A level set method without re-initialization
Bibliographic record
Abstract
Moving interface problems have practical applications in the fields of fluid mechanics, solid mechanics or medical imaging. There are several numerical methods in CFD to solve these problems. The volume of fluid method (VOF) of Hirt and Nichols set in 1981 is widely used for cases where the interface separates the domain into two distinct areas as in the case of two-phase flows [1]. The mixture model is based on the principle of averaged equations. The method of averaging is a wise choice to do [2] when the scale of one of the phases is too small compared to the others [3, 4]. The Level set method was first introduced by Osher and Sethian [5] to capture a moving front. This method was originally developed for the simulation of a phase change problem governed by a diffusion equation. Other applications followed in image analysis [6]. Earlier, the Level set equation was solved with the method of finite differences. Recently, a new variational formulation has been developed in order to remove the re-initialization process which is a reset phase [7]. Moreover, in most cases we are interested in having greater accuracy at the interface and not in the entire domain. Hence, the Level set method restricted to a narrow band around the zero level set was developed [8]. In this paper, a new stabilized finite element formulation is introduced to solve the level set equation without re-initialization. This method is compared with the one introduced in [7] on a timereversed flow field case [10]. 1. THE LEVEL SET METHOD The Level set method was introduced in 1988 by Osher and Sethian [5]. The starting point of this method is the definition of a level set scalar function . The zero value of the level set function is the interface that is transported by the velocity field. The contours of the level set function initialized as a distance function may move away from the level set distance function due to the accumulation of numerical errors, hence the need to reset the solution after a number of steps. The moving interface is the zero-level for the scalar function ( ): ( ) { ( ) }. For example in a two-phase flow the domain is divided into two subdomains using the sign of the level set function:
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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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.009 | 0.003 |
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".