A Conjugate Gradient Method for the Solution of the Inverse Heating Problem in Thermoforming
Bibliographic record
Abstract
Finding the setpoints of heater temperatures such that the sheet will achieve the desired temperature at the end of the heating cycle is known as the inverse heating problem (IHP) in thermoforming. Although a major portion of the heat is transferred from the oven to the sheet by radiation, conduction and convection also have a significant contribution in sheet heating. This makes the inverse heating problem more complex. In addition, the IHP is often ill-posed in a thermoforming process, making the solution unstable. In this paper, a conjugate gradient method is used to solve the IHP in the control of sheet temperature. In developing the method, computational cost is considered such that it can be implemented as a real-time algorithm in the controller. The performance of the proposed method for solving IHP, and thereby the corresponding sheet temperature controller, are tested in simulation at different operating conditions and compared with the conventional method of solving IHP based on the pseudo-inverse of the view factor matrix. The efficiency of the proposed method for the estimation of heater setpoints and the accuracy of the controller are evidenced by the results, showing promise for prospective real-time applications.
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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 | 0.001 |
| 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) | 0.003 | 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".