The statistical finite element method (statFEM) for coherent synthesis of observation data and model predictions

Abstract

The increased availability of observation data from engineering systems in operation poses the question of how to incorporate this data into finite element models. To this end, we propose a novel statistical construction of the finite element method that provides the means of synthesising measurement data and finite element models. The Bayesian statistical framework is adopted to treat all the uncertainties present in the data, the mathematical model and its finite element discretisation. From the outset, we postulate a statistical generating model which additively decomposes data into a finite element, a model misspecification and a noise component. Each of the components may be uncertain and is considered as a random variable with a respective prior probability density. The prior of the finite element component is given by a conventional stochastic forward problem. The prior probabilities of the model misspecification and measurement noise, without loss of generality, are assumed to have a zero-mean and a known covariance structure. Our proposed statistical model is hierarchical in the sense that each of the three random components may depend on one or more non-observable random hyperparameters with their own corresponding probability densities. We use Bayes rule to infer the posterior densities of the three random components and the hyperparameters from their known prior densities and a data dependent likelihood function. Because of the hierarchical structure of our statistical model, Bayes rule is applied on three different levels in turn. On level one, we determine the posterior densities of the finite element component and the true system response using the prior finite element density given by the forward problem and the data likelihood. In this step, approximating the prior finite element density with a multivariate Gaussian distribution allows us to obtain a closed-form expression for the posterior. On the next level, we infer the hyperparameter posterior densities from their respective priors and the marginal likelihood of the first inference problem. These posteriors are sampled numerically using the Markov chain Monte Carlo (MCMC) method. Finally, on level three we use Bayes rule to choose the most suitable finite element model in light of the observed data by computing the respective model posteriors. We demonstrate the application and versatility of statFEM with one and two-dimensional examples.