Signorini问题的无网格投影迭代算法

发布时间：2018-09-01 10:11

【摘要】：许多物理现象,如海滩渗流问题、电镀问题以及自由边界问题等都可以采用Signorini问题为数学模型.在Signorini问题中,Signorini边界上的Dirichlet边界条件和Neumann边界条件是采用相互交替的方式出现,并且交替出现的位置是未知的,这使得求解Signorini问题更加复杂.本文第一章对无网格方法和Signorini问题的发展历史和研究现状进行了综述.第二章给出了关于无网格方法的三种形函数—移动最小二乘近似法(MLS)、改进的移动最小二乘近似法(IMLS)以及改进的插值型移动最小二乘法(IIMLS)的推导过程.第三章是本文的主要工作,首先是利用投影迭代算子构造出了一个显式投影迭代格式,将Signorini边界条件转化为Neumann边界条件,从而把Signorini问题转化为与之等价的线性边值问题,并给出了用改进的插值型边界无单元法求解该问题的具体步骤.第四章是用本文的方法借助计算机进行编程求解几个经典的算例,并与其他方法进行比较,对本文方法的可行性和有效性进行数值验证.本文提出了一种数值求解Signorini问题的无网格投影迭代算法.首先,基于不动点方程提出了一个新的显式投影迭代算子,将Signorini边界条件转换成Neumann边界条件.其次,利用构造的投影迭代算子将Signorini问题转化为一般的椭圆边值问题.最后,用改进的插值型边界无单元法求解该椭圆边值问题.在整个问题的求解中,只需要计算一次系数矩阵,在后面的迭代中不需要更新该系数矩阵.
[Abstract]:Many physical phenomena, such as beach seepage problem, electroplating problem and free boundary problem, can be modeled by Signorini problem. In the Signorini problem, the Dirichlet boundary condition and the Neumann boundary condition on the Signorini boundary appear alternately, and the alternate positions are unknown, which makes the solution of the Signorini problem more complicated. In the first chapter, the history and research status of meshless methods and Signorini problems are reviewed. In the second chapter, the derivation process of three shape functions for meshless method-moving least square approximation (MLS),) modified moving least squares approximation (IMLS) and interpolating moving least square method (IIMLS) is given. The third chapter is the main work of this paper. Firstly, an explicit projective iterative scheme is constructed by using the projective iteration operator. The Signorini boundary condition is transformed into the Neumann boundary condition, and the Signorini problem is transformed into an equivalent linear boundary value problem. The concrete steps of solving the problem with the improved interpolation boundary free element method are also given. In the fourth chapter, we use the method of this paper to solve several classical examples by computer programming, and compare with other methods to verify the feasibility and effectiveness of this method. In this paper, a meshless projection iterative algorithm for numerical solution of Signorini problem is presented. Firstly, based on the fixed point equation, a new explicit projection iteration operator is proposed, which transforms the Signorini boundary condition into the Neumann boundary condition. Secondly, the Signorini problem is transformed into a general elliptic boundary value problem by using the constructed projection iteration operator. Finally, an improved interpolation boundary element free method is used to solve the elliptic boundary value problem. In the solution of the whole problem, it is necessary to calculate the coefficient matrix only once, and it is not necessary to update the coefficient matrix in subsequent iterations.
【学位授予单位】：重庆师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.82

【相似文献】