Failure Probability Function Research Articles

Probabilistic risk assessment of civil aircraft associated failures under condition-based maintenance

Maintenance can improve an aircraft system's reliability over a long operation period or when a component has failed. However, inappropriate maintenance inspection intervals will cause latent failures to be covered or undetected, leading to a large number of unplanned flight disruptions for airlines. In this paper, we present a two-stage framework to assess the associated failure risk of civil aircraft under condition-based maintenance. In the first stage of the framework, the probability of primary functional failure across the lifecycles of the monitored component is determined by analyzing whether the current inspection interval prevents the component from progressing from latent failure to functional failure. In the second stage of the framework, the associated failure probability between components and related systems is formulated by the adjacency matrix. The structure and performance of the proposed model were tested on a case study by run-to-failure data associated with aircraft engines from a large airline. Focusing on the scenario of turbine disk cracking leading to fragment penetration of the fuel tank and causing fire as the consequential fault impact path, the results show that the risk of aircraft fire caused by turbine disk fragments falls within an acceptable range, necessitating the completion of inspections and subsequent monitoring within the stipulated timeframe. The method can be used to readjust the inspection interval, optimize the operation plan, improve the on-time performance of flights, and reduce the risk of aviation accidents.

Adaptive support vector machine for time-variant failure probability function estimation

Time variant reliability analysis introduces additional complexity due to the inclusion of time. When the time-variant failure probability function (TFPF) of the structure is of interest, it inherently involves sequential evaluations of the failure probabilities of series systems varied with time in discretized space, posing a challenge to reliability analysis. An efficient approach for the evaluation of the TFPF, called ‘Time-dependent Adaptive Support Vector Machine combined with Monte Carlo Simulation’ (TASVM-MCS), is presented to reduce the corresponding computational cost. Based on the samples from Monte Carlo simulation (MCS), an iterative strategy is proposed to actively extract the most valuable sample points from the sample pool and iteratively update the support vector machine (SVM) model. In particular, an active learning function is proposed to take into account the diversity of samples and time simultaneously. In this way, the built SVM will be more suitable for the evaluation of TFPF other than a point-wise failure probability. The proposed TASVM-MCS method is relatively less sensitive to the dimensionality of the input variables, making it a powerful and promising approach for time-variant reliability computations. Four representative examples are given to demonstrate the significant effectiveness and efficiency of the proposed method.

Quantitative analysis of failure performance uncertainty based on multiscale mechanical analysis of PEMFCs’ C/C composite bipolar plates

In order to quantify the failure performance of PEMFCs’ C/C composite bipolar plates (BPs) under considering the uncertainty in multiscale sizes and loads, the failure probability function (FPF) of BPs at cold start is estimated. A three-stage multiscale simulation method is proposed for mechanical behavior analysis of multi-component heterogeneous composites. And the efficient estimation of FPF for C/C composite BPs is achieved via the proposed multiscale simulation method and decoupling sparse integration method, in which the unified density weight is constructed to decouple the double-loop framework of analyzing FPF, thereby reducing the computational complexity. Results indicate that the dangerous positions of BPs are around the gas flow channels and bolt holes. And the dangerous components are the carbon matrix and the interface between the matrix and fiber. The radius of elliptical fibers affects the distribution of failure factors on microscopic components, thereby affecting the variation of failure probability with fiber sizes significantly. The distribution parameter of composite layer thickness in BPs has a significant effect on structural safety, and the FPF result can be used to effectively guide the design of structure. Therefore, the proposed method is of great significance for guiding the rapid parameter design of composite BPs.

A novel cross-entropy-based importance sampling method for cumulative time-dependent failure probability function

Cumulative time-dependent failure probability function (C-T-FPF) can reflect the effects of distribution parameters of random inputs and the upper bound of service time interval, which vary in their design domains, on time-dependent failure probability. However, solving C-T-FPF involves a time-consuming triple-layer framework. Thus, we propose an efficient method by combining cross-entropy-based importance sampling (IS) with adaptive Kriging model (CE-IS-AK). The innovations of CE-IS-AK include two aspects. Firstly, we construct a space augmented by both distribution parameters and the upper bound of service time interval, on which the triple-layer framework is decoupled to a single-layer one. And in the augmented space, an optimal IS density is proposed to reduce the required candidate sample size for estimating C-T-FPF. Secondly, we employ Gaussian mixture model (GMM) to approximate the optimal IS density, and the parameters in GMM are determined by minimizing the cross entropy of GMM and the optimal IS density. Moreover, to reduce the model evaluations in the proposed method, Kriging model is adaptively embedded to replace the actual model, and a first failure instant based learning function is proposed to train Kriging model adaptively. Due to the proposed single-layer framework and IS strategy assisted by the Kriging model, the efficiency is greatly improved for estimating C-T-FPF, which is validated by several examples.

