PSI - Issue 80

Mengke Zhuang et al. / Procedia Structural Integrity 80 (2026) 299–309 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 During local refinement, GEALF is applied within the importance sampling domain to identify regions of high uncertainty. The iterative process terminates when the coefficient of variation satisfies ( ) = √ [ ]/ < 0.05 . The final failure probability is estimated using importance sampling Tabandeh et al. (2022): = 1 ∑ ( ) ( ) ℎ ( ) =1 (19) where ℎ ( ) denotes the importance sampling density and is the importance sampling pool. 3. Numerical example To demonstrate the method’s effectiveness in reducing HF cost while maintaining accurate failure -probability estimates, first, a two-dimensional analytical reliability problem which allows for direct comparison with Monte Carlo simulation results, followed by a numerical investigation of a shallow shell structure under cabin pressure is presented in this section. The stopping criterion of the following example are set as COV( ) less than 0.05 following Bichon et al. (2008). 3.1. 2D Multimodal Function A 2D highly non-linear multimode function is used to demonstrate the GEALF method and MF model in estimating the failure probability. The HF model is defined as Bichon et al. (2008): ( 1 , 2 )= − ( 12 +4)( 2 20 −1) + ( 5 1 2 ) (20) where 1 and 2 are independent standard normal random variables such that 1 ∼(1.5,1) and 2 ∼(2.5,1) . The failure domain is defined as ( 1 , 2 ) ≤ 0 . The corresponding LF model is constructed by introducing a correlation term to the HF model such that Zhang et al. (2025): ( 1 , 2 )= − ( 12 +4 )( 2 20 −1− ) + ( 5 1 2 − )+ ( 2 5 2 ( 1 + 2 2 )+ 5 4 ) (21) where =1 is used in the work. The LF model introduces slight modifications to the coefficients, creating a discrepancy between the two fidelity levels with a cost ratio of = / = 10 Zhang et al. (2025). The result of active learning and MF method is shown in Figure 1. The test evaluates two different values of the constant c allowing changes in the failure probability. The contour plots illustrate the complex nature of the limit state function. The initial samples of 6 HF and 16 LF were selected using the Latin Hypercube Sampling (LHS) method and are sparsely distributed across the design space, while the adaptively added samples concentrate around the limit state boundaries. It can be seen that the MF Kriging approximation shows good agreement with the true limit state, particular in the region shows high nonlinearity. Compare to the failure region shown in Figure 1a), when c = 4 Figure 1b), the failure regions shrink significantly, resulting in a smaller failure probability. The adaptive sampling strategy responds by focusing computational resources more tightly around the reduced failure domains. Similarly, the LF model provides a rough approximation of the general LSF, enabling efficient initial exploration. The HF samples are then added to refine the accuracy in critical regions.

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