SIAM J. SCI. COMPUT. c  2012 Society for Industrial and Applied Mathematics Vol. 34, No. 4, pp. B421–B446 DUAL QUADRATIC MORTAR FINITE ELEMENT METHODS FOR 3D FINITE DEFORMATION CONTACT ∗ A. POPP † , B. I. WOHLMUTH ‡ , M. W. GEE § , AND W. A. WALL † Abstract. Mortar finite element methods allow for a flexible and efficient coupling of arbitrary nonconforming interface meshes and are by now quite well established in nonlinear contact analysis. In this paper, a mortar method for three-dimensional (3D) finite deformation contact is presented. Our formulation is based on so-called dual Lagrange multipliers, which in contrast to the standard mortar approach generate coupling conditions that are much easier to realize, without impinging upon the optimality of the method. Special focus is set on second-order interpolation and on the construction of novel discrete dual Lagrange multiplier spaces for the resulting quadratic interface elements (8-node and 9-node quadrilaterals, 6-node triangles). Feasible dual shape functions are obtained by combining the classical biorthogonality condition with a simple basis transformation procedure. The finite element discretization is embedded into a primal-dual active set algorithm, which efficiently handles all types of nonlinearities in one single iteration scheme and can be inter- preted as a semismooth Newton method. The validity of the proposed method and its efficiency for 3D contact analysis including Coulomb friction are demonstrated with several numerical examples. Key words. mortar finite element methods, dual Lagrange multipliers, contact, finite deforma- tions, second-order interpolation AMS subject classifications. 65N22, 65N30, 74B20, 74M15, 74S05 DOI. 10.1137/110848190 1. Introduction. Computational contact analysis in the regime of finite defor- mations has received much attention in recent years owing to its great relevance in many fields of engineering and the applied sciences. Among the most important chal- lenges that have to be met with respect to finite element discretization is the question of how to treat arbitrarily nonmatching meshes at the interfaces between different bodies. Mortar methods, which were originally introduced as an abstract domain de- composition technique [2, 4], have proved to serve as a very convenient computational framework for contact analysis, especially when considering large deformations and sliding motions. The characteristic feature of mortar methods is the imposition of interface constraints in a weak sense instead of a strong, pointwise enforcement. In the context of contact analysis this allows for a variationally consistent treatment of nonpenetration and frictional sliding constraints using Lagrange multipliers despite the inevitably nonmatching interface meshes. One crucial ingredient of any mortar scheme is the definition of a suitable dis- crete Lagrange multiplier space. In the vast majority of publications, this space is simply based upon the trace space of the underlying finite element discretization, and this choice will be termed standard Lagrange multipliers in the following. Without ∗ Submitted to the journal’s Computational Methods in Science and Engineering section Septem- ber 16, 2011; accepted for publication (in revised form) April 24, 2012; published electronically July 26, 2012. http://www.siam.org/journals/sisc/34-4/84819.html † Institute for Computational Mechanics, Technische Universit¨at M¨ unchen, D-85748 Garching b. M¨ unchen, Germany (popp@lnm.mw.tum.de, wall@lnm.mw.tum.de). The first author’s research was supported by the TUM Graduate School of Technische Universit¨ at M¨ unchen. ‡ Institute for Numerical Mathematics, Technische Universit¨at M¨ unchen, D-85748 Garching b. M¨ unchen, Germany (wohlmuth@ma.tum.de). § Mechanics and High Performance Computing Group, Technische Universit¨at M¨ unchen, D-85748 Garching b. M¨ unchen, Germany (gee@lnm.mw.tum.de). B421