Bayesian evidence for Finite element model updating

This paper considers the problem of model selection within the context of finite element model updating. Given that a number of FEM updating models, with different updating parameters, can be designed, this paper proposes using the Bayesian evidence statistic to assess the probability of each updating model. This makes it possible then to evaluate the need for alternative updating parameters in the updating of the initial FE model. The model evidences are compared using the Bayes factor, which is the ratio of evidences. The Jeffrey’s scale is used to determine the differences in the models. The Bayesian evidence is calculated by integrating the likelihood of the data given the model and its parameters over the a priori model parameter space using the new nested sampling algorithm. The nested algorithm samples this likelihood distribution by using a hard likelihood-value constraint on the sampling region while providing the posterior samples of the updating model parameters as a by-product. This method is used to calculate the evidence of a number of plausible finite element models. Nomenclature θ Model Parameters D Real measured system data H Finite element model (Mathematical model) Z Evidence N Number of samples Max Maximum number of iterations L Likelihood probability π Prior Probability c Damping matrix k Stiffness matrix m Mass matrix ( ) t x& & Node acceleration ( ) t x& Node velocity ( ) t x Node displacement w Natural frequency vector φ Mode shape vector FEM Finite element model FEMUP Finite element model updating problem PDF Probability Distribution Function MCMC Markov Chain Monte Carlo Proceedings of the IMAC-XXVII February 9-12, 2009 Orlando, Florida USA ©2009 Society for Experimental Mechanics Inc.