一种计算对称反对称及周期性问题的改进无网格法

分析对称反对称或周期性问题时，通常只需要取一半或一个周期的局部模型进行离散求解．由于无网格法存在较为显著的边界截断效应，与整体模型相比局部模型的无网格解答精度会有所下降，在边界附近更为显著．为消除这种边界效应，提出了一种改进的无网格近似方法．该方法考虑了局部模型外的节点（虚拟节点）对于模型内各节点（真实节点）的近似影响，并利用对称反对称或周期性关系压缩虚拟节点的自由度，在不增加额外自由度的条件下能够使局部模型和整体模型得到完全相同的结果．通过求解对称反对称和周期性问题算例验证了该方法的有效性．最后对一般性问题的无网格边界效应也进行了简要的讨论．

An enhanced meshfree treatment for symmetric/antisymmetric and periodic problems

Most of the existing meshfree methods are based on moving least square (MLS) or reproducing kernel (RK) approximation. One noticeable property of the MLS/RK approximation is the boundary truncation effect. An enhanced treatment of the special conditions of symmetry, antisymmetry and periodicity in meshfree methods was proposed such that the reduced model was able to yield identical solutions as the full model. In the proposed method, the exterior nodes (dummy nodes) beyond the partial model (discretized with physical nodes) were included into the construction of MLS/RK approximation. Furthermore, using the kinematic relationship the extra dummy nodes were linked to those physical nodes on the partial problem domain being modeled and thus no additional degrees of freedom came into the later computation. It is shown that with this straightforward modification, the resulting meshfree approximation based on the partial model is exactly equivalent to that constructed from the full model. Then the truncation effect is totally removed. The superior performance of the proposed approach compared to the original meshfree formulation is verified through several symmetric, antisymmetric and periodic examples. A possible strategy for removing general boundary effect was also presented.