Novel Bayesian updating based interpolation method for estimating failure probability function in the presence of random-interval uncertainty

Under random-interval uncertainty, the failure probability function (FPF) represents the failure probability variation as a function of the random input distribution parameter. To quickly capture the effect of the distribution parameters on failure probability and decouple the reliability-based design optimization, a novel Bayesian updating method is proposed to efficiently estimate the FPF. In the proposed method, the prior augmented failure probability (AFP) is first estimated in the space spanned by random input and distribution parameter vectors. Subsequently, by treating the distribution parameter realization as an observation, the FPF can be estimated using posterior AFP based on Bayesian updating. The main novelty of this study is the elaborate treatment of the distribution parameter realization as an observation, whereby the FPF is transformed into the posterior AFP based on Bayesian updating, and can be obtained by sharing the prior AFP simulation samples. The computational cost of the proposed method is the same as that of estimating the prior AFP. To improve the efficiency of recognizing the sample state, and improve AFP and in turn FPF estimation, the adaptive Kriging model for random-interval uncertainty was inserted into the proposed method. The feasibility and novelty of the proposed method were verified on several examples.

Assessment of leak-tightness for nuclear reactor containment under overpressure conditions considering aging effects

This paper assesses the aging effects on the leak-tightness of containment under overpressure conditions. A global containment model and the detailed sub-models for the three main penetration regions in the containment are established. The main aging forms and mechanisms of containment are clarified, and corresponding simulation schemes are provided. Through numerical simulation, the impacts of different aging forms on the failure pressure of containment are discussed from a deterministic perspective. Finally, the fragility and functional failure probability of containment under different aging conditions are evaluated. When 60 years of concrete degradation and prestress loss are considered, the pressure capacities of equipment hatch, personnel airlock, and pipe penetration are reduced by approximately 5%. Upon further considering steel liner corrosion, when the corrosion degree reaches 30%, the pressure capacities of these regions are reduced by 24.37%, 25.52%, and 22.83%, respectively. Within the scope of this study, the impact of steel liner corrosion on the risk of containment leakage is the most pronounced, whereas the impact of concrete degradation is minimal. If steel liner corrosion occurs simultaneously in three penetration regions, the whole containment will fail to meet the probabilistic performance goal when the corrosion degree reaches 30%.

Efficient estimation procedure for failure probability function by an augmented directional sampling

AbstractFailure probability function (FPF) can reflect quantitative effects of random input distribution parameter (DP) on failure probability, and it is significant for decoupling reliability‐based design optimization (RBDO). But the FPF estimation is time‐consuming since it generally requires repeated reliability analyses at different DPs. For efficiently estimating FPF, an augmented directional sampling (A‐DS) is proposed in this paper. By using the property that the limit state surface (LSS) in physical input space is independent of DP, the A‐DS establishes transformation of LSS samples in standard normal spaces corresponding to different DPs. By the established transformation in different standard normal spaces, the LSS samples obtained by DS at a given DP can be transformed to those at other DPs. After simple interpolation post‐processing on those transformed samples, the failure probability at other DPs can be estimated by DS simultaneously. The main novelty of A‐DS is that a strategy of sharing DS samples is designed for estimating the failure probability at different DPs. The A‐DS avoids repeated reliability analyses and inherits merit of DS suitable for solving problems with multiple failure modes and small failure probability. Compared with other FPF estimation methods, the examples sufficiently verify the accuracy and efficiency of A‐DS.

Adaptive importance sampling approach for structural time-variant reliability analysis

A novel sampling approach, called adaptive importance sampling (AIS), is proposed to efficiently perform time-variant reliability analysis. In practice, structures are generally subject to time-variant deterioration processes and external loads, and the Time-variant Failure Probability Function (TFPF), which is the failure probability as a function of time, is a critical quantity of interest in engineering applications. The proposed approach leverages an adaptive strategy and an optimal combination algorithm to further improve the accuracy and efficiency of TFPF estimation using the importance sampling approach. The adaptive strategy is to seek for the best setting of importance sampling components to iteratively obtain estimator components of the TFPF. The optimal combination algorithm is to collect all these adaptive estimator components to form an overall estimator by its coefficient of variation (C.o.V.). The proposed approach outperforms traditional importance sampling methods in the sense that it ensures the convergence with minimal computational cost, specifically the C.o.V. of the TFPF estimator is below a predetermined threshold over the entire time domain. Therefore, the proposed approach offers an extension to traditional importance sampling methods for time-variant reliability assessment. Numerical examples are provided to demonstrate the effectiveness of the proposed approach in accurately estimating the TFPF of structures subjected to time-variant loads and deterioration processes.

