A novel locking-free virtual element method for linear elasticity problems

TL;DR

This paper introduces a new locking-free virtual element method for planar linear elasticity that achieves optimal convergence rates and is robust with respect to the Lamé constant, demonstrated through numerical tests.

Contribution

It proposes a novel virtual element space based on polygon subdivision, enabling a locking-free and optimally convergent method for linear elasticity problems.

Findings

Method is uniformly convergent with optimal rates in H^1 and L^2 norms.

Numerical tests confirm the theoretical convergence and robustness.

The approach effectively handles polygonal meshes in elasticity simulations.

Abstract

This paper devises a novel lowest-order conforming virtual element method (VEM) for planar linear elasticity with the pure displacement/traction boundary condition. The main trick is to view a generic polygon $K$ as a new one $\widetilde{K}$ with additional vertices consisting of interior points on edges of $K$, so that the discrete admissible space is taken as the $V_1$ type virtual element space related to the partition $\{\widetilde{K}\}$ instead of $\{K\}$. The method is shown to be uniformly convergent with the optimal rates both in $H^1$ and $L^2$ norms with respect to the Lam\'{e} constant $\lambda$. Numerical tests are presented to illustrate the good performance of the proposed VEM and confirm the theoretical results.