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.

我们获得了一些结果，这些结果涉及以下界面的代表性现象：“Belousov-Zhabotinsky 反应”、“粘性指法”和“吸收介质中的支撑行为”。 1）“奇异极限法”、“分岔理论” ”，“适当的粘度解决方案”和“水平设置方法”的开发使我们能够分析此类现象出现的机制。例如，行波和螺旋波的出现，以及描述晶体外延生长中观察到的一些聚束现象的模型方程解的全局存在性。 2)构建了基于奇异极限法的差分格式并使我们能够数值计算3维Stefan问题。3）手指图案的增长可以通过考虑流变特性的非线性来解释； 4)提出了一种实现吸收介质流动支撑行为的数值方法,并通过数学分析证明了支撑的分裂和不分裂现象。 5)构造了一种跟踪水平集的多精度算法数值方法，可以捕获行波解。6)构造了带有误差估计的有限元方法和域分解方法，使我们能够实现动态脑脊液的行为以及地幔对流，这是无法直接观测到的。此外，提出了一种高性能域分解方法，用于沿域界面使用无网格虚拟节点进行并行有限元分析。

项目成果

Y.Giga, Rybka: "Quasi-static evolution of 3-D crystals growth from supersaturated vapor"Differential and Integral Equations. 15. 1-15 (2002)

Y.Giga、Rybka：“过饱和蒸气中 3D 晶体生长的准静态演化”微分方程和积分方程。

D.Hilhorst, M.Iida, M.Mimura, H.Ninomiya: "A competition-diffusion system approximation to the classical two-phase Stefan problem"Japan J.Industrial and Appl.Mathematics. Vol.18. 161-180 (2001)

D.Hilhorst、M.Iida、M.Mimura、H.Ninomiya：“近似经典两相 Stefan 问题的竞争扩散系统”Japan J.Industrial and Appl.Mathematics。

Y.Giga: "On the two-dimensional nonstationary vorticity equations"Proc.of the international conference on "Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics"(ed.H.Fujita, S.T.Kuroda and H.Okamoto)(Univ.of Tokyo Press, Tokyo

Y.Giga：“关于二维非平稳涡度方程”国际会议“Tosio Katos Method and Principle for Evolution Equations in MathematicalPhysics”（ed.H.Fujita，S.T.Kuroda 和 H.Okamoto）（Univ）

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

T.Nakaki, K.Tomoeda: "Numerical approach to the waiting time for the one-dimensional porous medium equation"Quart. Appl. Math.. (to appear).

T.Nakaki、K.Tomoeda：“一维多孔介质方程等待时间的数值方法”Quart。