Cette thèse considère la solution numérique d'un problème aux valeurs propres de grandes dimensions, dans lequel l'opérateur est dérivé d'un problème de transfert radiatif. Ainsi, cette thèse étudie l'utilisation de matrices hiérarchiques, une représentation efficace de tableaux, très intéressante pour une utilisation avec des problèmes de grandes dimensions. Les matrices sont des représentations hiérarchiques de structures de données efficaces pour les matrices denses, l'idée de base étant la division d'une matrice en une hiérarchie de blocs et l´approximation de certains blocs par une matrice de petite caractéristique. Son utilisation permet de diminuer la mémoire nécessaire tout en réduisant les coûts informatiques. L'application de l'utilisation de matrices hiérarchique est analysée dans le contexte de la solution numérique d'un problème aux valeurs propres de grandes dimensions résultant de la discrétisation d'un opérateur intégral. L'opérateur est de convolution et est défini par la première fonction exponentielle intégrale, donc faiblement singulière. Pour le calcul informatique, nous avons accès à HLIB (Hierarchical matrices LIBrary) qui fournit des routines pour la construction de la structure hiérarchique des matrices et des algorithmes pour les opérations approximative avec ces matrices. Nous incorporons certaines routines comme la multiplication matrice-vecteur ou la decomposition LU, en SLEPc (Hierarchical matrices LIBrary) pour explorer les algorithmes existants afin de résoudre les problèmes de valeur propre. Nous développons aussi des expressions analytiques pour l'approximation des noyaux dégénérés utilisés dans la thèse et déduire ainsi les limites supérieures d'erreur pour ces approximations. Les résultats numériques obtenus avec d'autres techniques pour résoudre le problème en question sont utilisés pour la comparaison avec ceux obtenus avec la nouvelle technique, illustrant l'efficacité de ce dernier / In this thesis, we consider the numerical solution of a large eigenvalue problem in which the integral operator comes from a radiative transfer problem. It is considered the use of hierarchical matrices, an efficient data-sparse representation of matrices, especially useful for large dimensional problems. It consists on low-rank subblocks leading to low memory requirements as well as cheap computational costs. We discuss the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank discretization of an integral operator. The operator is of convolution type, it is defined through the first exponential-integral function and hence it is weakly singular. We access HLIB (Hierarchical matrices LIBrary) that provides, among others, routines for the construction of hierarchical matrix structures and arithmetic algorithms to perform approximative matrix operations. Moreover, it is incorporated the matrix-vector multiply routines from HLIB, as well as LU factorization for preconditioning, into SLEPc (Scalable Library for Eigenvalue Problem Computations) in order to exploit the available algorithms to solve eigenvalue problems. It is also developed analytical expressions for the approximate degenerate kernels and deducted error upper bounds for these approximations. The numerical results obtained with other approaches to solve the problem are used to compare with the ones obtained with this technique, illustrating the efficiency of the techniques developed and implemented in this work

Les travaux de recherche présentés dans cette thèse concernent des contributions à l’étude de stabilité des systèmes linéaires à retards avec contrôleurs d’ordre réduit. Cette mémoire est partagée en trois parties.La première partie est axée sur l’étude des systèmes linéaires à retard mono-entré /mono-sortie, bouclées avec un contrôleur de type PID. Inspiré par l’approche géométrique développée par Gu et al. Nous avons proposé une méthode analytique pour trouver la région (ou les régions) de tous les contrôleurs de type PID stabilisant pour le système à retard. Basée sur cette même approche, on a développé un algorithme pour calculer le dégrée de fragilité d’un contrôleur donné de type PID (PI, PD et PID).La deuxième partie de la thèse est axée sur l’étude de stabilité sous une approche NCS (pour son acronyme en anglais : Networked Control System). Plus précisément, nous avons d’abord étudié le problème de la stabilisation en tenant compte des retards induit par le réseau et les effets induits par la période d’échantillonnages. Pour mener une telle analyse nous avons adopté une approche basée sur la théorie des perturbations. Finalement, dans la troisième partie de la thèse nous abordons certains problèmes concernant le comportement des zéros d’une certaine classe de systèmes échantillonnés mono-entré /mono-sortie. Plus précisément, étant