Bayesian Linear Structural Model Updating using Gibbs Sampler with Modal Data

Abstract

This paper presents a new Bayesian model updating approach for linear structural models based on the Gibbs sampler, a stochastic simulation method. We show that with incomplete modal data (modal frequencies and incomplete modeshapes of some lower modes), and with appropriate choices of conjugate priors, the uncertain stiffness and mass parameters of the linear structural model can be decomposed into three groups so that the sampling from any one group is possible when conditional on the other groups and the modal data. Such decomposition provides a major advantage for the Gibbs sampler: even if the number of uncertain parameters is large, the effective dimension for the Gibbs sampler is always three in this application. Therefore, high-dimensional parameter spaces that are fatal to most model-updating techniques are handled by the Gibbs sampler. The approach inherits the advantages of Bayesian techniques: it updates the joint probability distribution of the model parameters and so it not only gives the optimal estimates (most probable values) but also quantifies the associated uncertainties. The approach is illustrated by applying it to an example of structural health monitoring, where the goal is to detect and quantify any damage using modal data obtained from the undamaged and possibly damaged structure. Two cases are considered for the modal data: one where the model parameters are globally identifiable and one where they are unidentifiable. We also consider Bayesian model updating at a higher level: we address the issue of how to find the most probable model class among several different candidates. We explain the results for the model class selection problem in terms of concepts from information theory.