Ce travail concerne le développement d’un outil numérique pour résoudre des problèmes de plaques et de coques mécaniques et piézo-électriques multicouches. En d’autres termes, nous développons des éléments finis basés sur le modèle de Reissner-Mindlin pour l’analyse des problèmes mécaniques et piézoélectriques de plaques et de coques multicouches. Cet outil doit être le moins coûteux en termes de degrés de liberté, simple à utiliser pour modéliser le problème, sans aucune pathologie numérique classique et satisfaisant :vitesse de convergence et efficacité.Dans le domaine des éléments finis, il existe une théorie appelée La « théorie des déformations du cisaillement du premier ordre »(FSDT), utilisée pour résoudre des problèmes purement mécanique prenant en compte l’effet des contraintes en cisaillement transverse. Dans cette théorie, cinq degrés de liberté (trois translations et deux rotations), sont à chercher.Prenant en compte l’hypothèse d’une approximation du déplacement au niveau de la surface moyenne, en utilisant la loi constitutive 2D.Dans ce travail, nous présentons l’extension piézoélectrique de cette théorie pour des éléments finis quadratiques à huit nœuds pour résoudre des problèmes de plaques et de coques multicouches. En effet, la nouveauté dans ce travail est celle de considérer une loi constitutive 3D afin de calculer non seulement le déplacement transversal au niveau du plan moyen de chaque couche mais aussi les déplacements transversaux en surface supérieure et en surface inférieure de chaque couche. Cette approximation de l’épaisseur fait intervenir deux nouveaux degrés de liberté, qui seront très importants dans l’étude des plaques et des coques épaisses et semi épaisses.Le déplacement mécanique est approximé en utilisant une approche Equivalent Single Layer(ESL) et le potentiel électrique est , quant à lui, approximé par une approche LayerWise (LW).Cette évolution est proposée afin d’acquérir un bon compromis entre le minimum de degrés de liberté et le maximum d’efficacité. D’une part, l’approximation par élément fini pour le potentiel électrique, respectant les coordonnées de l’épaisseur est représenté par une variation linéaire au niveau de chaque couche. D’autres part ce potentiel est constant par élément en chaque interface, ce qui réduit le nombre d’inconnus pour la recherche du potentiel électrique résultant. Plusieurs tests numériques sont présentés dans le but d’évaluer les éléments mécanique et piézoélectrique de plaque PQ8P7 et PQ8P7PZ et de coque CQ8P8 et CQ8P8PZ ainsi que leurs capacités de résoudre les problèmes physiques auxquels ils sont dédiés, en comparant nos résultats aux solutions de référence. / The aim of this work is to develop a computational tool for a multilayred piezolelectic plates and shells: a low cost tool, simple to use and very efficient for both convergence velocity and accuracy, without any classical numerical pathologies. In the field of finite elements, there is atheory, called "First-order-shear deformation theories (FSDT) used for the mechanical part in much reserches, taking in account the transverse shear stress effects. There are only five unknows generalized displacement, taking the hypothesis of the midsurface displacement approximation using a 2D constitutive law. In This work we present the piezo electric extension of the FSDT eight nodes plate/shell finite element using seven unknows genralized displacement. Two new mechanical unknowns are added to approximate the thickness in the top and the bottom of the /shell, considering a 3D constitutive law, wich is very interesting study especialy for the thick plate/shell modelisation. The mechanical displacement is approximated using the equivalent single layer approch (ESL) and the electric potential is approximated using the layerwise approch (LW).An evolution is proposed in order to access the best compromise between minimum number of degrees of freedom and maximum efficiency. On one hand, a finite element approximation for the electric potential with respect to the thickness coordinate are presented a linear variation in each layer. On the other hand, the in-plane variation is constant on the elementary domain at each interface layer. The use of a constant value reduces the number of unknown electric potentials. Furthermore, at the post processing level, the transverse shear stress are deduced using the equilibrium equations.Numerous tests are presented in order to evoluate that capability of these electric potential approximations to give accurate results with respect to piezoelasticity or finite element reference solutions.

