No description available.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23680号 / 理博第4770号 / 新制||理||1683(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 平岡 裕章, 教授 泉 正己, 教授 坂上 貴之 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

In a recent study of the geometric analysis of various masterpieces of many periods dating from early Egyptian to contemporary times, the author noted with particular interest the structure of the paintings of the German Renaissance masters. It seemed that the Germans used a simpler geometric plan in their compositions than did the Italian Renaissance painters. The writer was inspired to make further investigation to determine if such a theory were true.

John Lott berechnete eine Signatur mit ganzzahligen Werten für den Orbitraum einer kompakten, orientierbaren (4k + 1)-Mannigfaltigkeit mit einer halbfreien S1-Wirkung. Diese Signatur ist eine Homotopieinvariante für den Orbitraum. Allerdings konstruierte er keinen Operator vom Dirac-Typ, der die Signatur als Index besitzt. In dieser Arbeit konstruieren wir einen solchen Operator auf dem Orbitraum der S1-Wirkung, einem Thom-Mather stratifizierten Raum mit einem singulären Stratum von positiver Dimension, und wir zeigen, dass der Operator im wesentlichen eindeutig bestimmt ist. Ferner zeigen wir, dass sein Index mit Lotts Signatur übereinstimmt, zumindest wenn der stratifizierte Raum die sogenannte Witt-Bedingung erfüllt. Wirnennendiesen Operator den induzierten Dirac-Schrödinger Operator. Unsere Konstruktionsstrategie ist es, einen geeigneten S1-invarianten transversal elliptischen Operator erster Ordnung auf den S1-invarianten Differentialformen zu definieren, der den gesuchten Operator auf den Differentialformen des Orbitraums induziert. Die Witt-Bedingung, eine topologische Bedingung, welche in diesem Fall von der Kodimension der betrachteten Punktmenge abhängt, lässt verschiedene analytische Schlussfolgerungen zu. Insbesondere ist, wenn die Bedingung nicht erfüllt ist, der Hodge-de Rham Operator auf dem Quotientenraum nicht notwendigerweise essentiell selbstadjungiert und die Wahl einer Randbedingung ist daher notwendig. Diese Wahlfreiheit erscheint unnatürlich in Anbetracht der Tatsache, dass Lotts Signatur unabhängig von der Witt-Bedingung wohldefiniert ist. Der Dirac-Schrödinger Operator, der in dieser Arbeit konstruiert wird, unterschei- det sich vom Hodge-de Rham Operator durch einen Term nullter Ordnung, welcher sicherstellt, dass der Operator wesentlich selbstadjungiert ist. Außerdem antikommutiert dieser Term nullter Ordnung mit der Signatur-Involution, wodurch der gesamte Operator zerfällt und so der Index berechnet werden kann, auch wenn die Witt-Bedingung nicht erfüllt ist. / John Lott has computed an integer-valued signature for the orbit space of a compact orientable (4k + 1) manifold with a semi-free S1-action, which is a homotopy invariant of that space, but he did not construct a Dirac type operator which has this signature as its index. In this Thesis, we construct such operator on the orbit space, a Thom-Mather stratified space with one singular stratum of positive dimension, and we show that it is essentially unique and that its index coincides with Lott’s signature, at least when the stratified space satisfies the so called Witt condition. We call this operator the induced Dirac-Schrödinger operator. The strategy of the construction is to “push down” an appropriate S1-invariant first order transversally elliptic operator to the quotient space. The Witt condition, a topological condition which in this case depends on the codi- mension of the fixed point set, has various analytic consequences. In particular, when not satisfied, the Hodge-de Rham operator on the quotient space does not need to be essentially self-adjoint and therefore a choice of boundary conditions is required. This choice freedom is not natural in view of the fact that Lott’s signature is well defined independently of the Witt condition. The Dirac-Schrödinger operator constructed in this Thesis differs from the Hodge-de Rham operator by a zero order term which ensures it to be essentially self-adjoint. Moreover, this zero order term anti-commutes with the chirality involution allowing the whole operator to split so that the index can be computed even if the Witt condition is not satisfied.