Bayesian updating model of failure probability function and its solution

When new observations are collected, the failure probability function (FPF) should be calibrated to match the new observations. However, there is a lack of FPF updating model and corresponding solution at present. Therefore, this paper devotes to the corresponding research. The main contribution of this paper includes two aspects. The first is constructing a Bayesian updating model of FPF, and the second is proposing two methods, i.e., combination sampling (CS) method and combination importance sampling (CIS) method, to solve the FPF updating model. In the constructed FPF updating model, the likelihood function, which approximately describes the probability of the observation error, is combined with the prior information to calibrate the prior FPF by Bayesian theory. And by sharing the sample information of the constructed CS density or the CIS density, the complete FPF can be updated through a single simulation run. Moreover, to enhance the efficiency of the CS and CIS, the adaptive Kriging model of performance function is nested in the CS and CIS methods. Five examples show that the two proposed methods need much less computational cost than the competitive methods under the similar accuracy of calibrating FPF, and the proposed CIS method is more efficient than the proposed CS method.

Augmented line sampling and combination algorithm for imprecise time-variant reliability analysis

Assessment of imprecise time-variant reliability in engineering is a critical task when accounting for both the variability of structural properties and loads over time and the presence of uncertainties involved in the ambiguity of parameters simultaneously. To estimate the Imprecise Time-variant Failure Probability Function (ITFPF) and derive the imprecise reliability results as a byproduct, Adaptive Combination Augmented Line Sampling (ACALS) is proposed. It consists of three integrated features: Augmented Line Sampling (ALS), adaptive strategy, and the optimal combination. ALS is adopted as an efficient analysis tool to obtain the failure probability function w.r.t. imprecise parameters. Then, the adaptive strategy iteratively applies ALS while considering both imprecise parameters and time simultaneously. Finally, the optimal combination algorithm collects all result components in an optimal manner to minimize the Coefficient of Variance (C.o.V.) of the ITFPF estimate. Overall, the proposed ACALS method outperforms the original ALS method by efficiently estimating the ITFPF while guaranteeing a minimal C.o.V. Thus, the proposed approach can serve as an effective tool for imprecise time-variant reliability analysis in real engineering applications. Several examples are presented to demonstrate the superiority of the proposed approach in addressing the challenges of estimating the ITFPF.

Meta model-based importance sampling combined with adaptive Kriging method for estimating failure probability function

The failure probability function (FPF) can provide the entire effect of the distribution parameter in the concerned region on failure probability (FP), which is important for decoupling the optimal parameter search and constraint analysis in the FP-constrained design optimization. To improve the efficiency of solving the FPF, this paper combines meta model-based importance sampling with adaptive Kriging (Meta-IS-AK). First, based on the theoretical optimal importance sampling density (ISD) for estimating the FP (ISD-FP) at the given distribution parameter, the theoretical optimal ISD for solving FPF (ISD-FPF) is constructed using an integral operator. Then, a Kriging model of the performance function is adaptively trained to establish the quasi-optimal ISD-FPF for approximating the optimal performance function, and a simple rejection sampling (SRS) method is designed to extract the samples of the quasi-optimal ISD-FPF. For recognizing the state of the ISD-FPF sample and completing the IS estimation of FPF, the Kriging model is secondly updated to establish ISD-FPF in the sample pool comprising these ISD-FPF samples. Using the proposed method, the whole FPF corresponding to arbitrary distribution parameters can be obtained using one IS reliability analysis based on the constructed ISD-FPF. Four examples and an application illustrate the advantages of the proposed method.

A novel and efficient stochastic collocation method for estimating failure probability function in one-dimensional reduced space

Failure probability function (FPF) is an important index when measuring the effect of random input distribution parameters on the safety of structures. It can decouple reliability-based design optimization. However, efficient estimation of the FPF is challenging. This study proposes a novel stochastic collocation method for estimating the FPF. The novelty of the proposed method lies in three aspects: First, an equivalent integral expression is proposed for the FPF in one-dimensional reduced space, enabling FPF estimation using an efficient stochastic collocation method. Secondly, a unified sampling density function is constructed to remove the coupling between the computational cost of FPF estimation and the number of distribution parameter realizations. Thirdly, a stochastic collocation point set is derived and shared to estimate the whole FPF using arbitrary distribution parameter realization. The proposed method is particularly suitable for the strength-stress and resistance-load models used in the field of engineering. The efficiency and accuracy of the proposed method are verified by several examples.