Ingeniero Civil Mecánico / En la práctica de la ingeniería el método de elementos finitos o MEF se utiliza para realizar cálculos cuando resulta difícil o ineficiente obtener una solución analítica. En los últimos años se ha popularizado el uso de los métodos sin malla ya que estos entregan soluciones más exactas además de ser menos sensibles a las distorsiones de los elementos. Esto último se debe a que los métodos sin malla utilizan un vecindario de nodos para construir la aproximación de las variables, eliminando la necesidad de conexiones elementales (mallado) entre los nodos. La placa de Reissner-Mindlin (RM) se utiliza para el análisis de esfuerzos y deformaciones en placas gruesas. Este modelo supone que la deformación a lo largo del espesor varía en forma lineal además supone que la componente σzz del esfuerzo es nula.Sin embargo, la placa RM presenta el problema de bloqueo por corte, donde la rigidez de la placa aumenta drásticamente a medida que se disminuye el espesor. El objetivo de este trabajo es solucionar el bloqueo por corte mediante la proyección nodal, la cual ya ha sido utilizada para solucionar el bloqueo volumétrico en elasticidad incompresible (análogo al bloqueo por corte). Para este propósito se formula la proyección nodal para el problema de la placa RM. La proyección nodal se formula a partir de la formulación mixta eliminando los grados de libertad de corte S a partir de la discretización de los desplazamientos adyacentes disminuyendo el número de ecuaciones por resolver. Se implementa el esquema para la proyección nodal sin malla con funciones de base de máxima entropía utilizando el software Matlab. Se realizaron experimentos numéricos sobre problemas cuya solución analítica es conocida y se evaluó la convergencia y sensibilidad a distorsiones para la proyección nodal y métodos tradicionales. La proyección nodal converge óptimamente para la norma L2 del error y para la semi-norma H1 del error. La proyección nodal es menos sensible a distorsiones geométricas en la malla base que métodos MEF tradicionales. Se concluye que la proyección nodal soluciona el problema de bloqueo por corte utilizando métodos sin malla. Se consigue una mejor convergencia con respecto a los métodos tradicionales además de disminuir la sensibilidad a distorsiones geométricas.

Ce travail est consacré à l’étude d’estimateurs d'erreur a posteriori de type flux équilibrés et résiduels pour la résolution des équations de Reissner-Mindlin par la méthode des éléments finis. Le mémoire débute par l'introduction du problème aux limites et de son analyse de convergence a priori par la méthode des éléments finis. Nous construisons alors pour une discrétisation conforme un estimateur a posteriori de type flux équilibrés fiable, efficace et robuste en l'épaisseur de la plaque t. Nous obtenons finalement une constante multiplicative égale à 1 pour la fiabilité. Des tests numériques illustrent nos résultats pour différents maillages. Puis nous abordons le cas d’une discrétisation non-conforme, où nous proposons un estimateur a posteriori de type résiduel, utilisant une régularisation de la solution discrète. Des tests numériques illustrent également nos résultats. La suite du travail reprend la discrétisation conforme en construisant un estimateur a posteriori défini à partir de la résolution de problèmes localisés sur les patchs de la triangulation, menant à un choix plus consistant avec le problème aux limites. Le dernier chapitre est consacré à l'estimation a posteriori pour le problème aux valeurs propres de Reissner-Mindlin. L’estimateur obtenu est fiable et efficace pour la norme de l'erreur entre les vecteurs propres, permettant également de majorer l’erreur commise entre les valeurs propres. Des tests numériques illustrent nos résultats. / This work is devoted to the study of equilibrated fluxes and residual a posteriori error estimators for the finite element resolution of the Reissner-Mindlin system. This report begins by the introduction of the boundary value problem and of its a priori convergence analysis in the finite element method context. Then, an equilibrated fluxes a posteriori estimator is built for a conform discretization, which is proven to be reliable, efficient and robust on the plate thickness t. We finally obtain a multiplicative constant equal to 1 for the reliability. Numerical tests illustrate our results on different meshes. Then, we address the non-conforming discretization case, where a residual a posteriori estimator is proposed using a regularisation of the discrete solution. Numerical tests also illustrate our results. Next we come back to the conform discretization by building an a posteriori estimator defined from localised problems resolution on stars, leading to a consistent choice with the boundary value problem. The last chapter is devoted to an a posteriori estimation for the Reissner-Mindlin eigenvalues problem. The obtained estimator is reliable and efficient for the error norm between the eigenvectors, also allowing to evaluate the error between the eigenvalues. Numerical tests illustrate our results.

