【作者】 刘海波；

【导师】 宋晓琳； 姜潮；

【作者基本信息】 湖南大学 ， 机械工程， 2019， 博士

【摘要】 工程产品在其设计、生产到报废的整个生命周期内都充满了各种不确定性参数。这些不确定性参数虽然在大多数情况下数值较小,但诸多不确定性因素传播、累积放大可能引起产品性能不稳定、可靠性降低,甚至导致灾难性事故。因此,在产品设计及使用的全生命周期过程中,采用有效的方法进行不确定性传播分析对于保证产品的可靠性、安全性甚至经济性均具有非常重要的意义。实际上,根据不确定性产生机理和物理意义的不同,工程中的不确定性大致可以分为两类:随机不确定性和认知不确定性。随机不确定性表示自然界或物理现象中存在的随机性,设计者无法控制或减少这类随机性,也称为统计不确定性或客观不确定性。随机不确定性的分析方法一般为概率理论、数理统计和随机过程等,其理论和应用研究均较为完善。认知不确定性是由于设计者的主观认识不足、数据缺乏或信息不完全所引起的,通常也称为主观不确定性。认知不确定性的分析方法一般为非概率理论,主要包含模糊集、区间理论和证据理论等。在不确定性分析领域中,随机不确定性和认知不确定性均发挥了重要的作用,且取得了丰硕的理论成果,但是在实际工程中,除了单一不确定性的情况,许多情况下则是多种不确定性共存的情况。因此,亟需一种能融合现有多种不确定性的模型。概率盒(p-box)模型可以同时描述随机和认知的混合不确定性问题,现有的证据结构、概率分布和概率区间混合模型等都可以转化为概率盒的形式。更重要的是,概率盒可以看作是概率理论和区间理论的混合,容易被了解概率和区间理论的工程技术人员理解和使用。鉴于这种特性,概率盒不确定性分析的研究越来越受到国内外专家学者的重视,并取得了一些重要进展。然而概率盒模型的不确定性研究整体还处于初级阶段,仍存在许多亟待解决的关键科学问题。其中,大规模计算以及复杂相关性等问题是限制概率盒在工程中更广泛应用的主要难点。为此,本文有针对性地开展了如下几个方面的研究工作:(1)针对传统采样方法计算效率低的问题,构建了一种基于降维积分和Johnson分布的概率盒不确定性传播分析方法。首先,构建了一种基于单变量降维积分的优化方法求得响应函数统计矩的区间;然后采用Johnson分布拟合响应的所有可能概率分布;最后,基于响应前四阶统计矩的区间,提出了矩匹配方法获得响应的概率分布边界曲线,完成基于概率盒模型的不确定性传播分析。(2)针对响应函数含有强交互项作用的问题,提出了一种基于稀疏网格数值积分和鞍点逼近理论的概率盒不确定性传播分析方法。首先,构建了一种基于稀疏网格数值积分的优化方法分别求解响应函数统计矩和累计量的区间;然后根据鞍点逼近理论,提出了一种优化策略计算响应的概率分布边界,所提方法对于存在强交互项作用的响应函数问题具有较好的计算精度,同时能获得精确的响应尾部概率分布信息。(3)针对复杂费时的大规模计算问题,发展出了一种基于稀疏分解的混沌多项式展开(polynomial chaos expansion,PCE)方法进行概率盒的不确定性传播分析。首先,为处理随机不确定性,提出一种基于稀疏分解的基选择策略自动判别和选择混沌多项式中重要的基函数;然后,为处理认知不确定性,将稀疏分解混沌多项式的系数处理成关于区间分布参数的二次多项式函数,最终获得响应函数的区间均值、区间标准差以及累计概率分布边界。所提方法为求解复杂工程中含概率盒不确定性的传播问题提供一种有效的解决方案。(4)针对输入变量之间具有复杂相关性的问题,给出了一种针对相关概率盒模型的结构不确定性传播分析方法。该方法首先根据有限的实验数据构建概率盒模型;然后,采用AIC准则选择最优的Copula函数,并进一步求得输入变量的联合概率分布函数,再通过Rosenblatt变换将变量从相关空间转换为独立的正态空间;最后基于稀疏网格数值积分计算响应函数的统计矩区间。(5)针对多失效模式问题,提出了一种高效的含概率盒不确定性的系统可靠性分析方法。首先,基于一个高效求解方法获得单失效模式下结构的最小可靠度指标;再针对多失效模式下含p-box不确定性问题建立了系统可靠性分析模型;考虑各失效模式之间的相关性,通过线性相关度计算方法求得相关系数矩阵;最后提出了串联体系和并联体系可靠度求解方法,该方法具有较高的系统可靠性计算效率,能够满足实际工程需求。更多还原

