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.

[EN] The neutron diffusion equation is an approximation of the neutron transport equation that describes the neutron population in a nuclear reactor core. In particular, we will consider here VVER-type reactors which use the neutron diffusion equation discretized on hexagonal meshes. Most of the simulation codes of a nuclear power reactor use the multigroup neutron diffusion equation to describe the neutron distribution inside the reactor core.To study the stationary state of a reactor, the reactor criticality is forced in artificial way leading to a generalized differential eigenvalue problem, known as the Lambda modes equation, which is solved to obtain the dominant eigenvalues of the reactor and their corresponding eigenfunctions. To discretize this model a finite element method with h-p adaptivity is used. This method allows to use heterogeneous meshes, and allows different refinements such as the use of h-adaptive meshes, reducing the size of specific cells, and p-refinement, increasing the polynomial degree of the basic functions used in the expansions of the solution in the different cells. Once the solution for the steady state neutron distribution is obtained, it is used as initial condition for the time integration of the neutron diffusion equation. To simulate the behaviour of a nuclear power reactor it is necessary to be able to integrate the time-dependent neutron diffusion equation inside the reactor core. The spatial discretization of this equation is done using a finite element method that permits h-p refinements for different geometries. Transients involving the movement of the control rod banks have the problem known as the rod-cusping effect. Previous studies have usually approached the problem using a fixed mesh scheme defining averaged material properties and many techniques exist for the treatment of the rod cusping problem. The present work proposes the use of a moving mesh scheme that uses spatial meshes that change with the movement of the control rods avoiding the necessity of using equivalent material cross sections for the partially inserted cells. The performance of the moving mesh scheme is tested studying different benchmark problems. For reactor calculations, the accuracy of a diffusion theory solution is limited for for complex fuel assemblies or fine mesh calculations. To improve these results a method that incorporates higher-order approximations for the angular dependence, as the simplified spherical harmonics (SPN ) method must be employed. In this work an h-p Finite Element Method (FEM) is used to obtain the dominant Lambda mode associated with a configuration of a reactor core using the SPN approximation. The performance of the SPN (N= 1, 3, 5) approximations has been tested for different reactor benchmarks. / [ES] La ecuación de la difusión neutrónica es una aproximación de la ecuación del transporte de neutrones que describe la población de neutrones en el núcleo de un reactor nuclear. En particular, consideraremos reactores de tipo VVER y para simular su comportamiento se utilizará la ecuación de la difusión neutrónica para cuya discretización se hace uso de mallas hexagonales. La mayoría de los códigos de simulación de reactores nucleares utilizan aproximación multigrupo de energía de la ecuación de la difusión neutrónica para describir la distribución de neutrones en el interior del núcleo del reactor. Para estudiar el estado estacionario del reactor, es posible forzar la criticidad del reactor de forma artificial modificando las secciones eficaces de forma que se obtiene un problema de valores propios diferencial, conocido como el problema de los Modos Lambda, que se resuelve para obtener los valores propios dominantes del reactor y sus correspondientes funciones propias. Para discretizar este modelo se ha hecho uso de un método de elementos finitos con adaptabilidad h-p. Este método permite el uso de mallas heterogéneas, y de diferentes refinamientos como el uso mallas h-adaptativas, reduciendo el tamaño de los distintos nodos, y el p-refinado, aumentando el grado del polinomio de las funciones básicas utilizado en los desarrollos de la solución en los diferentes nodos. Se ha desarrollado un código basado en un método de elementos finitos de alto orden para resolver el problema de los Modos Lambda en un reactor con geometría hexagonal y se han obtenido los Modos dominantes para distintos problemas de referencia. Una vez que se ha obtenido la solución para la distribución de neutrones en estado estacionario, ésta se utiliza como condición inicial para la integración de la ecuación de difusión neutrónica dependiente del tiempo. Para simular el comportamiento de un reactor nuclear para un determinado transitorio, es necesario ser capaz de integrar la ecuación de la difusión neutrónica dependiente del tiempo en el interior del núcleo del reactor. La discretización espacial de esta ecuación se hace usando un método de elementos finitos de alto orden que permite refinados de tipo h-p para distintas geometrías. Los transitorios que implican el movimiento de los bancos de las barras de control tienen el problema conocido como el efecto 