A new method called the Direct Method is developed to solve for the propagating modes in waveguides via the finite-element method. The variational form of the Direct method is derived to ensure that an extremum is reached. The Direct method uses Maxwell's equations directly, both zero and first-order, scalar and vector bases that are used in the finite-element formulation. The direct solution method solves for both the magnetic and electric fields simultaneously. Comparisons are made with the traditionally used vector-Helmholtz equation set. The advantages and disadvantages of the newly developed method is described as well as several results displayed using the WR-90 waveguide and a circular waveguide as test waveguides. Results include a partially filled dielectric loaded rectangular waveguide. The effects of including the divergence of the fields in the functional as penalty terms on the quality of results obtained by the Direct method and the vector-Helmholtz method is explored. The quality of results is gauged on the accuracy of the computed modes as well as the elimination or a significant reduction in the number of 'spurious modes' that are often encountered in solutions to waveguide problems. It is shown that computational time for the solution and computer storage requirements exceed the typically used Helmholtz equation method but the results obtained can be more accurate. Future work may include developing a sparse eigenvalue solution method that could reduce the solution time and storage requirements significantly. The Direct method of solution in dynamics resulted after an initial search in magnetostatics for methods to solve for the magnetic field without using the magnetic-vector potential using finite-element methods. A variational derivation that includes the boundary conditions is developed for the magnetic-vector potential method. Several techniques that were used to attempt accurate solutions for the magnetostatic fields with multiple materials and without the use of the magnetic-vector potential are described. It was found that some of the newly developed general techniques for magnetostatics are only accurate when homogeneous media are present. A method using two curl equations is developed which is a Direct method in magnetostatics and reveals the interaction between the bases used. The transition from magnetostatics to dynamics is made and similar Direct methods are applied to the waveguide problem using different bases. / Ph. D.

Neste trabalho apresenta-se inicialmente algumas considerações sobre a difração de raios-X, suas principais leis, fatores de correção dos dados experimentais, uma revisão sobre os principais Métodos Diretos de resolução da estrutura, e considerações sobre o refinamento da estrutura obtida. A seguir descreve-se a determinação da estrutura cristalina do produto natural 5,4\'-Dihidroxi-3\', 5\' -dimetoxi- 6,7 -(2\",2\" -dimetilpirano) flavona de fórmula molecular C22H20O<sub<7, isolado de plantas da espécie Neoraputia paraensis, que cristaliza-se no sistema monoclínico, grupo espacial C2/c, com os seguintes parâmetros de cela unitária: a = 13,651(1), b= 23,428(2),c= 13,725(1) &#197; &#946= 119,528(4)&#176, V=3819,6(5)A&#9463 , Dc = 1,366g cm-3 e Z =8 moléculas por cela unitária. A estrutura foi resolvida através da aplicação de Métodos Diretos. Os índices de discordância finais são: R= 0,0509, Rw = 0,0530 para 1743 reflexões com I &#8805 3&#963(I) e Ra11=0,157. A estrutura foi refinada fazendo uso dos cálculos de Fourier Diferença e pelo método de mínimos quadrados usando matriz completa. A molécula apresenta duas ligações de hidrogênio intramoleculares, de força média (distâncias O-O 2,558(3) e 2,674(4) &#946). O empacotamento cristalino apresenta duas outras ligações de hidrogênio intermoleculares, mais fracas, sendo feitas com as moléculas geradas pelo espelho c (distâncias O-O 2,830(3) e 2,992(3) &#946). O efeito destas ligações intermoleculares é o da formação de cadeias ao longo da direção [101]. / Initially, some considerations about X-ray diffraction, its laws, the factors for the correction of experimental data, a revision of the main Direct Methods for structure resolution, and comments on the refinement of the resulting structure are presented. Next, the structure determination of the compound 5,4\' -dihydroxy-3\'-5\'- dimethoxy-6,7(2\",2\' \')dimethylpyran)flavone is described The compound is isolated from plants of the species Neuroputia paraenesis and has the molecular formula C22H20O<sub<7. It crystallizes in the monoclinic system, space group C2/c, with the following unit cell parameters: a = 13.651(1), b= 23.428(2), c= 13.725(1) &#197; &#946= 119,528(4)&#176, V=3819,6(5)A&#9463 , Dc = 1,366g cm-3 e Z =8molecules per unit cell. The structure was solved applying Direct Methods. The final disagreement indices are: R= 0.0509, Rw= 0.0530 for 1743 reflections with &#8805 3&#963(I) and Ra11=0,157. The structure was refined applying Fourier difference calculations and full matrix least squares methods. The molecule shows two intramolecular hydrogen bonds of medium strength (distances O-O 2.558(3) and 2.674(4) &#946). The crystal packing shows also two weaker hydrogen bonds; these are formed between the molecules generated by the c mirror (distances O-O 2.830(3) and 2.992(3) &#946). The result of these intermolecular hydrogen bonds is the formation of chains in [10 1] direction.

