Reliability analysis of structures based on a probability‐uncertainty hybrid model

The traditional reliability analysis method based on probabilistic method requires probability distributions of all the uncertain parameters. However, in practical applications, the distributions of some parameters may not be precisely known due to the lack of sufficient sample data. The probabilistic theory cannot directly measure the reliability of structures with epistemic uncertainty, ie, subjective randomness and fuzziness. Hence, a hybrid reliability analysis (HRA) problem will be caused when the aleatory and epistemic uncertainties coexist in a structure. In this paper, by combining the probability theory and the uncertainty theory into a chance theory, a probability‐uncertainty hybrid model is established, and a new quantification method based on the uncertain random variables for the structural reliability is presented in order to simultaneously satisfy the duality of random variables and the subadditivity of uncertain variables; then, a reliability index is explored based on the chance expected value and variance. Besides, the formulas of the chance theory‐based reliability and reliability index are derived to uniformly assess the reliability of structures under the hybrid aleatory and epistemic uncertainties. The numerical experiments illustrate the validity of the proposed method, and the results of the proposed method can provide a more accurate assessment of the structural system under the mixed uncertainties than the ones obtained separately from the probability theory and the uncertainty theory.     

Fault trees analysis using expert opinion based on fuzzy‐bathtub failure rates

The main objective of fault tree analysis method is to estimate the “Top Event occurrence probability”. This requires determination of failure time distribution functions also known as “Bathtub Curves” for each of the system elements/events. This paper introduces a novel method to determine the failure time distribution functions using possibility theory. For this purpose, fuzzy‐bathtub distributions using expert opinions are generated for basic events and fuzzy formulas are derived for static and dynamic gates fault tree constructions. This process completed by proposed fuzzy Monte Carlo simulation throughout the preferred operational time and uses the actual time‐to‐failure data. Accordingly, the Top Event failure curve and the reliability profile of the system are depicted based on the defuzzificated basic‐events' bathtub‐failure‐rates. The results show that the proposed method not only is feasible and powerful but can also be accurate more than the other probabilistic and possibilistic techniques because of the component failure rates follow the real failure distributions.     

Determination of bounds on failure probability in the presence of hybrid uncertainties

A fundamental component of safety assessment is the appropriate representation and incorporation of uncertainty. A procedure for handling hybrid uncertainties in stochastic mechanics problems is presented. The procedure can be used for determining the bounds on failure probability for cases where failure probability is a monotonic function of the fuzzy variables. The procedure is illustrated through an example problem of safety assessment of a nuclear power plant piping component against stress corrosion cracking, considering the stochastic evolution of stress corrosion cracks with time. It is found that the bounds obtained enclose the values of failure probability obtained from probabilistic analyses.     

Functional failure analysis of a thermal–hydraulic passive system by means of Line Sampling

Assessing the failure probability of a thermal–hydraulic (T–H) passive system amounts to evaluating the uncertainties in its performance. Two different sources of uncertainties are usually considered: randomness due to inherent variability in the system behavior (aleatory uncertainty) and imprecision due to lack of knowledge and information on the system (epistemic uncertainty).In this paper, we are concerned with the epistemic uncertainties affecting the model of a T–H passive system and the numerical values of its parameters. Due to these uncertainties, the system may find itself in working conditions that do not allow it to accomplish its functions as required. The estimation of the probability of these functional failures can be done by Monte Carlo (MC) sampling of the epistemic uncertainties affecting the model and its parameters, followed by the computation of the system function response by a mechanistic T–H code.Efficient sampling methods are needed for achieving accurate estimates, with reasonable computational efforts. In this respect, the recently developed Line Sampling (LS) method is here considered for improving the MC sampling efficiency. The method, originally developed to solve high-dimensional structural reliability problems, employs lines instead of random points in order to probe the failure domain of interest. An “important direction” is determined, which points towards the failure domain of interest; the high-dimensional reliability problem is then reduced to a number of conditional one-dimensional problems which are solved along the “important direction”. This allows to significantly reduce the variance of the failure probability estimator, with respect to standard random sampling.The efficiency of the method is demonstrated by comparison to the commonly adopted Latin Hypercube Sampling (LHS) and first-order reliability method (FORM) in an application of functional failure analysis of a passive decay heat removal system in a gas-cooled fast reactor (GFR) of literature.     

