Estimation Of Failure Probability Research Articles

An ensemble model‐based method for estimating failure probability function with application in reliability‐based optimization

The failure probability function (FPF) is a function of failure probability that varies with distribution parameters of random inputs, and is required in reliability-based optimization. To estimate the failure probability at each possible value of distribution parameters, it is necessary to conduct a great quantity of model evaluations. This results in a huge computational burden especially in practical engineering applications. The introduction of surrogate models effectively improves the computational efficiency by constructing a cheap-to-evaluate metamodel to approximate the true performance function. However, it is tough to determine which surrogate model is a prior when the performance function is extremely complex or even implicit. Based on the Bayes’ rule and augmented theory, this paper proposes an ensemble model method for estimating the FPF. Since the ensemble model utilizes each individual surrogate's capacity to predict outcomes by using a weighted form to combine numerous surrogate models and eliminates the trouble of seeking a proper individual model, it possesses better estimation accuracy and generalization ability than the individual surrogate model. Several examples have proved that the proposed ensemble model method for estimating the FPF not only further improves the computational efficiency but also gets sufficiently accurate estimation. Finally, the reliability-based optimization of a hydraulic pipeline system under random excitation is solved effectively by the proposed method.

Probabilistic Evaluations of Prescribed Safety Margins in Eurocode 7 for Spread Foundations

According to the design code Eurocode 7, analysis procedures require reaching a prescribed safety margin, based on the conditions of levelling design action and design resistance. Such semi-probabilistic procedures do not result in a consistent equivalent value of the Overall Factor of Safety (OFS), neither in individual analysis nor in different tasks in geotechnical engineering. Furthermore, the implementation of different calculation approaches in Eurocode 7 also does not guarantee an equal probability of the occurrence of a relevant limit state. A comparative analysis is conducted for an example of a centrically loaded spread foundation on homogenous, isotropic, and coarse-grained soil, according to procedures in Eurocode 7, Design Approach 3. An algorithm is developed to estimate failure probability, taking into consideration the relevant statistical characteristics of each calculation parameter. A significant influence of the statistical characteristics of the relevant sample is emphasized. The degree of required modification of the equivalent Overdesign Factor (ODF) and the Overall Factor of Safety (OFS), based on the criterion of the required reliability index β and failure probability pf , is quantified.

Stochastic reduced order modelling and analysis of rotating bladed discs

A computational finite-element (FE)-based technique is proposed for developing a stochastic reduced order model for rotating bladed disc with spatial random inhomogeneities. The spatial inhomogeneities imply the system to be randomly mistuned. The formulation assumes the availability of a high fidelity FE model for the tuned system. The corresponding FE matrices are antisymmetric on account of the Coriolis forces due to rotation. The spatial inhomogeneities, available from limited point measurements on the blades, are modelled as non-Gaussian random fields with arbitrary distributions. A low order stochastic computational model is developed by projecting the FE model onto a reduced dimensional state space defined in terms of specified observable nodal points and expressing the stochasticity through an arbitrary polynomial chaos (aPC) basis. This model enables probabilistic quantification of the variabilities in the system response and estimating failure probabilities. The methodology enables drastic reduction in the state space and stochastic dimensions, addresses the practical difficulties with having limited measurable data points, antisymmetric FE matrices, aPC representation in complex irregular geometries and carrying out probabilistic analyses on industrial systems, at significantly reduced computational costs. The methodology is illustrated through an academic rotor and an industrial rotor blade.

Entropy-based adaptive design for contour finding and estimating reliability

In reliability analysis, methods used to estimate failure probability are often limited by the costs associated with model evaluations. Many of these methods, such as multifidelity importance sampling (MFIS), rely upon a computationally efficient surrogate model like a Gaussian process (GP) to quickly generate predictions. The quality of the GP fit, particularly in the vicinity of the failure region(s), is instrumental in supplying accurately predicted failures for such strategies. We introduce an entropy-based GP adaptive design that, when paired with MFIS, provides more accurate failure probability estimates and with higher confidence. We show that our greedy data acquisition strategy better identifies multiple failure regions compared to existing contour-finding schemes. We then extend the method to batch selection, without sacrificing accuracy. Illustrative examples are provided on benchmark data as well as an application to an impact damage simulator for National Aeronautics and Space Administration (NASA) spacesuits.