The study of geodesics is of intrinsic significance in the study of the geometry of space-time. In this thesis null, space-like and time-like geodesics are studied in the case of the space-times of Schwarzschild, Reissner-Nordstrouml;m and Kerr black holes. These space-times have been investigated with varying degrees of thoroughness in many articles and some books. However, there are some significant gaps in these treatments and the central aim of this thesis is to fill these gaps where necessary. Moreover, the following topics are covered for the first time. 1. In Chapter 4 a thorough treatment of the space-like geodesics of the Schwarzschild solutions has been given. These geodesics are the trajectories of Tachyons (faster than light particles) and are treated in a complete manner. This has been done by obtaining exact solutions and solving them numerically. 2. In Part II all solutions for geodesics for a Reissner-Nordstrouml;m black hole have been given in complete detail, i.e. time-like, null and space-like geodesics and orbit of a charged particle. 3. In Chapter 14 all solutions for geodesics in the equatorial plane of a Kerr black hole have been given in complete detail, i.e. time-like, null and space-like geodesics. 4. The study of special types of non-equatorial geodesics for a Kerr black hole have been given in complete detail, i.e. time-like (Chapter 17), null (Chapter 15) and space-like (Chapter 16). This has been done in order to distinguish the qualitatively different types of solutions. Calculation of the explicit formulas, which describe these geodesics, as well as numerically computed diagrams representing the geodesics have been incorporated in these studies. The following subjects have been also treated: 5. Solutions for the geodesics in Reissner-Nordstrouml;m black holes with |Q_*| gt;= M, which are black holes with one (|Q_*| = M) or no horizon (|Q_*|gt; M) (Chapter 8). 6. Solutions of geodesics in extreme and fast Kerr black holes, i.e. black holes with a = M (extreme) and a gt; M (fast). As in the case of |Q_*| gt; M, fast black holes have naked singularities (Chapter 14). 7. Some general observations about orbit types of the Kerr black holes regarding relationships between parameters such as angular momentum, energy, Carter constant and mass and angular momentum of black holes (Chapter 13). 8. Some corrections to errors found in the literature. While it has not been possible to cover all different cases which occur for possible relations amongst the parameters specifying a general black hole, interesting geodesics have, however, been studied and a more thorough presentation of the properties of geodesics has now been given.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnologico / Made available in DSpace on 2016-01-08T20:09:55Z (GMT). No. of bitstreams: 1 99776.pdf: 2337217 bytes, checksum: 50b3b8f3605103420611e871f3f74cf4 (MD5) Previous issue date: 1995 / Formulação e análise de problemas elastoplásticos de cascas utilizando o método dos elementos finitos. A teoria de cascas semi-espessas ou teoria de Mindlin-Reissner é utilizada, e aversão langrangiana atualizada do princípio variacional de Hill é adotada para o tratamento das não-linearidades dos tipos material e geométrica. O fenômeno de travamento é controlado utilizando a regra de subintegração uniforme e os modos espúrios decorrentes deste procedimento são estabilizados utilizando um operador projeção construído especialmente para este fim.

Este trabalho consiste em analisar os esforços em tabuleiros de pontes por meio de superfícies de influência. Para isto, o método dos elementos finitos (MEF) é utilizado e os resultados são comparados com os das tabelas de Rüsch. Os elementos finitos de barra, representando longarinas e transversinas, e placa, as lajes do tabuleiro, são implementados no código SIPlacas. Estes elementos finitos são formulados pelas teorias de viga Timoshenko e placa Reissner-Mindlin, respectivamente. Estes apresentam problema de travamento de força cortante (Shear Locking), que é contornado por duas propostas: o artifício matemático da integração reduzida e elementos finitos com campo assumido de deformação de força cortante (CADFC). Verifica-se que os elementos com aproximações quadráticas para os deslocamentos e com CADFC são os que melhor se adequam à proposta de análise da presente pesquisa. Tais elementos apresentam convergência de resultados considerando estruturas com baixa discretização. Os resultados analisados foram o deslocamento, momento fletor e força cortante. Posteriormente realiza-se um estudo de caso de uma ponte em viga. O tabuleiro da ponte é calculado utilizando-se as tabelas de Rüsch e o código SIPlacas. O cálculo dos esforços pelo SIPlacas é realizado de três maneiras. Na primeira consideram-se os painéis de lajes do tabuleiro isolados; na segunda o tabuleiro está sobre apoios não deslocáveis; e na terceira, o tabuleiro apresenta-se com vigas acopladas. Foi concluído que a terceira configuração, cuja representação melhor se aproxima da estrutura real de análise, apresentou os menores esforços internos. / This work aims at the analysis of bridge deck stresses through influence surfaces. The finite element method (FEM) is used and the results are compared with those of Rüsch\'s tables. The bar and plate finite elements represent stringers, cross beams and slabs bridge deck. These finite elements are implemented in the SIPlacas code and the theories of Timoshenko beam and Reissner-Mindlin plate are used to theirs formulation. The Shear Locking problem is solved by two proposals: reduced integration and definition of element with transversal shear strain assumed (TSSA). The elements with quadratic approximations for the displacements and TSSA are the best suited to the proposed analysis of this research. Such elements have convergence of results considering structures with low discretization. Displacement, bending moment and shear force were the results analyzed. Subsequently a case study on a beam bridge was carried out. The bridge deck is calculated using Rüsch\'s tables and SIPlacas code. The calculation of the internal forces by SIPlacas is performed in three ways. The first one considers the slabs isolated panels; the second, the slab deck is on a rigid support; and third, the slab deck is on deformable supports. It was concluded that the third configuration showed the lowest internal forces. This configuration is the optimum representation to the structure analysis.

