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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Process (AMFGP) method Lu et al. (2024) introduces a comprehensive learning function incorporating variance contribution, computational cost, correlation structure, and spatial distribution of training points through a density penalty term. More recently, Zhang et al. (2025) proposed an approach utilizing a global-error active learning function (GEALF). This method incorporates cost-balancing considerations in both the learning function and stopping criteria, creating bias towards LF evaluations once satisfactory accuracy is achieved. Upon meeting the MF model error threshold, the method transitions to an importance sampling phase, allowing selective HF evaluations to refine the final probability estimate. This approach demonstrates reliability estimates within 1% error compared to Monte Carlo benchmarks, maintaining accuracy even for small failure probabilities while significantly reducing HF model evaluations compared to traditional AK-MCS methods. Building upon these advances, the present study employs the active learning GEALF method with multi-fidelity modelling to estimate small failure probabilities in shell structures subjected to cabin pressure loading. This approach is particularly suited for aerospace applications where computational efficiency is critical while maintaining high accuracy for rare event estimation. The remainder of this paper is organized as follows: Section 2 presents the methodological framework, including the boundary element method (BEM) for stress intensity factor evaluation and the GEALF-based multi-fidelity reliability analysis approach. Section 3 demonstrates the proposed methodology through two examples: first, a two-dimensional analytical reliability problem for validation purposes, followed by a numerical investigation of a shallow shell structure under cabin pressure. Finally, Section 4 provides conclusions and future work. 2. Methodology 2.1. Evaluation of the Stress Intensity Factor using the DBEM The Dual Boundary Element Method (DBEM) is employed for modelling crack propagation in the shallow shell structure. The boundary element method provides an efficient alternative to finite element analysis, requiring discretization of only the structural boundaries rather than the entire domain. DBEM, specifically developed for automated crack propagation analysis Portela et al. (1992), Aliabadi (2002), treats crack surfaces as internal boundaries within the computational domain. Crack advancement is achieved by incrementally adding boundary elements in front of the previous crack tip, eliminating the remeshing requirements inherent in FEM approaches. This characteristic results in substantial computational efficiency gains for fatigue crack growth simulations, making DBEM particularly suitable for the reliability analysis framework. The DBEM formulation for shallow shells was established by Dirgantara and Aliabadi (2001) and subsequently refined by Wen et al. (1999). These developments incorporated the dual reciprocity method (DRM) to transform domain integrals arising from body forces and thermal loads into equivalent boundary integrals, maintaining the boundary only discretization advantage. Stress intensity factors (SIFs) are computed using the crack surface displacement extrapolation (CSDE) technique Dirgantara and Aliabadi (2002). For shallow shell structures under combined membrane and bending loads, the stress intensity factors for modes I, II, and III are expressed as: = 1 ℎ 1 + 6 ℎ 2 1 (1) = 1 ℎ 2 + 6 ℎ 2 2 (2) = 2 3 ℎ [1−( 2 3 ℎ ) 2 ] 3 (3) where Ω represents a geometric coupling parameter defined as = 1+ 3 (1/ 1 +1/ 2 )/2 , accounting for shell curvature effects. The 3 denotes the coordinate along the thickness direction, with the stress intensity factors achieving maximum values at the shell surfaces. Consequently, SIF evaluations in the subsequent numerical analysis are performed at 3 = ℎ/2 , corresponding to the upper surface. The parameters ℎ , 1 , and 2 denote the shell