Made available in DSpace on 2014-07-29T15:18:19Z (GMT). No. of bitstreams: 1 Dissertacao Marco Antonio Borges Traldi.pdf: 492440 bytes, checksum: 93d641fe65bccb8763ec9b2e879faee7 (MD5) Previous issue date: 2010-08-13 / Topology optimization (TO) is an optimization technique that aims to help the designer in the phase of structural conception, seeking a better distribution of one fixed amount of material in a extended design domain. It combines a method of structural analysis with a mathematical programming (MP) algorithm. In this work, structural analysis is done by Finite Element Method (FEM) and the solution algorithm employed is the Optimality Criteria (OC). One of the main themes of research in TO refer to the regularization techniques, needed to make the solutions free of the classic numerical instabilities of the problem: the instability of the checkerboard and mesh dependency. The goal of this work is study the traditional regularization techniques found in literature as the sensitivity filter by Sigmund, Continuous Approximation of Material Distribution (CAMD) and projection techniques. A new projection technique is proposed in order to obtain solutions close to the void-solid form. The objective function is the mean compliance of the structure, and is imposed a restriction on the total volume of material to be distributed. Is used the model Solid Isotropic Material with Penalization (SIMP) to parameterize the distribution of isotropic materials, and their adaptation FGM-SIMP for functionally graded materials (FGM). To illustrate the behavior of these techniques examples of application are shown, fruits of the computational implementation. / A otimização de topologia (OT) é uma técnica de otimização que visa auxiliar o projetista na fase de concepção estrutural, buscando a melhor distribuição de uma quantidade fixa de material em um domínio estendido de projeto. Combina um método de análise estrutural com um algoritmo de programação matemática (PM). Neste trabalho, a análise estrutural é feita pelo Método dos Elementos Finitos (MEF) e o algoritmo de solução utilizado é o Critério de Otimalidade (CO). Um dos principais temas de pesquisa em OT remete às técnicas de regularização, necessárias para tornar as soluções livres das instabilidades numéricas clássicas do problema: a instabilidade do tabuleiro de xadrez (checkerboard) e a dependência de malha. O objetivo deste trabalho é estudar as técnicas de regularização tradicionais encontradas na literatura como o filtro de sensibilidade de Sigmund, Continuous Approximation of Material Distribution (CAMD) e técnicas de projeção. Uma nova técnica de projeção é proposta visando a obtenção de soluções próximas à forma vazio-sólido. A função objetivo é a flexibilidade média da estrutura, sendo imposta uma restrição ao volume total de material a ser distribuído. Utiliza-se o modelo Solid Isotropic Material with Penalization (SIMP) para parametrizar a distribuição de materiais isotrópicos, e sua adaptação FGM-SIMP para materiais com gradação funcional (FGM). Para ilustrar o comportamento das técnicas estudadas são apresentados exemplos de aplicação, frutos das implementações computacionais realizadas.

O desenvolvimento de novas técnicas capazes de produzir modelos de predição com erros de generalização relativamente baixos é uma constante em aprendizado de máquina e áreas correlatas. Nesse sentido, a composição de um conjunto de modelos no denominado ensemble merece destaque por seu potencial teórico e empírico de minimizar o erro de generalização. Diversos métodos para construção de ensembles de modelos são encontrados na literatura. Dentre esses, o método baseado em rotação (RB) tem apresentado desempenho superior a outros clássicos. O método RB utiliza a técnica de extração de características da análise de componentes principais (PCA) como estratégia de rotação para provocar acurácia e diversidade entre os modelos componentes. Contudo, essa estratégia não assegura que a direção resultante será apropriada para a técnica de aprendizado supervisionado (SLT) escolhida. Adicionalmente, o método RB não é adequado com SLTs invariantes à rotação e não foi amplamente validado com outras estáveis. Esses aspectos tornam-no inadequado e/ou restrito a algumas SLTs. Nesta tese, é proposta uma nova abordagem de extração baseada na concatenação de rotação e projeção otimizadas em prol da SLT (denominada roto-projeção otimizada). A abordagem utiliza uma metaheurística para otimizar os parâmetros da transformação de roto-projeção e minimizar o erro da técnica diretora da otimização. Mais enfaticamente, propõe-se a roto-projeção otimizada como parte fundamental de um novo método de ensembles, denominado ensemble baseado em roto-projeção otimizada (ORPE). Os resultados obtidos mostram que a roto-projeção otimizada pode reduzir a dimensionalidade e a complexidade dos dados e do modelo, além de aumentar o desempenho da SLT utilizada posteriormente. O método ORPE superou, com relevância estatística, o RB e outros com SLTs estáveis e instáveis em bases de classificação e regressão de domínio público e privado. O ORPE mostrou-se irrestrito e altamente eficaz assumindo a primeira posição em todos os ranqueamentos