Study of virtual element method (VEM)

Benchmark test with L-shaped element using VEM
Virtual element method (VEM), which is recently proposed by L. Beirão da Veiga [1], is well-developed since its birth and becomes a powerful tool in the field of computational mechanics. Different from classical finite element method (FEM), VEM has the following features
  • VEM can be applied to arbitrary polygonal mesh
  • VEM is able to handle hanging nodes on element edge
A number of researches are implemented to exploit the effectiveness and application of VEM, including elastic analysis [2], finite deformation analysis[3], contact analysis [4], fracture analysis [5] and structural topology optimization [6]. Based on Ref. [4], the study contact analysis with non-matching contact interface by using second-order VEM is investigated as in Fig.1. The results in Fig.2 show that even with irregular element shape and non-matching interface the patch test is also passed without biased stress distribution along the contact interface.
FIG.1: Model of Example 1; (left) Geometry models of two elastic blocks, (right) Second-order VEM discretization
FIG.2: Results of patch test; (left) Deformed shape, (right) Stress contour of $ \sigma_{22} $

There are still a lot more fascinating and interesting properties of VEM to be explored, and right now I would like to dig more on VEM in the areas of structural topology optimization and nonlinear and multiscale analysis.

[1] L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, A. Russo, Basic principles of virtual element methods, Math. Model. Methods Appl. Sci. 23 (2013) 199–214.
[2] L. Beirão da Veiga, F. Brezzi, L.D. Marini, Virtual elements for linear elasticity problems, SIAM J. Numer. Anal. 51 (2013) 794–812.
[3] H. Chi, L.B. da Veiga, G.H. Paulino, Some basic formulations of the virtual element method (VEM) for finite deformations, Comput. Methods Appl. Mech. Eng. 318 (2017) 148–192.
[4] P. Wriggers, W.T. Rust, B.D. Reddy, A virtual element method for contact, Comput. Mech. 58 (2016) 1039–1050.
[5] V.M. Nguyen-Thanh, X. Zhuang, H. Nguyen-Xuan, T. Rabczuk, P. Wriggers, A virtual rlement method for 2D linear elastic fracture analysis, Comput. Methods Appl. Mech. Eng. 340 (2018) 366–395.
[6] H. Chi, A. Pereira, I.F.M. Menezes, G.H. Paulino, Virtual element method (VEM)-based topology optimization: an integrated framework, Struct. Multidiscip. Optim. 62 (2020) 1089–1114.
Wei Shen
Wei Shen
Phd student