Implicit Performance Functions Research Articles

A double-loop adaptive relevant vector machine combined with Harris Hawks optimization-based importance sampling

PurposeEnsuring the safety of structures is important. However, when a structure possesses both an implicit performance function and an extremely small failure probability, traditional methods struggle to conduct a reliability analysis. Therefore, this paper proposes a reliability analysis method aimed at enhancing the efficiency of rare event analysis, using the widely recognized Relevant Vector Machine (RVM).Design/methodology/approachDrawing from the principles of importance sampling (IS), this paper employs Harris Hawks Optimization (HHO) to ascertain the optimal design point. This approach not only guarantees precision but also facilitates the RVM in approximating the limit state surface. When the U learning function, designed for Kriging, is applied to RVM, it results in sample clustering in the design of experiment (DoE). Therefore, this paper proposes a FU learning function, which is more suitable for RVM.FindingsThree numerical examples and two engineering problem demonstrate the effectiveness of the proposed method.Originality/valueBy employing the HHO algorithm, this paper innovatively applies RVM in IS reliability analysis, proposing a novel method termed RVM-HIS. The RVM-HIS demonstrates exceptional computational efficiency, making it eminently suitable for rare events reliability analysis with implicit performance function. Moreover, the computational efficiency of RVM-HIS has been significantly enhanced through the improvement of the U learning function.

Meta model-based and cross entropy-based importance sampling algorithms for efficiently solving system failure probability function

The multi-mode system failure probability function (SFPF) can quantify how the distribution parameters of the random input vector affect the system safety and decouple the system reliability-based design optimization model. However, for a problem with a time-consuming implicit performance function and rare failure domain, efficiently solving the SFPF remains significantly challenging. Therefore, in this study, two efficient algorithms are proposed, namely, the meta model-based importance sampling and cross entropy-based importance sampling. The contributions of this study are twofold. The first is constructing a single-loop optimal importance sampling density (SL-OISD) method to decouple the double-loop framework for analyzing the SFPF. The second is establishing two methods to efficiently approximate the SL-OISD and complete the SFPF estimation. The first method is based on the meta model of the system performance function, which is abbreviated as SL-Meta-IS. The second method is based on minimizing the cross entropy between the Gaussian mixture density model and SL-OISD, which is abbreviated as SL-CE-IS. To reduce the number of evaluating the system performance function when approximating the SL-OISD, sampling the SL-OISD, and identifying the state of the samples for completing the SFPF estimation, an adaptive Kriging model of the system performance function is introduced into SL-Meta-IS and SL-CE-IS. Owing to decoupling the double-loop framework into a single-loop framework, replacing the time-consuming system performance function with the economic Kriging model, and employing importance sampling variance reduction techniques to address issues related to the rare failure domain, the proposed SL-Meta-IS and SL-CE-IS methods greatly enhance the efficiency of SFPF estimations. The numerical and practical examples demonstrate that the two proposed methods are superior to the existing algorithms; moreover, the efficiency of SL-CE-IS is higher than that of SL-Meta-IS.

Time-Variant Reliability Analysis with an Adaptive Single-Loop Kriging Combined with Directional Sampling Method

The difficulty in calculating time-variant reliability lies in the large number of performance function calls. This paper proposes an efficient method for time-variant reliability analysis by incorporating the directional sampling method (DS). In this method, within the framework of single-loop active learning Kriging (SL–AK), the burden of fitting the Kriging surrogate model is alleviated by constructing a novel candidate sample pool. While reducing the computational burden, the accuracy of the Kriging surrogate model remains unaffected. Unlike the traditional SL–AK, the proposed method obtains the failure probability through an improved DS method. The improved DS utilizes a bisection method for root-finding, thereby circumventing the impact of interpolation coefficients on the computation results. This method of computing failure probability not only alleviates computational burden but also enhances the precision of the estimated failure probability. Furthermore, this paper introduces global sensitive analysis (GSA) into time-variant reliability analysis, thereby expanding the applicability of GSA. The accuracy and high efficiency of the proposed method are verified through three numerical examples and two engineering examples with implicit performance function.

