INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2007; 69:524–543 Published online 5 June 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1779 A fast and robust iterative solver for nonlinear contact problems using a primal-dual active set strategy and algebraic multigrid S. Brunssen 1, ∗, † , F. Schmid 2, ‡ , M. Sch¨ afer 2, § and B. Wohlmuth 1, ¶ 1 Institut f¨ ur Angewandte Analysis und Numerische Simulation, Universit¨ at Stuttgart, 70569 Stuttgart, Germany 2 Department of Numerical Methods in Mechanical Engineering, Technische Universit¨ at Darmstadt, 64287 Darmstadt, Germany SUMMARY For extending the usability of implicit FE codes for large-scale forming simulations, the computation time has to be decreased dramatically. In principle this can be achieved by using iterative solvers. In order to facilitate the use of this kind of solvers, one needs a contact algorithm which does not deteriorate the condition number of the system matrix and therefore does not slow down the convergence of iterative solvers like penalty formulations do. Additionally, an algorithm is desirable which does not blow up the size of the system matrix like methods using standard Lagrange multipliers. The work detailed in this paper shows that a contact algorithm based on a primal-dual active set strategy provides these advantages and therefore is highly efficient with respect to computation time in combination with fast iterative solvers, especially algebraic multigrid methods. Copyright 2006 John Wiley & Sons, Ltd. Received 8 November 2005; Revised 3 April 2006; Accepted 17 April 2006 KEY WORDS: dual Lagrange multipliers; algebraic multigrid; contact; penalty method; active set; non- linear material 1. INTRODUCTION The efficient treatment of contact problems is crucial to the performance of FE codes in the context of metal forming. Efficient contact algorithms have been developed in recent years, see References [1–6] and the references therein. For more complex contact problems where the contact area is ∗ Correspondence to: S. Brunssen, Institut f¨ ur Angewandte Analysis und Numerische Simulation, Universit¨ at Stuttgart, 70569 Stuttgart, Germany. † E-mail: brunssen@ians.uni-stuttgart.de ‡ E-mail: schmid@fnb.tu-darmstadt.de § E-mail: schaefer@fnb.tu-darmstadt.de ¶ E-mail: wohlmuth@mathematik.uni.stuttgart.de Contract/grant sponsor: German Research Foundation (DFG); contract/grant numbers: SCHA814/10-1, WO671/4-1 Copyright 2006 John Wiley & Sons, Ltd.