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Mathematical Models of Interfacial Motion of Crystalline Materials

晶体材料界面运动的数学模型

基本信息

批准号：
16340021
负责人：
KOBAYASHI Ryo
金额：
$ 8万
依托单位：
Hiroshima University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

Kobayashi had benn investigated the problem for the mathematical model of grain boundary in polycrystalline materials since the middle of 90's, and completed 2D model already. The main goal of this research is to extend Kobayashi-Warren-Carter model to 3D model, which includes essential difficulties because the orientation variable must be extended from SO(2)-valued to SO(3)-valued. Therefore we intended to formuLate the 3D model mathematically, study mathematical theories concerning to it, ・and develop the technique of numerical simulation.We first construct the SO(3)-valued singular diffusivity equation in one dimensional and two dimensional space, and simulation codes. Then we compare the 2 expressions for SO(3); local coordinates (Rodrigues, Vector) and imbedding to 9 dimensional Euclidean space R^{9}, and concluded that the imbedding method is the way we have to go. According to this decision we first developed a 3D numerial code in which the values of orientation variables are kept in SO(3) by means of the normal projection to SO(3) and some special penalty, function. By coupling this orientation equation with the phase field equation which describes the solid-liquid phase dynamics, we completed the model and 3D numerical code which can describe the while process from nucleation, solidification, formation of grain boundaries and growth of the grains accompanied by the motion of grain boundaries.Finally in order to include the symmetries of crystalline structure to our model, we took the orientation space as the homogeneous space (the space obtained by dividing SO(3) by symmetry group) and tried to imbed.it to some (higher dimensional) Euclidian space. However we found it is very difficult to obtain the concrete imbedding in the situation where we are interested in.Our research has interesting side effect, which is a construction of mathematical model of cleavage which can be considered (in some sense) to be a reverse process of coarsening of grains.

小林从90年代中期开始研究多晶材料晶界的数学模型问题，并已完成二维模型。本研究的主要目的是将Kobayashi-Warren-Carter模型推广到三维模型，其中的难点在于方向变量必须从SO（2）值推广到SO（3）值。为此，我们对三维模型进行了数学表述，研究了与之相关的数学理论，发展了数值模拟技术，首先构造了一维和二维空间的SO（3）值奇异扩散方程，并编制了模拟程序。然后我们比较了SO（3）的两种表达方式：局部坐标（Rodrigues，Vector）和嵌入到9维欧氏空间R^{9}，得出嵌入方法是我们必须走的路。根据这一决定，我们首先发展了一种三维数值码，它通过SO（3）的法向投影和一些特殊的罚函数，使方向变量的值保持在SO（3）中。通过将该取向方程与描述固液相动力学的相场方程相耦合，完成了描述从形核、凝固、晶界形成到晶粒长大并伴随晶界运动的三维数值模拟，最后为了将晶体结构的对称性纳入到我们的模型中，我们把定向空间当作齐次空间（SO（3）除以对称群得到的空间），并试图把它推广imbed.it到某些（高维）欧氏空间.然而我们发现在我们感兴趣的情况下很难获得混凝土的嵌入。我们的研究有一个有趣的副作用，那就是建立了解理的数学模型，在某种意义上，解理可以被认为是晶粒粗化的逆过程。