Modified Bayesian Kriging for Noisy Response Problems for Reliability Analysis

Abstract

This paper develops a new modified Bayesian Kriging (MBKG) surrogate modeling method for problems in which simulation analyses are inherently noisy and thus standard Kriging approaches fail to properly represent the responses. The purpose is to develop a method that can be used to carry out reliability analysis to predict probability of failure. The formulation of the MBKG surrogate modeling method is presented, and the full conditional distributions of the unknown MBKG parameters are presented. Using the full conditional distributions with a Gibbs sampling algorithm, Markov chain Monte Carlo is used to fit the MBKG surrogate model. A sequential sampling method that uses the posterior credible sets for inserting new design of experiment (DoE) sample points is proposed. The sequential sampling method is developed in such a way that the newly added DoE sample points will provide the maximum amount of information possible to the MBKG surrogate model, making it an efficient and effective way to reduce the number of DoE sample points needed. Therefore, the proposed method improves the posterior distribution of the probability of failure efficiently. To demonstrate the developed MBKG and sequential sampling methods, a 2-D mathematical example with added random noise is used. It is shown how, with the use of the sequential sample method, the posterior distribution of the probability of failure converges to capture the true probability of failure. A 3-D multibody dynamics (MBD) engineering block-car example illustrates an application of the new method to a simple engineering example for which standard Kriging methods fail.