几何非线性问题的无网格法分析

用局部Petrov-Galerkin方法求解几何非线性问题，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；只包含巾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力学问题。在涉及几何非线性问题的数值方法中，通常都采用增量和迭代分析的方法。本文从虚功原理出发，用移动最小二乘近似函数的权函数替代虚位移，并在整个分析过程中所有变量的表达格式都是采用全拉格朗日格式。数值算例表明，无网格局部Petrov-Galerkin方法在求解几何非线性问题时仍具有很好的精度。

ANALYSIS OF THE GEOMETRICALLY NONLINEAR PROBLEM BY MESHLESS METHOD

The local Petrov-Galerkin method (LPGM) is applied to solving geometrically nonlinear problems. It is a new truly meshless method. The local Petrov-Galerkin method uses the moving least square approximation as a trial function, and involves only integrations over a regular local sub-domain and on a local sub-boundary centered at the node in question. A banded sparse system matrix is obtained. These special properties lead to a more convenient formulation in dealing with nonlinear problems and non-homogeneous medium problems. An incremental and iterative solution procedure using modified Newton-Raphson iterations is used to solve the geometrically nonlinear problem. Formulations for the geometrically nonlinear problem are obtained from virtual work principle by using the weighted function of the moving least square approximation as a virtual displacement. All measures are related back to the original configuration. This technique is named as the total Lagrangian method. Several examples are given to show that in solving the geometrically nonlinear problem, the local Petrov-Galerkin method still has a good accuracy.