Reliability measure based on failure probability function and its solution by conditional probability simulation method

The solution of the failure probability as a function of the distribution parameters is the key problem in reliability-based optimization. A novel method is proposed to obtain the failure probability function and a new reliability measure, which is the statistics characteristic value of the failure probability function. The key idea of the proposed method is using the conditional probability Markov chain simulation and the third order maximum entropy method to obtain the failure probability function. The conditional probability Markov chain simulation firstly transforms the failure probability into the product of a feature ratio factor and the probability of an introduced linear failure region, and then Markov chain algorithm is adopted to calculate the ratio factor by directly simulating the samples of the failure regions, the probability of the introduced linear failure region can be calculated easily. The third order maximum entropy method is implemented to estimate the conditional density function based on failure samples and obtain the failure probability function finally. The accuracy, efficiency and applicability of the proposed method are demonstrated by several examples. The results show that the proposed method can efficiently estimate the failure probability with high accuracy. The proposed method should be valuable for reliability-based optimization and reliability sensitivity analysis.