In this study, variational Bayesian inference (VBI) with Gaussian mixture models is applied to update models of nonlinear structures, and then, the calibrated model is employed to estimate the failure probability of structures using a subset simulation (SS) algorithm. To improve the computation efficiency of probabilistic nonlinear model updating, a Gaussian Process (GP) model is used to construct a surrogate likelihood function in Bayesian inference using an active learning algorithm, and then, Gaussian mixture models (GMMs) are employed to approximate the unknown posterior probabilistic density functions (PDFs) of model parameters. The optimized hyperparameters of GMMs can be obtained by maximizing the evidence lower bound (ELBO), and the stochastic gradient search method is used to solve this optimization problem. Based on the optimized hyperparameters, the posterior distributions of model parameters can be approximated using a combination of multiple Gaussian components. Subsequently, the SS algorithm is used to calculate the earthquake-induced failure probability of structures based on the calibrated nonlinear model. To verify the feasibility and effectiveness of the proposed method, a numerical simulation of a two-span bridge structure subjected to seismic excitations was developed. Moreover, the proposed strategy is further applied to estimate the failure probability of a scaled monolithic column structure subjected to bi-directional earthquake excitations. Both numerical and experimental results indicate that the proposed method is feasible and effective for probabilistic nonlinear model updates, and the updated model can significantly enhance the accuracy of structural failure probability predictions.
Hou et al. (Fri,) studied this question.