联合分布函数构造的Copula函数方法及结构可靠度分析

不完备概率信息条件下变量联合分布函数的确定及其对结构可靠度的影响还缺少系统地研究。为此,提出了基于Copula函数的变量联合概率分布函数构造方法,并分析了不同Copula函数类型对结构可靠度的影响规律。首先,简要介绍了基于Copula函数的变量联合分布函数构造方法。其次,提出了构件失效概率计算的直接积分方法。最后以构件可靠度问题为例研究了Copula函数的类型对结构可靠度的影响规律。结果表明:不完备概率信息条件下构件可靠度是不唯一的,表征变量间相关性的Copula函数类型对构件可靠度具有明显的影响,不同Copula函数计算的构件失效概率存在明显的差别,这种差别随构件可靠指标的增大(或失效概率的减小)而增大。Copula函数尾部相关性对结构可靠度具有重要的影响。当功能函数的失效区域位于Copula函数尾部时,计算的失效概率明显比没有尾部相关性的Copula函数的失效概率大。基于功能函数的均值和标准差计算的可靠指标不能反映Copula函数的类型对结构可靠度的影响,而基于功能函数实际分布求得的失效概率则可以有效反映不同Copula函数对结构可靠度的影响。     

基于Copula函数的并联结构系统可靠度分析

不完备概率信息条件下变量联合分布函数的确定及其对结构系统可靠度的影响还缺少系统地研究,该文目的在于研究表征变量间相关性的Copula函数对结构系统可靠度的影响规律。首先,简要介绍了变量联合分布函数构造的Copula函数方法。其次,提出了并联系统失效概率计算方法,并推导了相应的计算公式。最后以几种典型Copula函数为例研究了Copula函数类型对结构并联系统可靠度的影响规律。结果表明:表征变量间相关性的Copula函数类型对结构系统可靠度具有明显的影响,不同Copula函数计算的系统失效概率存在明显的差别,这种差别随构件失效概率的减小而增大。当并联系统的失效区域位于Copula函数尾部时,Copula函数的尾部相关性对系统可靠度有明显的影响,计算的失效概率比没有尾部相关性的Copula函数的失效概率大。当组成并联系统的两构件功能函数间正相关时,系统失效概率随相关系数的增大而增加;当构件功能函数间负相关时,系统失效概率随相关系数的增大而减小。此外,无论构件失效概率和变量间相关系数如何变化,Copula函数计算的失效概率都位于系统失效概率的上下限内。     

Data-driven reliability assessment with scarce samples considering multidimensional dependence

This study proposes a data-driven method for assessing reliability, based on the scarce input dataset with multidimensional correlation. Since considering the distribution parameters estimated from the scarce dataset as those of the population may lead to epistemic uncertainty, the bootstrap resampling algorithm is adopted to infer the distribution parameters as interval parameters. To account for the variable dependence, vine copula theory is utilized to construct the joint probability density function (PDF) of input variables, and maximum likelihood estimation (MLE) and Akaike information criterion (AIC) analysis are employed to select optimal copulas based on the samples for the vine structure. Subsequently, the failure probability bounds of a response function are calculated based on the constructed joint PDF with interval distribution parameters by the active learning Kriging (AK) method combining the sparse grid integration (SGI) method. Finally, several examples are provided to demonstrate the feasibility and efficiency of the proposed method.     

Multivariate Degradation Modeling of Smart Electricity Meter with Multiple Performance Characteristics via Vine Copulas

For smart electricity meter with multiple performance characteristics (PCs) with coupling relationships because of amounts of components experiencing multiple deteriorating processes, we develop a multivariate degradation modeling method via vine copulas to estimate the reliability of products with multiple PCs reflecting degradation states. In the multivariate model, pair‐copula construction and vine graphical representation are used to describe the mutual relationship among PCs to overcome the lack of multivariate copula in high‐dimensional cases. Each PC model of smart electricity meter is built by using drift Brownian motion to describe degradation processes of each PC on the basis of degradation mechanism analysis, and parameters are estimated by using likelihood estimation method. The Pearson correlation coefficient, Kendall's τ and product information are used to analyze correlation among those PCs. Furthermore, on the basis of conditional probability theory, the vine graph is used to construct a multivariate copula which can be decomposed into pair copulas. Akaike information criterion principle is utilized to choose the forms of pair‐copula functions in the correlation model. Finally, the reliability joint distribution of all PCs of smart electricity meter is obtained with combining all PCs' marginal distributions and copula functions. Copyright © 2016 John Wiley & Sons, Ltd.     

