A Nitsche finite element method for dynamic contact: 1. Space semi-discretization and time-marching schemes

¹ Laboratoire de Mathématiques de Besançon – UMR CNRS 6623, Université de Franche Comté, 16 route de Gray, 25030 Besançon cedex, France
franz.chouly@univ-fcomte.fr
² Institut de Mathématiques de Toulouse – UMR CNRS 5219, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 9, France
patrick.hild@math.univ-toulouse.fr
³ Université de Lyon, CNRS, INSA-Lyon, ICJ UMR5208, LaMCoS UMR5259, 69621 Villeurbanne, France
yves.renard@insa-lyon.fr

Received: 13 March 2014
Revised: 1 September 2014

Abstract

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.].