弹性力学中鞍点问题的高效预处理方法与理论

项目名称： 弹性力学中鞍点问题的高效预处理方法与理论

项目编号： No.11301290

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 曹阳

作者单位： 南通大学

项目金额： 22万元

中文摘要： 无网格方法是一种新兴的求解偏微分方程的数值计算方法，是当前计算数学与计算力学领域的热门课题，也是对有限元等传统数值分析方法的重要补充和发展。本项目采用无网格方法中精度最高、应用最广的基于Lagrange乘子法施加本质边界条件的无单元Galerkin方法来研究弹性力学问题，分析离散大型稀疏鞍点矩阵的谱性质，构造高效的预处理子并分析求解方法的收敛性和误差。为此，我们首先对几类弹性力学模型（如平面应力应变问题、薄板小挠度问题、屈曲问题、壳问题等）进行离散得到鞍点型矩阵，并分析这类矩阵的谱性质。其次，为达到快速求解这类鞍点问题的目的，根据离散得到的大型稀疏鞍点型线性方程组系数矩阵的基本特点，寻求高效的预处理方法，并分析预处理矩阵的谱性质和迭代法的收敛性。最后，根据弹性力学的无单元Galerkin方法的收敛性分析和鞍点问题预处理方法的收敛速度研究弹性力学中鞍点问题数值解法的收敛性。

中文关键词： 弹性力学；无网格方法；鞍点问题；预处理；收敛性

英文摘要： Meshfree (or meshless) methods are a class of newly developed numerical methods for solving partial differential equations. It is currently the hot topic in areas of computational mathematics and computational mechanics, and is also an important complement and development of traditional numerical methods, such as finite element method. In this project, the element free Galerkin method with the Lagrange multiplier being imposed on the essential boundary conditions, which is the most widely used meshless method, is adopted to study elastic mechanics. The spectral properties of discretized saddle point matrices will be analyzed, some efficient preconditioners will be constructed， convergence of the preconditioning methods and error estimate will be studied. More specifically, we first use the element free Galerkin method to discretize several models (such as plane stress strain problems, thin plate problems, buckling problems and shell problems) in elastic mechanics and analyze the eigenproperty of the discretized saddle point matrices. Then, in order to solve the discretized saddle point linear systems quickly, we seek for efficient preconditioned iterative methods based on the characteristics of the coefficient matrices; properties of the preconditioned matrices and convergence of these iterative methods will be

英文关键词： elastic mechanics；meshless methods；saddle point problems；preconditioning；convergence