面向线弹性问题的虚拟元方法
Virtual Element Method for Linear Elasticity Problem
摘要: 本文研究基于虚拟元方法求解带混合边界条件的线弹性问题。首先给定弹性力学的基本方程和二维混合弱对称形式的线弹性问题，并通过变分原理得到原问题的变分形式。其次通过定义局部刚体运动空间，构造虚拟元空间。然后对于给定的每一个变量的近似得到方法的收敛性，并通过已知的引理和不等式验证收敛性，同时给出原问题的误差估计。最后根据悬臂梁问题验证理论分析的有效性与可行性。
Abstract: In this paper, the virtual element method is used to solve the linear elastic problem with mixed boundary conditions. Firstly, the basic equations of elasticity and the two dimensional mixed weak-ly symmetric formulation linear elasticity problems are given. The variational form of the original problem is obtained by the variational principle. Secondly, the virtual element space is obtained by defining the local rigid motion space. Then the convergence of the method is obtained for the ap-proximation of each given variable, and the convergence is verified by the known lemmas and ine-qualities. At the same time, the error estimate of the original problem is given. Finally, the validity and feasibility of the theoretical analysis are verified by the cantilever beam problem.
文章引用：程新博, 梁晓坤, 马俊驰. 面向线弹性问题的虚拟元方法[J]. 应用数学进展, 2022, 11(12): 9103-9111. https://doi.org/10.12677/AAM.2022.1112960

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