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An Overview of Recent Results on Nitsche’s Method for Contact Problems

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Geometrically Unfitted Finite Element Methods and Applications

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 121))

Abstract

We summarize recent achievements in applying Nitsche’s method to some contact and friction problems. We recall the setting of Nitsche’s method in the case of unilateral contact with Tresca friction in linear elasticity. Main results of the numerical analysis are detailed: consistency, well-posedness, fully optimal convergence in H 1(Ω)-norm, residual-based a posteriori error estimation. Some numerics and some recent extensions to multi-body contact, contact in large transformations and contact in elastodynamics are presented as well.

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Acknowledgements

The authors thank Erik Burman, the editorial board and Springer for the invitation to write this paper for their special volume. Moreover they thank the two anonymous referees for their constructive comments that helped to improve the paper. They thank also Thomas Boiveau and Susanne Claus for the GUFEM meeting and discussions. The first author thanks Région Bourgogne Franche-Comté for partial funding (“Convention Région 2015C-4991. Modèles mathématiques et méthodes numériques pour l’élasticité non-linéaire”), as well as Erik Burman and Miguel A. Fernández for some inspiring discussions on Nitsche’s method.

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Authors and Affiliations

  1. Laboratoire de Mathématiques de Besançon - UMR CNRS 6623, Université Bourgogne Franche–Comté, 16 route de Gray, 25030 Besançon Cedex, France

    Franz Chouly

  2. EPFL SB MATHICSE (Bt. MA), Station 8, CH 1015, Lausanne, Switzerland

    Mathieu Fabre

  3. Istituto di Matematica Applicata e Tecnologie, Informatiche “E. Magenes” del CNR, via Ferrata 1, 27100, Pavia, Italy

    Mathieu Fabre

  4. Institut de Mathématiques de Toulouse - UMR CNRS 5219, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex 9, France

    Patrick Hild

  5. CNRS, INSA-Lyon, ICJ UMR5208, Université de Lyon, F-69621 Villeurbanne, France

    Rabii Mlika, Jérôme Pousin & Yves Renard

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  1. Franz Chouly
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  2. Mathieu Fabre
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Correspondence to Franz Chouly .

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Editors and Affiliations

  1. Faculté des Sciences, de la Technologie et de la Communication, Université du Luxembourg, Luxembourg

    Stéphane P. A. Bordas

  2. Department of Mathematics, University College London, United Kingdom

    Erik Burman

  3. Department of Mathematics and Mathematical Statistics, Umeå University, Sweden

    Mats G. Larson

  4. Department of Mathematics, University of Houston, USA

    Maxim A. Olshanskii

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Chouly, F., Fabre, M., Hild, P., Mlika, R., Pousin, J., Renard, Y. (2017). An Overview of Recent Results on Nitsche’s Method for Contact Problems. In: Bordas, S., Burman, E., Larson, M., Olshanskii, M. (eds) Geometrically Unfitted Finite Element Methods and Applications. Lecture Notes in Computational Science and Engineering, vol 121. Springer, Cham. https://doi.org/10.1007/978-3-319-71431-8_4

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