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The reliability analysis results of complex structures based on the Kriging model were highly dependent on the fitting accuracy of the Kriging model. In the constructing processes of the Kriging model, the selection of different correlation and regression functions affected the accuracy of the model. In order to solve the impacts of model uncertainty on the reliability analysis results, while considering computational efficiency and accuracy, based on the Kriging model and Monte Carlo simulation(MCS) method, a structural reliability calculation method combining adaptive ensemble strategy and active learning function was proposed. Considering the modeling uncertainty of Kriging models, combined with multiple Kriging models, this method constructed an active learning function that comprehensively considered sample point contribution and sample point distance. The ensemble Kriging model was iteratively updated through the active learning function until the convergence conditions were satisfied. Finally the structural reliability analysis was carted out by the constructed ensemble Kriging model and MCS method. The validity of the proposed method was vetted by numerical and engineering examples, and the results show that the proposed method more robust than other major methods, and the computational efficiency higher while ensuring the computational accuracy. [ABSTRACT FROM AUTHOR]

Strongly nonlinear structural systems exhibit high computational errors when dependability is calculated using conventional approaches such as the primary second-order method of moments and the secondary second-order method of moments. The combination of the proxy model and Monte Carlo simulation is an effective method to solve the structural failure probability problem. However, existing studies on the active learning methods of proxy models for reliability calculation mainly focus on the kriging model, while for radial basis interpolation, the existing research results are relatively few. Based on the above analysis, this paper proposes to combine the cross-validation method with multiple kernel functions to evaluate the uncertainty at the prediction points. The mathematical expression of the active learning function considering three factors is proposed: the linear combination of the distance from the surface of limit state and the uncertainty of the predicted value of the proxy model as the optimization objective function, and the distance between the sample to be selected and the initial sample point as the constraint condition. Meanwhile, using the idea of the penalty function, the constrained problem is transformed into the unconstrained problem to get the final active learning function PLF. Finally, the efficiency, accuracy, and robustness of the PRBFM method are verified by classical cases and compared with other methods. It provides a new method and a new idea for the reliability analysis of complex structures. [ABSTRACT FROM AUTHOR]

In the existing method combining adaptive Kriging and fuzzy simulation (AK–FS) for estimating the failure possibility, the AK is required to correctly identify the state of each sample point in the candidate sample pool of FS. However, the states of some sample points have no contributions to the accuracy of estimating the failure possibility, and identifying their states by the existing AK–FS only results in a loss of efficiency. Therefore, this paper proposes a new probability learning strategy to improve the efficiency of the existing AK–FS. According to the theory of failure possibility, when taking the product of the failure domain indicator function and the joint membership function as the failure pointer of the point, failure possibility is the maximum, and the corresponding point is called fuzzy design point. Based on the probabilistic prediction property of Kriging, the proposed strategy establishes a global strategy. The probability of the failure indicator at arbitrary candidate point greater than the predicted maximum failure pointer is analytically derived for the single and multiple failure modes, respectively, and this probability represents the degree of the point to be fuzzy design point, thus it can be used to measure the contribution of the point to improve the accuracy. By adaptively adding the points with the largest contribution to the training sample set, the real failure possibility can be approximated gradually. The efficiency and accuracy of the proposed strategy are verified by several examples. [ABSTRACT FROM AUTHOR]

The Bayesian support vector regression (BSVR) metamodel is widely used in various engineering fields to analyze the uncertainty arising from uncertain parameters. However, the accuracy of the BSVR metamodel based on the traditional one-shot sampling method fails to meet the requirements of the uncertainty analysis of complex systems. To this end, an error-pursing adaptive uncertainty analysis method based on the BSVR metamodel is presented by combining a new adaptive sampling scheme. This new sampling scheme was improved by a new error-pursuing active learning function that is named, herein, adjusted mean square error (AMSE), which guides the adaptive sampling of the BSVR metamodel's design of experiments (DoE). During the sampling process, AMSE combines mean square error and leave-one-out cross-validation error to estimate the prediction error of the metamodel in the entire design space. Stepwise refinement of the metamodel was achieved by placing the sampled regions at locations with large prediction errors. Six benchmark analytical functions featuring different dimensions were used to validate the proposed method. The effectiveness of the method was then further illustrated by a more realistic application of an overhung rotor system. [ABSTRACT FROM AUTHOR]

