Analysis of nonlinear phenomena describing hysteresis operators

描述磁滞算子的非线性现象分析

基本信息

批准号：
14540169
负责人：
AIKI Toyohiko
金额：
$ 2.18万
依托单位：
Gifu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

项目摘要

[Shape memory alloy problems] : In the dynamics of shape memory alloy materials the relationship between the strain and the stress can not be described by a usual function, and is represented by a hysteresis operator. Here, the main idea of this research project is to describe the relationship by using a general stop operator. Moreover, we consider the ordinary differential equation, which is equivalent to the generalized stop operator, and propose a system describing the dynamics of shape memory alloys. The system consists of partial differential equations and the ordinary differential equation and is studied in this project. We consider the one-dimensional problem, in which we dealt the ordinary differential equation without an approximation. In this case the regularity of a solution to the ordinary differential equation is not enough in space so that it is impossible to prove the existence of a classical solution. Then we showed the existence and the uniqueness of a weak solution. Also, we considered a shape memory alloy problem in three dimensions. In 3-d case the regularity of a solution is not good. Hence, we proved the well-posed ness for only approximated problem. Here, we note that we also have a plan to analyze our mathematical model, numerically. Now, we do not success to get a numerical solution of a complete problem and have an algorithm for solving a kinetic equation, which is most difficult to compute in our process.[magnetization in ferromagnetic] : By using a generalized Duhem model we propose a mathematical model for dynamics of ferromagnetic and prove the well-posedness.

[形状内存合金问题]：在形状内存合金材料的动力学中，应变和应力之间的关系无法用通常的功能描述，并且由滞后算子表示。在这里，该研究项目的主要思想是使用通用停止操作员来描述这种关系。此外，我们考虑了普通的微分方程，该方程等同于广义停止操作员，并提出了一个描述形状内存合金动力学的系统。该系统由部分微分方程和普通微分方程组成，并在该项目中进行了研究。我们考虑了一维问题，在这种问题中，我们在没有近似值的情况下处理了普通的微分方程。在这种情况下，对普通微分方程的解决方案的规律性在空间中不足，因此不可能证明存在经典解决方案。然后我们展示了弱解决方案的存在和独特性。另外，我们考虑了三个维度的形状内存合金问题。在3-D情况下，解决方案的规律性不好。因此，我们仅出于近似问题证明了良好的NESS。在这里，我们注意到我们还计划以数值分析我们的数学模型。现在，我们没有成功获得一个完整问题的数值解决方案，并且具有解决动力学方程的算法，这在我们的过程中最难计算。铁磁动力学的数学模型并证明了良好的性能。