Geodesic
A Nitsche finite element method for dynamic contact: 1. Space semi-discretization and time-marching schemes
Chouly, Franz1  ; Hild, Patrick2  ; Renard, Yves3 
1 Laboratoire de Mathématiques de Besançon – UMR CNRS 6623, Université de Franche Comté, 16 route de Gray, 25030 Besançon cedex, France
2 Institut de Mathématiques de Toulouse – UMR CNRS 5219, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 9, France
3 Université de Lyon, CNRS, INSA-Lyon, ICJ UMR5208, LaMCoS UMR5259, 69621 Villeurbanne, France
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 481-502

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This paper presents a new approximation of elastodynamic frictionless contact problems based both on the finite element method and on an adaptation of Nitsche’s method which was initially designed for Dirichlet’s condition. A main interesting characteristic is that this approximation produces well-posed space semi-discretizations contrary to standard finite element discretizations. This paper is then mainly devoted to present an analysis of the space semi-discretization in terms of consistency, well-posedness and energy conservation, and also to study the well-posedness of some time-marching schemes (θ-scheme, Newmark and a new hybrid scheme). The stability properties of the schemes and the corresponding numerical experiments can be found in a second paper [F. Chouly, P. Hild and Y. Renard, A Nitsche finite element method for dynamic contact. 2. Stability analysis and numerical experiments. ESAIM: M2AN 49 (2015) 503–528.].

Reçu le :
DOI : 10.1051/m2an/2014041
Classification : 65N12, 65N30, 74M15
Keywords: Unilateral contact, elastodynamics, finite elements, Nitsche’s method, time-marching schemes, stability

Chouly, Franz 1 ; Hild, Patrick 2 ; Renard, Yves 3

1 Laboratoire de Mathématiques de Besançon – UMR CNRS 6623, Université de Franche Comté, 16 route de Gray, 25030 Besançon cedex, France
2 Institut de Mathématiques de Toulouse – UMR CNRS 5219, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 9, France
3 Université de Lyon, CNRS, INSA-Lyon, ICJ UMR5208, LaMCoS UMR5259, 69621 Villeurbanne, France 
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     author = {Chouly, Franz and Hild, Patrick and Renard, Yves},
     title = {A {Nitsche} finite element method for dynamic contact: 1. {Space} semi-discretization and time-marching schemes},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {481--502},
     publisher = {EDP-Sciences},
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     zbl = {1311.74113},
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     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2014041/}
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Chouly, Franz; Hild, Patrick; Renard, Yves. A Nitsche finite element method for dynamic contact: 1. Space semi-discretization and time-marching schemes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 481-502. doi: 10.1051/m2an/2014041

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