Proceedings in Applied Mathematics and Mechanics, 21 May 2008 Preconditioned Uzawa Algorithm for Elastic Contact Problems Nicolae POP ∗1 1 North University of Baia Mare, ROMANIA After finite element discretization of the elastic contact problems with friction, we obtain a big sparse non-symmetric and nonlinear systems of equations, and in many cases ill-conditioned. Solving these systems by direct methods or classical iterative methods are non efficient and with bad convergence properties. One way to overcome these difficulties is to use the preconditioned Uzawa-type algorithms. On this paper we focus on the transformation of the generalized Signorini elastic contact problems into a saddle point problem of some augmented Lagrangian functional and give a preconditioning technique for Uzawa algorithm. Copyright line will be provided by the publisher 1 Introduction This paper presents a preconditioning technique, for Uzawa algorithm, which uses incomplet decomposition of Gramm matrix of the finite element basis functions. We apply solvers to an unilateral contact problems with friction. Modelling with the Coulomb friction law contact problems between an elastic body and a rigid obstacle or between two elastic bodis, leads to variational inequalities of the form: find u ∈ H 1 (0,T ; V ) with a(u(t),v − ˙ u(t)) + j (u(t),v) − j (u(t), ˙ u(t)) ≥〈f (t),v − ˙ u(t)〉 , ∀ v ∈ V, (1) where V is a Banach space, ˙ u denotes the time derivative of the quantity u(t, x) and j (v 1 ,v 2 ) is a nonsmooth functional defined on V × V . This nonsmooth functional can be regularized by means of smooth functionals j ρ (ρ> 0) and considering a nonlocal Coulomb friction laws. 2 Lagrangian and Saddle Point Formulation for Approximated Version of the Contact Problems Using standard finite approximate element procedures and special contact element for contact area, approximate version of problem (1) can be constructed in finite dimensional spaces V h (⊂ V ⊂ V ′ ) resulting one discrete problem (P k ) h . Every discrete problem (P k ) h is a static one, it requires approximate updating of the displacements and loads after each increment. We define the Lagrangian of (P k ) h : L(v,λ 1 ,λ 2 ) = (1/2)v T Kv − F T v + λ T 2 Tv + λ T 1 (Nv − d), where λ 1 ∈ Λ 1 , λ 2 ∈ Λ 2 are the Lagrange multipliers and represent the normal and tangential contact stress on the contact part of the boundary, respectively Λ 1 = {λ 1 ∈ R m |λ 1 ≥ 0}, Λ 2 = {λ 2 ∈ R m |λ 2 ≤ g}. We have K ∈ R n×n positive definite and symmetric stiffness matrix, F ∈ R n is the vector of nodal forces, g ∈ R m contains the nodal slip bounds for the contact nodes, the matrices N, T ∈ R m×n contain the rows of the normal and tangential vectors in the contact nodes, respectively, and d ∈ R m is the vector of distances between the contact nodes and the rigid foundation. The saddle-point formulation, equivalent with (P k ) h is: find (w, λ 1 ,λ 2 ) ∈ V × Λ 1 × Λ 2 s.t. L(w, µ 1 ,µ 2 ) ≤L(w, λ 1 ,λ 2 ) ≤L(v,λ 1 ,λ 2 ), ∀ (v,µ 1 ,µ 2 ) ∈ V × Λ 1 × Λ 2 . (2) After a standard assembly procedure that can be used to add the contact contributions of each contact element to the global tangent stiffness and residual matrix, the saddle point problem of (3) is equivalent with the matriceal form:  K B B T B K C  ΔU Δp  =  R B R C  (3) where Δp = (Δλ 1 , Δλ 2 ), K B , R B are mechanical global tangent stiffness matrices and residual vector; K C , R C are mechanical contributions of all contact nods. Starting with an initial approximation Δp 0 of Δp, Uzawa algorithm constructs a sequence of approximation ΔU k and Δp k as follows: * Corresponding author: e-mail: nicpop@gmail.com, Phone: +40 262 276 059, Fax: +40 262 276 368 Copyright line will be provided by the publisher