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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]

Aiming at the problems of implicit and highly nonlinear limit state function in the process of reliability analysis of mechanical products, a reliability analysis method of mechanical structures based on Kriging model and improved EGO active learning strategy is proposed. For the problem that the traditional EGO method cannot effectively select points in the limit state surface region, an improved EGO method is proposed. By dealing with the predicted values of sample point model with absolute values and assume that the distribution state of response values remains the same, the work focus of active learning selection points is moved to the vicinity, where the points are with larger prediction variance or close to the limit state surface. Three examples show that, compared with the classical active learning method, the proposed method has good global and local search ability, and can estimate the exact failure probability value under the condition of less calculation of the limit state function.

Metamodels in lieu of time-demanding performance functions can accelerate the reliability analysis effectively. In this paper, we propose an efficient collaborative active learning strategy-based augmented radial basis function metamodel (CAL-ARBF), for reliability analysis with implicit and nonlinear performance functions. For generating the suitable samples, a CAL function is first designed to constrain the new samples being generated in sensitivity region, near limit state surface and keep certain distances mutually. Then by adjusting the adjustment coefficient of CAL function, the CAL-ARBF is mathematically modeled and the corresponding reliability analysis theory is developed. The effectiveness of the proposed approach is validated by four numerical samples, including global nonlinear problem, local nonlinear problem, nonlinear oscillator and truss structure. Through comparison of several state-of-the-art methods, the proposed CAL-ARBF is demonstrated to possess the computational advantages in efficiency and accuracy for reliability analysis.

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.

A significant challenge of surrogate model-based structural reliability analysis (SRA) is to construct an accurate approximated model of the nonlinear limit state function (LSF) with high order and high dimension effectively. As one of the sequential update-strategies of design of experiment (DoE), the active learning method is more attractive in recent years due to greatly reduces the burden of reliability analysis. Although the active learning method based on information entropy learning function H and the line simulation (AK-LS) is a powerful tool of SRA, the computational burden from the iterative algorithm is still large during the learning process. In this research, an improved learning criterion, named the weight information entropy function (WH), is developed to update the DoE of Kriging-based reliability analysis. The WH learning function consists of the information entropy function and an adaptive weight function (W). Locations in the variable space and probability densities of the samples are taken accounted into the WH learning function, which is the most important difference from the H learning function. The samples that are closer to the LSF and has a greater probability density can be preferentially selected into the DoE comparing to others by changing the weight of information entropy during the learning process. The WH learning function can efficiently match the limit state function in an important domain rather than the entire variable space. Consequently, the approximated model of LSF via Kriging interpolation can be constructed more effectively. The new active learning method is developed based on Kriging model, in which WH learning function and Monte Carlo simulation (MCS) are employed. Finally, several engineering examples with high non-linearity are analyzed. Results shown that the new method are very efficient when dealing with intractable problems of SRA. [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.

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]

• An adaptive Relevant Vector Machine is proposed for reliability assessments. • A new active learning function for RVM is proposed. • An efficient stopping criterion is adopted to terminate the active learning process. • The proposed method is applied to five illustrative examples, showing high accuracy and efficiency. In this study, an adaptive relevant vector machine, which is developed within a probabilistic Bayesian learning framework, is combined with Monte Carlo simulation (MCS) to perform reliability analysis with high efficiency and accuracy. A new active learning function is proposed that comprehensively considers the probability of mis-predictions and spatial locations of the existing sampling points to search for the best next point to enrich the training samples. To avoid redundant original model evaluations, an efficient stopping criterion is adopted to terminate the active learning process in a timely manner. The proposed method is tested with five examples involving non-differentiability, high non-linearity, and practical engineering problems. The proposed method can obtain highly accurate failure probability predictions with a small number of evaluations of the original model in comparison with the direct MCS and other methods in the literature. In conclusion, the proposed method is recommended for structural reliability analyses in engineering practice. [ABSTRACT FROM AUTHOR]

