A SECOND-ORDER NEWTON-RAPHSON METHOD FOR IMPROVED NUMERICAL STABILITY IN THE DETERMINATION OF DROPLET SIZE DISTRIBUTIONS IN SPRAYS
Bibliographic record
Abstract
The maximum entropy principle method has been very popular, and it has achieved reasonable success predicting droplet size and velocity distribution in sprays in the past two decades. The recently proposed method, maximization of entropy generation, takes into account the irreversibility during the atomization process, and is more consistent with the physics involved. Both of these methods generate models consisting of implicit, highly nonlinear equations involved with exponential functions and integrals. The classical Newton s method has traditionally been adopted as the solver; however, its inherent disadvantage is the requirement that the initial guess for the successive iteration in the numerical solution process be sufficiently close to the solution, otherwise the iteration may diverge rapidly. This study introduces a modification to the classical Newton's method with the Newton's second-order method and the successive under-relaxation (SUR) technique. Three other algorithms based on the Newton's method are also compared with the above methods. Results show that the proposed second-order Newton's method and the SUR technique can greatly improve the numerical stability and, indeed, relinquish the strict requirement on the initial guess.
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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 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| 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".