In my dissertation, I developed mixed hp-finite element methods for linear elasticity with weakly imposed symmetry, which is based on Arnold-Falk-Winther's stable mixed finite elements. I have proved the h-stability of my method for meshes with arbitrary variable orders. In order to show the h-stability, I need an upper limit of the highest order of meshes, which can be an arbitrary nonnegative integer. / text

Swelling of a polymer gel is a kinetic process coupling mass transport and mechanical deformation. A comparison between a nonlinear theory for polymer gels and the classical theory of linear poroelasticity is presented. It is shown that the two theories are consistent within the linear regime under the condition of a small perturbation from an isotropically swollen state of the gel. The relationships between the material properties in the linear theory and those in the nonlinear theory are established by a linearization procedure. Both linear and nonlinear solutions are presented for swelling kinetics of substrate-constrained and freestanding hydrogel layers. A new procedure is suggested to fit the experimental data with the nonlinear theory. A nonlinear, transient finite element formulation is presented for initial boundary value problems associated with swelling and deformation of hydrogels, based on nonlinear continuum theories for hydrogels with compressible and incompressible constituents. The incompressible instantaneous response of the aggregate imposes a constraint to the finite element discretization in order to satisfy the LBB condition for numerical stability of the mixed method. Three problems of practical interests are considered: constrained swelling, flat-punch indentation, and fracture of hydrogels. Constrained swelling may lead to instantaneous surface instability. Indentation relaxation of hydrogels is simulated beyond the linear regime under plane strain conditions, and is compared with two elastic limits for the instantaneous and equilibrium states. The effects of Poisson’s ratio and loading rate are discussed. On the study of hydrogel fracture, a method for calculating the transient energy release rate for crack growth in hydrogels, based on a modified path-independent J-integral, is presented. The transient energy release rate takes into account the energy dissipation due to diffusion. Numerical simulations are performed for a stationary center crack loaded in mode I, with both immersed and non-immersed chemical boundary conditions. Both sharp crack and blunted notch crack models are analyzed over a wide range of applied remote tensile strains. Comparisons to linear elastic fracture mechanics are presented. A critical condition is proposed for crack growth in hydrogels based on the transient energy release rate. The applicability of this growth condition for simulating concomitant crack propagation and solvent diffusion in hydrogels is discussed. / text

In this dissertation, we study the mixed finite element method for the linear plane elasticity problem and iterative solvers for the resulting discrete system. We use the Arnold-Winther Element in the mixed finite element discretization. An overlapping Schwarz preconditioner and a multigrid preconditioner for the discrete system are developed and analyzed. We start by introducing the mixed formulation (stress-displacement formulation) for the linear plane elasticity problem and its discretization. A detailed analysis of the Arnold-Winther Element is given. The finite element discretization of the mixed formulation leads to a symmetric indefinite linear system. Next, we study efficient iterative solvers for the symmetric indefinite linear system which arises from the mixed finite element discretization of the linear plane elasticity problem. The preconditioned Minimum Residual Method is considered. It is shown that the problem of constructing a preconditioner for the indefinite linear system can be reduced to the problem of constructing a preconditioner for the H(div) problem in the Arnold-Winther finite element space. Our main work involves developing an overlapping Schwarz preconditioner and a multigrid preconditioner for the H(div) problem. We give condition number estimates for the preconditioned systems together with supporting numerical results.

The miscible displacement equations provide the mathematical model for simulating the displacement of a mixture of oil and miscible fluid in underground reservoirs during the Enhance Oil Recovery(EOR) process. In this thesis, I propose a stable numerical scheme combining a mixed finite element method and space-time discontinuous Galerkin method for solving miscible displacement equations under low regularity assumption. Convergence of the discrete solution is investigated using a compactness theorem for functions that are discontinuous in space and time. Numerical experiments illustrate that the rate of convergence is improved by using a high order time stepping method. For petroleum engineers, it is essential to compute finely detailed fluid profiles in order to design efficient recovery procedure thereby increase production in the EOR process. The method I propose takes advantage of both high order time approximation and discontinuous Galerkin method in space and is capable of providing accurate numerical solutions to assist in increasing the production rate of the miscible displacement oil recovery process.