de dominância realizados / The development of new techniques capable of inducing predictive models with low generalization errors has been a constant in machine learning and other related areas. In this context, the composition of an ensemble of models should be highlighted due to its theoretical and empirical potential to minimize the generalization error. Several methods for building ensembles are found in the literature. Among them, the rotation-based (RB) has become known for outperforming other traditional methods. RB method applies the principal components analysis (PCA) for feature extraction as a rotation strategy to provide diversity and accuracy among base models. However, this strategy does not ensure that the resulting direction is appropriate for the supervised learning technique (SLT). Moreover, the RB method is not suitable for rotation-invariant SLTs and also it has not been evaluated with stable ones, which makes RB inappropriate and/or restricted to the use with only some SLTs. This thesis proposes a new approach for feature extraction based on concatenation of rotation and projection optimized for the SLT (called optimized roto-projection). The approach uses a metaheuristic to optimize the parameters from the roto-projection transformation, minimizing the error of the director technique of the optimization process. More emphatically, it is proposed the optimized roto-projection as a fundamental part of a new ensemble method, called optimized roto-projection ensemble (ORPE). The results show that the optimized roto-projection can reduce the dimensionality and the complexities of the data and model. Moreover, optimized roto-projection can increase the performance of the SLT subsequently applied. The ORPE outperformed, with statistical significance, RB and others using stable and unstable SLTs for classification and regression with databases from public and private domains. The ORPE method was unrestricted and highly effective holding the first position in every dominance rankings

Ce travail concerne l’étude d’écoulements incompressibles soumis à la rotation avec un solveur haute précision dans des géométries semi-complexes. La technique numérique mise en œuvre combine des schémas compacts, une méthode de projection multi domaine directe et un traitement efficace de la singularité à l’axe basé sur des conditions de parité dans l’espace de Fourier. Le solveur a été parallélisé avec une approche hybride MPI-OpenMP pour réduire les temps de calcul. Dans un premier temps, les précisions spatiales et temporelles de la méthode numérique et la scalabilité du solveur ont été vérifiées. La capacité du solveur à traiter des écoulements plus complexes a été évaluée en considérant des écoulements de type éclatement tourbillonnaire et un écoulement turbulent en conduite cylindrique. Dans un second temps, plusieurs écoulements typiques des machines tournantes ont été étudiés. Le premier écoulement est un écoulement turbulent incompressible isotherme dans un étage simplifié d’un compresseur haute pression d’une turbine à gaz. Les simulations menées ont mises en évidence l’effet de la rotation sur l’écoulement, notamment sur les instabilités se développant le long des parois et sur les différentes structures cohérentes. Le second cas traité est un écoulement turbulent de jet impactant un disque en rotation avec un fort confinement et transfert thermique. Une attention particulière a été portée sur les champs hydrodynamiques et thermique le long du rotor. Enfin, une étude préliminaire d’un jet turbulent impactant un disque fixe d’épaisseur non nulle dans une configuration moins contrainte avec prise en compte du couplage conduction-convection a été réalisée. / This work deals with the study of rotating incompressible flows with a high accurate solver in semi complex geometries. The numerical method used in this work combines compact schemes, a direct multidomain projection method and an efficient axis treatment based on parity conditions in Fourier space. The use of cylindrical coordinates introduces this mathematical singularity. In order to reduce the calculation time, the solver was parallelized with an hybrid MPI-OpenMP parallelization. First, the spatial and temporal accuracies of the numerical method and the scalability of the solver were checked. Then, the capability of the algorithm to deal with more complex flows was verified. Vortex breakdown flows and turbulent pipe flow were studied. In the second step, typical flows of turbomachineries and rotating systems were considered. The first flow was an incompressible isothermal turbulent flow in a high pressure compressor of gas turbine. The different simulations highlighted the rotation effects on the flows, especially on the instabilities appearing along the walls and the coherent structures. The second considered flow was a turbulent impinging jet on a rotating disk with heat transfer in a small aspect ratio cavity. The hydrodynamic fields and heat transfer near the rotor were analyzed in detail. Finally, a preliminary investigation of an impinging jet on a non-rotating disk in a larger aspect ratio cavity with a coupling between conduction and convection transfer was carried out.

