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Stochastic model updating and model selection with application to structural dynamics

Doctoral thesis, 2016

Uncertainty induced by our incomplete state of knowledge about engineering systems and their surrounding environment give rise to challenging problems in the process of building predictive models for the system behavior. One such challenge is the model selection problem, which arises due to the existence of invariably multiple candidate models with different mathematical forms to represent the system behavior, and so there is a need to assess their plausibility based on experimental data. However, model selection is a non-trivial problem since it involves a trade-off between predictive power and simplicity. Another challenge is the model updating problem, which refers to the process of inference of the unknown parameters of a specific model structure based on experimental data so that it makes more accurate predictions of the system behavior. However, the existence of modeling errors and uncertainties, e.g., the measurement noise and variability in material properties, along with sparsity of data regarding the parameters often make model updating an ill-conditioned problem. In this thesis, probabilistic tools and methodologies are established for model updating and selection of structural dynamic systems that can deal with the uncertainty arising from missing information, with special attention given to systems which can have high-dimensional uncertain parameter vector. The model updating problem is first formulated in the \textit{Frequentist} school of statistical inference. A framework for stochastic updating of linear finite element models and the uncertainty associated to their parameters is developed. It uses the techniques of damping equalization to eliminate the need for mode matching and bootstrapping to construct uncertainty bounds on the parameters. A combination of ideas from bootstrapping and unsupervised machine learning algorithms lead to an automated modal updating algorithm suitable for identification of large-scale systems with many inputs and outputs. The model updating problem is then formulated in the \textit{Bayesian} school of statistical inference. A recently appeared multi-level Markov chain Monte Carlo algorithm, ABC-SubSim, for approximate Bayesian computation is used to solve Bayesian model updating for dynamic systems. ABC-SubSim exploits the Subset Simulation method to efficiently draw samples from posterior distributions with high-dimensional parameter spaces. Formulating a dynamic system in form of a general hierarchical state-space model opens up the possibility of using ABC-SubSim for Bayesian model selection. Finally, to perform the exact Bayesian updating for dynamic models with high-dimensional uncertainties, a new multi-level Markov chain Monte Carlo algorithm called Sequential Gauss-Newton algorithm is proposed. The key to success for this algorithm is the construction of a proposal distribution which locally approximates the posterior distribution while it can be readily sampled.

Finite element model

Bayesian model selection

Bayesian model updating

stochastic simulation

Bootstrapping

Subspace system identification

Uncertainty quantification