Application of the B-Spline Method to Solve Nonlinear Problem of Heat Conduction With Radiation Boundary Conditions Using Kirchhoff Transformation
Bibliographic record
Abstract
Abstract This paper concerns the joint application of the B-spline method and the Kirchhoff transformation to solve the nonlinear problem of thermal conduction with radiation type boundary conditions. The proposed method requires few iterations, sometimes none, for solids subjected to prescribed temperature boundary conditions. This method can be deployed by other numerical approaches (boundary element method, finite element method, finite element method, etc.) for the resolution of the heat conduction equation (linear or nonlinear), in terms of the Kirchhoff transformation θ. For numerical implementation, the steady-state finite element method is considered. The numerical validation was performed for a hollow aluminum cylinder whose outer surface is subjected to radiation. Three types of thermal conductivities are considered: (i) constant, (ii) linear, and (iii) nonlinear. As an application, we studied the thermal response of an aluminum reactor, in the form of an annular disk with cooling tubes, exposed to thermal radiation.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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".