Patrick Laborde

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.

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.

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.

An existence theorem for slightly compressible materials in nonlinear elasticity

We show that if a hyperelastic material is slightly compressible, in this case the stored energy function is a function of the “modified invariants,” then the existence results of Ball are still valid. We then study the behavior of the solutions when the compressibility tends to zero.

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.

Study of Some Optimal XFEM Type Methods

The XFEM method in fracture mechanics is revisited. A first improvement is considered using an enlarged fixed enrichment subdomain around the crack tip and a bonding condition for the corresponding degrees of freedom. An efficient numerical integration rule is introduced for the nonsmooth enrichment functions. The lack of accuracy due to the transition layer between the enrichment aera and the rest of the domain leads to consider a pointwise matching condition at the boundary of the subdomain. An optimal numerical rate of convergence is then obtained using such a nonconformal method.

On the consistent tangent operator algorithm for thermo-plastic problems

Abstract In order to solve the finite element problem of thermo-elasto-plasticity , we consider an algorithm defined from a (pseudo)-linearization of the implicit time-discretized model. This procedure extends the so-called consistent tangent operator method introduced by Simo and Taylor in standard plasticity. Numerical simulations prove that the algorithm is more efficient than the well-known ‘tangent stiffness matrix’ method in computational plasticity, which is obtained by linearizing the continuous model.

Numerical formulation for solving soil/tool interaction problem involving large deformation

Purpose – The aim of this work is to provide a global 3D finite element (FE) model devoted to the modelling of superficial soil ploughing in the large deformation range and for a vast class of soil treatment tools. Design/methodology/approach – We introduced soil constitutive equation in a FE software initially designed for the metal forming. We performed the numerical integration of the non‐linear ploughing problem. Non‐linearities encountered by the problem can be summed up: as soil constitutive equation (idealized with non‐associated compressible plastic law), unilateral frictional contact conditions (with a rigid body), geometrical non‐linearities (the ploughing tool) and large deformation range. To handle such difficulties we performed several numerical methods as implicit temporal scheme, Newton‐Raphson, non‐symmetric iterative solver, as well as proper approximation on stress and strain measures. Findings – Main results deal with the validation of the integration of the non‐linear constitutive equation in the code and a parametric study of the ploughing process. The influence of tool geometric parameters on the soil deformation modes and on the force experienced on the tools had been point out. As well, the influence of soil characteristics as compressibility had been analyzed. Research limitations/implications – This research is devoted to perform a numerical model applicable for a large range of soil treatment tools and for a large class of soil. However, taking into account all kind of soil is utopist. So limitations met are essentially related to the limit of the accuracy of the elasto‐plastic idealization for the soil. Practical implications – In practice the numerical model exposed in the paper can clearly help to improve and optimize any process involving superficial soil submitted to the mechanical action of a rigid body. Originality/value – The original value of the paper is to provide a global and an applicable numerical model able to take into account the main topics related to the ploughing of superficial soils. Industrials in geotechnics, in agriculture or in military purposes can benefit in using such numerical model.

Analysis of the strain-stress relation in plasticity with non-associated laws

Abstract We study the strain-stress relation in the theory of Plasticity with non-associated laws. The search of a stress response to a path of strain leads to a mathematical problem of differential inclusion with a non-monotone operator. We obtain a solution by a viscoplastic regularization.