This thesis deals with the introduction of function of two variables and differential calculus of this function. This work should serve as a textbook for students of elementary school's teacher. Each chapter contains a summary of basic concepts and explanations of relationships, then solved model exercises of the topic and finally the exercises, which should solve the student himself. Thesis have transmit to students basic knowledges of differential calculus of functions of two variables, including practical knowledges.

This thesis includes solving the basic examples of differential calculus (especially two and three variables). The work covers examples of the solutions of the domain, the first and second partial derivatives, determining the functions of local extremes (both explicitly and implicitly given) and find the equation of the tangent plane to a point in the graphs of functions. The examples are sorted by difficulty.

Este trabalho traz uma proposta de atividades, a serem desenvolvidas em sala de aula, com o objetivo de introduzir o conceito de derivadas para os alunos da primeira série do Ensino Médio. Antes das atividades, estão presentes algumas breves pesquisas. O histórico da presença de tópicos do Cálculo Diferencial e Integral no Ensino Médio no Brasil, assim como a análise de alguns livros didáticos, serve para mostrar como o assunto já foi e está sendo tratado no país. Também são exibidos aspectos sobre o Ensino Médio na Alemanha e nos Estados Unidos, países onde o cálculo está presente na Escola Secundária, embora de formas bastante diferentes. Um capítulo sobre a preparação adequada para as aulas também foi incluído, uma vez que a simples inserção da derivada poderia causar problemas de tempo para o cumprimento do cronograma e não trazer os resultados esperados. São necessários algum grau de adequação dos conteúdos ministrados e de cooperação com professores de Física. As atividades visando o ensino dos conceitos iniciais de derivada são motivadas por um problema físico de movimento. O foco é dado na intuição e na visualização de gráficos, para que haja uma melhor compreensão dos conceitos envolvidos. A utilização de um software de geometria dinâmica é requerida em boa parte do tempo, como importante ferramenta de apoio pedagógico / This paper presents a proposal of activities to be developed in the classroom, with the goal of introducing the concept of derivative for students in the first grade of secondary school. Before the activities, some brief researches are presented. The historical presence of the topics of Differential and Integral Calculus in brazilian High Schools, as well as the analysis of some textbooks, serves to show how it has been and is being treated in the country. Aspects of the High School are also shown in Germany and the United States, countries where the calculus is present in High School, though in quite different ways. A chapter about the proper preparation for these classes was also included, since the simple insertion of the derivative could cause problems for meeting the schedule and could not bring the expected results. Some degree of adequacy of the contents and cooperation with Physics teachers are needed. The activities aiming at teaching the initial concepts of derivatives are motivated by a physical problem of motion. The focus is given on intuition and visualization of graphs, so there is a better understanding of the concepts involved. The use of a dynamic geometry software is required for much of the time, as an important tool for pedagogical support

