Efficient reliability analysis based on Bayesian framework under input variable and metamodel uncertainties

In the reliability analysis, input variables as well as the metamodel uncertainties are often encountered in practice. The input uncertainty includes the statistical uncertainty of the distribution parameters due to the lack of knowledge or insufficient data. Metamodel uncertainty arises when the response function is approximated by a surrogate function using a finite number of responses to reduce the costly computations. In this study, a reliability analysis procedure is proposed based on a Bayesian framework that can incorporate these uncertainties in an integrated manner into the form of posterior PDF. The PDF, often expressed by arbitrary functions, is evaluated via Markov Chain Monte Carlo (MCMC) method, which is an efficient simulation method to draw random samples that follow the distribution. In order to avoid the nested computation in the full Bayesian approach, a posterior predictive approach is employed, which requires only a single loop of reliability analysis. Gaussian process model is employed for the metamodel. Mathematical and engineering examples are used to demonstrate the proposed method. In the results, comparing with the full Bayesian approach, the predictive approach provides much less information, i.e., only a point estimate of the probability. Nevertheless, the predictive approach adequately accounts for the uncertainties with much less computation, which is more advantageous in the design practice. The smaller the data are provided, the higher the statistical uncertainty, leading to the higher (or lower) failure probability (or reliability).