Augmented Space Integral Approach for Structural Reliability-Based Optimization

An efficient, fully decoupled approach is proposed to solve the structural reliability-based design optimization (RBDO) problem. The proposed approach utilizes augmented space integral and importance sampling (ASI-IS) to efficiently evaluate a functional relationship between the probability of failure and the design parameters, namely the so-called failure probability function. ASI-IS allows for the avoidance of both the intractable density fitting task associated with augmented space methods and the time-consuming repeated reliability evaluations associated with surrogate model approaches. The resulting functional relationship can be used to completely decouple the original RBDO problem into a deterministic one. Then, an iteration mechanism is constructed with gradually smaller design domains to enhance the efficiency of the optimization process. Furthermore, a sample reuse algorithm is proposed to improve the performance of the proposed approach by collecting the samples generated in previous iterations and reusing them in the current iteration in order to produce a better estimator of the failure probability function. Numerical and engineering examples, including a turbine blade and an aircraft inner flap, are given to demonstrate the efficiency and feasibility of the proposed approach.

Meta model-based and cross entropy-based importance sampling algorithms for efficiently solving system failure probability function

The multi-mode system failure probability function (SFPF) can quantify how the distribution parameters of the random input vector affect the system safety and decouple the system reliability-based design optimization model. However, for a problem with a time-consuming implicit performance function and rare failure domain, efficiently solving the SFPF remains significantly challenging. Therefore, in this study, two efficient algorithms are proposed, namely, the meta model-based importance sampling and cross entropy-based importance sampling. The contributions of this study are twofold. The first is constructing a single-loop optimal importance sampling density (SL-OISD) method to decouple the double-loop framework for analyzing the SFPF. The second is establishing two methods to efficiently approximate the SL-OISD and complete the SFPF estimation. The first method is based on the meta model of the system performance function, which is abbreviated as SL-Meta-IS. The second method is based on minimizing the cross entropy between the Gaussian mixture density model and SL-OISD, which is abbreviated as SL-CE-IS. To reduce the number of evaluating the system performance function when approximating the SL-OISD, sampling the SL-OISD, and identifying the state of the samples for completing the SFPF estimation, an adaptive Kriging model of the system performance function is introduced into SL-Meta-IS and SL-CE-IS. Owing to decoupling the double-loop framework into a single-loop framework, replacing the time-consuming system performance function with the economic Kriging model, and employing importance sampling variance reduction techniques to address issues related to the rare failure domain, the proposed SL-Meta-IS and SL-CE-IS methods greatly enhance the efficiency of SFPF estimations. The numerical and practical examples demonstrate that the two proposed methods are superior to the existing algorithms; moreover, the efficiency of SL-CE-IS is higher than that of SL-Meta-IS.

A Decoupling Algorithm for Efficient Estimation of Failure Probability Function Based on Statistical Moment Functions

Abstract Failure probability function (FPF) is an important index that reflects the influence of designable distribution parameters on the safety degree of a structure, and it can be used for decoupling reliability optimization models. Thus, its efficient solution is expected. A decoupling algorithm based on statistical moment functions (SMFs) of performance function is proposed to solve the FPF efficiently in this paper. The proposed algorithm first constructs an extended density weight function (EDWF), which can cover the interested region of the distribution parameters and is independent of the distribution parameters so that the statistical moment integrals corresponding to different realizations of the distribution parameters can share the same EDWF. Then, using the same EDWF, a strategy is dexterously designed to estimate the SMFs by sharing a set of integral characteristic nodes. Finally, the FPF is approximated by the SMFs, which varies with the distribution parameters in the interested design region. In addition, the proposed algorithm introduces the Box–Cox transformation of the performance function to guide the high accuracy of FPF approximated by the SMFs. The main contribution of the proposed algorithm is constructing the EDWF to decouple the dependence of solving SMFs on the realizations of the distribution parameters over the interested region and designing the strategy of estimating the SMFs by sharing the same integral characteristic nodes. Since the proposed algorithm employs a point estimation method to evaluate the FPF, it has higher efficiency than the competitive methods. Numerical and engineering examples demonstrate the superiority of the proposed algorithm.

