Le but de cette thèse est de définir une méthode d'optimisation de formes qui conjugue l'utilisation de la dérivée seconde de forme et la méthode des lignes de niveaux pour la représentation d'une forme.On considèrera d'abord deux cas plus simples : un cas d'optimisation paramétrique et un cas d'optimisation discrète.Ce travail est divisé en quatre parties.La première contient le matériel nécessaire à la compréhension de l'ensemble de la thèse.Le premier chapitre rappelle des résultats généraux d'optimisation, et notamment le fait que les méthodes d'ordre deux ont une convergence quadratique sous certaines hypothèses.Le deuxième chapitre répertorie différentes modélisations pour l'optimisation de formes, et le troisième se concentre sur l'optimisation paramétrique puis l'optimisation géométrique.Les quatrième et cinquième chapitres introduisent respectivement la méthode des lignes de niveaux (level-set) et la méthode des éléments-finis.La deuxième partie commence par les chapitres 6 et 7 qui détaillent des calculs de dérivée seconde dans le cas de l'optimisation paramétrique puis géométrique.Ces chapitres précisent aussi la structure et certaines propriétés de la dérivée seconde de forme.Le huitième chapitre traite du cas de l'optimisation discrète.Dans le neuvième chapitre on introduit différentes méthodes pour un calcul approché de la dérivée seconde, puis on définit un algorithme de second ordre dans un cadre général.Cela donne la possibilité de faire quelques premières simulations numériques dans le cas de l'optimisation paramétrique (Chapitre 6) et dans le cas de l'optimisation discrète (Chapitre 7).La troisième partie est consacrée à l'optimisation géométrique.Le dixième chapitre définit une nouvelle notion de dérivée de forme qui prend en compte le fait que l'évolution des formes par la méthode des lignes de niveaux, grâce à la résolution d'une équation eikonale, se fait toujours selon la normale.Cela permet de définir aussi une méthode d'ordre deux pour l'optimisation.Le onzième chapitre détaille l'approximation d'intégrales de surface et le douzième chapitre est consacré à des exemples numériques.La dernière partie concerne l'analyse numérique d'algorithmes d'optimisation de formes par la méthode des lignes de niveaux.Le Chapitre 13 détaille la version discrète d'un algorithme d'optimisation de formes.Le Chapitre 14 analyse les schémas numériques relatifs à la méthodes des lignes de niveaux.Enfin le dernier chapitre fait l'analyse numérique complète d'un exemple d'optimisation de formes en dimension un, avec une étude des vitesses de convergence / The main purpose of this thesis is the definition of a shape optimization method which combines second-order differentiationwith the representation of a shape by a level-set function. A second-order method is first designed for simple shape optimization problems : a thickness parametrization and a discrete optimization problem. This work is divided in four parts.The first one is bibliographical and contains different necessary backgrounds for the rest of the work. Chapter 1 presents the classical results for general optimization and notably the quadratic rate of convergence of second-order methods in well-suited cases. Chapter 2 is a review of the different modelings for shape optimization while Chapter 3 details two particular modelings : the thickness parametrization and the geometric modeling. The level-set method is presented in Chapter 4 and Chapter 5 recalls the basics of the finite element method.The second part opens with Chapter 6 and Chapter 7 which detail the calculation of second-order derivatives for the thickness parametrization and the geometric shape modeling. These chapters also focus on the particular structures of the second-order derivative. Then Chapter 8 is concerned with the computation of discrete derivatives for shape optimization. Finally Chapter 9 deals with different methods for approximating a second-order derivative and the definition of a second-order algorithm in a general modeling. It is also the occasion to make a few numerical experiments for the thickness (defined in Chapter 6) and the discrete (defined in Chapter 8) modelings.Then, the third part is devoted to the geometric modeling for shape optimization. It starts with the definition of a new framework for shape differentiation in Chapter 10 and a resulting second-order method. This new framework for shape derivatives deals with normal evolutions of a shape given by an eikonal equation like in the level-set method. Chapter 11 is dedicated to the numerical computation of shape derivatives and Chapter 12 contains different numerical experiments.Finally the last part of this work is about the numerical analysis of shape optimization algorithms based on the level-set method. Chapter 13 is concerned with a complete discretization of a shape optimization algorithm. Chapter 14 then analyses the numerical schemes for the level-set method, and the numerical error they may introduce. Finally Chapter 15 details completely a one-dimensional shape optimization example, with an error analysis on the rates of convergence