AK-SYS-IE: A novel adaptive Kriging-based method for system reliability assessment combining information entropy

Structural reliability assessment is a popular topic in engineering problems, particularly for the larger and more complex systems with implicit performance functions, also called black-box problems. The reliability assessment for a black-box problem must continuously be computed using simulation models, such as the finite element model, a highly time-consuming process with a high computational cost. The adaptive Kriging has gained considerable attention over the past decade. The Kriging-based reliability assessment method reduces the computational cost to a great extent, on the premise of ensuring the accuracy of reliability assessment. However, many of the currently published system reliability assessment methods, construct adaptive Kriging models by reducing the probability of incorrect prediction of the system state, and do not make full use of the uncertainty of the system state prediction information. To this end, a new Kriging-based method for structural system reliability assessment is proposed in this study. First, the probabilities of incorrect or correct system state predictions were understood from the perspective of information entropy. Second, an active learning strategy is proposed based on information entropy theory. Finally, the advantages of the proposed method are demonstrated and highlighted through several numerical examples. The results show that the proposed method achieves a good balance between the accuracy and computational cost, and the numerical magnitude effect does not affect the computational cost. Moreover, this is an effective method for assessing the reliability of complex systems.

A Kriging-based adaptive adding point strategy for structural reliability analysis

Aiming at the problem of structural reliability analysis with complex performance function in practical engineering, an adaptive adding point strategy for structural reliability analysis is proposed by combining Kriging surrogate model and learning function. In the process of structural reliability analysis, in order to reduce the computational cost, the surrogate model is usually used to fit the implicit performance function. Existing learning functions rarely take into account the problem of sample point aggregation due to too many sample points selected from the failure domain (or safety domain) during the fitting process. In order to overcome this defect, a new method can ensure that the added sample points are evenly distributed on both sides of the limit state function, preventing the aggregation of sample points and causing information redundancy, thereby improving the fitting accuracy of the model, accelerating the convergence speed of the sample points and saving the sample space. Through the analysis of four-branch series system, nonlinear oscillator and truss structure, the results show that the algorithm needs less samples than other methods, and the reliability calculation accuracy is higher, which verifies the correctness and efficiency of the proposed method.

Probabilistic assessment of heavy-haul railway track using multi-gene genetic programming

This study presented a probabilistic assessment of heavy-haul railway track using a high-performance computational model called multi-gene genetic programming (MGGP). A reliability analysis (RA) method based on MGGP and the first-order second-moment method (FOSM) has been proposed in this study. First, GP was used to map the implicit performance functions; therefore, arriving at GP-based explicit performance functions. Subsequently, the developed GP model was used to perform RA of a soil slope of heavy-haul railway track under both seismic and non-seismic conditions. Using the FOSM, soil uncertainties were mapped based on the concepts of probability theory and statistics, and a ready-made expression was developed. Simulated results demonstrate that the GP-based FOSM approach can predict the probability of failure (POF) of slope with rational accuracy. The probabilistic analysis against bearing capacity failure was also investigated in this study to ensure serviceability of the soil slope. Based on the outcomes, it can be deduced that the coefficient of variation of soil properties affects the POF of slope significantly. With the aid of the developed expression, the POF of the soil slope of heavy-haul railway track can be assessed rationally and efficiently.

A dynamic Gaussian process surrogate model based on the grasshopper optimization algorithm for non-probabilistic reliability analysis of complex structures

In practical complex engineering structures, the performance function (PF) for reliability analysis is commonly implicit and highly nonlinear. Commonly used surrogate models are appropriate for structural reliability analysis with an implicit PF. However, these methods require hundreds of PF values, which are time-consuming to obtain by adopting numerical analysis, such as finite element analysis (FEA). Therefore, since non-probabilistic reliability analysis does not require a large number of samples, it has great development potential. In this paper, a dynamic Gaussian process (GP) surrogate model based on the grasshopper optimization algorithm (DGP-GOA) is proposed for the non-probabilistic reliability analysis. First, with the help of the scale factor of the convex set model, the non-probabilistic reliability analysis problem is transformed into an unconstrained optimization problem. Second, the DGP-GOA fits the PF by constructing a GP surrogate model with a small dataset. Third, the GOA is used to search for the global optimal solution to obtain a non-probabilistic reliability index. Then, a dynamic retraining strategy is proposed to improve the fitting accuracy and efficiency. The results demonstrate that the proposed method is highly applicable to the non-probabilistic structural reliability analysis of complex engineering structures.