基于鞍点逼近的机械结构可靠性稳健优化设计

将可靠性优化设计理论与可靠性灵敏度分析方法相结合,讨论了机械零部件稳健优化设计的问题.系统地推导了基于鞍点逼近的可靠性灵敏度公式,并把可靠性灵敏度计算结果融入可靠性稳健优化设计模型之中,将可靠性稳健优化设计归结为满足可靠性要求的多目标优化问题.在基本随机参数概率分布已知的前提下,应用鞍点逼近技术,得到极限状态函数的分布函数与概率密度函数,并且将此结果应用到机械零部件的可靠性灵敏度分析中,进而实现了机械零部件的可靠性稳健优化设计.通过与Monte-Carlo方法计算所得的结果相比可知,应用鞍点逼近技术可以迅速、准确地得到机械零部件可靠性稳健设计信息．     

基于矩方法的特高压输电塔抗风可靠度分析

该文研究特高压输电塔抗风可靠度。基于等效随机静风荷载模型,引入矩方法分析特高压输电塔的抗风体系可靠度。该方法基于等价极值事件,利用统计矩点估计法求解得到等价极值变量前四阶统计矩信息后,即可方便地获得特高压输电塔体系可靠度指标及相应失效概率。1000kV级特高压SZT2钢管塔的数值算例分析表明:1) 矩方法简单、高效,将其运用于特高压输电塔抗风可靠度分析切实可行,具有重要的理论意义和工程实用价值;2) 在采用点估计法进行等价极值变量统计矩估计时,所取估计点数目应依据失效概率收敛情况而定,并非任意选取或取得越多越好。     

Estimation in coherent reliability systems through copulas

The problem of estimating the parameter of a common distribution of components' lifetimes from system's lifetime data is of interest and importance in reliability engineering. The present paper deals with this problem when the common component distribution is exponential with mean μ and the lifetimes of components have an exchangeable joint distribution which is constructed by the help of Archimedean copula. In particular we obtain moment estimator of μ for Clayton and Ali-Mikhail-Haq copulas. We illustrate the findings of the paper for a special class of coherent systems called consecutive k-within-m-out-of-n:F system. A simulation study is performed to investigate the properties of the moment estimator. The method presented in this paper can be applied to all coherent systems.     

Reliability of dynamic systems in random environment by extreme value theory

A practical method is developed for estimating the performance of highly reliable dynamic systems in random environment. The method uses concepts of univariate extreme value theory and a relatively small set of simulated samples of system states. Generalized extreme value distributions are fitted to state observations and used to extrapolate Monte Carlo estimates of reliability and failure probability beyond data. There is no need to postulate functional forms of extreme value distributions since they are selected by the estimation procedure. Our approach can be viewed as an alternative implementation of the method in [7] for estimating system reliability. Numerical examples involving Gaussian and non-Gaussian system states are used to illustrate the implementation of the proposed method and assess its accuracy.     

联合分布函数蒙特卡罗模拟及结构可靠度分析

针对复杂极限状态方程可靠度计算问题,提出了基于理论联合分布函数以及2 种近似联合分布函数的结构失效概率蒙特卡罗模拟方法,并给出了计算流程图.采用2 个算例证明了所提方法的有效性.结果表明:所提的失效概率模拟方法的计算精度很高,尤其适用于复杂极限状态方程的可靠度计算问题.2 种联合分布函数近似构造方法得到的失效概率精度相当,近似方法与精确方法结果的差异随失效概率的减小而增大,而且随着变量间相关性的增加而增加.当失效概率小于10-3时,近似方法的失效概率误差较大.     

