Local Petrov-Galerkin Method for a Thin Plate
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摘要: 利用薄板控制微分方程的等效积分对称弱形式和对变量(挠度)采用移动最小二乘近似函数进行插值，研究了薄板弯曲问题的无网格局部Petrov-Galerkin方法．这是一种真正的无网格方法，它不需要任何有限元或边界元网格，不管这种网格是用于能量积分还是进行插值的目的．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．数值例子表明，无网格局部Petrov-Galerkin法不但能够求解二阶微分方程的边值问题，而且求解四阶微分方程的边值问题也很有效，也具有收敛快、稳定性好、对挠度和内力都具有精度高的特点．
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关键词:
- 薄板 /
- 无网格局部Pertov-Galerkin方法 /
- 移动最小二乘近似 /
- 微分方程的等效积分对称弱形式
Abstract: The meshless local Petrov-Galerkin(MLPG)method for solving the bending problem of the thin plate were presented and discussed.The method used the moving least-squares approximation to interpolate the solution variables,and employed a local symmetric weak form.The present method was a truly meshless one as it did not need a finite element or boundary element mesh,either for purpose of interpolation of the solution,or for the integration of the energy.All integrals could be easily evaluated over regularly shaped domains(in general,spheres in three-dimensional problems)and their boundaries.The essential boundary conditions were enforced by the penalty method.Several numerical examples were presented to illustrate the implementation and performance of the present method.The numerical examples presented show that high accuracy can be achieved for arbitrary grid geometries for clamped and simply-supported edge conditions.No post processing procedure is required to computer the strain and stress,since the original solution from the present method,using the moving least squares approximation,is already smooth enough. -
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