Virtual Elements for Elasticity Problems

One of the first applications of virtual elements in engineering was related to elasticity. It had the aim to construct a general methodology that could be applied to problems involving two-dimensional linear elastic solids under general loading conditions, see Beirão da Veiga et al. (2013). After these first developments and formulations of the new method many follow up contributions appeared. An extension to three-dimensional problems was then provided in Gain et al. (2014). A paper that explained the application of the virtual element method in elasticity in more details is due to Artioli et al. (2017a). In this line of work fits the paper by Mengolini et al. (2019) who discussed implementation issues and compared virtual and finite elements. A mixed approach based on the Hellinger-Reissner functional was presented in Artioli et al. (2017c, 2020c) for linear elasticity in two dimensions and in Dassi et al. (2020) for three dimensional applications. A mixed-enhanced virtual element formulation was developed in D’Altri et al. (2021). Chen and Sukumar (2023a, b) designed a stabilization free formulation of a two-dimensional linear virtual element for linear elasticity. In the same line is the paper of Lamperti et al. (2023) who used the Hu-Washizu variational approach to design sel-stabilized virtual elements for linear elastic solids.