Cette thèse propose un cadre méthodologique général et unifié adapté à l’étude de la locomotion d'une large gamme de robots, en particulier bio-inspirés. L'objectif de cette thèse est double. Tout d'abord, elle contribue à la classification des robots locomoteurs en adoptant les outils mathématiques mis en place par l'école américaine de mécanique géométrique. Deuxièmement,en profitant de la nature récursive de la formulation de Newton-Euler, elle propose de nouveaux outils efficaces sous la forme d'algorithmes aptes à résoudre les dynamiques externe directe et interne inverse de tout robot locomoteur approximable par un système multicorps mobile. Ces outils génériques peuvent aider l’ingénieur ou le chercheur dans la conception, la commande, la planification de mouvement des robots locomoteurs ou manipulateurs comprenant un grand nombre de degrés de liberté internes. Des algorithmes effectifs sont proposés pour les robots discrets ainsi que continus. Ces outils méthodologiques sont appliqués à de nombreux exemples illustratifs empruntés à la robotique bio-inspirée tels les robots serpents, chenilles et autres snake-board… / This thesis proposes a general and unified methodological framework suitable for studying the locomotion of a wide range of robots, especially bio-inspired. The objective of this thesis is twofold. First, it contributes to the classification of locomotion robots by adopting the mathematical tools developed by the American school of geometric mechanics.Secondly, by taking advantage of the recursive nature of the Newton-Euler formulation, it proposes numerous efficient tools in the form of computational algorithms capable of solving the external direct dynamics and the internal inverse dynamics of any locomotion robot considered as a mobile multi-body system. These generic tools can help the engineers or researchers in the design, control and motion planning of manipulators as well as locomotion robots with a large number of internal degrees of freedom. The efficient algorithms are proposed for discrete and continuous robots. These methodological tools are applied to numerous illustrative examples taken from the bio-inspired robotics such as snake-like robots, caterpillars, and others like snake-board, etc.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2015-12-16T16:23:08Z No. of bitstreams: 1 jhonatannascimentodacosta.pdf: 4337030 bytes, checksum: 25cf98ccb23a326195b9a1a90c7d43df (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2015-12-17T11:06:06Z (GMT) No. of bitstreams: 1 jhonatannascimentodacosta.pdf: 4337030 bytes, checksum: 25cf98ccb23a326195b9a1a90c7d43df (MD5) / Made available in DSpace on 2015-12-17T11:06:06Z (GMT). No. of bitstreams: 1 jhonatannascimentodacosta.pdf: 4337030 bytes, checksum: 25cf98ccb23a326195b9a1a90c7d43df (MD5) Previous issue date: 2015-02-27 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho são propostos dois índices eficientes baseados em Equivalentes de Thévenin para avaliação da segurança de tensão de Sistemas Elétricos de Potência de grande porte. Estes índices são denominados de Índice de Estabilidade de Tensão e Índice de Perda de Controle de Tensão e baseiam-se na característica de máxima transferência de potência de circuitos elétricos lineares. Neste sentido, propõe-se uma nova metodologia para a estimação da impedância de Thévenin baseada na técnica de Análise de Sensibilidade da matriz Jacobiana do problema de Fluxo de Potência. O Índice de Estabilidade de Tensão proposto pode ser calculado para todas as barras do sistema em um dado ponto de operação, fornecendo uma estimativa rápida do ponto de vista computacional da Margem de Carregamento e uma indicação das barras críticas do sistema. Por outro lado, o Índice de Perda de Controle de Tensão é calculado somente para as barras do tipo PV, fornecendo uma indicação dos geradores críticos para o controle de tensão da região em análise. Os índices propostos são avaliados através do estudo de sistemas tutoriais, de sistemas de médio porte e de um sistema de grande porte baseado no Sistema Interligado Nacional brasileiro. Sempre que possível procura-se validar os resultados obtidos através de comparações com as técnicas do vetor tangente do Método da Continuação e de menor Margem de Potência Reativa das Curvas V-Q, que são técnicas já consagradas para análise de estabilidade de tensão de Sistemas Elétricos de Potência. / In this work are proposed two efficient indexes based on Thévenin equivalent for assessment of voltage safety of large Electric Power Systems. These indexes are called Voltage Stability Index and Voltage Control Loss Index and are based on the characteristic of maximum power transfer of linear electrical circuits. In this sense, we propose a new methodology to estimate the Thévenin impedance based on the Sensitivity Analysis technique of the Jacobian matrix of power flow problem. The Voltage Stability Index proposed can be calculated for all system buses at a given operating point, providing a quick estimate of the computational point of view of the Loading Margin and an indication of the critical buses of the system. On the other hand, the Voltage Control Loss Index is calculated only for the PV type buses and provides an indication of the critical generators for control of voltage of the region in analisys. The proposed indexes are evaluated by studying tutorials systems, medium size systems and a large system based on the brazilian National Interconnected System. Whenever possible looking up validate the results obtained through comparisons with the techniques of the tangent vector of the Continuation Method and of the smaller Reactive Power Margin of the V-Q curves, which already are established techniques of voltage stability analysis of Electric Power Systems.