Summary from the epistemic uncertainty workshop: consensus amid diversity

The ‘Epistemic Uncertainty Workshop’ sponsored by Sandia National Laboratories was held in Albuquerque, New Mexico, on 6–7 August 2002. The workshop was organized around a set of Challenge Problems involving both epistemic and aleatory uncertainty that the workshop participants were invited to solve and discuss. This concluding article in a special issue of Reliability Engineering and System Safety based on the workshop discusses the intent of the Challenge Problems, summarizes some discussions from the workshop, and provides a technical comparison among the papers in this special issue. The Challenge Problems were computationally simple models that were intended as vehicles for the illustration and comparison of conceptual and numerical techniques for use in analyses that involve: (i) epistemic uncertainty, (ii) aggregation of multiple characterizations of epistemic uncertainty, (iii) combination of epistemic and aleatory uncertainty, and (iv) models with repeated parameters. There was considerable diversity of opinion at the workshop about both methods and fundamental issues, and yet substantial consensus about what the answers to the problems were, and even about how each of the four issues should be addressed. Among the technical approaches advanced were probability theory, Dempster–Shafer evidence theory, random sets, sets of probability measures, imprecise coherent probabilities, coherent lower previsions, probability boxes, possibility theory, fuzzy sets, joint distribution tableaux, polynomial chaos expansions, and info-gap models. Although some participants maintained that a purely probabilistic approach is fully capable of accounting for all forms of uncertainty, most agreed that the treatment of epistemic uncertainty introduces important considerations and that the issues underlying the Challenge Problems are legitimate and significant. Topics identified as meriting additional research include elicitation of uncertainty representations, aggregation of multiple uncertainty representations, dependence and independence, model uncertainty, solution of black-box problems, efficient sampling strategies for computation, and communication of analysis results.     

Unified reliability analysis by active learning Kriging model combining with random‐set based Monte Carlo simulation method

Reliability analysis with both aleatory and epistemic uncertainties is investigated in this paper. The aleatory uncertainties are described with random variables, and epistemic uncertainties are tackled with evidence theory. To estimate the bounds of failure probability, several methods have been proposed. However, the existing methods suffer the dimensionality challenge of epistemic variables. To get rid of this challenge, a so‐called random‐set based Monte Carlo simulation (RS‐MCS) method derived from the theory of random sets is offered. Nevertheless, RS‐MCS is also computational expensive. So an active learning Kriging (ALK) model that only rightly predicts the sign of performance function is introduced and closely integrated with RS‐MCS. The proposed method is termed as ALK‐RS‐MCS. ALK‐RS‐MCS accurately predicts the bounds of failure probability using as few function calls as possible. Moreover, in ALK‐RS‐MCS, an optimization method based on Karush–Kuhn–Tucker conditions is proposed to make the estimation of failure probability interval more efficient based on the Kriging model. The efficiency and accuracy of the proposed approach are demonstrated with four examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Hyperstatic and redundancy thresholds in truss topology optimization considering progressive collapse due to aleatory and epistemic uncertainties

Optimization leads to specialized structures which are not robust to disturbance events like unanticipated abnormal loading or human errors. Typical reliability-based and robust optimization mainly address objective aleatory uncertainties. To date, the impact of subjective epistemic uncertainties in optimal design has not been comprehensively investigated. In this paper, we use an independent parameter to investigate the effects of epistemic uncertainties in optimal design: the latent failure probability. Reliability-based and risk-based truss topology optimization are addressed. It is shown that optimal risk-based designs can be divided in three groups: (A) when epistemic uncertainty is small (in comparison to aleatory uncertainty), the optimal design is indifferent to it and yields isostatic structures; (B) when aleatory and epistemic uncertainties are relevant, optimal design is controlled by epistemic uncertainty and yields hyperstatic but nonredundant structures, for which expected costs of direct collapse are controlled; (C) when epistemic uncertainty becomes too large, the optimal design becomes redundant, as a way to control increasing expected costs of collapse. The three regions above are divided by hyperstatic and redundancy thresholds. The redundancy threshold is the point where the structure needs to become redundant so that its reliability becomes larger than the latent reliability of the simplest isostatic system. Simple truss topology optimization is considered herein, but the conclusions have immediate relevance to the optimal design of realistic structures subject to aleatory and epistemic uncertainties.     

