Bound state systems consisting of three nonrelativistic particles are numerically studied. Calculations are performed employing two-body and three-body forces as input in the Hamiltonian in order to study the role or contribution of three-body forces to the binding in these systems. The resulting differential Faddeev equations are solved as three-dimensional equations in the two Jacobi coordinates and the angle between them, as opposed to the usual partial wave expansion approach. By expanding the wave function as a sum of the products of spline functions in each of the three coordinates, and using the orthogonal collocation procedure, the equations are transformed into an eigenvalue problem. The matrices in the aforementioned eigenvalue equations are generally of large order. In order to solve these matrix equations with modest and optimal computer memory and storage, we employ the iterative Restarted Arnoldi Algorithm in conjunction with the so-called tensor trick method. Furthermore, we incorporate a polynomial accelerator in the algorithm to obtain rapid convergence. We applied the method to obtain the binding energies of Triton, Carbon-12, and Ozone molecule. / Physics / M.Sc (Physics)

Orientador: Eliana Xavier Linhares de Andrade / Banca: Alagacone Sri Ranga / Banca: Andre Piranhe da Silva / Resumo: Este trabalho trata do estudo da localização dos zeros dos polinômios gerados por uma determinada relação de recorrência de três termos. O objetivo principal é estudar limitantes, em termos dos coeficientes da relação de recorrência, para as regiões onde os zeros estão localizados. Os zeros são explorados atravé do problema de autovalor associado a uma matriz de Hessenberg. As aplicações são consideradas para polinômios de Szeg"o fSng, alguns polinômios para- ortogonais ½Sn(z) + S¤n (z) 1 + Sn(0) ¾ e ½Sn(z) ¡ S¤n (z) 1 ¡ Sn+1(0) ¾, especialmente quando os coeficientes de reflexão são reais. Um outro caso especial considerado são os zeros do polinômio Pn(z) = n Xm=0 bmzm, onde os coeficientes bm; para m = 0; 1; : : : ; n, são complexos e diferentes de zeros. / Abstract: In this work we studied the localization the zeros of polynomials generated by a certain three term recurrence relation. The main objective is to study bounds, in terms of the coe±cients of the recurrence relation, for the regions where the zeros are located. The zeros are explored through an eigenvalue representation associated with a Hessenberg matrix. Applications are considered to Szeg}o polynomials fSng, some para-orthogonal polyno- mials ½Sn(z) + S¤n (z) 1 + Sn(0) ¾and ½Sn(z) ¡ S¤n (z) 1 ¡ Sn+1(0) ¾, especially when the re°ection coe±cients are real. As another special case, the zeros of the polynomial Pn(z) = n Xm=0 bmzm, where the non-zero complex coe±cients bm for m = 0; 1; : : : ; n, were considered. / Mestre

A investigação dos mecanismos de instabilidade pode ter um papel importante no entendimento do processo laminar para turbulento de um escoamento. Análise de instabilidade de uma camada limite de uma linha de estagnação compressível foi realizada no contexto de teoria linear BiGlobal. O estudo dos mecanismos de instabilidade deste escoamento pode proporcionar uma visão útil no desenho aerodinâmico das asas. Um novo procedimento foi desenvolvido e implementado computacionalmente de maneira sequencial e paralela para o estudo de instabilidade BiGlobal. O mesmo baseia-se em formar a matriz esparsa associada ao problema discretizado por dois métodos: pontos de colocação de Chebyshev-Gauss-Lobatto e diferenças finitas, além das combinações destes métodos. Isto permitiu o uso de bibliotecas computacionais eficientes para resolver o sistema linear associado ao problema de autovalor utilizando o algoritmo de Arnoldi. O desempenho do método numérico e código computacional proposto são analisados do ponto de vista do uso de métodos de ordenação dos elementos da matriz, coeficientes de preenchimento, memória e tempo computacional a fim de determinar a solução mais eficiente para um problema físico geral com técnicas de matrizes esparsas. Um estudo paramétrico da instabilidade da camada limite de uma linha de estagnação foi realizado incluindo o estudo dos efeitos de compressibilidade. O excelente desempenho código computacional permitiu obter as curvas neutras e seus respectivos valores críticos para a faixa de número de Mach 0 \'< OU =\' Ma \'< OU =\' 1. Os resultados confirmam a teoria assintótica apresentada por (THEOFILIS; FEDOROV; COLLIS, 2004) e mostram que o incremento do número de Mach reduz o numero de Reynolds crítico e a faixa instável do número de ondas. / Investigation of linear instability mechanisms is essential for understanding the process of transition from laminar to turbulent flow. An algorithm for the numerical solution of the compressible BiGlobal eigenvalue problem is developed. This algorithm exploits the sparsity of the matrices resulting from the spatial discretization of the enigenvalue problem in order to improve the performance in terms of both memory and CPU time over previous dense algebra solutions. Spectral collocation and finite differences spatial discretization methods are implemented, and a performance study is carried out in order to determine the best practice for the efficient solution of a general physical problem with sparse matrix techniques. A combination of spectral collocation and finite differences can further improve the performance. The code developed is then applied in order to revisit and complete the parametric analyses on global instability of the compressible swept Hiemenz flow initiated in (THEOFILIS; FEDOROV; COLLIS, 2004) and obtain neutral curves of this flow as a function of the Mach number in the 0 \'< OU =\' Ma \'< OU =\' 1 range. The present numerical results fully confirm the asymptotic theory results presented in (THEOFILIS; FEDOROV; COLLIS, 2004). This work presents a complete parametric study of the instability properties of modal three dimensional disturbances in the subsonic range for the flow conguration at hand. Up to the subsonic maximum Mach number value studied, it is found that an increase in this parameter reduces the critical Reynolds number and the range of the unstable spanwise wavenumbers.