Este trabalho consiste em analisar os esforços em tabuleiros de pontes por meio de superfícies de influência. Para isto, o método dos elementos finitos (MEF) é utilizado e os resultados são comparados com os das tabelas de Rüsch. Os elementos finitos de barra, representando longarinas e transversinas, e placa, as lajes do tabuleiro, são implementados no código SIPlacas. Estes elementos finitos são formulados pelas teorias de viga Timoshenko e placa Reissner-Mindlin, respectivamente. Estes apresentam problema de travamento de força cortante (Shear Locking), que é contornado por duas propostas: o artifício matemático da integração reduzida e elementos finitos com campo assumido de deformação de força cortante (CADFC). Verifica-se que os elementos com aproximações quadráticas para os deslocamentos e com CADFC são os que melhor se adequam à proposta de análise da presente pesquisa. Tais elementos apresentam convergência de resultados considerando estruturas com baixa discretização. Os resultados analisados foram o deslocamento, momento fletor e força cortante. Posteriormente realiza-se um estudo de caso de uma ponte em viga. O tabuleiro da ponte é calculado utilizando-se as tabelas de Rüsch e o código SIPlacas. O cálculo dos esforços pelo SIPlacas é realizado de três maneiras. Na primeira consideram-se os painéis de lajes do tabuleiro isolados; na segunda o tabuleiro está sobre apoios não deslocáveis; e na terceira, o tabuleiro apresenta-se com vigas acopladas. Foi concluído que a terceira configuração, cuja representação melhor se aproxima da estrutura real de análise, apresentou os menores esforços internos. / This work aims at the analysis of bridge deck stresses through influence surfaces. The finite element method (FEM) is used and the results are compared with those of Rüsch\'s tables. The bar and plate finite elements represent stringers, cross beams and slabs bridge deck. These finite elements are implemented in the SIPlacas code and the theories of Timoshenko beam and Reissner-Mindlin plate are used to theirs formulation. The Shear Locking problem is solved by two proposals: reduced integration and definition of element with transversal shear strain assumed (TSSA). The elements with quadratic approximations for the displacements and TSSA are the best suited to the proposed analysis of this research. Such elements have convergence of results considering structures with low discretization. Displacement, bending moment and shear force were the results analyzed. Subsequently a case study on a beam bridge was carried out. The bridge deck is calculated using Rüsch\'s tables and SIPlacas code. The calculation of the internal forces by SIPlacas is performed in three ways. The first one considers the slabs isolated panels; the second, the slab deck is on a rigid support; and third, the slab deck is on deformable supports. It was concluded that the third configuration showed the lowest internal forces. This configuration is the optimum representation to the structure analysis.