Addressing imperfect maintenance modelling uncertainty in unavailability and cost based optimization

Optimization of testing and maintenance activities performed in the different systems of a complex industrial plant is of great interest as the plant availability and economy strongly depend on the maintenance activities planned. Traditionally, two types of models, i.e. deterministic and probabilistic, have been considered to simulate the impact of testing and maintenance activities on equipment unavailability and the cost involved. Both models present uncertainties that are often categorized as either aleatory or epistemic uncertainties. The second group applies when there is limited knowledge on the proper model to represent a problem, and/or the values associated to the model parameters, so the results of the calculation performed with them incorporate uncertainty. This paper addresses the problem of testing and maintenance optimization based on unavailability and cost criteria and considering epistemic uncertainty in the imperfect maintenance modelling. It is framed as a multiple criteria decision making problem where unavailability and cost act as uncertain and conflicting decision criteria. A tolerance interval based approach is used to address uncertainty with regard to effectiveness parameter and imperfect maintenance model embedded within a multiple-objective genetic algorithm. A case of application for a stand-by safety related system of a nuclear power plant is presented. The results obtained in this application show the importance of considering uncertainties in the modelling of imperfect maintenance, as the optimal solutions found are associated with a large uncertainty that influences the final decision making depending on, for example, if the decision maker is risk averse or risk neutral.     

基于概率的高层建筑地震需求模型与风险评估EI北大核心CSCD

提出了基于贝叶斯理论的地震风险评估方法,综合考虑了地震危险性模型、输入地震动记录、结构参数和需求模型的不确定性,并以云南大理地区1970年-2017年间的地震数据为研究基础进行了详细讨论。在传统基于概率地震危险性分析方法的基础上,提出了基于贝叶斯理论的地震危险性分析方法,通过贝叶斯更新准则,确定了地震概率模型中未知参数的后验概率分布;通过贝叶斯理论建立了基于概率的地震需求模型,并在易损性中考虑了需求模型认知不确定性的影响;以42层钢框架-RC核心筒建筑为例,开展了地震作用下的风险评估。研究表明:基于贝叶斯理论的地震危险性分析方法,能够获得更为合理的危险性模型;忽略需求模型中参数不确定性的影响,将错误估计结构的地震易损性;不同加载工况将对高层建筑的地震风险产生显著影响。提出的概率风险评估方法,提供了可以考虑固有不确定性和认知不确定性的有效途径,有助于推动高性能结构地震韧性评价和设计理论的发展。     

Quantification of epistemic and aleatory uncertainties in level-1 probabilistic safety assessment studies

There will be simplifying assumptions and idealizations in the availability models of complex processes and phenomena. These simplifications and idealizations generate uncertainties which can be classified as aleatory (arising due to randomness) and/or epistemic (due to lack of knowledge). The problem of acknowledging and treating uncertainty is vital for practical usability of reliability analysis results. The distinction of uncertainties is useful for taking the reliability/risk informed decisions with confidence and also for effective management of uncertainty. In level-1 probabilistic safety assessment (PSA) of nuclear power plants (NPP), the current practice is carrying out epistemic uncertainty analysis on the basis of a simple Monte-Carlo simulation by sampling the epistemic variables in the model. However, the aleatory uncertainty is neglected and point estimates of aleatory variables, viz., time to failure and time to repair are considered. Treatment of both types of uncertainties would require a two-phase Monte-Carlo simulation, outer loop samples epistemic variables and inner loop samples aleatory variables. A methodology based on two-phase Monte-Carlo simulation is presented for distinguishing both the kinds of uncertainty in the context of availability/reliability evaluation in level-1 PSA studies of NPP.     

Hybrid reliability analysis of structures with multi-source uncertainties

A new hybrid reliability analysis technique based on the convex modeling theory is developed for structures with multi-source uncertainties, which may contain randomness, fuzziness, and non-probabilistic boundedness. By solving the convex modeling reliability problem and further analyzing the correlation within uncertainties, the structural hybrid reliability is obtained. Considering various cases of uncertainties of the structure, four hybrid models including the convex with random, convex with fuzzy random, convex with interval, and convex with other three are built, respectively. The present hybrid models are compared with the conventional probabilistic and the non-probabilistic models by two typical numerical examples. The results demonstrate the accuracy and effectiveness of the proposed hybrid reliability analysis method.     

