About time-variant reliability analysis with reference to passive systems assessment

This study addresses the time-variant reliability assessment in relation to systems presenting a non-stationary random process during their operation, such as thermal-hydraulic passive systems for advanced reactors, resting on natural circulation.The reliability assessment efforts conducted so far do not address this specific aspect: the dependence upon time is usually ignored, or at most the system unavailability is intended to be assessed per mission time, during which the parameter values, as t–h parameters for instance, are assumed as constant quantities.The study endeavours to present a consistent approach to model the natural circulation passive systems, in terms of time-variant performance parameters, as for instance mass flow-rate and thermal power, to cite any.     

General principles of the use of safety factors in design and assessment

Any structure or component can be made to fail if it is subjected to loadings in excess of its strength. Structural integrity is achieved by ensuring that there is an adequate safety margin or reserve factor between strength and loading effects. The basic principles of ‘allowable stress’ and ‘limit state’ design methods to avoid failure in structural and pressure vessel components are summarised. The use of risk as a means of defining adequate safety is introduced where risk is defined as the product of probability of failure multiplied by consequences of failure. The concept of acceptable ‘target’ levels of risk is discussed. The use of structural reliability theory to determine estimates of probability of failure and the use of the reliability index β are described. The need to consider the effects of uncertainties in loading information, calculation of stresses, input data and material properties is emphasised. The way in which the effect of different levels of uncertainty can be dealt with by use of partial safety factors in limit state design is explained. The need to consider all potential modes of failure, including the unexpected, is emphasised and an outline given of safety factor treatments for crack tip dependent and time dependent modes. The relationship between safety factors appropriate for the design stage and for assessment of structural integrity at a later stage is considered. The effects of redundancy and system behaviour on appropriate levels of safety factors are discussed.     

Uncertainty analysis and reliability improvement of planetary roller screw mechanism using active learning Kriging model

The uncertainties in the geometry, material and operation conditions may cause structural failure of the planetary roller screw mechanism (PRSM). The uncertainty analysis model is the key to the reliability assessment of the PRSM, however, the relevant studies have been rarely reported in the past. This paper focuses on establishing a preliminary mathematical model of the PRSM considering uncertain factors. The quasi-Monte Carlo (QMC) method is introduced to improve the solving efficiency of the multidimensional and nonlinear implicit limit state function (LSF). Then, the parameter sensitivities of the uncertain factors to the load distribution and contact characteristics are comprehensively ranked by the design of experiment (DoE). The computational cost for constructing the active learning Kriging (ALK) model of PRSM is decreased by only selecting the most sensitive variables. Moreover, the ALK model and QMC method (ALK-QMC) are combined to explore how the main factors affect the structural reliability of PRSM, which further guides the implementation of multi-objective optimization to improve the reliability by the developed NSGA-II-Downhill algorithm. Finally, the theoretical model and optimization results are verified by the finite element method.     

An optimization method for learning statistical classifiers in structural reliability

Monte Carlo simulation is a general and robust method for structural reliability analysis, affected by the serious efficiency problem consisting in the need of computing the limit state function a very large number of times. In order to reduce this computational effort the use of several kinds of solver surrogates has been proposed in the recent past. Proposals include the Response Surface Method (RSM), Neural Networks (NN), Support Vector Machines (SVM) and several other methods developed in the burgeoning field of Statistical Learning (SL). Many of these techniques can be employed either for function approximation (regression approach) or for pattern recognition (classification approach). This paper concerns the use of these devices for discriminating samples into safe and failure classes using the classification approach, because it constitutes the core of Monte Carlo simulation as applied to reliability analysis as such. Due to the flexibility of most SL methods, a critical step in their use is the generation of the learning population, as it affects the generalization capacity of the surrogate. To this end it is first demonstrated that the optimal population from the information viewpoint lies around in the vicinity of the limit state function. Next, an optimization method assuring a small as well as highly informative learning population is proposed on this basis. It consists in generating a small initial quasi-random population using Sobol sequence for triggering a Particle Swarm Optimization (PSO) performed over an iteration-dependent cost function defined in terms of the limit state function. The method is evaluated using SVM classifiers, but it can be readily applied also to other statistical classification techniques because the distinctive feature of the SVM, i.e. the margin band, is not actively used in the algorithm. The results show that the method yields results for the probability of failure that are in very close agreement with Monte Carlo simulation performed on the original limit state function and requiring a small number of learning samples.     

