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An a posteriori error analysis based on equilibrated stresses for finite element approximations of frictional contact

Ilaria Fontana, Daniele A. Di Pietro

Abstract

We consider the unilateral contact problem between an elastic body and a rigid foundation in a description that includes both Tresca and Coulomb friction conditions. For this problem, we present an a posteriori error analysis based on an equilibrated stress reconstruction in the Arnold--Falk--Winther space that includes a guaranteed upper bound distinguishing the different components of the error. This analysis is the starting point for the development of an adaptive algorithm including a stopping criterion for the generalized Newton method. This algorithm is then used to perform numerical simulations that validate the theoretical results.

An a posteriori error analysis based on equilibrated stresses for finite element approximations of frictional contact

Abstract

We consider the unilateral contact problem between an elastic body and a rigid foundation in a description that includes both Tresca and Coulomb friction conditions. For this problem, we present an a posteriori error analysis based on an equilibrated stress reconstruction in the Arnold--Falk--Winther space that includes a guaranteed upper bound distinguishing the different components of the error. This analysis is the starting point for the development of an adaptive algorithm including a stopping criterion for the generalized Newton method. This algorithm is then used to perform numerical simulations that validate the theoretical results.
Paper Structure (16 sections, 11 theorems, 97 equations, 15 figures, 3 tables, 1 algorithm)

This paper contains 16 sections, 11 theorems, 97 equations, 15 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setting
  3. Continuous problem
  4. Discrete problem
  5. A posteriori error analysis
  6. Basic a posteriori error estimate
  7. Separating the error components
  8. Comparison with the energy norm
  9. Equilibrated stress reconstruction
  10. Numerical results
  11. Tresca friction
  12. Coulomb friction
  13. A test case from literature
  14. Efficiency of the estimators
  15. Proof of the local efficiency
  16. ...and 1 more sections

Key Result

Theorem 3

Let $\boldsymbol{u}_h$ be the solution of eq:Nitsche-based_method, $\mathcal{R}(\boldsymbol{u}_h)$ the residual defined by eq:residual definition, and $\boldsymbol{\sigma}_h$ an equilibrated stress reconstruction in the sense of Definition def:equilibrated stress reconstruction. Then,

Figures (15)

  • Figure 1: Example of domain $\Omega$ with $d=2$. The boundary $\partial\Omega$ is subdivided into $\Gamma_{\rm D}$ (in green), $\Gamma_{\rm N}$ (in red), and $\Gamma_{\rm C}$ (in blue).
  • Figure 2: Regularized operators for $d=2$ and constant Tresca friction.
  • Figure 3: Illustration of a patch $\omega_{\boldsymbol{a}}$ around an inner node $\boldsymbol{a}\in\mathcal{V}_{h}^{\rm i}$ and around a boundary node $\boldsymbol{a}\in\mathcal{V}_{h}^{\rm b}$.
  • Figure 4: Rectangular domain of the numerical cases of Section \ref{['sec:numerical results:tresca rectangular']} and \ref{['sec:numerical results:coulomb rectangular domain']} with representation of internal and lateral forces, and division of the domain's boundary. In particular, a uniform load $\boldsymbol{g}_{\rm N}$ is enforced on $\Gamma_{{\rm N},1}$, while homogeneous Neumann conditions are enforced on $\Gamma_{{\rm N},2}$. The portion of the boundary $\Gamma_{\rm D}$ is fixed, while contact is possible on $\Gamma_{\rm C}$.
  • Figure 5: Vertical (left) and horizontal displacement (right) in the deformed configuration for the Tresca test case of Section \ref{['sec:numerical results:tresca rectangular']}.
  • ...and 10 more figures

Theorems & Definitions (29)

  • Remark 1: Choice of normal contact conditions
  • Definition 2: Equilibrated stress reconstruction
  • Theorem 3: A posteriori error estimate for the dual norm of the residual
  • proof
  • Remark 4: Possible regularization of projection operators
  • Theorem 6: A posteriori error estimate distinguishing the error components
  • proof
  • Remark 7: Local stopping criterion
  • Theorem 8: Control of the energy norm
  • proof
  • ...and 19 more