SOH-DLF: A double-loop framework for time-dependent system reliability analysis

Time-dependent reliability analysis quantifies the evolution of system failure probabilities under time-varying uncertainties, such as dynamic loading and material degradation. However, surrogate model-based methods still face challenges in sampling efficiency and modeling accuracy, particularly in high-dimensional problems and scenarios involving multiple failure modes. To address these challenges, this study introduces a double-loop time-dependent system reliability analysis (TSRA) framework based on the Safe Optimal Hypersphere (SOH) method, referred to as SOH-DLF. The main contributions of this study are as follows: First, a SOH-based Kriging-guided stratified learning strategy is developed, which automatically partitions the Monte Carlo sample space, reduces the candidate sample pool, and simultaneously parallelizes new samples to update the surrogate model, thereby improving sampling efficiency and modeling accuracy. Second, an enhanced SOH search strategy is implemented, integrating performance function information to provide better directional guidance and accelerate the convergence of the β-iteration search. Finally, by tightly integrating SOH-based sampling, Kriging surrogate modeling, and failure probability estimation, a closed-loop process is established. The effectiveness of the SOH-DLF framework is demonstrated through three numerical examples and one black-box engineering application, confirming its accuracy, computational efficiency, and broad applicability to complex time-dependent structural systems.