INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2004; 61:2615–2638 Published online 28 October 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1173 Symmetry preserving algorithm for large displacement frictionless contact by the pre-discretization penalty method D. Gabriel 1 , J. Plešek 1, ∗, † and M. Ulbin 2 1 Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Praha, Czech Republic 2 Faculty of Mechanical Engineering, University of Maribor, Maribor, Slovenia SUMMARY A three-dimensional contact algorithm based on the pre-discretization penalty method is presented. Using the pre-discretization formulation gives rise to contact searching performed at the surface Gaussian integration points. It is shown that the proposed method is consistent with the continuum formulation of the problem and allows an easy incorporation of higher-order elements with midside nodes to the analysis. Moreover, a symmetric treatment of mutually contacting surfaces is preserved even under large displacement increments. The proposed algorithm utilizes the BFGS method modified for constrained non-linear systems. The effectiveness of quadratic isoparametric elements in contact analysis is tested in terms of numerical examples verified by analytical solutions and experimental measurements. The symmetry of the algorithm is clearly manifested in the problem of impact of two elastic cylinders. Copyright  2004 John Wiley & Sons, Ltd. KEY WORDS: contact; Gauss point search; higher-order elements 1. INTRODUCTION In the finite element method the contact constraints can be introduced either before or after the finite element discretization has been performed, which is called, respectively, the pre- discretization or post-discretization formulation [1]. The standard nodal algorithm belongs to the latter group whereas the pre-discretization formulation has little been explored so far. The reason probably is that the idea of discretizing a body by means of structural elements first and only then paying attention to the idealization of boundary conditions is extremely appealing to engineer’s turn of mind, for it naturally fits into the typical stages of a design process. Once the body has been replaced with a set of discrete entities such as elements, edges, nodes, ∗ Correspondence to: J. Plešek, Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Dolejškova 5, 182 00 Praha 8, Czech Republic. † E-mail: plesek@it.cas.cz Contract/grant sponsor: Grant Agency of the Czech Republic; contract/grant numbers: 101/02/0072; 101/03/0331; 101/03/D153; AVOZ 2076919 Received 20 December 2002 Revised 8 May 2004 Copyright  2004 John Wiley & Sons, Ltd. Accepted 10 May 2004