O principal objetivo deste trabalho 6 estudar o comportamento dos zeros de polinômios ortogonais e similares. Inicialmente, consideramos uma relação entre duas sequências ele polinômios ortogonais, onde as medidas associadas estão relacionadas entre si. Usamos esta relação para estudar as propriedades de monotonicidade dos zeros dos polinômios ortogonais relacionados a uma medida obtida através da generalização da medida associada a uma outra sequência de polinômios ortogonais. Apresentamos, como exemplos, os polinômios ortogonais obtidos a partir da generalização das medidas associadas aos polinômios de Jacobi, Laguerre e Charlier. Em urna segunda etapa, consideramos polinômios gerados por uma certa relação de recorrência de três termos com o objetivo de encontrar limitantes, em termos dos coeficientes da relação de recorrência, para as regiões onde os zeros estão localizados. Os zeros são estudados através do problema de autovalor associado a uma matriz de Hessenberg. Aplicações aos polinômios de Szegó, polinômios para-ortogonais e polinômios com coeficientes complexos não-nulos são consideradas. / The main purpose of this work is to study the behavior of the zeros of orthogonal and similar polynomials. Initially, we consider a relation between two sequences of orthogonal polynomials, where the associated measures are related to each other. We use this relation to study the monotonicity propertios of the zeros of orthogonal polynomials related with a measure obtained through a generalization of the measure associated with other sequence of orthogonal polynomials. As examples, we consider the orthogonal polynomials obtained in this way from the measures associated with the Jacobi, Laguerre and Charlier polynomials. We also consider the zeros of polynomials generated by a certain three term recurrence relation. Here, the main objective is to find bounds, in terms of the coefficients of the recurrence relation, for the regions where the zeros are located. The zeros are explored through an eigenvalue representation associated with a Hessenberg matrix. Applications to Szegõ polynomials, para-orthogonal polynomials anti polynomials with non-zero complex coefficients are considered.

L'objectif de cette thèse est de fournir des bornes d'erreur pour des problèmes aux valeurs propres linéaires et non linéaires issus du calcul de structure électronique, en particulier celui de l'état fondamental avec la théorie de la fonctionnelle de la densité. Ces bornes d'erreur reposent principalement sur des estimations a posteriori. D'abord, nous étudions un phénomène de compensation d'erreur de discrétisation pour un problème linéaire aux valeurs propres, grâce à une analyse a priori de l'erreur sur l'énergie. Ensuite, nous présentons une analyse a posteriori pour le problème du laplacien aux valeurs propres discrétisé par une large classe d'éléments finis. Les bornes d'erreur proposées pour les valeurs propres simples et leurs vecteurs propres associés sont garanties, calculables et efficaces. Nous nous concentrons alors sur des problèmes aux valeurs propres non linéaires. Nous proposons des bornes d'erreur pour l'équation de Gross-Pitaevskii, valables sous des hypothèses vérifiables numériquement, et pouvant être séparées en deux composantes venant respectivement de la discrétisation et de l'algorithme itératif utilisé pour résoudre le problème non linéaire aux valeurs propres. L'équilibrage de ces composantes d'erreur permet d'optimiser les ressources numériques. Enfin, nous présentons une méthode de post-traitement pour le problème de Kohn-Sham discrétisé en ondes planes, améliorant la précision des résultats à un faible coût de calcul. Les solutions post-traitées peuvent être utilisées soit comme solutions plus précises du problème, soit pour calculer une estimation de l'erreur de discrétisation, qui n'est plus garantie, mais néanmoins proche de l'erreur. / The objective of this thesis is to provide error bounds for linear and nonlinear eigenvalue problems arising from electronic structure calculation. We focus on ground-state calculations based on Density Functional Theory, including Kohn-Sham models. Our bounds mostly rely on a posteriori error analysis. More precisely, we start by studying a phenomenon of discretization error cancellation for a simple linear eigenvalue problem, for which analytical solutions are available. The mathematical study is based on an a priori analysis for the energy error. Then, we present an a posteriori analysis for the Laplace eigenvalue problem discretized with finite elements. For simple eigenvalues of the Laplace operator and their corresponding eigenvectors , we provide guaranteed, fully computable and efficient error bounds. Thereafter, we focus on nonlinear eigenvalue problems. First, we provide an a posteriori analysis for the Gross-Pitaevskii equation. The error bounds are valid under assumptions that can be numerically checked, and can be separated in two components coming respectively from the discretization and the iterative algorithm used to solve the nonlinear eigenvalue problem. Balancing these error components allows to optimize the computational resources. Second, we present a post-processing method for the Kohn-Sham problem, which improves the accuracy of planewave computations of ground state orbitals at a low computational cost. The post-processed solutions can be used either as a more precise solution of the problem, or used for computing an estimation of the discretization error. This estimation is not guaranteed, but in practice close to the real error.

