A lowest-order locking-free nonconforming virtual element method based on the reduced integration technique for linear elasticity problems

Abstract

We develop a lowest-order nonconforming virtual element method for planar linear elasticity, which can be viewed as an extension of the idea in Falk (1991) to the virtual element method (VEM), with the family of polygonal meshes satisfying a very general geometric assumption. The method is shown to be uniformly convergent for the nearly incompressible case with optimal rates of convergence. The crucial step is to establish the discrete Korn's inequality, yielding the coercivity of the discrete bilinear form. We also provide a unified locking-free scheme both for the conforming and nonconforming VEMs in the lowest order case. Numerical results validate the feasibility and effectiveness of the proposed numerical algorithms.

A node-based uniform strain virtual element method for compressible and nearly incompressible elasticity

“…Moreover, in the interior of the element the basis functions are not known. With these particular assumptions for the basis functions, the mean element nodal displacements (8), and the element averages ( 9) and (10), can be computed in terms of the element degrees of freedom.…”

“…No additional degrees of freedom are introduced (displacement-based formulation). In the existing VEM approaches, the volumetric locking is alleviated by the B-bar formulation [1,2], mixed formulation [3], enhanced strain formulation [4], hybrid formulation [5], nonconforming formulations [6][7][8], or addition of degrees of freedom related to the normal components of the displacement field on the element's edges to satisfy the inf-sup condition [9].…”

mVEM: A MATLAB Software Package for the Virtual Element Methods

“…where V nc 1 (K) and V 1 (K) are the lowest-order nonconforming and conforming virtual element spaces, respectively. In this case, the computation for k = 1 reads 1 C01xe = Ne (: ,1) '; C01ye = Ne (: ,2) '; We also develop a lowest-order nonconforming virtual element method for planar linear elasticity in [53], which can be viewed as an extension of the idea in [32] to the virtual element method, with the family of polygonal meshes satisfying a very general geometric assumption. The method is shown to be uniformly convergent for the nearly incompressible case with optimal rates of convergence.…”

“…In addition, we provide a unified locking-free scheme both for the conforming and nonconforming VEMs in the lowest order case. The implementation and the numerical test can be found in [53]. The test scripts are main elasticityVEM NCreducedIntegration.m and main elasticityVEM NCUniformReducedIntegration.m, respectively.…”