donné un système à temps continu, on obtient les intervalles d’échantillonnage garantissant l’invariance du nombre de zéros instables dans chaque intervalle. Pour développer cette analyse, nous adoptons une approche basée sur la perturbation aux valeurs propres. / The research work presented in this thesis concern to the stability analysis of linear time-delay systems with low-order controllers. This thesis is divided into three parts.The first part of the thesis focus on the study of linear SISO (single-input/single-output) systems with input/output delays, where the feedback loop is closed with a controller of PID-type. Inspired by the geometrical approach developed by Gu et al. we propose an analytical method to find the stability regions of all stabilizing controllers of PID-type for the time-delay system. Based on this same approach, we propose an algorithm to calculate the degree of fragility of a given controller of PID- type (PI, PD and PID).The second part of the thesis focuses on the stability analysis of linear systems under an NCS (Networked System Control) based approach. More precisely, we first focus in the stabilization problem by taking into account the induced network delays and the effects induced by the sampling period. To carry out such an analysis we have adopted an eigenvalue perturbation-based approach.Finally, in the third part of the thesis we tackle certain problems concerning to the behavior of the zeros of a certain class of sampled-data SISO systems. More precisely, given a continuous-time system, we obtain the sampling intervals guaranteeing the invariance of the number of unstable zeros in each interval. To perform such an analysis, we adopt an eigenvalue perturbation-based approach.

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.

In dieser Arbeit werden Techniken beschrieben, die es erlauben (höhere) Ableitungen und Taylorapproximationen solcher Computerprogramme effizient zu berechnen. Auch inbesondere dann, wenn die Programme Algorithmen der numerischen linearen Algebra (NLA) enthalten. Im Gegensatz zur traditionellen algorithmischen Differentiation (AD), bei der die zugrunde liegenden Algorithmen um zusätzliche Befehlere erweitert werden, sind in dieser Arbeit die Zerlegungen durch definierende Gleichungen charakterisiert. Basierend auf den definierenden Gleichungen werden Strukturausnutzende Algorithmen hergeleitet. Genauer, neuartige Algorithmen für die Propagation von Taylorpolynomen durch die QR, Cholesky und reell-symmetrischen Eigenwertzerlegung werden präsentiert. Desweiteren werden Algorithmen für den Rückwärtsmodus der AD hergeleitet, welche im Wesentlichen nur die Faktoren der Zerlegungen benötigen. Im Vergleich zum traditionellen Ansatz, bei dem alle Zwischenergebnisse gespeichert werden, ist dies eine Reduktion von O(N^3) zu O(N^2) für Algorithmen mit O(N^3) Komplexität. N ist hier die Größe der Matrix. Zusätzlich kann bestehende, hoch-optimierte Software verwendet werden. Ein Laufzeitvergleich zeigt, dass dies im Vergleich zum traditionellen Ansatz zu einer Beschleunigung in der Größenordnung 100 führen kann. Da die NLA Funktionen als Black Box betrachtet werden, ist desweiteren auch der Berechnungsgraph um Größenordnungen kleiner. Dies bedeutet, dass Software, welche Operator Overloading benutzt, weniger Overhead hervorruft und auch weniger Speicher benötigt. / This thesis provides a framework for the evaluation of first and higher-order derivatives and Taylor series expansions through large computer programs that contain numerical linear algebra (NLA) functions. It is a generalization of traditional algorithmic differentiation (AD) techniques in that NLA functions are regarded as black boxes where the inputs and outputs are related by defining equations. Based on the defining equations, structure-exploiting algorithms are derived. More precisely, novel algorithms for the propagation of Taylor polynomials through the QR, Cholesky,- and real-symmetric eigenvalue decomposition are shown. Recurrences for the reverse mode of AD, which require essentially only the returned factors of the decomposition, are also derived. Compared to the traditional approach where all intermediates of an algorithm are stored, this is a reduction from O(N^3) to O(N^2) for algorithms with O( N^3) complexity. N denotes the matrix size. The derived algorithms make it possible to use existing high-performance implementations. A runtime comparison shows that the treatment of NLA functions as atomic can be more than one order of magnitude faster than an automatic differentiation of the underlying algorithm. Furthermore, the computational graph is orders of magnitudes smaller. This reduces the additional memory requirements, as well as the overhead, of operator overloading techniques to a fraction.

