单侧问题的无网格投影迭代法

发布时间：2018-10-09 12:54

【摘要】：单侧问题是一类含有变分不等式的数学物理问题。科学工程中诸如电镀问题、障碍问题、自由水坝问题等都被统称为单侧问题。在数值计算中,无网格方法不需要生成网格,适用性强,它是一种新的数值求解偏微分方程边界问题的方法。本论文就是用无网格投影迭代法来解决单侧问题,具体研究工作如下:本论文第1章绪论介绍了单侧问题的研究背景和研究现状,引出了本文的研究目的是用无网格投影迭代法和无网格边界积分方程方法结合起来,尝试用无网格插值边界无单元投影迭代法来解决单侧问题。第2章是本文的核心部分,系统的阐述了单侧问题的无网格投影迭代算法。首先,介绍了用改进的插值型移动最小二乘方法来构造形函数,接着引入投影迭代算子和隐式投影格式,提出了单侧问题的插值边界无单元法,总结了相应的算法步骤,并对该算法的收敛性进行了推导和证明。第3章采用本文的算法解决分析了三个数值算例,经过对比分析,发现该算法收敛性好,迭代次数少,计算时间也相对较短,并且数值解与精确解很吻合,进一步表明本算法具有较好的可行性与有效性。第4章给出了相关总结,说明该无网格投影迭代法是解决单侧问题模型的一种有效方法,有利于工程应用。
[Abstract]:Unilateral problems are a class of mathematical and physical problems with variational inequalities. In scientific engineering, such as electroplating problem, obstacle problem, free dam problem and so on are all referred to as unilateral problems. Meshless method is a new numerical method for solving boundary problems of partial differential equations. In this paper, the meshless projection iteration method is used to solve the unilateral problem. The specific research work is as follows: the first chapter of this paper introduces the background and research status of the unilateral problem. The purpose of this paper is to combine the meshless projection iteration method with the meshless boundary integral equation method and try to solve the unilateral problem by using the meshless interpolation boundary element free projection iteration method. Chapter 2 is the core of this paper. Firstly, an improved interpolation moving least square method is introduced to construct the shape function. Then the projection iterative operator and implicit projection scheme are introduced, and the interpolation boundary element free method for the one-sided problem is proposed, and the corresponding algorithm steps are summarized. The convergence of the algorithm is deduced and proved. In chapter 3, three numerical examples are solved and analyzed by using the algorithm presented in this paper. Through comparison and analysis, it is found that the algorithm has good convergence, few iterations and relatively short calculation time, and the numerical solution is in good agreement with the exact solution. It further shows that this algorithm is feasible and effective. In chapter 4, the relevant summary is given, which shows that the meshless projection iterative method is an effective method to solve the one-sided problem model, which is beneficial to engineering application.
【学位授予单位】：重庆师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O176;O241.8

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