Bibliographic record
Abstract
Tetrahedral finite elements are widely used in 3D electromagnetics. They are the simplest shape into which a 3D region can be broken, and are well-suited to automatic mesh generation. Hierarchical elements are finite elements which have the useful property that elements with different polynomial orders can be used together in the same mesh without causing discontinuities. This is highly desirable, because it permits polynomial order to be used to control the distribution of the degrees of freedom. This paper introduces a new hierarchical tetrahedral element in which the basis functions are constructed from orthogonal polynomials (Jacobi polynomials), allowing mixing of polynomial orders up to three. Explicit basis functions are given in addition to the description of the linear independence property. As was the case for regular elements the pre-calculation of universal matrices will yield faster and more accurate results. The derivation and the corresponding universal matrices for the new elements are also shown. The new elements are used to solve for the electrostatic potential in a 3D region (where there is no analytical solution).
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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 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| 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 | 0.002 |
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".