An a Posteriori Error Estimator for Two-Body Contact Problems on Non-Matching Meshes

Abstract A posteriori error estimates for two-body contact problems are established. The discretization is based on mortar finite elements with dual Lagrange multipliers. To define locally the error estimator, Arnold–Winther elements for the stress and equilibrated fluxes for the surface traction are used. Using the Lagrange multiplier on the contact zone as Neumann boundary conditions, equilibrated fluxes can be locally computed. In terms of these fluxes, we define on each element a symmetric and globally H(div)-conforming approximation for the stress. Upper and lower bounds for the discretization error in the energy norm are provided. In contrast to many other approaches, the constant in the upper bound is, up to higher order terms, equal to one. Numerical examples illustrate the reliability and efficiency of the estimator.

[1]  Douglas N. Arnold,et al.  Mixed finite elements for elasticity , 2002, Numerische Mathematik.

[2]  Jaroslav Haslinger,et al.  Numerical methods for unilateral problems in solid mechanics , 1996 .

[3]  H. Blum,et al.  An adaptive finite element discretisation¶for a simplified Signorini problem , 2000 .

[4]  Anthony T. Patera,et al.  Domain Decomposition by the Mortar Element Method , 1993 .

[5]  Pierre Ladevèze,et al.  ERROR ESTIMATION AND MESH OPTIMIZATION FOR CLASSICAL FINITE ELEMENTS , 1991 .

[6]  S. Ohnimus,et al.  Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems , 1999 .

[7]  R. Hoppe,et al.  Adaptive multilevel methods for obstacle problems , 1994 .

[8]  Patrice Coorevits,et al.  Mixed finite element methods for unilateral problems: convergence analysis and numerical studies , 2002, Math. Comput..

[9]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[10]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis: Oden/A Posteriori , 2000 .

[11]  J.-P. Pelle,et al.  A posteriori error estimation for unilateral contact with matching and non-matching meshes , 2000 .

[12]  Barbara I. Wohlmuth,et al.  Discretization Methods and Iterative Solvers Based on Domain Decomposition , 2001, Lecture Notes in Computational Science and Engineering.

[13]  R. Verfürth.  A review of a posteriori error estimation techniques for elasticity problems , 1999 .

[14]  J. T. Oden,et al.  A posteriori error estimation of h-p finite element approximations of frictional contact problems , 1994 .

[15]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[16]  Mark Ainsworth,et al.  A posteriori error estimators for second order elliptic systems part 2. An optimal order process for calculating self-equilibrating fluxes , 1993 .

[17]  Serge Nicaise,et al.  A posteriori error estimations of residual type for Signorini's problem , 2005, Numerische Mathematik.

[18]  J. Oden,et al.  Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods , 1987 .

[19]  Peter Wriggers,et al.  Adaptive Finite Elements for Elastic Bodies in Contact , 1999, SIAM J. Sci. Comput..

[20]  Yanqiu Wang,et al.  Preconditioning for the mixed formulation of linear plane elasticity , 2005 .

[21]  Barbara Wohlmuth.  A COMPARISON OF DUAL LAGRANGE MULTIPLIER SPACES FOR MORTAR FINITE ELEMENT DISCRETIZATIONS , 2002 .

[22]  Patrick Hild,et al.  Numerical Implementation of Two Nonconforming Finite Element Methods for Unilateral Contact , 2000 .

[23]  Ralf Kornhuber,et al.  Adaptive finite element methods for variational inequalities , 1993 .

[24]  Frédéric Hecht,et al.  Error indicators for the mortar finite element discretization of the Laplace equation , 2002, Math. Comput..

[25]  Erwin Stein,et al.  A posteriori error estimation in large-strain elasticity using equilibrated local Neumann problems , 1998 .

[26]  Barbara I. Wohlmuth,et al.  An Optimal A Priori Error Estimate for Nonlinear Multibody Contact Problems , 2005, SIAM J. Numer. Anal..

[27]  F. B. Belgacem,et al.  EXTENSION OF THE MORTAR FINITE ELEMENT METHOD TO A VARIATIONAL INEQUALITY MODELING UNILATERAL CONTACT , 1999 .

[28]  Barbara Wohlmuth,et al.  A primal–dual active set strategy for non-linear multibody contact problems , 2005 .

