Error Estimation and Error Reduction in Separable Monte-Carlo Method

Reliability-based design often uses the Monte-Carlo method as a sampling procedure for predicting failure. The combination of designing for very small failure probabilities (similar to 10(-8) - 10(-6)) and using computationally expensive finite element models, makes Monte-Carlo simulations very expensive. This paper uses an improved sampling procedure for calculating the probability of failure, called separable Monte-Carlo method. The separable Monte-Carlo method can improve the accuracy of the traditional crude Monte-Carlo when response and capacity are independent. In previous research, accuracy of separable Monte-Carlo for a simple limit state was estimated via expectation calculus for a simple form of the limit state. In this paper, error estimates for a general limit state are developed through bootstrapping, and it is demonstrated that the estimates are reasonably accurate. Separable Monte-Carlo allows us to choose different sample sizes of the response and capacity in the limit state, and the paper demonstrates that bootstrapping may be used to estimate the contribution of the response and capacity to the total error. When the accuracy of the probability of failure is not good enough, the paper proposes reformulation of the limit state as another way to reduce uncertainty associated with the expensive random variable (usually the response). The accuracy of the bootstrapping estimates and the effectiveness of regrouping is demonstrated with an example of prediction of failure in a composite laminate with the Tsai-Wu failure criterion.