Enhanced reliability analysis method for multistate systems with epistemic uncertainty based on evidential network

Evidential network is considered to have superiority in conducting reliability analysis for complex engineering systems with epistemic uncertainty. However, existing methods tend to result in combinational explosion when multistate systems are involved in the reliability analysis, which means the reliability analysis cost increases exponentially with the number of components and that of functioning states. Therefore, an enhanced reliability analysis method is proposed in this paper for reliability analysis and performance evaluation of multistate systems with epistemic uncertainty, through which the combination explosion can be significantly alleviated. Firstly, the functioning states of each component are sequenced according to utility functions. Secondly, the basic belief assignment (BBA) of each component is reassigned in terms of commonality function, through which the BBA defined in the power set space is represented by two extreme BBA distributions defined in the frame of discernment. Thirdly, the reliability intervals of the system states are calculated through evidential network, and the system performance level is computed. Two multistate system numerical examples are investigated to demonstrate the effectiveness and efficiency of the proposed method.     

Lifetime cost optimization with time-dependent reliability

Product lifetime cost is largely determined by product lifetime reliability. In product design, the former is minimized while the latter is treated as a constraint and is usually estimated by statistical means. In this work, a new lifetime cost optimization model is developed where the product lifetime reliability is predicted with computational models derived from physical principles. With the physics-based reliability method, the state of a system is indicated by computational models, and the time-dependent system reliability is then predicted for a given set of distributions and stochastic processes in the model input. A sampling approach to extreme value distributions of input stochastic processes is employed to make the system reliability analysis efficient and accurate. The physics-based reliability analysis is integrated with the lifetime cost model. The integration enables the minimal lifetime costs including those of maintenance and warranty. Two design examples are used to demonstrate the proposed model.     

A STATISTICAL INVESTIGATION OF THE FATIGUE LIVES OF Q235 STEEL-WELDED JOINTS

An investigation into the fitting of six assumed distributions (three-parameter Weibull, two-parameter Weibull, extreme minimum value, extreme maximum value, normal and lognormal distributions) of 23 groups of fatigue life data for Q235 steel-welded joints is performed in terms of linear regression analyses. The results reveal that the fatigue life distribution shapes mostly tend to be positively skewed. Therefore, the extreme minimum value and normal distributions are not the most appropriate distributions to assume for a fatigue life evaluation. The three-parameter Weibull distribution may give misleading results in fatigue reliability analyses because the shape parameter is often lesss than 1. This means that the hazard rate decreases with fatigue cycling. This is contrary to the general understanding of the behaviour of welded joints. Reliability analyses may also be affected by slightly non-conservative evaluations in tail regions of the three-parameter Weibull distribution. The two-parameter Weibull distribution does not give as good a fit as either the extreme maximum value distribution or the lognormal distribution. On the other hand, the extreme maximum value and lognormal distributions can be safely assumed in reliability analyses due to the good total fit effects and the conservative evaluations in tail regions. In addition, the extreme maximum value distribution is in good agreement with the general physical understanding of the structural behaviour of welded joints.     

Adaptive surrogate model coupled with stochastic configuration network strategies for time-dependent reliability assessment

Time-dependent reliability assessment is crucial in enhancing product development economics and product performance sustainability throughout the lifecycle. It is still a challenge to accurately and efficiently evaluate the time-dependent reliability of engineering systems. This paper proposes a novel adaptive surrogate model method combining stochastic configuration network (SCN) and Kriging strategies to evaluate time-dependent reliability. SCN has accurate approximation ability and learning efficiency for strongly nonlinear systems that can overcome the conventional time-dependent reliability calculation, which is time-consuming and characterized by low accuracy. The proposed method first applies SCN to establish the response model of the performance function with respect to time and obtain the extreme value of the performance function. Then, Kriging is used to establish the extreme value model of the performance function with respect to the random variables based on the extreme value of performance function. The adaptive process considering the characteristics of random variables samples is adopted to update the extreme value model until the model meets the confidence target. Lastly, Monte Carlo simulation is employed for time-dependent reliability assessment based on the established extreme value model. Three example studies are used to demonstrate the effectiveness of the proposed approach for time-dependent reliability assessment.     

