Quantification of margins and uncertainties: Conceptual and computational basis

In 2001, the National Nuclear Security Administration of the U.S. Department of Energy in conjunction with the national security laboratories (i.e., Los Alamos National Laboratory, Lawrence Livermore National Laboratory and Sandia National Laboratories) initiated development of a process designated Quantification of Margins and Uncertainties (QMU) for the use of risk assessment methodologies in the certification of the reliability and safety of the nation's nuclear weapons stockpile. This presentation discusses and illustrates the conceptual and computational basis of QMU in analyses that use computational models to predict the behavior of complex systems. The following topics are considered: (i) the role of aleatory and epistemic uncertainty in QMU, (ii) the representation of uncertainty with probability, (iii) the probabilistic representation of uncertainty in QMU analyses involving only epistemic uncertainty, and (iv) the probabilistic representation of uncertainty in QMU analyses involving aleatory and epistemic uncertainty.     

Quantification of margins and uncertainties of complex systems in the presence of aleatoric and epistemic uncertainty

Performance assessment of complex systems is ideally done through full system-level testing which is seldom available for high consequence systems. Further, a reality of engineering practice is that some features of system behavior are not known from experimental data, but from expert assessment, only. On the other hand, individual component data, which are part of the full system are more readily available. The lack of system level data and the complexity of the system lead to a need to build computational models of a system in a hierarchical or building block approach (from simple components to the full system). The models are then used for performance prediction in lieu of experiments, to estimate the confidence in the performance of these systems. Central to this are the need to quantify the uncertainties present in the system and to compare the system response to an expected performance measure. This is the basic idea behind Quantification of Margins and Uncertainties (QMU). QMU is applied in decision making—there are many uncertainties caused by inherent variability (aleatoric) in materials, configurations, environments, etc., and lack of information (epistemic) in models for deterministic and random variables that influence system behavior and performance. This paper proposes a methodology to quantify margins and uncertainty in the presence of both aleatoric and epistemic uncertainty. It presents a framework based on Bayes networks to use available data at multiple levels of complexity (i.e. components, subsystem, etc.) and demonstrates a method to incorporate epistemic uncertainty given in terms of intervals on a model parameter.     

Probabilistic bounding analysis in the Quantification of Margins and Uncertainties

The current challenge of nuclear weapon stockpile certification is to assess the reliability of complex, high-consequent, and aging systems without the benefit of full-system test data. In the absence of full-system testing, disparate kinds of information are used to inform certification assessments such as archival data, experimental data on partial systems, data on related or similar systems, computer models and simulations, and expert knowledge. In some instances, data can be scarce and information incomplete. The challenge of Quantification of Margins and Uncertainties (QMU) is to develop a methodology to support decision-making in this informational context. Given the difficulty presented by mixed and incomplete information, we contend that the uncertainty representation for the QMU methodology should be expanded to include more general characterizations that reflect imperfect information. One type of generalized uncertainty representation, known as probability bounds analysis, constitutes the union of probability theory and interval analysis where a class of distributions is defined by two bounding distributions. This has the advantage of rigorously bounding the uncertainty when inputs are imperfectly known. We argue for the inclusion of probability bounds analysis as one of many tools that are relevant for QMU and demonstrate its usefulness as compared to other methods in a reliability example with imperfect input information.     

Quantification of margins and uncertainties: Example analyses from reactor safety and radioactive waste disposal involving the separation of aleatory and epistemic uncertainty

In 2001, the National Nuclear Security Administration (NNSA) of the U.S. Department of Energy (DOE) in conjunction with the national security laboratories (i.e., Los Alamos National Laboratory, Lawrence Livermore National Laboratory, and Sandia National Laboratories) initiated development of a process designated quantification of margins and uncertainties (QMU) for the use of risk assessment methodologies in the certification of the reliability and safety of the nation's nuclear weapons stockpile. A previous presentation, “Quantification of Margins and Uncertainties: Conceptual and Computational Basis,” describes the basic ideas that underlie QMU and illustrates these ideas with two notional examples. The basic ideas and challenges that underlie NNSA's mandate for QMU are present, and have been successfully addressed, in a number of past analyses for complex systems. To provide perspective on the implementation of a requirement for QMU in the analysis of a complex system, three past analyses are presented as examples: (i) the probabilistic risk assessment carried out for the Surry Nuclear Power Station as part of the U.S. Nuclear Regulatory Commission's (NRC's) reassessment of the risk from commercial nuclear power in the United States (i.e., the NUREG-1150 study), (ii) the performance assessment for the Waste Isolation Pilot Plant carried out by the DOE in support of a successful compliance certification application to the U.S. Environmental Agency, and (iii) the performance assessment for the proposed high-level radioactive waste repository at Yucca Mountain, Nevada, carried out by the DOE in support of a license application to the NRC. Each of the preceding analyses involved a detailed treatment of uncertainty and produced results used to establish compliance with specific numerical requirements on the performance of the system under study. As a result, these studies illustrate the determination of both margins and the uncertainty in margins in real analyses.     

