Virtual sensing based on Hierarchical Bayesian Modeling framework using a Laplace-based Gibbs sampler

This paper presents a Bayesian-based approach to realize virtual strain sensing and quantify the uncertainty in these estimates using sparse output-only measurements. This method relies on output-only measurements to identify the modal coordinates of the structure and realizes the virtual sensing according to the modal superposition method. A robust Hierarchical Bayesian modeling (HBM) framework is developed to fully account for the uncertainties arising from modeling errors and measurement noise. Moreover, the HBM framework has the ability to quantify the uncertainty of parameters in the statistical model by introducing an additional layer of the prior distribution. The Gibbs sampler is adopted to obtain the marginal posterior distributions of the hyper-parameters, with the implementation of Gibbs methods relying on the derived analytical expressions of the full conditional posterior distributions. The Laplace asymptotic approximation simplified the form of the full conditional posterior distributions, thus improving the computational efficiency. Given the hyper-parameters' marginal posterior distribution, the dynamic parameters' marginal posterior distribution can be calculated according to the total probability theorem. Finally, one can propagate the uncertainty to make predictions of quantities of interest (QoI). Three case studies, including two numerical examples and a laboratory model test, are carried out to verify the accuracy and efficiency of the proposed algorithm. The results show that the estimates of the parameters are close to the true values. Besides, the prediction results of QoI also show high accuracy, verifying the effectiveness of the proposed HBM method in virtual sensing.