非高斯随机过程的短期极值估计:复合Hermite模型

Hermite模型自20世纪80年代后期开始被广泛应用于非高斯随机过程的短期极值估计。当随机过程的非高斯性很强时,尤其是偏度很大时,常用的3阶Hermite模型不足以表征出极值分布的尾端特征。工程中,样本统计矩的不确定性使得更高阶的Hermite模型不宜使用。基于此,该文提出了同时基于中心矩与线性矩的复合Hermite模型,有效地将Hermite模型由3阶拓展到4阶。该文以对数正态模型作为非线性系统的研究对象,对比分析了在解析条件下和在使用蒙特卡洛模拟获得样本数据条件下,各类Hermite模型与传统的Gumbel法以及平均条件穿越率（ACER）法用于极值分析的表现。结果表明,对于大偏度强非高斯随机过程的极值预测,复合Hermite模型具有更好的精确度和鲁棒性。     

一维机械强度参数概率模型的合理重构

发展了一维机械强度参数概率模型的Monte Carlo模拟重构方法,以实现任意概率水平可靠性分析。方法用于解决参数仅以特定存活概率(P)值或特定P与置信度(C)值给出时,除特定值外无法实现其它概率水平可靠性分析的问题。为避免模拟样本过大使结果脱离实践、预测偏于危险,建议材料小试样7个―20个样本、结构试样至多10个、还原统计参数误差≤5%的模拟策略。发展了包含6种统计分布模型即正态、对数正态、三参数Weibull、两参数Weibull、极大值和极.小值分布的重构方法与实施细节。10种工程材料疲劳极限的重构实践,验证了方法的有效性与可用性。     

Extreme value distributions for nonlinear transformations of vector Gaussian processes

Approximations are developed for the marginal and joint probability distributions for the extreme values, associated with a vector of non-Gaussian random processes. The component non-Gaussian processes are obtained as nonlinear transformations of a vector of stationary, mutually correlated, Gaussian random processes and are thus, mutually dependent. The multivariate counting process, associated with the number of level crossings by the component non-Gaussian processes, is modelled as a multivariate Poisson point process. An analytical formulation is developed for determining the parameters of the multivariate Poisson process. This, in turn, leads to the joint probability distribution of the extreme values of the non-Gaussian processes, over a given time duration. For problems not amenable for analytical solutions, an algorithm is developed to determine these parameters numerically. The proposed extreme value distributions have applications in time-variant reliability analysis of randomly vibrating structural systems. The method is illustrated through three numerical examples and their accuracy is examined with respect to estimates from full scale Monte Carlo simulations of vector non-Gaussian processes.     

基于Legendre正交多项式逼近法的结构可靠性分析

提出了结构可靠性分析的Legendre正交多项式逼近法。主要是基于数值逼近原理,以Legendre正交函数族做基,利用功能函数的高阶矩信息,通过计算功能函数概率密度函数的逼近表达式,然后根据工程结构可靠性的一般表达式来计算结构的失效概率,进行可靠性分析。通过数值检验,证明该方法可以很好地逼近各种经典理论分布曲线(正态分布、指数分布等6种经典分布)。文后给出了结构构件失效概率的实例计算,并和其它几种常用方法进行对比,进一步表明了Legendre正交多项式数值逼近法在结构可靠性分析理论上的正确性和实用性。     

Space–time extreme value statistics of a Gaussian random field

This paper describes a new method for estimating the extreme values of a Gaussian random field in both space and time. The method relies on the use of data provided by measurement or Monte Carlo simulation combined with a technique for estimating the extreme value distribution of a recorded time series. The time series in question represents the spatial extremes of the random field at each point in time. The time series is constructed by sampling the available realization of the random field over a suitable grid defining the domain in question and extracting the extreme value. This is done for each time point of a suitable time grid. Thus, the time series of spatial extremes is produced. This time series provides the basis for estimating the extreme value distribution using available techniques for time series, which results in an accurate practical procedure for solving a very difficult problem. This is demonstrated by comparison with the results obtained from analytically derived expressions for the extreme values of a Gaussian random field. Properties of these analytical formulas are also discussed.     

An integrated reliability-based design optimization of offshore towers

After recognizing the uncertainty in the parameters such as material, loading, geometry and so on in contrast with the conventional optimization, the reliability-based design optimization (RBDO) concept has become more meaningful to perform an economical design implementation, which includes a reliability analysis and an optimization algorithm. RBDO procedures include structural analysis, reliability analysis and sensitivity analysis both for optimization and for reliability. The efficiency of the RBDO system depends on the mentioned numerical algorithms. In this work, an integrated algorithms system is proposed to implement the RBDO of the offshore towers, which are subjected to the extreme wave loading. The numerical strategies interacting with each other to fulfill the RBDO of towers are as follows: (a) a structural analysis program, SAPOS, (b) an optimization program, SQP and (c) a reliability analysis program based on FORM. A demonstration of an example tripod tower under the reliability constraints based on limit states of the critical stress, buckling and the natural frequency is presented.     

