Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM

© EDP Sciences, SMAI 2011This paper is concerned with the dual formulation of the interface problem consisting of a linear partial differential equation with variable coefficients in some bounded Lipschitz domain Ω in Rn (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc := Rn\￣Ω. The two problems are coupled by transmission and Signorini contact conditions on the interface Γ = ∂Ω. The exterior part of the interface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequality with a linear operator. Then we treat the corresponding numerical scheme and discuss an approximation of the NtD mapping with an appropriate discretization of the inverse Poincar´e-Steklov operator. In particular, assuming some abstract approximation properties and a discrete inf-sup condition, we show unique solvability of the discrete scheme and obtain the corresponding a-priori error estimate. Next, we prove that these assumptions are satisfied with Raviart- Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory

Hybrid discretization of the Signorini problem with Coulomb friction. Theoretical aspects and comparison of some numerical solvers

International audienceThe purpose of this work is to present in a general framework the hybrid discretization of unilateral contact and friction conditions in elastostatics. A projection formulation is developed and used. An existence and uniqueness results for the solutions to the discretized problem is given in the general framework. Several numerical methods to solve the discretized problem are presented (Newton, SOR, fixed points, Uzawa) and compared in terms of the number of iterations and the robustness with respect to the value of the friction coefficient

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

Space-time adaptive finite elements for nonlocal parabolic variational inequalities

This article considers the error analysis of finite element discretizations and adaptive mesh refinement procedures for nonlocal dynamic contact and friction, both in the domain and on the boundary. For a large class of parabolic variational inequalities associated to the fractional Laplacian we obtain a priori and a posteriori error estimates and study the resulting space-time adaptive mesh-refinement procedures. Particular emphasis is placed on mixed formulations, which include the contact forces as a Lagrange multiplier. Corresponding results are presented for elliptic problems. Our numerical experiments for $2$-dimensional model problems confirm the theoretical results: They indicate the efficiency of the a posteriori error estimates and illustrate the convergence properties of space-time adaptive, as well as uniform and graded discretizations.Comment: 47 pages, 20 figure

Error estimates for Stokes problem with Tresca friction condition

In this work we propose and study a three field mixed formulation for solving the Stokes problem with Tresca-type non-linear boundary conditions. Two Lagrange multipliers are used to enforce div(u)=0 constraint and to regularize the energy functional. The resulting problem is discretised using "P1 bubble/P1-P1" finite elements. Error estimates are derived and several numerical studies are achieved

A priori error for unilateral contact problems with Lagrange multiplier and IsoGeometric Analysis

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 two- and three-dimensional and in small and large deformation demonstrate the accuracy of the proposed method

Fixed point strategies for elastostatic frictional contact problems

International audienceSeveral fixed point strategies and Uzawa algorithms (for classical and augmented Lagrangian formulations) are presented to solve the unilateral contact problem with Coulomb friction. These methods are analyzed, without introducing any regularization, and a theoretical comparison is performed. Thanks to a formalism coming from convex analysis, some new fixed point strategies are presented and compared to known methods. The analysis is first performed on continuous Tresca problem and then on the finite dimensional Coulomb problem derived from an arbitrary finite element method