Bayesian updating of failure probability curves with multiple performance functions of nonlinear structural dynamic systems

System failure often involves multiple failure modes which require considering multiple performance functions. Based on measured system response data, Bayesian updating of multiple failure probability curves needs to be performed. In this paper, a new approach based on extending ISS is proposed which can update multiple failure probability curves by simultaneously considering all the multiple performance functions in one run. A new scheme of how the generated samples are used is proposed to improve the estimation of failure probability curves. Discontinuity issues on the failure curves in ISS are resolved by the proposed method. The proposed method is applied to two illustrative examples involving uncertain model parameters and nonlinear structural dynamic systems subjected to future uncertain earthquake excitations. The computational efficiency of the proposed method is compared with direct application of Subset Simulation. Significant computational savings are observed while the coefficient of variation of the failure probability estimators is kept at the same level.

Enhanced Adaptive Kriging Combined Adaptive Radial-Based Importance Sampling Method for Reliability Analysis

The surrogate model-based sampling method has been an active area of research due to its evident efficiency over direct sampling method. The fusion of the adaptive kriging (AK) model with the adaptive radial-based importance sampling (ARBIS) method is developed as an improvement of the AK model combined with Monte Carlo simulation (MCS) method especially for estimating the small failure probabilities. In the AK–ARBIS method, the number of candidate samples in each updating process of the kriging model is reduced through partitioning the whole MCS sample domain into different subdomains and removing the samples inside the last subdomain from the participating candidate sampling pool, which can avoid the time-consuming matrix operation involved in kriging predictions in order to find the best training sample in each updating step. In the existing AK–ARBIS method, kriging model stops updating according to the learning function-based stopping criterion, which cannot directly reflect the error of the estimated failure probability. In this regard, this paper proposes the error-based stopping criterion (ESC) enhanced AK–ARBIS method, where the processes of finding the optimal hypersphere and accurately identifying the sign of candidate samples are decoupled in order to successfully embed the ESC. In addition to embedding the ESC into the AK–ARBIS method, two target-oriented candidate sampling reduction strategies are investigated, respectively, to improve the efficiency of finding the optimal hypersphere and accurately identifying the sign of the participating candidate samples. Based on the candidate sampling pool reduction strategies and ESC, both the computational time and the required number of calls to the real limit state function are reduced over the existing AK–ARBIS method. Three case studies are analyzed to demonstrate the accuracy and efficiency of the proposed enhanced AK–ARBIS method, especially for estimating the small failure probabilities.

Industrial performance, reusability and mechanical reliability of mesoporous gamma alumina packed bed fabricated through boehmite extrusion for removal of reactive dyes from textile wastewaters

The reusable γ-alumina packed bed was fabricated by extrusion of boehmite as rods for the removal of reactive dyes from the industrial textile wastewaters in a dynamic system. The appropriate calcination temperature, and time were determined to maximize the removal efficiency, ∼90%, in the acidic environments, pH 2–3. On the other hand, the current contribution addressed the estimation of failure probability which is a challenge for the reliable design of packed beds. Therefore, the normal, and Weibull distributions were used to evaluate the reliability of rods through Akaike information criterion (AIC), and Anderson–Darling test (ADT). The results revealed that the Weibull distribution possesses the higher accuracy in the analysis of compression strength scatter. The calcination of rods at 850 °C within 90 min not only led to fabricate a reliable packed bed with a mean strength about 860 kPa but also brought the better mesoporous structure, 8 nm, which is the main reason for the development of active sites. The thermal stability of rods provided an industrial potential for the regeneration of packed bed at 500 °C, without efficiently losing the adsorptive performance, even after reuse for ten times.

