An innovative estimation of failure probability function based on conditional probability of parameter interval and augmented failure probability

The failure probability function (FPF) is defined as a function of failure probability varying with the design parameters of random inputs, and it is usually estimated in advance to transform reliability-based design optimization into ordinary one. The existing method on Bayes formula can transform the FPF as the estimation of the augmented failure probability (AFP) and conditional joint probability density function (PDF) of design parameters on the failure event. Although the existing method can estimate the FPF by one Monte Carlo simulation (MCS), it suffers the limited capability of the existing PDF estimation techniques in capturing the tail characteristic and being extended to multi-dimensional problem. To alleviate this issue, an innovative strategy is presented by introducing a small interval of parameters to approximate the parameter, on which the FPF can be transformed into the estimation of the AFP and the conditional probability of the parameter interval by the Bayes formula. After the criterion of selecting the small intervals of the parameters is determined to satisfy the approximation precision, a single MCS and the adaptive Kriging nested MCS (AK-MCS) are organized to estimate AFP and the conditional probability of the parameter interval simultaneously. By introducing a small interval to approximate the parameter and replacing the conditional PDF estimation with the conditional probability one, the applicability and accuracy of the proposed method are greatly improved to the existing method. Two examples are employed to verify the accuracy and efficiency of the proposed method. The results show that the proposed method is more accurate than the existing method, and the AK-MCS is more efficient than the direct MCS, the small enough computational cost helps the proposed method be used in actual engineering application.