In this dissertation, we study the mixed finite element method for the linear plane elasticity problem and iterative solvers for the resulting discrete system. We use the Arnold-Winther Element in the mixed finite element discretization. An overlapping Schwarz preconditioner and a multigrid preconditioner for the discrete system are developed and analyzed. We start by introducing the mixed formulation (stress-displacement formulation) for the linear plane elasticity problem and its discretization. A detailed analysis of the Arnold-Winther Element is given. The finite element discretization of the mixed formulation leads to a symmetric indefinite linear system. Next, we study efficient iterative solvers for the symmetric indefinite linear system which arises from the mixed finite element discretization of the linear plane elasticity problem. The preconditioned Minimum Residual Method is considered. It is shown that the problem of constructing a preconditioner for the indefinite linear system can be reduced to the problem of constructing a preconditioner for the H(div) problem in the Arnold-Winther finite element space. Our main work involves developing an overlapping Schwarz preconditioner and a multigrid preconditioner for the H(div) problem. We give condition number estimates for the preconditioned systems together with supporting numerical results.

The main focus of this work lies in the simulation of the deformation of mechanical components which consist of nonlinear elastic, incompressible material and that are subject to large deformations. Starting from a nonlinear formulation one can derive a discrete problem by using linearisation techniques and an adaptive mixed finite element method. This turns out to be a saddle point problem that can be solved via a Bramble-Pasciak conjugate gradient method. With some modifications the simulation can be improved.

Les simulations en mécanique du solide présentent des difficultés comme le traitement de l'incompressibilité ou les non-linéarités dues aux grandes déformations, aux lois de comportement et de contact. L'objectif principal de ce travail est de proposer des méthodes éléments finis capables de gérer l'incompressibilité en grandes déformations en utilisant des éléments de faible ordre. Parmi les approches de la littérature, les formulations mixtes offrent un cadre théorique intéressant. Dans ce travail, une formulation mixte à trois champs (déplacements, pression, gonflement) est introduite. Dans certains cas, cette formulation peut être condensée en formulation à deux champs. Cependant, il est connu que le problème discret obtenu par une approche éléments finis de type Galerkin n'hérite pas automatiquement de la condition de stabilité “inf-sup” du problème continu : les éléments finis utilisés, et notamment les ordres d'interpolation doivent être choisis de sorte à vérifier cette condition de stabilité. Cependant, il est possible de s'affranchir de cette contrainte en ajoutant des termes de stabilisation à la formulation EF Galerkin. Cette approche permet entre autres d'utiliser des ordres d'interpolation égaux. Dans ce travail, des éléments finis stables de type P2/P1 sont utilisés comme référence, et comparés à une formulation P1/P1, stabilisée soit avec une fonction bulle, soit avec une méthode VMS (Variational Multi-Scale) basée sur un espace sous-grille orthogonal à l'espace EF. Combinées à un modèle grandes déformations basé sur des déformations logarithmiques, ces approches sont d'abord validées sur des cas académiques puis sur des cas industriels. / Simulations in solid mechanics exhibit difficulties as dealing with incompressibility or nonlinearities due to finite strains, constitutive laws and contact. The basic motivation of our work is to propose efficient finite element methods capable of dealing with incompressibility in finite strain context, and using low order elements. Among the approaches in the literature, mixed formulations offer an interesting theoretical framework. In this work, a three-field mixed formulation (displacement, pressure, volumetric strain) is investigated. In some cases, this formulation can be condensed in a two-field formulation. However, it is well-known that the discrete problem given by the Galerkin finite element technique, does not inherit the “inf-sup” stability condition from the continuous problem: the finite elements used, and in particular the interpolation orders must be chosen so as to satisfy this stability condition. However, it is possible to circumvent it, by adding terms stabilizing the FE Galerkin formulation. The latter approach allows the use of equal order interpolation. In this work, stable finite elements of type P2/P1 are used as reference, and compared to a P1/P1 formulation, stabilized with a bubble function, or with a VMS method (Variational Multi-Scale) based on a sub-grid-space orthogonal to the FE space. Combined to a finite strain model based on logarithmic strain, these approaches are first validated on academic cases and then on industrial cases.

