Data-driven weaker mixed formulation for diffusion problems

Kuliková, Adriana (2025) Data-driven weaker mixed formulation for diffusion problems. PhD thesis, University of Glasgow.

Abstract

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, the models carry uncertainty about the values of these parameters, which we lose by choosing a single value per parameter, which often leads to inaccuracy or overconfidence 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, which employs a weaker DD formulation, is designed to ensure adherence to conservation laws and boundary conditions. The temperature and its gradient are approximated in the L2 (discontinuous) space while the heat flux is approximated in the H(div) space, which enforces normal flux continuity across internal boundaries and provides a posteriori error estimates. An algorithm for adaptive mesh and order refinement is proposed, guided by error indicators and the proximity of the computed fields to the material dataset, assessing the suitability of the material dataset to the problem at hand while minimising computational effort. In contrast to the original “stronger” DD approach, weaker mixed DD formulation results in problems with fewer unknowns and searches through the material dataset while reaching the same accuracy.

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

The developed framework reformulates the original DD approach and can be further extended to include more complex material datasets and physical phenomena. Allowing for the control and minimisation of FE errors before quantifying the uncertainty of the results propagating from the dataset, the framework can be applied to challenging industrial problems.

This thesis is one of the steps towards the development of a data-driven finite element framework for the analysis of graphite bricks in the advanced gas cooled nuclear reactors (AGRs), which is a complex and evolving environment. The current work introduces and tests the "weaker" DD approach on diffusion problems and provides a foundation for future research and development of the DD framework for more complex problems, such as fracture analysis in nuclear graphite bricks. To this end, a preliminary study is performed on a nuclear graphite brick slice with synthetically generated material datasets.

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] for the reproducibility of the results presented in this thesis.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: This work was supported by EDF and EPSRC.
• Subjects: T Technology > T Technology (General)
• Colleges/Schools: College of Science and Engineering > School of Engineering
• Funder's Name: EDF, Engineering and Physical Sciences Research Council (EPSRC)
• Supervisor's Name: Pearce, Professor Chris, Kaczmarcyk, Dr. Łukasz and Shvarts, Dr. Andrei
• Date of Award: 2025
• Depositing User: Theses Team
• Unique ID: glathesis:2025-85306
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 07 Jul 2025 11:06
• Last Modified: 07 Jul 2025 11:13
• Thesis DOI: 10.5525/gla.thesis.85306
• URI: https://theses.gla.ac.uk/id/eprint/85306