A variance reduction sampling method to efficiently estimate the probability-of-failure for damage-tolerant structures

A common metric in a structural reliability analysis is the probability-of-failure (POF), the probability that a structure does not meet a design requirement. In general, when the failure domain is complex the POF is estimated using sampling-based methods. The most recognized method is Monte Carlo simulation, even though it is a straightforward method it is inefficient for obtaining accurate estimates of the POF when the POF is small. Increasing the efficiency of the sampling method can be accomplished by using a variance reduction technique such as importance sampling; however, determining the importance sampling density is a nontrivial task. Furthermore, in time dependent problems such as fatigue crack growth explored here, the POF changes in time requiring an approach for updating the importance sampling density as the problem evolves. In this work a quantitative approach is developed to estimate the importance sampling density in order to reduce the variance in the POF estimate. The method uses the Kullback-Leibler cross-entropy (KL-CE) to describe the distance between the optimal importance sampling density and a proposal importance sampling density named the near-optimal density. A modified KL-CE method is developed in this research to handle functions that are computed using the conditional expectation method. Since the importance sampling density evolves as a function of time, a strategy is proposed and implemented to calibrate the parameters of the near-optimal importance sampling density periodically. Additionally, a modified weighted branch integration method is used to update the damage detection information and repairs at different times. The proposed methodology is applied to several fracture mechanics problems and is integrated with current probabilistic software, which includes crack growth packages.