Structural reliability analysis has an inherent contradiction between efficiency and accuracy. The metamodel can significantly reduce the computational cost of reliability analysis by a simpler approximation. Therefore, it is crucial to build a metamodel, which achieves the minimum simulations and accurate estimation for reliability analysis. Aiming at this, an effective adaptive metamodel based on the combination of radial basis function (RBF) model and Monte Carlo simulation (MCS) is proposed. Different shape parameters are first used to generate the weighted prediction variance, and the search for new training samples is guided by the active learning function that achieves a tradeoff of (1) being close enough to limit state function (LSF) to have a high reliability sensitivity; (2) keeping enough distance between the existing samples to avoid a clustering problem; and (3) being in the sensitive region to ensure the effectiveness of the information obtained. The performance of the proposed method for a nonlinear, non-convex, and high dimensional reliability analysis is validated by three numerical cases. The results indicate the high efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a novel MRGP-SS method is proposed to deal with the reliability analysis problems under multiple failure modes. First, a random moving quadrilateral grid sampling (RMQGS) method is proposed to improve the randomness and uniformity of initial samples. Second, an adaptive procedure, which combines the multiple response Gaussian process (MRGP) model and the novel active learning functions, is proposed to efficiently and accurately produce surrogate models for failure surfaces. In this regard, two novel learning functions are introduced to adapt to different iterative cycles, one is employed to correct the quality of samples, and the other is used to search for the samples closest to the limit state surface. Third, the subset simulation (SS) is integrated into the adaptive MRGP model to estimate the failure probability under multiple failure modes with fewer function calls and time consumption. Numerical and engineering case studies are finally provided to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

To enhance the efficiency of slope reliability analysis, this paper presents an improved active learning Kriging method integrated with the Monte Carlo simulation. The turbulent flow of water-based optimization (TFWO) algorithm is used to estimate the correlation parameter of Kriging method, enhancing the accuracy of the Kriging surrogate modeling (KSM) used to approximate the complex and implicit performance functions. An active learning procedure is applied to update the KSM with the aid of the best sample points determined by the active learning function. For illustration and validation, three cases with different layered soil are examined using the improved method. The results show that, compared with other reliability methods, the improved method not only has better computational efficiency, but also maintains the accuracy. For the complex multiple-layered slope problems, accurate results can be obtained in only 0.19 h. Moreover, the active learning function EFF and U have a fine performance for slope reliability compared to other functions used in reliability evaluation. [ABSTRACT FROM AUTHOR]

• A new reliability analysis method via approximating parallel strategy is proposed. • A novel kriging updating strategy is developed by integrating an intermediate model. • Sample point acceptance and rejection domains are set by an improved u function. • The performance of the new method is verified by one attitude control engine case. With the ever-increasing complexity of engineering problems, active learning functions fused with Kriging models are receiving significant attention and are extensively applied in various reliability analysis methods. It is argued unfavorably that the existing fusion-type methods usually obtain only a single optimal sample point during model updating process. Meanwhile, the information of the used sample points is not sufficiently utilized. To these gaps, this study proposes an adaptive Kriging reliability analysis method based on an approximate parallel computing strategy, namely AP-AK for short. To start with, a new Kriging model update strategy is proposed, where the first selected sample points help to introduce the intermediate model and further search for optimal points closer to the limit state surface. An improved learning function is then proposed to limit the locations of points based on the information of used sample points. The new function is characterized by setting the acceptance and rejection domains of the sample points, which prevents computational waste incurred by local sample point aggregation. Superior performance of the AP-AK method is demonstrated against other Kriging-based methods through five numerical cases and one solid attitude orbit control engine case. The comparison results show that the proposed method exhibits high efficiency and robustness for solving complex reliability analysis problems. [ABSTRACT FROM AUTHOR]

• This study proposes an AK-Gibbs method for small failure probabilities with nonlinear and time-consuming performance function. • EALF function directly linked to global error is constructed. • IEALF function based on proposed EALF and joint probability density function is proposed. • Gibbs importance sampling algorithm is derived based on Gibbs algorithm aims to effectively establish the candidate importance sample pool. • The AK-Gibbs is more efficient and accuracy than other prevailing methods for small failure probabilities. In engineering practices, it is a time-consuming procedure to estimate the small failure probability of highly nonlinear and dimensional limit state functions and Kriging-based methods are more effective representatives. However, it is an important challenge to construct the candidate importance sample pool for Kriging-based small failure probability analysis methods with multiple input random variables when the Metropolis-Hastings (M-H) algorithm with acceptance-rejection sampling principle is employed. To address the challenge and estimate the reliability of structures in a more efficient and accurate way, an active learning Kriging model based on the Gibbs importance sampling algorithm (AK-Gibbs) is proposed, especially for the small failure probabilities with nonlinear and high-dimensional limit state functions. A new active learning function that can be directly linked to the global error is first constructed. Weighting coefficients of the joint probability density function in the new active learning function are then determined to select the most probable points (MPPs) and update samples efficiently and accurately. The Gibbs importance sampling algorithm is derived based on the Gibbs algorithm to effectively establish the candidate importance sample pool. An improved global error-based stopping criterion is finally constructed to avoid pre-mature or late-mature for the estimation of small failure probabilities with complicated failure domains. Two numerical and four engineering examples are respectively employed to elaborate and validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Slope reliability analysis can effectively account for uncertainties involved in a slope system. However, commonly used slope reliability analysis methods often require huge computational cost, especially in large-scale problems, which hinders its wide application to engineering practice. This paper proposes an efficient slope reliability analysis method based on the active learning support vector machine (SVM) and Monte Carlo simulation (MCS). The proposed method makes use of an active learning function and cross-validation techniques to select the most suitable training samples to update the SVM model. The selected training samples are associated with a small distance to the limit state surface of the slope stability model and a large local uncertainty, which are more informative to gradually tune the SVM model to approximate the actual slope performance function. As a result, the proposed method can estimate the slope reliability with a small number of evaluations of the slope performance function, thus improving the efficiency significantly. Four slope examples are employed to demonstrate the effectiveness of the proposed method. The presented approach is also compared with some other commonly used surrogate models in slope reliability analysis. It is shown that the proposed method performs better in terms of computational efficiency to obtain similar estimation accuracy of the failure probability for the investigated examples. [ABSTRACT FROM AUTHOR]

