Assessing small failure probabilities by AK–SS: An active learning method combining Kriging and Subset Simulation

Assessing small failure probability by importance splitting method and its application to wind turbine extreme response prediction

An adaptive parallel learning dependent Kriging model for small failure probability problems

In the practical engineering, to solve the small failure probabilities with correlated high-dimensional variables, the subset simulation (SS) is combined together with the Monte Carlo and importance sampling (IS) method. The samples from the probability density functions (PDF) of the importance sampling are used to construct the intermediate failure events, by which the small failure probabilities are turned into a Markov chain (MC), which is a continuous product made of a series large failure probability or conditional failure probability (CFP) which is easily answered, on which the structural reliability can be efficiently simulated by directly obtaining the samples with correlated ones. Finally, the 3 planet carriers of 3 grade planetary reducers in shield tunneling machine(STM) are as examples to check the algorithm above, the results show that the SS of the IS with correlated variables can highly simulate failure probability.

Structural reliability analysis for small failure probabilities remains a challenging task, despite the significant progress made by the active learning reliability methods (ALRMs) represented by AK-MCS (ALRMs combining adaptive Kriging and Monte Carlo simulations). In order to address this issue, advanced ALRMs with improved computational efficiency than AK-MCS have been proposed, however at the expense of sacrificing the generality and simplicity of AK-MCS. Therefore, a novel ALRM combining adaptive Kriging and spherical decomposition-MCS (AK-SDMCS) was proposed to alleviate the computational cost and preserve the advantages of AK-MCS simultaneously. The key of AK-SDMCS lies in the spherical decomposition-MCS (SDMCS), which not only decomposes the computationally intensive active learning of AK-MCS into a series of easy-to-compute active learning in the divided subsets but also reduces the total number of candidate samples by focusing the sampling region on the most probable failure regions (MPFRs) and adaptively allocating samples in the divided subregions. As a result, AK-SDMCS significantly alleviates the computational cost of active learning of Kriging model. Since AK-SDMCS is built in the framework of AK-MCS, the advantages of AK-MCS can be preserved. Besides, the active learning strategy is modified to further increase the computational efficiency. Thus, AK-SDMCS provides an efficient, easy-to-implement and generalized option for reliability analysis of small failure probabilities. The application and efficiency of AK-SDMCS are demonstrated by three academic examples.

An improved adaptive kriging-based importance technique for sampling multiple failure regions of low probability

Assessing small failure probabilities by combined subset simulation and Support Vector Machines

An enhanced method for improving the accuracy of small failure probability of structures

Methodical aspects of the reliability-based structural optimisation using stochastic quasigradient methods are considered. For an example of the simply supported reinforced concrete beam, the employment of the Lagrange multiplier method that belongs to the class of stochastic quasigradient methods is demonstrated. The classical optimum design goal to minimise structural cost or weight under the constraint on the structural failure probability is taken for consideration. Optimisation problems solved with the Lagrangemultiplier method are formulated in form of general stochastic programming problem. The mathematical expectation of the concrete volume reduced with respect to the in-place cost of the beam materials is taken as the objective function. Constraint function is the limitation placed on the beam failure probability. The beam is considered as a series structural system. Values of the prescribed allowable failure probability belongs to the interval in which the estimation of the failure probabilities by the simple Monte-Carlomethod is possible with an acceptable confidence. The time-independent case as well as the time-dependent one is considered in the optimisation problems. The generalisation on the time-dependent case is undertaken through the introduction into the constraint function of the quasi-linear distribution law of the random variables. In the time-dependent case, the objective function is associated with beginning and the constraint function with end of the service period. An expression of the stochastic gradient based on the differentiation under the integral sign is used for calculations with the Lagrange multiplier method. The stochastic gradient used is computationally more effective in comparison with stochastic finite-difference formulae usual in stochastic quasigradient methods because it requires only one computation of the structure in search iteration of the optimisation process. Three rules based on statistical argumentation are used for the stopping of the seat according to the procedure of the Lagrange multiplier method. The optimising of the beam shows that the Lagrange multiplier method is applicable for the optimal design of structures in that cases when the structural reliability can be estimated by means of the simple Monte-Carlo method. Additional research is needed for integration in the Lagrange multiplier method of statistical simulation techniques for the estimation of small structural failure probabilities.

