Das Thema dieser Dissertation ist die Herleitung und numerische Analyse von finiten Elementen für ein Problem in der Elastoplastizität mit Kontaktbedingungen. Die hergeleiteten finite Elemente Verfahren basieren auf einer Formulierung als Sattelpunktproblem und der Nutzung von Polynomen höherer Ordnung. Die Analyse der vorgestellten Verfahren beginnt mit dem Zeigen der Wohldefiniertheit und der Konvergenz. Im nächsten Schritt werden a priori Abschätzungen der Konvergenzraten gezeigt. Weiterhin führt die Einführung von Lagrange Multiplikatoren zu einem einheitlichen Ansatz zur a posteriori Abschätzung des Diskretisierungfehlers unter der Verwendung von Elementen höherer Ordnung. Zusätzlich ermöglicht es der Zugang über Lagrange Multiplikatoren die Äquivalenz der Diskretisierungsfehler in den Spannungen und in den Energien für finite Elemente niederer Ordnung zu zeigen, was insbesondere neu für Viereckselemente ist. Diese Äquivalenz wiederum erlaubt nun den Beweis der Konvergenz von adaptiven finiten Elementen niederer Ordnung. Für Dreieckselemente wird sogar die optimale Konvergenz bewiesen. Die theoretischen Erkenntnisse werden durch numerische Experimente bestätigt. / The topic of this thesis is the derivation and analysis of some finite element schemes for a contact problem in elastoplasticity. These schemes are based on the formulation of the models as saddle point problems and use finite element spaces of arbitrary polynomial degrees. In this thesis, these new approaches with higher-order finite elements are shown to be well defined and convergent. Moreover, some a~priori estimates on the rates of convergences are proven. The use of Lagrange multipliers in the saddle point formulation yields a coherent approach to reliable a~posteriori error estimates for the proposed higher-order schemes. Additionally, the Lagrange multipliers are used to show the equivalence of the errors of the stresses and the energies, for low order finite elements using triangular or quadrilateral cells. For the first time, this allows for a proof of convergence for quadrilateral-based adaptive finite elements. Furthermore, the approach based on triangular cells is shown to be of optimal convergence. The theoretical findings are confirmed by numerical experiments.

O Método dos Elementos Finitos quando aplicado à análise de estruturas, em sua forma usual, conduz a sistemas de equações que, no caso não-linear, exigem algoritmos iterativos que realizam, em essência, uma linearização a cada passo de carga. Por outro lado, o Método da Energia formula o problema de análise estrutural na forma de uma minimização, podendo apresentar restrições sobre a função deslocamento, por exemplo. Nesse caso, os algoritmos de programação matemática proporcionam a maneira mais consistente para a obtenção da solução. O presente trabalho de mestrado trata, essencialmente, da aplicação das técnicas de otimização como ferramenta para a análise do comportamento não-linear de estruturas, que pode ser decorrente de condições de vinculação. Os problemas estruturais são formulados via Método da Energia, que resulta na minimização de funções quadráticas sujeitas a um conjunto de restrições. São discutidos os métodos do tipo Gradiente, Newton e Quase-Newton, com a descrição dos seus algoritmos básicos e apresentação da regra de busca unidimensional adotada (Regra de Armijo ou Exata). Devido ao fato do Método de Newton ter apresentado uma melhor convergência em relação aos demais algoritmos estudados, optou-se por combiná-lo com uma estratégia de conjuntos ativos para o caso de minimização com variáveis canalizadas. / The finite element method when applied to structural analysis, in its usual form, it drives the equations systems that, in the nonlinear case, they demand algorithms repetitive that accomplish, in essence, a linear programming to each load step. However, the Energy Method formulates the problem of structural analysis in the form of the minimizing, could present restrictions on the displacement function, for example. In that case, the algorithms of mathematical programming provide the most consistent way for obtaining of the solution. The present work negotiates, essentially, of the application in mathematical programming as a form to analyze the nonlinear structures behavior, that can be current of boundary conditions. The structural problems are formulated through Energy Method, that results in the mathematical programming of quadratic functions subject to a group of restrictions. The methods of the type Gradient are discussed, of Newton and Quasi-Newton, with the description of its basic algorithms and presentation of the rule of search adopted unidimensional (Rule of Armijo or Exact). Due to the fact of Newton\'s Method to have presented a better convergence in relation to the other studied algorithms, it was opted for combining it with a \"strategy of the active groups\" for the case of mathematical programming with restricted variables.

