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A posteriori error estimator for contact problems

Project description

In this project, we consider error estimators for contact problems. The focus is on error estimators which are defined in terms of H(div) conforming stress approximations and equilibrated fluxes. It can be shown that the error is bounded by the estimator with a constant one plus a higher order data oscillation term plus a term arising from the contact that is shown numerically to be of higher order. Moreover bounds for the Lagrange multiplier and efficiency of the estimator are derived.

setting solution LM
Hertz-Problem: setting, solution (stress and Lagrange multiplier).

level 6 level 6 level 6
Hertz-Problem: adaptive meshes after 6,9 and 12 refinement steps.

ad/uniform high order term
Hertz-Problem: adaptive vs. uniform refinement and influence of higher order contact term.
Moreover, control-based refinement strategies are analyzed that guarantee a strict energy decay for the series of refined meshes. Here, different refinement strategies are compared.

comparison factor
Hertz-Problem: energy decay for different refinement strategies and factor between exact energy.
The estimator has been implemented and applied to various examples including frictional contact with Coulomb friction, two-body contact problem with non-matching meshes and low regularity problems. Here, optimal convergence is achieved using adaptive mesh refinement.

grid-mat1.jpg grid-mat3.jpg grid-mat5.jpg
sol-mat1.jpg sol-mat3.jpg sol-mat5.jpg
2-body Hertz problem with Coulomb friction and different material parameters.

grid-l3.jpg grid-l5.jpg grid-l5.jpg
fehler-l3.jpg fehler-l5.jpg fehler-l7.jpg
Low regularity problems: Adaptive refinement preserves optimality.


Author(s) Title Appeared In Year Type Download
Stefan Hüeber, Barbara Wohlmuth Equilibration techniques for solving contact problems with Coulomb friction Comput. Meth. Appl. Mech. Eng., Vol. 205-208, (2012), pp. 29-45 2012 Article in Journal Abstract
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Alexander Weiss, Alexander Weiss, Barbara Wohlmuth, Barbara Wohlmuth A posteriori error estimator and error control for contact problems Math. Comp. 78, 1237-1267 2009 Article in Journal Abstract
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Barbara Wohlmuth An a Posteriori Error Estimator for Two-Body Contact Problems on Non-Matching Meshes J. Sci. Comp. 33, 25-45 2007 Article in Journal Abstract
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Stefan Hüeber, Barbara Wohlmuth An optimal a priori error estimate for nonlinear multibody contact problems SIAM J. Numer. Anal. 43, 157-173 2005 Article in Journal Abstract
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