Consider an under-determined system of nonlinear equations F(x)=0, F:R^m→R^n, where F is continuously differentiable and m > n. This system appears in a variety of applications, including parameter-dependent systems, dynamical systems with periodic solutions, and nonlinear eigenvalue problems. Robust, efficient numerical methods are often required for the solution of this system. Newton's method is an iterative scheme for solving the nonlinear system of equations F(x)=0, F:R^n→R^n. Simple to implement and theoretically sound, it is not, however, often practical in its pure form. Inexact Newton methods and globalized inexact Newton methods are computationally efficient variations of Newton's method commonly used on large-scale problems. Frequently, these variations are more robust than Newton's method. Trust region methods, thought of here as globalized exact Newton methods, are not as computationally efficient in the large-scale case, yet notably more robust than Newton's method in practice. The normal flow method is a generalization of Newton's method for solving the system F:R^m→R^n, m > n. Easy to implement, this method has a simple and useful local convergence theory; however, in its pure form, it is not well suited for solving large-scale problems. This dissertation presents new methods that improve the efficiency and robustness of the normal flow method in the large-scale case. These are developed in direct analogy with inexact-Newton, globalized inexact-Newton, and trust-region methods, with particular consideration of the associated convergence theory. Included are selected problems of interest simulated in MATLAB.

La maîtrise du bruit et des vibrations est un objectif fréquemment rencontré dans le domaine industriel. Qu’il s’agisse de questions de confort ou de sécurité, les domaines d’applications sont nombreux et variés : transport, BTP, ingénierie civile et militaire… Dans cette thèse, un problème de vibroacoustique interne avec couplage fluide-structure est étudié. Il s’agit d’une cavité remplie de fluide dont les parois sont constituées d’une structure sandwich viscoélastique. Les difficultés numériques associées à ce modèle portent sur la non linéarité du matériau et sur les propriétés des opérateurs matriciels manipulés (conditionnement, non symétrie). Le calcul des vibrations du système dissipatif couplé nécessite une valeur initiale, choisie comme la solution du problème conservatif. Cette solution n’étant pas aisée à déterminer, deux solveurs aux valeurs propres basés sur la Méthode Asymptotique Numérique (MAN) sont proposés pour résoudre le problème des vibrations libres du système conservatif. Associant des techniques de perturbation d'ordre élevé et de continuation, la MAN permet de transformer le problème non linéaire de départ en une suite de problèmes linéaires, plus simples à résoudre. Les solutions obtenues sont ensuite utilisées comme point initial pour déterminer la réponse libre du système dissipatif. Un solveur de Newton d’ordre élevé, basé sur les techniques d’homotopie et de perturbation est développé pour résoudre ce problème. Enfin, le régime forcé est étudié. Pour toutes les configurations envisagées, les résultats obtenus mettent en évidence des performances numériques améliorées par rapport aux méthodes classiquement utilisées (Arpack, Newton…). / Noises and vibrations control is a common objective in the industrial field. Whether it is a question of comfort or safety, the fields of application are numerous and varied: transport, building, civil and military engineering… In this thesis, a vibroacoustics interior problem with fluid-structure coupling is studied. A cavity filled of fluid whose walls are made of a sandwich viscoelastic structure is considered. The numerical difficulties associated with this model relate to the non-linearity of the viscoelastic material and the properties of the matrix operators used (conditioning, non-symmetry). The calculation of the vibrations of the coupled dissipative system requires an initial value, chosen as the solution to the conservative problem. Since this solution is difficult to determine, two eigenvalue algorithms based on the Asymptotic Numerical Method (ANM) are proposed to solve the problem of free vibrations of the conservative system. Combining high order perturbation and continuation techniques, ANM transforms the initial non-linear problem into a set of linear problems that are easier to solve. The solutions obtained are then used as the initial point to determine the free vibrations of the dissipative problem. A high order Newton solver, based on homotopy and perturbation techniques, is developed to solve this problem. Finally, the forced harmonic response of the damped system is computed. For all the configurations tested, the results obtained show improved numerical performance compared to the methods conventionally used (Arpack solver, Newton algorithm…).

