Research on the structure and stability of the dynamical system describing phase transition phenomena

描述相变现象的动力系统的结构和稳定性研究

基本信息

批准号：
13440052
负责人：
KENMOCHI Nobuyuki
金额：
$ 7.36万
依托单位：
Chiba University
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 2004
项目状态：
已结题

项目摘要

The subjects of this project are the following :(1)Modelings of phase transition phenomena with their mathematical theory(2)Formulations of the related optimal control problems and numerical simulations(3)Return of the research results to the school educationConcerning (1), on the basis of the fundamental laws of the thermo-mechanics, we investigated the structure (phase change, component separation, damage evolution, ordering of atoms) of materials from the view-point of the theory of dymanical systems. Especially, during the last two years (2003-2004) of the project, we treated a class of models for irreversible phase change phenomena, taking account of the process of damage, and evolved its mathematical theory. In our research the subdifferential theory, which have been accumulated for these 15 years, worked very well and we had a big progress more than we expected. Concerning (2), proposing some formulations of optimal control problems in which control parameters are described by a class of hysteresis functions, we finished almost the theoretical part of the problems and tried partially their numerical tests. It was pointed out that there are still some questions which should be improved. Many papers treating subjects (1)and (2)were published and we organized two international conferences during this research project.As far as (3)is concerned, many practices were reported in the meetings "Mathematical analysis of phase transitions and their related mathematics education" which we organized every year. We paid our attention to time-dependent phenomena noticed easily in our daily lives, verifying very carefully their value-added educational aspects as teaching materials of mathematics or sciences for the secondary school. We expect that this research enables to provide new teaching materials by which students can perceive usefulness of mathematics.

该项目的主题如下：（1）相关的数学理论（2）相关最佳控制问题和数值模拟的模型（3）研究结果回报到学校教育（1）的基础上，基于热力学的基本法律，我们研究了材料的理论，损坏的理论，损坏的理论，损坏的理论，损害的理论，损害的理论，损害的理论，相互分化，相互分裂，相互分化，相互分裂，相互分化，相互作用。堤防系统。尤其是，在该项目的过去两年（2003-2004）中，我们处理了一类用于不可逆的相变现象的模型，考虑了损害的过程，并进化了其数学理论。在我们的研究中，这15年来积累的细分理论效果很好，我们取得的进步比我们预期的要多。关于（2），提出了一些最佳控制问题的公式，其中控制参数由一类滞后功能描述，我们几乎完成了问题的理论部分，并部分尝试了其数值测试。有人指出，仍然有一些问题应该得到改善。许多论文处理了受试者（1）和（2），我们在此研究项目中组织了两次国际会议。就（3）而言，在会议上报告了许多实践“对相位过渡及其相关数学教育的数学分析”，我们每年都会组织这些实践。我们关注的是与时间相关的现象在我们的日常生活中很容易注意到，并非常仔细地验证了他们作为中学数学或科学的教学材料的增值教育方面。我们希望这项研究使学生能够通过这些新的教材来感知数学的有用性。