Solution of frictional contact problems by an EBE preconditioner P. Alart, M. Barboteu, F. Lebon Abstract We present an Element by Element (EBE) pro- cedure to solve non symmetric linear systems arising from the solution of contact with friction problems. Hybrid formulation is introduced and different types of contact elements are reviewed. To deal with large scale problems the EBE method has proved to be a strongly parallel al- gorithm. Numerical experiments described in this paper con®rmed the ef®ciency of this speci®c solver. 1 Introduction Frictional contact phenomena are frequent in structural analysis problems. These problems are dif®cult to for- mulate and even more to solve because they are governed by multivalued tribological laws and some numerical res- olutions can lead to unsymmetric operators. The non symmetry becomes crucial for very large problems in- volving three dimensional discretization and time evolu- tion. This paper shows how to use a simple hybrid formu- lation together with an ef®cient Element-By-Element (EBE) preconditioned generalized conjugate gradient al- gorithm when dealing with frictional contact problems. The augmented Lagrangian approach given by Alart and Curnier (1991) generates non linear and non-differ- entiable systems whose unknowns are node displacements and Lagrangian multipliers identi®ed with contact forces. Because the tangent matrix of the system is non sym- metric, non-positive de®nite, ill-conditioned and with zeros on the diagonal, we used generalized conjugate gradient methods introduced by Sonnenveld, Wesseling and de Zeeuw (1985). Appropriate preconditioners are necessary. In this paper, a special EBE procedure for elasticity and another one for frictional contact problems are presented. In Sect. 2, we give a contact ®nite element approach: we review the necessary mathematical background and we in- troduce the mixed formulation for the frictional contact problem. The main contact elements are described in detail. Section 3 is devoted to conjugate gradient squared methods and to a special EBE preconditioner for only elasticity. In Sect. 4, we extend this last EBE method for frictional contact problems, characterized by a simple ``node on rigid plan'' element. Finally, in Sect. 5, classical test problems are pre- sented such as the square test and the dovetail assembly; we describe the numerical features of the EBE preconditioner; its ef®ciency is discussed and a comparison between stan- dard and EBE preconditioners is made. 2 Contact finite element approach 2.1 Contact and friction by hybrid formulation In the following, only a 2D discretized body is considered and the mechanical laws are written for one contact node. Unilateral contact and friction laws are given by multi- valued relationships between static and kinematic vari- ables expressed with respect to a local frame noted t ; n. Their graphs are drawn in the Fig. 1. The normal contact distance of the node from the obstacle is denoted d n ; d t is the tangential slip increment, k n and k t are the normal and tangential components of the contact nodal force. The adopted contact and friction laws derive from pseudo- (i.e. non-differentiable) potentials, i.e. the two following simi- lar inclusions, d n 2 oW R k n d t 2 oW Ck n k t : 1 W D denotes the indicator function of a convex set D and oW D the sub-differential of this function. The set Ck n (the section of the Coulomb's cone) is de®ned as follows with friction coef®cient l, Ck n fk t : kk t klk n g k n 0 : 2 As in a previous paper, based on an augmented Lagrang- ian approach (Alart and Curnier 1991), the equilibrium of a discretized elastic body in frictional contact with an obstacle or of elastic bodies in frictional contact is gov- erned by the following equation: Gu Fu; k 0 3 where Gu is de®ned by: Gu F int u F ext 0 ; 4 F int u and F ext denote the internal and the external forces. Fu; k de®nes the continuous frictional contact operator which may be written for several contact elements as, Computational Mechanics 20 (1997) 370±378 Ó Springer-Verlag 1997 370 Communicated by S. N. Atluri, 15 February 1997 P. Alart, M. Barboteu, F. Lebon Laboratoire de Me Âcanique et Ge Ânie Civil Universite  Montpellier 2 Pl. E. Bataillon, F-34095 Montpellier Cedex 5 France E-mail:fred@lmgc.univ-montp2.fr Correspondence to: F. Lebon