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Hyperbolic threshold dynamics: applications and analysis

Research Project

Project/Area Number 17K14229
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Foundations of mathematics/Applied mathematics
Research InstitutionMeiji University

Principal Investigator

Ginder Elliott  明治大学, 総合数理学部, 専任准教授 (30648217)

Project Period (FY) 2017-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
KeywordsThreshold dynamics / interfacial motion / approximation methods / curvature flow / threshold dynamics / hyperbolic pde / MBO / Interfacial motions
Outline of Final Research Achievements

The main result of this research is the discovery of a generalized MBO algorithm (GMBO) using hyperbolic threshold dynamics. The GMBO was found to be an approximation method for the damped hyperbolic mean curvature flow. The result enables one to approximate oscillatory interfacial motions, as well as mean curvature flow. In particular, by changing the base PDE used in the MBO to the wave equation, we were showed that the initial velocity field can be used to control the propagation of interfaces. Interestingly, this clarified that damped interfacial motions are not obtained through additional damping terms in the PDE, but by encoding normal velocity fields within level set functions at each time step. We also established working numerical methods for performing computational analyses and simulations. Here we developed an energy preserving minimizing movement for treating the hyperbolic mean curvature flow, and we confirmed the method’s properties through computational investigations.

Academic Significance and Societal Importance of the Research Achievements

今までの Threshold Dynamics 研究では,本研究のHMBO以外,MBO法に厳守されていた.MBOは,Level Set 法を生み出したものとして知られていたが,本研究の初期段階で作成したHMBOは,慣性の影響を含む振動する界面においてTDの適応範囲を広げることができた.GMBOはMBOとHMBOを同時に取り扱えるため,応用数学と産業に関する課題が1つのTD法で表現することが可能となった.また,今までのMinimizing Movementsの研究において,エネルギー保存するのものが不在だったため,本研究のCrank-Nicholson MMは異分野にも応用があると期待している.

Report

(4 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (8 results)

All 2020 2019 2018 2017 Other

All Journal Article (3 results) (of which Int'l Joint Research: 3 results,  Open Access: 2 results,  Peer Reviewed: 1 results) Presentation (2 results) (of which Invited: 2 results) Remarks (2 results) Funded Workshop (1 results)

  • [Journal Article] A Crank-Nicolson type minimization scheme for a hyperbolic free boundary problem2020

    • Author(s)
      Y. Akagawa, E. Ginder, S. Koide, S. Omata, K. Svadlenka
    • Journal Title

      arXiv (submitted, under revision)

      Volume: 2004.07458 Pages: 1-24

    • Related Report
      2019 Annual Research Report
    • Open Access / Int'l Joint Research
  • [Journal Article] On the inclusion of damping terms in the hyperbolic MBO algorithm2019

    • Author(s)
      E. Ginder, A. Katayama
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: 81 Pages: 1-12

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A variational approach to the inverse imaging of composite elastic materials2019

    • Author(s)
      E. Ginder, R. Kanai
    • Journal Title

      arXiv (submitted, under revision)

      Volume: 1903.05835 Pages: 1-20

    • Related Report
      2019 Annual Research Report
    • Open Access / Int'l Joint Research
  • [Presentation] Approximation Methods for oscillatory and constrained interfacial dynamics2018

    • Author(s)
      Elliott Ginder
    • Organizer
      MSJ SI 2018
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] 減衰項付き双曲型平均曲率流の近似解法について2018

    • Author(s)
      Elliott Ginder
    • Organizer
      表面・界面ダイナミクスの数理15
    • Related Report
      2018 Research-status Report
    • Invited
  • [Remarks] 研究室のホームページ

    • URL

      http://amth.mind.meiji.ac.jp/

    • Related Report
      2019 Annual Research Report
  • [Remarks] Research Homepage

    • URL

      http://amth.mind.meiji.ac.jp/

    • Related Report
      2018 Research-status Report
  • [Funded Workshop] Free boundary problems and nonlinear PDEs 20172017

    • Related Report
      2017 Research-status Report

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Published: 2017-04-28   Modified: 2021-02-19  

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