求解线弹性双孔问题的虚拟元方法

Solving the Linear Elastic Porous Problem on Virtual Element Method

摘要: 本文研究虚拟元方法求解线弹性双孔板问题，该方法可以有效解决孔边缘出现的应力集中现象，能够克服线弹性问题的非物理上的压力震荡现象，而且具有剖分过程简单、网格适应性强、可以借助悬点优化边界问题、基函数不需要显式表达、借助特殊的投影算子及自由度求出刚度矩阵的优势，进而奠定了虚拟元方法在处理同类问题的地位。最后通过误差分析和数值实验，证明了虚拟元方法求解线弹性双孔板问题的有效性与可行性，因此，该方法可以为检验耦合杂交混合元、Galerkin法和有限元方法等其它近似方法提供参考。

Abstract: In this paper, the virtual element method is used to study the linear elastic porous plate problem, it can effectively solve the phenomenon of stress concentration at the edge of the hole and overcome the non-physical pressure shock phenomenon of linear elastic problem. Moreover, this method has the advantages of simple subdivision process, strong grid adaptability, optimized boundary problem with suspension points, no explicit expression of basis function, and stiffness matrix obtained with special projection operators and degrees of freedom, thus establishing the position of virtual element method in dealing with similar problems. Finally, through error analysis and numerical experiments, the validity and feasibility of the virtual element method for the problem of linear elastic porous plate are proved. Therefore, the method can provide a reference for testing other approximate methods such as a coupled of hybrid element, Galerkin method and finite element method.