Στο πέρασμα των αιώνων, οι άνθρωποι αναζητούσαν νόμους που να περιγράφουν τα φαινόμενα του φυσικού κόσμου. Το 1744 ο Γάλλος επιστήμονας Pierre Louis Moreau de Maupertious έθεσε την αρχή ότι η φύση ενεργεί πάντα με τέτοιο τρόπο ώστε να ελαχιστοποιείται κάποια ποσότητα που ο ίδιος ονόμασε «δράση». Στη μαθηματική θεμελίωση της σχετικής θεωρίας των μεγίστων και ελαχίστων των βαθμωτών ποσοτήτων συνέβαλλε ο Ελβετός μαθηματικός Leonard Euler. Στα προβλήματα του Λογισμού των Μεταβολών μελετάμε παραστάσεις που περιέχουν μία ή περισσότερες άγνωστες πραγματικές συναρτήσεις μιας ή περισσοτέρων πραγματικών μεταβλητών. Έτσι, αναζητούμε μια συνάρτηση που να δίνει στη συγκεκριμένη παράσταση μέγιστη ή ελάχιστη τιμή. Οι παραστάσεις αυτές ονομάζονται συναρτησιακά και αποτελούν μια γενίκευση της έννοιας της συνάρτησης. H διπλωματική αυτή εργασία αποτελεί μια βιβλιογραφική επισκόπηση των άμεσων μεθόδων που χρησιμοποιούνται στην επίλυση των προβλημάτων του λογισμού των μεταβολών.Κάτω από αυτό το πρίσμα, παρουσιάζονται οι προσεγγιστικές λύσεις, η εφαρμογή τους καθώς και οι πρόσφατες βελτιώσεις τους. Συγκεκριμένα, στο κεφάλαιο 1 αναφέρονται κάποιες βασικές έννοιες του λογισμού των μεταβολών . Στο κεφάλαιο 2 γίνεται λόγος για τέσσερις άμεσες μεθόδους επίλυσης συναρτησιακών προβλημάτων. Στο κεφάλαιο 3 περιγράφονται τέσσερις μέθοδοι, οι οποίες προσεγγίζουν τη λύση μη γραμμικών προβλημάτων του λογισμού των μεταβολών με ακρίβεια 2ης τάξης ως συνάρτηση του μήκους βήματος. Στο κεφάλαιο 4 παρουσιάζονται κάποιες νέες τεχνικές επίλυσης των προβλημάτων που συζητήθηκαν. Στο κεφάλαιο 5 αναφέρεται η χρήση υποπρογραμμάτων (“packages”) που χρησιμοποιούνται μέσω συστημάτων λογισμικού όπως το Maple και το Mathematica. / Over the centuries, people seeking laws that describe the phenomena of the natural world. In 1744 the French scientist Pierre Louis Moreau de Maupertious established the principle that nature always acts in such a way as to minimize a quantity he called ‘action’. The Swiss mathematician Leonard Euler helped the mathematical foundations of the theory of maximum and minimum of the scalar quantities. Problems of the Calculus of Variations contain unknown functions of one or more real variables. So, the aim is to find a function that maximizes an integral. This project deals with direct methods which give approximate solutions to functional problems. Furthermore, their implementation and their recent improvements are described. Specifically, in Chapter 1 some basic concepts of calculus of variations are discussed. In chapter 2 four direct methods of solving functional problems are illustrated. In the chapter four methods which approximate the solution of nonlinear problems in the calculus of variations with second order accuracy in terms of the step length are described and some results are pointed out. Chapter 4 presents some new techniques to solve these problems. The chapter 5 refers the use of “packages” used by software systems such as Maple and Mathematica.

