In this work, a space-time virtual element method is presented for the discretization of the two-dimensional elastodynamics in a space-time cylinder. The basic idea of the space-time scheme is to treat time as an additional dimension. This approach has been extensively applied to numerous initial value problems based on the finite element method and virtual element method. However, there has been no report on the application of the space-time virtual element method in elastodynamics. The variational formulation and space-time bilinear format are derived based on Hamilton’s principle to achieve this. The discretization space is obtained by the virtual element method in space and the upwind finite element method in time. The space-time mesh is obtained as a tensor product of space and time meshes, then the element stiffness matrix for the elastodynamic problem can be obtained based on the Kronecker product directly. Some two-dimensional examples solved by the developed space-time virtual element method are given to demonstrate the accuracy and stabilization. MATLAB codes for the space-time virtual element and finite element methods can be downloaded from https://github.com/Qinxiaoye/VEM-spcae-time-dynamic.

中文翻译：

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弹性动力学的时空虚元方法：理论、应用和代码开发


在这项工作中，提出了一种时空虚元方法，用于时空圆柱体中二维弹性动力学的离散化。时空方案的基本思想是将时间视为一个附加维度。该方法已广泛应用于许多基于有限元法和虚元法的初始值问题。然而，目前还没有关于时空虚元法在弹性动力学中的应用的报道。变分公式和时空双线性格式是根据哈密尔顿原理推导出来的，以实现这一点。离散化空间在空间上通过虚元法和在时间上通过逆风有限元法得到。将时空网格作为空间和时间网格的张量积得到，然后可以直接基于 Kronecker 积得到弹性动力学问题的单元刚度矩阵。给出了一些由开发的时空虚元方法求解的二维算例，以验证其准确性和稳定性。时空虚元和有限元方法的 MATLAB 代码可以从 https://github.com/Qinxiaoye/VEM-spcae-time-dynamic 下载。