PSI - Issue 80

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

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Fig. 3. DBEM mesh showing the detailed boundary element discretization around the crack area for (a) LF model and (b)HF model.

Table 3 presents the reliability analysis results for the shallow shell structure with varying initial crack sizes compared with the MCS results. It can be seen that the GEALF method demonstrates good agreement with Monte Carlo simulation across a wide range of failure probabilities. For moderate to large failure probabilities ( > 10 −5 ), the relative errors remain below 5%, indicating that the multi-fidelity approach accurately captures the limit state function. The computational efficiency of the proposed method is evident from the number of function evaluations. The GEALF approach typically requires only 30-39 high-fidelity evaluations and 72-88 low-fidelity evaluations to achieve convergence. Compared to the 267 high-fidelity evaluations required to pre-train the HF Kriging model, the multi-fidelity active learning approach achieves similar accuracy with around 40 high-fidelity calls, demonstrating the method’s computational efficiency. Table 3. Comparison of failure probability estimates for the shallow shell structure between the GEALF MF model and MCS. a (mm) (MCS) (GEALF) Number of MF calls Error(%) 30 8.0× 10 −6 8.434× 10 −6 HF:39 LF:88 5.43 32 3.8× 10 −5 3.947× 10 −5 HF:37 LF:83 3.87 34 7.2× 10 −5 7.533× 10 −5 HF:35 LF:77 4.63 36 5.27× 10 −4 5.130× 10 −4 HF:33 LF:73 2.66 38 1.783× 10 −3 1.735× 10 −3 HF:31 LF:72 2.69 40 4.827× 10 −3 4.941× 10 −3 HF:30 LF:73 2.36 While the relative error increases for smaller failure probabilities, it remains within acceptable engineering tolerances. The maximum error of 5.43% occurs at ∼ 10 −6 . For practical reliability-based design applications, such accuracy is generally sufficient, particularly considering the significant computational savings. Moreover, the error stabilizes for failure probabilities above 10−5, suggesting that the method is well -suited for typical aerospace reliability requirements where target failure probabilities range from 10 −3 to 10 −5 . The reliability index and the with varying crack length is shown in Figure 4. The consistent performance across different crack sizes demonstrates the robustness of the multi-fidelity framework. The adaptive sampling strategy GEALF successfully allocates computational resources where needed, maintaining efficiency while ensuring accuracy in the failure probability estimates.