Magnetodynamic H–$\phi$ Formulation for Improving the Convergence and Speed of Numerical Simulations of Superconducting Materials
Bibliographic record
Abstract
The growing interest in fast and accurate simulation methods of the electromagnetic behavior of high-temperature superconducting materials has led to many exciting developments in the last decade. Although the H formulation implemented in the finite-element method has proven to be very robust for electromagnetic calculations, the dummy resistivity and vector-dependent variables required in air domains lead to long computation times and spurious currents in certain applications. Thus, the H–$\phi$formulation has recently gained significant traction in order to speed up simulations. In our previous work, we implemented the H–$\phi$formulation in COMSOL Multiphysics using the divergence-free condition in air domains and showed that the computation times were reduced by two and three in 2-D and 3-D, respectively. However, high-order shape functions were required in order to obtain sufficient accuracy when compared to the full H formulation. In this article, we implemented a magnetodynamic H–$\phi$formulation in COMSOL by using Faraday's law in air domains to improve the coupling between the H and$\phi$physics and correctly represent time-varying phenomena in superconducting materials. We show that this formulation leads to better convergence, more accurate solutions, and a slight speed advantage compared to the previously used H–$\phi$formulation. The magnetodynamic H–$\phi$formulation results are shown to be nearly identical to the full H formulation even with linear shape functions, while reducing the computation times by a factor of up to four for a given mesh.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".