A multi-mode failure boundary exploration and exploitation framework using adaptive kriging model for system reliability assessment

Various adaptive reliability analysis methods based on surrogate models have recently been developed. A multi-mode failure boundary exploration and exploitation framework (MFBEEF) was proposed for system reliability assessment using the adaptive kriging model based on sample space partitioning to reduce computational cost and use the characteristics of the failure boundary in multiple failure mode systems. The efficiency of the adaptive construction of kriging model can be improved by using the characteristics of the center sample of the small space to represent the characteristics of all samples in the small space. This method proposes a failure boundary exploration and exploitation strategy and a convergence criterion based on the maximum failure probability error for a system with multiple failure modes to adaptively approximate the failure boundary of a system with multiple failure modes. A multiple-failure-mode learning function was used to identify the optimal training sample to gradually update the kriging model during the failure boundary exploration and exploitation stages. In addition, a complex failure boundary-oriented adaptive hybrid importance sampling method was developed to improve the applicability of the MFBEEF method to small failure probability assessments. Finally, the MFBEEF method was proven to be effective using five system reliability analysis examples: a series system, a parallel system, a series–parallel hybrid system, a multi-dimensional series system with multiple failure modes, and an engineering problem with multiple implicit performance functions.

Adaptive Kriging-based Bayesian updating of model and reliability

Bayesian updating is a powerful tool to reassess and calibrate models and their reliability as new observations emerge, and the Bayesian updating with structural reliability method (BUS) is an efficient approach that reformulates it as a structural reliability problem. However, the efficiency and accuracy of BUS depend on a constant c determined by the maximum of likelihood function. To efficiently complete Bayesian updating with new observations related to implicit performance function, a method that combines adaptive Kriging with Bayesian updating is proposed. The proposed method involves three stages. Firstly, an innovatively advanced expected improvement (AEI) learning function is proposed to train the Kriging model of the likelihood function for estimating c, in which the convergence criterion and the strategy of selecting new training point guarantee the accuracy and efficiency of estimating c. Secondly, a new learning function based on expectation and variance of contribution uncertainty function (EVCUF) is proposed to adaptively train the Kriging model of the performance function constructed in BUS to extract posterior samples and complete Bayesian updating of model. By simultaneously taking the expectation and variance of the contribution of the candidate sample to improving accuracy of the Kriging model into consideration, the EVCUF learning function ensures the robust and efficient convergence of the Kriging model. Finally, based on the training points of the previous two stages, the traditional U learning function is employed to subsequentially update Kriging model of the performance function for classifying posterior samples and completing Bayesian updating of reliability. Additionally, a reduction strategy of the candidate sample pool is proposed to improve the efficiency of the proposed method. After demonstrating the basic principle and advantage of the proposed method, three examples are introduced to verify the efficiency and accuracy of the proposed method.

Response Surface Method for Reliability Analysis Based on Iteratively-Reweighted-Least-Square Extreme Learning Machines

A response surface method for reliability analysis based on iteratively-reweighted-least-square extreme learning machines (IRLS-ELM) is explored in this paper, in which, highly nonlinear implicit performance functions of structures are approximated by the IRLS-ELM. Monte Carlo simulation is then carried out on the approximate IRLS-ELM for structural reliability analysis. Some numerical examples are given to illustrate the proposed method. The effects of parameters involved in the IRLS-ELM on accuracy in reliability analysis are respectively discussed. The results exhibit that a proper number of samples and neurons in hidden layer nodes, an appropriate regularization parameter, and the number of iterations for reweighting are of important assurance to obtain reasonable precision in estimating structural failure probability.

Reliability analysis with correlated random variables based on a novel transformation, adaptive dimension-reduction and maximum entropy method

In this paper, structural reliability analysis including correlated random variables is implemented based on a novel transformation and fractional exponential moments-based maximum entropy method (FEM-MEM) with a new adaptive dimension reduction. First, a novel transformation, which does not require the computation of correlation matrix in correlated standard normal space, is first presented to transform correlated random variables to be independent standard normal ones, which is quite easy and simple to implement. Then, an adaptive dimension-reduction model is developed for efficient FEMs estimation, where the contribution-degree analysis is performed and a hybrid integration formula is established accordingly. Further, the FEM-MEM, in which the proposed transformation and hybrid integration formula are embedded, is applied to derive the unknown probability distribution of the performance function with correlated random variables, which overcomes the shortcomings of traditional fractional moments-based MEM and significantly enhances the robustness, accuracy and efficiency. Four numerical examples including both the explicit and practical implicit performance functions are investigated to validate the proposed method, where pertinent Monte Carlo simulation and Nataf transformation are employed for comparisons.

