A machine learning-based stabilized finite element formulation for the mean stress computation in linear elasticity and coupled poroelasticity
Bibliographic record
Abstract
This work addresses numerical instabilities that can appear when computing the mean stress in linear elasticity and coupled poroelasticity problems discretized with low-order finite elements. The linear elasticity and coupled poroelasticity models are solved using both primal and mixed finite element formulations. Stabilization is obtained by enriching the finite element approximation with an approximation of the Laplacian of displacements. This Laplacian is then evaluated with the Physical Influence Scheme (PIS) by leveraging the underlying governing equation. A key step in the proposed stabilization is the calculation of a parameter h , often computed in the literature as a characteristic length of the element. In this work, we calculate h by solving an optimization problem at the element level. To avoid the high computational cost associated with this procedure, a machine learning model is proposed to predict the optimal h . The benefit of combining PIS with an appropriate computation of h is that the resulting stabilization scheme does not rely on any type of heuristic or user-specified tuning parameter, as often required in other stabilization methods. The results show that the proposed stabilization strategy can effectively remove both saddle-point and Gibbs mean stress oscillations in linear elasticity. We also report, for the first time, that mean stress oscillations can also appear when solving coupled poroelasticity problems, and, differently from pore pressure oscillations (which naturally vanish with time), mean stress instabilities are persistent throughout the whole simulation time, unless deliberately removed. The proposed stabilized mixed formulation is able to remove both pore pressure and mean stress oscillations in coupled poroelasticity problems. Finally, the calculation of h is shown to be critical for the quality of the stabilization, with the machine learning-based approach providing the best compromise between numerical diffusion and accuracy.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".