The main focus of this work lies in the simulation of the deformation of mechanical components which consist of nonlinear elastic, incompressible material and that are subject to large deformations. Starting from a nonlinear formulation one can derive a discrete problem by using linearisation techniques and an adaptive mixed finite element method. This turns out to be a saddle point problem that can be solved via a Bramble-Pasciak conjugate gradient method. With some modifications the simulation can be improved.:1. Introduction 2. Basics 3. Mixed variational formulation 4. Solution method 5. Error estimation 6. LBB conditions 7. Improvement suggestions

Effiziente und zuverlässige a posteriori Fehlerabschätzungen sind eine Hauptzutat für die effiziente numerische Berechnung von Lösungen zu Variationsungleichungen durch die Finite-Elemente-Methode. Die vorliegende Arbeit untersucht zuverlässige und effiziente Fehlerabschätzungen für beliebige Finite-Elemente-Methoden und drei Variationsungleichungen, nämlich dem Hindernisproblem, dem Signorini Problem und dem Bingham Problem in zwei Raumdimensionen. Die Fehlerabschätzungen hängen vom zum Problem gehörenden Lagrange Multiplikator ab, der eine Verbindung zwischen der Variationsungleichung und dem zugehörigen linearen Problem darstellt. Effizienz und Zuverlässigkeit werden bezüglich eines totalen Fehlers gezeigt. Die Fehleranschätzungen fordern minimale Regularität. Die Approximation der exakten Lösung erfüllt die Dirichlet Randbedingungen und die Approximation des Lagrange Multiplikators ist nicht-positiv im Falle des Hindernis- und Signoriniproblems, und hat Betrag kleiner gleich 1 für das Bingham Problem. Dieses allgemeine Vorgehen ermöglicht das Einbinden nicht-exakter diskreter Lösungen, welche im Kontext dieser Ungleichungen auftreten. Aus dem Blickwinkel der Anwendungen ist Effizienz und Zuverlässigkeit im Bezug auf den Fehler der primalen Variablen in der Energienorm von großem Interesse. Solche Abschätzungen hängen von der Wahl eines effizienten diskreten Lagrange Multiplikators ab. Im Falle des Hindernis- und Signorini Problems werden postive Beispiele für drei Finite-Elemente Methoden, der konformen Courant Methode, der nicht-konformen Crouzeix-Raviart Methode und der gemischten Raviart-Thomas Methode niedrigster Ordnung hergeleitet. Partielle Resultate liegen im Fall des Bingham Problems vor. Numerischer Experimente heben die theoretischen Ergebnisse hervor und zeigen Effizienz und Zuverlässigkeit. Die numerischen Tests legen nahe, dass der aus den Abschätzungen resultierende adaptive Algorithmus mit optimaler Konvergenzrate konvergiert. / Efficient and reliable a posteriori error estimates are a key ingredient for the efficient numerical computation of solutions for variational inequalities by the finite element method. This thesis studies such reliable and efficient error estimates for arbitrary finite element methods and three representative variational inequalities, namely the obstacle problem, the Signorini problem, and the Bingham problem in two space dimensions. The error estimates rely on a problem connected Lagrange multiplier, which presents a connection between the variational inequality and the corresponding linear problem. Reliability and efficiency are shown with respect to some total error. Reliability and efficiency are shown under minimal regularity assumptions. The approximation to the exact solution satisfies the Dirichlet boundary conditions, and an approximation of the Lagrange multiplier is non-positive in the case of the obstacle and Signorini problem and has an absolute value smaller than 1 for the Bingham flow problem. These general assumptions allow for reliable and efficient a posteriori error analysis even in the presence of inexact solve, which naturally occurs in the context of variational inequalities. From the point of view of the applications, reliability and efficiency with respect to the error of the primal variable in the energy norm is of great interest. Such estimates depend on the efficient design of a discrete Lagrange multiplier. Affirmative examples of discrete Lagrange multipliers are presented for the obstacle and Signorini problem and three different first-order finite element methods, namely the conforming Courant, the non-conforming Crouzeix-Raviart, and the mixed Raviart-Thomas FEM. Partial results exist for the Bingham flow problem. Numerical experiments highlight the theoretical results, and show efficiency and reliability. The numerical tests suggest that the resulting adaptive algorithms converge with optimal convergence rates.