Este trabalho trata da análise da propagação de fissuras, independente do tempo, em domínios bidimensionais utilizando uma formulação alternativa do método dos elementos de contorno (MEC). O MEC vem sendo utilizado com sucesso na análise de diversos problemas de engenharia. Considerando problemas de mecânica da fratura, o MEC é especialmente eficiente devido à redução da dimensionalidade de sua malha, o que permite a simulação do crescimento das fissuras sem as dificuldades do processo de remalhamento. Nesta pesquisa, desenvolvem-se formulações não lineares do MEC para a análise da propagação de fissuras em materiais quase frágeis. Nesses materiais, a zona de processo à frente da ponta da fissura introduz efeitos fisicamente não lineares no comportamento estrutural. Assim, para a simulação da presença da zona de processo, modelos não lineares são necessários. Classicamente a formulação dual do MEC é utilizada para modelar propagação de fissuras na quais equações singulares e hipersingulares são escritas para elementos definidos ao longo das faces das fissuras. O presente trabalho propõe uma segunda formulação utilizando um campo de tensões iniciais para a representação da zona coesiva. Nesta formulação, o termo de domínio da equação integral clássica do MEC é degenerado, de forma a atuar somente ao longo do caminho de crescimento das fissuras, sendo que esse procedimento dá origem a uma nova variável denominada dipolo, responsável por garantir o atendimento das condições de contorno. Em conjunto com essa nova formulação, se propõe o uso do operador tangente (OT), que é deduzido no trabalho, a fim de acelerar o processo de convergência da solução. Os resultados obtidos, por meio da formulação alternativa, são comparados tanto com dados experimentais quanto com o MEC dual, ambos disponíveis na literatura. As respostas encontradas foram satisfatórias no sentido de conseguir reproduzir o comportamento real da estrutura explorando as vantagens computacionais proporcionadas pelo OT. / This work presents a time-independent crack propagation analysis, in two-dimensional domains, using an alternative boundary element method (BEM) formulation. BEM has been used successfully to analyze several engineering problems. Considering fracture mechanics problems, BEM is especially efficient due to its mesh reduction aspects, which allows the simulation of crack growth without remeshing difficulties. In this research, nonlinear BEM formulations are develop in order to analyze crack propagation in quasi-brittle materials. Considering these materials, the process zone ahead of the crack tip leads to nonlinear effects related to structural behavior. Thus, nonlinear models are required for simulating the presence of the process zone. Classically, the dual BEM is used for modeling the crack propagation, in which singular and hyper-singular equations are written for elements defined along the crack faces. This work proposes an alternative formulation using the initial stress field to represent the cohesive zone. In this formulation, the classic domain integral term is degenerated in order to be non-null only at the crack growth path. This procedure leads the creation of new variable called dipole, which is responsible for ensuring the compliance of the boundary conditions. In addition to this new formulation, it is proposed the use of the tangent operator (TO), which is derived in this work, in order to accelerate the convergence. The results obtained using the new formulation, are compared with experimental data and dual BEM results available in the literature. The responses were found satisfactory in reproducing the behavior of real structures exploiting the computational advantages provided by the TO.

O presente trabalho trata do desenvolvimento de formulações numéricas para avaliar o comportamento mecânico de estruturas compostas planas, no contexto de elasticidade linear e mecânica da fratura não linear. As formulações propostas são baseadas no Método dos Elementos de Contorno (MEC), por meio das representações integrais singular e hiper singular dos problemas elastostáticos. A técnica de multi-regiões é considerada para acoplar a interface de sólidos multifásicos. O MEC é uma técnica numérica robusta e precisa para analisar o fenômeno da fratura em sólidos. Esse método numérico apresenta uma natural redução na dimensionalidade do problema, tornando mais simples a modelagem das superfícies de fratura. Além disso, essa redução de dimensionalidade faz também com que o tratamento de interfaces materiais em estruturas compostas seja uma tarefa menos árdua. Com o uso da solução fundamental de Kelvin nas representações integrais, materiais isotrópicos podem ser considerados para constituir as estruturas compostas. Por outro lado, utilizando a solução fundamental de Cruse & Swedlow, também é possível lidar, de maneira geral, com materiais anisotrópicos em estruturas compostas. Nessas estruturas, as fraturas são assumidas como ocorrendo ao longo das interfaces e o comportamento não linear é introduzido pelo modelo coesivo de fratura, o qual é aplicável a materiais quase frágeis. Nessas análises, o sistema não linear de equações pode ser solucionado utilizando dois distintos algoritmos de resolução iterativa. O primeiro sempre leva em consideração a rigidez elástica da estrutura e é, portanto denominado Operador Constante (OC). Já o segundo é denominado Operador Tangente (OT), pois considera uma rigidez tangente à resposta estrutural não linear, o que resulta em melhores taxas de convergência em comparação ao OC. Como aplicações das formulações, estruturas compostas teóricas foram analisadas em regime elástico. Além disso, testes experimentais de fratura em espécimes de concreto e madeira também foram simulados. A comparação dos resultados com as referências demonstrou que, as formulações foram efetivas e precisas para avaliar respostas mecânicas de estruturas, seja em regime elástico linear ou nos testes de fratura quase frágil. / The present work deals the development of numerical formulations to evaluate the mechanical behaviour of plane composed structures, in the context of linear elasticity and nonlinear fracture mechanics. The proposed formulations are based on the Boundary Element Method (BEM), through its classical singular and hyper singular integral equations. The multi-region technique is adopted to couple the interfaces of non-homogeneous multiphase bodies. The BEM is a robust and accurate numerical technique to analyse fracture phenomena in solids. This numerical method presents a mesh dimensionality reduction, which makes easier the modelling of cracks surfaces. Besides, this dimensionality reduction also makes the treatment of interfaces in composed structures a less complex task. Considering the use of Kelvin fundamental solutions at the integrals equations, isotropic materials can be represent as parts of the composed structures. On the other hand, using Cruse & Swedlow fundamental solution it is also possible to deal with general anisotropic materials. At the composed structures, cracks can propagate along the materials interfaces and the cohesive crack model is responsible for the nonlinear structural behaviour of the quasi-brittle failures. The nonlinear system of equations at the fracture analyses is solved using two different algorithms for iterative resolution. The first always takes into account the structure elastic strength and, hence it is called Constant Operator (CO). On the other hand, the second is denominated Tangent Operator (TO) due to the fact that it considers strengths at the tangent directions of the nonlinear structural response. Therefore, convergence rates are faster when compared with the CO. As applications, composed structures were analysed with the developed formulations in linear elastic range. In addition, experimental fracture testes performed in concrete and wood specimens were also analysed. The confront of obtained results with the reference ones show that, the formulation was effective and accurate to evaluate the mechanical responses of composed structures in linear elastic range, and also to perform nonlinear quasi-brittle fracture tests.

