Data-driven weaker mixed formulation for diffusion problems
Creators: name: Kulikova, Adriana
Publisher: name: Zenodo
Year: 2025
Subjects: Data-driven mechanics
Finite element method
Mixed formulation
Nonlinear diffusion problem
hp-refinement

description: This thesis presents a data-driven mixed finite element framework for solving elliptic equations, i.e., nonlinear diffusion problems, focusing on heat transfer in porous media such as nuclear graphite. Traditional computational approaches rely on phenomenological material models, which contain empirically estimated parameters. However, these models carry uncertainty regarding the values of these parameters. By selecting a single value for each parameter, the uncertainty is often neglected, leading to inaccuracies or overconfidence, particularly with complex or inherently stochastic materials.

In contrast, the data-driven (DD) approach, as first introduced by Kirchdoerfer and Ortiz [2016], leverages material data directly, avoiding the need for fitted models and allowing uncertainty estimation from data imperfections, such as noise or missing values.

The framework employs a weaker DD formulation designed to ensure adherence to conservation laws and boundary conditions. The temperature and its gradient are approximated in the L2L2 (discontinuous) space, while the heat flux is approximated in the H(div)H(div) (Raviart-Thomas) space. This formulation enforces normal flux continuity across internal boundaries and provides a posteriori error estimates. An adaptive algorithm for mesh and order refinement is proposed, guided by error indicators and the proximity of the computed fields to the material dataset. This approach assesses the suitability of the material dataset to the problem at hand while minimising computational effort. Unlike the original “stronger” DD approach, the weaker DD mixed formulation reduces the number of unknowns and searches through the material dataset while achieving the same accuracy.

The developed framework is applied to a nonlinear problem where the heat flux depends on both the temperature and its gradient. Uncertainty quantification is performed by perturbing the resulting fields and repeating the iterative process, similar to Monte Carlo simulations, to obtain the standard deviation of the results. By analysing the standard deviation and the distances of the resulting fields from the dataset, information about the quality of the material dataset for the problem is obtained. This analysis highlights where the material data does not cover the ranges of the fields in the analysis or where the data is too noisy or incomplete.

The developed framework reformulates the original DD approach and can be extended to include more complex material datasets and physical phenomena. By controlling and minimising finite element errors before quantifying the uncertainty of the results propagated from the dataset, the framework can be applied to challenging industrial problems.

This thesis is written for engineers, and all of the work is open and reproducible. The implemented framework is an independently developed module [Kulikova, 2024a] in MoFEM, an open-source parallel finite element library, and all examples are collected in another open repository [Kulikova, 2024b] to ensure the reproducibility of the results presented in this thesis.