Estimation Of Failure Probability Research Articles

Time varying reliability analysis of corroded gas pipelines using copula and importance sampling

Ensuring the safety of gas pipelines is crucial for the reliable and efficient transportation of natural gas. This study introduces a methodology for evaluating the time-varying reliability of corroded gas pipelines. The proposed approach employs both importance sampling and copula theory to effectively address the uncertainties associated with random variables involved and provide efficient estimates of failure probability. Small leak and burst failure modes are specifically investigated within this study. The methodology first establishes a step-by-step procedure for analysis. Subsequently, the study investigates the failure probability of corroded gas pipelines, taking into account both small leak and burst failures, under scenarios involving single and multiple defects. Furthermore, the study examines the influence of correlation strength and dependence structure among the involved random variables using the copula theory. Additionally, the generation of newly formed defects is thoroughly investigated, along with its impact on failure probability. Numerical examples are provided to evaluate the performance of the proposed methodology, comparing it with the benchmark failure probability estimated through Monte Carlo simulation. The results validate the precision and efficiency of the proposed methodology, while offering practical and insightful suggestions for reliability analysis of real-world gas pipelines.

Reliability evaluation in the zero‐failure Weibull case based on double‐modified hierarchical Bayes

AbstractHierarchical Bayes, or E‐Bayes, is frequently used to estimate the failure probability when solving a zero‐failure reliability evaluation model; however, the accuracy of the reliability estimation using these methods is not very good in practice. Due to this, a novel double‐modified hierarchical Bayes (DMH‐Bayes) is proposed for Weibull characteristic data in this study to enhance failure probability estimation and improve reliability point estimation accuracy. Meanwhile, in order to guarantee the preservation of the assessment findings' consistency and confidence level, the parametric Bootstrap method (P‐Bootstrap) and the L‐moment estimation method based on point estimation are introduced to obtain reliability confidence interval estimates. Based on Monte–Carlo simulation testing and analysis of a gyroscope bearing, the new model is confirmed to have better applicability and robustness while improving the accuracy of reliability assessment.

Bayesian Safety Validation for Failure Probability Estimation of Black-Box Systems

Estimating the probability of failure is an important step in the certification of safety-critical systems. Efficient estimation methods are often needed due to the challenges posed by high-dimensional input spaces, risky test scenarios, and computationally expensive simulators. This work frames the problem of black-box safety validation as a Bayesian optimization problem and introduces a method that iteratively fits a probabilistic surrogate model to efficiently predict failures. The algorithm is designed to search for failures, compute the most-likely failure, and estimate the failure probability over an operating domain using importance sampling. We introduce three acquisition functions that aim to reduce uncertainty by covering the design space, optimize the analytically derived failure boundaries, and sample the predicted failure regions. Results show this Bayesian safety validation approach provides a more accurate estimate of failure probability with orders of magnitude fewer samples and performs well across various safety validation metrics. We demonstrate this approach on three test problems, a stochastic decision-making system, and a neural-network-based runway detection system. This work is open-sourced (https://github.com/sisl/BayesianSafetyValidation.jl) and is currently being used to supplement the FAA certification process of the machine learning components for an autonomous cargo aircraft.

An active learning method using deep adversarial autoencoder-based sufficient dimension reduction neural network for high-dimensional reliability analysis

Reliability analysis often requires time-consuming evaluations, especially when dealing with high-dimensional and nonlinear problems. To address this challenge, surrogate model methods are frequently employed. One way to improve the efficiency of surrogate model methods involves selecting informative samples that significantly enhance the accuracy of the surrogate model. This paper introduces a novel approach to facilitate the construction of surrogate models and selection of informative samples in high-dimensional reliability analysis, through an active learning method based on a deep adversarial autoencoder-based sufficient dimension reduction (AAE-SDR) neural network. The AAE-SDR neural network serves as a surrogate model, transforming complex high-dimensional variables into tractable, low-dimensional embeddings relevant to the target. These embeddings are Gaussian-distributed with a distinct latent limit state boundary. A new sampling strategy is proposed to select informative misclassified samples by iteratively identifying candidate samples near the latent limit state boundary and uniformly sampling from the candidate sample dataset based on the latent Gaussian distribution. The effectiveness of the proposed approach is demonstrated through two high-dimensional numerical examples and a cable-stayed bridge case study. Results show that the proposed method simplifies complex high-dimensional reliability problems and provides a relatively accurate estimated failure probability with a limited number of samples.

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.

