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An adaptive Kriging method based on K-means clustering and sampling in n-ball for structural reliability analysis

Jinsheng Wang (Department of Bridge Engineering, Southwest Jiaotong University, Chengdu, China)
Zhiyang Cao (Department of Bridge Engineering, Southwest Jiaotong University, Chengdu, China)
Guoji Xu (Department of Bridge Engineering, Southwest Jiaotong University, Chengdu, China)
Jian Yang (School of Civil Engineering, Central South University, Changsha, China)
Ahsan Kareem (Nathaz Modeling Laboratory, University of Notre Dame, Notre Dame, Indiana, USA)

Engineering Computations

ISSN: 0264-4401

Article publication date: 28 February 2023

Issue publication date: 19 April 2023




Assessing the failure probability of engineering structures is still a challenging task in the presence of various uncertainties due to the involvement of expensive-to-evaluate computational models. The traditional simulation-based approaches require tremendous computational effort, especially when the failure probability is small. Thus, the use of more efficient surrogate modeling techniques to emulate the true performance function has gained increasingly more attention and application in recent years. In this paper, an active learning method based on a Kriging model is proposed to estimate the failure probability with high efficiency and accuracy.


To effectively identify informative samples for the enrichment of the design of experiments, a set of new learning functions is proposed. These learning functions are successfully incorporated into a sampling scheme, where the candidate samples for the enrichment are uniformly distributed in the n-dimensional hypersphere with an iteratively updated radius. To further improve the computational efficiency, a parallelization strategy that enables the proposed algorithm to select multiple sample points in each iteration is presented by introducing the K-means clustering algorithm. Hence, the proposed method is referred to as the adaptive Kriging method based on K-means clustering and sampling in n-Ball (AK-KBn).


The performance of AK-KBn is evaluated through several numerical examples. According to the generated results, all the proposed learning functions are capable of guiding the search toward sample points close to the LSS in the critical region and result in a converged Kriging model that perfectly matches the true one in the regions of interest. The AK-KBn method is demonstrated to be well suited for structural reliability analysis and a very good performance is observed in the investigated examples.


In this study, the statistical information of Kriging prediction, the relative contribution of the sample points to the failure probability and the distances between the candidate samples and the existing ones are all integrated into the proposed learning functions, which enables effective selection of informative samples for updating the Kriging model. Moreover, the number of required iterations is reduced by introducing the parallel computing strategy, which can dramatically alleviate the computation cost when time demanding numerical models are involved in the analysis.



This work has been supported in part from the National Natural Science Foundation of China (No. 52078425) and the NSF grant CMMI-1562244, which is greatly appreciated. All the findings in this study are those of the writers and do not necessarily represent those of the sponsors.


Wang, J., Cao, Z., Xu, G., Yang, J. and Kareem, A. (2023), "An adaptive Kriging method based on K-means clustering and sampling in n-ball for structural reliability analysis", Engineering Computations, Vol. 40 No. 2, pp. 378-410.



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