Numerische Mathematik

, Volume 139, Issue 3, pp 593–631

# An unbiased Nitsche’s approximation of the frictional contact between two elastic structures

• Franz Chouly
• Rabii Mlika
• Yves Renard
Article

## Abstract

Most of the numerical methods dedicated to the contact problem involving two elastic bodies are based on the master/slave paradigm. It results in important detection difficulties in the case of self-contact and multi-body contact, where it may be impractical, if not impossible, to a priori nominate a master surface and a slave one. In this work we introduce an unbiased finite element method for the finite element approximation of frictional contact between two elastic bodies in the small deformation framework. In the proposed method the two bodies expected to come into contact are treated in the same way (no master and slave surfaces). The key ingredient is a Nitsche-based formulation of contact conditions, as in Chouly et al. (Math Comput 84:1089–1112, 2015). We carry out the numerical analysis of the method, and prove its well-posedness and optimal convergence in the $$H^1$$-norm. Numerical experiments are performed to illustrate the theoretical results and the performance of the method.

## Mathematics Subject Classification

74M10 65N30 74M15

## Notes

### Acknowledgements

We would like to sincerely thank the company “Manufacture Française des Pneumatiques Michelin” for the financial and technical support. We thank, as well, Région Franche-Comté for partial funding (Convention Région 2015C-4991 “Modèles mathématiques et méthodes numériques pour l’élasticité non-linéaire”).

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