A Bayesian probabilistic framework for avalanche modelling based on observations

Applied avalanche models are based on parameters which cannot be measured directly. As a consequence, these models are associated with large uncertainties, which must be addressed in risk assessment. To this end, we present an integral probabilistic framework for the modelling of avalanche hazards. The framework is based on a deterministic dynamic avalanche model, which is combined with an explicit representation of the different parameter uncertainties. The probability distribution of these uncertainties is then determined from observations of avalanches in the area under investigation through Bayesian inference. This framework facilitates the consistent combination of physical and empirical avalanche models with the available observations and expert knowledge. The resulting probabilistic spatial model can serve as a basis for hazard maping and spatial risk assessment. In this paper, the new model is applied to a case study in a test area located in the Swiss Alps.     

The interval estimation of reliability for probabilistic and non-probabilistic hybrid structural system

The aim of this paper is to improve evaluation of the reliability of probabilistic and non-probabilistic hybrid structural system. Based on the probabilistic reliability model and interval arithmetic, a new model of interval estimation for reliability of the hybrid structural system was proposed. Adequately considering all uncertainties affecting the hybrid structural system, the lower and upper bounds of reliability for the hybrid structural system were obtained through the probabilistic and non-probabilistic analysis. In the process of non-probabilistic analysis, the interval truncation method was used. In addition, a recognition method of the main failure modes in the hybrid structural system was presented. A five-bar statically indeterminate truss structure and an intermediate complexity wing structure were used to demonstrate the new model is more suitable for analysis and design of these structural systems in comparison with the probabilistic model. The results also show that the method of recognition of main failure modes is effective. In addition, range obtained through interval estimation is shown to be more credible than certain results of other reliability models.     

Fatigue reliability assessment of turbine discs under multi‐source uncertainties

Hot section components of aircraft engines like high pressure turbine (HPT) discs usually operate under complex loadings coupled with multi‐source uncertainties. The effect of these uncertainties on structural response of HPT discs should be accounted for its fatigue life and reliability assessment. In this study, a probabilistic framework for fatigue reliability analysis is established by incorporating FE simulations with Latin hypercube sampling to quantify the influence of material variability and load variations. Particularly, variability in material response is characterized by combining the Chaboche constitutive model with Fatemi‐Socie criterion. Results from fatigue reliability and sensitivity analysis of a HPT disc indicated that dispersions of basic variables must be taken into account for its fatigue reliability analysis. Moreover, the proposed framework based on the strength‐damage interference provides more reasonably correlations with its field number of flights rather than the load‐life interference one.     

基于Bayesian-MCMC方法的深受弯构件受剪概率模型研究

考虑主观、客观不确定性因素的影响,以深受弯构件受剪分析模型为研究对象,基于引入马尔科夫链-蒙特卡洛（MCMC）高效采样方法,通过R语言对深受弯构件概率模型参数进行MCMC随机模拟,给出参数的最优估计值及其对应的可信度,在先验模型基础上建立钢筋混凝土深受弯构件受剪承载力概率模型,完成模型前后的对比分析,并根据不同置信水平确定了深受弯构件受剪承载力的特征值。结果表明:基于MCMC方法得到的受剪承载力概率模型是在50000次迭代分析后产生的结果,能合理地解释影响参数的不确定性,可信度较高;后验概率模型计算结果与试验结果吻合良好,较先验模型更接近试验值,且离散性小。     

钢筋混凝土梁抗剪承载力计算的概率模型

传统的钢筋混凝土（RC）梁抗剪承载力模型属于确定性模型,难以有效考虑几何尺寸、材料特性、边界约束条件等因素存在的客观（物理）不确定性和在模型推导过程中存在的主观（模型）不确定性的影响,导致计算结果的离散性较大,计算精度和适用性有限。鉴于此,该文首先结合修正压力场理论和考虑剪跨比影响的临界斜裂缝倾角模型,建立了RC梁的确定性抗剪承载力模型;然后综合考虑主观不确定性和客观不确定性因素的影响,结合贝叶斯理论和马尔科夫链蒙特卡洛法（MCMC）,建立了RC梁抗剪承载力计算的概率模型;最后通过与试验数据和传统确定性计算模型的对比分析,验证了该模型的有效性和适用性。分析结果表明,所建立的概率模型不仅可以合理地描述RC梁抗剪承载力的概率分布特性,而且可以校准传统确定性计算模型的计算精度和置信水平,还可以根据预定的置信水平确定RC梁抗剪承载力的概率特征值,具有良好的计算精度和适用性。     

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.     

