2023 Fiscal Year Research-status Report
Surface PDE: a minimizing movement approach
| Project/Area Number |
22K03440
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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, 専任教授 (30648217)
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| Project Period (FY) |
2022-04-01 – 2025-03-31
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| Keywords | surface PDE / minimizing movements / interfacial dynamics / level set method / approximation methods |
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| Outline of Annual Research Achievements |
This year, we focused on furthering our surface-type minimizing movments and their application to constrained interfacial motions. We successfully formulated surface MBO and surface HMBO algorithms in the minimizing movement framework and, by extending techniques from the level set method to the surface PDE setting, we generalized these algorithms to incorporate area-constraints. Additionally, our algorithms were formalized as approximation methods, and their numerical error analyses were performed.
Our computational methods, those corresponding to the surface-type minimizing movement for heat-type problems, were used to illustrate the numerical behavior of area-constrained motion of interfaces moving under surface-constrained curvature flow in the two phase setting. Similarly, our wave-type minimizing movement was used to visualize the behavior of area-constrained hyperbolic mean curvature flow on surfaces. These results revealed new novel interfacial motions, and suggest the wide-ranging applicability of the closest point method.
We also integrated the SDVF into our surface-type minimizing movements and conducted numerous experiments involving multiphase geometries. These experiments demonstrated that, as in the flat setting, penalty terms can be employed in surface-type minimizing movements. Moreover, given that computations involving the SDVF are conducted on point clouds, interpolation techniques that precisely locate interfaces were required. In response, we devised and tested a technique to address this challenge.
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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
Our research has progressed according to our original plan. Namely, this year we successfully extended our surface-type minimizing movements to be able to treat area-constrained interfacial motions. We also performed corresponding numerical error analyses, which was one of our main research goals. Also, the computational speed of the numerical methods used to construct the signed distance vector fields were improved. Although they are still computationally heavy, these improvements have enabled us to apply our methods to solve surface PDE on complex geometries. Our work also uncovered a new area for research. In particular, we have found that the notion of "point correspondence" could be added into our HMBO algorithm.
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| Strategy for Future Research Activity |
This year, using the signed distance vector field, we aim to show that surface-type minimizing movements can approximate multiphase area-constrained interfacial motions on surfaces. Taking surface MCF and surface HMCF as primary examples, this will be done by adding auxillary penalties onto energy functionals for each of the area constraints. Although new level set techniques for computing multiphase areas and lengths will be required, we expect that the approaches in the flat setting will carry over to the case of surfaces. We also anticipate that the interpolation methods used within the closest point method will need refining, especially near junctions. Regarding the newly understood point correspondence, we will first inquire about its usage in the flat setting.
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| Causes of Carryover |
Funds allotted for travel were preserved due to online participation and discussions. Transferred funds will be used to supplement travel plans.
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