L'étude proposée porte sur l'adaptation d'une méthode de mesure optique à lamesure de la topologie d'objet de grandes dimensions et à une distance de travail proche. Laméthode optique utilisée est la projection de franges car elle permet l'étude de grands objets.Dans un premier temps, des essais expérimentaux ont été réalisés pour évaluer l'exactitudedu développement actuel ; ce dernier utilisant une analyse de franges s'appuyant sur lacombinaison d'une méthode quasi-hétérodyne utilisant une transformation de Fourier etd'une méthode de code gray.Après avoir quantifié les erreurs et déterminé leurs sources, le choix dudéveloppement d'une procédure d'étalonnage et de nouvelles équations associées à cetteprocédure se sont imposés. Le nouvel étalonnage est quant à lui basé sur une interpolationpolynomiale de points définissant un volume de grandes dimensions. Un objet étalon a étéspécialement conçu pour cette procédure de calibration. Pour évaluer l'erreur du à lacalibration, une étude systématique de cas de polynômes dont le plus haut degré varie de 1à 4 a été effectué. Cette approche a permis de déterminer le degré optimal du polynôme àutiliser. Dans le meilleur cas, l'estimation de l'erreur a permis d'évaluer la précision del'étalonnage à 1 mm sur un objet de 2 m évalué à une distance de 2 m.La méthode a été par la suite appliquée, dans un cadre industriel à l'étude deflotteurs et dans un cadre médical à l'étude du relief de la paroi abdominale. D'un point devue médical, cette méthode permet d'obtenir rapidement et facilement la morphologie ducorps humain. Elle permet aussi d'effectuer un meilleur suivi des pathologiesmorphologiques des patients. / The proposed study deals with the adaptation of an optical method to themeasurement of large objects at a low working distance. The optical method used is thefringe projection technique allowing the study of large objects. At first, experimental trials hasbeen used to evaluate the accuracy of the actual development combining the phase shiftingmethod using a Fourier transform and the gray code technique.When the errors have been quantified and their origins determined, the developmentof a calibration procedure and new associated equations have been chosen. The newcalibration is based on polynomial interpolation of points defining a volume of largedimensions. A tested object was designed specifically for this calibration procedure. Toestimate the calibration error, a systematic study of polynomials cases is performed. Thehighest degree of those cases varies from 1 to 4. This approach allows the determination theoptimal polynomial degree to be used. In the best case, the estimation of the error allows theevaluation of the calibration accuracy of about 1 mm for an object of 2 m large, measured ata distance of 2 m.The method has been subsequently applied, in an industrial setting, to the study offloats and in a medical setting to the study of the relief of the abdominal wall. From a medicalpoint of view, this method gives a rapid and easy access to the topology of human body. Itallows a better follow-up of the patient pathology.

In this thesis there can be found various examples from Monge projection. The theory is summarized in the beginning, which is important of understanding the projection and for solving the examples. There are also examples of solving axial affinity and central collineation. Then there is a chapter about the projection of all types of angular and rotational solids, which are solved at the secondary schools. Then follows a chapter, where the sections of these solids are constructed. In the last chapter, there are solved intersection of solids from each type. Powered by TCPDF (www.tcpdf.org)

La méthode IIM (Immersed Interface Method) permet d'étendre certaines méthodes numériques à des problèmes présentant des discontinuités. Elle est utilisée ici pour étudier un fluide incompressible régi par les équations de Navier-Stokes, dans lequel est immergée une membrane exerçant une force singulière. Nous utilisons une méthode de projection dans une grille de différences finies de type MAC. Une dérivation très complète des conditions de saut dans le cas où la viscosité est continue est présentée en annexe. Deux exemples numériques sont présentés : l'un sans membrane, et l'un où la membrane est immobile. Le cas général d'une membrane mobile est aussi étudié en profondeur. / The Immersed Interface Method allows us to extend the scope of some numerical methods to discontinuous problems. Here we use it in the case of an incompressible fluid governed by the Navier-Stokes equations, in which a membrane is immersed, inducing a singular force. We use a projection method and staggered (MAC-type) finite difference approximations. A very complete derivation for the jump conditions is presented in the Appendix, for the case where the viscosity is continuous. Two numerical examples are shown : one without a membrane, and the other where the membrane is motionless. The general case of a moving membrane is also thoroughly studied.

