An adaptive importance sampling algorithm and its application for multiple failure modes

For failure probability calculation of system with multiple failure modes,an adaptive importance sampling algorithm is developed. The importancesampling function for each failure mode is sought and optimized by means ofthe simulated annealing method. During the optimization of the importancesampling function, the variance of the failure probability evaluation isdecreased. For the system with multiple failure modes, a weighted mixingimportance sampling function is proposed, in which the contribution of eachfailure mode to the system failure probability is represented appropriately.When not all basic variables are included in the limit state equation ofsome failure modes, an extended algorithm is presented to unify the basicvariables in all failure modes, hence the weighted mixing importancesampling can be implemented successfully in the case. The variances and thecoefficients of variation are derived for the failure probabilityevaluation. The feasibility and the rationality of the presented method areillustrated by numerical example and engineering example.