The aim of the thesis is to compare the iterative methods which program RFEM 5 uses the non-linear calculations of structures, namely the analysis of large deformations and post critical analysis. Comparison should serve as a basis for which calculation method is the most accurate, fastest and most reliable in terms of getting results. Time-consuming will be judged according to the calculation of the solution and the time needed to compute one iterativ. Robustness we will compare the reliability of methods in in normal use. Accuracy of the calculation will be determined by comparing the maximum deformation structures. Comparison will be made with examples from practice.

Si la pensée des lumières a parfois été célébrée par l’historiographie pour sa contribution significative au développement culturel et scientifique de la modernité, elle a le plus souvent été critiquée pour sa superficialité spéculative, ainsi que pour sa confiance aveugle dans les résultats des enquêtes scientifiques. La figure de Pierre-Louis Moreau de Maupertuis (1698-1759), à laquelle notre travail est consacré, rentre de plein droit dans ce cadre interprétatif : mentionné dans quelques récits historiques comme scientifique de premier plan, notamment pour sa contribution décisive à la diffusion des théories newtoniennes en France, il est relégué au second plan dans les travaux d’histoire de la philosophie et de l’épistémologie. L’un des objectifs principaux de notre travail est précisément de proposer une relecture de l’œuvre de Maupertuis qui emphatise l’intérêt spéculatif de ses doctrines, afin de montrer la place éminente qu’il occupe dans l’histoire de la pensée moderne. Nous nous pencherons ainsi sur quelques textes importants de Maupertuis, en soulignant le rôle capital qu’y jouent les questions épistémologiques et métaphysiques. L’analyse critique de la pensée scientifique et philosophique de Maupertuis n’est pourtant pas le seul aspect significatif de notre recherche : notre ambition est de proposer un récit plus vaste que la simple étude des travaux de Maupertuis, qui puisse contextualiser son œuvre dans les dynamiques de la République des Lettres au début du XVIIIe siècle. En ce sens, nous attribuons une importance toute particulière à l’étude des controverses de Maupertuis avec d’autres savants et philosophes. Il sera notamment question de deux controverses. La première, qui concerne la forme de la Terre, éclate au sein de l’Académie des Sciences de Paris entre 1733 et 1740. La deuxième, qui porte sur les monades et la philosophie wolffienne, a pour cadre l’Académie de Berlin entre 1746 et 1750. L’intérêt du travail sur les controverses de Maupertuis est double. Tout d’abord, il s’agit d’un aspect de la vie et de l’œuvre du savant que les commentateurs ont peu étudié : certes, il existe quelques études particulières sur les controverses maupertuisiennes, notamment de celle qui porte sur la figure de la Terre, mais la question n’est jamais thématisée de manière systématique dans les monographies publiées jusqu’ici. Ensuite, les controverses de Maupertuis nous paraissent intéressantes comme étude de cas pour comprendre le fonctionnement concret des controverses scientifiques et philosophiques. Nous ne prétendons évidemment pas présenter les conclusions que nous allons tirer de l’examen de ces controverses en tant que normes universelles pour l’étude de toute controverse scientifique ou philosophique du passé. Toutefois, comme les historiens travaillant sur les controverses privilégient souvent les scientifiques et les philosophes du XVIIe siècle, il nous semble intéressant de présenter une étude de cas provenant du XVIIIe siècle, qui appartient à un contexte intellectuel et social profondément différent de celui du siècle précédent. Nous pourrons ainsi apporter de nouveaux éléments de réflexion sur la nature du désaccord scientifique et philosophique, qui puissent intégrer et compléter les travaux existants. Ce que nous présentons ici, en définitive, n’est pas seulement une étude sur Maupertuis, mais un travail qui, à travers l’analyse de sa pensée et des controverses qu'elle a suscitées, vise à la reconstruction des rapports et des échanges qui constituent le réseau de la République des Lettres au début du XVIIIe siècle. L’œuvre de Maupertuis est ainsi envisagée comme un prisme, à travers lequel nous observons les lignes principales de développement de la science et de la philosophie de son époque. / Whereas historians have sometimes acknowledged the momentous contribution the Enlightenment provided to the cultural and scientific development of the modern world, more often they have criticized its speculative shallowness, and its overconfidence in the results of the scientific inquiry. The figure of Pierre-Louis Moreau de Maupertuis (1698-1759), holding a central place in our study, suffered the same fate as many