PSI - Issue 80

Sadjad Naderi et al. / Procedia Structural Integrity 80 (2026) 77–92 Sadjad Naderi et al. / Structural Integrity Procedia 00 (2025) 000–000

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4.1. Parameter inference and evolution Fig. 5 shows the temporal evolution of hierarchical parameters ( ! C , !D ), Paris’ law parameters , and over sequential iterations. The mean parameter ( ! C ) exhibits rapid convergence within the first 5 iterations across all schemes, with final values of 0.343 mm (Scheme I), 0.362 mm (Scheme II), and 0.339 mm (Scheme III). These values remain stable throughout the evolution range, indicating fast convergence even with limited data. Physically, these values reflect manufacturing-induced micro-defects characteristic of high-quality aluminum specimens used in controlled fatigue testing. The hierarchical uncertainty parameter ( !D ) shows markedly different behaviour between schemes. Scheme I maintains relatively stable uncertainty (~2.0 mm), while Schemes II and III exhibit systematic increases from approximately 2.0 to 5.5 mm. This divergence reflects the integration of heterogeneous data sources: as online measurements accumulate, the difference between online and historical datasets becomes pronounced, manifesting as increased posterior uncertainty in the hierarchical structure rather than parameter instability. The increase (approaching 2.75 for hybrid schemes) do not indicate methodological limitations but rather capture legitimate sample-to-sample variations inherent in the hierarchical Bayesian framework. The effective distribution remains within physically meaningful ranges despite this apparent uncertainty magnification.

Fig. 5. Parameter inference results for ( ' ), and and under different updating schemes.