This Master’s thesis deals with the computational modeling of complex natural frequencies of the tram wheels during cornering. The aim of this work is to determine eigenvalues, perform analysis of the influence of different parameters on eigenvalues and perform harmonic response analysis in order to find surface velocities that can be used in future noise emission analysis.

L'optimisation topologique vise à concevoir une structure en recherchant la disposition optimale du matériau dans un espace de conception donné, permettant ainsi de proposer des designs optimaux innovants. Cette thèse est centrée sur l'optimisation topologique pour des problèmes de conception de structures prenant en compte des contraintes de flambage. Dans une large variété de domaines de l'ingénierie, la conception innovante de structures est cruciale. L'allègement des structures lors la phase de conception tient une place prépondérante afin de réduire les coûts de fabrication. Ainsi l'objectif est souvent la minimisation de la masse de la structure à concevoir. En ce qui concerne les contraintes, en plus des contraintes mécaniques classiques (compression, tension), il est nécessaire de prendre en compte des phénomènes dits de flambage, qui se caractérisent par une amplification des déformations de la structure et une potentielle annihilation des capacités de la structure à supporter les efforts appliqués. Dans le but d'adresser un large panel de problèmes d'optimisation topologique, nous considérons les deux types de représentation d'une structure : les structures treillis et les structures continues. Dans le cadre de structures treillis, l'objectif est de minimiser la masse en optimisant le nombre d'éléments de la structure et les dimensions des sections transversales associées à ces éléments. Nous considérons les structures constituées d'éléments poutres et nous introduisons une formulation du problème comme un problème d'optimisation non-linéaire en variables mixtes. Afin de prendre en compte des contraintes de manufacturabilité, nous proposons une fonction coût combinant la masse et la somme des seconds moments d'inertie de chaque poutre. Nous avons développé un algorithme adapté au problème d'optimisation considéré. Les résultats numériques montrent que l'approche proposée mène à des gains de masses significatifs par rapport à des approches existantes. Dans le cas des structures continues, l'optimisation topologique vise à discrétiser le domaine de conception et à déterminer les éléments de ce domaine discrétisé qui doivent être composés de matière, définissant ainsi un problème d'optimisation discret. [...] / Topology optimization aims to design a structure by seeking the optimal material layout within a given design space, thus making it possible to propose innovative optimal designs. This thesis focuses on topology optimization for structural problems taking into account buckling constraints. In a wide variety of engineering fields, innovative structural design is crucial. The lightening of structures during the design phase holds a prominent place in order to reduce manufacturing costs. Thus the goal is often the minimization of the mass of the structure to be designed. Regarding the constraints, in addition to the conventional mechanical constraints (compression, tension), it is necessary to take into account buckling phenomena which are characterized by an amplification of the deformations of the structure and a potential annihilation of the capabilities of the structure to support the applied efforts. In order to adress a wide range of topology optimization problems, we consider the two types of representation of a structure: lattice structures and continuous structures. In the framework of lattice structures, the objective is to minimize the mass by optimizing the number of elements of the structure and the dimensions of the cross sections associated to these elements. We consider structures constituted by a set of frame elements and we introduce a formulation of the problem as a mixed-integer nonlinear problem. In order to obtain a manufacturable structure, we propose a cost function combining the mass and the sum of the second moments of inertia of each frame. We developed an algorithm adapted to the considered optimization problem. The numerical results show that the proposed approach leads to significant mass gains over existing approaches. In the case of continuous structures, topology optimization aims to discretize the design domain and to determine the elements of this discretized domain that must be composed of material, thus defining a discrete optimization problem. [...]