other Enlightenment thinkers: mentioned in some historical narratives as a leading scientist and experimenter, most notably for the decisive contribution he gave to the circulation of Newtonian theories in France, he is relegated to the background in histories of modern philosophy and epistemology. One of the main objectives of the present research is to advance an interpretation of Maupertuis’ works which insists on the speculative depth of his doctrines, in order to display the preeminent place epistemological and metaphysical stakes occupy therein. The critical analysis of Maupertuis’ texts, however, is not the only important aspect of our study: we aim indeed at creating a wider narrative, which might contribute to clarify Maupertuis’ place within the intellectual network of the early 18th-century Republic of Letters. In this sense, we attribute a special importance to the study of Maupertuis’ controversies with other scientists and philosophers. The two controversies on which we concentrate are the controversy over the shape of the Earth, which took place at the Paris Academy of Sciences between 1733 and 1740, and the controversy over monads and Christian Wolff’s philosophy, taking place at the Berlin Academy of Sciences between 1746 and 1750. The interest of studying Maupertuis’ controversies is twofold. Firstly, it is an aspect of Maupertuis’ life and work commentators have overlooked: although there are a few essays dealing with Maupertuis’ controversies, which are for the most part focused on the question of shape of the Earth, monographs hitherto published do not discuss this aspect in detail. Secondly, Maupertuis’ controversies are an interesting case study to understand how scientific and philosophical controversies concretely work. We do not intend to claim that our study of Maupertuis’ controversies might function as a universally valid model for the study of each and every controversy in the history of science and philosophy. However, since historians studying controversies privilege case studies drawn from the 17th century, it seems useful to discuss an episode from another century, which belongs therefore to a different intellectual and social context. In this way, we will provide new elements of reflection on the nature of philosophical and scientific disagreement, which might integrate the existing literature on the subject. Ultimately, our study is not only a work on Maupertuis’ thought, but a survey that, through the analysis of Maupertuis’ texts and controversies, reconstructs the network of relations and exchanges which constitute the early 18th-century Republic of Letters. The work of Maupertuis functions therefore as a prism through which we observe the main lines of development of science and philosophy of his time.

Die Dissertation beschäftigt sich mit Parameteridentifikationsproblemen, wie sie häufig in Fragestellungen der Festkörpermechanik zu finden sind. Hierbei betrachten wir die Identifikation von Materialparametern -- die typischerweise die Eigenschaften der zugrundeliegenden Materialien repräsentieren -- aus gemessenen Verformungen oder Belastungen eines Testkörpers. In mathematischem Sinne entspricht dies der Lösung von Identifikationsproblemen, die eine spezielle Klasse von inversen Problemen bilden. Der Inhalt der Dissertation ist folgendermaßen gegliedert. Nach dem einführenden Abschnitt 1 wird in Abschnitt 2 ein Überblick von Optimierungs- und Regularisierungsverfahren zur stabilen Lösung nichtlinearer inverser Probleme diskutiert. In Abschnitt 3 betrachten wir die Identifikation von skalaren und stückweise konstanten Parametern in linearen elliptischen Differentialgleichungen. Hierbei werden zwei Testprobleme erörtert, die Identifikation von Diffusions- und Reaktionsparameter in einer allgemeinen elliptischen Differentialgleichung und die Identifikation der Lame-Konstanten in einem Modell der linearisierten Elastizität. Die zugrunde liegenden PDE-Modelle und Lösungszugänge werden erläutert. Insbesondere betrachten wir hier Newton-artige Algorithmen, Gradientenmethoden, Multi-Parameter Regularisierung and den evolutionären Algorithmus CMAES. Abschließend werden Ergebnisse einer numerischen Studie präsentiert. Im Abschnitt 4 konzentrieren wir uns auf die Identifikation von verteilten Parametern in hyperelastischen Materialmodellen. Das nichtlineare Elastizitätsproblem wird detailiert erläutert und verschiedene Materialmodelle werden diskutiert (linear elastisches St.