Made available in DSpace on 2016-04-27T16:57:30Z (GMT). No. of bitstreams: 1 Joelma Iamac Nomura.pdf: 7399337 bytes, checksum: 3b1b78708c15a38620c94201d8ab977e (MD5) Previous issue date: 2014-03-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The objective of this research harnesses to the results obtained in the Master's Dissertation defended in September 2008 in Postgraduate Studies Program in Mathematics Education at PUC - SP. In this same essay, issues related to teaching and learning of linear algebra sought to answer and find new ways of targeting and perspectives of students in a graduate in Electrical Engineering, asking Why and How should it be taught the discipline of linear algebra on a course with this profile? Among the results, we identified that the interdisciplinarity inherent to the topics of Linear Algebra and specific content of engineering or applied constituted an essential factor for the recognition of mathematical disciplines as theoretical and conceptual basis. Interdisciplinarity reflected in specific mathematical objects of linear algebra and practical situations of engineering materials for the formation of conceptual and general engineer seeking the theoretical foundation and basic justification for the technological improvement of its area. Based on a scenario and results envisioned in the dissertation we propose to investigate the cognitive structures involved in the construction of mathematical object eigenvalue and eigenvector in the initial and final student education phases in Engineering courses, showing the cognitive schemes in their mathematical minds. For this, the following issues are highlighted: ( 1 ) What conceptions (action - process -object- schema ) are evidenced in students after studying the mathematical object eigenvalue and eigenvector in the initial and final phases of their academic training courses in Engineering? and ( 2 ) these same phases, which concept image and concept definition are highlighted in the study of eigenvalue and eigenvector mathematical object? Substantiated by the theoretical contributions of Dubinsky (1991), on the APOS Theory and Vinner (1991), about the concept image and concept definition, we consider the cognitive processes involved in the construction of mathematical object, identifying the nature of their cognitive entities portrayed in mathematical mind. The discussion focuses on mathematical mind both the mathematical structure that is designed and shared by the community as the design in which each mental biological framework handles such ideas. To do so, we consider the relationship between the ideas which constitute the APOS theory, concepts image and definition and some aspects of Cognitive Neuroscience. Characterized as multiple case studies, data collection covered the speech of students in engineering courses in various training contexts, established by the institutions. The analysis of the specific mathematical concept called genetic decomposition led to this concept, which was proposed by System Dynamic Discrete problem, described by the difference equation K K x A.x 1 = + , (K = 0,1,2 , ... ) . Based on the ideas of Stewart (2008) and Trigueros et al. (2012) it was possible to us to identify some characteristics of showing the different conceptions of the students. Moreover, we consider some ideas that characterize the concept image and concept definition according Vinner (1991) and Domingos (2003). As a result of our investigation, we identified that the students of the first case study, at different stages of training, present the design process and the concept image on an instrumental level mathematical object eigenvalue and eigenvector. Have students in the second case, particularly, all of the first phase, and two of the second, showed signs of action and concept image incipient level. As a student of the second phase, have also highlighted the design process and the concept image on an instrumental level as the subject of the first case study. Therefore, we find no significant evolution between the inherent APOS Theory concepts and the concepts image of the object of study. We show that all students presented their speeches in relations between the Linear Algebra course and other courses in the program, such as Numerical Calculation, Electrical Circuits , Computer Graphics and Control Systems, with lesser or greater degree of depth and knowledge. We realize that students attach importance to mathematical disciplines in its formations and seek for a new approach to teaching that address the relationships between them and the disciplines of Engineering / O objetivo desta pesquisa atrela-se aos resultados obtidos na Dissertação de Mestrado defendida em setembro de 2008 no Programa de Estudos Pós-Graduados em