Bayesian Hierarchical Models for Uncertainty Quantification in Structural Dynamics

The Bayesian framework for hierarchical modeling is applied to quantify uncertainties, arising mainly due to manufacturing variability, for a group of identical structural components. Parameterized Gaussian models of uncertainties in structural model parameters are introduced that depend on a set of hyperparameters. Laplace methods of asymptotic approximation and, more accurate, stochastic simulation algorithms are used for estimating posterior joint and marginal distributions of structural model parameters and hyper-parameters. The posterior distribution of the hyperparameters is formulated as the product of as many multi-dimensional integrals as the number of structural components tested, with the dimension in each integral equal to the number of physical uncertain model parameters. Novel analytical approximations of these integrals are proposed that are shown to be computationally efficient, accurate and highly parallelized. A parallelized blocked Gibbs sampler is also proposed to efficiently generate samples simultaneously along the different uncertain parameter subspaces associated with the different number of structural component tested. Theoretical and computational developments are illustrated using simulated data for modal frequencies obtained from a number of identical bar structures.