Estimation Of Failure Probability Research Articles

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.

CSP-free adaptive Kriging surrogate model method for reliability analysis with small failure probability

In the field of reliability engineering, the active learning reliability method that amalgamates Kriging model and Monte Carlo simulation has been devised and proven to be efficacious in reliability analysis. Nevertheless, the performance of this method is sensitive to the magnitude of candidate sample pool, particularly for systems with small failure probabilities. To surmount these limitations, this paper proposes an active learning method that obviates the need for candidate sample pools. The proposed method comprises two stages: construction of surrogate model and Monte Carlo simulation for failure probability estimation. During the surrogate model construction stage, the surrogate model is iteratively refined based on the representative samples selected by solving the optimization problem facilitated by the particle swarm optimization algorithm. To achieve an optimal balance between solution accuracy and efficiency, the penalty intensity control and the density control for the experimental design points are incorporated to modify the objective function in optimization. The performance of the proposed method is evaluated using numerical examples, and results indicate that by leveraging an optimization algorithm to select representative samples, the proposed method overcomes the limitations of traditional active learning methods based on candidate sample pool and exhibits exceptional performance in addressing small failure probabilities.

Line sampling for time-variant failure probability estimation using an adaptive combination approach

An efficient sampling approach ‘Adaptive Combined Line Sampling’ is proposed for evaluating the ‘time-variant failure probability function’ (TFPF) of structures. Line Sampling is implemented in an adaptive and iterative way, where each individual Line Sampling run is carried out based on adaptively selected important directions, in order to ensure a sufficiently precise estimation of the TFPF over the whole time interval of analysis. An adaptive strategy and an optimal combination algorithm are developed for the practical implementation of the Line Sampling process. The adaptive strategy allows to determine the optimal important direction which is then used in the next Line Sampling run. The combination strategy allows to collect all these adaptive sampling runs together in an optimal way, which aims at minimising the coefficient of variation (C.o.V.) of the TFPF estimate. Due to these strategies, the proposed approach can estimate the TFPF in a more efficient way than the traditional Line Sampling, while guaranteeing that the C.o.V. of the estimate remains below a prescribed threshold over the whole time of analysis. Thus it can be seen as an extended version of classical Line Sampling specially tailored for time-variant reliability analysis. Examples are given to illustrate the performance of the proposed approach.

Failure Probability Prediction for Offshore Floating Structures Using Machine Learning

Summary Accurately estimating the failure probability is crucial in designing civil infrastructure systems, such as floating offshore platforms for oil and gas processing/production, to ensure their safe operation throughout their service periods. However, as a system becomes complex, the evaluation of a limit state function may involve the use of an external computer solver, resulting in a significant computational burden to perform Monte Carlo simulations (MCS). Moreover, the high-dimensionality of the limit state function may limit efficient sampling of input variables due to the “curse of dimensionality.” To address these issues, an efficient machine learning framework is proposed, combining polynomial chaos expansion (PCE) and active subspace. This will enable the accurate and efficient evaluation of the failure probability of an offshore structure, which typically involves a large number of uncertain parameters. Unlike conventional PCE schemes that use the original random variable space or the auxiliary variable space for building a surrogate model, the proposed method utilizes a reduced-dimension space to circumvent the “curse of dimensionality.” An appropriate coordinate transformation is first sought so that most of the variability of a limit state function can be accounted for. Next, a PCE surrogate limit state function is constructed on the derived low-dimensional “active subspace.” The Gram-Schmidt orthogonalization process is used for making basis polynomial functions, which is particularly effective when input random parameters do not follow the Askey scheme and/or when a dependence structure between the input parameters exists. Therefore, a nonlinear iso-probabilistic transformation, which makes the convergence of a surrogate to the true model difficult, is not required, unlike traditional PCE. Numerical examples, including limit state functions related to structural dynamics problems, are presented to illustrate the advantages of the proposed method in estimating failure probabilities for complex structural systems. Specifically, the method exhibits significantly improved efficiency in estimating the failure probability of an offshore floating structure without compromising accuracy as compared to traditional PCE and MCS.

