Stochastic Analysis, Model and Reliability Updating of Complex Systems with Applications to Structural Dynamics

Abstract

In many engineering applications, it is a formidable task to construct mathematical models that are expected to produce accurate predictions of the behavior of a system of interest. During the construction of such predictive models, errors due to imperfect modeling and uncertainties due to incomplete information about the system and its environment (e.g., input or excitation) always exist and can be accounted for appropriately by using probability logic. To assess the system performance subjected to dynamic excitations, a stochastic system analysis considering all the uncertainties involved has to be performed. In engineering, evaluating the robust failure probability (or its complement, robust reliability) of the system is a very important part of such stochastic system analysis. The word 'robust' is used because all uncertainties, including those due to modeling of the system, are taken into account during the system analysis, while the word 'failure' is used to refer to unacceptable behavior or unsatisfactory performance of the system output(s). Whenever possible, the system (or subsystem) output (or maybe input as well) should be measured to update models for the system so that a more robust evaluation of the system performance can be obtained. In this thesis, the focus is on stochastic system analysis, model and reliability updating of complex systems, with special attention to complex dynamic systems which can have high-dimensional uncertainties, which are known to be a very challenging problem. Here, full Bayesian model updating approach is adopted to provide a robust and rigorous framework for these applications due to its ability to characterize modeling uncertainties associated with the underlying system and to its exclusive foundation on the probability axioms. First, model updating of a complex system which can have high-dimensional uncertainties within a stochastic system model class is considered. To solve the challenging computational problems, stochastic simulation methods, which are reliable and robust to problem complexity, are proposed. The Hybrid Monte Carlo method is investigated and it is shown how this method can be used to solve Bayesian model updating problems of complex dynamic systems involving high-dimensional uncertainties. New formulae for Markov Chain convergence assessment are derived. Advanced hybrid Markov Chain Monte Carlo simulation algorithms are also presented in the end. Next, the problem of how to select the most plausible model class from a set of competing candidate model classes for the system and how to obtain robust predictions from these model classes rigorously, based on data, is considered. To tackle this problem, Bayesian model class selection and averaging may be used, which is based on the posterior probability of different candidate classes for a system. However, these require calculation of the evidence of the model class based on the system data, which requires the computation of a multi-dimensional integral involving the product of the likelihood and prior defined by the model class. Methods for solving the computationally challenging problem of evidence calculation are reviewed and new methods using posterior samples are presented. Multiple stochastic model classes can be created even there is only one embedded deterministic model. These model classes can be viewed as a generalization of the stochastic models considered in Kalman filtering to include uncertainties in the parameters characterizing the stochastic models. State-of-the-art algorithms are used to solve the challenging computational problems resulting from these extended model classes. Bayesian model class selection is used to evaluate the posterior probability of an extended model classe and the original one to allow a data-based comparison. The problem of calculating robust system reliability is also addressed. The importance and effectiveness of the proposed method is illustrated with examples for robust reliability updating of structural systems. Another significance of this work is to show the sensitivity of the results of stochastic analysis, especially the robust system reliability, to how the uncertainties are handled, which is often ignored in past studies. A model validation problem is then considered where a series of experiments are conducted that involve collecting data from successively more complex subsystems and these data are to be used to predict the response of a related more complex system. A novel methodology based on Bayesian updating of hierarchical stochastic system model classes using such experimental data is proposed for uncertainty quantification and propagation, model validation, and robust prediction of the response of the target system. Recently-developed stochastic simulation methods are used to solve the computational problems involved. Finally, a novel approach based on stochastic simulation methods is developed using current system data, to update the robust failure probability of a dynamic system which will be subjected to future uncertain dynamic excitations. Another problem of interest is to calculate the robust failure probability of a dynamic system during the time when the system is subjected to dynamic excitation, based on real-time measurements of some output from the system (with or without corresponding input data) and allowing for modeling uncertainties; this generalizes Kalman filtering to uncertain nonlinear dynamic systems. For this purpose, a novel approach is introduced based on stochastic simulation methods to update the reliability of a nonlinear dynamic system, potentially in real time if the calculations can be performed fast enough.

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