Neste trabalho apresenta-se inicialmente algumas considerações sobre a difração de raios-X, suas principais leis, fatores de correção dos dados experimentais, uma revisão sobre os principais Métodos Diretos de resolução da estrutura, e considerações sobre o refinamento da estrutura obtida. A seguir descreve-se a determinação da estrutura cristalina do produto natural 5,4\'-Dihidroxi-3\', 5\' -dimetoxi- 6,7 -(2\",2\" -dimetilpirano) flavona de fórmula molecular C22H20O<sub<7, isolado de plantas da espécie Neoraputia paraensis, que cristaliza-se no sistema monoclínico, grupo espacial C2/c, com os seguintes parâmetros de cela unitária: a = 13,651(1), b= 23,428(2),c= 13,725(1) &#197; &#946= 119,528(4)&#176, V=3819,6(5)A&#9463 , Dc = 1,366g cm-3 e Z =8 moléculas por cela unitária. A estrutura foi resolvida através da aplicação de Métodos Diretos. Os índices de discordância finais são: R= 0,0509, Rw = 0,0530 para 1743 reflexões com I &#8805 3&#963(I) e Ra11=0,157. A estrutura foi refinada fazendo uso dos cálculos de Fourier Diferença e pelo método de mínimos quadrados usando matriz completa. A molécula apresenta duas ligações de hidrogênio intramoleculares, de força média (distâncias O-O 2,558(3) e 2,674(4) &#946). O empacotamento cristalino apresenta duas outras ligações de hidrogênio intermoleculares, mais fracas, sendo feitas com as moléculas geradas pelo espelho c (distâncias O-O 2,830(3) e 2,992(3) &#946). O efeito destas ligações intermoleculares é o da formação de cadeias ao longo da direção [101]. / Initially, some considerations about X-ray diffraction, its laws, the factors for the correction of experimental data, a revision of the main Direct Methods for structure resolution, and comments on the refinement of the resulting structure are presented. Next, the structure determination of the compound 5,4\' -dihydroxy-3\'-5\'- dimethoxy-6,7(2\",2\' \')dimethylpyran)flavone is described The compound is isolated from plants of the species Neuroputia paraenesis and has the molecular formula C22H20O<sub<7. It crystallizes in the monoclinic system, space group C2/c, with the following unit cell parameters: a = 13.651(1), b= 23.428(2), c= 13.725(1) &#197; &#946= 119,528(4)&#176, V=3819,6(5)A&#9463 , Dc = 1,366g cm-3 e Z =8molecules per unit cell. The structure was solved applying Direct Methods. The final disagreement indices are: R= 0.0509, Rw= 0.0530 for 1743 reflections with &#8805 3&#963(I) and Ra11=0,157. The structure was refined applying Fourier difference calculations and full matrix least squares methods. The molecule shows two intramolecular hydrogen bonds of medium strength (distances O-O 2.558(3) and 2.674(4) &#946). The crystal packing shows also two weaker hydrogen bonds; these are formed between the molecules generated by the c mirror (distances O-O 2.830(3) and 2.992(3) &#946). The result of these intermolecular hydrogen bonds is the formation of chains in [10 1] direction.

The aim of this thesis is to conduct a general investigation in the field of sparse matrices, to investigate and compare various techniques for handling sparse systems suggested in the literature, to develop some new techniques, and to discuss the feasibility of using sparsity techniques in the solution of overdetermined equations and the eigenvalue problem.

Stability analysis methods may be categorized by two major stability analysis methods: small-signal stability and transient stability analyses. Transient stability methods are further categorized into two major categories: numerical methods based on numerical integration, and direct methods. The purpose of this thesis is to study and investigate transient stability analysis using a combination of step-by-step and direct methods using Equal Area Criterion. The proposed method is extended for transient stability analysis of multi machine power systems. The proposed method calculates the potential and kinetic energies for all machines in a power system and then compares the largest group of kinetic energies to the smallest groups of potential energies. A decision based on the comparison can be made to determine stability of the power system. The proposed method is used to simulate the IEEE 39 Bus system to verify its effectiveness by comparison to the results obtained by pure numerical methods.

