For the multi-failure-mode system with multi-dimensional distribution parameter, estimating system failure probability function (SFPF) is essential to grasp the influence of the distribution parameter on system failure probability and decouple the reliability-based design optimization model constrained by system failure probability. However, there is still a significant challenge in efficiently estimating the SFPF at present. Therefore, an efficient and universal algorithm is proposed in this paper for estimating the SFPF. In the proposed algorithm, a unified sampling probability density function, which is independent with the distribution parameter, is originally constructed by the integration operation over the concerned design domain of the distribution parameter, on which a single-loop numerical simulation can be formulated to simultaneously estimate the SFPF at arbitrary realization of multi-dimensional distribution parameter. Since the single-loop method is used to replace the direct double-loop one in the proposed algorithm, the computational efficiency is greatly improved in estimating the SFPF. Additionally, the proposed algorithm has no restriction on the dimensionality and the concerned design domain of the distribution parameter, and an adaptive Kriging model of the system performance function is embedded to help the proposed algorithm further improve the computational efficiency. A new adaptive learning strategy, which considers the possible correlations among the multi-failure-mode Kriging models, is presented using the probability of the multi-failure-mode Kriging model misjudging the candidate sample state. The superiority of the proposed algorithm in terms of single-loop numerical simulation and the new learning strategy over the existing methods is fully demonstrated by numerical and engineering examples.
