We consider mixed and hybrid variational formulations to the linearizedelasticity system in domains with cracks. Inequality type conditions areprescribed at the crack faces which results in unilateral contact problems. Thevariational formulations are extended to the whole domain including the crackswhich yields, for each problem, a smooth domain formulation. Mixedfinite element methods such as PEERS or BDM methods are designed to avoidlocking for nearly incompressible materials in plane elasticity. We study andimplement discretizations based on such mixed finite element methods for thesmooth domain formulations to the unilateral crack problems. We obtainconvergence rates and optimal error estimates and we present some numericalexperiments in agreement with the theoretical results.

The contact between two membranes can be described by a system of variationalinequalities, where the unknowns are the displacements of the membranes and theaction of a membrane on the other one. We first perform the analysis of thissystem. We then propose a discretization, where the displacements areapproximated by standard finite elements and the action by alocal postprocessing. Such a discretization admits an equivalent mixedreformulation. We prove the well-posedness of the discrete problem and establishoptimal a priori error estimates.

Wilkins' method has been successfully used since early 60s for numericalsimulation of high velocity contact elastic-plastic flows. The present workproposes some effective modifications of this method including moresophisticated material model including the Baushinger effect; modification ofthe algorithm based on correction of the initial configuration of a solid; acontact algorithm based on the idea of a mild contact.

This paper is devoted to the introduction of a new variant of the extendedfinite element method (Xfem) for the approximation of elastostatic fractureproblems. This variant consists in a reduced basis strategy for the definitionof the crack tip enrichment. It is particularly adapted when the asymptoticcrack-tip displacement is complex or even unknown. We give a mathematical resultof quasi-optimal a priori error estimate and some computational tests includinga comparison with some other strategies.

We present new a posteriori error estimates for the finite volume approximationsof elliptic problems. They are obtained by applying functional a posteriorierror estimates to natural extensions of the approximate solution and its fluxcomputed by the finite volume method. The estimates give guaranteed upper boundsfor the errors in terms of the primal (energy) norm, dual norm (for fluxes), andalso in terms of the combined primal-dual norms. It is shown that the estimatesprovide sharp upper and lower bounds of the error and their practicalcomputation requires solving only finite-dimensional problems.

The paper deals with an iterative method for numerical solving frictionlesscontact problems for two elastic bodies. Each iterative step consists of aDirichlet problem for the one body, a contact problem for the other one and twoNeumann problems to coordinate contact stresses. Convergence is proved by theBanach fixed point theorem in both continuous and discrete case. Numericalexperiments indicate scalability of the algorithm for some choices of therelaxation parameter.

This work is concerned with the equilibrium configurations of elastic structuresin contact with Coulomb friction. We obtain a variational formulation of thisequilibrium problem. Then we propose sufficient conditions for the existence ofan infinity of equilibrium configurations with arbitrary small frictioncoefficients. We illustrate the result in two space dimensions with asimple example.

In this work we consider a stabilized Lagrange multiplier method in order toapproximate the Coulomb frictional contact model in linear elastostatics. Theparticularity of the method is that no discrete inf-sup condition is needed. Westudy the existence and the uniqueness of solution of the discrete problem.

In this note we give a result of convergence when time goes to infinity for aquasi static linear elastic model, the elastic tensor of which vanishes atinfinity. This method is applied to segmentation of medical images, and improvesthe 'elastic deformable template' model introduced previously.