A priori error for unilateral contact problems with Lagrange multipliers and IsoGeometric Analyis

In this paper, we consider unilateral contact problem without friction between a rigid body and deformable one in the framework of isogeometric analysis. We present the theoretical analysis of the mixed problem using an active-set strategy and for a primal space of NURBS of degree p and p − 2 for a dual space of B-Spline. A inf − sup stability is proved to ensure a good property of the method. An optimal a priori error estimate is demonstrated without assumption on the unknown contact set. Several numerical examples in twoand threedimensional and in small and large deformations demonstrate the accuracy of the proposed method. Introduction In the past few years, the study of contact problems in small and large deformations is increased. The numerical resolution of contact problems presents several difficulties as the computational cost, the high nonlinearity and the ill-conditioning. Contrary to many others problems in nonlinear mechanics, these problems can not be solved always at a satisfactory level of robustness and accuracy [22, 32] with the existing numerical methods. One of the reasons that make robustness and accuracy hard to achieve is that the computation of gap, i.e. the distance between the deformed body and the obstacle is indeed an ill-posed problem and its numerical approximation often introduce extra discontinuity that breaks the converge of the iterative schemes; see [1, 22, 32, 21] where a master-slave method is introduced to weaken this effect. To this respect, the use of NURBS or spline approximations within the framework of isogeometric analysis [19], holds great promises thanks to the increased regularity in the geometric description which makes the gap computation intrinsically easier. Isogeometric methods for frictionless contact problems have been introduced in [33, 29, 30, 12, 10, 9], see also with primal and dual elements [31, 18, 17, 26, 28]. Both point-to-segment and segment-to-segment (i.e, mortar type) algorithms have been designed and tested with an engineering prospective, showing that, indeed, the use of smooth geometric representation helps the design of reliable methods for contact problems. ∗EPFL SB MATHICSE MNS (Bât. MA), station 8, CH 1015 Lausanne (Switzerland). †Istituto di Matematica Applicata e Tecnologie Informatiche ’E. Magenes’ del CNR via Ferrata 1, 27100, Pavia (Italy). email: [email protected], [email protected], [email protected].