Nonlinear model updating through a hierarchical Bayesian modeling framework

Abstract

A new time-domain probabilistic technique based on hierarchical Bayesian modeling (HBM) framework is proposed for calibration and uncertainty quantification of hysteretic type nonlinearities of dynamical systems. Specifically, probabilistic hyper models are introduced respectively for material hysteretic model parameters as well as prediction error variance parameters, aiming to consider both the uncertainty of the model parameters as well as the prediction error uncertainty due to unmodeled dynamics. A new asymptotic approximation is developed to simplify the process of nonlinear model updating and substantially reduce the computational burden of the HBM framework. This asymptotic approximation is further employed to provide insightful expressions on the hyper parameters for both the model and prediction error variance parameters. Given a large number of data points within a dataset, the hyper model parameters are formulated to be independent of the hyper parameters for prediction error variance parameter. Two numerical examples are conducted to verify the accuracy and performance of the proposed method considering Bouc–Wen (BW) hysteretic type nonlinearities. Model error is manifested as uncertainty due to variability in the measured data from multiple datasets. Results from a five-story numerical structure indicate that the model error is the main source of error that can affect the uncertainty in the model parameters due to the variability in the experimental data. It is also demonstrated that the parameter uncertainty due to the variability arising from model error depends on the sensor locations. It is shown that the proposed approach is robust for not only quantifying uncertainties of structural parameters and prediction error parameters, but also predicting the system quantities of interests (QoI) with reasonable accuracy and providing reliable uncertainty bounds, as opposed to the conventional Bayesian approach which often severely underestimates the uncertainty bounds.