To improve the computational efficiency and accuracy in the reliability analysis of gear heat transfer error， an efficient reliability analysis method combining PC-Kriging and active learning function LIF is proposed. Polynomial-chaos-expansion (PCE) is adopted to replace the regression basis function of the traditional Kriging model to enhance its global approximation accuracy and its ability to capture local features. The least-angle regression (LAR) is used to construct the optimal polynomial quantity set of the regression basis function， and the Akaike information criterion (AIC) is utilized to determine the optimal truncated set. Furthermore， the active learning function LIF is employed to select the optimal sample during each iteration to improve the convergence efficiency of the PC-Kriging model. The application to gear heat transfer error shows that compared with the traditional Kriging model， the proposed method can significantly reduce the number of performance function evaluations while ensuring accuracy in the reliability analysis. [ABSTRACT FROM AUTHOR]

Surrogate models combined with Monte Carlo simulation (MCS) are effective ways to address failure probability problems, which involve time-consuming computer codes, in structural reliability analysis. Recently, many active learning functions based on the Kriging model have been developed to conduct adaptive sequential sampling due to its predicted value and variance. However, effective methods for learning functions based on other surrogate models to address failure probability problems remain sparse. Hence, this paper presents a new adaptive sampling function that combines the radial basis function (RBF) approximate model with MCS to calculate the failure probability of structures. The new learning function is named the active RBF method with multiple kernel functions and Monte Carlo simulation (ARBFM-MCS). It uses several RBF kernel functions to estimate the local uncertainty of the predicted values based on interquartile range (IQR) to formulate the active learning function. This method can establish several surrogate models simultaneously for application to different problems. The stopping criterion of this method is that if one of the errors calculated by the various RBF models meets the demand, the learning process stops automatically. Furthermore, the method is also compared with the k-fold cross-validation approach based on a single RBF kernel function and some other approaches presented in the literature. Six numerical examples are considered to verify the accuracy and the efficiency of the proposed method. The results reveal that the multiple kernel function method is effective and has the same accuracy level as other methods. Moreover, for these examples, the method often calls the limit state function in fewer times to obtain accurate failure probabilities. [ABSTRACT FROM AUTHOR]

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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]

Metamodel combined with simulation type reliability method is an effective way to determine the probability of failure (P f) of complex structural systems and reduce the burden of computational models. However, some existing challenges in structural reliability analysis are minimizing the number of calls to the numerical model and reducing the computational time. Most research work considers adaptive methods based on ordinary Kriging with a single point enrichment of the experimental design (ED). This paper presents an active learning reliability method using a hybrid metamodel with multiple point enrichment of ED for structural reliability analysis. The hybrid method (termed as APCKKm-MCS) takes advantage of the global prediction and local interpolation capability of Polynomial Chaos Expansion (PCE) and Kriging, respectively. The U learning function drives active learning in this approach, while K-means clustering is proposed for multiple point enrichment purposes. Two benchmark functions and two practical marine structural cases validate the performance and efficiency of the method. The results confirm that the APCKKm-MCS approach is efficient and reduces the computational time for reliability analysis of complex structures with nonlinearity, high dimension input random variables, or implicit limit state function. • An active learning structural reliability method with multiple point enrichment is proposed. • The proposed method takes advantage of the global prediction and local interpolation capability. • Two benchmark and two practical marine structural cases validate the performance and efficiency of the method. • The proposed method reduces the computational time for reliability analysis of complex structures with nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper, the active learning Kriging model (ALK), which has been studied extensively in recent years, has been expanded by combining with the directional importance sampling (DIS) method. The directional sampling method can reduce the dimensionality of the variable space by random sampling or interpolation in the direction of vector diameter, which can improve the efficiency of reliability analysis. It is especially suitable for the surfaces whose limit state is spherical or near-spherical. By introducing the control coefficient and constructing the directional importance sampling density function, the sampling efficiency can be further improved in the design point domain. A novel reliability analysis method called ALK-DIS method is proposed. The greatest advantage of the proposed method is its ability on great computational efficiency and dealing with small failure probability problem In addition, due to the excellent performance of directional sampling method in dealing with multi-failure model reliability problems, the ALK-DIS method has the advantage of being applied to system reliability analysis in this paper successfully. The applicability, feasibility and efficiency of the proposed method are proved on examples which contain linearity equation, non-linear numerical example, non-linear oscillator and system reliability engineering problems. • A brand-new method combined ALK model and DIS is proposed. • Application on small failure probability and system reliability is validated. • The performance of proposed method is compared with other existing methods. • The combination of ALK model and advanced simulation methods is expanded. [ABSTRACT FROM AUTHOR]