This thesis introduces fractal interpolation functions, exposes advantages of fractal interpolation of real world objects and presents some newly developed procedures, associated with fractal interpolation process. The work briefly presents the context needed for introduction of fractal approach and relevant definitions. Also, the detailed description of fractal generating algorithms (deterministic, random iteration, “escape time”) as well as fractal classifications is presented. Since the research object is theoretical and practical aspects of fractal interpolation function analysis, special attention is paid to geometric fractals, obtained using systems of iterated functions (IFS). The notion of a fractal interpolation function is introduced in the work. The author shows that it is possible to generate fractal interpolation functions for various types of data. The generated functions are “close” (in the sense of Housdorf dimension) to the data under processing, i.e., it is possible to ensure that the fractal interpolation graph dimension were equal to the fractal dimension of experimental data (graph). The random iteration algorithm is used for the analysis of fractal interpolation functions, since it is relatively simple and fast enough. The author makes an attempt to analyze and solve the problem of choosing interpolation points (general case). A few approaches are proposed, namely the uniform distribution of interpolation points (for the interactive use) and collage. On... [to full text]

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Les présents travaux de thèse envisagent une nouvelle technique d'optimisation au sens topologique dans des géométries de type canal vertical où se réalisent des transferts de chaleur conducto-convectifs en régime laminaire. Les équations qui décrivent l'écoulement du fluide et le transfert d'énergie sont discrétisées par la méthode des volumes finis. La première partie du mémoire présente une nouvelle technique d'optimisation et sa validation sur des cas d'études de la littérature (single pipe, bend pipe). Cette technique consiste à définir des fonctions d'interpolation de type sigmoïde et permet d'obtenir une amélioration de l'interface fluide-solide au cours du processus d'optimisation. La seconde partie met en évidence les phénomènes physiques dans le canal asymétriquement chauffé, notamment l'influence de la stratification thermique extérieure et du rayonnement de surface sur les quantités aérauliques et thermiques. Enfin, une nouvelle expression de la puissance mécanique pour contrôler les pertes de charge (malgré l'ajout de matière) dans le canal vertical combinée avec une nouvelle expression de la puissance thermique sont étudiées. Le problème ainsi posé est résolu pour un écoulement en convection naturelle. Pour les cas considérés, chacune des fonctions coût est optimisée sans détériorer l'autre. Nous comparons aussi les valeurs des puissances obtenues par notre algorithme avec celles couramment utilisées dans la littérature et montrons que ces nouvelles fonctionnelles sont performantes. / This thesis deals with topology optimization of mass and heat transfer in the framework of the vertical asymetrically-heated channel. The incompressible Navier-Stokes equations coupled to the convection-diffusion equation through the Boussinesq approximation are employed and are solved with the finite volume method. We first propose a new interpolation technique for heat transfer optimization and validate it on referenced cases such as the "single pipe" and the "bend pipe". This new technique consists in the introduction of sigmoid interpolation functions to obain a better definition of the interface between fluid and solid domains, during the optimization process. We study then physical phenomenon in the asymmetrically heated channel , in particular the influence of thermal stratification outside the channel and surface radiation on thermal and dynamic quantities. We thus highlight the size variation of reversed flow at the exit of the channel and the plug-effect linked on external thermal stratification. Finally, we propose a new expression of mechanical power in order to control charges losses (despite addition of material) in the vertical channel combined with the expression of thermal power. In all considered cases, our algorithm succeeds to enhance one of the phenomenon modelled by our new cost functions without deteriorating the other one. We also compare the values of standard cost functions from the litterature over iteration of our optimization algorithm and show that our new cost functions are effective.

[pt] Uma análise não linear geométrica de estruturas, utilizando o Método dos Elementos Finitos (MEF), depende de cinco aspectos: a teoria de flexão, da descrição cinemática, das relações entre deformações e deslocamentos, da metodologia de análise não linear e das funções de interpolação de deslocamentos. Como o MEF é uma solução numérica, a discretização da estrutura fornece grande influência na resposta dessa análise. Contudo, ao se empregar funções de interpolação correspondentes à solução homogênea da equação diferencial do problema, obtêm-se o comportamento exato da estrutura para uma discretização mínima, como ocorre em uma análise linear. Assim, este trabalho visa a integrar as soluções para o problema da não linearidade geométrica, de maneira a tentar reduzir essa influência e permitir uma discretização mínima da estrutura, considerando ainda grandes deslocamentos e rotações. Então, utilizando-se a formulação Lagrangeana atualizada, os termos de ordem elevada no tensor deformação, as teorias de flexão de Euler-Bernoulli e Timoshenko, os algoritmos para solução de problemas não lineares e funções de interpolação, que consideram a influência da carga axial, obtidas da solução da equação diferencial do equilíbrio de um elemento infinitesimal na condição deformada, desenvolve-se um elemento de pórtico espacial com uma formulação completa. O elemento é implementado no Framoop e sua resposta, utilizando-se uma discretização mínima da estrutura, é comparada com formulações usuais, soluções analíticas e com o programa Mastan2 v3.5. Os resultados evidenciam a eficiência da formulação desenvolvida para prever a carga crítica de estruturas planas e espaciais utilizando uma discretização mínima. / [en] A structural geometric nonlinear analysis, using the finite element method (FEM), depends on the consideration of five aspects: the bending theory, the kinematic description, the strain-displacement relations, the nonlinear solution scheme and the interpolation (shape) functions. As MEF is a numerical solution, the structure discretization provides great influence on the analysis response. However, applying shape functions calculated from the homogenous solution of the differential equation of the problem, the exact behavior of the structure is obtained for a minimum discretization, as for a linear analysis. Thus, this work aims to integrate the solutions for the formulations of the geometric nonlinearity problem, in order to reduce this influence and allow a minimum discretization of the structure, also considering, large displacements and rotations. Then, using an updated Lagrangian kinematic description, considering a higher-order Green strain tensor, The Euler-Bernoulli and Timoshenko beam theories, the nonlinear solutions schemes and the interpolation functions, that includes the influence of axial force, obtained directly from the solution of the equilibrium differential equation of an deformed infinitesimal element, a spatial bar frame element is developed using a complete formulation. The element was implemented in the Framoop, and their results, for a minimum discretization, were compared with conventional formulations, analytical solutions and with the software Mastan2 v3.5. Results clearly show the efficiency of the developed formulation to predict the critical load of plane and spatial structures using a minimum discretization.