Une estimation d'erreur quasi-optimale pour l'approximation par éléments finis du problème de Signorini bidimensionnel

International audienceThe aim of this Note is to present a quasi-optimal a priori error estimate for the linear finite element approximation of the so-called two-dimensional Signorini problem, i.e. the equilibrium of a plane linearly elastic body in contact with a rigid foundation. Previous works on that subject give either non-optimal estimates or with a more restrictive supplementary condition on the solution.On présente dans cette Note une estimation optimale de l'erreur d'approximation par éléments finis affines du problème de Signorini, c'est à dire du problème de l'équilibre d'un corps élastique en contact avec une fondation rigide. Les travaux précédents sur ce sujet donnent soit des résultats non optimaux, soit avec des conditions supplémentaires plus contraignantes sur la solution

On convergence of the penalty method for unilateral contact problems

We present a convergence analysis of the penalty method applied to unilateral contact problems in two and three space dimensions. We first consider, under various regularity assumptions on the exact solution to the unilateral contact problem, the convergence of the continuous penalty solution as the penalty parameter $\varepsilon$ vanishes. Then, the analysis of the finite element discretized penalty method is carried out. Denoting by $h$ the discretization parameter, we show that the error terms we consider give the same estimates as in the case of the constrained problem when the penalty parameter is such that $\varepsilon = h$

An error estimate for the Signorini problem with Coulomb friction approximated by finite elements

International audienceThe present paper is concerned with the unilateral contact model and the Coulomb friction law in linear elastostatics. We consider a mixed formulation in which the unknowns are the displacement field and the normal and tangential constraints on the contact area. The chosen finite element method involves continuous elements of degree one and continuous piecewise affine multipliers on the contact zone. A convenient discrete contact and friction condition is introduced in order to perform a convergence study. We finally obtain a first a priori error estimate under the assumptions ensuring the uniqueness of the solution to the continuous problem

A three-field augmented Lagrangian formulation of unilateral contact problems with cohesive forces

International audienceWe investigate unilateral contact problems with cohesive forces, leading to the constrained minimization of a possibly nonconvex functional. We analyze the mathematical structure of the mini- mization problem. The problem is reformulated in terms of a three-field augmented Lagrangian, and sufficient conditions for the existence of a local saddle-point are derived. Then, we derive and analyze mixed finite element approximations to the stationarity conditions of the three-field augmented La- grangian. The finite element spaces for the bulk displacement and the Lagrange multiplier must satisfy a discrete inf-sup condition, while discontinuous finite element spaces spanned by nodal basis functions are considered for the unilateral contact variable so as to use collocation methods. Two iterative algo- rithms are presented and analyzed, namely an Uzawa-type method within a decomposition-coordination approach and a nonsmooth Newton's method. Finally, numerical results illustrating the theoretical analysis are presented

An efficient and reliable residual-type a posteriori error estimator for the Signorini problem

We derive a new a posteriori error estimator for the Signorini problem. It generalizes the standard residual-type estimators for unconstrained problems in linear elasticity by additional terms at the contact boundary addressing the non-linearity. Remarkably these additional contact-related terms vanish in the case of so-called full-contact. We prove reliability and efficiency for two- and three-dimensional simplicial meshes. Moreover, we address the case of non-discrete gap functions. Numerical tests for different obstacles and starting grids illustrate the good performance of the a posteriori error estimator in the two- and three-dimensional case, for simplicial as well as for unstructured mixed meshes