剪切型钢筋混凝土柱抗剪承载力计算的概率模型

现有的钢筋混凝土（RC）柱抗剪承载力计算模型大多属于确定性模型,难以有效考虑几何尺寸、材料特性和外荷载等因素存在的不确定性,导致计算结果的离散性较大,且计算精度和适用性有限。鉴于此,该文结合变角桁架-拱模型和贝叶斯理论,研究建立了剪切型RC柱抗剪承载力计算的概率模型。首先基于变角桁架-拱模型理论,并考虑轴压力对临界斜裂缝倾角的影响,建立了剪切型RC柱抗剪承载力的确定性修正模型;然后考虑主观不确定性和客观不确定性因素的影响,结合贝叶斯理论和马尔科夫链蒙特卡洛（MCMC）法,建立了剪切型RC柱的概率抗剪承载力计算模型;最后通过与试验数据和现有模型的对比分析,验证了该模型的有效性和实用性。分析结果表明,该模型不仅可以合理描述剪切型RC柱抗剪承载力的概率分布特性,而且可以校准现有确定性计算模型的置信水平,并且可以确定不同置信水平下剪切型RC柱抗剪承载力的特征值。     

Reliability study of fracture mechanics based non‐probabilistic interval analysis model

In this paper, we advanced a new interval reliability analysis model for fracture reliability analysis. Based on the non‐probabilistic stress intensity factor interference model and the ratio of the volume of the safe region to the total volume of the region associated with the variation of the standardized interval variables is suggested as the measure of structural non‐probabilistic reliability. We use this theory to calculate the reliability of structure based on fracture criterion. This model needs less uncertain information, so it has less limitation for analysing an uncertain structure or system. Examples of practical application are given to explain the simplicity and practicability of this model by comparing the interval reliability analysis model with probabilistic reliability analysis model.     

Updating robust reliability using structural test data

The concept of robust reliability is defined to take into account uncertainties from structural modeling in addition to the uncertain excitation that a structure will experience during its lifetime. A Bayesian probabilistic methodology for system identification is integrated with probabilistic structural analysis tools for the purpose of updating the assessment of the robust reliability based on dynamic test data. Methods for updating the structural reliability for both identifiable and unidentifiable models are presented. Application of the methodology to a simple beam model of a single-span bridge with soil-structure interaction at the abutments, including a case with a tuned-mass damper attached to the deck, shows that the robust reliabilities computed before and after updating with “measured” dynamic data can differ significantly.     

Variation mode and effect analysis: an application to fatigue life prediction

We present an application of the probabilistic branch of variation mode and effect analysis (VMEA) implemented as a first‐order, second‐moment reliability method. First order means that the failure function is approximated to be linear around the nominal values with respect to the main influencing variables, while second moment means that only means and variances are taken into account in the statistical procedure. We study the fatigue life of a jet engine component and aim at a safety margin that takes all sources of prediction uncertainties into account. Scatter is defined as random variation due to natural causes, such as non‐homogeneous material, geometry variation within tolerances, load variation in usage, and other uncontrolled variations. Other uncertainties are unknown systematic errors, such as model errors in the numerical calculation of fatigue life, statistical errors in estimates of parameters, and unknown usage profile. By treating also systematic errors as random variables, the whole safety margin problem is put into a common framework of second‐order statistics. The final estimated prediction variance of the logarithmic life is obtained by summing the variance contributions of all sources of scatter and other uncertainties, and it represents the total uncertainty in the life prediction. Motivated by the central limit theorem, this logarithmic life random variable may be regarded as normally distributed, which gives possibilities to calculate relevant safety margins. Copyright © 2008 John Wiley & Sons, Ltd.     

结构控制系统的稳定可靠度分析

在控制系统中,存在着各种不确定因素,如质量、刚度、阻尼和时间滞后等,这些不确定因素可能使控制系统的控制效果大大降低,甚至造成失稳。因此如何在控制设计中考虑不确定因素,是将结构控制技术应用于实际工程所要解决的问题之一。可靠度分析方法提供了一个更合理１更实际的工具。应用这一分析方法,可以对不同控制系统,按统一标准进行控制效果和稳定性的分析评价。本文应用二阶矩可靠度理论,分析不同控制系统,在相同的不确定     

Reliability analysis in fracture mechanics using the first-order reliability method and Monte Carlo simulation

The development of a Windows‐based framework to undertake probabilistic fracture mechanics studies is reported. For a selective library of standard case problems, the reliability index of critical and sub‐critical (fatigue) problems with stochastic definition is evaluated. Both first‐order reliability method (FORM) as well as and Monte Carlo simulation method (MCS) techniques are used in critical crack growth, and only MCS is adopted for fatigue problems. Numerical predictions for the stress intensity factors (SIF) were validated with NASA/FLAGRO and reliability predictions were validated with both RELTRAN and VaP. With the advent of powerful and inexpensive personal computer, and the user‐friendliness of graphical user interface, programs such as the one developed will indeed make it possible for engineer to correctly account for the stochastic nature of most fracture problems they are confronted with.     

