PSI - Issue 80

Mengke Zhuang et al. / Procedia Structural Integrity 80 (2026) 299–309 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 thickness and principal radius of curvature in the and directions, respectively. The membrane stress resultant intensity factors and bending stress resultant intensity factors (where = I, II, III denotes the fracture mode) are obtained through the CSDE technique applied to the computed crack surface displacements. To characterize mixed-mode crack propagation in shallow shells, the effective stress intensity factor is evaluated using the effective energy release rate approach: = √ (4) where G eff is the effective energy release rate such that: = + ( + + + ) (5) =√ | | | |+| | (6) The individual contribution of the stress resultant can be separated as: = 2 , = 2 , = 2 3 , = 2 , = 8 (1+ ) 2 5 (7) 2.2. Active learning function GEALF This section presents the GEALF methodology adopted from the AEK-MIFS framework Zhang et al. (2025). While the original work proposed a novel expected-improvement model with weighted combinations of high fidelity and Multi-fidelity predictions, the present study implements GEALF with Co-Kriging-based multi-fidelity modelling. The Co-Kriging model is constructed using the ooDACE toolbox Couckuyt et al. (2014), which provides efficient implementation of multi-fidelity Gaussian process regression. The key equations are summarized below, with detailed derivations available in Zhang et al. (2025). The reliability analysis proceeds through a two-stage adaptive sampling strategy: • Global exploration: Initial sampling across the entire design space to identify potential failure regions • Local refinement: Importance sampling concentrated around identified failure domains The failure probability is evaluated through the performance function ( ) as ： =∫ ( ) = ∫ ( ) ( ) ( )≤0 (8) where ( ) denotes the probability density function of input parameters, and ( ) is the failure indicator function: ( ) = {10,, (( )) ≤> 00 (9) Initial training sets comprising 0 low-fidelity and 0 high-fidelity samples are generated using Latin hypercube sampling (LHS) for space-filling purpose. These samples enable construction of the initial surrogate models: a high fidelity Kriging model ̂ ( ) ∼ ( ( ), 2 ( )) and a multi-fidelity Co-Kriging model ̂ ( )∼ ( ( ), 2 ( )) , where and 2 represent the predicted mean and variance, respectively. A candidate pool of points is randomly generated to cover the design space for subsequent adaptive sampling. For any candidate point x, the probability of failure is estimated as in Zhang et al. (2025): 4

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