En physique statistique computationnelle, de bonnes techniques d'échantillonnage sont nécessaires pour obtenir des propriétés macroscopiques à travers des moyennes sur les états microscopiques. La principale difficulté est que ces états microscopiques sont généralement regroupés autour de configurations typiques, et un échantillonnage complet de l'espace configurationnel est donc typiquement très complexe à réaliser. Des techniques ont été proposées pour échantillonner efficacement les états microscopiques dans l'ensemble canonique. Un exemple important de quantités d'intérêt dans un tel cas est l'énergie libre. Le calcul d'énergie libre est très important dans les calculs de dynamique moléculaire, afin d'obtenir une description réduite d'un système physique complexe de grande dimension. La première partie de cette thèse est consacrée à une extension de la méthode adaptative de force biaisante classique (ABF), qui est utilisée pour calculer l'énergie libre associée à la mesure de Boltzmann-Gibbs et une coordonnée de réaction. Le problème de cette méthode est que le gradient approché de l'énergie libre, dit force moyenne, n'est pas un gradient en général. La contribution à ce domaine, présentée dans le chapitre 2, est de projeter la force moyenne estimée sur un gradient en utilisant la décomposition de Helmholtz. Dans la pratique, la nouvelle force gradient est obtenue à partir de la solution d'un problème de Poisson. En utilisant des techniques d'entropie, on étudie le comportement à la limite de l'équation de Fokker-Planck non linéaire associée au processus stochastique. On montre la convergence exponentielle vers l'équilibre de l'énergie libre estimée, avec un taux précis de convergence en fonction des constantes de l'inégalité de Sobolev logarithmiques des mesures canoniques conditionnelles à la coordonnée de réaction. L'intérêt de la méthode d'ABF projetée par rapport à l'approche originale ABF est que la variance de la nouvelle force moyenne est plus petite. On observe que cela implique une convergence plus rapide vers l'équilibre. En outre, la méthode permet d'avoir accès à une estimation de l'énergie libre en tout temps. La deuxième partie (voir le chapitre 3) est consacrée à étudier l'existence locale et globale, l'unicité et la régularité des solutions d'une équation non linéaire de Fokker-Planck associée à la méthode adaptative de force biaisante. Il s'agit d'un problème parabolique semilinéaire avec une non-linéarité non locale. L'équation de Fokker-Planck décrit l'évolution de la densité d'un processus stochastique associé à la méthode adaptative de force biaisante. Le terme non linéaire est non local et est utilisé lors de la simulation afin d'éliminer les caractéristiques métastables de la dynamique. Il est lié à une espérance conditionnelle qui définit la force biaisante. La preuve est basée sur des techniques de semi-groupe pour l'existence locale en temps, ainsi que sur une estimée a priori utilisant une sursolution pour montrer l'existence globale / In computational statistical physics, good sampling techniques are required to obtain macroscopic properties through averages over microscopic states. The main difficulty is that these microscopic states are typically clustered around typical configurations, and a complete sampling of the configurational space is thus in general very complex to achieve. Techniques have been proposed to efficiently sample the microscopic states in the canonical ensemble. An important example of quantities of interest in such a case is the free energy. Free energy computation techniques are very important in molecular dynamics computations, in order to obtain a coarse-grained description of a high-dimensional complex physical system. The first part of this thesis is dedicated to explore an extension of the classical adaptive biasing force (ABF) technique, which is used to compute the free energy associated to the Boltzmann-Gibbs measure and a reaction coordinate function. The problem of this method is that the approximated gradient of the free energy, called biasing force, is not a gradient. The contribution to this field, presented in Chapter 2, is to project the estimated biasing force on a gradient using the Helmholtz decomposition. In practice, the new gradient force is obtained by solving Poisson problem. Using entropy techniques, we study the longtime behavior of the nonlinear Fokker-Planck equation associated with the ABF process. We prove exponential convergence to equilibrium of the estimated free energy, with a precise rate of convergence in terms of the Logarithmic Sobolev inequality constants of the canonical measure conditioned to fixed values of