Comput Mech (2012) 49:1–20 DOI 10.1007/s00466-011-0623-4 ORIGINAL PAPER A mortar formulation for 3D large deformation contact using NURBS-based isogeometric analysis and the augmented Lagrangian method L. De Lorenzis · P. Wriggers · G. Zavarise Received: 3 June 2011 / Accepted: 23 June 2011 / Published online: 14 July 2011 © Springer-Verlag 2011 Abstract NURBS-based isogeometric analysis is applied to 3D frictionless large deformation contact problems. The contact constraints are treated with a mortar-based approach combined with a simplified integration method avoiding seg- mentation of the contact surfaces, and the discretization of the continuum is performed with arbitrary order NURBS and Lagrange polynomial elements. The contact constraints are satisfied exactly with the augmented Lagrangian formula- tion proposed by Alart and Curnier, whereby a Newton-like solution scheme is applied to solve the saddle point prob- lem simultaneously for displacements and Lagrange mul- tipliers. The numerical examples show that the proposed contact formulation in conjunction with the NURBS dis- cretization delivers accurate and robust predictions. In both small and large deformation cases, the quality of the con- tact pressures is shown to improve significantly over that achieved with Lagrange discretizations. In large deformation and large sliding examples, the NURBS discretization pro- vides an improved smoothness of the traction history curves. In both cases, increasing the order of the discretization is either beneficial or not influential when using isogeometric analysis, whereas it affects results negatively for Lagrange discretizations. Keywords Contact · Isogeometric analysis · Large deformation · Mortar method · NURBS L. De Lorenzis (B ) · G. Zavarise Department of Innovation Engineering, University of Salento, Lecce, Italy e-mail: laura.delorenzis@unisalento.it P. Wriggers Institute for Continuum Mechanics, Leibniz Universität Hannover, Hanover, Germany 1 Introduction The numerical solution of large deformation, large slip multi- body contact problems with the finite element method (FEM) presents several difficulties, including high non-linearity and non-smoothness, potential ill-conditioning, and heavy com- putational costs associated with contact detection. Although several improvements have been achieved in the past few years, contact problems still represent a significant challenge for the analyst and cannot yet be considered solved with the same level of robustness and accuracy of many other prob- lems in non-linear mechanics [10]. The research related to computational contact mechan- ics has taken several directions. One of these has been the development of smoothing techniques, aimed at reducing the drawbacks associated with the non-smooth discretiza- tion of the master surface. Several techniques are avail- able in the literature, including Bézier, Hermitian or other spline interpolations, Gregory patches, subdivision surfaces [12, 23, 26, 28, 31, 42], and more recently also NURBS inter- polations [24, 36]. These procedures generally improve the performance of the contact algorithms by enhancing the con- tinuity of the contact master surface, whereas, being typ- ically associated to node-to-surface contact formulations, they leave the geometrical smoothness of the slave sur- face unaltered. Due to the interaction of the bulk and surface discretizations in determining the smoothness of the traction history curves for large deformation and large sliding problems, the observed improvement in the qual- ity of the contact response is limited by the fact that the higher-order approximation does not involve the bulk behavior of the solid. Moreover, the introduction of a smoothened master surface in addition to the existing finite element mesh yields additional complications in the imple- mentation and data management, and can in some cases 123