S'il fallait résumer le sujet de cette thèse en une expression, cela pourrait être quelque chose comme: phénomènes de grande dimension (mais néanmoins finie) en théorie quantique de l'information. Cela étant dit, essayons toutefois de développer brièvement. La physique quantique a inéluctablement affaire à des objets de grande dimension. Partant de cette observation, il y a, en gros, deux stratégies qui peuvent être adoptées: ou bien essayer de ramener leur étude à celle de situations de plus petite dimension, ou bien essayer de comprendre quels sont les comportements universels précisément susceptibles d'émerger dans ce régime. Nous ne donnons ici notre préférence à aucune de ces deux attitudes, mais au contraire oscillons constamment entre l'une et l'autre. Notre but dans la première partie de ce manuscrit (Chapitres 5 et 6) est de réduire autant que possible la complexité de certains processus quantiques, tout en préservant, évidemment, leurs caractéristiques essentielles. Les deux types de processus auxquels nous nous intéressons sont les canaux quantiques et les mesures quantiques. Dans les deux cas, la complexité d'une transformation est mesurée par le nombre d'opérateurs nécessaires pour décrire son action, tandis que la proximité entre la transformation d'origine et son approximation est définie par le fait que, quel que soit l'état d'entrée, les deux états de sortie doivent être proches l'un de l'autre. Nous proposons des solutions universelles (basées sur des constructions aléatoires) à ces problèmes de compression de canaux quantiques et d'amenuisement de mesures quantiques, et nous prouvons leur optimalité. La deuxième partie de ce manuscrit (Chapitres 7, 8 et 9) est, au contraire, spécifiquement dédiée à l'analyse de systèmes quantiques de grande dimension et certains de leurs traits typiques. L'accent est mis sur les systèmes multi-partites et leurs propriétés ayant un lien avec l'intrication. Les principaux résultats auxquels nous aboutissons peuvent se résumer de la façon suivante: lorsque les dimensions des espaces sous-jacents augmentent, il est générique pour les états quantiques multi-partites d'être à peine distinguables par des observateurs locaux, et il est générique pour les relaxations de la notion de séparabilité d'en être des approximations très grossières. Sur le plan technique, ces assertions sont établies grâce à des estimations moyennes de suprema de processus gaussiens, combinées avec le phénomène de concentration de la mesure. Dans la troisième partie de ce manuscrit (Chapitres 10 et 11), nous revenons pour finir à notre état d'esprit de réduction de dimensionnalité. Cette fois pourtant, la stratégie est plutôt: pour chaque situation donnée, tenter d'utiliser au maximum les symétries qui lui sont inhérentes afin d'obtenir une simplification qui lui soit propre. En reliant de manière quantitative symétrie par permutation et indépendance, nous nous retrouvons en mesure de montrer le comportement multiplicatif de plusieurs quantités apparaissant en théorie quantique de l'information (fonctions de support d'ensembles d'états, probabilités de succès dans des jeux multi-joueurs non locaux etc.). L'outil principal que nous développons dans cette optique est un résultat de type de Finetti particulièrement malléable / If a one-phrase summary of the subject of this thesis were required, it would be something like: miscellaneous large (but finite) dimensional phenomena in quantum information theory. That said, it could nonetheless be helpful to briefly elaborate. Starting from the observation that quantum physics unavoidably has to deal with high dimensional objects, basically two routes can be taken: either try and reduce their study to that of lower dimensional ones, or try and understand what kind of universal properties might precisely emerge in this regime. We actually do not choose which of these two attitudes to follow here, and rather oscillate between one and the other. In the first part of this manuscript (Chapters 5 and 6), our aim is to reduce as much as possible the complexity of certain quantum processes, while of course still preserving their essential characteristics. The two types of processes we are interested in are quantum channels and quantum measurements. In both cases, complexity of a transformation is measured by the number of operators needed to describe its action, and proximity of the approximating transformation towards the original one is defined in terms of closeness between the two outputs, whatever the input. We propose universal ways of achieving our quantum channel compression and quantum measurement sparsification goals (based on random