Educação Matemática da PUC-SP. Nesta mesma dissertação, questões relacionadas ao ensino e aprendizagem de Álgebra Linear buscaram responder e encontrar novas formas de direcionamento e perspectivas de ensino em uma graduação em Engenharia Elétrica, indagando Por que e Como deve ser lecionada a disciplina de Álgebra Linear em um curso com este perfil? Dentre os resultados obtidos, identificou-se que a interdisciplinaridade inerente aos tópicos de Álgebra Linear e conteúdos específicos ou aplicados da Engenharia constituiu-se de fatores imprescindíveis para ao reconhecimento das disciplinas matemáticas, como base teórica e conceitual. A interdisciplinaridade refletida em objetos matemáticos específicos da Álgebra Linear e situações práticas da Engenharia prima pela formação do engenheiro conceitual e generalista que busca na fundamentação teórica e básica a justificativa para o aprimoramento tecnológico de sua área. Com base no cenário e resultados vislumbrados na defesa da dissertação, propusemonos investigar as estruturas cognitivas envolvidas na construção do objeto matemático autovalor e autovetor nas fases inicial e final de formação do aluno dos cursos de Engenharia, evidenciando os esquemas cognitivos e a mente matemática dos estudantes, sujeitos de nossa investigação. Para tanto, as seguintes questões são destacadas: (1) Quais concepções (ação-processo-objeto-esquema) são evidenciadas nos alunos, após o estudo do objeto matemático autovalor e autovetor nas fases inicial e final de sua formação acadêmica em cursos de Engenharia?; e (2) Nessas mesmas fases, quais conceitos imagem e definição são evidenciados no estudo do objeto matemático autovalor e autovetor? Fundamentados pelos aportes teóricos de Dubinsky (1991), sobre a Teoria APOS, e Vinner (1991) nos conceitos imagem e definição, foram considerados os processos cognitivos envolvidos na construção do objeto matemático, identificando a natureza de suas entidades cognitivas retratadas na mente matemática. A discussão sobre mente matemática foca-se tanto na estrutura matemática que é concebida e compartilhada pela comunidade como no delineamento em que cada estrutura biológica mental trata essas mesmas ideias. Para tanto, considerou-se a relação entre as ideias que constituem a Teoria APOS, os conceitos imagem e definição e alguns aspectos da Neurociência Cognitiva. A pesquisa caracterizada como estudos de caso múltiplos, identificou os dados a partir do discurso dos estudantes dos cursos de Engenharia em contextos diversos de formação, estabelecidos pelas instituições de ensino. A análise do conceito matemático específico levou à chamada decomposição genética desse conceito, que foi proposto pelo problema de Sistema Dinâmico Discreto, descrito pela equação de diferença K K x A.x 1 = + (K=0,1,2,...). Com base nas ideias de Stewart (2008) e Trigueros et al. (2012), foi possível identificar algumas características que evidenciassem as diferentes concepções dos estudantes. Além disso, foram consideradas algumas ideias que caracterizam o conceito imagem e definição de acordo com Vinner (1991) e Domingos (2003). Como resultado desta investigação, identificou-se que os alunos do primeiro estudo de caso, em fases distintas de formação, apresentam a concepção processo e o conceito imagem em nível instrumental do objeto matemático autovalor e autovetor. Já os alunos do segundo de caso, particularmente, todos os da primeira fase, e dois da segunda apresentaram indícios da concepção ação e conceito imagem em nível incipiente. Apenas um aluno da segunda fase também evidenciou ter a concepção processo e o conceito imagem em nível instrumental, como os sujeitos do primeiro estudo de caso. Portanto, constatou-se que não houve evolução significativa entre as concepções inerentes à Teoria APOS e os conceitos imagem do objeto de estudo. Evidenciou-se que todos os alunos apresentaram em seus discursos relações existentes entre a disciplina Álgebra Linear e demais disciplinas do curso, como Cálculo Numérico, Circuitos Elétricos, Computação Gráfica e Sistemas de Controle, com menor ou maior grau de profundidade e conhecimento. Percebe-se que os alunos atribuem relevância às disciplinas matemáticas em suas formações e buscam por um novo enfoque de ensino que contemple as relações entre as mesmas e as disciplinas da Engenharia

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The subject of this Thesis is to examine nontrivial solutions of boundary value problems for second order ODEs with unilateral jumping nonlinearities. Considered problems can be interpreted as models of a simple beam supported by three types of elastic obstacles. Couples of eigenvalues and eigenfunctions are found and discussed with respect to parameter which represents the strength of the obstacle.