ORIGINAL PAPER K. A. Fischer Æ P. Wriggers Frictionless 2D Contact formulations for finite deformations based on the mortar method Received: 4 August 2004 / Accepted: 31 January 2005 / Published online: 14 April 2005 Ó Springer-Verlag 2005 Abstract In this paper two different finite element for- mulations for frictionless large deformation contact problems with non-matching meshes are presented. Both are based on the mortar method. The first formulation introduces the contact constraints via Lagrange multi- pliers, the other employs the penalty method. Both formulations differ in size and the way of fulfilling the contact constraints, thus different strategies to determine the permanently changing contact area are required. Starting from the contact potential energy, the varia- tional formulation, the linearization and finally the matrix formulation of both methods are derived. In combination with different contact detection methods the global solution algorithm is applied to different two- dimensional examples. Keywords Contact mechanics Æ Mortar method Æ Large deformations Æ Finite element discretization Æ Lagrange multiplier Æ Penalty method 1 Introduction Several different methods for the enforcement of the contact constraints within the finite element method are discussed in the literature. The contact constraints can be included to the variational problem for example with the Lagrange multiplier method, the penalty method or the augmented Lagrange multiplier method, see e.g. Wriggers and Simo (1985), Oden (1981), Pietrzak (1997). For small changes in the contact geometry the contact constraints can be incorporated by node-to-node contact elements, see Stadter and Weiss (1979) or by using contact segments Semo et al. (1985). For the more general case, which allows large tangential sliding and deformations, the contact surface is mostly discretized with node-to-segment elements. Early implementations can be found in Hallquist (1979). This method fulfills the contact constraints strongly at each node. An overview with respect to different approaches to finite element contact descriptions can be found in Wriggers (2002a) and Laursen (2002). Recently, new discretizations schemes based on the domain decomposition technique for non-matching grids are applied to contact mechanics where the contact constraints are treated continuously. One of these methods is the mortar method where two different dis- cretized surfaces are connected using Lagrange multi- pliers. These multipliers are approximated by shape functions which have to match the displacement approximation in order to arrive at a stable discretiza- tion scheme. For introduction to this method, the mathematical literature, see Belgacem et al. (1999), who present examples for unilateral contact problems. Fur- thermore, the mortar method is mathematically ana- lyzed in Bernardi et al. (2001) and in Wohlmuth (2000). Another weak contact discretization technique is based on the Nitsche method Nitsche (1970). It is purely dis- placement-based and involves the material laws of the neighboring continuum elements, see Becker and Hansbo (1999), Wriggers (2002a). For unilateral small deformation contact problems McDevitt and Laursen (2000) introduce the mortar method with an intermediate contact surface to define the Lagrange multipliers on this surface. Another ap- proach is based on the assumption that one discretized surface, the non-mortar side, is the contact surface, see e.g. Krause (2001). All these techniques are applied within the geomet- rically linear theory. The extension of the mortar method to large deformations is still in development. For arbitrarily curved surfaces but fixed contact areas Puso (2004) derives a 3D mortar approach. Puso and Laursen (2004) extends this to large sliding problems using a non-symmetric implementation with linear shape Comput Mech (2005) 36: 226–244 DOI 10.1007/s00466-005-0660-y K. A. Fischer (&) Æ P. Wriggers Institut fu¨r Baumechanik und Numerische Mechanik, Universita¨t Hannover, E-mail: fischer@ibnm.uni-hannover.de