Model and parameter identification through Bayesian inference in solid mechanics

Predicting the behaviour of various engineering systems is commonly performed using mathematical models. These mathematical models include application-specific parameters that must be identified from measured data. The identification of model parameters usually comes with uncertainties due to model simplifications and errors in the experimental measurements. Quantifying these uncertainties can effectively improve the predictions as well as the performance of the engineering systems.

Bayesian inference provides a probabilistic framework for quantifying these uncertainties in parameter identification problems. In a Bayesian framework, the user's initial knowledge, which is represented by a probability distribution, is updated by measurement data through Bayes' theorem.

In the first two chapters of this thesis, Bayesian inference is developed for the identification of material parameters in elastoplasticity and viscoelasticity. The effect of the user's prior knowledge is systematically studied with respect to the number of measurements available. In addition, the influence of different types of experiments on the uncertainty is studied.

Since all mathematical models are simplifications of reality, uncertainties of the model itself may also be incorporated. The third chapter of this thesis presents a Bayesian framework for parameter identification in elastoplasticity in which not only the uncertainty of the experimental output is included (i.e. stress measurements), but also the uncertainty of the model and the uncertainty of the experimental input (i.e. strain). Three different formulations for describing the model uncertainty are considered: (1) a random variable which is taken from a normal distribution with constant parameters, (2) a random variable which is taken from a normal distribution with an input-dependent mean, and (3) a Gaussian random process with a stationary covariance function.

In the fourth chapter of this thesis, a Bayesian scheme is proposed to identify material parameter distributions, instead of material parameters. The application in this chapter are random fibre networks, in which the set of material parameters of each fibre is assumed to be a realisation from a material parameter distribution. The fibres behave either elastoplastically or in a perfectly brittle manner. The goal of the identification scheme is to avoid the experimentally demanding task of testing hundreds of constituents. Instead, only 20 fibres are considered.

In addition to their material randomness, the macroscale behaviours of these fibre networks are also governed by their geometrical randomness. Another question aimed to be answered in this chapter is therefore is `how precise the material randomness needs to be identified, if the geometrical randomness will also influence the macroscale behaviour of these discrete networks'.