基于双变量降维模型和Kriging近似的统计矩点估计法

对复杂随机系统进行统计矩分析时,双变量降维近似模型一定程度上可以缓解“维数灾难”。但当系统维数较高时,双变量分量函数较多,计算量仍然较大。为此,该文将降维近似和Kriging代理模型有机结合起来,提出了一类高效、合理的改进点估计法。充分考虑函数逼近和数值积分中积分点的特点,提出了“米”字形的选点策略,并基于此发展了双变量分量函数的Kriging近似模型;将此近似模型用于原函数和矩函数的双变量降维近似模型中双变量分量函数的近似,分别建立了基于原函数近似和矩函数近似的统计矩改进点估计法;通过多个算例对该文提出方法进行了效率和精度的分析。算例分析结果表明:基于“米”字形选点策略的双变量分量函数的Kriging近似具有较高的精度;相比于已有的基于双变量降维近似模型的统计矩点估计法,建议方法仅需较少的结构分析即可达到与已有方法相当的精度,能更好地体现精度和效率的平衡。     

FGM Copula的生成与拓展

从Copula的构造方法出发,本文引出四类性质不同,形态各异的Copulas。证明了FGM Cop-ula可以由它们生成,从而兼有这四类Copulas的性质。进一步,以FGM Copula及其联合生存函数为基础,应用Copula的几何构造方法拓展出一系列二参数三次幂的广义FGM Copulas。最后,从理论上说明FGMCopula是一个性质非常优良的Copula,在建模时被广泛应用。     

Reliability assessment models for dependent competing failure processes considering correlations between random shocks and degradations

This article develops reliability models for systems subject to two dependent competing failure processes, considering the correlation between additional damage size on degradation in soft failure process and stress magnitude of shock load in hard failure process, both of which are caused by the same kth random shock. The generalized correlative reliability assessment model based on copulas is proposed, which is then extended to three different shock patterns: (1) δ‐shock, (2) m‐shock, and (3) m‐run shocks. There are some statistical works to be introduced in reliability modeling, including data separation of total degradation amount, inferring the distribution of amount of aging continuous degradation at time t, and fitting copula to the specific correlation. The developed reliability models are demonstrated for an application example of a micro‐electro‐mechanical system.     

A Copula Approach in the Point Estimate Method for Reliability Engineering

In order to arrive at realistic results in statistical analysis, it is often advisable to consider involved uncertainties as random variables. An important aspect in this context is the evaluation of the importance of parameter uncertainty. Because of the complexity of computational models, the point estimate method is usually adopted as an easy‐run approach for approximating the statistical moments of a system/model in a reliability analysis. The efficiency of this method highly depends on the correlation coefficient. However, the complex nature of parameters in computational problems often exhibits a nonlinear relationship. This paper aims to develop an original and efficient point estimate method based on the copula approach for reliability engineering problems. The paper discusses the use of the copula theory in the point estimate method for computing the statistical moments of a function involving random variables. The study performs two engineering applications to demonstrate the benefits of this approach. The performance of this proposed method can significantly improve the quality of the results in using the point estimate method when a nonlinear relationship exists between the parameters. Copyright © 2015 John Wiley & Sons, Ltd.     

Reliability assessment for failure-dependent and uncertain systems: A Bayesian network based on copula method and probability-box

Managing failure dependence of complex systems with hybrid uncertainty is one of the hot problems in reliability assessment. Epistemic uncertainty is attributed to complex working environment, system structure, human factors, imperfect knowledge, etc. Probability-box has powerful characteristics for uncertainty analysis and can be effectively adopted to represent epistemic uncertainty. However, arithmetic rules on probability-box structures are mostly used among structures representing independent random variables. In most practical engineering applications, failure dependence is always introduced in system reliability analysis. Therefore, this paper proposes a developed Bayesian network combining copula method with probability-box for system reliability assessment. There are four main steps involved in the reliability computation process: marginal distribution identification and estimation, copula function selection and parameter estimation, reliability analysis of components with correlations and Bayesian forward analysis. The benefits derived from the proposed approach are used to overcome the computational limitations of n-dimensional integral operation, and the advantages of useful properties of copula function in reliability analysis of systems with correlations are adopted. To demonstrate the effectiveness of the developed Bayesian network, the proposed method is applied to a real large piston compressor.     

Reliability analysis of large structural systems

Brute force Monte Carlo simulation methods can, in principle, be used to calculate accurately the reliability of complicated structural systems, but the computational burden may be prohibitive. A new Monte Carlo based method for estimating system reliability that aims at reducing the computational cost is therefore proposed. It exploits the regularity of tail probabilities to set up an approximation procedure for the prediction of the far tail failure probabilities based on the estimates of the failure probabilities obtained by Monte Carlo simulation at more moderate levels. In this paper, the usefulness and accuracy of the estimation method is illustrated by application to a particular example of a structure with several thousand potentially critical limit state functions. The effect of varying the correlation of the load components is also investigated.     