A novel single-loop Kriging importance sampling method for estimating failure probability upper bound under random-interval mixed uncertainties

For reliability analysis under wide existing random and interval mixed uncertainties (RIMU), the failure probability upper bound (FPUB), a conservative index to quantify the reliability level of the structure, is the basis for structural safety assessment under RIMU. To accurately and efficiently estimate FPUB, this paper proposes a novel importance sampling method based single-loop Kriging model (IS-SLK), in which the IS variance reduction is combined with the SLK of the performance function, and a new most reduction learning function under RIMU (MRLF-RI) is established. By introducing the IS probability density function (PDF), the proposed IS-SLK firstly expresses the FPUB as the product of the approximate FPUB estimated by current SLK and the correction term, then it designs a strategy of two-step adaptive updating SLK to calibrate FPUB. The innovation of the IS-SLK includes two aspects. Firstly, it constructs a quasi optimal IS PDF to approach the optimal one based on adaptively updating SLK, and the simple reject sampling method is employed to extract the sample of IS PDF, on which the required size of candidate sample pool for obtaining the convergent FPUB can be reduced and the robustness of the algorithm is improved. Secondly, based on the comprehensive consideration of multiple factors, which include the misclassification probability of structural state, joint PDF and Kriging prediction at the IS candidate sample, as well as the logical relationship between the states of random input candidate samples and those at different realizations of interval input, the new learning function named as MRLF-RI is established to adaptively select the training sample contributing most to improve the estimation accuracy of FPUB. Moreover, by deriving the upper bound of the expected relative error, a convergence criterion of MRLF-RI is constructed to further improve the efficiency. Compared with the up-to-date methods, the proposed IS-SLK is more efficient, and its superiority is verified by four examples.

Risk-based maintenance and inspection of riverine flood defence systems

The condition of flood defence revetments is influenced by many different degradation processes such as animal burrowing, rutting and growth of weeds. Many of these processes are shock-based rather than progressive continuous. As shocks can cause a drop in performance, this means that the condition of a revetment can suddenly decrease, meaning that revetments can have significant initial damage at the beginning of a storm. Combined with the limited detection probability of common visual inspections of flood defences, this can have a significant influence on the reliability of flood defence systems, something typically not considered in reliability analysis. In this paper we study the reliability of a flood defence system subject to shock-based degradation. Various maintenance concepts are compared for a case study of a riverine flood defence of 20 kilometres length. This demonstrates that the current maintenance concept is insufficient to satisfy the reliability requirements for failure of the revetment. Overall, the joint influence of degradation and the existing maintenance concept leads to a 20 times higher failure probability estimate compared to a typical assessment without these aspects. Next, we demonstrate that both additional inspections, and targeted interventions to reduce the impact of for instance animal burrowing, can significantly reduce total cost and improve robustness of the considered flood defence system.

SS-MASVM: An advanced technique for assessing failure probability of high-dimensional complex systems using the multi-class adaptive support vector machine

The integrity of structures in contemporary engineering practices is heavily influenced by inherent uncertainties, which necessitates the development of reliable structural reliability analysis methods for uncertainty quantification and probabilistic analysis. This paper investigates the integration of surrogate models and subset simulation (SS) to accurately compute the failure probabilities of high-dimensional complex systems. Focus is placed on enhancing the computational efficiency and accuracy through using the multi-class adaptive support vector machine (MASVM). The proposed SS-MASVM method incorporates more efficient learning functions, integrates K-means clustering, and employs SS technology for precise estimation of failure probabilities of high-dimensional complex systems. Numerical and engineering structural examples validate the effectiveness of the proposed method and highlight its potential in practical reliability analysis, and the results disclose that the SS-MASVM is efficient and accurate for assessing the complex system reliability.

A multi-fidelity stochastic simulation scheme for estimation of small failure probabilities

Computing small failure probabilities is often of interest in the reliability analysis of engineering systems. However, this task can be computationally demanding since many evaluations of expensive high-fidelity models are often required. To address this, a multi-fidelity approach is proposed in this work within the setting of stratified sampling. The overall idea is to reduce the required number of high-fidelity model runs by integrating the information provided by different levels of model fidelity while maintaining accuracy in estimating the failure probabilities. More specifically, strata-wise multi-fidelity models are established based on Gaussian process models to efficiently predict the high-fidelity response and the system collapse from the low-fidelity response. Due to the reduced computational cost of the low-fidelity models, the multi-fidelity approach can achieve a significant speedup in estimating small failure probabilities associated with high-fidelity models. The effectiveness and efficiency of the proposed multi-fidelity stochastic simulation scheme are validated through an application to a two-story two-bay steel building under extreme winds.