Efficient Methods for Reliability Sensitivity Analysis of Distribution Parameters and Their Application

To efficiently evaluate the influence of the distribution parameters of the input variables on the failure probability of engineering structures and improve the reliability and safety of engineering structures in a targeted manner, new methods for the global reliability sensitivity analysis (RSA) of distribution parameters are proposed in this study based on the cubature formula (CF), surrogate sampling probability density function (SSPDF), and quasi-Monte Carlo (QMC) method. By introducing CF, the proposed methods can effectively improve the computational efficiency of the nested expectation and variance operators in the reliability sensitivity indices of the distribution parameters. Based on the concept of SSPDF, a surrogate importance sampling probability density function was developed. This not only overcomes the problem of the computational effort of propagating parameter uncertainty to the failure probability function (FPF), which depends on the dimensionality of the parameters; it also further improves the efficiency of the RSA of the parameters in the case of a small failure probability. Finally, by incorporating the idea of the QMC method, the process of calculating the reliability sensitivity indices of the parameters is reduced from a double-loop to a single-loop one. Three engineering examples are used in this study to demonstrate the efficiency and accuracy of the new algorithms.

Single-loop cross-entropy-based directional importance sampling for efficient estimation of failure probability function

Failure probability function (FPF) can not only reflect the effect of random input distribution parameter (DP) on structural safety degree, but also decouple reliability based design optimization. Nowadays, it still desires the efficient algorithms for FPF. Thus, this paper proposes an efficient and novel single-loop cross-entropy-based directional importance sampling method (SL-CE-DIS). The proposed SL-CE-DIS has two innovations. The first is constructing the transformation relationship (Tr-Re) of the samples on the limit state surface (LSS) in the standard normal space corresponding to different DP realizations. This Tr-Re is established by the property of the LSS in original physical space independent of DP, and it makes the LSS sample explored by single CE-DIS at a given DP realization can be shared by the CE-DIS at other DP realizations. The second is designing an inverse updating rule to reconstruct the quasi-optimal DIS density (DIS-D) corresponding to other DP realization, which is in the adjacent region of the given DP realization, based on CE principle, and this designed inverse updating rule can easily estimate the FPF values corresponding to other DP realization by matching the transformed LSS samples. Based on the designed LSS sample sharing strategy and inverse updating rule of DIS-D, the proposed SL-CE-DIS can use the LSS information explored by single CE-DIS at the given DP realization to estimate the FPF corresponding to the DP over the adjacent of the given DP realization, and the presented examples fully verify the superiority of the SL-CE-DIS over the other existing single-loop sampling methods for FPF.

A novel single-loop simulation algorithm combined with adaptive Kriging model for estimating the system failure probability function with multi-dimensional distribution parameter

A novel single-loop simulation algorithm combined with adaptive Kriging model for estimating the system failure probability function with multi-dimensional distribution parameter

Seismic performance and vulnerability analysis for bifurcated tunnels in soft soil

This paper investigates the seismic performance of complex comprehensive interchange underground structures. As an example, the seismic performance of Nanjing underground bifurcated tunnels constructed in soft soil is analyzed. A three-dimensional (3D) numerical finite element analysis model is established considering soil-tunnel interaction to analyze the seismic response characteristics and failure mechanism of the bifurcated tunnels. The results show that the tunnel damage degree is closely associated with the deformation response, and that the inter-story drift ratio of the lower layer is significantly greater than that of the upper layer. The pushover analysis method is used to obtain the seismic performance curve of the tunnel, and inter-story drift ratio limits are defined. The IDA procedure is then used to conduct multiple non-linear dynamic time history analyses based on the 3D finite element model, and IDA curve clusters are drawn with peak ground acceleration (PGA) as the earthquake intensity measures and the peak inter-story drift ratio as the damage measures of tunnels. The corresponding IDA percentiles lines are calculated, and the seismic fragility curves of the bifurcated tunnels are established. The post-earthquake functional failure probability of the bifurcated tunnels is obtained under different seismic fortification performance level.

Dynamic pruning-based Bayesian support vector regression for reliability analysis

Adaptive surrogate-based reliability analysis methods have garnered significant attention due to their potential to enhance computational efficiency in accurately estimating failure probability. However, the candidate sample pool remains constant for most surrogate-based reliability methods, and traversing the candidate sample pool one by one will reduce the efficiency of surrogate modeling. More importantly, maintaining a static sample pool may lead to the inclusion of samples that contribute minimally to the construction of the surrogate modeling, thereby impacting the accuracy and efficiency of the reliability analysis. Accordingly, this paper leverages the robust performance of Bayesian support vector regression to propose a dynamic pruning strategy for the candidate sample pool to estimate failure probability efficiently. A dynamic pruning strategy is presented to streamline the process further, iteratively reducing the candidate sample pool. An adaptive learning algorithm is then introduced, integrating the U function and the sparsity of training samples. This is complemented by a formulated convergence condition, contributing to an ideal surrogate model. The proposed approaches showcase superior efficiency and accuracy through illustrations using well-known benchmark problems and complex reliability analysis problems involving small failure probability and high-dimensional limit state function.