Cette thèse développe des estimations d’erreur a posteriori et critères d’arrêt pour les méthodes de décomposition de domaine avec des conditions de transmission de Robin optimisées entre les interfaces. Différents problèmes sont considérés: l’équation de Darcy stationnaire puis l’équation de la chaleur, discrétisées par les éléments finis mixtes avec un schéma de Galerkin discontinu de plus bas degré en temps pour le second cas. Pour l’équation de la chaleur, une méthode de décomposition de domaine globale en temps, avec mêmes ou différents pas de temps entre les différents sous domaines, est utilisée. Ce travail est finalement étendu à un modèle diphasique en utilisant une méthode de volumes finis centrés par maille en espace. Pour chaque modèle, un problème d’interface est résolu itérativement, où chaque itération nécessite la résolution d’un problème local dans chaque sous-domaine, et les informations sont ensuite transmises aux sous-domaines voisins. Pour les modèles instationnaires, les problèmes locaux dans les sous-domaines sont instationnaires et les données sont transmises par l’interface espace-temps. L’objectif de ce travail est, pour chaque modèle, de borner l’erreur entre la solution exacte et la solution approchée à chaque itération de l’algorithme de décomposition de domaine. Différentes composantes d’erreur en jeu de la méthode sont identifiées, dont celle de l’algorithme de décomposition de domaine, de façon à définir un critère d’arrêt efficace pour cette méthode. En particulier, pour l’équation de Darcy stationnaire, on bornera l’erreur par un estimateur de décomposition de domaine ainsi qu’un estimateur de discrétisation en espace. On ajoutera à la borne de l’erreur un estimateur de discrétisation en temps pour l’équation de la chaleur et pour le modèle diphasique. L’estimation a posteriori répose sur des techniques de reconstructions de pressions et de flux conformes respectivement dans les espaces H1 et H(div) et sur la résolution de problèmes locaux de Neumann dans des bandes autour des interfaces de chaque sous-domaine pour les flux. Ainsi, des critères pour arrêter les itérations de l’algorithme itératif de décomposition de domaine sont développés. Des simulations numériques pour des problèmes académiques ainsi qu’un problème plus réaliste basé sur des données industrielles sont présentées pour illustrer l’efficacité de ces techniques. En particulier, différents pas de temps entre les sous-domaines sont considérés pour cet exemple. / This work contributes to the developpement of a posteriori error estimates and stopping criteria for domain decomposition methods with optimized Robin transmission conditions on the interface between subdomains. We study several problems. First, we tackle the steady diffusion equation using the mixed finite element subdomain discretization. Then the heat equation using the mixed finite element method in space and the discontinuous Galerkin scheme of lowest order in time is investigated. For the heat equation, a global-in-time domain decomposition method is used for both conforming and nonconforming time grids allowing for different time steps in different subdomains. This work is then extended to a two-phase flow model using a finite volume scheme in space. For each model, the multidomain formulation can be rewritten as an interface problem which is solved iteratively. Here at each iteration, local subdomain problems are solved, and information is then transferred to the neighboring subdomains. For unsteady problems, the subdomain problems are time-dependent and information is transferred via a space-time interface. The aim of this work is to bound the error between the exact solution and the approximate solution at each iteration of the domain decomposition algorithm. Different error components, such as the domain decomposition error, are identified in order to define efficient stopping criteria for the domain decomposition algorithm. More precisely, for the steady diffusion problem, the error of the domain decomposition method and that of the discretization in space are estimated separately. In addition, the time error for the unsteady problems is identified. Our a posteriori estimates are based on the reconstruction techniques for pressures and fluxes respectively in the spaces H1 and H(div). For the fluxes, local Neumann problems in bands arround the interfaces extracted from the subdomains are solved. Consequently, an effective criterion to stop the domain decomposition iterations is developed. Numerical experiments, both academic and more realistic with industrial data, are shown to illustrate the efficiency of these techniques. In particular, different time steps in different subdomains for the industrial example are used.