Joint Failure Probability of Dams Based on Probabilistic Flood Hazard Analysis

Probabilistic risk methods are becoming increasingly accepted as a means of carrying out risk-informed decision making regarding the design and operation policy of structures such as dams. Probabilistic risk calculations require the quantification of epistemic and aleatory uncertainties not investigated through deterministic methodologies. In this hydrological study, a stochastic sampling methodology is employed to investigate the joint failure probability of three dams in adjacent similarly sized watersheds within the same hydrologic unit code (HUC) 6 basin. A probabilistic flood hazard analysis (PFHA) framework is used to simulate the hydrologic loading of a range of extreme precipitation events across the combined watershed area of the three studied dams. Precipitation events are characterized by three distinct storm types influential in the Tennessee Valley region with implications for weather variability and climate change. The stochastic framework allows for the simulation of hundreds of thousands of spillway outflows that are used to produce empirical bivariate exceedance probabilities for spillway discharge pairs at selected dams. System response curves that indicate the probability of failure given spillway discharge are referenced for each dam and applied to generate empirical bivariate failure probability (joint failure probability) estimates. The stochastic simulation results indicate the range of spillway discharges for each pair of dams that pose the greatest risk of joint failure. The estimate of joint failure considering the dependence of spillway discharges between dams is shown to be three to four orders of magnitude more likely (7.42 × 102 to 5.68 × 103) than estimates that assume coincident failures are the result of independent hydrologic events.

Time-Variant Reliability Analysis with an Adaptive Single-Loop Kriging Combined with Directional Sampling Method

The difficulty in calculating time-variant reliability lies in the large number of performance function calls. This paper proposes an efficient method for time-variant reliability analysis by incorporating the directional sampling method (DS). In this method, within the framework of single-loop active learning Kriging (SL–AK), the burden of fitting the Kriging surrogate model is alleviated by constructing a novel candidate sample pool. While reducing the computational burden, the accuracy of the Kriging surrogate model remains unaffected. Unlike the traditional SL–AK, the proposed method obtains the failure probability through an improved DS method. The improved DS utilizes a bisection method for root-finding, thereby circumventing the impact of interpolation coefficients on the computation results. This method of computing failure probability not only alleviates computational burden but also enhances the precision of the estimated failure probability. Furthermore, this paper introduces global sensitive analysis (GSA) into time-variant reliability analysis, thereby expanding the applicability of GSA. The accuracy and high efficiency of the proposed method are verified through three numerical examples and two engineering examples with implicit performance function.

Bayesian reinforcement learning reliability analysis

A Bayesian reinforcement learning reliability method that combines Bayesian inference for the failure probability estimation and reinforcement learning-guided sequential experimental design is proposed. The reliability-oriented sequential experimental design is framed as a finite-horizon Markov decision process (MDP), with the associated utility function defined by a measure of epistemic uncertainty about Kriging-estimated failure probability, referred to as integrated probability of misclassification (IPM). On this basis, a one-step Bayes optimal learning function termed integrated probability of misclassification reduction (IPMR), along with a compatible convergence criterion, is defined. Three effective strategies are implemented to accelerate IPMR-informed sequential experimental design: (i) Analytical derivation of the inner expectation in IPMR, simplifying it to a single expectation. (ii) Substitution of IPMR with its upper bound IPMRU to avoid element-wise computation of its integrand. (iii) Rational pruning of both quadrature set and candidate pool in IPMRU to alleviate computer memory constraint. The efficacy of the proposed approach is demonstrated on two benchmark examples and two numerical examples. Results indicate that IPMRU facilitates a much more rapid reduction of IPM compared to other existing learning functions, while requiring much less computational time than IPMR itself. Therefore, the proposed reliability method offers a substantial advantage in both computational efficiency and accuracy, especially in complex dynamic reliability problems.

Failure probability estimation of natural gas pipelines due to hydrogen embrittlement using an improved fuzzy fault tree approach

The estimation of failure probability is challenging in hydrogen embrittlement in steel pipelines due to the complexity of the synergistic effect of multiple factors. The present study proposed a hybrid methodology to estimate the failure probability of steel pipelines due to hydrogen embrittlement. The methodology integrates the fault tree analysis with a fuzzy comprehensive evaluation. Fault tree analysis captures the logical relationships between influencing indicators to develop a new assessment model of hydrogen embrittlement in steel pipelines. An improved fuzzy fault tree analysis method was proposed to process aleatoric and epistemic uncertainties to estimate the probability of each basic event due to the difficulty in obtaining the actual probabilities. The failure probability of blended hydrogen natural gas pipelines was estimated by considering the correlation of events. A case study demonstrated the applicability of the proposed method. Maintenance measures can be implemented according to the evaluation results to ensure pipeline safety.