constructions) and prove their optimality. Oppositely, the second part of this manuscript (Chapters 7, 8 and 9) is specifically dedicated to the analysis of high dimensional quantum systems and some of their typical features. Stress is put on multipartite systems and on entanglement-related properties of theirs. We essentially establish the following: as the dimensions of the underlying spaces grow, being barely distinguishable by local observers is a generic trait of multipartite quantum states, and being very rough approximations of separability itself is a generic trait of separability relaxations. On the technical side, these statements stem mainly from average estimates for suprema of Gaussian processes, combined with the concentration of measure phenomenon. In the third part of this manuscript (Chapters 10 and 11), we eventually come back to a more dimensionality reduction state of mind. This time though, the strategy is to make use of the symmetries inherent to each particular situation we are looking at in order to derive a problem-dependent simplification. By quantitatively relating permutation symmetry and independence, we are able to show the multiplicative behavior of several quantities showing up in quantum information theory (such as support functions of sets of states, winning probabilities in multi-player non-local games etc.). The main tool we develop for that purpose is an adaptable de Finetti type result

Nesta tese, apresenta-se o método dos elementos finitos posicional descrito em um referencial Lagrangiano total dedicado à análise de instabilidade de estruturas tridimensionais. Três tipos de elementos finitos são implementados e testados, a saber: os elementos de barra simples, casca e barra geral. A análise de instabilidade para o elemento de barra simples é efetuada determinando-se os pontos críticos ao longo da trajetória de equilíbrio em grandes deslocamentos. Para se determinar essas trajetórias são utilizados os algoritmos de Newton-Raphson e arc-length. Este tipo de análise é particularmente importante na definição de estruturas multi-estáveis de uso crescente na indústria mecânica e aeroespacial. Para o estudo da instabilidade empregando-se os elementos finitos de casca e barra geral realizam-se as análises para pequenos níveis de carga e deslocamentos por meio do cálculo dos autovalores e autovetores da matriz de rigidez da estrutura. Avaliam-se também as trajetórias de equilíbrio em grandes deslocamentos considerando-se pequenas imperfeições na geometria dos elementos estruturais. Quando os elementos de casca são utilizados na modelagem de perfis estruturais esbeltos surgem naturalmente modos de falha locais associados à mudança de forma da seção transversal. Com a finalidade de inserir essas mobilidades no elemento de barra geral propõem-se uma metodologia que considera os aprimoramentos na cinemática da barra. Esses aprimoramentos são tratados como parâmetros nodais generalizados e estão associadas a intensidade da mudança de forma de seção transversal, incluindo os modos de empenamento. Descreve-se originalmente uma metodologia de decomposição da matriz Hessiana usada para o cálculo dos valores e vetores próprios em pequenos deslocamentos. Essa metodologia possui importância adicional pois é utilizada na preparação e avaliação do parâmetro de carga em cinemáticas alternativas da formulação posicional. Utiliza-se o algoritmo de Lanczos na determinação das cargas e modos de falha realizando-se chamadas a biblioteca ARPACK. Os algoritmos são testados em exemplos modelados com os elementos finitos propostos. Próximo aos pontos críticos realiza-se a separação da matriz Hessiana procurando-se possíveis modos de colapso da estrutura. Além dos modos de falha globais é possível se identificar os modos de falha locais e distorcionais. O equilíbrio do sistema mecânico é garantido pelo princípio da estacionariedade da energia potencial total. Nas análises com os elementos de casca e barra geral, a solução do sistema não-linear é obtida empregando-se o método incremental iterativo de Newton-Raphson. Os aprimoramentos sugeridos nesta pesquisa são acoplados ao código computacional utilizado pelo grupo de mecânica computacional do departamento de engenharia de estruturas, onde diversas funcionalidades estão disponíveis, como análise dinâmica e não-linearidade material. Exemplos selecionados são apresentados ao longo da tese para demonstrar a eficiência dos elementos propostos e a aplicabilidade da técnica. Por fim, são realizadas comparações com estratégia de solução já consagradas, como por exemplo: o método das faixas finitas e a teoria generalizadas de vigas. Os resultados