Adaptive improved response surface method for reliability-based design optimization

Reliability-based design optimization (RBDO) requires the evaluation of probabilistic constraints (or reliability), which can be very time consuming. Therefore, a practical solution for efficient reliability analysis is needed. The response surface method (RSM) and dimension reduction (DR) are two well-known approximation methods that construct the probabilistic limit state functions for reliability analysis. This article proposes a new RSM-based approximation approach, named the adaptive improved response surface method (AIRSM), which uses the moving least-squares method in conjunction with a new weight function. AIRSM is tested with two simplified designs of experiments: saturated design and central composite design. Its performance on reliability analysis is compared with DR in terms of efficiency and accuracy in multiple RBDO test problems.     

Adjoint operator approach to functional reliability analysis of passive fluid dynamical systems

Reliability analysis of passive systems mainly involves quantification of the margin to safety limits in probabilistic terms. For systems represented by complex models, propagating input uncertainty to get the response uncertainty and hence probability information requires intensive computational effort. Here a computationally efficient method for the functional reliability analysis of passive fluid dynamical systems is presented. The approach is based on continuous adjoint operator technique to generate a response surface approximating the given system model from the sensitivity coefficients. A numerical application of this method to the reliability analysis of heat transport in an asymmetrical natural convection loop is demonstrated. Computational efficiency and accuracy compared with the direct Monte-Carlo and forward response surface methods.     

Enhanced high‐dimensional model representation for reliability analysis

This paper presents a new and alternative computational tool for predicting failure probability of structural/mechanical systems subject to random loads, material properties, and geometry based on high‐dimensional model representation (HDMR) generated from low‐order function components. HDMR is a general set of quantitative model assessment and analysis tools for capturing the high‐dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher‐order variable correlations are weak, allowing the physical model to be captured by the lower‐order terms and facilitating lower‐dimensional approximation of the original high‐dimensional implicit limit state/performance function. When first‐order HDMR approximation of the original high‐dimensional implicit limit state/performance function is not adequate to provide the desired accuracy to the predicted failure probability, this paper presents an enhanced HDMR (eHDMR) method to represent the higher‐order terms of HDMR expansion by expressions similar to the lower‐order ones with monomial multipliers. The accuracy of the HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input–output samples without directly invoking the determination of second‐ and higher‐order terms. The mathematical foundation of eHDMR is presented along with its applicability to approximate the original high‐dimensional implicit limit state/performance function for subsequent reliability analysis, given that conventional methods for reliability analysis are computationally demanding when applied in conjunction with complex finite element models. This study aims to assess how accurately and efficiently the eHDMR approximation technique can capture complex model output uncertainty. The limit state/performance function surrogate is constructed using moving least‐squares interpolation formula by component functions of eHDMR expansion. Once the approximate form of implicit response function is defined, the failure probability can be obtained by statistical simulation. Results of five numerical examples involving elementary mathematical functions and structural/solid‐mechanics problems indicate that the failure probability obtained using the eHDMR approximation method for implicit limit state/performance function, provides significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations. Copyright © 2008 John Wiley & Sons, Ltd.     

基于切平面布点的一种改进响应面方法

基于合理选择试验点的位置,该文提出一种改进响应面方法。该方法首先在经过设计点的切平面上布置试验点,然后沿切平面法向量方向移动试验点,并利用设计点和先前试验点的信息布置加强试验点。所布置的试验点既对设计点附近区域给予足够重视,同时又考虑极限状态函数在设计点附近区域的变化趋势,进而提高响应面函数在设计点附近区域的拟合精度。在响应面函数的拟合过程中,该文方法能够保证响应面函数在设计点处是无误差的,进一步提高失效概率的评估精度。算例表明,对于显式和隐式极限状态函数,该方法均具有较好的效率和精度。     

Reliability studies of a high-power proton accelerator for accelerator-driven system applications for nuclear waste transmutation

