Conservative data-driven finite element framework

Abstract

This paper presents a new data-driven finite element framework that is applicable to a broad range of engineering simulation problems. In the data-driven approach, the conservation laws and boundary conditions are satisfied by means of the finite element method, while the experimental data is used directly in numerical simulations, avoiding material models. Critically, we introduce a "weaker'" mixed finite element formulation, which relaxes the regularity requirements on the approximation space for the primary field. At the same time, the continuity of the normal flux component is enforced across inner boundaries, allowing the conservation law to be satisfied in the strong sense. The relaxed regularity of the approximation spaces makes it easier to observe how imperfections in the datasets, such as missing or noisy data, result in non-uniqueness of the solution. This can be quantified to predict the uncertainty of the results using methods such as Markov chain Monte Carlo. Furthermore, this formulation provides a posteriori error indicators tailored for the data-driven approach, providing confidence in the results and enabling efficient solution schemes via adaptive hp-refinement. The capabilities of the formulation are demonstrated on an example of the nonlinear heat transfer in nuclear graphite using synthetically generated material datasets. This work provides an essential component for numerical frameworks for complex engineering systems such as digital twins.