A locking-free mixed virtual element discretization for the elasticity eigenvalue problem

TL;DR

The paper addresses the elasticity eigenvalue problem in 2D under near-incompressible conditions, where locking can degrade primal discretizations. It proposes a locking-free mixed virtual element method based on a pseudostress formulation and computable bilinear forms, enabling robust spectrum approximation on general polygonal meshes. The authors prove that the discrete solution operator converges to the continuous one using compact-operator theory, and they derive eigenvalue/eigenfunction error estimates with a double order convergence for eigenvalues; numerical experiments on square, L-shaped, and circular domains corroborate the theory and demonstrate robustness to mesh geometry and stabilization. The approach facilitates accurate spectrum computations on arbitrary polygonal meshes and provides insight into incompressible limits akin to the Stokes problem, with practical implications for elasticity simulations and related mixed formulations.

Abstract

In this paper, we introduce a mixed virtual element method to approximate the eigenvalues and eigenfunctions of the two-dimensional elasticity eigenvalue problem. Under standard assumptions on the meshes, we prove the convergence of the discrete solution operator to the continuous one as the mesh size tends to zero. Using the theory of compact operators, we analyze the convergence of the method and derive error estimates for both the eigenvalues and eigenfunctions. We validate our theoretical results with a series of numerical tests, in which we compute convergence orders and show that the method is locking-free and capable of accurately approximating the spectrum independently of the shape of the polygons on the meshes.

Figures (4)

• Figure 1: Sample meshes: ${\mathcal{T}}_{h}^{1}$ top left), ${\mathcal{T}}_{h}^{2}$ (top middle), ${\mathcal{T}}_{h}^{3}$ (top right),${\mathcal{T}}_{h}^{4}$ (bottom left), ${\mathcal{T}}_{h}^{5}$ (bottom middle) and ${\mathcal{T}}_{h}^{6}$ (bottom right).
• Figure 2: Test 1. Plots of the computed first eigenfunction for ${\mathcal{T}}_{h}^{4}$(top left), second eigenfunction (top right) for ${\mathcal{T}}_{h}^{2}$, fourth eigenfunction for ${\mathcal{T}}_{h}^{3}$(bottom left) and fifth eigenfunction for ${\mathcal{T}}_{h}^{5}$(bottom right).
• Figure 3: Test 2. Plots of the computed first eigenfunction for ${\mathcal{T}}_{h}^{2}$(left), second eigenfunction (right) for ${\mathcal{T}}_{h}^{4}$, third eigenfunction for ${\mathcal{T}}_{h}^{5}$(bottom left) and fourth eigenfunction for ${\mathcal{T}}_{h}^{5}$(bottom right). All these plots were obtained for $\nu=0.49$.
• Figure 4: Test 3. Plots of the computed first and the second eigenfunction for ${\mathcal{T}}_{h}^{6}$ and $\nu=0.35$,

Theorems & Definitions (15)

• Lemma 2.1
• Remark 2.2
• Remark 2.4
• Theorem 2.5: Spectral characterization of $\boldsymbol{T}$
• Lemma 3.1
• Proof 1
• Theorem 4.1
• Proof 2
• Theorem 4.2
• Lemma 4.3