Fast convergence strategy for adaptive structural reliability analysis based on kriging believer criterion and importance sampling

The kriging-based adaptive structural reliability analysis method has become widely used, and various learning functions have been proposed. In this study, a fast convergence strategy (FCS) for adaptive structural reliability analysis is proposed based on the Kriging Believer criterion and importance sampling. FCS considers the improvement in the accuracy of the failure probability estimation instead of overemphasizing the approximation accuracy of the limit state function. Contribution of samples to the accuracy of failure probability estimation is quantified based on the Kriging Believer criterion. FCS can implement sequence and parallel additions. The optimal importance sampling function is constructed to further improve the efficiency of the FCS. Several examples are used to demonstrate that FCS can efficiently and accurately handle complex limit state function and the engineering problem of implicit functions.

Reliability-based design optimization using adaptive Kriging-A single-loop strategy and a double-loop one

With the development of the surrogate-assisted Reliability-Based Design Optimization (RBDO) methods in recent decade, efficiency has been continuously improved by various state-of-the-art learning schemes. However, ensuring sufficient accuracy and feasibility at the optimum is still challenging, especially for real scenarios with complex probabilistic constraints and high fidelity. In order to achieve a good balance between efficiency and accuracy and feasibility at the optimum, both a single-loop and a double-loop adaptive Kriging-based RBDO methods are proposed. By directly approximating the probabilistic constraints using Kriging models in the single-loop method, the original RBDO problem is converted into a trivial deterministic optimization, which can be solved by any type of optimization method. The real values of failure probabilities are estimated by Generalized Subset Simulation (GSS) at the initial construction or during each update of the Kriging models. To further raise efficiency in the double-loop adaptive Kriging-based method, GSS is substituted by an inner-loop Kriging-based failure probability estimation, which includes each local enrichment at the limit states corresponding to the active probabilistic constraints based on an initial global enrichment within the augmented reliability space. Once a certain Kriging model meets the requirement during the refining of all, only the others are kept updating until the accuracy of all are accepted. Four examples are used to demonstrate the performance of the proposed two methods.

Hybrid reliability analysis with incomplete interval data based on adaptive Kriging

Hybrid reliability analysis with mixed random and interval uncertainties is a significant challenge in the reliability assessment of engineering structures. The situation will be more intractable when involving incomplete interval data. To obtain reliable estimates of the failure probability limits, an effective parameter estimation method, integrating the quantile variant of the Expectation-Maximization algorithm and Kullback-Leibler divergence, is proposed to transform uncertain variables with incomplete data into random variables with distribution uncertainty. Then, an adaptive Kriging-assisted hybrid reliability analysis method is developed to ensure computational accuracy and efficiency. In this method, a candidate pool incorporating the distribution uncertainty is constructed and its size is adaptively reduced by removing the samples that violate the projection uniformity on input dimensions with exact distributions. Meanwhile, an improved U learning function and an error-based convergence criterion are defined to drive and stop the adaptive process. Then the failure probability limits are estimated by combining the refined Kriging model and Monte Carlo simulation. Four application examples are employed to verify the superiority of the proposed method. Comparison results show that the proposed method can significantly improve computational efficiency while ensuring the accuracy and reliability of the estimated failure probability interval under incomplete interval observations.

Probabilistic machine learning approach for structural reliability analysis

In recent years, the surrogate modeling technique has been increasingly employed for the failure probability estimation of civil structures. The surrogate modeling technique acts as a black box that predicts the performance of expensive-to-evaluate problems. However, the uncertainty of failure probability estimation due to the discrepancy between the surrogate model and physical model has not been well studied. This study introduces a new surrogate modeling technique for failure probability estimation of civil structures that utilizes a Bayesian neural network. Specifically, a fully connected neural network is constructed to approximate the limit state function. The hyperparameters in the neural network are obtained within the Bayesian paradigm, and the probability distributions are estimated through Laplace approximation. When the Bayesian neural network method is used for reliability analysis, the failure curve and the corresponding uncertainty can be estimated. Numerical studies on 2d truss, a simple support beam under moving load, and dynamic analysis of subway station considering the soil–structure interaction are conducted to validate the efficiency of the approach. Results with Monte Carlo simulation and subset simulation are also presented. The results demonstrate the proposed method’s potential for improving prediction accuracy by factoring in the uncertainty embedded in the surrogate model. Additionally, it exhibits higher levels of efficiency than traditional methods.