2018 Fiscal Year Research-status Report
双曲型 Threshold Dynamics：応用と数理解析
| Project/Area Number |
17K14229
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| Research Institution | Meiji University |
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Principal Investigator |
Ginder Elliott 明治大学, 総合数理学部, 専任准教授 (30648217)
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| Project Period (FY) |
2017-04-01 – 2020-03-31
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| Keywords | threshold dynamics / interfacial motion / hyperbolic pde / curvature flow / MBO |
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| Outline of Annual Research Achievements |
In 2018, this research project focused on continuing development of threshold dynamical (TD) algorithms and on analyzing physical systems for comparison with results from our mathematical models. Our previous algorithms were mainly designed to handle the motion of planar interfaces and so we have extended our interfacial tracking algorithms and level set methods to the three dimensional setting. This has allowed us to use the HMBO to compute the damped motion of interfaces by hyperbolic mean curvature flow in the three dimensional setting. This research also helped lead to a simple PDE based method for constructing Voronoi diagrams subject to periodic boundary conditions.
In addition, since many of our approximation methods are based on minimizing movements (MM), and since the standard MM are known to impart energy dissipation, we (together with K. Svadlenka and S. Omata) have designed new numerical methods based on Crank-Nicolson schemes which are expect to improve energy preservation properties. The methods were used to simulate the volume constrained, scalar membrane motion of droplets.
We also focused on finding an experimental system for recording the motion and other physical characteristics of thin membranes. Together with partners from industry (Beijing Taigeek Technology Co., Ltd.) we examined the use of a holographic system for measuring the thickness and height evolution of soap bubbles in a volume chamber. Experimental data from the device was also obtained.
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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
Construction of numerical counterparts for implementing Crank Nicolson type minimizing movements (MM) has provided a setting for comparing our simulations with real phenomena involving oscillating thin membranes. Connecting our TD algorithms to the modeling of natural phenomena had been a standing goal.
We have also come across a new potential application for the TD algorithms. In particular, together with Z. Wang and R. Zou, we expect that it may be possible to use holographic techniques to analyze fluid flow through thin membranes. Since the corresponding phenomena should be modeled using surface pde, such experiments could represent a good target application for surface constrained threshold dynamical algorithms.
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| Strategy for Future Research Activity |
We will aim to compare experimental results on membrane motion with those of our mathematical models. Here, our initial goal will be to simulate and compare the volume chamber experimental results. Simultaneously, we will aim to develop computational methods for investigating threshold dynamics (TD) on surfaces. This will require us to establish techniques for performing calculus on manifold and we would like to realize a method for solving the surface eikonal equation. This would allow us to define a framework for the surface-based level set method.
Such a framework would then allow us to perform a mathematical inquiry into interfacial motions on surfaces, beginning with the case of mean curvature flow. We expect that standard ideas from the level set can be carried over to the surface setting, and we anticipate that further numerical tests will be required. By using minimizing movements, we also intend to incorporate volume preservation into the interfacial motions. If we are successful, then we want to generalize our idea to the multiphase setting.
We hope that our research leads to the ability to simulate interfacial motions governed by surface constrained hyperbolic mean curvature flow. This has only been investigated in the flat setting and is a completely new area of research. The successful design of approximations methods is expected to carry significance in both theory and applications.
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| Causes of Carryover |
Delayed purchase of computational server and publishing fees. We expect to incur the corresponding fees in 2019.
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