Taoufik Sassi

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.

Domain decomposition with nonmatching grids: augmented Lagrangian approach

We propose and study a domain decomposition method which treats the constraint of displacement continuity at the interfaces by augmented La- grangian techniques and solves the resulting problem by a parallel version of the Peaceman-Rachford algorithm. We prove that this algorithm is equivalent to the fictitious overlapping method introduced by P.L. Lions. We also prove its linear convergence independently of the discretization step h , even if the finite element grids do not match at the interfaces. A new preconditioner using fictitious overlapping and well adapted to three-dimensional elasticity problems is also introduced and is validated on several numerical examples.

Méthode d'éléments finis avec hybridisation frontière pour les problèmes de contact avec frottement

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A Neumann-Neumanndomain decomposition algorithm for the Signorini problem

Abstract In this paper, we propose a Neumann-Neumann algorithm to approximate a frictionless static Signorini contact problem between two elastic bodies and we prove its convergence. The Neumann-Neumann algorithm is a parallel one, in which we have to solve a Dirichlet problem and then a Neumann one, simultaneously on each domain. The primary feature of this new algorithm is the retention the natural interface between the two bodies as a numerical interface for the domain decomposition.

Finite Element Analysis of a Contact With Friction Between an Elastic Body and a Thin Soft Layer

When studying a mechanical system involving contact between two bodies such as a disc and brake pad system, finite element simulations are often used to predict the phenomena involved. However, due to model size and calculation time problems, when modeling this type of mechanical system on a scale of about 100 mm, it is difficult to model as well a layer (for example a third body layer) on a scale of approximately 10 pm. In quasi-static problems it is possible to simulate the contact between an elastic body and a thin elastic layer bonded to a rigid surface, by considering the contact between this elastic body and a rigid surface with a specific contact law. This paper shows that it is possible to implement this specific contact law in a dynamic finite element code to simulate thin layers undergoing quasi-static and dynamic problems without physical contact instabilities. This specific contact law saves a large amount of calculation time. Once the specific contact law has been validated, the influence of the layer thickness is studied.

Méthode d'éléments finis hybrides en décomposition de domaines pour des problèmes de contact unilatéral

In this Note, we deal with the domain decomposition method with non-matching grids for solving unilateral contact problem without friction between two deformable bodies. The unilateral contact conditions at the interface are expressed with a mixed formulation. The space of Lagrange multiplier defined on the contact zone is approximated by piecewise constant functions. Optimal error bound of h3/4 is obtained.