Construction of time discrete geometric models based on discrete differential geometry

2017 Fiscal Year Final Research Report

• Project/Area Number: 16K13763
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Kyushu University
• Principal Investigator: Kajiwara Kenji 九州大学, マス・フォア・インダストリ研究所, 教授 (40268115)
• Co-Investigator(Renkei-kenkyūsha): MARUNO Ken-ichi 早稲田大学, 理工学術院, 教授 (80380674) ; KAKEI Saburo 立教大学, 理学部, 教授 (60318798) ; HIROSE Sampei 芝浦工業大学, デザイン工学部, 助教 (20743230)
• Research Collaborator: Broadbridge Philip ラ・トローブ大学, 教授
• Project Period (FY): 2016-04-01 – 2018-03-31
• Keywords: 離散微分幾何 / 可積分系 / 曲線 / ソリトン方程式 / 弾性曲線 / 相似幾何 / 変分原理 / 対数型美的曲線

Outline of Final Research Achievements

In this project, the following three topics have been studied:(a)discrete models of curve deformation with extension and external force, (b)time discrete model of deformation of layer, (c) discrete model of one-dimensional elastic rod. As to (a) we succeeded in constructing a new discrete model of the curve shortening equation and discrete local induction equation. For (b) we obtained the bilinearization of the complex Dym equation describing the Hele-Shaw flow, and also succeeded in discretization of the Broadbridge-White model for the one-dimensional soil water infiltration and its numerical simulation. Regarding (c) we succeeded in formulating the integrable discrete model of the Euler's elastic curves by the discrete variational principle. Further we have shown that the elastic curves in the similarity geometry are nothing but the log-aesthetic curves used in the industrial design, and gave a sound mathematical foundation for their generalization.

Free Research Field

可積分系，離散微分幾何