Modified subset simulation method for reliability analysis of structural systems

Structural reliability is an engineering method that focuses on the calculation and prediction of the probability of failure for a structural system. Structural reliability is used by many pipeline operators to assess the probability of failure, and these probability of failure estimates are then used to manage safety, environmental, and operational risks for threats such as corrosion, cracking, and third-party damage. In the structural reliability approach, the probability of failure is obtained from a multi-dimensional integral, for which the solution is commonly estimated numerically using Direct Monte Carlo (DMC) simulation. DMC is straight-forward and robust but requires a significant amount of computational effort to estimate small probabilities, which are typical of threats to pipeline integrity. Subset Simulation is an approach that improves on the efficiency issues of DMC by representing the rare event probability as the product of a number of more frequent events, which are each estimated separately. Previous work published at IPC in 2020 by Bandstra and Fraser [1] showed very close agreement between DMC and Subset Simulation for a single defect reference case with a single limit state. This study extends that initial work by applying Subset Simulation to various failure models for a variety of pipeline defects. Subset Simulation is applied to the CSA corrosion model, PRCI MAT-8 crack model, and the EPRG dent failure model, and the performance is evaluated by comparing the results to DMC. For each of these comparisons, simulations are performed across a large-scale grid of validation cases that consider a range of pipeline and defect sizes. An approach that utilizes Subset Simulation to handle multiple limit states is also presented and applied for relevant failure models and the results are evaluated against DMC.

Profust reliability analysis is investigated in this article. The main aim is to address problems with very small failure probabilities. First, a single-layer multimodal importance sampling (MMIS) method based on the most probable points (MPPs) in the fuzzy failure region is proposed. Secondly, to reduce function calls, an adaptive method fusing kriging model and MMIS is developed. In this method, a surrogate fuzzy failure region (SFFR) based on the kriging prediction information is defined. The MPPs in the SFFR are explored, the important samples are generated, the optimal training point is chosen, and the kriging model and the SFFR are updated. The learning process is executed iteratively until the SFFR converges to the true one. To find all MPPs and maintain unbiasedness, an intelligent multimodal optimization method based on multi-objective optimization is introduced. Four case studies are used to verify the advantages of the proposed method.

On the evaluation of multiple failure probability curves in reliability analysis with multiple performance functions

SummaryPolynomial chaos expansions (PCEs) have been widely employed to estimate failure probabilities in geotechnical engineering. However, PCEs suffer from two deficiencies: (a) PCE coefficients are solved by the least‐square minimization method which easily causes overfitting issues; (b) building a high order PCE is often computationally expensive. In order to overcome the aforementioned drawbacks, the Bayesian regression technique is employed to evaluate PCE coefficients, which not only provides a sparse solution but also avoids overfitting. With the aid of the predictive means and variances given by Bayesian analysis, a learning function is proposed to sequentially select the most informative samples that are critical to build a PCE. This sequential learning scheme can highly enhance the computational efficiency of PCEs. Besides, importance sampling (IS) is incorporated into the sequential learning (SL)‐PCEs to deal with geotechnical problems with small failure probabilities. The proposed method of SL‐PCE‐IS is applied to three illustrative examples, which shows that the improved PCE method is more effective and efficient than the common PCEs method, leading to accurate estimations of small failure probabilities using fewer training samples.

A layer assigned probability space partition method for structural small failure probability problem

AK-SEUR: An adaptive Kriging-based learning function for structural reliability analysis through sample-based expected uncertainty reduction