On New Triangle Quadrature Rules for the Locally Corrected Nyström Method Formulated on NURBS-Generated Bézier Surfaces in 3-D
Bibliographic record
Abstract
Nonuniform rational b-splines (NURBS) are the most widely used technique in today's geometric computer-aided design systems for modeling surfaces. Combining the locally corrected Nyström (LCN) method with NURBS requires formulating LCN on both quadrilateral and triangular Bézier surfaces, as a typical NURBS-generated Bézier mesh includes elements of both the types. While on quadrilateral elements the product of 1-D Gaussian quadrature rules can be applied to LCN effectively, Gaussian integration rules available for triangles cannot efficiently be applied to LCN for two reasons. First, they do not possess the same number of quadrature points as the number of functions in a complete set of polynomial basis at an arbitrary order. Second, they exacerbate the condition number of the resulting matrix equation at higher orders due to an increasing density of quadrature points near the edges and corners of triangles. In this paper, we construct a new set of quadrature rules for Bézier triangles (i.e., degenerate quadrilaterals) based on the Newton-Cotes (equidistant) quadrature rules and apply these rules to the LCN solution of the electric, magnetic, and combined field integral equations. Results show that the new family of quadrature rules overcomes both the aforementioned issues and can be applied to LCN effectively for orders from 0 to 9, inclusively.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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 | 0.000 | 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".