BAYESIAN NETWORKS FOR RELIABILITY ANALYSIS OF TECHNICAL SYSTEM

The works to provide reliability assurance and enhancement are most effective at the early stages of life cycle. The common requirement is fast and precise performance of reiterated reliability analysis, as far as the system design development and the complete data accumulation on system components reliability. Recently, Bayesian networks has been introduced in reliability engineering. Bayesian networks are a promising tool for reliability analysis. Today, Bayesian networks are actively used to fault diagnosis and reliability estimates. In this article, utilization of the Bayesian networks for reliability analysis at the early stages of life cycle is studied. The reliability can be calculated by converting block diagrams or fault trees into Bayesian networks. Fault trees can easily be converted into Bayesian networks. Therefore, in this article, a perspective approach to combine the Bayesian networks mechanism and Fault tree analysis method is studied for reliability assessment of a technical system. The system is an electronic fire sensor. As a result, the developed model of Bayesian networks provides the estimation of failure probability at the conditions of real-life scenarios of system behavior; the calculation of the probability of no-failure operation of the system under control and its sub-systems; the detection of the elements that contribute the most to the failure probability. The Bayesian networks model can be easily built from the circuit diagram analysis and the FMEA (Failure Modes and Effects Analysis). In general, this method has the best capability in this real instance compared to other methods.

Statistical estimation of random failure probability based on stress levels

Statistical estimation of random failure probability based on stress levels

Statistical estimation of random failure probability based on stress levels

Statistical estimation of random failure probability based on stress levels

Cross-entropy-based directional importance sampling with von Mises-Fisher mixture model for reliability analysis

In structural reliability analysis, the estimation of rare failure probability is a significant challenge. Directional importance sampling may address this challenge when a quasi-optimal sampling density is chosen. Directional importance sampling involves sampling on the unit hypersphere and one-dimensional reliability analysis. This paper develops a cross-entropy-based directional importance sampling (CE-DIS) for reliability analysis. The von Mises-Fisher mixture (vMFM) model is selected as sampling density to generate direction vectors on the unit hypersphere. The updating rules based on the cross-entropy method are derived for the vMFM model to find a quasi-optimal sampling density. In addition, the DBSCAN algorithm is used to choose the number of distributions in the vMFM model in case of no prior information of failure domain, and the intermediate failure event is introduced in case of not enough samples to update the distribution parameters. Several benchmark numerical examples and engineering applications are used to test the performance of the proposed CE-DIS method. The results demonstrate that the CE-DIS method significantly improves the computational efficiency compared to original directional sampling and other CE-based importance sampling methods. In addition, the CE-DIS method can be applied to different types of reliability problems, e.g., multiple design points, multiple failure modes and rare failure probability.

Probabilistic parameter estimation and reliability assessment of a simulated sheet pile wall system

This paper investigates the impact of structural health monitoring (SHM) on the failure probability of a simulated sheet pile wall system. We use Bayesian statistics to estimate non-directly observable soil parameters from measured structural responses. These estimates are used to calculate the failure probability of the structure against serviceability and ultimate limit states before and after measurements. We study the impact of sensor installation time and the contribution of various sensor types to the change in failure probability. Nested sampling is used to compute the posterior distributions and to estimate the failure probabilities. The results show that SHM largely decreases the failure probability estimates compared to the prior value. The largest failure probability reduction is obtained when the monitoring starts right after the sheet pile wall is driven into the ground: about 1017× and larger reductions for the considered ultimate limit state. With each delay in the sensor installation time, the failure probability reduction can substantially decrease, in some cases 103× and larger reductions are observed. The results imply that SHM coupled with Bayesian statistics is a promising approach to substantially reduce our uncertainty in modeling hydraulic structures and, in turn, to increase their calculated safety.

Integrated approach for pipe failure prediction and condition scoring in water infrastructure systems

Pipe failures in water distribution infrastructure have significant economic, environmental, and public health impacts. To alleviate these impacts, pipe deterioration modeling has been increasingly implemented to characterize and predict pipe failure patterns with the aim of prioritizing repair and replacement decisions. Logistic regression has been recognized in recent literature as a strong candidate for failure prediction modeling. However, previous studies have often been limited to demonstrating the application of logistic regression for estimating failure probabilities. This study builds on previous efforts by proposing an approach for implementing logistic regression into a holistic framework for asset management decision-making. This framework incorporates logistic regression modeling, with a flexible time-interval, into a practical condition scoring methodology that accounts for the attitude of water utilities towards risk. The developed framework is demonstrated and tested on a 20-year pipe failure dataset of a large metropolitan US city. The logistic regression model displayed high accuracy in estimating the probability of failure within different time intervals, and the scoring method showed a reasonable ability to predict the criticality of repair decisions for pipes based on their condition.

