虚单元计算方法的最新理论与应用进展

Virtual element method:Theory and applications

虚单元方法是近几年在计算领域迅速发展的一种先进数值方法,相比于有限元方法,该方法放松了对单元凸凹性的限制,可适用于任意形状的多边形单元,因而在处理悬挂节点、接触、多晶体变形等特定问题方面具有优势,是当前计算力学领域的国际前沿与热点方向.本文全面综述了虚单元方法的理论发展,通过介绍该方法在泊松方程、线弹性、非线性等问题中的应用,向读者展示了虚单元法的理论核心以及它和有限元方法的异同.尽管虚单元法的发展目前还处在起步阶段,但该方法在诸多的非线性问题、接触问题、裂纹扩展以及多场耦合等方面展现出了巨大潜力.通过对虚单元方法最新理论与应用进展的综述,为面临单元凸凹性等问题苦恼的计算领域科研工作者提供一种新的解决方案;同时为对工程科学计算感兴趣的青年科研人员提供关于虚单元方法的快速而系统的全面认知,以期青年学者能融会贯通,发展出适应我国计算力学需求的新型算法与高性能软件.

Virtual Element Method(VEM)is a recently-developed numerical method suitable for arbitrarily convex or concave cells.This benefits the VEM to handle hanging nodes,contacts and polycrystalline deformations.We here illustrate the theory of the VEM via the Poisson equation and the elastic problem,and summarize its applications to non-linear problems.Compared to the Finite Element Method(FEM),the characteristics of the VEM are explained in details.The VEM has demonstrated potentials to model contacts,cracks,coupling of multiple physics,and etc.We hope that this review can provides an alternative means for software developers in computational mechanics.