Cette thèse a pour but de développer un ensemble de méthodes permettant de gérer les problèmes de contact et de couplage de maillages dans le cadre de la méthode des éléments finis classiques et étendus. Ces problèmes d'interfaces sont traités le long de surfaces réelles et virtuelles, dites “surfaces immergées”. Le premier objectif est d’élaborer une formulation de Mortar tridimensionnelle, efficace et parfaitement cohérente en utilisant la méthode du Lagrangien augmenté monolithique (ALM) pour traiter les problèmes de contact et de frottement. Cet objectif est réalisé dans le cadre de la méthode des éléments finis classique. Divers aspects du traitement numérique du contact sont discutés : la détection, la discrétisation, l’évaluation précise des intégrales de Mortar (projections, découpage, triangulation), la parallélisation du traitement sur des architectures parallèles à mémoire distribuée et l’optimisation de la convergence pour les problèmes impliquant à la fois le contact/frottement et les non-linéarités de comportement des matériaux. Grâce aux formulations de Mortar tirées des méthodes de décomposition de domaines, les problèmes de couplage de maillage pour la classe des interfaces non-compatibles sont également présentés.En outre, une nouvelle méthode numérique a été élaborée en 2D : nous la dénommons “MorteX”, car elle rassemble à la fois des fonctionnalités de la méthode Mortar et de la méthode X-FEM (méthode des éléments finis étendus). Dans ce cas, le couplage des maillages entre des domaines qui se chevauchent ainsi que le contact frottant entre des surfaces réelles d'un solide et certaines surfaces immergées au sein du maillage d'un autre corps peuvent être traités efficacement. Cependant, la gestion du couplage/contact entre des géométries non conformes à l'aide de surfaces immergées pose des problèmes de stabilité numérique. Nous avons donc proposé une technique de stabilisation qui consiste à introduire une interpolation des multiplicateurs de Lagrange à grains grossiers. Cette technique a été testée avec succès sur des “patch-tests” classiques et elle s'est également avérée utile pour les méthodes Mortar classiques, ce qui est illustré par plusieurs exemples pratiques.La méthode MorteX est aussi utilisée pour traiter des problèmes d’usure en fretting. Dans ce cas, l’évolution des surfaces de contact qui résulte de l’enlèvement de matière dû à l’usure est modélisée comme une évolution de surface virtuelle qui se propage au sein du maillage existant. L’utilisation de la méthode MorteX élimine donc le besoin de recourir aux techniques complexes de remaillage. Les méthodes proposées sont développées et implémentées dans le logiciel éléments finis Z-set. De nombreux exemples numériques ont été considérés pour valider la mise en œuvre et démontrer la robustesse, la performance et la précision des méthodes Mortar et MorteX. / In this work we develop a set of methods to handle tying and contact problems along real and virtual (embedded) surfaces in the framework of the finite element method. The first objective is to elaborate an efficient and fully consistent three-dimensional mortar formulation using the monolithicaugmented Lagrangian method (ALM) to treat frictional contact problems. Variousaspects of the numerical treatment of contact are discussed: detection, discretization, accurate evaluation of mortar integrals (projections, clipping, triangulation), the parallelization on distributedmemory architectures and optimization of convergence for problems involving both contact and material non-linearities. With mortar methods being drawn from the domain decomposition methods, the mesh tying problems for the class on non-matching interfaces is also presented.A new two-dimensional MorteX framework, which combines features of the extended finite element method (X-FEM) and the classical mortar methods is elaborated. Within this framework, mesh tying between overlapping domains and contact between embedded (virtual) boundaries can be treated. However, in this setting, severe manifestation of mesh locking phenomenon can take place under specific problem settings both for tying and contact. Stabilization techniques such as automatic triangulation of blending elements and coarse-grained Lagrange multiplier spaces are proposed to overcome these adverse effects. In addition, the coarse graining of Lagrange multipliers was proven to be useful for classical mortar methods, which is illustrated with relevant numericalexamples.The MorteX framework is used to treat frictional wear problems. Within this framework the contact surface evolution as a result of material removal due to wear is modeled as an evolving virtual surface. Use of MorteX method circumvents the need for complex remeshing techniques to account for contact surface evolution. The proposed methods are developed and implemented in the in-house finite element suite Z-set. Numerous numerical examples are considered to validate the implementation and demonstrate the robustness, performance and accuracy of the proposed methods.