Ensemble of randomized soft decision trees for robust classification

For classification, decision trees have become very popular because of its simplicity, interpret-ability and good performance. To induce a decision tree classifier for data having continuous valued attributes, the most common approach is, split the continuous attribute range into a hard (crisp) partition having two or more blocks, using one or several crisp (sharp) cut points. But, this can make the resulting decision tree, very sensitive to noise. An existing solution to this problem is to split the continuous attribute into a fuzzy partition (soft partition) using soft or fuzzy cut points which is based on fuzzy set theory and to use fuzzy decisions at nodes of the tree. These are called soft decision trees in the literature which are shown to perform better than conventional decision trees, especially in the presence of noise. Current paper, first proposes to use an ensemble of soft decision trees for robust classification where the attribute, fuzzy cut point, etc. parameters are chosen randomly from a probability distribution of fuzzy information gain for various attributes and for their various cut points. Further, the paper proposes to use probability based information gain to achieve better results. The effectiveness of the proposed method is shown by experimental studies carried out using three standard data sets. It is found that an ensemble of randomized soft decision trees has outperformed the related existing soft decision tree. Robustness against the presence of noise is shown by injecting various levels of noise into the training set and a comparison is drawn with other related methods which favors the proposed method.     

Dynamic response of structures with uncertain parameters: A comparative study of probabilistic and fuzzy sets models

This paper presents the results of a study on the response of structures with uncertain properties such as mass, stiffness and damping. The effect of the uncertain parameters on the response and the effect of the modelling of the uncertainties on the response are investigated. In particular, two types of uncertainties are distinguished: random and fuzzy uncertainties. Two kinds of models are studied: probabilistic and fuzzy set models. The two approaches to uncertainty modelling are compared with regard to their impacts on the analysis and on the uncertain structural response obtained. The study considers free vibration, forced vibration with deterministic excitation, and forced vibration with Gaussian white noise excitation. It is concluded that, in general, fuzzy models are much easier to implement and the associated analysis easier to perform than their probabilistic counterparts. When the available data on the structural parameters are crude and do not support a rigorous probabilistic model, the fuzzy set approach should be considered in view of its simplicity.     

基于区间与概率的星载网状天线展开可靠性分析

针对大型星载网状天线展开过程的特点,该文采用区间与概率混合可靠性分析方法对星载网状天线的展开过程可靠性进行了分析和评估。首先,建立了星载网状天线的展开失效树模型,并对失效树模型中各底事件进行了归类;其次,将关键底事件中所涉及到的不确定量视其特点描述为随机变量或区间变量,并利用混合可靠性模型分析方法获得了相应底事件的失效概率;再者,制作了2 m口径试验天线与典型试验展开机构样件,通过试验获得了天线展开过程中伸缩杆滑动的失效概率,进而得到了星载网状天线总的展开失效概率和展开可靠度;最后,对基本底事件进行了重要度分析,找出了可能导致星载网状天线展开失效的薄弱环节。     

Mixed aleatory-epistemic uncertainty quantification with stochastic expansions and optimization-based interval estimation

Uncertainty quantification (UQ) is the process of determining the effect of input uncertainties on response metrics of interest. These input uncertainties may be characterized as either aleatory uncertainties, which are irreducible variabilities inherent in nature, or epistemic uncertainties, which are reducible uncertainties resulting from a lack of knowledge. When both aleatory and epistemic uncertainties are mixed, it is desirable to maintain a segregation between aleatory and epistemic sources such that it is easy to separate and identify their contributions to the total uncertainty. Current production analyses for mixed UQ employ the use of nested sampling, where each sample taken from epistemic distributions at the outer loop results in an inner loop sampling over the aleatory probability distributions. This paper demonstrates new algorithmic capabilities for mixed UQ in which the analysis procedures are more closely tailored to the requirements of aleatory and epistemic propagation. Through the combination of stochastic expansions for computing statistics and interval optimization for computing bounds, interval-valued probability, second-order probability, and Dempster-Shafer evidence theory approaches to mixed UQ are shown to be more accurate and efficient than previously achievable.     

