Title: Numerical modeling of free oscillations applied to superconducting-gravimeter data in a low-frequency seismic range Author: Eliška Zábranová Department: Department of Geophysics Supervisor: Doc. RNDr. Ctirad Matyska, DrSc. Abstract: Deformations and changes of the gravitational potential of prestressed selfgravitating elastic bodies caused by free oscillations are described by means of the momentum and Poisson equations and the constitutive relation. For spheri- cally symmetric bodies we transform the equations and boundary conditions into ordinary differential equations of the second order by the spherical harmonic de- composition and further discretize the equations by highly accurate pseudospectral difference schemes on Chebyshev grids. We thus receive a series of matrix eigenvalue problems for eigenfrequencies and eigenfunctions of the free oscillations. Since elas- tic parameters are frequency dependent, we solve the problem for several fiducial frequencies and interpolate the results. Both the mode frequencies and the eigen- functions are benchmarked against the output from the Mineos software package based on Runge-Kutta integration techniques. Subsequently, we use our method to calculate low-frequency synthetic accelerograms of the recent megathrust events and compare them with the observed...

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'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 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.

[pt] Neste trabalho, são apresentadas algumas aplicações da otimização topológica para problemas de autovalor onde o principal objetivo é maximizar um determinado autovalor, como por exemplo uma frequência natural de vibração ou uma carga crítica linearizada, usando elementos finitos poligonais em domínios bidimensionais arbitrários. A otimização topológica tem sido comumente utilizada para minimizar a flexibilidade de estruturas sujeitas a restrições de volume. A ideia desta técnica é distribuir uma certa quantidade de material em uma estrutura, sujeita a carregamentos e condições de contorno, visando maximizar a sua rigidez. Neste trabalho, o objetivo é obter uma distribuição ótima de material de maneira a maximizar uma determinada frequência natural (para mantê-la afastada da frequência de excitação externa, por exemplo) ou maximizar a menor carga crítica linearizada (para garantir um nível mais elevado de estabilidade da estrutura). Malhas poligonais construídas usando diagramas de Voronoi são empregadas na solução do problema de otimização topológica. As variáveis de projeto, i.e. as densidades do material, utilizadas no processo de otimização, são associadas a cada elemento poligonal da malha. Vários exemplos de otimização topológica, tanto para problemas de frequências naturais de vibração quanto para cargas críticas linearizadas, são apresentados para demonstrar a funcionalidade e a aplicabilidade da metodologia proposta. / [en] In this work, we present some applications of topology optimization for eigenvalue problems where the main goal is to maximize a specified eigenvalue, such as a natural frequency or a linearized buckling load using polygonal finite elements in arbitrary two-dimensional domains. Topology optimization has commonly been used to minimize the compliance of structures subjected to volume constraints. The idea is to distribute a certain amount of material in a given design domain subjected to a set of loads and boundary conditions such that to maximize its stiffness. In this work, the objective is to obtain the optimal material distribution in order to maximize the fundamental natural frequency (e.g. to keep it away from an external excitation frequency) or to maximize the lowest critical buckling load (e.g. to ensure a higher level of stability of the structures). We employ unstructured polygonal meshes constructed using Voronoi tessellations for the solution of the structural topology optimization problems. The design variables, i.e. material densities, used in the optimization scheme, are associated with each polygonal element in the mesh. We present several topology optimization examples for both eigenfrequency and buckling problems in order to demonstrate the functionality and applicability of the proposed methodology.

