Mathematical Models of Interfacial Motion of Crystalline Materials

Kobayashi Had Bennn调查了多晶体管材料中谷物边界的数学模型的问题，自90年代中期以来，已经完成了二维模型。本研究的主要目标是将Kobayashi-Warren-Carter模型扩展到3D模型，其中包括基本差异，因为方向变量必须扩展到SO（2）-值到SO（3）-值。因此，我们打算在数学上格式化三维模型，研究数学理论，讨论它，并开发数字模拟的技术。我们首先构建SO（3）--在一个维度和两个维度空间以及模拟代码中有价值的单一差异性等同。然后我们比较SO（3）的2个表达式;局部协调（Rodrigues， Vector），嵌入到9个维度的欧几里得空间R^{9}，并结合了嵌入方法是我们必须走的路。根据这一决定，我们首先开发了一个三维数字代码，其中方向变量的价值仍然保留在SO（3）中，这意味着正常投影对SO（3）和一些特殊处罚，函数。通过将这一方向equation与阶段场equation结合起来，描述固体-液体相位动力学，我们完成了模型和3D数字代码，当过程从核、固体、颗粒边界形成和颗粒接受运动的生长时，最后订购包括晶体结构到我们模型的对称性，we took the orientation space as the homogen？space （the space obtained by dividing SO（3） by symmetry group） and tried to imbed.it to some （higher dimensional） Euclidian space。我们如何发现这是非常困难的，可以在我们感兴趣的情况下实现混凝土嵌入，我们的研究有有趣的副作用，它是一个清理的数学模型的构造，它可以被认为是（在某些意义上），成为一个与谷物协调的反向过程。

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.