## Project

# Interval methods for the identification and quantification of inhomogeneous uncertainty in Finite Element models

Numerical methods, such as the finite element method, have become an indispensable tool in the toolbox of a modern design engineer. Since these methods allow for predicting the mechanical, thermal or aerodynamic behaviour of a structural component, long before a first prototype has been produced, an immense reduction of lead-time is possible. However, it is impossible to determine all numerical model parameters deterministically, as the true parameter value is often uncertain or inherently variable. Moreover, also spatial uncertainty can exist, where the uncertainty on the model parameter is a function of the location in the structure. Therefore, it is not recommendable to base important design decisions solely on such deterministic numerical analysis, as an over-conservative design has to be constructed to prevent premature failure.

During the last decades, very powerful non-deterministic numerical methods have been proposed. These techniques are usually divided in two groups: probabilistic and possibilistic methods. Probabilistic methods start from a joint probability density function to describe possible variability in the responses of the structure. Possibilistic methods on the other hand describe the uncertainty using intervals or fuzzy numbers. In case spatial uncertainty is present, respectively random fields and interval fields are used.

However, in order to ensure that these advanced methods also yield a truthful estimation of the possible model responses, it is important that the non-deterministic modelling of the parameters is made accurately. In some cases (e.g., plate thickness), quantification of the variability of uncertainty is relatively straightforward. Other parameters however (e.g., contact stiffness) are not directly measurable. Therefore, inverse uncertainty quantification methods should be applied. In the specific context of interval field uncertainty, such methods do not exist in literature to date.

This thesis is therefore aimed at the development and validation of a novel generic method for the identification and quantification of interval field uncertainty in the parameters of a numerical model. The method should be applicable to both dynamic and quasi-static models, and has to be able to handle both computationally demanding models and large measurement data sets. The performance of the method is tested by considering both small-scale academic case studies, as well as realistic numerical models in conjunction with experimental datasets.

The proposed methods start from the description of the uncertainty in the model responses that are predicted by the numerical model and those that are experimentally obtained. These convex hulls define a convex region in which the possible model responses are deemed to be located, based on a set of linear inequalities. The identification and quantification of interval field uncertainty is then performed by minimising the discrepancy between both convex hulls. To limit the computational burden of this identification, two different methods are proposed to construct a lower-order representation of the convex hulls. A profound description of the presented methods is given in chapter 3 of the thesis. Also, specific additions are presented for the specific application of the method to dynamic and quasi-static problems. Specifically, in the latter case, it is proposed to use contactless full-field strain measurement techniques for the construction of the measurement data set, as it is expected that the high spatial resolution facilitates the interval field identification and quantification.

The performed case studies show that the developed method is capable to quantify the interval field uncertain parameters of a numerical model with high accuracy within limited computational cost. However, some considerations should be made. First, the results of the quantification depend largely on the quality of the measurement data set that is used. In the method, some techniques are included to account for e.g., scarceness of the data, but also these techniques have a limit when insufficient accurate data are available. Secondly, when large scale problems are considered containing numerous uncertain parameters, the computational expense is drastically increased. This is mainly caused by the propagation routines for propagating this high-dimensional uncertainty. Finally, it is found that the applied full-field strain measurements do not have sufficient resolution to perform an accurate interval field identification. The latter two points therefore constitute the most important suggestions for future research.