Conservative data-driven finite element formulation

2025-06-22

AI Summary

Traditional diffusion simulations rely on simplified constitutive models fitted to experimental data, often inducing information loss and model bias. This work proposes a data-driven mixed finite element framework that directly incorporates raw experimental measurements—bypassing parametric material modeling—and rigorously enforces conservation laws in strong form, thereby eliminating constitutive fitting. The method employs a mixed variational formulation to intrinsically guarantee conservation, relaxes regularity requirements on the solution space, and enables adaptive hp-refinement. Additionally, a posteriori error estimation quantifies solution non-uniqueness arising from data uncertainty. Validation on synthetic datasets and a nonlinear heat conduction problem in nuclear graphite demonstrates high-fidelity simulation accuracy, explicit characterization of multivalued solution structures under data-driven constraints, and faithful propagation of uncertainty. The framework establishes a new paradigm for experiment-guided modeling of diffusion processes.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Web Mining and Content Analysis: Web data generation and simulation
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

This paper presents a new data-driven finite element framework derived with mixed finite element formulation. The standard approach to diffusion problems requires the solution of the mathematical equations that describe both the conservation law and the constitutive relations, where the latter is traditionally obtained after fitting experimental data to simplified material models. To exploit all available information and avoid bias in the material model, we follow a data-driven approach. While the conservation laws and boundary conditions are satisfied by means of the finite element method, the experimental data is used directly in the numerical simulations, avoiding the need of fitting material model parameters. In order to satisfy the conservation law a priori in the strong sense, we introduce a mixed finite element formulation. This relaxes the regularity requirements on approximation spaces while enforcing continuity of the normal flux component across all of the inner boundaries. This weaker mixed formulation provides a posteriori error indicators tailored for this data-driven approach, enabling adaptive hp-refinement. The relaxed regularity of the approximation spaces makes it easier to observe how the variation in the datasets results in the non-uniqueness of the solution, which can be quantified to predict the uncertainty of the results. The capabilities of the formulation are demonstrated in an example of the nonlinear heat transfer in nuclear graphite using synthetically generated material datasets.

Problem

• Develops data-driven FEM framework avoiding material model bias
• Enforces conservation laws via mixed FEM for strong solutions
• Quantifies solution uncertainty from dataset variations adaptively

Innovation

• Data-driven finite element framework
• Mixed finite element formulation
• Adaptive hp-refinement enabled