obtidos justificam as contribuições originais da presente pesquisa destacando-se a contribuição da formulação posicional ao estudo da instabilidade das estruturas. / This thesis presents the positional finite element method in a total Lagrangian framework dedicated to instability analysis of the three-dimensional structures. Three types of finite elements are implemented and tested, namely: truss, shells and frames. The instability analysis for truss element is computed using equilibrium path in large displacements. The critical points are computed using Newton-Raphson and arc-length algorithm. This analysis is particularly important in the definition of multi-stable and large displacements structures widely used in mechanical and aerospace industry. For shell and frame geometrically non-linear finite elements, the instability phenomenon is studied from the eigenvalues and eigenvectors analysis for small levels of loads and displacements. It is also evaluate the equilibrium trajectories for large displacements, considering small imperfections in the geometry of the structure. When using the shell elements to model the frames structures local failure modes associated with changing of the cross section shape arise. In order to consider the mobility in frame element new improvements are propose in the kinematic. These improvements are treated as generalized nodal parameters and are associated with the intensity of the cross-sectional change, including warping. The originally methodology of decomposition of the Hessian matrix are described and used for calculating eigenvalues and eigenvectors of the stiffness matrix. This methodology has additional importance because it is used in the preparation and evaluation of load parameter in kinematic alternatives of the positional formulation. The Lanczos algorithm is used to determining the loads and failure modes, through calls to ARPACK library for calculating eigenvalues and eigenvectors. The algorithms are tested on the examples modeled by proposed finite elements. Near the critical point takes place the separation of the Hessian matrix for possible identification of the failure modes. In addition to global failure methods, local and distortion failure are captured by this methodology. The balance of the mechanical system is guaranteed by the stationarity of the total potential energy principle. In the analysis using shells and frames elements the solution of the nonlinear system is calculated using the iterative incremental Newton-Raphson method. The improvements suggested in this research are coupled to the computer code used by computational mechanics group of the structures engineering department, where several features are available like dynamic and plasticity analysis. Selected examples are presented throughout the thesis to demonstrate the efficiency of the proposed elements and applicability of the technique. Finally, comparisons are carried out with already established solving strategy such as the finite strip methods and the generalized beam theory. The results justified the original contributions of this research to study of unstable structures.

Large-eddy simulation (LES) of wind and wave forced oceanic turbulent boundary layers is performed using the residual-based variational multiscale method (RBVMS) and near-wall modeling. Wind and surface gravity wave forcing generates Langmuir turbulence characterized by Langmuir circulation (LC) with largest scales consisting of streamwise vortices aligned in the direction of the wind, acting as a secondary flow structure to the primary wind-driven component of the flow. The LES here is representative of a shallow water continental shelf flow (10 to 30 meters in depth) far from lateral boundaries in which LC engulfs the full depth of the water column and disrupts the bottom log layer. Field measurements indicate that occurrence of full-depth LC is typical during the passage of storms. The RBVMS method with quadratic NURBS (Non-Uniform Rational B-splines) with near-wall resolution is shown to possess good convergence characteristics for this flow. The use of near-wall modeling facilitates simulations with expanded domains over horizontal directions. Thus, these simulations are able to resolve multiple Langmuir cells permitting analysis of the interaction between the cells. Results in terms of velocity statistics are presented from simulations performed with various domain sizes and distinct near-wall treatments: (1) the classical treatment based on prescription of the wall shear stress assuming a law of the wall and (2) a recent treatment based on weak imposition of the no-slip bottom boundary condition.