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Importance sampling enhanced by adaptive two-stage Kriging model and active subspace for analyzing rare probability with high dimensional input vector

For the high dimensional and small failure probability events widely existing in engineering, structural reliability analysis method still faces challenges at present. Since active subspace (AS) method can effectively reduce the dimension of input vector space, and metamodel-based importance sampling (IS) can efficiently estimate small failure probability, this paper combines AS with IS to propose an adaptive two-stage Kriging model method (AS-IS-AK2) for rare failure event with high dimensional problem. The proposed AS-IS-AK2 includes two stages. In the first stage, a new learning function is proposed to construct the Kriging model for roughly estimating the performance function values of all IS samples, and the gradient information is provided as a by-product for AS. In the second stage, an adaptive Kriging method is constructed in the reduced dimension space to accurately judge the states of all IS samples. Since AS reduces the dimension of input vector space, and IS reduces the required size of candidate sample pool for obtaining the convergent failure probability estimation, the computational complexity is reduced in reconstructing the Kriging model to judge the states of IS samples. The results of three examples verify the advantage of the proposed method for high dimensional problem with small failure probability.

Time-dependent reliability analysis of structural systems based on parallel active learning Kriging model

Time-dependent reliability analysis quantifies the failures of structural systems due to time-dependent uncertainties, such as material degradation and dynamic loads. The active learning Kriging model methods are widely used in structural reliability analysis to replace extensive time-consuming finite element simulations. However, they can only update one training sample and one failure mode per iteration, which limits their application to time-dependent, parallel computing, and multiple failure modes problems. In this study, we propose a new parallel active learning Kriging model for time-dependent reliability analysis, which can update multiple training samples and multiple failure modes per iteration. It includes the following strategies: (1) a novel parallel learning function is proposed, which combines the correlation function and U learning function to allow for the selection of multiple training samples per iteration; (2) an adaptive adjustment strategy for the number of parallel samples is proposed, which takes into account the prediction probability of parallel samples; (3) the proposed parallel learning function is integrated into time-dependent reliability analysis with multiple failure modes, enabling simultaneous updates of multiple training samples and failure modes, thus greatly reducing the number of iterations and computational time; and (4) a new stopping criterion is proposed to improve the efficiency of the estimation of failure probability. The proposed method can be applied to series or parallel time-dependent structural systems with multiple failure modes. We demonstrate the effectiveness of the proposed method through three examples, and the proposed method can achieve a balance between the computational time and function calls while maintaining a high level of accuracy in the estimation of time-dependent failure probability.

Refined parallel adaptive Bayesian quadrature for estimating small failure probabilities

Bayesian active learning methods have emerged for structural reliability analysis, showcasing more attractive features compared to existing active learning methods. The parallel adaptive Bayesian quadrature (PABQ) method, as a representative of them, allows to efficiently assessing small failure probabilities but faces the problem of empirically specifying several important parameters. The unreasonable parameter settings could lead to the inaccurate estimates of failure probability or the non-convergence of active learning. This study proposes a refined PABQ (R-PABQ) method by presenting three novel refinements to overcome the drawbacks of PABQ. Firstly, a sequential population enrichment strategy is presented and embedded into the importance ball sampling technique to solve the computer memory problem when involving large sample population. Secondly, an adaptive determination strategy for radius is developed to automatically adjust the sampling region during the active learning procedure. Lastly, an adaptive multi-point selection method is proposed to identify a batch of points to enable parallel computing. The effectiveness of the proposed R-PABQ method is demonstrated by four numerical examples. Results show that the proposed method can estimate small failure probabilities (e.g., 10−7∼10−9) with superior accuracy and efficiency over several existing active learning reliability methods.

Reliability Analysis of Crack Growth Occurrence for a Secondary Hull Component due to Vibration Excitation

Abstract Ship hull vibration is a significant contributor to fatigue crack growth and the major sources of vibrations are found to be the main engine vibration excitation, the wave-induced springing and whipping loads, and the action of the propeller. In the midship region, wave-induced loads and the main engine are the major contributors, whereas propeller excitation dominates in the aft region of the ship hull. No general method exists to solve all kinds of vibration problems and they need to be evaluated through a cost-by-case approach. The complex and uncertain aspects of hull vibration and fatigue crack growth motivate the need for a reliability-based scheme for assessing the resulting fatigue crack propagation. In the present paper, a probabilistic formulation for the failure probability of the occurrence of crack propagation of a secondary hull component is outlined. A generic cargo hold model is analyzed with engine excitation and wave-induced loading as vibration sources, and a stochastic model for vibration response is outlined. The limit state is formulated as the possible occurrence of fatigue crack growth. The secondary hull component considered is a pipe stack support, which is a supporting component that attaches the cargo pipes to the wall inside a cargo tank. Different initial crack sizes are implemented to evaluate the adequacy of the applied stochastic model for vibration response and the accuracy of the estimated failure probability is assessed.