-Venant-Kirchhoff Material und nichtlineare Neo-Hooke, Mooney-Rivlin und Modified-Fung Materialien. Zur Lösung des resultierenden Parameteridentifikationsproblems werden Lösungsansätze aus der optimalen Steuerung in Form eines Newton-Lagrange SQP Algorithmus verwendet. Die Resultate einer numerischen Studie werden präsentiert, basierend auf einem zweidimensionales Testproblem mit einer sogenannten Cook-Mebran. Abschließend wird im Abschnitt 5 die Verwendung adaptiver FEM für die Lösung von Parameteridentifikationsproblems kurz erörtert. / The dissertation is focussed on parameter identification problems arising in the context of structural mechanics. At this, we consider the identification of material parameters - which typically represent the properties of an underlying material - from given measured displacements and forces of a loaded test body. In mathematical terms such problems denote identification problems as a special case of general inverse problems. The dissertation is organized as follows. After the introductive section 1, section 2 is devoted to a survey of optimization and regularization methods for the stable solution of nonlinear inverse problems. In section 3 we consider the identification of scalar and piecewise constant parameters in linear elliptic differential equations and examine two test problems, namely the identification of diffusion and reaction parameters in a generalized linear elliptic differential equation of second order and the identification of the Lame constants in the linearized elasticity model. The underlying PDE models are introduced and solution approaches are discussed in detail. At this, we consider Newton-type algorithms, gradient methods, multi-parameter regularization, and the evolutionary algorithm CMAES. Consequently, numerical studies for a two-dimensional test problem are presented. In section 4 we point out the identification of distributed material parameters in hyperelastic deformation models. The nonlinear elasticity boundary value problem for large deformations is introduced. We discuss several material laws for linear elastic (St.-Venant-Kirchhoff) materials and nonlinear Neo-Hooke, Mooney-Rivlin, and Modified-Fung materials. For the solution of the corresponding parameter identification problem, we focus on an optimal control solution approach and introduce a regularized Newton-Lagrange SQP method. The Newton-Lagrange algorithm is demonstrated within a numerical study. Therefore, a simplified two-dimensional Cook membrane test problem is solved. Additionally, in section 5 the application of adaptive methods for the solution of parameter identification problems is discussed briefly.

<p>U doktorskoj diseratciji posmatrani su singularni nelinearni problemi. U prvom&nbsp;poglavlju predstavljene su oznake i osnovne definicije i teoreme koje se koriste u&nbsp;disertaciji. U drugom poglavlju prikazani su poznati postupci i njihovo pona&scaron;anje&nbsp;u slučajevima da je re&scaron;enje regularno ili singularno. Takođe su pokazane poznate&nbsp;modifikacije ovih postupaka kako bi se pobolj&scaron;ala konvergencija. Posebno su&nbsp;predstavljena četiri kvazi-Njutnova metoda i predložene njihove modifikacije u&nbsp;slučaju singularnosti re&scaron;enja. U trećem poglavlju predstavljeni su teorijski okvir&nbsp;pri definisanju graničnih sistema i neki poznati algoritmi za njihovo re&scaron;avanje i&nbsp;definisan je novi algoritam koji je podjednako efikasan ali jeftiniji za rad jer ne&nbsp;uključuje izračunavanje izvoda. Takođe, predložena je kombinacija definisanog&nbsp;algortitma sa metodom negativnog gradijenta, kao i algoritam koji predstavlja&nbsp;primenu poznatog algoritma na definisani granični sistem. U četvrtom poglavlju&nbsp;predstavljeni su numerički rezultati dobijeni primenom definisanih algoritama na&nbsp;relevantne primere i potvrđeni su teorijski dobijeni rezultati.</p> / <p>In this doctoral thesis nonlinear singular problems were observed. The first&nbsp;chapter presents basic definitions and theorems that are used in the thesis. The&nbsp;second chapter presents several methods that are commonly used and their&nbsp;behavior if the solution is regular or singular. Also, some known modifications to&nbsp;these methods are presented in order to improve convergence. In addition four&nbsp;quasi-Newton methods and their modifications in the case the singularity of the&nbsp;solution. The third chapter consists of the theoretical foundation for defining the&nbsp;bordered system, some known algorithms for solving them and new algorithm is&nbsp;defined to accelerate convergence to a singular solution. New algorithm is&nbsp;efficient but cheaper for the use since there is no derivative evaluations in it. It is&nbsp;presented synthesis of new algorithm with negative gradient method and using&nbsp;one of well known method on the bordered system as well. The fourth chapter&nbsp;presents the numerical results obtained by the defined algorithms on the relevant&nbsp;examples and theoretical results are confirmed.</p>