Le travail de cette thèse a été réalisé dans le cadre d'une collaboration entre EDF R&D et le LMA de Marseille (CNRS). Le but était de développer des outils théoriques et numériques pour le calcul de modes non-linéaires de structures industrielles possédant des non-linéarités localisées à jeu. La méthode de calcul utilisée est une combinaison de la méthode d'équilibrage harmonique (EH) et de la méthode asymptotique numérique (MAN), appelée EHMAN. Elle est réputée pour sa robustesse sur les problèmes réguliers. L'enjeu de ce travail de thèse est de l'appliquer sur des problèmes non-réguliers régularisés de type butée à jeu pour lequel un grand nombre d'harmonique est nécessaire. Des améliorations ont été apportées à la méthode de base pour rendre effectif le traitement de modèles à "grand" nombre de degrés de liberté (DDL). Les développements réalisés pendant la thèse ont été capitalisés par la création de nouveaux opérateurs dans Code_Aster.Une étude approfondie d'un système à 2 degrés de liberté a permis de faire émerger quelques caractéristiques des systèmes non-linéaires à jeu. Celles-ci ont servi entre autre à établir une méthodologie pour l'étude de systèmes à grand nombre de DDL. Pour finir, la potentialité des modes non-linéaires comme outil de diagnostic vibratoire est démontrée avec l'étude d'un tube cintré de générateur de vapeur. Le calcul des modes non-linéaires a monté l'existence d'une interaction entre un mode hors-plan (basse fréquence) et un mode plan (haute fréquence) expliquant des régimes vibratoires non-standards. Ce résultat, impossible à obtenir avec les outils de l'analyse modale linéaire, est confirmé expérimentalement. / This work is a collaboration between EDF R&D and the Laboratory of Mechanics and Acoustics. The objective is to develop theoretical and numerical tools to compute nonlinear normal modes (NNMs) of structures with localized nonlinearities.We use an approach combining the harmonic balance and the asymptotic numerical methods, known for its robustness principally for smooth systems. Regularization techniques are used to apply this approach for the study of nonsmooth problems. Moreover, several aspects of the method are improved to allow the computation of NNMs for systems with a high number of degrees of freedom (DOF). Finally, the method is implemented in Code_Aster, an open-source finite element solver developed by EDF R&D.The nonlinear normal modes of a two degrees-of-freedom system are studied and some original characteristics are observed. These observations are then used to develop a methodology for the study of systems with a high number of DOFs. The developed method is finally used to compute the NNMs for a model U-tube of a nuclear plant steam generator. The analysis of the NNMs reveals the presence of an interaction between an out-of-plane (low frequency) and an in-plane (high frequency) modes, a result also confirmed by the experiment. This modal interaction is not possible using linear modal analysis and confirms the interest of NNMs as a diagnostic tool in structural dynamics.