No estudo de estabilidade transitória, as não linearidades inerentes aos sistemas aliadas a grande dimensão do problema, contribuem para que as análises dos sistemas de potência sejam muito complexas. O estudo clássico de estabilidade transitória utiliza soluções numéricas iterativas de um conjunto de equações diferenciais associadas à dinâmica do sistema, visando a obtenção do tempo crítico de abertura. Porém, este não é o processo mais adequado à aplicações em tempo real devido ao esforço computacional exigido em tais iterações numéricas. Os métodos diretos são adequados para análises em tempo real, já que obtêm as informações necessárias sem a solução explícita de equações diferenciais. Dentre os métodos diretos existentes, as idéias de Lyapunov associadas ao princípio de invariância de LaSalle destacam-se por serem métodos energéticos e diretos adequados ao estudo de estabilidade em sistemas não lineares. Baseados em tais idéias, diversos métodos de estimativa da região de estabilidade foram propostos. Dentre estes, o BCU tem sido aceito como o mais eficiente para a determinação do tempo crítico de abertura. Apesar do BCU ser bastante eficiente, existem casos de falha do mesmo. Este trabalho visa eliminar dois problemas relacionados ao BCU: o primeiro problema advém do fato de que nem sempre o máximo de energia potencial ocorre nas vizinhanças do PEBS. O segundo problema está associado aos casos em que a trajetória do sistema gradiente reduzido não passa nas vizinhanças do ponto de equilíbrio de controle. Para solucionar estes dois problemas utiliza-se um método dinâmico para a deteção do “exit point" e o “Shadowing Method" para cálculo do ponto de equilíbrio de controle. Testes com os dois algoritmos mostram que eles encontram soluções para casos de falha do PEBS e BCU, porém problemas de deteção dos pontos de interesse ainda persistem. Alguns métodos alternativos, baseados nos algoritmos citados são propostos, bem como aspectos de melhoria de convergência dos mesmos. / In transient stability assessment, the nonlinearities of the systems and the dimension of the problem contribute to the complexity of the analysis in power systems. The classical numerical solution to obtain the critical clearing time (cct) has been used, but it is time-consuming and not adequate for real-time applications. The direct methods have the adequacy for real-time analysis because they get the necessary information for stability without the explicit solution of the set of differential equations associated to the system dynamics. Among the existing methods of power system transient stability analysis, Lyapunov ideas associated to LaSalle´s invariance principle are very important. Inspired by these ideas, methods to estimate the stability region have been obtained. One of them is the BCU, that has been accepted as the most efficient method in the determination of the critical cct. Despite BCU´s efficiency, it fails in many cases. The main objective of this research is to solve two problems associated to the BCU: the first problem is the fact that not always the maximum of potential energy occurs in the neighborhood of PEBS. The second problem is the fact that in many cases the trajectory of the reduced gradient system does not pass in the neighborhood of the controlling unstable equilibrium point. For the solution of these problems it will be used a dynamic method for the exit point detection and the Shadowing Method to calculate the controlling unstable equilibrium point. Tests have shown that the aforementioned robust algorithms find the solutions for many cases for which BCU and PEBS method fails, but many cases of fail persist. Some alternative methods, based in the mentioned algorithms are proposed, and so, aspects to improve its convergency.

Os métodos diretos são adequados à análise de estabilidade transitória em sistemas de potência, já que não requerem a resolução, integração numérica, do conjunto de equações diferenciais que representam o sistema. Os métodos diretos utilizam as idéias de Lyapunov associadas ao princípio de invariância de LaSalle para estimar a área de atração dos sistemas de potência. A grande dificuldade dos métodos diretos está em encontrar uma função auxiliar V, denominada função de Lyapunov que satisfaça as condições estabelecidas pelo Teorema de Lyapunov. Neste trabalho é realizada uma revisão bibliográfica das funções de Lyapunov utilizadas para análise de estabilidade transitória em sistemas de potência. Analisa-se o problema da existência de funções de Lyapunov quando as condutâncias de transferência são consideradas. Utilizando-se de uma extensão do princípio de Invariância de LaSalle, apresenta-se uma nova função a qual é uma função de Lyapunov no sentido mais geral da extensão do princípio de invariância de LaSalle quando as condutâncias de transferência da matriz admitância da rede reduzida são consideradas. Estudou-se também a existência de funções de Lyapunov no sentido mais geral de extensão do princípio de invariância de LaSalle para modelos que preserva a estrutura da rede. Neste caso, infelizmente não encontramos uma função satisfazendo todas as hipóteses requeridas. / The direct methods are well-suited for transient stability analysis to power systems, since they do not require the solution of the set of differential equations of the system model. The direct methods use the Lyapunov\'s ideas related to the LaSalle\'s invariance principle to estimate the power system attraction area. The great difficulty of the direct methods is to find an auxiliar function V, called Lyapunov function, which satisfies the conditions of Lyapunov\'s theorem. In this work, a bibliographic review of the Lyapunov functions used in transient stability analysis of power systems is done. The problem of existence of Lyapunov functions, when the transfer conductances are considered, is analysed. Using LaSalle\'s invariance principle extension, a Lyapunov function considering the transfer conductances is presented. The existence of Lyapunov functions for models that preserv the network structure was studied using the LaSalle\'s invariance principle. Unfortunately, in these cases, we did not find a function satisfing all the required hypothesis.