The ground state of a quantum system is the minimizer of the total energy of that system. The aim of this thesis is to present and numerically solve the Gross-Pitaevskii eigenvalue problem (GPE) as a physical model for the formation of ground states of dilute Bose gases at ultra-low temperatures in a disordered potential. The first part of the report introduces the quantum mechanical phenomenon that arises at ground states of the Bose gases; the Anderson localization, and presents the nonlinear eigenvalue problem and the finite element method (FEM) used to discretize the GPE. The numerical method used to solve the eigenvalue problem for the smallest eigenvalue is called the inverse power iteration method, which is presented and explained. In the second part of the report, the smallest eigenvalue of a linear Schrödinger equation is compared with the numerically computed smallest eigenvalue (ground state) in order to evaluate the accuracy of a linear numerical scheme constructed as first step for numerically solving the non-linear problem. In the next part of the report, the numerical methods are implemented to solve for the eigenvalue and eigenfunction of the (non-linear) GPE at ground state (smallest eigenvalue). The mathematical expression for the quantum energy and smallest eigenvalue of the non-linear system are presented in the report. The methods used to solve the GPE are the FEM and inverse power iteration method and different instances of the Anderson localization are produced. The study shows that the error of the smallest eigenvalue approximation for the linear case without disorder is satisfying when using FEM and Power iteration method. The accuracy of the approximation obtained for the linear case without disorder is satisfying, even for a low numbers of iterations. The methods require many more iterations for solving the GPE with a strong disorder. On the other hand, pronounced instances of Anderson localizations are produced in a certain scaling regime. The study shows that the GPE indeed works well as a physical model for the Anderson localization. / Syftet med denna avhandling är att undersöka hur väl Gross-Pitaevskii egenvärdesekvation (GPE) passar som en fysisk modell för bildandet av stationära elektronstater i utspädda Bose-gaser vid extremt låga temperaturer. Fenomenet som skall undersökas heter Anderson lokalisering och uppstår när potentialfältets styrka och störning i systemet är tillräckligt hög. Undersökningen görs i denna avhandling genom att numeriskt lösa GPE samt illustrera olika utfall av Anderson lokaliseringen vid olika numeriska värden. Den första delen av rapporten introducerar det icke-linjära matematiska uttrycket för GPE samt de numeriska metoderna som används för att lösa problemet numerisk: finita elementmetoden (FEM) samt egenvärdesalgoritmen som heter inversiiteration. Finita elementmetoden används för att diskretisera variationsproblemet av GPE och ta fram en enkel algebraisk ekvation. Egenvärdesalgoritmen tillämpas på den algebraiska ekvation för att iterativt beräkna egenfunktionen som motsvarar det minsta egenvärdet. Det minsta egenvärdet av en fullt definierad (linjär) Schrödinger ekvation löses i rapportens andra del. Den linjära ekvationen löses för att ta fram en förenklad numerisk algoritm att utgå ifrån innan den icke-linjära algoritmen tas fram. För att försäkra sig att den linjära algoritmen stämmer bra jämförs det exakta egenvärdet för problemet med ett numeriskt framtaget värde. Undersökningen av den linjära algoritmen visar att vi får en bra uppskattning av egenvärdet - även vid få iterationer. Vidare konstrueras den ickelinjära algoritmen baserat på den linjära. Ekvationen löses och undersökes. Egenfunktionen som motsvarar minsta egenvärdet framtas och beskriver kvantsystemet i lägsta energitillståndet, så kallade grundtillståndet. Undersökningen av GPE visar att de numeriska metoderna kräver många fler iterationer innan en tillräckligt bra uppskattning av egenvärdet fås. Å andra sidan fås markanta Anderson lokaliseringar för ett skalningsområde som beskrivs av styrkan av potentialfältet i relation till dess störning. Slutsatsen är att Gross-Pitaevskii egenvärdesekvation passar bra som en fysisk modell för detta kvantsystem.

Max-plus代数は，実数全体に無限小元を付加した集合に，加法として最大値をとる演算，乗法として通常の加法を考えた代数系である．本論文では，max-plus線形方程式に対するCramerの公式の類似物を用いて，線形方程式の解空間の基底が構成できることを示した．さらに固有値問題に関連して，max-plus行列の固有ベクトルの概念を2通りの観点から拡張した． / The max-plus algebra is the semiring with addition "max" and multiplication "+". In the present thesis, the author gives a combinatorial characterization of solutions of linear systems in terms of the max-plus Cramer's rule. Further, the author extends the concept of eigenvectors of max-plus matrices from two different perspectives. / 博士(理学) / Doctor of Philosophy in Science / 同志社大学 / Doshisha University

In this thesis, a method to calculate eigenpairs is implemented for the Multipsi library. While the standard implemtentations use the Davidson method with Rayleigh-Ritz extraction to calculate the eigenpairs with the lowest eigenvalues, the new method uses the harmonic Davidson method with the harmonic Rayleigh-Ritz extraction to calculate eigenpairs with eigenvalues near a chosen target. This is done for Configuration Interaction calculations and for Multiconfigurational methods. From calculations, it seems the new addition to the Multipsi library is worth investigating further as convergence for difficult systems with a lot of near-degeneracy was improved. / I denna avhandling implementeras en metod för att beräkna egenpar för Multipsi-biblioteket. Medan standardimplementeringarna använder Davidson-metoden med Rayleigh-Ritz-extraktion för att beräkna egenparen med de lägsta egenvärdena, använder den nya metoden den harmoniska Davidson-metoden med den harmoniska Rayleigh-Ritz-extraktionen för att beräkna egenparen med egenvärden nära ett valt mål. Detta görs för konfigurationsinteraktionsberäkningar och för multikonfigurationsmetoder. Utifrån beräkningarna verkar det nya tillskottet till Multipsi-biblioteket vara värt att undersöka vidare eftersom konvergensen för svåra system med mycket nära degenerering förbättrades.