用无网格局部Petrov-Galerkin方法分析Winkler弹性地基板

An Analysis of Plates on the Winkler Foundation with the Meshless Local Petrov-Galerkin Method

利用Winkler弹性地基板控制微分方程的等效积分对称弱形式,同时对变量(挠度)采用移动最小二乘近似函数进行插值,研究了无网格局部Petrov Galerkin方法在弹性地基板弯曲问题中的应用.它不需要任何形式的网格划分,所有的积分都在规则形状的子域及其边界上进行,并用罚因子法施加本质边界条件.数值算例说明,无网格局部Petrov Galerkin法不但能够求解弹性静力学问题,而且在求解弹性地基板问题时仍具有收敛快、稳定性好和精度高的特点.

The Meshless Local Petrov-Galerkin(MLPG) method was applied to solve the bending problems of a thin plate resting on the Winkler′s elastic foundation. The method adopted the moving least-square approximation to interpolate the solution variables, and employed a local symmetric weak form. The present method is a meshless one as it doesn′t need any meshgrids, and all integrals can be easily evaluated in regularly shaped domains (in general, a sphere in the three-dimensional problem) and their boundaries. The essential boundary conditions are enforced by a penalty method. Several examples given in this paper have shown that, in solving the bending problems of thin plates resting on the elastic foundation, the meshless local Petrov-Galerkin method has good stability, high accuracy and high convergence, just as in solving elastic static problems.