Bayesian damage identification from elastostatic data

Many problems in science and engineering consist of estimating some unknown parameters of a physical system from limited and noisy observations. In practical conditions, these problems are often unstable; hence, they should be treated in an appropriate framework in the theory of inverse problems. The regularisation framework for inverse problems is concerned with stabilisation of unstable problems. In this framework, the unknown is taken as a deterministic multi-dimensional quantity and the observation error is taken to be unknown without any probabilistic structure. In the Bayesian framework for inverse problems, which is the focus of this thesis, the unknown and observation error are explicitly taken and modelled as jointly distributed random variables. One of the advantages of the Bayesian framework is that it can quantify the uncertainty of a point estimate for the unknown, unlike the deterministic framework.
The problem of damage identification in the theory of elasticity is one of the practical examples of inverse problems. This problem, which usually involves high-order differential equations, is to estimate the damage occurred in a continuum body given some data, for example, on the elastic deformation of the body. When the deformation is vibrational, the deformation data is a rich source of information about the underlying physical system. However, when the deformation is in steady-state conditions (elastostatic deformation), the deformation data is a rather limited source of information about the underlying physical system. Hence, the available literature on the damage identification of continuum bodies from elastostatic data is limited.
In this thesis, we consider a paradigm for Bayesian damage identification of continuum bodies from limited and noisy elastostatic data. We consider three separate cases of continuum bodies including two one-dimensional cases and one two-dimensional case. For the one-dimensional cases, we use the Timoshenko beam theory to model the elastostatic deformation of the damaged bodies. For the two-dimensional case, we use the Kirchhoff- Love plate theory to do so.
For each case, we first derive a finite-dimensional forward model relating the parameters representing the damage (unknown) to the given data. The aim is to obtain a simple forward model that does not impose computationally expensive methods on the next steps of the solution method. Second, we propose a probabilistic structure for modelling a suitable prior density function for the unknown parameters. Third, we assume a multi-variate Gaussian density function for the observation error and derive an approximate posterior density function for the unknowns from the prior density function and the forward model. Fourth, we suggest to maximize the posterior density function to obtain a maximum a posteriori point estimate for the unknown. Finally, we quantify the uncertainty associated with the maximum a posteriori point estimate by using the characteristics of the posterior density function. The results suggest that the maximum a posteriori point estimate can efficiently predict the occurred damage even if the data is limited and noisy. Moreover, the actual damage field lies within the one-standard-deviation interval of the point estimate.