The statistical finite element method: A theoretical foundation for digital twins

Abstract

Digital twin modelling is now an established paradigm in data-centric engineering whereby measurements and models of assets are synthesized to create a digital representation. Underlying digital twin technologies are fundamental mechanistic models which can often be represented by differential equations. These systems of equations are solved numerically for generic tractability allowing for nonlinearity or complex geometries. A popular approach for solving these equations is the finite element method (FEM), which discretizes the governing differential equations using an accompanying mesh --- the discrete representation of the solution domain --- to yield approximations to the solution. FEM is a powerful deterministic tool but traditional use-cases fail to recognize the inherent uncertainty in the governing system, due to potentially unknown model parameters, boundary conditions, and so on. In this review, we introduce the statistical finite element method (statFEM) as a way to incorporate model uncertainty and construct solutions which, a posteriori, are calibrated with observed data. The statFEM is a fundamental, theoretically sound, tool with which digital twin models can be realized in practice. We cover background material, the basics of the methodology, and go over two case studies from previous works to illustrate the statFEM.