoff提出非概率可靠性概念,理论上正趋于完善,但也存在一些认识上的错误,在算法上,仍缺乏正确有效的算法。本文对区间和超椭球凸集模型非概率可靠性指标的算法进行了研究,对具有显式极限状态方程的可靠性分析采用优化算法中的梯度投影法,或者蒙特卡罗法,对隐式或者复杂极限状态方程的可靠指标求解,采用蒙特卡罗法,响应面法和支持向量机。不确定参数变量之间的独立性与相关性,决定非概率可靠性模型的建立,影响可靠指标的求解,本文也对非概率可靠性中的独立性与相关性进行了讨论。本文的研究内容和研究成果如下： (1)对非概率可靠性与传统可靠性进行比较研究,分析不同参数凸集合对可靠性的影响,给出原因。 (2)基于约束优化理论中的可行方向法,将可行方向法中的梯度投影法,运用于非概率可靠性指标的求解,针对非概率可靠指标的特征,在求解过程中,提出空间搜索算法,以达到算法的收敛性。 (3)蒙特卡罗法是一种直接的数值模拟技术,虽然计算量巨大,但计算结果可信度高,本文针对非概率可靠指标的不同形式,建立针对非概率可靠指标的蒙特卡罗法,为非概率可靠指标的其它算法提供检验方法。 (4)响应面法是针对具有复杂或隐式极限状态方程的结构可靠性分析方法,本文依据非概率可靠指标的特性,结合梯度投影算法,建立针对非概率可靠性的线性响应面法、线性加权响应法和二阶响应面法。 (5)支持向量机是一种建立在小样本的统计学习理论基础上的机器学习方法,本文分别利用支持向量回归机和支持向量分类机,建立适用于隐式极限状态方程的非概率可靠性问题,解决了蒙特卡罗法和响应面法的维数灾难,计算成本大、分析效率低下等问题。 (6)不确定参数的独立性与相关性是客观存在的,不因参数样本的多寡而改变；本文对非概率凸集合的独立性和相关性进行初步探讨,用凸集合表述不确定参数的独立性,在凸集合内部表述不确定参数之间的相关性,并依此进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.
【学位授予单位】：华中科技大学
【学位级别】：博士
【学位授予年份】：2013
【分类号】：TU311.2

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