The main emphasis of this thesis is to study a posteriori error analysis of discontinuous Galerkin (DG) methods for the elliptic variational inequalities. The DG methods have become very pop-ular in the last two decades due to its nature of handling complex geometries, allowing irregular meshes with hanging nodes and different degrees of polynomial approximation on different ele-ments. Moreover they are high order accurate and stable methods. Adaptive algorithms refine the mesh locally in the region where the solution exhibits irregular behaviour and a posteriori error estimates are the main ingredients to steer the adaptive mesh refinement. The solution of linear elliptic problem exhibits singularities due to change in boundary con-ditions, irregularity of coefficients and reentrant corners in the domain. Apart from this, the solu-tion of variational inequality exhibits additional irregular behaviour due to occurrence of the free boundary (the part of the domain which is a priori unknown and must be found as a component of the solution). In the lack of full elliptic regularity of the solution, uniform refinement is inefficient and it does not yield optimal convergence rate. But adaptive refinement, which is based on the residuals ( or a posteriori error estimator) of the problem, enhance the efficiency by refining the mesh locally and provides the optimal convergence. In this thesis, we derive a posteriori error estimates of the DG methods for the elliptic variational inequalities of the first kind and the second kind. This thesis contains seven chapters including an introductory chapter and a concluding chap-ter. In the introductory chapter, we review some fundamental preliminary results which will be used in the subsequent analysis. In Chapter 2, a posteriori error estimates for a class of DG meth-ods have been derived for the second order elliptic obstacle problem, which is a prototype for elliptic variational inequalities of the first kind. The analysis of Chapter 2 is carried out for the general obstacle function therefore the error estimator obtained therein involves the min/max func-tion and hence the computation of the error estimator becomes a bit complicated. With a mild assumption on the trace of the obstacle, we have derived a significantly simple and easily com-putable error estimator in Chapter 3. Numerical experiments illustrates that this error estimator indeed behaves better than the error estimator derived in Chapter 2. In Chapter 4, we have carried out a posteriori analysis of DG methods for the Signorini problem which arises from the study of the frictionless contact problems. A nonlinear smoothing map from the DG finite element space to conforming finite element space has been constructed and used extensively, in the analysis of Chapter 2, Chapter 3 and Chapter 4. Also, a common property shared by all DG methods allows us to carry out the analysis in unified setting. In Chapter 5, we study the C0 interior penalty method for the plate frictional contact problem, which is a fourth order variational inequality of the second kind. In this chapter, we have also established the medius analysis along with a posteriori analy-sis. Numerical results have been presented at the end of every chapter to illustrate the theoretical results derived in respective chapters. We discuss the possible extension and future proposal of the work presented in the Chapter 6. In the last chapter, we have documented the FEM codes used in the numerical experiments.

Os problemas de contato representam uma classe de problemas da mecânica dos sólidos para a qual a não-linearidade é introduzida pela alteração das condições de contorno, as quais só podem ser determinadas no decorrer do processo de resolução. O presente trabalho trata dos problemas de contato abordando aspectos de sua formulação e implementação numérica. Apresentam-se, em particular, as formulações de dois diferentes tipos de elemento de contato revendo-se, mais detalhadamente, o tratamento numérico das restrições decorrentes de contato. Algumas estratégias para resolução computacional desta classe de problemas, consistindo em técnicas de otimização, foram implementadas num programa computacional de elementos finitos e avaliadas comparativamente por meio de exemplos numéricos com diferentes graus de complexidade. / Contact problems represent a class of solid mechanics problems for which the nonlinear behavior is caused by the change of the boundary conditions during the solution process. The present work treats contact problems observing aspects of its formulation and numerical implementation. Specifically, the formulation for two different contact elements is presented, analyzing, in details, the numerical formulation that results from the contact. Some strategies for the computational solution of this class of problems, given by optimization techniques, were implemented in a finite element computational program and were compared and evaluated by numerical examples with different levels of complexity.

