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Volumetric locking-free Mixed Virtual Element Methods for Contact Problems

C. Lovadina, L. Molinari

TL;DR

This paper develops and analyzes mixed displacement/pressure Virtual Element Methods (VEMs) for 2D frictionless elastic contact, addressing volumetric locking in nearly incompressible materials by introducing a pressure variable $p^i = \lambda^i\,\mathrm{div}\,{\bf u}^i$ and using a coupled variational formulation. It presents two VEM schemes (k=1 and k=2) on polygonal meshes, including a detailed convergence theory with explicit stability constants and an explicit treatment of small-edge effects, and validates the theory through comprehensive numerical tests. The results show robustness to the incompressibility parameter $\lambda$ (no locking), stability even with small edges, and convergence rates consistent with theory (with occasional superconvergence for the quadratic scheme in some tests). The work provides a flexible, polygon-geometry-friendly framework for contact problems in elasticity and sets the stage for extensions to 3D, friction, and alternative VEM stabilizations.

Abstract

We consider the approximation of the 2D frictionless contact problem in elasticity using the Virtual Element Methods (VEMs). To overcome the volumetric locking phenomenon in the nearly incompressible case, we adopt a mixed displacement/pressure ($u/p$) variational formulation, where pressure is introduced as an independent unknown. We present the VEM discretization and develop a general error analysis, keeping explicit track of the constants involved in the error estimates, thus allowing to consider meshes with "small edges". As examples, we consider two possible VEM schemes: a first-order scheme and a second-order scheme. The numerical results confirm the theoretical predictions, specifically both schemes show: 1) robustness with respect to the volumetric parameter $λ$, thus preventing the occurrence of the volumetric locking phenomenon; 2) good behavior even in the presence of "small edges"; 3) achievement of the expected theoretical convergence rates.

Volumetric locking-free Mixed Virtual Element Methods for Contact Problems

TL;DR

This paper develops and analyzes mixed displacement/pressure Virtual Element Methods (VEMs) for 2D frictionless elastic contact, addressing volumetric locking in nearly incompressible materials by introducing a pressure variable and using a coupled variational formulation. It presents two VEM schemes (k=1 and k=2) on polygonal meshes, including a detailed convergence theory with explicit stability constants and an explicit treatment of small-edge effects, and validates the theory through comprehensive numerical tests. The results show robustness to the incompressibility parameter (no locking), stability even with small edges, and convergence rates consistent with theory (with occasional superconvergence for the quadratic scheme in some tests). The work provides a flexible, polygon-geometry-friendly framework for contact problems in elasticity and sets the stage for extensions to 3D, friction, and alternative VEM stabilizations.

Abstract

We consider the approximation of the 2D frictionless contact problem in elasticity using the Virtual Element Methods (VEMs). To overcome the volumetric locking phenomenon in the nearly incompressible case, we adopt a mixed displacement/pressure () variational formulation, where pressure is introduced as an independent unknown. We present the VEM discretization and develop a general error analysis, keeping explicit track of the constants involved in the error estimates, thus allowing to consider meshes with "small edges". As examples, we consider two possible VEM schemes: a first-order scheme and a second-order scheme. The numerical results confirm the theoretical predictions, specifically both schemes show: 1) robustness with respect to the volumetric parameter , thus preventing the occurrence of the volumetric locking phenomenon; 2) good behavior even in the presence of "small edges"; 3) achievement of the expected theoretical convergence rates.
Paper Structure (22 sections, 18 theorems, 177 equations, 12 figures, 1 table)

This paper contains 22 sections, 18 theorems, 177 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. A Mixed Formulation of the Contact Problem
  3. Virtual Element Discretization of the Contact Problem
  4. Virtual element meshes
  5. Discrete bilinear forms
  6. The Discrete Problem
  7. A general convergence result
  8. Two examples of Virtual Element Approximation
  9. A first order element (k=1): local spaces
  10. A quadratic element (k=2): local spaces
  11. Spaces and forms
  12. Convergence analysis of the quadratic virtual element
  13. Evaluation of the stability constants
  14. Evaluating the continuity and coercivity constants αh* and α*h
  15. Evaluating the inf-sup constant βh
  16. ...and 7 more sections

Key Result

Proposition 2.1

The contact problem (eq:contact_problem) has a unique solution $({\bf u},{\bf p}) \in {\bf K}\times {\bf Q}$ such that where $C$ is a positive constant independent from $\lambda^i$.

Figures (12)

  • Figure 1: Degrees of freedom for $k=1$ and $k=2$.
  • Figure 2: Reference configuration and boundary conditions.
  • Figure 3: Example of polygonal meshes: $\mathcal{Q}_{64}$, $\mathcal{H}_{64}$, $\mathcal{W}_{128}$.
  • Figure 4: Test with trigonometric displacement solution (linear element): Behavior of $\delta({\bf u})$ and $\delta({\bf p})$ for the sequence of meshes $\mathcal{Q}_h$.
  • Figure 5: Test with trigonometric displacement solution (linear element): Behavior of $\delta({\bf u})$ and $\delta({\bf p})$ for the sequence of meshes $\mathcal{H}_h$.
  • ...and 7 more figures

Theorems & Definitions (33)

  • Proposition 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 5.1
  • proof
  • ...and 23 more