[pt] A formulação híbrida dos elementos finitos, proposta por Pian, com base no princípio variacional de Hellinger-Reissner, mostrou-se uma ótima alternativa para a construção de elementos finitos eficientes que atendessem a condições tanto de compatibilidade como de equilíbrio. O potencial de Hellinger-Reissner consiste na aproximação de dois campos: um campo tensões que satisfaz, a priori, as equações diferenciais homogêneas de equilíbrio do problema, e um campo de deslocamentos que atende a compatibilidade ao longo do contorno. O conjunto de funções não-singulares que satisfazem as equações governantes de um problema é conhecido como soluções fundamentais ou soluções de Trefftz, e é a base para a interpolação do campo de tensões no método híbrido de elementos finitos. O presente trabalho apresenta uma metodologia geral para a formulação de uma família de elementos finitos híbridos poligonais de membrana para problemas de elasticidade bidimensional, assim como elementos finitos híbridos simples e eficientes a para análise numérica de problemas de placa de Kirchhoff e Mindlin-Reissner. Algumas contribuições conceituais são introduzidas nas soluções fundamentais para a correta concepção dos elementos híbridos em problemas de placa espessa. O desempenho dos elementos é avaliado através de alguns exemplos numéricos, os quais os resultados são confrontados com os de outros elementos encontrados na literatura. / [en] The hybrid finite element formulation, proposed by Pian, on the basis of the Hellinger-Reissner variational principle, has proved to be a good alternative for the development of efficient finite elements that best attend compatibility and equilibrium conditions. The Hellinger-Reissner potential assumes two trial fields: a stress field that satisfies the equilibrium homogenous differential equation in the domain and a displacement field that attends the compatibility along the boundary. The set of nonsingular functions that satisfy the governing equations of the problem is known as Trefftz or fundamental solutions. This work presents a general methodology for the formulation of a family of polygonal hybrid elements for plane strain problems, as well as simple and efficient plate elements for the numerical evaluation of Kirchhoff and Mindlin-Reissner plate problems. Conceptual approaches are introduced for the correct use of fundamental solutions in the plate elements formulation. The performance of the proposed hybrid elements is assessed by means of several numerical examples from the literature.

Desde a formulação da Teoria da Relatividade por Einstein no início do século XX, foram várias as tentativas para se unificar as diversas teorias físicas. Kaluza e Klein foram os primeiros a buscarem esta unificação. Atualmente baseado nas idéias de Kaluza e Klein o sonho de uma unificação retorna através das teorias de cordas e dos modelos de grandes dimensões extras. Neste trabalho faremos um estudo de dois desses modelos contendo grandes dimensões extras, mostrando como é possível se explicar o problema da hierarquia existente entre as interações fundamentais através de dimensões extras compactificadas. O foco, contudo, será o estudo da possibilidade de produção de mini buracos negros (MBN's) em laboratório talvez já na próxima geração de colisores, como o LHC. Dentre os possíveis tipos de buracos negros, estudaremos o de Schwarzschild (sem carga e com momento angular nulo) enfatizando o efeito da carga elétrica que dá origem ao buraco negro de Reissner-Nordström (carregado eletricamente e sem momento angular). Finalmente trataremos dos observáveis que os MBN's (se de fato existirem) poderão deixar nos detectores.

This thesis presents new boundary element formulations for solution of bending problems in plates and shells. Also presented are the dual boundary element formulations for analysis of crack problems in plates and shells. Reissner plate theory is adopted to represent the bending and shear, and two dimensional (2-D) plane stress is used to model the membrane behaviour of the plate. New set of boundary element formulations to solve bending problems of shear deformable shallow shells having quadratic mid-surface is derived based on the modified Reissner plate and two dimensional plane stress governing equations which are now coupled due to the curvature of the shell. Dual Boundary Element Methods (DBEM) for plates and shells are developed for fracture mechanics analysis of structures loaded in combine bending and tension. Five stress intensity factors, that is, two for membrane and three for bending and shear are computed. The JIntegral technique and Crack Surface Displacements Extrapolation (CSDE) technique are used to compute the stress intensity factors. Special shape functions for crack tip elements are implemented to represent mom accurately displacement fields close to the crack tip. Crack growth processes are simulated with an incremental crack extension analysis. During the simulation, crack growth direction is determined using the maximum principal stress criterion. The crack extension is modelled by adding new boundary elements to the previous crack boundaries. As a consequence remeshing of existing boundaries is not required, and using this method the simulation can be effectively performed. Finally, a multi-region boundary element formulation is presented for modelling assembled plate-structures. The formulation enforces the compatibility of translations and rotations as well as equilibrium of membrane, bending and shear tractions. Examples are presented for plate and shell structures with different geometry, loading and boundar-y conditions to demonstrate the accuracy of the proposed formulations. The results obtained are shown to be in good agreement with analytical and other numerical results. Also presented are crack growth simulations of flat and curved panels loaded in combine bending and tension. The DBEM results are in good agreement with existing numerical and experimental results. Assembled plate-structure and a non-shallow shell bending problems are also analysed using a multi-region formulation developed in this thesis.