'rod-cusping'. Estudios anteriores, por lo general, han abordado este problema utilizando una malla fija y definiendo propiedades promedio para los materiales correspondientes a las celdas donde se tiene la barra de control parcialmente insertada. En el presente trabajo se propone el uso de un esquema de malla móvil, de forma que en mallado espacial va cambiando con el movimiento de la barra de control, evitando la necesidad de utilizar secciones eficaces equivalentes para las celdas parcialmente insertadas. El funcionamiento de este esquema de malla móvil propuesto se estudia resolviendo distintos problemas tipo. La precisión obtenida mediante de la teoría de la difusión en los cálculos de reactores es limitada cuando se tienen elementos de combustible complejos o se pretenden realizar cálculos en malla fina. Para mejorar estos resultados, es necesario disponer de un método que incorpore aproximaciones de orden superior de la ecuación del transporte de neutrones. Una posibilidad es hacer uso de las ecuaciones PN simplificadas (SPN ). En este trabajo se utiliza un método de elementos finitos h-p para obtener los modos dominantes asociados con una configuración dada del núcleo de un reactor nuclear con geometría hexagonal usando la aproximación SPN . El funcionamiento de las aproximaciones SPN (N = 1, 3, 5) se ha estudiado para distintos problemas de referencia. / [CAT] L'equació de la difusió neutrònica és una aproximació de l'equació del transport de neutrons que descriu la població de neutrons en el nucli de un reactor nuclear. En particular, considerarem reactors de tipus VVER i per a simular el seu comportament s'utilitzarà l'equació de la difusió neutrónica que es discretitza fent ús de malles hexagonals. La majoria dels codis de simulació de reactors nuclears utilitzen l'aproximació multigrup d'energia de l'equació de la difusió neutrónica per a descriure la distribució de neutrons a l'interior del nucli del reactor. Per a estudiar l'estat estacionari del reactor, és possible forçar la seua criticitat de forma artificial modificant les seccions eficaces de manera que s'obté un problema de valors propis diferencial, conegut com el problema dels Modes Lambda, que es resol per a obtenir els valors propis dominants del reactor i les seues corresponents funcions pròpies. Per a discretitzar aquest model s'ha fet ús d'un mètode d'elements finits amb adaptabilitat h-p. Aquest mètode permet l'ús de malles heterogènies, i de diferents refinaments com l'ús malles h-adaptatives, reduint la grandària dels diferents nodes, i el p-refinat, augmentant el grau del polinomi de les funcions bàsiques utilitzat en els desenvolupaments de la solució en els diferents nodes. S'ha desenvolupat un codi basat en un mètode d'elements finits d'alt ordre per a resoldre el problema dels Modes Lambda en un reactor amb geometria hexagonal i s'han obtingut els Modes dominants per a diferents problemes de referència. Una vegada que s'ha obtingut la solució per a la distribució de neutrons en estat estacionari, aquesta s'utilitza com a condició inicial per a la integració de l'equació de difusió neutrònica depenent del temps. Per a simular el comportament d'un reactor nuclear per a un determinat transitori, és necessari ser capaç d'integrar l'equació de la difusió neutrónica depenent del temps a l'interior del nucli del reactor. La discretitzación espacial d'aquesta equació es fa usant un mètode d'elements finits d'alt ordre que permet refinats de tipus h-p per a diferents geometries. Els transitoris que impliquen el moviment dels bancs de les barres de control tenen el problema conegut com l'efecte 'rod-cusping'. Estudis anteriors, en general, han abordat aquest problema utilitzant una malla fixa i definint propietats equivalents per als materials corresponents a les cel·les on es té la barra de control parcialment inserida. En el present treball es proposa l'ús d'un esquema de malla mòbil, de manera que en mallat espacial va canviant amb el moviment de la barra de control, evitant la necessitat d'utilitzar seccions eficaces equivalents per a les cel·les parcialment inserides. El funcionament de aquest esquema de malla mòbil s'estudia resolent diferents problemes tipus. La precisió obtinguda mitjançant de la teoria de la difusió en els càlculs de reactors és limitada quan es tenen elements de combustible complexos o es pretenen realitzar càlculs en malla fina. Per a millorar aquests resultats, és necessari disposar d'un mètode que incorpore aproximacions d'ordre superior de l'equació del transport de neutrons. Una possibilitat és fer ús de les equacions PN simplificades (SPN ). En aquest treball s'utilitza un mètode d'elements finits h- p per a obtenir els modes dominants associats amb una configuració donada del nucli de un reactor amb geometria hexagonal usant l'aproximació SPN . El funcionament de les aproximacions SPN (N = 1, 3, 5) s'ha estudiat per a diferents problemes de referència. / Fayez Moustafa Moawad, R. (2016). Approximation of The Neutron Diffusion Equation on Hexagonal Geometries Using a h-p finite element method [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/65353 / TESIS

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.