无单元Galerkin方法施加本质边界条件研究进展  

Advances in Implementation of Essential Boundary Conditions for Element-Free Galerkin Method

作　　者：曹阳 陈莹婷 姚林泉 CAO Yang;CHEN Yingting;YAO Linquan(School of Transportation and Civil Engineering,Nantong University,Nantong 226019,Jiangsu,China;School of Science,Nantong University,Nantong 226019,Jiangsu,China;School of Rail Transportation,Soochow University,Suzhou 215006,Jiangsu,China)

机构地区：[1]南通大学交通与土木工程学院,江苏南通226019 [2]南通大学理学院,江苏南通226019 [3]苏州大学轨道交通学院,江苏苏州215006

出　　处：《力学季刊》2020年第4期591-612,共22页Chinese Quarterly of Mechanics

基　　金：国家自然科学基金(11771225,11572210);南通市第五期“226工程”培养对象科研项目;江苏省研究生科研与实践创新计划项目(KYCX19-2087)。

摘　　要：无网格方法是一种基于节点离散问题域的数值方法,已在许多科学计算和工程领域中得到广泛应用.基于移动最小二乘(MLS)近似的全局弱形式无单元Galerkin方法具有计算简单、精确度高等优点,是最著名的无网格方法之一.但由于MLS方法所构造的形函数一般不具备Kronecker delta函数性质,离散所得到的代数方程组的未知量是节点参数而非节点函数值,因而本质边界条件不易施加.本文以弹性力学方程为例,首先简单回顾了构造形函数的MLS方法和无单元Galerkin方法的计算过程,然后从求解问题步骤的四个方面,即求解区域的划分、变分原理的修正、形函数的构造、离散代数方程组的建立,对目前已提出的数十种关于如何方便准确地施加本质边界条件的方法进行归纳总结,比较了这些方法的优缺点,最后提出了展望.Meshfree method is a numerical method based on node discretization and has been widely used in many scientific computation and engineering fields. The element-free Galerkin(EFG) method, which is based on moving least squares(MLS) approximation and global weak form, is one of the most famous meshfree methods and has the advantages of simple calculation and high accuracy. Since the shape functions constructed by the MLS method generally do not have the properties of the Kronecker delta function, the unknowns of the discrete algebraic equations are node parameters rather than node function values. Thus, it is not easy to impose the essential boundary conditions. In this paper, by taking the elastic mechanics equation as an example, we first briefly review the calculation process of the MLS method for constructing the shape functions and the element-free Galerkin method, then summarize dozens of methods on how to conveniently and accurately apply essential boundary conditions from four aspects of the computational steps, i.e. the division of the solution area, the modification of variation principle, the construction of the shape function, and the establishment of discrete algebraic equations. The advantages and disadvantages of these methods are compared in details. Finally, we put forward some prospects for future research.

关 键 词：无网格方法 无单元Galerkin方法 移动最小二乘 本质边界条件 

分 类 号：O234[理学—运筹学与控制论]