The main effort of the present study is to analyze the availability and reliability of a high-performance linac (linear accelerator) conceived for Accelerator-Driven Systems (ADS) purpose and to suggest recommendations, in order both to meet the high operability goals and to satisfy the safety requirements dictated by the reactor system. Reliability Block Diagrams (RBD) approach has been considered for system modelling, according to the present level of definition of the design: component failure modes are assessed in terms of Mean Time Between Failure (MTBF) and Mean Time To Repair (MTTR), reliability and availability figures are derived, applying the current reliability algorithms. The lack of a well-established component database has been pointed out as the main issue related to the accelerator reliability assessment. The results, affected by the conservative character of the study, show a high margin for the improvement in terms of accelerator reliability and availability figures prediction. The paper outlines the viable path towards the accelerator reliability and availability enhancement process and delineates the most proper strategies. The improvement in the reliability characteristics along this path is shown as well.     

Developing safety performance functions incorporating reliability-based risk measures

Current geometric design guides provide deterministic standards where the safety margin of the design output is generally unknown and there is little knowledge of the safety implications of deviating from these standards. Several studies have advocated probabilistic geometric design where reliability analysis can be used to account for the uncertainty in the design parameters and to provide a risk measure of the implication of deviation from design standards. However, there is currently no link between measures of design reliability and the quantification of safety using collision frequency. The analysis presented in this paper attempts to bridge this gap by incorporating a reliability-based quantitative risk measure such as the probability of non-compliance (P_nc) in safety performance functions (SPFs). Establishing this link will allow admitting reliability-based design into traditional benefit–cost analysis and should lead to a wider application of the reliability technique in road design. The present application is concerned with the design of horizontal curves, where the limit state function is defined in terms of the available (supply) and stopping (demand) sight distances. A comprehensive collision and geometric design database of two-lane rural highways is used to investigate the effect of the probability of non-compliance on safety. The reliability analysis was carried out using the First Order Reliability Method (FORM). Two Negative Binomial (NB) SPFs were developed to compare models with and without the reliability-based risk measures. It was found that models incorporating the P_nc provided a better fit to the data set than the traditional (without risk) NB SPFs for total, injury and fatality (I + F) and property damage only (PDO) collisions.     

Reliability based design optimization using response surfaces in application to multidisciplinary systems

Traditionally, reliability based design optimization (RBDO) is formulated as a nested optimization problem. For these problems the objective is to minimize a cost function while satisfying the reliability constraints. The reliability constraints are usually formulated as constraints on the probability of failure corresponding to each of the failure modes or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. The difficulty in evaluating reliability constraints comes from the fact that modern reliability analysis methods are themselves formulated as an optimization problem. Solving such nested optimization problems is extremely expensive for large scale multidisciplinary systems which are likewise computationally intensive. In this research, a framework for performing reliability based multidisciplinary design optimization using approximations is developed. Response surface approximations (RSA) of the limit state functions are used to estimate the probability of failure. An outer loop is incorporated to ensure that the approximate RBDO converges to the actual most probable point of failure. The framework is compared with the exact RBDO procedure. In the proposed methodology, RSAs are employed to significantly reduce the computational expense associated with traditional RBDO. The proposed approach is implemented in application to multidisciplinary test problems, and the computational savings and benefits are discussed.     

Reliability assessment of passive isolation condenser system of AHWR using APSRA methodology

In this paper, a methodology known as APSRA (Assessment of Passive System ReliAbility) is used for evaluation of reliability of passive isolation condenser system of the Indian Advanced Heavy Water Reactor (AHWR). As per the APSRA methodology, the passive system reliability evaluation is based on the failure probability of the system to perform the design basis function. The methodology first determines the operational characteristics of the system and the failure conditions based on a predetermined failure criterion. The parameters that could degrade the system performance are identified and considered for analysis. Different modes of failure and their cause are identified. The failure surface is predicted using a best estimate code considering deviations of the operating parameters from their nominal states, which affect the isolation condenser system performance. Once the failure surface of the system is predicted, the causes of failure are examined through root diagnosis, which occur mainly due to failure of mechanical components. Reliability of the system is evaluated through a classical PSA treatment based on the failure probability of the components using generic data.     

