Mikaël Barboteu

ANALYSIS OF A CONTACT PROBLEM WITH NORMAL COMPLIANCE, FINITE PENETRATION AND NONMONOTONE SLIP DEPENDENT FRICTION

We consider a mathematical model which describes the frictional contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is modeled with normal compliance condition of such a type that the penetration is restricted with unilateral constraint. The friction is modeled with a nonmonotone law in which the friction bound depends both on the tangential displacement and on the value of the penetration. In order to approximate the contact conditions, we consider a regularized problem wherein the contact is modeled by a standard normal compliance condition without finite penetrations. For each problem, we derive a variational formulation and an existence result of the weak solutions of regularized problems is obtained. Next, we prove the convergence of the weak solutions of regularized problems to the weak solution of the initial nonregularized problem. Finally, we provide a numerical validation of this convergence result. To this end we introduce a discrete scheme for the numerical approximation of the frictional contact problems. The solution of the resulting nonsmooth and nonconvex frictional contact problems is found, basing on approximation by a sequence of nonsmooth convex programming problems. Some numerical simulation results are presented in the study of an academic two-dimensional example.

An analytical and numerical approach to a bilateral contact problem with nonmonotone friction

We consider a mathematical model which describes the contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is bilateral, i.e., there is no loss of contact. The friction is modeled with a nonmotonone law. The purpose of this work is to provide an error estimate for the Galerkin method as well as to present and compare two numerical methods for solving the resulting nonsmooth and nonconvex frictional contact problem. The first approach is based on the nonconvex proximal bundle method, whereas the second one deals with the approximation of a nonconvex problem by a sequence of nonsmooth convex programming problems. Some numerical experiments are realized to compare the two numerical approaches.

Solvability of a dynamic contact problem between a piezoelectric body and a conductive foundation

We consider a mathematical model which describes the dynamic process of contact between a piezoelectric body and an electrically conductive foundation. We model the materials behavior with a nonlinear electro-viscoelastic constitutive law; the contact is frictionless and is described with the normal compliance condition and a regularized electrical conductivity condition. We derive a variational formulation for the problem and then, under a smallness assumption on the data, we prove the existence of a unique weak solution to the model. We also investigate the behavior of the solution with respect the electric data on the contact surface and prove a continuous dependence result. Then, we introduce a fully discrete scheme, based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. We treat the contact by using a penalized approach and a version of Newtons method. We implement this scheme in a numerical code and, in order to verify its accuracy, we present numerical simulations in the study of two-dimensional test problems. These simulations provide a numerical validation of our continuous dependence result and illustrate the effects of the conductivity of the foundation, as well.

A frictionless contact problem for viscoelastic materials

We consider a mathematical model which describes the contact between a deformable body and an obstacle, the so-called foundation. The body is assumed to have a viscoelastic behavior that we model with the Kelvin-Voigt constitutive law. The contact is frictionless and is modeled with the well-known Signorini condition in a form with a zero gap function. We present two alternative yet equivalent weak formulations of the problem and establish existence and uniqueness results for both formulations. The proofs are based on a general result on evolution equations with maximal monotone operators. We then study a semi-discrete numerical scheme for the problem, in terms of displacements. The numerical scheme has a unique solution. We show the convergence of the scheme under the basic solution regularity. Under appropriate regularity assumptions on the solution, we also provide optimal order error estimates.

On the behavior of the solution of a viscoplastic contact problem

We consider a mathematical model which describes the frictionless contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic and the contact is modeled with normal compliance and unilateral constraint. We provide a mixed variational formulation of the model which involves a dual Lagrange multiplier, and then we prove its unique weak solvability. We also prove an estimate which allows us to deduce the continuous dependence of the weak solution with respect to both the normal compliance function and the penetration bound. Finally, we provide a numerical validation of this convergence result.

Numerical Analysis of a Hyperbolic Hemivariational Inequality Arising in Dynamic Contact

In this paper a fully dynamic viscoelastic contact problem is studied. The contact is assumed to be bilateral and frictional, where the friction law is described by a nonmonotone relation between the tangential stress and the tangential velocity. A weak formulation of the problem leads to a second order nonmonotone subdifferential inclusion, also known as a second order hyperbolic hemivariational inequality. We study both semidiscrete and fully discrete approximation schemes and bound the errors of the approximate solutions. Under some regularity assumptions imposed on the true solution, optimal order error estimates are derived for the linear element solution. This theoretical result is illustrated numerically.

Analysis of a contact problem with unilateral constraint and slip-dependent friction

We consider a mathematical model which describes the equilibrium of an elastic body in frictional contact with an obstacle. The contact is modelled with normal compliance and unilateral constraint, associated with a slip-dependent version of Coulomb’s law of dry friction. We present a weak formulation of the problem, then we state and prove an existence and uniqueness result of the solution. The proof is based on arguments of elliptic quasivariational inequalities. We also study the finite element approximations of the problem and derive error estimates. Finally, we provide numerical simulations which illustrate both the behaviour of the solution related to the frictional contact conditions and the convergence order of the error estimates.

Numerical Analysis of Elliptic Hemivariational Inequalities

This paper is devoted to a study of the numerical solution of elliptic hemivariational inequalities with or without convex constraints by the finite element method. For a general family of elliptic...

Analysis of two active set type methods to solve unilateral contact problems

In this work two active set type methods are considered in order to solve a mathematical problem which describes the frictionless contact between a deformable body and a perfectly rigid obstacle, the so-called Signorini Problem. These methods are the primal dual active set method and the projection iterative method. Our aim, here, is to analyze these two active set type methods and to carry out a comparison with the well-known augmented Lagrangian method by considering two representative contact problems in the case of large and small deformation. After presenting the mechanical formulation in the hyperelasticity framework, we establish weak formulations of the problem and the existence result of the weak solution is recalled. Then, we give the finite element approximation of the problem and a description of the numerical methods is presented. The main result of this work is to provide a convergence result for the projection iterative method. Finally, we present numerical simulations which illustrate the behavior of the solution and allow the comparison of the numerical methods.

A Contact Problem with Normal Compliance, Finite Penetration and Nonmonotone Slip Dependent Friction

In this work, we consider a static frictional contact problem between a linearly elastic body and an obstacle, the so-called foundation. This contact is described by a normal compliance condition of such a type that the penetration is restricted with unilateral constraint. The friction is modeled with a nonmonotone law. In order to approximate the contact conditions, we consider a regularized problem wherein the contact is modeled by a standard normal compliance condition without finite penetration. Next, we present a convergence result between the solution of the regularized problem and the original problem. Finally, we provide a numerical validation of this convergence result. To this end we introduce a discrete scheme for the numerical approximation of the frictional contact problems.