PSI - Issue 80

Mengke Zhuang et al. / Procedia Structural Integrity 80 (2026) 299–309 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

303

5

( ) = (− ( ) ( ) )

(10)

where Φ(⋅) denotes the standard normal cumulative distribution function. To quantify the potential information gain from different fidelity levels, across-correlation function is introduced Yi et al. (2021): ℎ( )= ( ) | ( )− ( )|+ ( ) (11) where subscript ∈ { , } indicates the fidelity level. This metric measures the expected uncertainty reduction when adding a sample at location with fidelity . To prevent clustering of training points and ensure efficient space coverage, a distance-based penalty function is incorporated: ( ) = ∈ ‖ − ‖ 2 (12) where represents the set of existing training points. The unified GEALF criterion combines these components Zhang et al. (2025): ( ) = ( )[1 − ( )] ⋅ ℎ( ) ⋅ ( ) ⋅ ) ( ) (13) The next training point is selected as = ( ) . The fidelity level is determined by a cost aware criterion Zhang et al. (2025): : >√ / (14) where and represent the computational cost of high-and low-fidelity evaluations, respectively. The surrogate model error is quantified through Zhang et al. (2025): = | ̂ − 1| ≤ | ̂ ) ( ̂ )±1.96√ [ ̂ ] −1| = (15) Where the mean and variance of the failure probability estimate are: ( ̂ )= 1 ∑ ( ) =1 (16) ( ̂ )= 1 2 ∑ ( )[1− ( )] =1 (17) The global exploration phase continues until < ℎ , with ℎ =0.05 in this study. Following global exploration, most probable points (MPPs) are identified by ranking candidate points according to their predicted performance function values: | ( 1 )| ≤ | ( 2 )| ≤ ⋯ ≤ | ( )| (18) Points with indices i ≤ | ⋅ | are designated as MPPs, where p = 0.2 following Zhang et al. (2025). The importance sampling density is then constructed around these MPPs for the local refinement stage.