Staged active-learning-based multi-fidelity Kriging method for reliability analysis with small failure probabilities

Estimating the small failure probability is crucial in reliability analysis of complex engineering systems. Although surrogate-based methods have been recognized as one of the most effective representatives, they are still less satisfactory in terms of computational efficiency and accuracy when applied to rare event problems involving nonlinear/dynamic finite element analyses. In this work, a staged active-learning-based multi-fidelity Kriging method is developed for assessing small failure probabilities. We firstly develop a new multi-fidelity Kriging model, in which a joint architecture is proposed to characterize the cross-fidelity correlations. Then, a multi-fidelity sampling scheme is established to assign different fidelity levels to a series of intermediate failure domains. With this sampling scheme, a staged active learning strategy is further developed to sequentially refine the constructed multi-fidelity model for reliability analysis. In our method, since the intermediate failure domains are assigned with varying low-fidelity levels, they can be approximated just with inexpensive low-fidelity evaluations to sequentially identify the critical regions. Within these identified regions, the accuracy of multi-fidelity surrogate can be effectively improved with just a few high-fidelity observations due to the joint architecture of the developed multi-fidelity model. Thus, the rare event probability can be estimated with high precision and significantly reduced computational burden. Four illustrative examples, including a nonlinear Duffing oscillator system and a transient contact spur gear dynamic system, are investigated to validate the developed method.