**Bayesian inference to identify material parameters in solid mechanics**

A commonly employed approach to identify material parameters in mechanics is the standard least squares method in which the sum of the squared residuals (i.e. the difference between the measurement and the model response) is minimised with respect to the parameters of interest. An alternative way is the Bayesian approach which can provide a quantification of uncertainty. Bayesian inference results in a probability density function (PDF), a so-called posterior distribution, which is a function of the material parameters of interest and represents our uncertainty about the material parameter. Once the posterior is established, the statistical summaries of the material parameters, e.g. the mean values of material parameters, the material properties at which the PDF is maximum (called the ‘maximum-a-posteriori-probability’ or ‘MAP’ point) and the standard deviation (i.e. a quantity that shows how much measurements for a group are different from their mean) can be obtained by analysing the posterior distribution. This analysis can be either by computing the statistical moments of the posterior (e.g. the first moment of a PDF is its mean value) or by simulating samples from the posterior. It is important to note that from a Bayesian perspective, probability distributions are not real things but are models chosen, or derived from modelling assumptions, to represent the user’s uncertainty about the values of quantities such as model parameters and measurement and model prediction errors.In this framework the prior knowledge (so-called prior distribution which chosen by the user) is updated by data through Bayes’ theorem.

**Aims**

Two major goals are defined in this project: (1) employing Bayesian inference for parameter identification in the solid mechanics and (2) developing a scheme that enables us to identify material parameters joint PDF (i.e. the PDFs that represents the material randomness) with** limited number** of experiments/experimental specimens.

The second case can be used to identify material randomness in structures of discrete fibres, yarns or struts. Material randomness can be considered as one of the factors that can characterise the structural behaviours of these structures at a small length scale. One possible way to identify the material randomness is to test numerous small fibres, yarns of struts but this entails a substantial amount of experimental efforts. In our study, we employ Bayes’ theorem to only test a **relatively small number** of specimens and use their results to infer the parameters of an initially assumed distribution.

**Team members
**

This work is done by Mr Hussein Rappel as part of his PhD thesis under supervision of Dr Lars Beex, Prof. Stéphane Bordas and assoc-Prof. Ludovic Noels .

**Financial support**

This project is financially supported by Computational Sciences Research Priority of the University of Luxembourg.

**Publication**

[1] H. Rappel, L. A. A. Beex, S. P. A. Bordas, Bayesian inference to identify parameters in viscoelasticity, Mechanics of Time-Dependent Materials. URL https://doi.org/10.1007/s11043-017-9361-0

[2] H. Rappel, L. A. A. Beex, J. S. Hale, L. Noels, S. P. A. Bordas, A tutorial on Bayesian inference to identify material parameters in solid mechanics, submitted for publication.

[3] H. Rappel, L. A. A. Beex, L. Noels, S. P. A. Bordas, Identifying elastoplastic parameters with Bayes theorem considering double error sources and model uncertainty, submitted for publication.

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