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

TL;DR

This paper introduces a lowest-order nonconforming virtual element method for planar linear elasticity problems, achieving uniform convergence and locking-free performance on polygonal meshes, validated by numerical experiments.

Contribution

It extends Falk's idea to the virtual element method, establishing a unified locking-free scheme for conforming and nonconforming VEMs with optimal convergence.

Findings

Uniform convergence for nearly incompressible cases.

Establishment of discrete Korn's inequality.

Numerical validation of the method's effectiveness.

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.