[pt] Neste trabalho focamos sobre alguns modelos matemáticos na área do petróleo, com o objetivo de propor um modelo inicial de simulador numérico de reservatórios. Inicialmente apresentamos uma EDP do calor não-linear com um termo fonte de calor constante, estudada para o domínio sendo uma placa plana quadrada homogênea e heterogênea, onde aplicamos soluções numéricas utilizando o método das diferenças finitas implícito. Abordamos o problema de refinamento da malha no entorno dos poços utilizando o método JFNK (Jacobian-Free Newton-Krylov), que aumenta a eficiência computacional através de uma aproximação para a matriz Jacobiana. Por fim resolvemos um sistema de EDPs não-lineares que representam o escoamento bifásico de água e óleo, constituído por equações de transporte em termos da pressão e da saturação. Fizemos simulações numéricas de alguns casos conhecidos e os resultados mostraram uma boa qualidade no nosso método. / [en] In this work we focus on the numerical approximation of some mathematical models in the oil field. First, we present a non-linear heat equation with a constant heat source term, studied for the domain of a homogeneous and heterogeneous square domain, where we apply numerical solutions using an implicit finite difference method. We approach the problem of mesh refinement around the wells using the JFNK (Jacobian- Free Newton-Krylov) method, which improves the computational efficiency through an approximation to the Jacobian matrix. Finally, we solve a system of non-linear EDPs that represent the two-phase flow of water and oil, consisting of equations of transport in terms of pressure and saturation. Numerical simulations for some known cases showed accurate approximation of our method.

The world population is expected to increase to approximately 11 billion by 2100. The ageing population (aged 60 and over) is projected to exceed the number of children in 2047. This will be a situation without precedent. The number of citizens with disorders of old age like Dementia will rise to 115 million worldwide by 2050. The estimated cost of Dementia will also increase, from $604 billion in 2010, to $1,117 billion by 2030. At the same time, medical expertise, evidence-driven policymaking and commissioning of services are increasingly evolving the definitive architecture of comprehensive long-term care to account for these changes. Technological advances, such as those provided by computational science and biomedical engineering, will allow for an expansion in our ability to model and simulate an almost limitless variety of complex problems that have long defied traditional methods of medical practice. Numerical methods and simulation offer the prospect of improved clinically relevant predictive information, and of course optimisation, enabling more efficient use of resources for designing treatment protocols, risk assessment and urgently needed management of a long term care system for a wide spectrum of brain disorders. Within this paradigm, the importance of the relationship of senescence of cerebrospinal fluid transport to dementia in the elderly make the cerebral environment notably worthy of investigation through numerical and computational modelling. Hydrocephalus can be succinctly described as the abnormal accumulation (imbalance between production and circulation) of cerebrospinal fluid (CSF) within the brain. Using hydrocephalus as a test bed, one is able to account for the necessary mechanisms involved in the interaction between cerebral fluid production, transport and drainage. The current state of knowledge about hydrocephalus, and more broadly integrative cerebral dynamics and its associated constitutive requirements, advocates that poroelastic theory provides a suitable framework to better understand the disease. In this work, Multiple-network poroelastic Theory (MPET) is used to develop a novel spatio-temporal model of fluid regulation and tissue displacement in various scales within the cerebral environment. The model is discretised in a variety of formats, through the established finite difference method, finite difference – finite volume coupling and also the finite element method. Both chronic and acute hydrocephalus was investigated in a variety of settings, and accompanied by emerging surgical techniques where appropriate. In the coupled finite difference – finite volume model, a key novelty was the amalgamation of anatomically accurate choroid plexuses with their feeding arteries and a simple relationship relaxing the constraint of a unique permeability for the CSF compartment. This was done in order to account for Aquaporin-4 sensitisation. This model is used to demonstrate the impact of aqueductal stenosis and fourth ventricle outlet obstruction. The implications of treating such a clinical condition with the aid of endoscopic third (ETV) and endoscopic fourth ventriculostomy (EFV) are considered. It was observed that CSF velocity in the aqueduct, along with ventricular displacement, CSF pressure, wall shear stress and pressure difference between lateral and fourth ventricles