AK-HMC-IS: A Novel Importance Sampling Method for Efficient Reliability Analysis Based on Active Kriging and Hybrid Monte Carlo Algorithm

Sampling methods are powerful tools for structural reliability analysis with complex failure domains due to their stability and accuracy. One of the most frequently used sampling methods is the importance sampling (IS) method, which can markedly reduce the sampling variance and computational costs. The pivotal problem in IS method is the determination of the IS probability density function (ISPDF), which influences the accuracy and efficiency of reliability analysis greatly. This study proposes an effective method for constructing the ISPDF, combining the hybrid Monte Carlo algorithm (HMC) with the Gaussian mixture model. The HMC is superior to the common Markov chain Monte Carlo algorithm in convergence, which is helpful in improving sampling efficiency. Our ISPDF is generated adaptively and does not require the most probable failure point (MPFP); therefore, it can also work well for multiple MPFPs and high-nonlinear problems. To release the computational burden further, the performance function is replaced with the Kriging model, and the well-known U criterion is used for its refinement. In the proposed method, the process of the refinement of the Kriging model is coupled with the HMC sampling for constructing the ISPDF, which is the difference between some common methods; thus, no samples are vain. We verify the proposed method using three classical numerical examples and one practical engineering problem. Results show that the proposed method is accurate and superior to common IS methods in efficiency.