AKSE: A novel adaptive Kriging method combining sampling region scheme and error-based stopping criterion for structural reliability analysis

The reliability analysis of complex structures usually involves implicit performance function and expensive-to-evaluate computational models, which pose a great challenge for the estimation of failure probability. In this paper, an adaptive Kriging-based method is proposed for the efficient estimation of failure probability with high accuracy. Starting from a small set of initial design of experiments (DoE), a Kriging model is constructed and iteratively refined by adding judiciously selected sample points to the DoE. To effectively select more sample points for updating the Kriging model, a new learning function is proposed for the selection of representative samples following the idea of the penalty function method in optimization. Besides, the inclusion of the term that measures the distance between the candidate samples and those in DoE controls the density of samples, and thus enables efficient exploration of the regions of interest. To improve the efficiency of the algorithm, this learning function is integrated with a sampling region scheme to filter out sample points in regions with rather low probability density from the candidate sampling pool. Moreover, a convergence criterion based on the error-based stopping criterion is developed to terminate the learning process at an appropriate stage. Hence, the proposed method is referred to as the Adaptive Kriging method combining Sampling region scheme and Error-based stopping criterion (AKSE). To further explore the possible schemes for the improvement of the proposed method, the combinations of AKSE with a dynamic Kriging method and the bootstrap variance are also investigated. For rare failure events, the Monte Carlo simulation (MCS) in AKSE is replaced by the importance sampling (IS) to establish an improved version of AKSE, namely the AKSE-IS. The performance of the proposed algorithm is evaluated through five numerical examples and the results demonstrate the superior performance of the proposed method both in terms of accuracy and efficiency.

Structural dominant failure modes searching method based on deep reinforcement learning

The dominant failure modes (DFMs) of a structural system are significant for structural analysis and failure probability estimation. However, existing failure modes (FMs) searching methods often face problems of combinatorial explosion. To address this issue, a deep reinforcement learning (DRL)-based method is proposed for DFMs searching, which transforms the probability-based failure component selection process into a sequential decision process. First, the failure stages and the selected failure components of a structural system are transformed to be the states and actions in the DRL. Second, a deep neural network (DNN) is established to observe the failure stages and select failure components. Finally, a new reward function is designed to guide the network to learn the failure component selection policy. The proposed method was tested through a roof truss structure and a truss bridge structure. It was demonstrated that the trained DNN could learn to observe the failure stages and select the most critical components in a completely unknown testing set. High accuracy of the identified DFMs can be achieved. In comparison with the calculation results of Monte Carlo Simulation (MCS) and β-unzipping method, this proposed method shows significant computational efficiency advantages with high accuracy in dealing with combinatorial explosion.

An improved AK-MCS for reliability analysis by an efficient and simple reduction strategy of candidate sample pool

Adaptive Kriging combined with Monte Carlo simulation (AK-MCS) is popularly used to estimate the failure probability, but its candidate sample pool (CSP) size should be large enough to guarantee the accuracy of failure probability estimation, especially for small failure probability. To select training point for updating the Kriging model, AK-MCS needs to traverse the CSP to calculate the U-learning function of every point, which is time-demanding. Thus, an enhanced AK-MCS (shorten as EAK-MCS Ⅰ) is proposed by use of an efficient CSP reduction strategy. In EAK-MCS Ⅰ, the candidate samples are classified into two types, i.e., the samples with accurately judged states and the other samples. Obviously, removing the first type samples from CSP does not need extra computational cost, which has no effect on the precision of estimating failure probability and can improve the efficiency of training Kriging model. Hence, EAK-MCS Ⅰ is more efficient than AK-MCS in estimating failure probability. Additionally, for the risk of misjudging the states of the samples by the Kriging model trained in the reduced CSP for complicated problem, another version of EAK-MCS Ⅰ (shorten as EAK-MCS Ⅱ) is proposed by nesting an adaptive strategy to calibrate the Kriging model. The proposed methods improve the efficiency of AK-MCS and inherit all advantages of AK-MCS.