increased with applied stenosis. The application of ETV reduced the aqueductal velocity, ventricular displacement, CSF pressure, wall shear stress and pressure difference within nominal levels. The greatest reversal of the effects of atresia come by opting for ETV rather than the more complicated procedure of EFV. For the finite difference model incorporating nonlinear permeability, qualitatively similar results were obtained in comparison to the pertinent literature, however, there was an overall amplification of ventriculomegaly and transparenchymal pressure difference using this model. A quantitative and qualitative assessment is made of hydrocephalus cases involving aqueductal stenosis, along with the effects to CSF reabsorption in the parenchyma and subarachnoid space. The finite element discretisation template produced for the n^th- dimensional transient MPET system allowed for novel insight into hydrocephalus. In the 1D formulation, imposing the breakdown of the blood-CSF barrier responsible for clearance resulted in an increase in ventricular displacement, transparenchymal venous pressure gradient and transparenchymal CSF pressure gradient, whilst altering the compliance proved to markedly alter the rate of change of displacement and CSF pressure gradient. The influence of Poisson's ratio was investigated through the use of the dual-grid solver in order to distinguish between possible over or under prediction of the ventricular displacement. In the 2D model based on linear triangles, the importance of the MPET boundary conditions is acknowledged, along with the quality of the underlying mesh. Interesting results include that the fluid content is highest in the periventricular region and the skull, whilst after longer time scales, the peak CSF content becomes limited to the periventricular region. Venous fluid content is heavily influenced by the Biot-Willis constant, whilst both the venous and CSF/ISF compartments show to be strongly influenced by breakdown in the blood-CSF barrier. Increasing the venous compliance effects the arterial, capillary and venous compartments. Decreasing the venous compliance shows an accumulation of fluid, possibly helping to explain why the ventricles can be induced to compress rather than expand under decreased compliance. Finally, a successful application of the 3D-MPET template is shown for simple geometries. It is envisaged that future observations into the biology of cerebral fluid flow (such as perivascular CSF-ISF fluid exchange) and its interaction with the surrounding parenchyma, will demand the evolution of the MPET model to reach a level of complexity that could allow for an experimentally guided exploration of areas that would otherwise prove too intricate and intertwined under conventional settings.

Neste estudo, apresentamos conteúdos matemáticos adaptáveis tanto para os anos finais do ensino fundamental quanto para o ensino médio. Iniciamos com um conjunto de ideias preliminares: indução matemática, triângulo de Pascal, Binômio de Newton e relações trigonométricas, para a obtenção de fórmulas de somas finitas, em que os valores das parcelas são computados sobre números inteiros consecutivos, e da técnica de transformação de soma finita em telescópica. Enunciamos Progressões Aritméticas e Geométricas como sequências numéricas e suas propriedades, obtendo a soma de seus n primeiros termos, associando com propriedades do triângulo de Pascal. Por fim, descrevemos Funções Aritméticas, Funções Aritméticas Totalmente Multiplicativas e Fortemente Multiplicativas, como sequências de números naturais, com suas operações e propriedades, direcionando ao objetivo de calcular o número de divisores naturais de n, a soma de todos os divisores naturais de n, e assim por diante. Como consequência, exibimos a fórmula de contagem do número de polinômios mônicos irredutíveis. / In this study, we present mathematical content that is adaptable to both of the final years of elementary school and to high school. We start with a set of preliminary ideas: mathematical induction, Pascal\'s triangle, Newton\'s binomial and trigonometric relations, to obtain finite sum formulas, where the parts are computed on consecutive integers, and the technique for transforming a finite sum in telescopic one. We state the Arithmetic and Geometric Progressions as numerical sequences and study their properties, obtaining the sum of their n first terms, associating with properties of the Pascal\'s triangle. Finally, we describe the Arithmetic, Totally Multiplicative and Strongly Multiplicative Arithmetic Functions, as sequences of natural numbers, with their operations and properties, as a way to calculating the number of natural divisors of n, the sum of all natural divisors of n, and so on. As a consequence, we obtain the counting formula of the number of irreducible mononical polynomials.