The Frontier of Numerical Analysis for Dynamics of Interfaces and Developments in Sciences and Engineering

本文研究了典型动力界面中出现的现象的数值计算方法：1)反应扩散系统中的模式动力学，2)Hele-Shaw单元中的粘指现象，3)蒸发过程中水所占区域的动力学行为。结果表明：1)建立了反应扩散系统的TCD （Threshold Competition Dynamics）数值计算方法，实现了R^n （n= 1,2,3）范围内自由边界的动力学行为。该方法的思想是基于“奇异极限法”的理论。2)将晶体生长的数学模型考虑为具有对流效应的反应-扩散方程，得到了有趣的数学结果。3)在粘指现象下，研究了Hele-Shaw胞内气泡上升的浮力驱动路径不稳定性。作为一种有趣的现象，出现了一种类似彗星的尾迹。然而，这种尾迹还没有在数值方法中实现。4)建立了基于晶体均匀化方法的多尺度有限元方法，预测了板料成形过程中界面的动力学特性。5)研究了蒸发过程中支架的重复劈裂连接特性，其中支架为水所占区域。建立了这一过程的数值计算方法，并明确地得到了这一性质出现的初始分布的形状。

We were concerned with the following numerical methods to the phenomena appearing in the repre-sentative dynamical interfaces :i) Pattern dynamics in the reaction-diffusion system,ii) Viscous fingering phenomena in Hele-Shaw Cell,iii) Dynamical behavior of the region occupied by the water in the process of evaporation.We obtained several results :1) The TCD (Threshold Competition Dynamics) method is developed for the numerical computation in reaction-diffusion system, and enables us to realize the dynamical behavior of free boundary in R^n (n=1, 2, 3.). The idea of this method is based on the theory of "Singular limit method".2) The mathematical model for the crystal growth is considered in the, form of the reaction-diffusion equation with the effect of a convection, and gives us interesting mathematical results.3) In viscous fingering phenomena, the buoyancy-driven path instabilities of bubble rising in Hele-Shaw Cell are examined. As an interesting phenomenon there appears a wake which is similar to a comet. However, such a wake is not realized in numerical method yet.4) Multi-scale FEM based on crystallographic homogenization method is developed to predict the dynamics of interfaces in the formability of sheet metal.5) The repeated support splitting and connecting property in the process of evaporation is investigated, where the the support means the region occupied by the water. The numerical methods for this process are established and the shape of the initial distribution for which such a property appear is explicitly obtained.