The field of statistics is becoming increasingly more important as the amount of data in the world grows. This thesis studies the Growth Curve model in multivariate statistics which is a model that is not widely used. One difference compared with the linear model is that the Maximum Likelihood Estimators are more complicated. That makes it more difficult to use and to interpret which may be a reason for its not so widespread use. From this perspective this thesis will compare the traditional mean estimator for the Growth Curve model with the unweighted mean estimator. The unweighted mean estimator is simpler than the regular MLE. It will be proven that the unweighted estimator is in fact the MLE under certain conditions and examples when this occurs will be discussed. In a more general setting this thesis will present conditions when the un-weighted estimator has a smaller covariance matrix than the MLEs and also present confidence intervals and hypothesis testing based on these inequalities.

The SHIRA method of Mehrmann and Watkins belongs among the structure preserving Krylov subspace methods for solving skew-Hamiltonian eigenvalue problems. It can also be applied to Hamiltonian eigenproblems by considering a suitable transformation. Structure induced shift-and-invert techniques are employed to steer the algorithm towards the interesting region of the spectrum. However, the shift cannot be altered in the middle of the computation without discarding the information that has been accumulated so far. This paper shows how SHIRA can be combined with ideas from Ruhe's Rational Krylov algorithm to yield a method that permits an adjustment of shift after every step of the computation, adding greatly to the flexibility of the algorithm. We call this new method rational SHIRA. A numerical example is presented to demonstrate its efficiency.

Ce mémoire traite de l’harmonicitié d’instruments de musique à travers la géométrie spectrale. Nous y présentons, en premier lieu, les résultats connus concernant la corde de guitare, le tambour circulaire et puis le tabla ; le premier est harmonique, le deuxième ne l’est pas et puis le dernier s’en approche. Le cas de la timbale est ce qui constitue la majeure partie de notre travail. L’ingénieur-physicien Robert E. Davis en avait déjà étudié la quasi-harmonicité et nous faisons ici une relecture mathématique de sa démarche. En alliant les méthodes analytiques et numériques, nous montrons que la caisse de résonance de la timbale permet à la fois d’ajuster les fréquences de vibration de la forme ω_(i1) , avec 1 ≤ i ≤ 5, afin qu’elles s’approchent du rapport idéal 2 : 3 : 4 : 5 : 6, et elle permet aussi d’étouffer certains autres modes dissonants. Pour ce faire, nous élaborons un modèle simplifié de timbale cylindrique basé sur la physique et sur ce que propose Davis dans sa thèse. Ce modèle nous fournit un système d’équations divisé en trois parties : la vibration de la peau et la pression à l’intérieur et à l’extérieur de la timbale. Nous utilisons la méthode des fonctions de Green pour trouver les expressions des deux pressions. Nous nous servons de celles-ci ainsi que d’un développement en série de Fourier-Bessel modifiée pour résoudre les équations de la vibration de la peau. La résolution de ces équations se ramène finalement à celle d’un système matriciel infini dont nous faisons l’analyse numériquement. À l’aide de Mathématica et de ce système matriciel, nous trouvons les fréquences de vibration de la timbale, ce qui nous permet d’analyser l’harmonicité de l’instrument. Grâce à une mesure de dissonance, nous optimisons l’harmonicité de la timbale en fonction du rayon du cylindre, de sa hauteur et de la tension. / This thesis deals with the harmonicity of musical instruments through spectral geometry. First, we present the known results concerning the guitar string, the circular drum and the tabla ; the first is harmonic, the second is not, and the last is somewhere in between. The case of the timpani constitutes the major part of our work. The physicist-engineer Robert E. Davis had already studied its quasi-harmonicity and here we undergo a mathematical proofreading of his approach. By combining analytical and numerical methods, we show that the sound box of the timpani allows an adjustement of the vibration frequencies of the form ω_(i1) , with 1 ≤ i ≤ 5, so that they get close to the ideal 2 : 3 : 4 : 5 : 6 ratio, while it also stifles some other dissonant modes. To do so, we develop a simplified model of a cylindrical timpani based on physics and on what Davis suggests in his thesis. This model provides a system of equations divided into three parts : the vibration of the skin and the pressure inside and outside the timpani. We use the method of Green’s functions to find the expressions of the pressures. We use these together with a modified Fourier-Bessel series development to solve the equations of the vibration of the skin. In the end, the solving of these equations is reduced to an infinite matrix system that we analyze numerically. Using Mathematica and this matrix system, we find the vibrational frequencies of the timpani, which allows us to analyze the harmonicity of the instrument. Thanks to a measure of dissonance, we optimize the harmonicity of different timpani models with different cylinder radii, heights and tensions.