IE-AK: A novel adaptive sampling strategy based on information entropy for Kriging in metamodel-based reliability analysis

This article focuses on the adaptive Kriging metamodel-based reliability analysis for reducing a sequential number of calls of the complex original functions. To avoid the repetitive and tedious deterministic response analysis with stochastic simulation method (including Monte Carlo Simulation and its various improvement, such as importance sampling, subset simulation) in reliability analysis, herein a novel sequential sampling strategy related to Kriging metamodel is proposed, which is implemented based on information entropy theory. In addition, the generalized F-discrepancy method is simultaneously quoted to further optimize the candidate pool to improve the effectiveness of the training metamodel. Finally, a new structural reliability analysis method is proposed, which continuously reduces the number of deterministic analysis of structures without sacrificing accuracy. To highlight the applicability of the method and verify its accuracy and effectiveness, a series of typical examples are tested and compared, including highly nonlinear limit state functions, high-dimension performance function with analytic expressions and dynamic reliability analysis of nonlinear engineering structures subject to seismic excitation with implicit performance function. Numerical results show that significant computational savings and desired accuracy can be achieved when dealing with different reliability analysis cases.

A Grasshopper Optimization Algorithm-Based Response Surface Method for Non-Probabilistic Structural Reliability Analysis with an Implicit Performance Function

Non-probabilistic reliability analysis has great developmental potential in the field of structural reliability analysis, as it is often difficult to obtain enough samples to construct an accurate probability distribution function of random variables based on probabilistic theory. In practical engineering cases, the performance function (PF) is commonly implicit. Monte Carlo simulation (MCS) is commonly used for structural reliability analysis with implicit PFs. However, MCS requires the calculation of thousands of PF values. Such calculation could be time-consuming when the structural systems are complicated, and numerical analysis procedures such as the finite element method have to be adopted to obtain the PF values. To address this issue, this paper presents a grasshopper optimization algorithm-based response surface method (RSM). First, the method employs a quadratic polynomial to approximate the implicit PF with a small set of the actual values of the implicit PF. Second, the grasshopper optimization algorithm (GOA) is used to search for the global optimal solution of the scaling factor of the convex set since the problem of solving the reliability index is transformed into an unconstrained optimal problem. During the search process in the GOA, a dynamic response surface updating strategy is used to improve the approximate accuracy near the current optimal point to improve the computing efficiency. Two mathematical examples and two engineering structure examples that use the proposed method are given to verify its feasibility. The results compare favorably with those of MCS. The proposed method can be non-invasively combined with finite element analysis software to solve non-probabilistic reliability analysis problems of structures with implicit PF with high efficiency and high accuracy.

An active learning Kriging-based method combining the weight information entropy function and the adaptive candidate sample pool

The reliability analysis methods based on the Kriging model have been explored to update the design of experiments (DoE), but the number of calls to the performance function is high. In this paper, a new active learning method combining the weight information entropy function (WH) and the adaptive candidate sample pool is proposed to improve the reliability analysis efficiency. Based on the information entropy function (H), the learning function WH is constructed to consider Kriging variance and the joint probability density function (PDF). In this proposed method, the sample points can be assigned different weight by the learning function WH according to the important degrees of sample points. The point which not only nears the limit state function (LSF), but also has a high probability density function value and large Kriging variance is assigned more weight than others. To select the sample points with lower confidence level, the adaptive candidate sample pool is generated by Markov Chain Monte Carlo (MCMC) simulation. The Kriging model can be updated efficiently by the proposed method. Four numerical examples and an engineering example with implicit performance function are used to verify the efficiency and accuracy of the proposed method. The results show that the proposed method can significantly improve the computational efficiency of the reliability analysis without losing accuracy.

HUT‐based method for structural reliability considering the non‐normal and unknown distributions

AbstractStructural reliability analysis involving the non‐normal random variables are commonly encountered in practical engineering. Furthermore, some random variables are often unknown probability distributions, and the probabilistic characteristic of these variables may be expressed using only statistical moments. In this contribution, an efficient algorithm for structural reliability analysis considering the random variables with non‐normal and unknown probability distributions is proposed, in which high‐order unscented transformation (HUT) and fourth‐moment methods are adopted to efficiently evaluate the structural reliability index. By introducing the Rosenblatt transformation and third‐order polynomial transformation (TPT) techniques, random variables with non‐normal or unknown distributions are transformed into standard normal distributions. Accordingly, the performance functions can be reconstructed using standard normal random variables, and HUT is then employed to efficiently estimate the first four moments (i.e., mean, standard deviation, skewness, and kurtosis) of the performance function. Based on the Hermite polynomial model, the complete expression of fourth‐moment reliability index is derived for estimating the reliability index using the first four moments of the performance function. The efficiency and accuracy of the proposed algorithm are demonstrated through six numerical examples. The results show that the proposed algorithm is applicable to estimating reliability index of both static and dynamic reliability problems involving explicit and implicit performance functions.

