Quantification of Aleatory Uncertainty in Modal Updating Problems using a New Hierarchical Bayesian Framework

Identification of structural damage requires reliable assessments of damage-sensitive quantities, including natural frequencies, mode shapes, and damping ratios. Lack of knowledge about the correct value of these parameters introduces a particular sort of uncertainty often referred to as epistemic uncertainty. This class of uncertainty is reducible in a sense that it can be decreased by enhancing the modeling accuracy and collecting new information. On the contrary, such damage-sensitive parameters might also have intrinsic randomness arising from unknown phenomena and effects, which gives rise to an irreducible category of uncertainty often referred to as aleatory uncertainty. The present Bayesian modal updating methodologies can produce reasonable quantification of the epistemic uncertainties, while they often fail to account for the aleatory uncertainties. In this paper, a new multilevel (hierarchical) probabilistic modeling framework is proposed to bridge this significant gap in uncertainty quantification and propagation of structural dynamics inverse problems. Since multilevel model calibration schemes establish a complicated model structure associated with additional parameters and variables, their computational costs are often considerable, if not prohibitive. To reduce the computational costs, the modal updating procedure is simplified using a second-order Taylor expansion approximation. This approximation is combined with a Markov chain Monte-Carlo (MCMC) sampling method to compute marginal posterior distributions of quantities of interest. The proposed framework is illustrated using one simple experimental example. As a result, it is demonstrated that the proposed framework surpasses the present Bayesian modal updating methods as it accounts for both the aleatory and epistemic uncertainties.Financial support from the Hong Kong research grants councils under grant numbers 16234816 and 16212918 is gratefully appreciated. The last author gratefully acknowledges the European Commission for its support of the Marie Sklodowska Curie program through the ETN DyVirt project (GA 764547). This paper is completed as a part of the second authors PhD dissertation conducted jointly at Sharif University of Technology and the Hong Kong University of Science and Technology. The second author would like to gratefully appreciate kind support and supervision of Professor Fayaz R. Rofooei at Sharif University of Technology. We would also like to express our sincere appreciation to Professor Chih-chen Chang for generously sharing sensors, prototypes, and laboratory facilities