【Abstract】 There are various kinds of uncertainties in the whole life cycle from design,production to scrap of engineering products.In general,these uncertainties are small in most cases,but their spread and amplification may cause large fluctuation of product peformance,poor reliability,and even cause catastrophic accidents.Therefore,throughout the whole life cycle of the product,from design to use,it is of great significance to adopt uncertainty propagation analysis to ensure the reliability,safety and even economy of products.Actually,according to the different generation mechanisms and physical meanings,uncertainties can be divided into two different categories: aleatory uncertainty and epistemic uncertainty.Aleatory uncertainty,known as statistical uncertainty or objective uncertainty,which refers to the r andomness existing in nature or physical phenomena,can not be controlled or reduced by any designer.The analysis methods of aleatory uncertainty include probability theory,mathematical statistics and random process,whose theory and application research es are relatively mature and perfect.Epistemic uncertainty,generally called subjective uncertainty,is caused by insufficient subjective knowledge,data or incomplete information of the designer.The uncertainty analysis methods corresponding to epistemic uncertainty are non-probabilistic theories,which include fuzzy set,interval analysis theory and evidence theory etc.In fact,both aleatory uncertainty and epistemic uncertainty play an important role in the field of uncertainty analysis,and have obtained a lot of research achievements on theories.But in practice,there exists multiple uncertainty models rather than a single uncertainty model in practical engineering.Thus a new model is needed to integrate the existing multiple uncertainty models.Probabiligy box(p-box)is the model that can describe both aleatory uncertainty and epistemic uncertainty problems simultaneously,in gereral,the existing Dempster-Shafer structures,probability distribution,hybrid probability and interval model,etc.can be transformed into probability box.More importantly,probability box can be regarded as the mixture of probability theory and interval theory,which can be easily understood and accepted by engineers who are familiar with probability theory and interval theory.Due to this characteristic,the uncertainty analysis of p-box is getting more and more domestic and international attention in recent years,and some progressive achievements have been made.However,the research of uncertainty analysis based on probability box is still in the primary stage.A series of key scientific issues are required to be solved.Especailly,the large-scale computing and complex correlation problems hinder the applicability of probability box in practical engineering problems.Therefore,to deal with such challenges,the following aspects are carried out in this paper:(1)For the low computational efficiency of traditional sampling method,an efficient uncertainty propagation method based on univariate dimension reduction method(UDRM)and Johnson distribution is constructed for parameterized p-box inputs.Firstly,an optimization method based on univariate dimension reduction method is constructed to calculate the bounds on statistical moments of response function.Then,the Johnson distribution function is utilized for fitting the possible distribution functions of response.Finally,the moment matching method is presented based on the interval-valued moments,by which the probability bounds of the response p-box can be successfully obtained,and the uncertainty propagation analysis based on parameterized p-box is completed at the same time.(2)For the response function with strong interaction terms problems,a p-box uncertainty propagation analysis method based on sparse grid numerical integration and saddlepoint approximation is presented.Firstly,an optimization method based on the sparse grid numerical integration is presented to calculate the bounds on the statistical moments of the response function and the cumulants of the cumulant generating function,respectively.Then,an optimization strategy based on the saddlepoint approximation is proposed to calculate the probability bounds of reponse.The proposed method has high computational accuracy for high dimension and strong interaction problems,and can obtain a good tail probability distribution information.(3)For parameterized p-boxes variables,an uncertainty propagation analysis method based on sparse-decomposition-based polynomial chaos expansion is developed.Firstly,a sparse-decomposition-based polynomial chaos expansion(PCE)method is presented to process the aleatory uncertainty,in which a basis selec tion strategy that based on sparse decomposition is devised for automatically detecting the significant basis set of PCE.Then,to deal with the epistemic uncertainty,the coefficients of the sparse-decomposition-based PCE are treated as quadratic polynomial functions of the intervalvalued distribution parameters.Finally,the interval-valued mean,interval-valued standard deviation and probability bounds of the response function can be successfully obtained.The proposed method provides a new solution for the uncertainty propagation of probability boxes.(4)An uncertainty propagation method for correlated probability-boxes is given for input variables with complex nonlinear dependence.The parameterized p-box model is constructed first based on limited experimental data.Then,the AIC criterion is adopted to select the optimal copula function,by which the joint cumulative distribution function is acquired for the correlated input variables.Then the correlated variables are further transformed into independent normal variables through rosenblatt transformation.Finally,the bounds on statistic momens of the response function can be obtained based on the sparse grid numerical integration.(5)For the multiple failure modes problem,an efficient system reliability analysis method for structures with probability boxes uncertainties is proposed.Firstly,the minimum reliability index of each failure mode is obtained based on an efficient solution method.Then the system reliability model under multiple failure modes with probability box uncertainties is constructed.Considering the dependence between different failure modes of systems,a correlation coefficient matrix is obtained by the linear correlati on calculated method.Finally,the maximum failure probabilities are calculated for series and parallel system,respecively.The presented method has high computational efficiency and accuracy,which can meet the practical engineering requirements.更多还原