Application of the Gaussian process for fatigue life prediction under multiaxial loading

Fatigue life prediction with failure probability estimation for materials subjected to multiaxial loading is an important task in engineering design because it reduces the financial cost by eliminating expensive experimental tests. Owing to the complex deterioration mechanisms of fatigue failure, existing fatigue life prediction models are of the empirical or semi-empirical type, and their applicability is limited to validating the loading condition and material. Improper selection of the fatigue model could result in non-conservative life prediction with high financial and catastrophic consequences. To solve this problem, an innovative approach based on the Gaussian process for fatigue life prediction under multiaxial loading is presented. The inherent features of the Gaussian process predispose its application to fatigue life prediction under multiaxial loading as an efficient and practical approach to avoid the problem in selecting adequate semi-empirical parametric fatigue models. The model was validated on two sets of experimental data obtained by fatigue testing of S355N steel and 2124 T851 aluminum alloy under uniaxial and multiaxial loadings. In the model training process, only data obtained under uniaxial and pure torsion cyclic loadings were applied. Owing to the implemented physics-based input data, which are related to the stress components acting on the critical plane of crack initiation, the model accurately predicted the fatigue lives of the two tested materials by implementing the squared exponential covariance function. In addition, the fatigue lives were also computed using four parametric fatigue strength criteria (Crossland, Matake, Carpinteri et al., and Papuga–Růžička models). The best results obtained by the parametric models exhibited a lower fatigue prediction performance than the results using the Gaussian process-based model.

Robust optimization of a dynamic Black-box system under severe uncertainty: A distribution-free framework

In the real world, a significant challenge faced in designing critical systems is the lack of available data. This results in a large degree of uncertainty and the need for uncertainty quantification tools so as to make risk-informed decisions. The NASA-Langley UQ Challenge 2019 seeks to provide such setting, requiring different discipline-independent approaches to address typical tasks required for the design of critical systems.This paper addresses the NASA-Langley UQ Challenge by proposing 4 key techniques to provide the solution to the challenge: (1) a distribution-free Bayesian model updating framework for the calibration of the uncertainty model; (2) an adaptive pinching approach to analyse and rank the relative sensitivity of the epistemic parameters; (3) the probability bounds analysis to estimate failure probabilities; and (4) a Non-intrusive Stochastic Simulation approach to identify an optimal design point.

Assessment of the probability of the fatigue fracture of structural components subjected to deterministic and stochastic loading taking into account the scatter in the initial crack size

An analytical approach which provides conservative estimates of the probability of fatigue brittle fracture of structural components of technical systems taking into account the scatter of the initial dimensions of crack-like defects described by the exponent probabilistic distribution is presented. The operational loading is considered both as deterministic (with loading cycles of constant amplitude and frequency) and random (stationary narrow-band Gaussian random loading) process. The crack growth kinetics is described by the modified Paris equation that takes into account the stress ratio effects. The parameters of the Paris law are considered deterministic values. An example of the assessment of the probability of fatigue failure of tubular structural component with an axial crack on the inner surface of a tube subjected to internal pressure is presented. A comparative analysis of the results obtained with and without taking into account the random nature of the operational loading is carried out. It is shown that neglection of the random nature of the operational loading leads to non-conservative estimates of the fatigue failure probability, which may differ by an order of magnitude compared to the calculation data obtained with allowance for the stochastic nature of the loading process. The developed method can be used in the implementation of probabilistic and risk-based approaches to ensuring strength, service life and safety of technical systems in real operation conditions and in adjusting standard operation programs in terms of choosing the frequency and scope of non-destructive testing.

Performance based assessment of transmission lines to seismic events

Typical transmission line steel lattice towers are not designed for seismic excitations, based on the assumption that their ultimate limit state is governed by climatic actions (wind and ice). However, it is not unusual for line inspection reports to indicate the observation of moderate to severe structural damages after intense ground motions, which can ultimately reduce the capacity of tower elements. Performance-Based Earthquake Engineering (PBEE) assessment is a modern technique to earthquake resistant design frequently adopted for building frame structures, but not yet employed in the design of transmission line supports, to the best of the authors’ knowledge. The present article evaluates the seismic performance of two typical 230kV transmission towers in a PBEE framework, considering artificial and actual ground motions based on Chilean seismicity. Time-domain nonlinear dynamic analysis is employed to numerically obtain tower responses considering distinct earthquake loading scenarios according to two model configurations: stand-alone and tower-line section. Significant structural damage was detected even for moderate ground motions, which might reduce the tower capacity to future climatic events. The estimated failure probability for the ultimate limit state was found greater than the recommended values in the standard IEC 60826 (2017).