The aim of the diploma thesis is to propose a modification of the road I / 52 between the junction with the D1 motorway and Rajhrad on the D52 motorway in a way that a continuous highway can be created without the need to create an alternative route over the southern tangent in route Wien-Prague and D2 and also to design routes for transport which cannot enter the highway. The other aim is to define the problem elements and create a variant solution.

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This study proposes novel optimization methodologies for the optimization of problems that reveal non-physical step discontinuities. More specifically, it is proposed to use gradient-only techniques that do not use any zeroth order information at all for step discontinuous problems. A step discontinuous problem of note is the shape optimization problem in the presence of remeshing strategies, since changes in mesh topologies may - and normally do - introduce non-physical step discontinuities. These discontinuities may in turn manifest themselves as non-physical local minima in which optimization algorithms may become trapped. Conventional optimization approaches for step discontinuous problems include evolutionary strategies, and design of experiment (DoE) techniques. These conventional approaches typically rely on the exclusive use of zeroth order information to overcome the discontinuities, but are characterized by two important shortcomings: Firstly, the computational demands of zero order methods may be very high, since many function values are in general required. Secondly, the use of zero order information only does not necessarily guarantee that the algorithms will not terminate in highly unfit local minima. In contrast, the methodologies proposed herein use only first order information, rather than only zeroth order information. The motivation for this approach is that associated gradient information in the presence of remeshing remains accurately and uniquely computable, notwithstanding the presence of discontinuities. From a computational effort point of view, a gradient-only approach is of course comparable to conventional gradient based techniques. In addition, the step discontinuities do not manifest themselves as local minima. / Thesis (PhD)--University of Pretoria, 2010. / Mechanical and Aeronautical Engineering / unrestricted

This thesis comprises the study of two moduli spaces of piecewise J-holomorphic curves. The main scheme is to consider a subdivision of the 2-sphere into a collection of small domains and to study collections of J-holomorphic maps into a symplectic manifold. These maps are coupled by Lagrangian boundary conditions. The work can be seen as finding a 2-dimensional analogue of the finite-dimensional path space approximation by piecewise geodesics on a Riemannian manifold (Q,g). For a nice class of target manifolds we consider tangent bundles of Riemannian manifolds and symplectizations of unit tangent bundles. Via polarization they provide a rich set of Lagrangians which can be used to define appropriate boundary value problems for the J-holomorphic pieces. The work focuses on existence theory as a pre-stage to global questions such as combinatorial refinement and the quality of the approximation. The first moduli space of lifted type is defined on a triangulation of the 2-sphere and consists of disks in the tangent bundle whose boundary projects onto geodesic triangles. The second moduli space of punctured type is defined on a circle packing domain and consists of boundary punctured disks in the symplectization of the unit tangent bundle. Their boundary components map into single fibers and at punctures the disks converge to geodesics. The coupling boundary conditions are chosen such that the piecewise problem always is Fredholm of index zero and both moduli spaces only depend on discrete data. For both spaces existence results are established for the J-holomorphic pieces which hold true on a small scale. Each proof employs a version of the implicit function theorem in a different setting. Here the argument for the moduli space of punctured type is more subtle. It rests on a connection to tropical geometry discovered by T. Ekholm for 1-jet spaces. The boundary punctured disks are constructed in the vicinity of explicit Morse flow trees which correspond to the limiting objects under degeneration of the boundary condition.

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