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Numerical analysis of variational-hemivariational inequalities with applications in contact mechanics

Weimin Han, Fang Feng, Fei Wang, Jianguo Huang

TL;DR

The paper develops a rigorous numerical framework for stationary variational-hemivariational inequalities (VHIs) arising in contact mechanics, proving a general Galerkin convergence result under minimal solution regularity and Céa-type error estimates. It then specializes the theory to three representative contact problems and shows optimal-order error bounds for linear finite elements and for the virtual element method, supported by numerical experiments that confirm predicted convergence. The work bridges abstract VHI theory and practical, robust numerical schemes, enabling reliable simulations of nonsmooth, frictional contact problems with both standard and polygonal meshes. These results have important implications for the accurate and efficient numerical treatment of complex contact phenomena in engineering and applied mechanics.

Abstract

Variational-hemivariational inequalities are an important mathematical framework for nonsmooth problems. The framework can be used to study application problems from physical sciences and engineering that involve non-smooth and even set-valued relations, monotone or non-monotone, among physical quantities. Since no analytic solution formulas are expected for variational-hemivariational inequalities from applications, numerical methods are needed to solve the problems. This paper focuses on numerical analysis of variational-hemivariational inequalities, reporting new results as well as surveying some recent published results in the area. A general convergence result is presented for Galerkin solutions of the inequalities under minimal solution regularity conditions available from the well-posedness theory, and Céa's inequalities are derived for error estimation of numerical solutions. The finite element method and the virtual element method are taken as examples of numerical methods, optimal order error estimates for the linear element solutions are derived when the methods are applied to solve three representative contact problems under certain solution regularity assumptions. Numerical results are presented to show the performance of both the finite element method and the virtual element method, including numerical convergence orders of the numerical solutions that match the theoretical predictions.

Numerical analysis of variational-hemivariational inequalities with applications in contact mechanics

TL;DR

The paper develops a rigorous numerical framework for stationary variational-hemivariational inequalities (VHIs) arising in contact mechanics, proving a general Galerkin convergence result under minimal solution regularity and Céa-type error estimates. It then specializes the theory to three representative contact problems and shows optimal-order error bounds for linear finite elements and for the virtual element method, supported by numerical experiments that confirm predicted convergence. The work bridges abstract VHI theory and practical, robust numerical schemes, enabling reliable simulations of nonsmooth, frictional contact problems with both standard and polygonal meshes. These results have important implications for the accurate and efficient numerical treatment of complex contact phenomena in engineering and applied mechanics.

Abstract

Variational-hemivariational inequalities are an important mathematical framework for nonsmooth problems. The framework can be used to study application problems from physical sciences and engineering that involve non-smooth and even set-valued relations, monotone or non-monotone, among physical quantities. Since no analytic solution formulas are expected for variational-hemivariational inequalities from applications, numerical methods are needed to solve the problems. This paper focuses on numerical analysis of variational-hemivariational inequalities, reporting new results as well as surveying some recent published results in the area. A general convergence result is presented for Galerkin solutions of the inequalities under minimal solution regularity conditions available from the well-posedness theory, and Céa's inequalities are derived for error estimation of numerical solutions. The finite element method and the virtual element method are taken as examples of numerical methods, optimal order error estimates for the linear element solutions are derived when the methods are applied to solve three representative contact problems under certain solution regularity assumptions. Numerical results are presented to show the performance of both the finite element method and the virtual element method, including numerical convergence orders of the numerical solutions that match the theoretical predictions.

Paper Structure

This paper contains 23 sections, 21 theorems, 284 equations, 13 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sample problems from contact mechanics
  4. Notation
  5. Three contact problems
  6. Numerical analysis of an abstract variational-hemivariational inequality
  7. The abstract variational-hemivariational inequality
  8. Galerkin method for the abstract VHI
  9. Convergence under basic solution regularity
  10. Error estimation
  11. Studies of the contact problems
  12. Studies of Problem \ref{['prob:cont1']}
  13. Studies of Problem \ref{['prob:cont2']}
  14. Studies of Problem \ref{['prob:cont3']}
  15. Virtual element method for variational-hemivariational inequality
  16. ...and 8 more sections

Key Result

Proposition 2.2

Assume that $\Psi \colon U\to \hbox{$\mathbb{R}$}$ is a locally Lipschitz function. Then the following statements are valid. (i) $\Psi^0(u;\lambda\,v)= \lambda\, \Psi^0(u; v)$$\forall\,\lambda\ge 0$, $u\in U$, $v\in V$. $\Psi^0(u; -v) = (-\Psi)^0(u; v)$$\forall\,u\in U$, $v\in V$. (ii) $\Psi^0 (u; v (v) For every $u\in U$, $\partial \Psi(u)$ is nonempty, convex, and weak-ly$^{\,*}$ compact in $V^*

Figures (13)

  • Figure 1: Physical setting
  • Figure 2: Example \ref{['nexample1']}: Numerical solutions with $N$ elements: $N=256$ (upper left), $N=1024$ (upper right), $N=4096$ (bottom left) and $N=16384$ (bottom right).
  • Figure 3: Example \ref{['nexample1']}: (a) Initial mesh with $N=8000$; (b) deformed meshes with $N=8000$
  • Figure 4: Example \ref{['nexample1']}: Tangential displacement on $\Gamma_3$ for (a) FEM solution; (b) VEM solution on square mesh
  • Figure 5: Example \ref{['nexample1']}: Numerical convergence orders for (a) FEM; (b) VEM on the square mesh
  • ...and 8 more figures

Theorems & Definitions (27)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Lemma 4.2
  • Lemma 4.3
  • ...and 17 more