This study considers the elastostatic contact problem of a semi-infinite cylinder. The cylinder is compressed against a layer lying on a rigid foundation. There is a sharp-edged circular hole in the middle of the foundation. It is assumed that all the contacting surfaces are frictionless and only compressive normal tractions can be transmitted through the interfaces. The contact along interfaces of the elastic layer and the rigid foundation forms a circular area of which outer diameter is unknown. The problem is converted into the singular integral equations of the second kind by means of Hankel and Fourier integral transform techniques. The singular integral equations are then reduced to a system of linear algebraic equations by using Gauss-Lobatto and Gauss-Jacobi integration formulas. This system is then solved numerically. In this study, firstly, the extent of the contact area between the layer and foundation are evaluated. Secondly, contact pressure between the cylinder and layer and contact pressure between the layer and foundation are calculated for various material pairs. Finally, stress intensity factor on the edge of the cylinder and in the end of the sharp-edged hole are calculated.

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Os problemas de contato representam uma classe de problemas da mecânica dos sólidos para a qual a não-linearidade é introduzida pela alteração das condições de contorno, as quais só podem ser determinadas no decorrer do processo de resolução. O presente trabalho trata dos problemas de contato abordando aspectos de sua formulação e implementação numérica. Apresentam-se, em particular, as formulações de dois diferentes tipos de elemento de contato revendo-se, mais detalhadamente, o tratamento numérico das restrições decorrentes de contato. Algumas estratégias para resolução computacional desta classe de problemas, consistindo em técnicas de otimização, foram implementadas num programa computacional de elementos finitos e avaliadas comparativamente por meio de exemplos numéricos com diferentes graus de complexidade. / Contact problems represent a class of solid mechanics problems for which the nonlinear behavior is caused by the change of the boundary conditions during the solution process. The present work treats contact problems observing aspects of its formulation and numerical implementation. Specifically, the formulation for two different contact elements is presented, analyzing, in details, the numerical formulation that results from the contact. Some strategies for the computational solution of this class of problems, given by optimization techniques, were implemented in a finite element computational program and were compared and evaluated by numerical examples with different levels of complexity.

Orientador: Alberto Luiz Serpa / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 2018-08-12T19:17:55Z (GMT). No. of bitstreams: 1 Silva_GabrielHattorida_M.pdf: 1021601 bytes, checksum: 4cc8fdf21ddc6d423c4f3264c5c4a0f2 (MD5) Previous issue date: 2009 / Resumo: Este projeto tem o objetivo de estudar o efeito dos principais parâmetros que afetam a solução do problema de contato entre corpos elásticos. Foi utilizado o software comercial ANSYS 11.0 para realizar as análises de contato. A influência dos principais parâmetros considerados pelo ANSYS no problema de contato, tais como a rigidez de contato normal, o limite de penetração, os algoritmos de contato e métodos de solução, é investigada no trabalho. Observou-se que a rigidez de contato normal influi diretamente na convergência e nos resultados obtidos. Foram estudados alguns exemplos com resultados conhecidos (analíticos ou numéricos) para uma comparação com a solução do ANSYS, e exemplos de maior interesse prático, como o problema de contato do olhal menor de uma biela automotiva. A partir dos casos analisados, algumas recomendações foram feitas para a escolha dos parâmetros de contato. No entanto, existem parâmetros que dependem do conhecimento do usuário ou da realização de testes preliminares, o que requer em muitas situações um maior tempo para se obter os resultados. Como alternativa, foi investigado o potencial das redes neurais para contornar esta limitação. As redes neurais foram treinadas com resultados obtidos da solução do problema de contato (penetração e variação da pressão de contato) de modelos simplificados, tendo como saída da rede a rigidez de contato normal, que é então usada para estimar a rigidez de contato normal de problemas mais complexos. Foi usada a implementação de redes neurais do software MATLAB 7.0 para o treinamento e a simulação das redes neurais / Abstract: The objective of this project is to study the effect of the main contact parameters that affect the solution of the elastic bodies contact problem. The commercial software ANSYS 11.0 was used to run the contact analysis. The influence of ANSYS main parameters in the contact problem, such as normal contact stiffness, penetration limit, contact algorithms and solvers, is investigated in this work. The normal contact stiffness acts directly in convergence and in the obtained results. Some examples with known results (analytic or numeric) were studied to be compared with ANSYS solution, and some examples of more practical interest, as the connecting rod small end contact problem, were also studied. With the analysed cases, some recommendations were done to the choice of the contact parameters. However, there are parameters that depend on the user's knowledge or it is necessary to run some preliminary tests. As an alternative, it was investigated the neural networks potential to overcome this limitation. The neural networks were trained with obtained results of the contact problem solution (penetration and contact pressure variation) of simplified models. The normal contact stiffness was used as output of the network, which was used to estimate the normal contact stiffness of more complex problems. It was used the neural network implementation of the softwareMATLAB 7.0 to the training and simulation of the neural networks / Mestrado / Mecanica dos Solidos / Mestre em Engenharia Mecânica

