Kirchhoff板弯曲问题的改进的复变量无单元Galerkin方法  

The improved complex variable element-free Galerkin method for the analysis of Kirchhoff plates

作　　者：王斌骅 马永其[1,2] 冯伟[1] 程玉民[1] WANG BinHua MA YongQi FENG Wei CHENG YuMin(Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China Department of Mechanics, Shanghai University, Shanghai 200444, China)

机构地区：[1]上海大学上海市应用数学和力学研究所,上海200072 [2]上海大学力学系,上海200444

出　　处：《中国科学：物理学、力学、天文学》2017年第9期28-40,共13页Scientia Sinica Physica,Mechanica & Astronomica

基　　金：国家自然科学基金(编号:11571223;U1433104)资助项目

摘　　要：基于改进的复变量移动最小二乘法,建立了Kirchhoff板弯曲问题的改进的复变量无单元Galerkin方法.相对于移动最小二乘法,改进的复变量移动最小二乘法采用一维基函数建立二维问题的逼近函数,提高了形函数计算效率.由改进的复变量移动最小二乘法建立Kirchhoff板的挠度逼近函数,根据Kirchhoff板弯曲问题的Galerkin弱形式建立离散方程,并应用罚函数法施加本质边界条件,推导了Kirchhoff板弯曲问题的改进的复变量无单元Galerkin法的公式.通过对4个典型算例进行计算和分析,说明了本文建立的Kirchhoff板弯曲问题的改进的复变量无单元Galerkin方法的有效性,并通过分析数值解的精度对本文方法中如何选取合适的基函数、权函数、影响域比例参数、节点分布和罚因子进行了讨论.数值算例说明了本文方法具有较好的收敛性和较高的计算精度.Based on the improved complex variable moving least-squares （ICVMLS） approximation, the improved complex variable element-free Galerkin （ICVEFG） method for the bending problem of Kirchhoff plate is presented. Compared with the moving least-squares （MLS） approximation, in the ICVMLS approximation, the approximation ftmction of two-dimensional problems can be obtained with one-dimensional basis function, then the computational efficiency of the shape functions is higher. Compared with the meshless methods based on the MLS approximation, under the same node distributions, the ones using the ICVMLS approximation can obtain the solutions with higher computational accuracy; and under the similar computational accuracy, the ones using the ICVMLS approximation have higher computational efficiency. The ICVMLS approximation is used to form the approximation function of the deflection of a Kirchhoff plate, the Galerkin weak form of the bending problem of Kirchhoff plates is adopted to obtain the discretized system equations, and the penalty method is employed to enforce the essential boundary conditions, then the corresponding formulae of the ICVEFG method for the bending problem of Kirchhoff plates are presented. By computing and analyzing four typical examples, it is shown that the ICVEFG method of Kirchhoff plates in this paper is efficient. And the computational precision of the numerical solutions is analyzed to select the basis function, weight function, scaling factor, node distribution and penalty factor in the ICVEFG method. Numerical examples show that the method in this paper has better convergence and higher accuracy. When the quadratic polynomial basis function and the cubic or quartic spline weight function are used, and dm~ ^-- 4.2-4.4, the ICVEFG method of Kirchhoffplates in this paper can obtain the solutions with high computational accuracy. And more nodes are distributed in the problem domain, higher computational accuracy of the solution will obtained, which shows that the method in this pape

关 键 词：无网格方法 改进的复变量移动最小二乘法 复变量无单元Galerkin方法 改进的复变量无单元 GALERKIN方法 Kirchhoff板 

分 类 号：O241.8[理学—计算数学]