Toward applications of the crystalline mean curvature flow
| Project/Area Number |
23K03212
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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| Research Institution | Kanazawa University |
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Principal Investigator |
POZAR Norbert 金沢大学, 数物科学系, 准教授 (00646523)
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| Project Period (FY) |
2023-04-01 – 2026-03-31
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| Project Status |
Granted (Fiscal Year 2023)
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| Budget Amount *help |
¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2025: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2024: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2023: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | crystalline curvature / adaptive mesh refinement / free boundary / mesh refinement / numerical method / viscosity solution |
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| Outline of Research at the Start |
The goal of this project is to bring the crystalline mean curvature flow closer to its applicability to real-world modeling, in particular by developing an efficient numerical method in three dimensions. This flow appears in models of the growth of small crystals, formation of snow crystals, and other phase transitions, as well as in methods in image processing. Since a large number of materials are crystalline (for example metals and semiconductors), the understanding and accurate modeling of crystal growth has direct benefits to development of new materials and manufacturing processes.
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| Outline of Annual Research Achievements |
As this is the first year of the project, the work on the main proposed research, efficient numerical method for the crystalline mean curvature flow, is still ongoing.
In a related work, we investigated a model of droplet motion on a surface with contact line hysteresis: the advancing and receding contact angles differ. The evolution of the contact line of the droplet is driven by a boundary condition on a time scale that is assumed to be much slower than the time required for the droplet shape to reach equilibrium. Therefore the evolution is quasistatic and rate-independent. The theory of the solutions is mathematically challenging. We investigated the existence and uniqueness of a few different notions of solutions and showed that they give equivalent evolution if there are no jumps in the contact line evolution. A numerical method was developed to illustrate the evolution. The paper has been submitted.
The cause of the contact angle hysteresis is still not completely understood. One hypothesis is that it arises from microscopic scale inhomogeneity of the droplet contact angle. If the inhomogeity is periodic, it appears to result in a droplet dynamics with anisotropic contact angle. We submitted a paper on the comparison principle of a simplified model of a stationary droplet (Bernoulli problem) with anisotropic contact angle.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
In the first of year of the project, we continued building the required code for solving crystalline mean curvature flow and related free boundary problems using adaptive mesh refinement on quadtrees and octrees. This code is now being tested on relatively simpler free boundary problems like Hele-Shaw and Stefan problems (models of phase transitions) and droplet evolution, and fluid problems like the shallow water equations. The octree code for three dimensional computation has been implemented and successfully applied to the Stefan problem.
A paper on the application to the Stefan problem (efficiently estimating average velocity of phase transition in an inhomogeneous medium) is under preparation with a Ph.D. student.
We are also working with another Ph.D. student on the application for the shallow water equation.
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| Strategy for Future Research Activity |
The project is progressing mostly as planned. We are at the stage where the main steps for the adaptive mesh refinement are mostly implemented in both two and three dimensions. From now on we plan to investigate how to apply it to make the computation of numerical solutions of various important free boundary problems more efficient and accurate, with the focus on the crystalline mean curvature flow. However, the approach can be adapted to other free boundary problems or fluid dynamics problems with sharp transitions and we plan to search for such applications.
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Report
(1 results)
Research Products
(4 results)
