Probabilistic hierarchical Bayesian framework for time-domain model updating and robust predictions

A new time-domain hierarchical Bayesian framework is proposed to improve the performance of Bayesian methods in terms of reliability and robustness of estimates particularly for uncertainty quantification and propagation in structural dynamics. The proposed framework provides a reliable approach to account for the variability of the inference results observed when using different data sets. The proposed formulation is compared with a state-of-the-art Bayesian approach using numerical and experimental examples. The results indicate that the hierarchical Bayesian framework provides a more realistic account of the uncertainties, whereas the non-hierarchical Bayesian approach severely underestimates them. Moreover, the proposed hierarchical framework predicts the system output quantities of interest with reasonable accuracy producing reliable uncertainty bounds, as opposed to the non-hierarchical approach which yields unrealistically narrow uncertainty bounds, although the model error is considerable. It is found that in the hierarchical approach the response prediction uncertainties are dominated by the uncertainties in the model parameters. As a result, the propagation of uncertainty can be performed using only the uncertainty of structural model parameters. This feature allows it making robust predictions of system output quantities of interest, especially where no information about the statistics of prediction error parameters is available. The hierarchical framework is proposed herein in the time-domain when incomplete input-output data is available. However, it has great potential to be applied to different forms of inference problems met in various disciplines of science and engineering.