Parametric and nonparametric inference for the reliability of copula-based stress-strength models

This paper provides a general treatment of statistical inference for the reliability in copula-based stress-strength models. Most of the current literature is either focused on specific models that yield clean formulas or restricted to estimation and engineering aspects without addressing statistical inference. We present two general frameworks, one parametric, one nonparametric, for the estimation of the reliability. The parametric methodology is presented under the general framework of estimating equations, mostly as a combination of existing methodologies from the fields of multivariate analysis, reliability, and econometrics, with some new results. The nonparametric methodology is a novel application based on an existing bivariate kernel method combined with Monte Carlo estimation of the reliability without specification of the copula or the margins. We present results from a small simulation study designed to assess the robustness of the methods discussed in terms of model misspecification. We used geotechnical data and data from the Brazilian Household Survey to illustrate the proposed methodologies in the estimation of factors of safety and financial fragility.     

基于改进统计矩点估计法和最大熵原理的结构整体可靠度分析

结构整体可靠度评估一直以来是结构可靠度领域研究的热点与难点。该文将结构整体可靠度分类,并给出其对应功能函数的统一描述;结合提出的有效维度两步分析法和共轭无迹变换法,发展了改进统计矩点估计法;结合最大熵原理和改进统计矩点估计法,提出了适用于两类结构整体可靠度的统一分析方法;通过2个数值算例对该文方法进行了验证。算例分析结果表明:同一精度水平下,该文方法的计算效率较传统的三变量降维近似统计矩点估计法高2.3倍~2.6倍;该文方法具有高的精度水平,其最大相对误差低于2%,适用于结构整体可靠度评估。     

Effect of blow-holes on reliability of cast component

This paper presents a study on the effect of blow-holes on the reliability of a cast component. The most probable point (MPP) based univariate response surface approximation is used for evaluating reliability. Crack geometry, blow-hole dimensions, external loads and material properties are treated as independent random variables. The methodology involves novel function decomposition at a most probable point that facilitates the MPP-based univariate response surface approximation of the original multivariate implicit limit state/performance function in the rotated Gaussian space. Once the approximate form of the original implicit limit state/performance function is defined, the failure probability can be obtained by Monte Carlo simulation (MCS), importance sampling technique, and first- and second-order reliability methods (FORM/SORM). FORTRAN code is developed to automate calls to ABAQUS for numerically simulating responses at sample points, to construct univariate response surface approximation, and to subsequently evaluate the failure probability by MCS, importance sampling technique, and FORM/SORM.     

含间隙非线性的惯容橡胶复合隔振系统可靠性分析

大多数惯容系统的研究未考虑间隙非线性的影响,有研究表明,大间隙的产生对系统响应的影响不可忽略。该文建立了含间隙非线性的惯容-橡胶复合隔振系统的随机微分方程,基于随机非线性分析方法,推导了系统响应的统计矩,计算了系统响应的概率密度函数,利用首超可靠性分析理论求得了系统的失效概率,并分析了间隙对系统响应的统计特性及可靠性的影响。同时,也考虑了非平稳激励下间隙非线性对系统响应及可靠性的影响。结果表明,间隙值变大时,系统响应的统计矩变大,概率密度函数曲线快速发散,系统的失效概率迅速增加,这与确定性分析得到的结果不同,在设计隔振器时应当考虑间隙对系统动力可靠性的影响。     

利用矩点估计法简化响应面可靠度指标的计算

针对响应面可靠度指标计算方法的缺陷,将罗森布鲁斯统计矩点估计法引入到经典响应面方法中对其进行改进。改进后响应面方法,考虑基本随机变量偏态情况,改变了模拟试验中随机变量抽样值的计算方法;直接采用了统计矩计算可靠度指标,使可靠度指标的计算在真实极限状态曲面的某些特殊点上进行;在计算过程中不需拟合近似极限状态曲面和对非线性方程进行线性化处理,不产生迭代误差和累积误差,计算过程简洁明了。最后分别利用改进的响应面方法和经典响应面方法分析了某一矿山大型工程的稳定可靠性,并以蒙特卡洛法计算结果作为准精确解进行了比较,计算结果满足工程要求。