Modified Bayesian Kriging for noisy response problems and Bayesian confidence-based reliability-based design optimization

Abstract

The objective of this study is to develop a new modified Bayesian Kriging (MBKG) surrogate modeling method that can be used to carry out confidence-based reliability-based design optimization (RBDO) for problems in which simulation analyses are inherently noisy and standard Kriging approaches fail. The formulation of the MBKG surrogate modeling method is presented, and the full conditional distributions of the unknown MBKG parameters are derived and coded into a Gibbs sampling algorithm. Using the coded 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 new DoE sample points added 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, it improves the posterior distribution of the probability of failure efficiently. Finally, a confidence-based RBDO method using the posterior distribution of the probability of failure is developed. The confidence-based RBDO method is developed so that the uncertainty of the MBKG surrogate model is included in the optimization process. A 2-D mathematical example was used to demonstrate fitting the MBKG surrogate model and the developed sequential sampling method that uses the posterior credible sets for inserting new DoE. A detailed study on how the posterior distribution of the probability of failure changes as new DoE are added using the developed sequential sampling method is presented. Confidence-based RBDO is carried out using the same 2-D mathematical example. Three different noise levels are used for the example to compare how the MBKG surrogate modeling method, the sequential sampling method, and the confidence-based RBDO method behave for different amounts of noise in the response. A comparison of the optimization results for the three different noise levels for the same 2-D mathematical example is presented. A 3-D multibody dynamics (MBD) engineering block-car example is presented. The example is used to demonstrate using the developed methods to carry out confidence-based RBDO for an engineering problem that contains noise in the response. The MBD simulations for this example were done using the commercially available MBD software package RecurDyn. Deterministic design optimization (DDO) was first done using the MBKG surrogate model to obtain the mean response values, which then were used with standard Kriging methods to obtain the sensitivity of the responses. Confidence-based RBDO was then carried out using the DDO solution as the initial design point.

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