The spans of precast prestressed concrete bridge girders have become longer to provide more economical and safer transportation structures. As the spans have increased, so has the depth of the girders which in turn have increased the slenderness of the girders. Slenderness in a beam or girder would increase the likelihood that a stability failure would occur. Stability failures could pose a danger to construction personnel due to the sudden nature in which a stability failure would occur. Furthermore, stability failures of prestressed concrete girders during construction would cause a detrimental economic impact due to the costs associated with the failure of the girder, the ensuing construction delays, damage to construction equipment and potential closures to highways over which the bridge was being constructed. An experimental and analytical study was performed to determine the stability behavior of prestressed concrete beams. Two stability phenomenons were investigated: (1) lateral-torsional buckling and (2) global stability. An emphasis was placed on the effects of initial imperfections on the stability behavior; the effect elastomeric bearing pads and support rotational stiffness was investigated. The experimental study involved testing six rectangular prestressed concrete beams for lateral-torsional buckling, a PCI BT-54 for thermal deformations and the same PCI BT-54 for global stability. The 32-ft. long rectangular beams were 4-in. wide and 40-in. deep. The PCI BT-54 had a 100-ft. long span. A material and geometric nonlinear, incremental load analysis was performed on the six rectangular beams. The nonlinear analyses matched the experimental load versus lateral displacement and load versus rotation behavior, and the analysis predicted the experimental maximum load within an error of 2%. The nonlinear analysis was extrapolated to several different initial imperfection conditions to parametrically study the effect of initial lateral displacement and initial rotation on the inelastic lateral-torsional buckling load. A simplified expression for lateral-torsional stability of beams with initial imperfections was developed. The data from the parametric study were used to develop reduction parameters for both initial sweep and initial rotation. The rollover stability behavior of the PCI BT-54 was investigated experimentally, and it was found that support end rotations and the elastomeric bearing pads had an adverse effect on the global stability. The nonlinear analysis was employed with the addition of a bearing pad model. It was found that the behavior was sensitive to the bearing pad stiffness properties and the assumption of uniform bearing. From the research, it was apparent that rollover stability was the controlling stability phenomenon for precast prestressed concrete bridge girders, not lateral-torsional buckling.

Nesta tese, apresenta-se o método dos elementos finitos posicional descrito em um referencial Lagrangiano total dedicado à análise de instabilidade de estruturas tridimensionais. Três tipos de elementos finitos são implementados e testados, a saber: os elementos de barra simples, casca e barra geral. A análise de instabilidade para o elemento de barra simples é efetuada determinando-se os pontos críticos ao longo da trajetória de equilíbrio em grandes deslocamentos. Para se determinar essas trajetórias são utilizados os algoritmos de Newton-Raphson e arc-length. Este tipo de análise é particularmente importante na definição de estruturas multi-estáveis de uso crescente na indústria mecânica e aeroespacial. Para o estudo da instabilidade empregando-se os elementos finitos de casca e barra geral realizam-se as análises para pequenos níveis de carga e deslocamentos por meio do cálculo dos autovalores e autovetores da matriz de rigidez da estrutura. Avaliam-se também as trajetórias de equilíbrio em grandes deslocamentos considerando-se pequenas imperfeições na geometria dos elementos estruturais. Quando os elementos de casca são utilizados na modelagem de perfis estruturais esbeltos surgem naturalmente modos de falha locais associados à mudança de forma da seção transversal. Com a finalidade de inserir essas mobilidades no elemento de barra geral propõem-se uma metodologia que considera os aprimoramentos na cinemática da barra. Esses aprimoramentos são tratados como parâmetros nodais generalizados e estão associadas a intensidade da mudança de forma de seção transversal, incluindo os modos de empenamento. Descreve-se originalmente uma metodologia de decomposição da matriz Hessiana usada para o cálculo dos valores e vetores próprios em pequenos