加权最小二乘无网格法

MESHLESS WEIGHTED LEAST-SQUARE METHOD

在最小二乘法和移动最小二乘近似的基础上提出了加权最小二乘无网格法。该方法除节点外又引入了一些辅助点,控制方程在所有节点和辅助点处的残差用最小二乘法予以消除,边界条件用罚函数法引入。另外对移动最小二乘近似进行了改进,并给出了最小二乘法中泛函的简化格式,因而提高了计算效率。与配点法相比,新方法精度高,稳定性好,并且系数矩阵是对称正定矩阵。与Galerkin法相比,该方法不需要进行高斯积分,因而计算量小。算例表明该方法具有效率高、精度高和稳定性好等优点,并且易于实现。

Two methods of discretization, collocation method and Galerkin method, have been dominant in existing meshless methods. Although Galerkin method possesses several advantages, one of the major difficulties in the implementation of Galerkin based meshless method is how to evaluate integrals. In the Galerkin method, derivatives in domain integrals are lowered by using the divergence theorem to establish the weak form. The inaccuracy in integration will result in significant error in the solution. However, the shape functions in meshless method are very complex, so a large number of quadrature points must generally be used to integrate the weak form as accurate as possible. As a consequence, the Galerkin based meshless methods are much more expensive than FEM. In contrast, collocation based meshless methods are truly meshless methods, and are very efficient. However, equilibrium conditions are satisfied only at nodes within domain, even not at nodes on boundary, so significant error could be resulted in. These methods also suffer from instability. Based on the weighted least-square method and the moving least-square approximation, a new efficient meshless method, Meshless Weighted Least-Square Method (MWLS), is developed. Except for nodes, a number of auxiliary points are used to eliminate the residual of governing equations in a weighted least-square sense. The trial functions are constructed by using the moving least-square approximation, and the boundary conditions are imposed by using the penalty method. The coefficient matrix obtained is symmetric and positive definite. To investigate the accuracy of the proposed method, a patch test, a cantilerver beam and an infinite plate with a central circular hole are analyzed in detail. Numerical studies show that the new method possesses several advantages, such as high accuracy, high stability, and high efficiency.