Yves Renard

Hybrid finite element methods for the Signorini problem

We study three mixed linear finite element methods for the numerical simulation of the two-dimensional Signorini problem. Applying Falks Lemma and saddle point theory to the resulting discrete mixed variational inequality allows us to state the convergence rate of each of them. Two of these finite elements provide optimal results under reasonable regularity assumptions on the Signorini solution, and the numerical investigation shows that the third method also provides optimal accuracy.

A New Fictitious Domain Approach Inspired by the Extended Finite Element Method

The purpose of this paper is to present a new fictitious domain approach inspired by the extended finite element method introduced by Moes, Dolbow, and Belytschko in [Internat. J. Numer. Methods Engrg., 46 (1999), pp. 131-150]. An optimal method is obtained thanks to an additional stabilization technique. Some a priori estimates are established and numerical experiments illustrate different aspects of the method. The presentation is made on a simple Poisson problem with mixed Neumann and Dirichlet boundary conditions. The extension to other problems or boundary conditions is quite straightforward.

A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics

In this work we consider a stabilized Lagrange (or Kuhn–Tucker) multiplier method in order to approximate the unilateral contact model in linear elastostatics. The particularity of the method is that no discrete inf-sup condition is needed in the convergence analysis. We propose three approximations of the contact conditions well adapted to this method and we study the convergence of the discrete solutions. Several numerical examples in two and three space dimensions illustrate the theoretical results and show the capabilities of the method.

On the discretization of contact problems in elastodynamics

In this work, we will presente a comparison of two formulation for the discretization of elastodynamic contact problems. The first approach consists on a midpoint scheme and a contact condition expressed in terms of velocity. This approach gives an energy conserving scheme. The second one we propose is a new distribution of the solid mass. The problem expressed with the new mass matrix is well posed, energy conserving and has a lipschitz solution. Finally, some numerical results are presented.

A uniqueness criterion for the Signorini problem with Coulomb friction

The purpose of this paper is to study the solutions to the Signorini problem with Coulomb friction (the so-called Coulomb problem). Some optimal a priori estimates are given, and a uniqueness criterion is exhibited. Recently, nonuniqueness examples have been presented in the continuous framework. It is proved, here, that if a solution satisfies a certain hypothesis on the tangential displacement and if the friction coefficient is small enough, it is the unique solution to the problem. In particular, this result can be useful for the search of multisolutions to the Coulomb problem because it eliminates a lot of uniqueness situations.

Symmetric and non-symmetric variants of Nitsche's method for contact problems in elasticity: theory and numerical experiments

A general Nitsche method, which encompasses symmetric and non-symmetric variants, is proposed for frictionless unilateral contact problems in elasticity. The optimal convergence of the method is established both for two and three-dimensional problems and Lagrange affine and quadratic finite element methods. Two and three-dimensional numerical experiments illustrate the theory.

The singular dynamic method for constrained second order hyperbolic equations: Application to dynamic contact problems

The purpose of this paper is to present a new family of numerical methods for the approximation of second order hyperbolic partial differential equations submitted to a convex constraint on the solution. The main application is dynamic contact problems. The principle consists in the use of a singular mass matrix obtained by the mean of different discretizations of the solution and of its time derivative. We prove that the semi-discretized problem is well-posed and energy conserving. Numerical experiments show that this is a crucial property to build stable numerical schemes.

Spider XFEM: an extended finite element variant for partially unknown crack-tip displacement

In this paper, we introduce a new variant of the extended finite element method (Xfem) allowing an optimal convergence rate when the asymptotic displacement is partially unknown at the crack tip. This variant consists in the addition of an adapted discretization of the asymptotic displacement. We give a mathematical result of quasi-optimal a priori error estimate which allows to analyze the potentialities of the method. Some computational tests are provided and a comparison is made with the classical Xfem.

An Error Estimate for the Signorini Problem with Coulomb Friction Approximated by Finite Elements

The 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.

An Improved a Priori Error Analysis for Finite Element Approximations of Signorini's Problem

The present paper is concerned with the unilateral contact model in linear elastostatics, the so-called Signorini problem. (Our results can also be applied to the scalar Signorini problem.) A standard continuous linear finite element approximation is first chosen to approach the two-dimensional problem. We develop a new error analysis in the