Non-Probabilistic Structural Reliability Analysis Integrating the PSO and the Kriging Model

In view of the problems with implicit performance functions and limited experimental data in the reliability analysis of complex structures, a non-probabilistic reliability method combining the particle swarm optimization (PSO) with the Kriging model was presented. A multidimensional ellipsoid was first used to characterize the uncertain parameters of structures. The Kriging model was then constructed for the implicit performance function, wherein its optimal related parameters was determined through the PSO. Based on the proposed model the reliability analysis was explicitly conducted. The results of 3 numerical examples show that, the proposed method is of effectiveness and feasibility, and has higher accuracy and efficiency than those based on the traditional Kriging model.

An equivalent expectation evaluation method for approximating the probability distribution of performance functions

This paper proposes an equivalent expectation evaluation method for approximating the probability distribution of performance functions. The innovation of the proposed method is to formulate the probability distribution of the performance function as an expectation regardless of the dimensionality and complexity of the problem. When a monotonic inverse function (MIF) of the performance function with respect to a certain random variable can be obtained, the probability distribution can be exactly formulated as an expectation related to the distribution of this variable and the MIF. While if such a MIF does not exist or cannot be easily obtained, by constructing two simple auxiliary functions whose MIF can be obtained, a generalized method is proposed to formulate the probability distribution of the performance function as an expectation related to the distribution of the auxiliary functions. Then, the Sobol sequence and point estimate method are employed to determine the expectation. The efficiency and accuracy of the proposed method are demonstrated through four numerical examples. The results show that the proposed method is applicable to approximating probability distributions of both static and dynamic reliability problems involving high-dimensional, nonlinear, or implicit performance functions.

Adaptive relevance vector machine combined with Markov-chain-based importance sampling for reliability analysis

Many engineering systems involve complex implicit performance functions, and evaluating the failure probability of these systems usually requires time-consuming finite element simulations. In this research, a new reliability method is proposed by combining relevance vector machine and Markov-chain-based importance sampling (RVM-MIS), which improves computational efficiency by decreasing the number of expensive model simulations. Relevance vector machine (RVM) is a machine learning method based on the concept of probabilistic Bayesian learning framework. It is worth noting that RVM provides predicted value of the sample and corresponding variance. Due to this important feature, various active learning functions can be applied to improve the accuracy of RVM to approximate real performance functions. In addition, Markov-chain-based importance sampling (MIS) is utilized to generate important samples covering areas that significantly contribute to failure probability. The important samples are then predicted by a well-constructed RVM to obtain failure probability, rather than being evaluated using real performance functions, so the computation time is drastically decreased. RVM-MIS reduces the number of calls to real performance function while ensuring the accuracy of results. Four academic examples and a bearing statics problem with an implicit performance function are performed to verify the accuracy and efficiency of the proposed method.

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 Reliability Assessment Based on Enhanced Conjugate Unscented Transformation and Improved Maximum Entropy Method

Estimation of statistical moments remains one of the main topics of stochastic analysis whose accuracy greatly affects reliability analysis results. In this work, conjugate unscented transformation (CUT) methods, which can balance accuracy and efficiency, are introduced for the statistical moment estimation of responses. Because of the drawbacks of existing CUT methods, a family of enhanced conjugate unscented transformation (ECUT) methods, including ECUT-4, ECUT-6, and ECUT-8 methods, is proposed for statistical moment estimation by combining the original CUT, variable transformation, and exact dimension reduction method. Then the probability density function of a performance function is reconstructed by the improved maximum entropy method (IMEM) with the available statistical moments as constraints. To demonstrate the accuracy and effectiveness of the proposed methods, five numerical examples, including linear and nonlinear, low-dimensional and high-dimensional, and explicit and implicit performance functions, are investigated, in which the results obtained from the proposed methods are compared with other existing moment methods and a Monte Carlo simulation (MCS) method. The results of the examples show that the proposed ECUT-6 and ECUT-8 methods have fairly high accuracy and efficiency for structural reliability analysis.