The Bayesian support vector regression (BSVR) metamodel is widely used in various engineering fields to analyze the uncertainty arising from uncertain parameters. However, the accuracy of the BSVR metamodel based on the traditional one-shot sampling method fails to meet the requirements of the uncertainty analysis of complex systems. To this end, an error-pursing adaptive uncertainty analysis method based on the BSVR metamodel is presented by combining a new adaptive sampling scheme. This new sampling scheme was improved by a new error-pursuing active learning function that is named, herein, adjusted mean square error (AMSE), which guides the adaptive sampling of the BSVR metamodel’s design of experiments (DoE). During the sampling process, AMSE combines mean square error and leave-one-out cross-validation error to estimate the prediction error of the metamodel in the entire design space. Stepwise refinement of the metamodel was achieved by placing the sampled regions at locations with large prediction errors. Six benchmark analytical functions featuring different dimensions were used to validate the proposed method. The effectiveness of the method was then further illustrated by a more realistic application of an overhung rotor system.

In this paper, a novel MRGP-SS method is proposed to deal with the reliability analysis problems under multiple failure modes. First, a random moving quadrilateral grid sampling (RMQGS) method is proposed to improve the randomness and uniformity of initial samples. Second, an adaptive procedure, which combines the multiple response Gaussian process (MRGP) model and the novel active learning functions, is proposed to efficiently and accurately produce surrogate models for failure surfaces. In this regard, two novel learning functions are introduced to adapt to different iterative cycles, one is employed to correct the quality of samples, and the other is used to search for the samples closest to the limit state surface. Third, the subset simulation (SS) is integrated into the adaptive MRGP model to estimate the failure probability under multiple failure modes with fewer function calls and time consumption. Numerical and engineering case studies are finally provided to demonstrate the effectiveness of the proposed method.

Structural reliability analysis has an inherent contradiction between efficiency and accuracy. The metamodel can significantly reduce the computational cost of reliability analysis by a simpler approximation. Therefore, it is crucial to build a metamodel, which achieves the minimum simulations and accurate estimation for reliability analysis. Aiming at this, an effective adaptive metamodel based on the combination of radial basis function (RBF) model and Monte Carlo simulation (MCS) is proposed. Different shape parameters are first used to generate the weighted prediction variance, and the search for new training samples is guided by the active learning function that achieves a tradeoff of (1) being close enough to limit state function (LSF) to have a high reliability sensitivity; (2) keeping enough distance between the existing samples to avoid a clustering problem; and (3) being in the sensitive region to ensure the effectiveness of the information obtained. The performance of the proposed method for a nonlinear, non-convex, and high dimensional reliability analysis is validated by three numerical cases. The results indicate the high efficiency and accuracy of the proposed method.

• A reliability-based design optimization under multiple simulation models is proposed. • An active learning function is proposed by estimating how updates on simulation model affect reliability. • A concept of activity function is proposed, which estimates how active constraint functions are. • The proposed active learning function is modified to apply to problems where the cost of each simulation model is different. • Two numerical and one engineering examples are investigated to validate the proposed method. This paper proposes an effective active learning strategy for reliability-based design optimization (RBDO) problems in which the constraint functions are acquired from multiple simulation models. To achieve this goal, a new active learning function (ALF) is derived by estimating the increased reliability of active constraint functions after adding one point to the train points of constraint functions in each simulation model. The proposed ALF distinguishes possibly active constraint functions that seem active near the current optimum and considers how the constraint functions are active. In the proposed RBDO method, a Kriging model is iteratively updated by adding the best point to the train points of constraint functions included in the crucial simulation model until the optimum converges and the Kriging model is sufficiently accurate. The best point and the crucial simulation model are obtained by comparing the proposed ALF. The ALF is further modified to apply to problems where the cost of each simulation model is different. To verify the effectiveness of the proposed method, two numerical and one engineering examples are analyzed. The results show that the proposed method efficiently and accurately obtains the RBDO optimum involving multiple simulation models, regardless of simulation cost. [ABSTRACT FROM AUTHOR]

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. • A novel adaptive Kriging method is proposed for efficient structural reliability analysis. • An effective learning function is developed for selecting the informative sample at each iteration. • A new convergence criterion is proposed for terminating the learning process at appropriate stage. • Two potential schemes are investigated to further enhance the performance of the proposed method. • High accuracy, efficiency, and robustness are observed in all test cases. [ABSTRACT FROM AUTHOR]