Fire safety assessment and optimal design of passive fire protection for offshore structures

The article presents a unified probabilistic approach to fire safety assessment and optimal design of passive fire protection on offshore topside structures. The methodology was developed by integrating quantitative risk analysis (QRA) techniques with the modem methods of structural system reliability analysis (SRA) and reliability based design optimisation (RBDO). Reliability analysis methodologies are presented for both plated (e.g. fire and blast walls) and skeletal structures (deck framing), which take into account uncertainties in fire and blast loading, thermal and mechanical properties of the steel and insulation. Probability of component and system failure are evaluated using first- and second-order reliability methods (FORM/SORM). The optimisation of passive fire protection is performed such that the total expected cost of the protection system is minimised while satisfying reliability constraints.     

Geotechnical system reliability of slopes

In slope stability analysis it is customary to search for the critical slip surface considering the conventional factor of safety as an index of stability. With the development of reliability analysis approaches within a probabilistic framework, alternative definitions of the critical slip surface can be adopted. Thus one may define a critical slip surface as one with the lowest reliability index or one with the highest probability of failure. However, it is important to consider the slope stability problem in terms of a system of many potential slip surfaces. For such a system, the calculation of system reliability is appropriate and desirable. In this paper, system reliability bounds are calculated within a probabilistic framework. The ‘system reliability’ or the ‘system probability of failure’ must be estimated for comparison with the corresponding reliability or probability of failure with respect to a ‘critical’ slip surface. The general slope stability problem involving non-zero internal friction angle involves a non-linear performance function. Moreover, the expression for factor of safety is usually inexplicit except for the ordinary method of slices which is not accurate except when ‘ø = 0’. This paper addresses the system reliability for inexplicit and non-linear performance functions as well as for linear and explicit ones. Any version of the method of slices may be used although the proposed approach is presented on the basis of the Bishop simplified method. It is shown that the upper bound system failure probability is higher than the failure probability associated with a critical slip surface. The difference increases as the coefficient of variation of the shear strength parameters increases.     

Response surface method based on uniform design and weighted least squares for non-probabilistic reliability analysis

The non-probabilistic reliability theory is a promising methodology for implementing structural reliability analysis in case of scarce statistical data. One of the main obstacles to implement non-probabilistic reliability analysis is the implication of the limit state function (LSF) for complex structures. This paper aims to establish a surrogate model of the LSF with higher simulation precision, and whereby proposes a response surface method based on the combination of uniform design (UD) and weighted least squares (WLS). At first, the UD method is selected as the sampling method of interval variables to realize the uniform space-filling of the initial samples, and the sample set is updated by gradually adding the approximate optimal points to increase the sampling density of critical domain. Then, the WLS method is applied to improve the precision of the response surface by adjusting the importance of samples to the function fitting. Finally, a method of constructing sample weights is developed. Two examples are applied to validate the feasibility and efficiency of the proposed method. Results show that the proposed method is effective for non-probabilistic reliability analysis of complex structures owning to high computational precision and low computational cost in both numerical and case study.     

Developing safety performance functions incorporating reliability-based risk measures

Current geometric design guides provide deterministic standards where the safety margin of the design output is generally unknown and there is little knowledge of the safety implications of deviating from these standards. Several studies have advocated probabilistic geometric design where reliability analysis can be used to account for the uncertainty in the design parameters and to provide a risk measure of the implication of deviation from design standards. However, there is currently no link between measures of design reliability and the quantification of safety using collision frequency. The analysis presented in this paper attempts to bridge this gap by incorporating a reliability-based quantitative risk measure such as the probability of non-compliance (P_nc) in safety performance functions (SPFs). Establishing this link will allow admitting reliability-based design into traditional benefit–cost analysis and should lead to a wider application of the reliability technique in road design. The present application is concerned with the design of horizontal curves, where the limit state function is defined in terms of the available (supply) and stopping (demand) sight distances. A comprehensive collision and geometric design database of two-lane rural highways is used to investigate the effect of the probability of non-compliance on safety. The reliability analysis was carried out using the First Order Reliability Method (FORM). Two Negative Binomial (NB) SPFs were developed to compare models with and without the reliability-based risk measures. It was found that models incorporating the P_nc provided a better fit to the data set than the traditional (without risk) NB SPFs for total, injury and fatality (I + F) and property damage only (PDO) collisions.     

