An importance bubble sampling method for small failure probability

In reliability analysis, numerical simulation methods have been widely adopted due to their simplicity and versatility. However, engineering practices frequently encounter small failure probability problems, which necessitate computationally intensive sampling to obtain the solutions with enough accuracy. To address this challenge, this study proposes an importance bubble sampling method grounded in the principles of importance sampling. The method establishes the structural MPP (Most Probable Point) as the sampling center, constructs bubbles using the shortest distance from the random samples to the limit state function as the radius These bubbles selectively filter stochastic samples within the bubble-constrained domain, thereby avoiding the computational burden associated with large sample sizes while enhancing the efficiency of reliability analysis. Numerical case studies demonstrate that the proposed importance bubble sampling method achieves sufficiently accurate failure probability with significantly reduced samples compared to conventional approaches.