Time-dependent earthquake-fire coupling fragility analysis under limited prior knowledge: A perspective from type-2 fuzzy probability

Earthquake-triggered fire domino scenarios (E-FDSs) arise frequently from the interaction between earthquakes and chemical installations, resulting in catastrophic multi-hazard coupling events. The complicated mutually amplified phenomena between natural disasters and chemical accidents significantly aggravates the escalation of domino accidents, which has posed great challenges for modeling and preventing E-FDSs. Under this impetus, this work proposes an advanced type-2 fuzzy probabilistic methodology to obtain the time-dependent failure probability of steel cylindrical tanks (SCTs) subjected to the earthquake-fire sequence. To cope with the limited prior knowledge on E-FDSs, a basic universal is established to describe the fire resistance attenuation caused by the seismic damage. The coupling failure criterion of SCTs is formulated by a type-2 fuzzy time-dependent limit state equation. A credibility-based stochastic simulation algorithm is developed for the hybrid uncertainty analysis (combining ambiguity and stochasticity). The proposed methodology is validated by case studies of a 5000 m3 fixed roof tank. Compared to the existing accident probability model, the proposed methodology can not only capture the fire resistance attenuation caused by the seismic damage but also provide a dynamic estimation of tank failure probability with respect to the fire exposure time. The proposed methodology can effectively and dynamically capture the accident evolution process, which in turn helps mitigate and prevent the spatiotemporal propagation of domino effects.

An investigation into Markov chain Monte Carlo algorithms for Subset simulation: Emphasizing uncertainty analysis

Subset simulation (SuS) has emerged as a crucial method for estimating failure probability, striking an efficient balance between accuracy and computational cost. This paper introduces an uncertainty analysis method for the Markov chain Monte Carlo (MCMC) algorithm within the SuS framework by examining three key aspects: Effective Sample Size (ESS) ratio, intra-chain correlation, and inter-chain correlation during SuS iterations. The first two affect the standard deviation and bias of the results when MCMC algorithms are run robustly, respectively, while the latter indicates whether the sample distribution generated from MCMC algorithms is stationary or not. Based on these insights, three MCMC algorithms are discussed: Component-wise adaptive Metropolis-Hastings (CW-aMH), Aadaptive Conditional sampling (aCS), and Elliptical Slice sampling (ES). It is demonstrated that better control of the low decreasing rate of ESS ratio in aCS results in a smaller standard deviation compared to CW-aMH in estimation. On the other hand, ES offers better unbiasedness and robustness owing to its low intra-chain correlation and inter-chain correlation. This method provides new insights into MCMC uncertainty analysis, establishing a potential statistical tool for future research in this area.

Failure probability estimation of dynamic systems employing relaxed power spectral density functions with dependent frequency modeling and sampling

This work addresses the critical task of accurately estimating failure probabilities in dynamic systems by utilizing a probabilistic load model based on a set of data with similar characteristics, namely the relaxed power spectral density (PSD) function. A major drawback of the relaxed PSD function is the lack of dependency between frequencies, which leads to unrealistic PSD functions being sampled, resulting in an unfavorable effect on the failure probability estimation. In this work, this limitation is addressed by various methods of modeling the dependency, including the incorporation of statistical quantities such as the correlation present in the data set. Specifically, a novel technique is proposed, incorporating probabilistic dependencies between different frequencies for sampling representative PSD functions, thereby enhancing the realism of load representation. By accounting for the dependencies between frequencies, the relaxed PSD function enhances the precision of failure probability estimates, opening the opportunity for a more robust and accurate reliability assessment under uncertainty. The effectiveness and accuracy of the proposed approach is demonstrated through numerical examples, showcasing its ability to provide reliable failure probability estimates in dynamic systems.

Sampling-based adaptive design strategy for failure probability estimation

Sampling-based adaptive design strategy for failure probability estimation

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.

A Metamodel-Based Multi-Scale Reliability Analysis of FRP Truss Structures under Hybrid Uncertainties.

This study introduces a Radial Basis Function-Genetic Algorithm-Back Propagation-Importance Sampling (RBF-GA-BP-IS) algorithm for the multi-scale reliability analysis of Fiber-Reinforced Polymer (FRP) composite structures. The proposed method integrates the computationally powerful RBF neural network with GA, BP neural network and IS to efficiently calculate inner and outer optimization problems for reliability analysis with hybrid random and interval uncertainties. The investigation profoundly delves into incorporating both random and interval parameters in the reliability appraisal of FRP constructs, ensuring fluctuating parameters within designated boundaries are meticulously accounted for, thus augmenting analytic exactness. In application, the algorithm was subjected to diverse structural evaluations, including a seven-bar planar truss, an architectural space dome truss, and an intricate nonlinear truss bridge. Results demonstrate the algorithm's exceptional performance in terms of model invocation counts and accurate failure probability estimation. Specifically, within the seven-bar planar truss evaluation, the algorithm exhibited a deviation of 0.08% from the established failure probability benchmark.