Hazard maps and global probability as a way to transfer standard fracture results to reliable design of real components

Design and lifetime prediction of structural and mechanical components require the assessment of the global probability of failure to be determined from stress and strain distributions obtained by FEM, as well as calculation of hazard maps in order to facilitate redesign and recognition of critical parts to be inspected regularly. The so-called generalized probabilistic local approach (GPLA), developed by the authors, allows the primary failure cumulative distribution function (PFCDF) owning to a certain failure type to be determined for a given material from experimental data and used subsequently for probabilistic design. The approach ensures a realistic safety margin provided that the failure criterion represented by an adequate generalized parameter (GP) and the corresponding failure criterion is properly recognized as a reference variable to be considered in the failure assessment. The way in which the results of such a reliability analysis are interpreted encompasses a variety of concepts under which failure can be understood and may be classified as global probability of failure and hazard maps, the former providing the conclusive failure probability for definitive design, and the latter representing, presumably, a risk of local failure that facilitates the possible component redesign but without providing the global probability of failure. In order to promote the implementation of the methodology proposed, an application is exemplary presented for the particular case of experimental results of glass plates. A finite element subroutine is developed for calculation of hazard maps and global probabilities of failure.     

基于ELM的可折支撑锁机构可靠性及其敏感度分析

 针对大多可靠性工程问题中机构极限状态函数为隐式的情况,提出了一种基于极限学习机（ELM）回归近似极限状态方程的可靠性及灵敏度分析的新方法.通过极限学习机与蒙特卡洛法相结合,利用极限学习机快速学习的能力,将复杂或隐式极限状态方程近似等价为显式极限状态方程,运用蒙特卡洛法计算出机构的失效概率,然后由高精度的显式极限状态方程进行各随机变量参数的灵敏度分析.该方法采用了基于单隐层前馈神经网络极限学习算法,因而在拟合非线性极限状态方程上表现优越,计算精度和效率高.最后以某型起落架中可折支撑锁机构为对象,进行了机构的可靠性及敏感度分析.结果表明：该方法具有高精度和高效率的优点,在工程应用上具有一定的价值．     

Set theoretic formulation of performance reliability of multiple response time‐variant systems due to degradations in system components

This paper presents a design stage method for assessing performance reliability of systems with multiple time‐variant responses due to component degradation. Herein the system component degradation profiles over time are assumed to be known and the degradation of the system is related to component degradation using mechanistic models. Selected performance measures (e.g. responses) are related to their critical levels by time‐dependent limit‐state functions. System failure is defined as the non‐conformance of any response and unions of the multiple failure regions are required. For discrete time, set theory establishes the minimum union size needed to identify a true incremental failure region. A cumulative failure distribution function is built by summing incremental failure probabilities. A practical implementation of the theory can be manifest by approximating the probability of the unions by second‐order bounds. Further, for numerical efficiency probabilities are evaluated by first‐order reliability methods (FORM). The presented method is quite different from Monte Carlo sampling methods. The proposed method can be used to assess mean and tolerance design through simultaneous evaluation of quality and performance reliability. The work herein sets the foundation for an optimization method to control both quality and performance reliability and thus, for example, estimate warranty costs and product recall. An example from power engineering shows the details of the proposed method and the potential of the approach. Copyright © 2006 John Wiley & Sons, Ltd.     

一种基于自适应集成学习代理模型的结构可靠性分析方法

结构可靠性分析需要精确计算结构或系统的失效概率,当结构失效概率低时,运算量大且操作困难。可采用代理模型替代原始性能函数,结合自适应实验设计,在保证准确率的同时大幅减少原始模型的总运行次数。该文提出了基于自适应集成学习代理模型的结构可靠性分析方法,将适应性较广的Kriging与最近发展的PC-Kriging代理模型集成;利用代理模型提供预测点的方差特征,提出新的集成学习函数,识别高预测误差区域,实现高效拟合失效边界;通过主动学习算法在预测误差大和接近极限状态的区域添加采样,迭代更新集成代理模型。通过3个算例,验证了该文方法与单一代理模型结构可靠性分析方法的优势,与AK-MCS+U和AK-MCS+EFF相比,所提方法计算成本低、准确度高。