结构风振响应极值估计方法比较

为考察基于高斯分布假定的工程极值估计方法对结构风振响应极值估计的准确性,以单层球面网壳结构为例,首先通过重复采样风洞实验和重复风振响应分析,获得了超过1000个以上的结构风振响应时程样本;其次利用经典极值理论根据大量样本确定结构风振响应的极值准确值,作为评价标准;然后分别采用几种常用的工程极值估计方法,包括峰值因子法、改进Hermite模型法和Sadek-Simiu法,仅根据单个样本确定极值;最后通过对众多极值估计值进行统计分析,系统考察了不同方法极值估计结果的稳定性和准确性。结果表明,峰值因子法估计结果的离散性较小,但数值多数偏低;改进Hermite模型法估计结果的离散性较大,且数值多数偏高;Sadek-Simiu法的离散性介于二者之间,且估计结果与准确值较为接近,因而建议在进行工程极值估计时采用。     

A new method for selection of population distribution and parameter estimation

An optimization method for parameter estimation is presented with the Kolmogorov-Smirnov distance used as the objective. A step-by-step implementation procedure is given. The method is demonstrated by estimating the parameters for three-parameter Weibull distributions from three different samples (with different sample sizes). A comparison of the proposed method and the usual methods such as the least-squares method, the matching moments method and the maximum likelihood method shows that more reasonable estimates of the parameters are given by the proposed optimization method. Then, the proposed method is successfully extended to estimate the parameters for the sum of two three-parameter Weibull distributions. Based on these findings, a new procedure for selection of population distribution and parameter estimation is presented.     

Estimation of density functions and upcrossing-rates from samples

This paper assesses techniques for estimating univariate probability density functions, and highlights strengths and weaknesses of several methods. Applications are given for non-Gaussian distributions typical of the dynamic response of nonlinear systems. It is concluded that of the methods considered the maximum likelihood method is generally the most suitable when predicting extreme values. The moment and maximum likelihood methods are adapted for estimation of upcrossing-rate functions.     

Reliability-based evaluation of automotive wind noise quality

This paper proposes an efficient method for the reliability analysis of a vehicle body–door subsystem with respect to an important quality issue—wind noise. A nonlinear seal model is constructed for the automotive wind noise problem and the limit state function is evaluated using finite element analysis. A multi-modal adaptive importance sampling (AIS) method is developed for reliability analysis at system level, to improve the efficiency of the Monte Carlo simulation. The method can easily handle implicit and time-dependent limit-state functions, with variables of any statistical distributions. Existing analytical as well as simulation-based methods are also investigated to solve the car door wind noise problem. It is demonstrated through this industrial application problem that the multi-modal AIS method is superior to existing methods in terms of efficiency and accuracy. A generalized framework for reliability estimation is then established for series system reliability problems with large numbers of random variables and complicated, implicit limit states.     

Direct control method for improving stability and reliability of nonlinear stochastic dynamical systems

Optimal control for improving the stability and reliability of nonlinear stochastic dynamical systems is of great significance for enhancing system performances. However, it has not been adequately investigated because the evaluation indicators for stability (e.g. maximal Lyapunov exponent) and for reliability (e.g. mean first-passage time) cannot be explicitly expressed as the functions of system states. Here, a unified procedure is established to derive optimal control strategies for improving system stability and reliability, in which a physical intuition-inspired separation technique is adopted to split feedback control forces into conservative components and dissipative components, the stochastic averaging is then utilized to express the evaluation indicators of performances of controlled system, the optimal control strategies are finally derived by minimizing the performance indexes constituted by the sigmoid function of maximal Lyapunov exponent (for stability-based control)/the reciprocal of mean first-passage time (for reliability-based control), and the mean value of quadratic form of control force. The unified procedure converts the original functional extreme problem of optimal control into an extremum value problem of multivariable function which can be solved by optimization algorithms. A numerical example is worked out to illustrate the efficacy of the optimal control strategies for enhancing system performance.     