Cette thèse est consacrée à l’analyse mathématique du problème du mouvement d’un nombre fini de corps rigides homogènes au sein d’un fluide visqueux incompressible homogène. Les fluides visqueux sont classés en deux catégories: les fluides newtoniens et les fluides non newtoniens. En premier lieu, nous considérons le système formé par les équations de Navier Stokes incompressible couplées aux lois de Newton pour décrire le mouvement de plusieurs disques rigides dans un fluide newtonien visqueux homogène dans l’ensemble de l’espace R^2. Nous montrons que ce problème est bien posé jusqu’à l’apparition de la première collision. Ensuite, nous éliminons tous les types de contacts pouvant survenir si le domaine fluide reste connexe à tout moment. Avec cette hypothèse, le système considéré est globalement bien posé. Dans la deuxième partie de cette thèse, nous montrons la non-unicité des solutions faibles au problème d’interaction fluide-solide 3D, dans le cas d’un fluide newtonien, après collision. Nous montrons qu’il existe des conditions initiales telles que nous pouvons étendre les solutions faibles après le temps pour lequel le contact a eu lieu de deux manières différentes. Enfin, dans la dernière partie, nous étudions le mouvement bidimensionnel d’un nombre fini de disques immergés dans une cavité remplie d’un fluide viscoélastique tel que des solutions polymériques. Les équations de Navier Stokes incompressible sont utilisées pour modéliser le solvant, dans lesquelles un tenseur de contrainte élastique supplémentaire apparaît comme un terme source. Dans cette partie, nous supposons que le tenseur de contrainte supplémentaire satisfait la loi différentielle d’Oldroyd ou sa version régularisée. Dans les deux cas, nous prouvons l’existence et l’unicité des solutions fortes locales en temps du problème considéré. / This thesis is devoted to the mathematical analysis of the problem of motion of afinite number of homogeneous rigid bodies within a homogeneous incompressible viscous fluid. Viscous fluids are classified into two categories: Newtonian fluids, and non-Newtonian fluids. First, we consider the system formed by the incompressible Navier-Stokes equations coupled with Newton’s laws to describe the movement of several rigid disks within a homogeneous viscous Newtonian fluid in the whole space R^2. We show the well-posedness of this system up to the occurrence of the first collision. Then we eliminate all type of contacts that may occur if the fluid domain remains connected at any time. With this assumption, the considered system is well-posed globally in time. In the second part of this thesis, we prove the non-uniqueness of weak solutions to the fluid-rigid body interaction problem in 3D in Newtonian fluid after collision. We show that there exist some initial conditions such that we can extend weak solutions after the time for which contact has taken place by two different ways. Finally, in the last part, we study the two-dimensional motion of a finite number of disks immersed in a cavity filled with a viscoelastic fluid such as polymeric solutions. The incompressible Navier–Stokes equations are used to model the flow of the solvent, in which the elastic extra stress tensor appears as a source term. In this part, we suppose that the extra stress tensor satisfies either the Oldroyd or the regularized Oldroyd constitutive differential law. In both cases, we prove the existence and uniqueness of local-in-time strongsolutions of the considered moving-boundary problem.

