Patrick Hild

Numerical Implementation of Two Nonconforming Finite Element Methods for Unilateral Contact

We consider the finite element approximation of the unilateral contact problem between elastic bodies. We are interested in a practical problem which often occurs in finite element computations concerning two independently discretized bodies in unilateral contact. It follows that the nodes of both bodies located on the contact surface do not fit together. We present two different approaches in order to define unilateral contact on nonmatching meshes. The first is an extension of the mortar finite element method to variational inequalities that defines the contact in a global way. On the contrary, the second one expresses local node-on-segment contact conditions. In both cases, the theoretical approximation properties are given. Then, we implement and compare the two methods.

The mortar finite element method for contact problems

The purpose of this paper is to describe a domain decomposition technique: the mortar finite element method applied to contact problems between two elastic bodies. This approach allows the use of no-matching grids and to glue different discretizations across the contact zone in an optimal way, at least for bilateral contact. We present also an adaptation of this method to unilateral contact problems.

EXTENSION OF THE MORTAR FINITE ELEMENT METHOD TO A VARIATIONAL INEQUALITY MODELING UNILATERAL CONTACT

The purpose of this paper is to extend the mortar finite element method to handle the unilateral contact model between two deformable bodies. The corresponding variational inequality is approximated using finite element meshes which do not fit on the contact zone. The mortar technique allows one to match these independent discretizations of each solid and takes into account the unilateral contact conditions in a convenient way. By using an adaptation of Falks lemma and a bootstrap argument, we give an upper bound of the convergence rate similar to the one already obtained for compatible meshes.

Mixed finite element methods for unilateral problems: convergence analysis and numerical studies

In this paper, we propose and study different mixed variational methods in order to approximate with finite elements the unilateral problems arising in contact mechanics. The discretized unilateral conditions at the candidate contact interface are expressed by using either continuous piecewise linear or piecewise constant Lagrange multipliers in the saddle-point formulation. A priori error estimates are established and several numerical studies corresponding to the different choices of the discretized unilateral conditions are achieved.

Quadratic finite element methods for unilateral contact problems

The present paper is concerned with the frictionless unilateral contact problem between two elastic bodies in a bidimensional context. We consider a mixed formulation in which the unknowns are the displacement field and the contact pressure. We introduce a finite element method using quadratic elements and continuous piecewise quadratic multipliers on the contact zone. The discrete unilateral non-interpenetration condition is either an exact non-interpenetration condition or only a nodal condition. In both cases, we study the convergence of the finite element solutions and a priori error estimates are given. Finally, we perform the numerical comparison of the quadratic approach with linear finite elements.

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.

Approximation du problème de contact unilatéral par la méthode des éléments finis avec joints

Abstract The purpose of this Note is to extend the mortar finite element method to handle the unilateral contact model between two deformable bodies. The corresponding variational inequality is approximated using finite elements with meshes which do not fit on the contact zone. The mortar technique allows us to match (independent) discretizations within each solid and to express the contact conditions in a satisfying way. Then, we carry out a numerical analysis of the algorithm and, using a bootstrap argument, we give an upper bound of the convergence rate similar to that already obtained for compatible grids.

A Nitsche-based method for unilateral contact problems: numerical analysis

We introduce a Nitsche-based formulation for the finite element discretization of the unilateral contact problem in linear elasticity. It features a weak treatment of the non-linear contact conditions through a consistent penalty term. Without any additional assumption on the contact set, we can prove theoretically its fully optimal convergence rate in the H1(Ω)-norm for linear finite elements in two dimensions, which is O(h^(1/2+ν)) when the solution lies in H^(3/2+ν)(Ω), 0 < ν ≤ 1/2. An interest of the formulation is that, conversely to Lagrange multiplier-based methods, no other unknown is introduced and no discrete inf-sup condition needs to be satisfied.

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.

A posteriori error estimation for unilateral contact with matching and non-matching meshes

In this paper, we consider the unilateral contact problem between elastic bodies. We propose an error estimator based on the concept of error in the constitutive relation in order to evaluate the finite element approximation involving matching and non-matching meshes on the contact zone. The determination of the a posteriori error estimate is linked to the building of kinematically-admissible stress fields and statically-admissible stress fields. We then propose a finite element method for approximating the unilateral contact problem taking into account matching and non-matching meshes on the contact zone; then, we describe the construction of admissible fields. Lastly, we present optimized computations by using both the error estimates and a convenient mesh adaptivity procedure. ” 2000 Elsevier Science S.A. All rights reserved.