Development of Numerical Methods for Dynamics of Interfaces and its Applications to Experiments in Science and Engineering

界面动力学数值方法的发展及其在科学与工程实验中的应用

• 批准号: 13440038
• 负责人: TOMOEDA Kenji
• 金额: $ 9.22万
• 依托单位: Osaka Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440038/
• 关键词: Free boundary; Belousov-Zhabotinsky; Viscous fingering; Level set approach; Proper viscosity solutions; Support splitting phenomena; Singular limit method; Helical waves; ベル-ソフ・ザボチンスキ-反応; サポートの分裂と再融合; 反応拡散; 渦の相互作用; サポート分離現象; 電気流体問題; 特性有限要素法

We obtained several results, which are concerned with the following representative phenomena of inter-faces : "Belousov-Zhabotinsky reaction", "Viscous fingering" and "Behavior of support in an absorbingmedium".1)"Singular limit method", "Bifurcation theory", "Proper viscosity solutions" and "Level set method"are developed and enable us to analyze the mechanism of the appearance of such phenomena. For example, the appearance of a travelling wave and a helical wave, and the global existence of the solution of the model equation describing some bunching phenomena observed in the epitaxial growth of crystals.2)A difference scheme based on the singular limit method is constructed and enables us to compute the 3-dimensional Stefan problem numerically.3)The growth of finger patterns can be interpreted by taking into account the non-linearity in the rheological properties ; that is, it becomes easy to construct the mathematical model.4)A numerical method which realizes the behavior of the support of the flow through an absorbing medium is developed, and the support splitting and non-splitting phenomena are justified by the mathematical analysis.5)A numerical method with the multiple precision arithmetic for tracking the level set is constructed, and can capture the travelling wave solutions.6)A finite element method with error estimates and a domain decomposition method are constructed, and enable us to realize the dynamical behavior of the cerebrospinal fluid and the Earth's mantle convection, which are unable to be observed directly. Furthermore, a high-performance domain decomposition method is proposed for the parallel finite element analysis using meshless virtual nodes along the domain interface.

我们得到了几个有关界面的典型现象：“Belousov-Zhabotinsky反应”、“粘性指进”和“吸附介质中的支撑行为”的结果。1）发展了“奇异极限方法”、“分歧理论”、“固有粘性解”和“水平集方法“，使我们能够分析这些现象出现的机理。例如，行波和螺旋波的出现，2）构造了一个基于奇异极限方法的差分格式，并对三维Stefan问题进行了数值计算; 3）在考虑晶体生长过程中的非线性效应的情况下，可以解释指状结构的生长，并给出了数值结果。流变性能的线性; 4）建立了一种数值方法，实现了通过吸收介质的流动的支撑行为，并通过数学分析证明了支撑分裂和非分裂现象的合理性。5）建立了一种多精度算法跟踪水平集的数值方法，6）构造了一种带误差估计的有限元方法和一种区域分解方法，使我们能够实现无法直接观测的脑脊液和地幔对流的动力学行为。在此基础上，提出了一种基于无网格虚拟节点的高性能区域分解并行有限元方法。

项目成果

H.Murakawa, T.Nakaki: "A singular limit method for the Stefan problem"The proceedings of European Conference on Numerical Mathematics and Advanced Applications, Springer-Verlag. (to appear). 6 (2004)

H.Murakawa、T.Nakaki：“Stefan 问题的奇异极限方法”欧洲数值数学和高级应用会议记录，Springer-Verlag。

Hiroyuki Kuramae, Akitsugu Shimano, Akinori Kimura, Masahide Matsumoto, Yoshiki Furuno, Kohji Kamejima: "Integration of User Account Database on Multiple OS Environment"Proc.of IEEE Region 10 Conference on Computers, Communications, Control And Power Engi

Hiroyuki Kuramae、Akitsugu Shimano、Akinori Kimura、Masahide Matsumoto、Yoshiki Furuno、Kohji Kamejima：“多操作系统环境上的用户帐户数据库的集成”Proc.of IEEE Region 10 计算机、通信、控制和电力工程会议

K.Tomoeda: "Numerical approach to support splitting phenomena in some diffusion equations with strong absorption"Proceedings of Fifth China-Japan Seminar on Numerical Mathematics, Edited by Zhong-Ci Shi and Hideo Kawarada, Siences Press. 211-219 (2002)

K.Tomoeda：“支持强吸收扩散方程中分裂现象的数值方法”第五届中日数值数学研讨会论文集，史忠慈、河原田秀雄主编，科学出版社。

S.-I.Ei, M.M.Mimura, M.Nagayama: "Pulse-pulse interaction in reaction-diffusion systems"Physica D. 165. 176-198 (2002)

S.-I.Ei、M.M.Mimura、M.Nagayama：“反应扩散系统中的脉冲-脉冲相互作用”Physica D. 165. 176-198 (2002)

M.Kawaguchi, S.Yamazaki, T.Kato: "Polymer Depletion-Induced Instability of Silica Suspensions in the Presence of Flexible and Semiflexible Polymers"J.Colloid Interface Sci.. 254. 396-401 (2002)

M.Kawaguchi、S.Yamazaki、T.Kato：“在柔性和半柔性聚合物存在下，聚合物消耗引起的二氧化硅悬浮液的不稳定性”J.Colloid Interface Sci.. 254. 396-401 (2002)