On propagating uncertainty from stochastic subcomponents using the statistical finite element method

The ability to explore the effects of unknown properties (e.g. material properties) on finite element models is of particular interest to practitioners; offering valuable insight and aiding decision-making; this is particularly true in the context of structural health monitoring (SHM), where one may want to understand the effect that a a reduction in stiffness in one component might have over the possible set of solutions. The statistical finite element method (statFEM) has emerged as a new approach to address this challenge, where the associated uncertainty over the possible set of finite element solutions can be suitably quantified in a Bayesian framework. Despite its effectiveness, statFEM requires significant resource to compute the uncertainty over the entire domain. This challenge has highlighted the need for alternative approaches that can reduce this burden and make statFEM suitable for SHM. In this work, the authors consider the problem of stochastic subcomponents, where only part of the domain is represented by a random field. Under the existing statFEM approach, the covariance over the solution requires consideration of the entire domain. In the proposed method, the domain is separated into stochastic and deterministic subdomains, and block matrix algebra used to compute the uncertainty. This approach allows the covariance over the stochastic domain to be computed on a reduced set of elements and propagated using affine transformations to the full domain solution. This method is demonstrated here on a simulated case study of a steel-reinforced concrete beam cross-section.