Extensions of the uncertainty quantification of common cause failure rates

Uncertainties in common cause event observation, documentation and interpretation are taken into account by conditional probabilities and generalized impact vector weights that separate single and double events of a specific multiplicity in a single observation. Distributions and moments of common cause failure (CCF) rates of a system are obtained in terms of the weights by using probability generating functions, combining assessment uncertainties and statistical uncertainties. These results are then used to generate effective plant-specific input data to general empirical Bayes estimation methods to combine data from many plants. The posterior output yields CCF probabilities for standby safety system fault tree analysis or probabilistic safety assessments of a target plant.     

Group-based product scheme-screening decision method based on fuzzy AHP and evidential reasoning theory

Selecting the favourable product scheme is the first step to successful new product development (NPD). There are usually large numbers of uncertainties in product scheme evaluation and screening process of NPD due to lack of or incomplete reliable information. Considering fully the uncertainties and then conducting correct reasoning could guarantee reliability and rationality of scheme-screening results. As an extension of analytic hierarchy process (AHP), fuzzy AHP inherits multi-merits of the AHP approach and is capable of dealing with fuzzy information effectively, but it still has two weaknesses. One is the well-known ranking reversal problem. Although several researchers have analysed the reasons, we think the root cause for ranking reversal problem is due to the fact that AHP treats weights of attribute criteria and performance scores of alternatives in the same way. Therefore, we intend to deal with attribute weights and performance scores of alternatives separately and introduce evidential reasoning (ER) theory, which is good at uncertain reasoning, into fuzzy AHP to calculate the performance scores of alternatives. On the other hand, in view of the difficulty in resolution for fuzzy weights from fuzzy comparison matrix, a linear goal-programming model is proposed to calculate fuzzy weights, whose objective is to minimise the inconsistency degree of comparison matrix. By combining fuzzy AHP with ER, a group-based hybrid decision model FAHP-ER is developed. The hybrid model not only gets a great improvement in the capability of dealing with uncertainty, but also reflects the most real decision scenario and thinking process of the decision maker. Finally, a case study for schemes screening of the rotor and bearing system in the turbine generator is presented to demonstrate the application of the hybrid decision method.     

Reliability Analysis of a conveyor system using hybrid data

The quantification of a fault tree is difficult without an exact probability value for all of the basic events of the tree. To overcome this difficulty in this paper, we propose a methodology which employs ‘hybrid data’ as a tool to analyse the fault tree. The proposed methodology estimates the failure probability of basic events using the statistical analysis of field recorded failures. Under these circumstances, where there is an absence of past failure records, the method follows a fuzzy set based theoretical evaluation based on the subjective judgement of experts for the failure interval. The proposed methodology has been applied to a conveyor system. The results of the analysis reveal the effectiveness of the proposed methodology and the instrumental role played by the experience of experts in providing reliability oriented information. Copyright © 2006 John Wiley & Sons, Ltd.     

Treatment of uncertainties in space nuclear risk assessment with examples from Cassini mission applications

By means of several examples from a recent comprehensive space nuclear risk analysis of the Cassini mission, a scenario and consequence representational framework is presented for risk analysis of space nuclear power systems in the context of epistemic and aleatory uncertainties. The framework invites the use of probabilistic models for the calculation of both event probabilities and scenario consequences. Each scenario is associated with a frequency that may include both aleatory and epistemic uncertainties. The outcome of each scenario is described in terms of an end state vector. The outcome of each scenario is also characterized by a source term. In this paper, the source term factors of interest are number of failed clads in the space nuclear power system, amount of fuel released and amount of fuel that is potentially respirable. These are also subject to uncertainties. The 1990 work of Apostolakis is found to be a useful formalism from which to derive the relevant probabilistic models. However, an extension to the formalism was necessary to accommodate the situation in which aleatory uncertainty is represented by changes in the form of the probability function itself, not just its parameters. Event trees that show reasonable alternative accident scenarios are presented. A grouping of probabilities and consequences is proposed as a useful structure for thinking about uncertainties. An example of each category is provided. Concluding observations are made about the judgments involved in this analysis of uncertainties and the effect of distinguishing between aleatory and epistemic uncertainties.