<p>A time-discrete formulation of the variational principle of mechanics is used to provide a consistent theoretical framework for the construction and analysis of low order integration methods. These are applied to mechanical systems subject to mixed constraints and dry frictional contacts and impacts---machines. The framework includes physics motivated constraint regularization and stabilization schemes. This is done by adding potential energy and Rayleigh dissipation terms in the Lagrangian formulation used throughout. These terms explicitly depend on the value of the Lagrange multipliers enforcing constraints. Having finite energy, the multipliers are thus massless ghost particles. The main numerical stepping method produced with the framework is called SPOOK.</p><p>Variational integrators preserve physical invariants globally, exactly in some cases, approximately but within fixed global bounds for others. This allows to product realistic physical trajectories even with the low order methods. These are needed in the solution of nonsmooth problems such as dry frictional contacts and in addition, they are computationally inexpensive. The combination of strong stability, low order, and the global preservation of invariants allows for large integration time steps, but without loosing accuracy on the important and visible physical quantities. SPOOK is thus well-suited for interactive simulations, such as those commonly used in virtual environment applications, because it is fast, stable, and faithful to the physics.</p><p>New results include a stable discretization of highly oscillatory terms of constraint regularization; a linearly stable constraint stabilization scheme based on ghost potential and Rayleigh dissipation terms; a single-step, strictly dissipative, approximate impact model; a quasi-linear complementarity formulation of dry friction that is isotropic and solvable for any nonnegative value of friction coefficients; an analysis of a splitting scheme to solve frictional contact complementarity problems; a stable, quaternion-based rigid body stepping scheme and a stable linear approximation thereof. SPOOK includes all these elements. It is linearly implicit and linearly stable, it requires the solution of either one linear system of equations of one mixed linear complementarity problem per regular time step, and two of the same when an impact condition is detected. The changes in energy caused by constraints, impacts, and dry friction, are all shown to be strictly dissipative in comparison with the free system. Since all regularization and stabilization parameters are introduced in the physics, they map directly onto physical properties and thus allow modeling of a variety of phenomena, such as constraint compliance, for instance.</p><p>Tutorial material is included for continuous and discrete-time analytic mechanics, quaternion algebra, complementarity problems, rigid body dynamics, constraint kinematics, and special topics in numerical linear algebra needed in the solution of the stepping equations of SPOOK.</p><p>The qualitative and quantitative aspects of SPOOK are demonstrated by comparison with a variety of standard techniques on well known test cases which are analyzed in details. SPOOK compares favorably for all these examples. In particular, it handles ill-posed and degenerate problems seamlessly and systematically. An implementation suitable for large scale performance and accuracy testing is left for future work.</p>

A time-discrete formulation of the variational principle of mechanics is used to provide a consistent theoretical framework for the construction and analysis of low order integration methods. These are applied to mechanical systems subject to mixed constraints and dry frictional contacts and impacts---machines. The framework includes physics motivated constraint regularization and stabilization schemes. This is done by adding potential energy and Rayleigh dissipation terms in the Lagrangian formulation used throughout. These terms explicitly depend on the value of the Lagrange multipliers enforcing constraints. Having finite energy, the multipliers are thus massless ghost particles. The main numerical stepping method produced with the framework is called SPOOK. Variational integrators preserve physical invariants globally, exactly in some cases, approximately but within fixed global bounds for others. This allows to product realistic physical trajectories even with the low order methods. These are needed in the solution of nonsmooth problems such as dry frictional contacts and in addition, they are computationally inexpensive. The combination of strong stability, low order, and the global preservation of invariants allows for large integration time steps, but without loosing accuracy on the important and visible physical quantities. SPOOK is thus well-suited for interactive simulations, such as those commonly used in virtual environment applications, because it is fast, stable, and faithful to the physics. New results include a stable discretization of highly oscillatory terms of constraint regularization; a linearly stable constraint stabilization scheme based on ghost potential and Rayleigh dissipation terms; a single-step, strictly dissipative, approximate impact model; a quasi-linear complementarity formulation of dry friction that is isotropic and solvable for any nonnegative value of friction coefficients; an analysis of a splitting scheme to solve frictional contact complementarity problems; a stable, quaternion-based rigid body stepping scheme and a stable linear approximation thereof. SPOOK includes all these elements. It is linearly implicit and linearly stable, it requires the solution of either one linear system of equations of one mixed linear complementarity problem per regular time step, and two of the same when an impact condition is detected. The changes in energy caused by constraints, impacts, and dry friction, are all shown to be strictly dissipative in comparison with the free system. Since all regularization and stabilization parameters are introduced in the physics, they map directly onto physical properties and thus allow modeling of a variety of phenomena, such as constraint compliance, for instance. Tutorial material is included for continuous and discrete-time analytic mechanics, quaternion algebra, complementarity problems, rigid body dynamics, constraint kinematics, and special topics in numerical linear algebra needed in the solution of the stepping equations of SPOOK. The qualitative and quantitative aspects of SPOOK are demonstrated by comparison with a variety of standard techniques on well known test cases which are analyzed in details. SPOOK compares favorably for all these examples. In particular, it handles ill-posed and degenerate problems seamlessly and systematically. An implementation suitable for large scale performance and accuracy testing is left for future work.