deslocamentos. Essa metodologia possui importância adicional pois é utilizada na preparação e avaliação do parâmetro de carga em cinemáticas alternativas da formulação posicional. Utiliza-se o algoritmo de Lanczos na determinação das cargas e modos de falha realizando-se chamadas a biblioteca ARPACK. Os algoritmos são testados em exemplos modelados com os elementos finitos propostos. Próximo aos pontos críticos realiza-se a separação da matriz Hessiana procurando-se possíveis modos de colapso da estrutura. Além dos modos de falha globais é possível se identificar os modos de falha locais e distorcionais. O equilíbrio do sistema mecânico é garantido pelo princípio da estacionariedade da energia potencial total. Nas análises com os elementos de casca e barra geral, a solução do sistema não-linear é obtida empregando-se o método incremental iterativo de Newton-Raphson. Os aprimoramentos sugeridos nesta pesquisa são acoplados ao código computacional utilizado pelo grupo de mecânica computacional do departamento de engenharia de estruturas, onde diversas funcionalidades estão disponíveis, como análise dinâmica e não-linearidade material. Exemplos selecionados são apresentados ao longo da tese para demonstrar a eficiência dos elementos propostos e a aplicabilidade da técnica. Por fim, são realizadas comparações com estratégia de solução já consagradas, como por exemplo: o método das faixas finitas e a teoria generalizadas de vigas. Os resultados obtidos justificam as contribuições originais da presente pesquisa destacando-se a contribuição da formulação posicional ao estudo da instabilidade das estruturas. / This thesis presents the positional finite element method in a total Lagrangian framework dedicated to instability analysis of the three-dimensional structures. Three types of finite elements are implemented and tested, namely: truss, shells and frames. The instability analysis for truss element is computed using equilibrium path in large displacements. The critical points are computed using Newton-Raphson and arc-length algorithm. This analysis is particularly important in the definition of multi-stable and large displacements structures widely used in mechanical and aerospace industry. For shell and frame geometrically non-linear finite elements, the instability phenomenon is studied from the eigenvalues and eigenvectors analysis for small levels of loads and displacements. It is also evaluate the equilibrium trajectories for large displacements, considering small imperfections in the geometry of the structure. When using the shell elements to model the frames structures local failure modes associated with changing of the cross section shape arise. In order to consider the mobility in frame element new improvements are propose in the kinematic. These improvements are treated as generalized nodal parameters and are associated with the intensity of the cross-sectional change, including warping. The originally methodology of decomposition of the Hessian matrix are described and used for calculating eigenvalues and eigenvectors of the stiffness matrix. This methodology has additional importance because it is used in the preparation and evaluation of load parameter in kinematic alternatives of the positional formulation. The Lanczos algorithm is used to determining the loads and failure modes, through calls to ARPACK library for calculating eigenvalues and eigenvectors. The algorithms are tested on the examples modeled by proposed finite elements. Near the critical point takes place the separation of the Hessian matrix for possible identification of the failure modes. In addition to global failure methods, local and distortion failure are captured by this methodology. The balance of the mechanical system is guaranteed by the stationarity of the total potential energy principle. In the analysis using shells and frames elements the solution of the nonlinear system is calculated using the iterative incremental Newton-Raphson method. The improvements suggested in this research are coupled to the computer code used by computational mechanics group of the structures engineering department, where several features are available like dynamic and plasticity analysis. Selected examples are presented throughout the thesis to demonstrate the efficiency of the proposed elements and applicability of the technique. Finally, comparisons are carried out with already established solving strategy such as the finite strip methods and the generalized beam theory. The results justified the original contributions of this research to study of unstable structures.

No description available.