Adaptive importance sampling algorithm and its application for multiple failure modes

For failure probability of a system with multiple failure modes, an adaptive importance sampling algorithm is developed. The importance sampling function for each failure mode is found and optimized by means of the simulated annealing method. During the optimization of the importance sampling function, the variance of the failure probability evaluation is reduced. For the system with multiple failure modes, a weighted mixed importance sampling function is proposed, in which the contribution of each failure mode to the system failure probability is represented appropriately. When not all basic variables are included in the limit state equation of some failure modes, an extended algorithm is presented to unify the basic variables in all failure modes, hence the weighted mixed importance sampling can be implemented successfully in the case. The variances and the coefficients of variation are derived for the failure probability evaluation. The feasibility and the validity of the presented method are illustrated by numerical and engineering examples.