Least-squares stabilized augmented Lagrangian multiplier method for elastic contact.pdfVIP

Finite Elements in Analysis and Design 116 (2016) 32–37 Contents lists available at ScienceDirect Finite Elements in Analysis and Design journal homepage: /locate/?nel Least-squares stabilized augmented Lagrangian multiplier method for elastic contact Peter Hansbo, Asim Rashid n, Kent Salomonsson Department of Mechanical Engineering, J?nk?ping University, SE-55111 J?nk?ping, Sweden article info Article history: Received 11 November 2015 Received in revised form 5 March 2016 Accepted 27 March 2016 Available online 11 April 2016 Keywords: Lagrange multiplier Stabilization Contact Augmented Lagrangian abstract In this paper, we propose a stabilized augmented Lagrange multiplier method for the ?nite element solution of small deformation elastic contact problems. We limit ourselves to friction-free contact with a rigid obstacle, but the formulation is readily extendable to more complex situations. 2016 Elsevier B.V. All rights reserved. 1. Introduction This paper is motivated by the recent work by Chouly and Hild [9] on Nitsches method for contact problems. In their work, the contact conditions were incorporated into the bilinear form to transform the variational inequality describing the contact problem to a nonlinear variational equality. We here point out that the same can be done for a more standard stabilized Lagrange multiplier method if we augment the Lagrangian using the standard approach of, e.g., Alart and Curnier [1]. Using a multiplier method has an advantage compared to the Nitsche method in that there is an increased freedom in choosing the multiplier space. For example, using a continuous multiplier with nodes coinciding with the displacement nodes on the surface we can use nodal quadrature schemes to emulate point Lagrange multipliers (at least for low order elements). For such schemes, contact will be checked at the nodes as is usually done in engineering practice. This is not possible following the Nitsche approach. A basic issue when using Lagrange mu