The failures of mechanical structure featuring high nonlinearity, non-normal and non-independent are implicit function and small-probability events. This normally results in low computational efficient and accuracy for gradient algorithm scenario, which can hardly calculate models for large complex structure and flexible systems. To deal with the above constraints, an efficient and accurate reliability numerical method named adaptive reliability index importance sampling-based extended domain PSO (ARIIS-EDPSO) is proposed to combine the reliability numerical simulation and the particle swarm optimization (PSO) algorithm. The reliability index and limit state equation in ARIIS-EDPSO are regarded as the objective function and the constraint function. The Nataf transformation is adopted to complete the conversion process from an original variable space to an independent standard normal space, which only requires the marginal probability density function and the correlation coefficient among the random variables. To verify the effectiveness of the proposed ARIIS-EDPSO, experimental studies are conducted with five case studies. The results indicate that the constraint conflict function obtained via ARIIS-EDPSO is smaller than that is obtained via the other methods, and its convergence can be guaranteed. Also, the accuracy of the ARIIS-EDPSO is superior to the other methods for nonlinear reliability calculation. Furthermore, the ARIIS-EDPSO can accurately predict the failure probability. This approach exhibits advantageous global search ability, high efficiency and high accuracy in solving constrained reliability engineering problems.