[29]  Wenbin Liu,et al.  A Posteriori Error Estimators for a Class of Variational Inequalities , 2000, J. Sci. Comput..

[30]  Franz-Theo Suttmeier,et al.  On a direct approach to adaptive FE-discretisations for elliptic variational inequalities , 2005, J. Num. Math..

[31]  Faker Ben Belgacem,et al.  Numerical Simulation of Some Variational Inequalities Arisen from Unilateral Contact Problems by the Finite Element Methods , 2000, SIAM J. Numer. Anal..

[32]  J. Tinsley Oden,et al.  Local a posteriori error estimators for variational inequalities , 1993 .

[33]  Weimin Han,et al.  A posteriori error analysis for finite element solutions of a frictional contact problem , 2006 .

[34]  Dietrich Braess,et al.  A posteriori error estimators for obstacle problems – another look , 2005, Numerische Mathematik.

[35]  I. Babuska,et al.  The finite element method and its reliability , 2001 .

[36]  D. W. Kelly,et al.  Procedures for residual equilibration and local error estimation in the finite element method , 1989 .

[37]  Maurizio Falcone,et al.  Numerical methods for viscosity solutions and applications , 2006 .

[38]  D. Kelly,et al.  The self‐equilibration of residuals and complementary a posteriori error estimates in the finite element method , 1984 .

[39]  Faker Ben Belgacem,et al.  Hybrid finite element methods for the Signorini problem , 2003, Math. Comput..

[40]  Peter Wriggers,et al.  Different a posteriori error estimators and indicators for contact problems , 1998 .

[41]  S. Ohnimus,et al.  Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems , 2001 .

[42]  Mohammed Hjiaj,et al.  A Posteriori Error Control of Finite Element Approximations for Coulomb's Frictional Contact , 2001, SIAM J. Sci. Comput..

[43]  C. Bernardi,et al.  A New Nonconforming Approach to Domain Decomposition : The Mortar Element Method , 1994 .

[44]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[45]  R. Glowinski,et al.  Third International Symposium on Domain Decomposition Methods for Partial Differential Equations , 1990 .

[46]  Pierre Ladevèze,et al.  A general method for recovering equilibrating element tractions , 1996 .

[47]  Claes Johnson,et al.  ADAPTIVE FINITE ELEMENT METHODS FOR THE OBSTACLE PROBLEM , 1992 .

[48]  Carsten Carstensen,et al.  Convergence analysis of a conforming adaptive finite element method for an obstacle problem , 2007, Numerische Mathematik.

[49]  Andreas Veeser.  On a posteriori error estimation for constant obstacle problems , 2001 .

[50]  J. H. Bramble,et al.  A second order finite difference analog of the first biharmonic boundary value problem , 1966 .

[51]  Weimin Han,et al.  A posteriori error estimation and adaptive solution of elliptic variational inequalities of the second kind , 2005 .

[52]  F. Ben.  NUMERICAL SIMULATION OF SOME VARIATIONAL INEQUALITIES ARISEN FROM UNILATERAL CONTACT PROBLEMS BY THE FINITE ELEMENT METHODS , 2000 .

[53]  Weimin Han,et al.  A Posteriori Error Analysis Via Duality Theory: With Applications in Modeling and Numerical Approximations , 2004 .

[54]  Pierre Ladevèze,et al.  Error Estimate Procedure in the Finite Element Method and Applications , 1983 .

[55]  Ricardo H. Nochetto,et al.  Fully Localized A posteriori Error Estimators and Barrier Sets for Contact Problems , 2004, SIAM J. Numer. Anal..

[56]  Marc Garbey,et al.  Asymptotic and numerical methods for partial differential equations with critical parameters , 1993 .

[57]  Patrick Hild,et al.  Quadratic finite element methods for unilateral contact problems , 2002 .

[58]  Barbara I. Wohlmuth,et al.  A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier , 2000, SIAM J. Numer. Anal..

[59]  H. Rentz-Reichert,et al.  UG – A flexible software toolbox for solving partial differential equations , 1997 .

[60]  Adaptive Finite Element Discretization in Elasticity and Elastoplasticity by Global and Lokal Error Estimators using Local Neumann‐Problems , 1999 .