L'objectif ultime de ce travail est de fournir des méthodologies robustes et efficaces sur le plan des calculs pour la modélisation et la résolution des problèmes de contact adhésifs basés sur le potentiel de Lennard-Jones (LJ). Pour pallier les pièges théoriques et numériques du modèle LJ liés à ses caractéristiques nondéfinies et non-bornées, une méthode d'adaptativité en modèle est proposée pour résoudre le problème purement-LJ comme limite d'une séquence de problèmes multiniveaux construits de manière adaptative. Chaque membre de la séquence consiste en une partition modèle entre le modèle microscopique LJ et le modèle macroscopique de Signorini. La convergence de la méthode d'adaptativité est prouvée mathématiquement sous certaines hypothèses physiques et réalistes. D'un autre côté, la méthode asymptotique numérique (MAN) est adaptée et utilisée pour suivre avec précision les instabilités des problèmes de contact à grande échelle et souples. Les deux méthodes sont incorporées dans le cadre multiéchelle Arlequin pour obtenir une résolution précise, tout en réduisant les coûts de calcul. Dans la méthode d'adaptativité en modèle, pour capturer avec précision la localisation des zones d'intérêt (ZDI), une stratégie en deux résolutions est suggérée : une résolution macroscopique est utilisée comme une première estimation de la localisation de la ZDI. La méthode Arlequin est alors utilisée pour obtenir une résolution microscopique en superposant des modèles locaux aux modèles globaux. En outre, dans la stratégie MAN, la méthode Arlequin est utilisée pour supprimer les oscillations numériques, améliorer la précision et réduire le coût de calcul. / The ultimate goal of this work is to provide computationally efficient and robust methodologies for the modelling and solution of a class of Lennard-Jones (LJ) potential-based adhesive contact problems. To alleviate theoretical and numerical pitfalls of the LJ model related to its non-defined and nonbounded characteristics, a model-adaptivity method is proposed to solve the pure-LJ problem as the limit of a sequence of adaptively constructed multilevel problems. Each member of the sequence consists of a model partition between the microscopic LJ model and the macroscopic Signorini model. The convergence of the model-adaptivity method is proved mathematically under some physical and realistic assumptions. On the other hand, the asymptotic numerical method (ANM) is adapted to track accurately instabilities for soft contact problems. Both methods are incorporated in the Arlequin multiscale framework to achieve an accurate resolution at a reasonable computational cost. In the model-adaptivity method, to capture accurately the localization of the zones of interest (ZOI), a two-step strategy is suggested: a macroscopic resolution is used as the first guess of the ZOI localization, then the Arlequin method is used there to achieve a fine scale resolution. In the ANM strategy, the Arlequin method is also used to suppress numerical oscillations and improve accuracy.

This thesis deals with the design of a dynamic model of a harmonic drive. It includes a theoretical study aimed at the analysis of the harmonic drive gearing principle and its nonlinear properties. The first part of the practical section deals with the analytical calculation of the nondeformed geometry of the Flexspline. Based on these results, several simulations in ANSYS are created to identify torsional characteristics of a harmonic drive. These simulation models are further enhanced by the analysis of clearance, backlash and inaccuracies and their impact on torsional properties. By using MATLAB /Simulink, several dynamic submodels are created representing the individual characteristics of nonlinearities in harmonic drives. Furthermore, a comprehensive dynamic model is created of the mechatronic system which is describing all nonlinearities and kinematic error of the transmission. The dynamic model is also experimentally verified based on its damping properties.

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