Time-variant reliability analysis using stratified Beta-sphere sampling and active learning

Estimating the failure probability of structures subjected to both time-invariant and time-variant stochastic inputs has long been reorganized as one of the most challenging tasks in structural engineering. Despite there are many developments for this problem, it still faces challenges in terms of accuracy and efficiency, especially for problems with small failure probability, highly nonlinearity and multiple disconnected failure domains that evolve over time. To fill this gap, a state-of-the-art stochastic simulation method utilizing stratified Beta-sphere sampling scheme is used to efficiently, accurately and robustly estimate the time-variant failure probability. Several novel developments, including a scheme to search the optimal training point, a single-layer strategy to train the Gaussian process regression (GPR) model, an adaptive filtering scheme to tackle the challenges caused by the potentially multiple failure domains, and remarkably, a new acquisition function for saving computational cost, have been presented in this work. The new acquisition function, called Time-variant Expected Integrated Error Reduction (TEIER) function, admits a prospective view as it measures the expected reward from refining the GPR model with a new point, and is capable of substantially reducing the required number of function calls. The superiority of the proposed methods in terms of efficiency, accuracy and robustness are demonstrated with numerical and engineering examples.