No estudo de estabilidade transitória, as não linearidades inerentes aos sistemas aliadas a grande dimensão do problema, contribuem para que as análises dos sistemas de potência sejam muito complexas. O estudo clássico de estabilidade transitória utiliza soluções numéricas iterativas de um conjunto de equações diferenciais associadas à dinâmica do sistema, visando a obtenção do tempo crítico de abertura. Porém, este não é o processo mais adequado à aplicações em tempo real devido ao esforço computacional exigido em tais iterações numéricas. Os métodos diretos são adequados para análises em tempo real, já que obtêm as informações necessárias sem a solução explícita de equações diferenciais. Dentre os métodos diretos existentes, as idéias de Lyapunov associadas ao princípio de invariância de LaSalle destacam-se por serem métodos energéticos e diretos adequados ao estudo de estabilidade em sistemas não lineares. Baseados em tais idéias, diversos métodos de estimativa da região de estabilidade foram propostos. Dentre estes, o BCU tem sido aceito como o mais eficiente para a determinação do tempo crítico de abertura. Apesar do BCU ser bastante eficiente, existem casos de falha do mesmo. Este trabalho visa eliminar dois problemas relacionados ao BCU: o primeiro problema advém do fato de que nem sempre o máximo de energia potencial ocorre nas vizinhanças do PEBS. O segundo problema está associado aos casos em que a trajetória do sistema gradiente reduzido não passa nas vizinhanças do ponto de equilíbrio de controle. Para solucionar estes dois problemas utiliza-se um método dinâmico para a deteção do “exit point” e o “Shadowing Method” para cálculo do ponto de equilíbrio de controle. Testes com os dois algoritmos mostram que eles encontram soluções para casos de falha do PEBS e BCU, porém problemas de deteção dos pontos de interesse ainda persistem. Alguns métodos alternativos, baseados nos algoritmos citados são propostos, bem como aspectos de melhoria de convergência dos mesmos. / In transient stability assessment, the nonlinearities of the systems and the dimension of the problem contribute to the complexity of the analysis in power systems. The classical numerical solution to obtain the critical clearing time (cct) has been used, but it is time-consuming and not adequate for real-time applications. The direct methods have the adequacy for real-time analysis because they get the necessary information for stability without the explicit solution of the set of differential equations associated to the system dynamics. Among the existing methods of power system transient stability analysis, Lyapunov ideas associated to LaSalle´s invariance principle are very important. Inspired by these ideas, methods to estimate the stability region have been obtained. One of them is the BCU, that has been accepted as the most efficient method in the determination of the critical cct. Despite BCU´s efficiency, it fails in many cases. The main objective of this research is to solve two problems associated to the BCU: the first problem is the fact that not always the maximum of potential energy occurs in the neighborhood of PEBS. The second problem is the fact that in many cases the trajectory of the reduced gradient system does not pass in the neighborhood of the controlling unstable equilibrium point. For the solution of these problems it will be used a dynamic method for the exit point detection and the Shadowing Method to calculate the controlling unstable equilibrium point. Tests have shown that the aforementioned robust algorithms find the solutions for many cases for which BCU and PEBS method fails, but many cases of fail persist. Some alternative methods, based in the mentioned algorithms are proposed, and so, aspects to improve its convergency.

We consider the design of finite element methods for inverse problems with full-field data governed by elliptic forward operators. Such problems arise in applications in inverse heat conduction, in mechanical property characterization, and in medical imaging. For this class of problems, novel finite element methods have been proposed (Barbone et al., 2010) that give good performance, provided the solutions are in the H^1(Ω) function space. The material property distributions being estimated can be discontinuous, however, and therefore it is desirable to have formulations that can accommodate discontinuities in both data and solution. Toward this end, we present a mixed variational formulation for this class of problems that handles discontinuities well. We motivate the mixed formulation by examining the possibility of discretizing using a discontinuous discretization in an irreducible finite element method, and discuss the limitations of that approach. We then derive a new mixed formulation based on a least-square error in the constitutive equation. We prove that the continuous variational formulations are well-posed for applications in both inverse heat conduction and plane stress elasticity. We derive a priori error bounds for discretization error, valid in the limit of mesh refinement. We demonstrate convergence of the method with mesh refinement in cases with both continuous and discontinuous solutions. Finally we apply the formulation to measured data to estimate the elastic shear modulus distributions in both tissue mimicking phantoms and in breast masses from data collected in vivo.