弹性力学问题的局部Petrov-Galerkin方法  

A LOCAL PETROV-GALERKIN METHOD FOR THE ELASTICITY PROBLEM

作　　者：龙述尧[1] 

机构地区：[1]湖南大学工程力学系,长沙410082

出　　处：《力学学报》2001年第4期508-518,共11页Chinese Journal of Theoretical and Applied Mechanics

摘　　要：提出了弹性力学平面问题的局部Ｐｅｔｒｏｖ－Ｇａｌｅｒｋｉｎ方法，这是一种真正的无网格方法．　　　　这种方法采用移动最小二乘近似函数作为试函数，并且采用移动最小二乘近似函数的权函数作　　　　为加权残值法加权函数；同时这种方法只包含中心在所考虑点处的规则局部区域上以及局部边　　　　界上的积分；所得系统矩阵是一个带状稀疏矩阵，该方法可以容易推广到求解非线性问题以及　　　　非均匀介质的力学问题．还计算了两个弹性力学平面问题的例子，给出了位移和能量的索波列　　　　夫模及其相对误差．所得计算结果证明：该方法是一种具有收敛快、精度高、简便有效的通用　　　　方法；在工程中具有广阔的应用前景．The basic concept and numerical implementation of a local Petrov-Galerkin method for solving the elasticity problem have presented in the present paper. It is a new truly meshless method, because the numerical implementation of the method leads to an efficient meshless discrete model. It uses the moving least square approximation as a trial function, and uses the weighted function of the moving least square approximation as a test function of the weighted residual method. The essential boundary conditions in the present formulation are imposed by a penalty factor method. It involves only integrations over a regular local subdomain and on a local subboundary centered at the node in question. It possesses a great flexibility in dealing with the numerical model of the elasticity plane problems under various boundary conditions with arbitrary shapes. Convergence studies in the numerical examples show that the present method possesses an excellent rate of convergence and reasonably accurate results for both the unknown displacement and strain energy, as the original approximated trial solutions have good continuity and smoothness. The numerical results also show that using both linear and quadratic bases as well as spline and Gaussian weighted functions in approximation functions can give quite accurate numerical results. Compared with the element-free Galerkin method based on a global Galerkin formulation, the present approach is found to have the following advantages. i) Absolutely no elements or cells are needed in the present formulation, either for interpolation purposes or for integration purposes, while regular cells are required in the element-free Galerkin method to evaluate volume integrals. ii) No special special integration scheme is needed to evaluate the volume and boundary integrals. The integrals in the present method are evaluated only over regularly-shaped subdomains and their boundaries. The local subdomain in general is a sphere for three dimensional problem or a circle for two dimensional problem

关 键 词：局部Petrov-Galerkin方程 移动最小二乘近似函数 索波列夫模 带状稀疏矩阵 非线性力学 平面问题 弹性力学 

分 类 号：O343.1[理学—固体力学]