A Risk‐based Approach to Managing Performance‐based Maintenance Contracts

The global trend towards performance‐based maintenance contracting has presented new challenges to maintenance service providers as they are compensated or penalized based on performance outcomes instead of time and materials consumed during maintenance service. The problem becomes more complex when uncertainties exist in reliability performance and maintenance activities of technical systems. In this paper, a general framework for managing performance‐based maintenance contract under risks is proposed. We illustrate our approach with an application in a multi‐echelon multi‐system spare parts control problem. Several different performance measures are considered and a probabilistic constrained optimization problem is formulated from the perspective of the service provider. Hybrid simulation/analytic heuristics are proposed to solve the problem based on the monotonic properties of performance measures. This approach is flexible and can be applied to a wide range of problems with similar properties. Numerical example shows that the probability of violating performance requirements is high if the risk is overlooked. We also provide guidelines on how to apply this approach in practice. Copyright © 2016 John Wiley & Sons, Ltd.     

Handling Uncertainties in Fault Tree Analysis by a Hybrid Probabilistic–Possibilistic Framework

Fault tree analysis is a method largely used in probabilistic risk assessment. Uncertainties should be properly handled in fault tree analyses to support a robust decision making. While many sources of uncertainties are considered, dependence uncertainties are not much explored. Such uncertainties can be labeled as ‘epistemic’ because of the way dependence is modeled. In practice, despite probability theory, alternative mathematical structures, including possibility theory and fuzzy set theory, for the representation of epistemic uncertainty can be used. In this article, a fuzzy β factor is considered to represent the failure dependence uncertainties among basic events. The relationship between β factor and system failure probability is analyzed to support the use of a hybrid probabilistic–possibilistic approach. As a result, a complete hybrid probabilistic–possibilistic framework is constructed. A case study of a high integrity pressure protection system is discussed. The results show that the proposed method provides decision makers a more accurate understanding of the system under analysis when failure dependencies are involved. Copyright © 2015 John Wiley & Sons, Ltd.     

A critical reliability evaluation of fibre reinforced composite materials based on probabilistic micro and macro-mechanical analysis

In probabilistic composite mechanics, uncertainty modelling may be introduced at a constituent (micro-scale), ply (meso-scale) or component (macro-scale) level. Each of these approaches has particular advantages/limitations and appropriate fusing and benchmarking is desirable in order to improve confidence in probabilistic performance estimates of composite structures. In the present study, random variable based micro and macro-scale reliability analyses are critically compared through a limit state formulation based on the analytical stress tensor components of a rectangular simply supported orthotropic FRP composite plate and the Tsai–Hill failure criterion. The study aims to promote cross-fertilisation of alternative uncertainty modelling approaches in a multi-scale analysis framework. Propagation of uncertainty from micro to macro-scale, and the corresponding influence of changes in random variability on the reliability estimates is quantified. The importance of benchmarking experimentally-based probability distributions of mechanical properties through micro-scale modelling is illustrated, and the confidence that can be placed on reliability estimates is quantified through a series of numerical examples.     

Free vibration and reliability of composite cantilevers featuring uncertain properties

The problem of free vibration and reliability of cantilever composite beams featuring structural uncertainties is analyzed. The random structural uncertainties involve material properties, thickness and fiber orientation of the individual constituent laminae. Such uncertainties undoubtedly affect the achievable performance as well as their structural reliabilities. In order to investigate the effects of random structural uncertainties on free vibration problem, a stochastic eigenvalue problem of self-adjoint systems is formulated to provide first and second moments of eigenvalues, i.e., their mean and variance. In this context, a stochastic finite element method based on the mean-centered-second-moment method and first-order perturbation technique are employed during the probabilistic discretization of uncertain distributed-parameter structural systems.Sensitivity and reliability analyses for the uncertain beam when subjected to an external oscillatory load are performed. In addition, in order to mitigate the detrimental effects of uncertainties and so, to render the structure more robust to such effects, the structural tailoring technique is implemented and its beneficial effects are revealed.