The geometrically nonlinear constant moment triangle based on the von Karman theory of thin plates is first described. This finite element, which is believed to be the simplest possible element to pass the totality of the von Karman patch test, is employed throughout the present work. It possesses the special characteristic of providing a tangent stiffness matrix which is accurate and without approximation. The stability of equilibrium of discrete conservative systems is discussed. The criteria which identify the critical points (limit and bifurcation), and the method of determination of the stability coefficients are presented in a simple matrix formulation which is suitable for computation. An alternative formulation which makes direct use of higher order directional derivatives of the total potential energy is also presented. Continuation along the stable equilibrium solution path is achieved by using a recently developed Newton method specially modified so that stable points are points of attraction. In conjunction with this solution technique, a branch switching method is introduced which directly computes any intersecting branches. Bifurcational buckling often exhibits huge structural changes and it is believed that the computation of the required switch procedure is performed here, and for the first time, in a satisfactory manner. Hence, both limit and bifurcation points can be treated without difficulty and with continuation into the post buckling regime. In this way, the ability to compute the stable equilibrium path throughout the load-deformation history is accomplished. Two numerical examples which exhibit bifurcational buckling are treated in detail and provide numerical evidence as to the ability of the employed techniques to handle even the most complex problems. Although only relatively coarse finite element meshes are used it is evident that the technique provides a powerful tool for any kind of thin elastic plate and shell problem. The thesis concludes with a proposal for an algorithm to automate the computation of the unknown parameter in the branch switching method.

Made available in DSpace on 2016-12-12T20:15:52Z (GMT). No. of bitstreams: 1 Rafael Marques.pdf: 14630401 bytes, checksum: 352c48d8c01e0db8ba6b41f5e8317ec8 (MD5) Previous issue date: 2015-03-11 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The main subject developed in this dissertation is the characterization of the dynamics of high-dimensional conservative systems using different statistical approaches. Looking at the conservative system phase-space, we can find chaotic and regular regions that are characterized by a random distribution of points and periodic structures formed by closed orbits, respectively. The nonlinearity parameter has a fundamental hole to the occurrence of chaotic trajectories that can get stuck for a finite time on the vicinity of regular regions. This phenomenon is known as stickiness effect and can be identified using different tools as the spectrum of finite time Lyapunov exponents or the recurrence time statistics (RTS), e.g. Throughout this dissertation, we propose to characterize this effect using such approaches and also apply a new methodology which uses the time series of the spectrum of finite time Lyapunov exponents to separate the dynamics in different regimes of motion. For this purpose, we study two conservative systems that are derived from standard map, a symplectic map extensively used to investigate the transition from regular to chaotic dynamic. The first system consists in a chain of coupled standard maps that originates a 2N-dimensional system, where N is the number of coupled maps. Using this system, from the definition of regimes of motion, we obtained the cumulative distribution of the consecutive time that the trajectory spends in a particular regime, which reproduces with a good precision the results obtained when using the RTS. The second system studied was the Modified Standard Map, which is obtained adding an action variable to the standard map. The coupling with an extra dimension allows the penetration of the regular structures by the trajectories, what was forbidden for the two-dimensional case. The application of the method of separation of regimes in this system enables a more detailed analysis of the stickiness effect, showing that only the trajectories located near the regular structures have Local Lyapunov exponents about zero. Thus, the development of this research contributes to a better understanding of the stickiness effect in high-dimensional conservative systems. / O tema principal desenvolvido nesta dissertação de Mestrado está relacionado com o estudo da dinâmica de sistemas conservativos, utilizando diferentes abordagens estatísticas. Ao analisarmos o espaço de fases de um sistema dinâmico pertencente a esta classe, podemos encontrar regiões caóticas e regulares que são caracterizadas pela distribuição aleatória de pontos e por estruturas periódicas formadas por órbitas fechadas, respectivamente. O parâmetro de não-linearidade tem um papel fundamental na existência de trajetórias caóticas que podem ser aprisionadas por um tempo finito nas proximidades das regiões regulares. Este fenômeno é conhecido como efeito stickiness, e pode ser identificado através da utilização de diferentes abordagens como, por exemplo, o espectro de Lyapunov calculado a tempo finito ou a estatística dos tempos de recorrência de Poincaré (ETR). No decorrer desta dissertação, propomos caracterizar o efeito stickiness utilizando tais abordagens, além de aplicar uma nova metodologia que consiste em analisar séries temporais do espectro de expoentes de Lyapunov afim de definir diferentes regimes de movimento. Para isso, estudamos dois sistemas conservativos multidimensionais derivados do mapa padrão, um mapa simplético muito utilizado para a investigação da transição da dinâmica regular para caótica. O primeiro deles consiste em uma rede de mapas padrão acoplados que dá origem a um sistema de 2N-dimensões, sendo N o número de mapas acoplados. Utilizando este sistema, a partir da definição de regimes de movimento, foi possível determinar a distribuição cumulativa do tempo consecutivo que a trajetória permanece em um determinado regime, sendo que os resultados obtidos por meio da análise desta quantidade podem reproduzir de forma satisfatória aqueles obtidos quando utilizamos a ETR. O segundo sistema estudado foi o Mapa Padrão Modificado (MPM), resultante do acoplamento entre uma variável ação extra e o mapa padrão tradicional. O acoplamento com uma dimensão extra permite que trajetórias penetrem nas regiões de regularidade, o que antes era proibido para o caso bidimensional. A aplicação da técnica de separação de regimes neste sistema permite uma análise mais detalhada do efeito stickiness, mostrando que apenas trajetórias que se encontram em torno das estruturas de regularidade possuem expoentes de Lyapunov Locais com valores próximos a zero. Desta forma, o desenvolvimento desta pesquisa contribui para o melhor entendimento do efeito stickiness em sistemas conservativos de alta dimensionalidade.