the reaction coordinate. The interest of this projected ABF method compared to the original ABF approach is that the variance of the new biasing force is smaller, which yields quicker convergence to equilibrium. The second part, presented in Chapter 3, is dedicated to study local and global existence, uniqueness and regularity of the mild, Lp and classical solution of a nonlinear Fokker-Planck equation, arising in an adaptive biasing force method for molecular dynamics calculations. The partial differential equation is a semilinear parabolic initial boundary value problem with a nonlocal nonlinearity and periodic boundary conditions on the torus of dimension n, as presented in Chapter 3. The Fokker-Planck equation rules the evolution of the density of a given stochastic process that is a solution to Adaptive biasing force method. The nonlinear term is non local and is used during the simulation in order to remove the metastable features of the dynamics

Les techniques avancées pour l’approximation de rang faible des matrices sont des outils de réduction de dimension fondamentaux pour un grand nombre de domaines du calcul scientifique. Les approches hiérarchiques comme les matrices H2, en particulier la méthode multipôle rapide (FMM), bénéficient de la structure de rang faible par bloc de certaines matrices pour réduire le coût de calcul de problèmes d’interactions à n-corps en O(n) opérations au lieu de O(n2). Afin de mieux traiter des noyaux d’interaction complexes de plusieurs natures, des formulations FMM dites ”kernel-independent” ont récemment vu le jour, telles que les FMM basées sur l’interpolation polynomiale. Cependant elles deviennent très coûteuses pour les noyaux tensoriels à fortes dimensions, c’est pourquoi nous avons développé une nouvelle formulation FMM efficace basée sur l’interpolation polynomiale, appelée Uniform FMM. Cette méthode a été implémentée dans la bibliothèque parallèle ScalFMM et repose sur une grille d’interpolation régulière et la transformée de Fourier rapide (FFT). Ses performances et sa précision ont été comparées à celles de la FMM par interpolation de Chebyshev. Des simulations numériques sur des cas tests artificiels ont montré que la perte de précision induite par le schéma d’interpolation était largement compensées par le gain de performance apporté par la FFT. Dans un premier temps, nous avons étendu les FMM basées sur grille de Chebyshev et sur grille régulière au calcul des champs élastiques isotropes mis en jeu dans des simulations de Dynamique des Dislocations (DD). Dans un second temps, nous avons utilisé notre nouvelle FMM pour accélérer une factorisation SVD de rang r par projection aléatoire et ainsi permettre de générer efficacement des champs Gaussiens aléatoires sur de grandes grilles hétérogènes. Pour finir, nous avons développé un algorithme de réduction de dimension basé sur la projection aléatoire dense afin d’étudier de nouvelles façons de caractériser la biodiversité, à savoir d’un point de vue géométrique. / Advanced techniques for the low-rank approximation of matrices are crucial dimension reduction tools in many domains of modern scientific computing. Hierarchical approaches like H2-matrices, in particular the Fast Multipole Method (FMM), benefit from the block low-rank structure of certain matrices to reduce the cost of computing n-body problems to O(n) operations instead of O(n2). In order to better deal with kernels of various kinds, kernel independent FMM formulations have recently arisen such as polynomial interpolation based FMM. However, they are hardly tractable to high dimensional tensorial kernels, therefore we designed a new highly efficient interpolation based FMM, called the Uniform FMM, and implemented it in the parallel library ScalFMM. The method relies on an equispaced interpolation grid and the Fast Fourier Transform (FFT). Performance and accuracy were compared with the Chebyshev interpolation based FMM. Numerical experiments on artificial benchmarks showed that the loss of accuracy induced by the interpolation scheme was largely compensated by the FFT optimization. First of all, we extended both interpolation based FMM to the computation of the isotropic elastic fields involved in Dislocation Dynamics (DD) simulations. Second of all, we used our new FMM algorithm to accelerate a rank-r Randomized SVD and thus efficiently generate multivariate Gaussian random variables on large heterogeneous grids in O(n) operations. Finally, we designed a new efficient dimensionality reduction algorithm based on dense random projection in order to investigate new ways of characterizing the biodiversity, namely from a geometric point of view.