We use Darcy's law and conservation of mass to model the flow of a fluid through a porous medium. It is a second order elliptic system with a heterogeneous coefficient. We consider the equations written in mixed form. In the heterogeneous case, we define a new multiscale mortar space that incorporates purely local information from homogenization theory to better approximate the solution along the interfaces with just a few degrees of freedom. In the case of a locally periodic heterogeneous coefficient of period epsilon, we prove that the new method achieves both optimal order error estimates in the discretization parameters and good approximation when epsilon is small. Moreover, we present numerical examples to assess its performance when the coefficient is not obviously locally periodic. We show that the new mortar method works well, and better than polynomial mortar spaces. On the other hand, we also propose to use multiscale mortars as a coarse component to construct a two-level preconditioner for the saddle point linear system arising from the fine scale discretization of the mixed finite element system. The two-level preconditioners are constructed based on the interfaces. We propose a framework to define the interpolation operators for the face based two-level preconditioners for different combination of coarse and fine scale mortar spaces for matching and nonmatching grids. In this dissertation, we show that for quasi-homogeneous problems and matching grids, the condition number of the preconditioned interface operator is bounded by (log(H/h))², which is the same as the traditional two-level preconditioners, for quasi-homogeneous problems. We show several numerical examples to demonstrate that for the strongly heterogeneous porous media, it is often desirable and even necessary to use a higher dimensional coarse mortar space to construct the coarse preconditioner to achieve convergence. We apply our ideas to study slightly compressible single phase and two-phase flow in a porous medium. We find that for the nonlinear single phase problem, the two-level preconditioners could be successfully applied to the symmetrized linear system. For the two-phase problem, using the fine scale, instead of multiscale, velocity solutions from the flow problem can greatly benefit the transport problem. / text

Isogeometric analysis is aimed to mitigate the gap between Computer-Aided Design (CAD) and analysis by using a unified geometric representation. Thanks to the exact geometry representation and high smoothness of adopted basis functions, isogeometric analysis demonstrated excellent mathematical properties and successfully addressed a variety of problems. In particular, it allows to solve higher order Partial Differential Equations (PDEs) directly omitting the usage of mixed approaches. Unfortunately, complex CAD geometries are often constituted by multiple Non-Uniform Rational B-Splines (NURBS) patches and cannot be directly applied for finite element analysis.parIn this work, we presents a dual mortaring framework to couple adjacent patches for higher order PDEs. The development of this formulation is initiated over the simplest 4th order problem-biharmonic problem. In order to speed up the construction and preserve the sparsity of the coupled problem, we derive a dual mortar compatible C1 constraint and utilize the Bezier dual basis to discretize the Lagrange multipler spaces. We prove that this approach leads to a well-posed discrete problem and specify requirements to achieve optimal convergence. After identifying the cause of sub-optimality of Bezier dual basis, we develop an enrichment procedure to endow Bezier dual basis with adequate polynomial reproduction ability. The enrichment process is quadrature-free and independent of the mesh size. Hence, there is no need to take care of the conditioning. In addition, the built-in vertex modification yields compatible basis functions for multi-patch coupling.To extend the dual mortar approach to couple Kirchhoff-Love shell, we develop a dual mortar compatible constraint for Kirchhoff-Love shell based on the Rodrigues' rotation formula. This constraint provides a unified formulation for both smooth couplings and kinks. The enriched Bezier dual basis preserves the sparsity of the coupled Kirchhoff-Love shell formulation and yields accurate results for several benchmark problems.Like the dual mortaring formulation, locking problem can also be derived from the mixed formulation. Hence, we explore the potential of Bezier dual basis in alleviating transverse shear locking in Timoshenko beams and volumetric locking in nearly compressible linear elasticity. Interpreting the well-known B projection in two different ways we develop two formulations for locking problems in beams and nearly incompressible elastic solids. One formulation leads to a sparse symmetric symmetric system and the other leads to a sparse non-symmetric system.