Two-dimensional extreme distribution for estimating mechanism reliability under large variance

The effective estimation of the operational reliability of mechanism is a significant challenge in engineering practices, especially when the variance of uncertain factors becomes large. Addressing this challenge, a novel mechanism reliability method via a two-dimensional extreme distribution is investigated in the paper. The time-variant reliability problem for the mechanism is first transformed to the time-invariant system reliability problem by constructing the two-dimensional extreme distribution. The joint probability density functions (JPDFs), including random expansion points and extreme motion errors, are then obtained by combining the kernel density estimation (KDE) method and the copula function. Finally, a multidimensional integration is performed to calculate the system time-invariant reliability. Two cases are investigated to demonstrate the effectiveness of the presented method.The full text can be downloaded at https://link.springer.com/article/10.1007/s40436-020-00311-4     

A generic method for estimating system reliability using Bayesian networks

This study presents a holistic method for constructing a Bayesian network (BN) model for estimating system reliability. BN is a probabilistic approach that is used to model and predict the behavior of a system based on observed stochastic events. The BN model is a directed acyclic graph (DAG) where the nodes represent system components and arcs represent relationships among them. Although recent studies on using BN for estimating system reliability have been proposed, they are based on the assumption that a pre-built BN has been designed to represent the system. In these studies, the task of building the BN is typically left to a group of specialists who are BN and domain experts. The BN experts should learn about the domain before building the BN, which is generally very time consuming and may lead to incorrect deductions. As there are no existing studies to eliminate the need for a human expert in the process of system reliability estimation, this paper introduces a method that uses historical data about the system to be modeled as a BN and provides efficient techniques for automated construction of the BN model, and hence estimation of the system reliability. In this respect K2, a data mining algorithm, is used for finding associations between system components, and thus building the BN model. This algorithm uses a heuristic to provide efficient and accurate results while searching for associations. Moreover, no human intervention is necessary during the process of BN construction and reliability estimation. The paper provides a step-by-step illustration of the method and evaluation of the approach with literature case examples.     

Forecasting reliability growth

This paper describes a method for estimating and forecasting reliability from attribute data, using the binomial model, when reliability requirements are very high and test data are limited. Integer data—specifically, numbers of failures — are converted into non-integer data. The rationale is that when engineering corrective action for a failure is implemented, the probability of recurrence of that failure is reduced; therefore, such failures should not be carried as full failures in subsequent reliability estimates. The reduced failure value for each failure mode is the upper limit on the probability of failure based on the number of successes after engineering corrective action has been implemented. Each failure value is less than one and diminishes as test programme successes continue. These numbers replace the integral numbers (of failures) in the binomial estimate. This method of reliability estimation was applied to attribute data from the life history of a previously tested system, and a reliability growth equation was fitted. It was then ‘calibrated’ for a current similar system's ultimate reliability requirements to provide a model for reliability growth over its entire life-cycle. By comparing current estimates of reliability with the expected value computed from the model, the forecast was obtained by extrapolation.     

System Reliability and Component Importance Under Dependence: A Copula Approach

System reliability and component importance are of great interest in reliability modeling, especially when the components within the system are dependent. We characterize the influence of dependence structures on system reliability and component importance in coherent systems with discrete marginal distributions. The effects of dependence are captured through copula theory. We extend our framework to coherent multi-state system. Applications of the derived results are demonstrated using a Gaussian copula, which yields simple interpretations. Simulations and two examples are presented to demonstrate the importance of modeling dependence when estimating system reliability and ranking of component importance. Proofs, algorithms, code, and data are provided in supplementary materials available online.     

Mixing Bayes and empirical Bayes inference to anticipate the realization of engineering concerns about variant system designs

Mixing Bayes and Empirical Bayes inference provides reliability estimates for variant system designs by using relevant failure data - observed and anticipated - about engineering changes arising due to modification and innovation. A coherent inference framework is proposed to predict the realization of engineering concerns during product development so that informed decisions can be made about the system design and the analysis conducted to prove reliability. The proposed method involves combining subjective prior distributions for the number of engineering concerns with empirical priors for the non-parametric distribution of time to realize these concerns in such a way that we can cross-tabulate classes of concerns to failure events within time partitions at an appropriate level of granularity. To support efficient implementation, a computationally convenient hypergeometric approximation is developed for the counting distributions appropriate to our underlying stochastic model. The accuracy of our approximation over first-order alternatives is examined, and demonstrated, through an evaluation experiment. An industrial application illustrates model implementation and shows how estimates can be updated using information arising during development test and analysis.