Ce travail est divisé en deux parties: Dans la première partie, on présente un résultat d'intégration dans les espaces localement convexes valable pour une longe classe des fonctions non-convexes. Cela nous permet de récupérer l'enveloppe convexe fermée d'une fonction à partir du sous-différentiel convexe de cette fonction. Motivé par ce résultat, on introduit la classe des espaces ``Subdifferential Dense Primal Determined'' (SDPD). Ces espaces jouissent des conditions nécessaires permettant d'appliquer le résultat ci-dessus. On donne aussi une interprétation géométrique de ces espaces, appelée la Propriété Radon-Nikod'ym de Faces (FRNP). Dans la seconde partie, on étudie dans le contexte d'espaces d'Hilbert, la relation entre la lissité de la frontière d'un ensemble prox-régulier et la lissité de sa projection métrique. On montre que si un corps fermé possède une frontière $mathcal{C}^{p+1}$-lisse (avec $pgeq 1$), alors sa projection métrique est de classe $mathcal{C}^p$ dans le tube ouvert associé à sa fonction de prox-régularité. On établit également une version locale du même résultat reliant la lissité de la frontière autour d'un point à la prox-régularité en ce point. On étudie par ailleurs le cas où l'ensemble est lui-même une $mathcal{C}^{p+1}$-sous-variété. Finalement, on donne des réciproques de ces résultats. / This work is divided in two parts: In the first part, we present an integration result in locally convex spaces for a large class of nonconvex functions which enables us to recover the closed convex envelope of a function from its convex subdifferential. Motivated by this, we introduce the class of Subdifferential Dense Primal Determined (SDPD) spaces, which are those having the necessary condition which allows to use the above integration scheme, and we study several properties of it in the context of Banach spaces. We provide a geometric interpretation of it, called the Faces Radon-Nikod'ym property. In the second part, we study, in the context of Hilbert spaces, the relation between the smoothness of the boundary of a prox-regular set and the smoothness of its metric projection. We show that whenever a set is a closed body with a $mathcal{C}^{p+1}$-smooth boundary (with $pgeq 1$), then its metric projection is of class $mathcal{C}^{p}$ in the open tube associated to its prox-regular function. A local version of the same result is established as well, namely, when the smoothness of the boundary and the prox-regularity of the set are assumed only near a fixed point. We also study the case when the set is itself a $mathcal{C}^{p+1}$-submanifold. Finally, we provide converses for these results.

This project studied two different ways of imposing boundary conditions weakly with the finite difference summation-by-parts (SBP) operators. These operators were combined with the boundary handling methods of simultaneous-approximation-terms (SAT) and the Projection to impose homogeneous Neumann and Dirichlet boundary conditions. The convergence rate of both methods was analyzed for different boundary conditions for the one-dimensional (1D) Schrödinger equation, without potential, which resulted in both methods performing similarly. A multi-block discretization was then implemented and different combinations of SBP-SAT and SBP-Projection were applied to impose inner boundary conditions of continuity between the blocks. A convergence study of the different methods of imposing the inner BC:s was conducted for the 1D Schrödinger equation without potential. The resulting convergence was the same for all methods and it was concluded that they performed similarly. Methods involving SBP-Projection had the slight advantage of faster computation time. Finally, the 1D Gross-Pitaevskii equation (GPE) and the 1D Schrödinger equation were analyzed with a step potential. The waves propagating towards the potential barrier were in both cases partially transmitted and partially reflected. The waves simulated with the Schrödinger equation dispersed, while the solitons simulated with the GPE kept their shape due to the equations reinforcing non-linear term. The bright soliton was partly transmitted and partly reflected. The dark soliton was either totally reflected or totally transmitted.