Many critical systems are based on the combination of components from different physical domains (e.g. mechanical, electrical, hydraulic), and are mathematically modeled as Switched Multi-Domain Kirchhoff Networks (SMDKN). In this thesis, we tackle a major obstacle to formal verification of SMDKN, namely devising a global model amenable to verification in the form of a Hybrid Automaton. This requires the combination of the local dynamics of the components, expressed as Differential Algebraic Equations, according to Kirchhoff's laws, depending on the (exponentially many) operation modes of the network. We propose an automated SMT-based method to analyze networks from multiple physical domains, detecting which modes induce invalid (i.e. inconsistent) constraints, and to produce a Hybrid Automaton model that accurately describes, in terms of Ordinary Differential Equations, the system evolution in the valid modes, catching also the possible non-deterministic behaviors. The experimental evaluation demonstrates that the proposed approach allows several complex multi-domain systems to be formally analyzed and model checked against various system requirements.

This thesis is concerned with the efficient numerical simulation of finite element based flexible multibody systems. Scaling operations are systematically applied to the governing index-3 differential algebraic equations in order to solve the problem of ill conditioning for small time step sizes. The importance of augmented Lagrangian terms is demonstrated. The use of fast sparse solvers is justified for the solution of the linearized equations of motion resulting in significant savings of computational costs. Three time stepping schemes for the integration of the governing equations of flexible multibody systems are discussed in detail. These schemes are the two-stage Radau IIA scheme, the energy decaying scheme, and the generalized-α method. Their formulations are adapted to the specific structure of the governing equations of flexible multibody systems. The efficiency of the time integration schemes is comprehensively evaluated on a series of test problems. Formulations for structural and constraint elements are reviewed and the problem of interpolation of finite rotations in geometrically exact structural elements is revisited. This results in the development of a new improved interpolation algorithm, which preserves the objectivity of the strain field and guarantees stable simulations in the presence of arbitrarily large rotations. Finally, strategies for the spatial discretization of beams in the presence of steep variations in cross-sectional properties are developed. These strategies reduce the number of degrees of freedom needed to accurately analyze beams with discontinuous properties, resulting in improved computational efficiency.

Nesta pesquisa são analisados os mecanismos que levam um sistema elétrico de potência (SEP) à instabilidade proveniente de uma perturbação de grande porte e as formas de se avaliar diretamente a margem de estabilidade desse sistema quando o mesmo é modelado preservando a estrutura da rede de transmissão. O sistema foi matematicamente modelado por um conjunto de equações algébrico-diferenciais (EAD), que permite modelagens mais compreensivas da carga e da rede e possibilita melhor avaliação da estabilidade de um sistema quando comparado com o modelo de equações diferenciais ordinárias (EDO) utilizado tradicionalmente para o estudo de estabilidade transitória. A avaliação direta da margem de estabilidade do sistema de potência modelado por conjuntos de EAD foi realizada usando métodos diretos de análise com base no conceito de ponto de equilíbrio instável de controle (PEIC). Tais métodos permitem a obtenção da margem deforma local e rápida, sem requerer a integração numérica de equações diferenciais do modelo pós-falta. No entanto, existem alguns problemas abertos para se alcançar a completa fundamentação do método PEIC para modelos de EAD. Assim, neste estudo dá-se um passo nesta direção, mostrando que as definições existentes de PEIC e de outros pontos de interesse podem ser falhas, principalmente quando a trajetória de falta do sistema alcança superfícies singulares. Neste sentido, são propostos a correção destas definições e um método direto de detecção do PEIC. O método proposto é adequado para análise direta de estabilidade angular e de tensão de curto-termos devido a grandes perturbações e capaz de fornecer corretos tempos críticos de abertura e a identificação dos mecanismos de instabilização do sistema de EAD, mesmo quando as trajetórias do sistema alcançam superfícies singulares. / This thesis addresses to the mechanisms that lead an electric power system to instability due to large disturbances and to the methods to assess directly the stability margin when the system is modeled preserving the network structure. The system is modeled by a set of differential-algebraic equations (DAE) that permits more comprehensive models for the load and network and provides a better stability margin assessment when compared to the model of ordinary differential equations (ODE) traditionally used for transient stability analysis. The direct assessment of the stability margin was realized using direct methods based on the controlling unstable equilibrium point (CUEP) concept and permits to assess the margin in a local and fast manner, without requires the time integration of the post-fault system differential equations. Nevertheless, some open problems remain to be solved in order to provide a complete foundation of the CUEP method for DAE power system models. In this research a further step is given in this direction, showing that the existent definitions for the CUEP and other interest points may fail, mainly when the fault-on trajectory reaches singular surfaces. In this sense, it is proposed the correction of these definitions and a new CUEP method that is adequate to the angular and voltage short-term direct stability assessment due to large disturbances; capable to provide precise critical clearing times and the identification of the instability mechanisms for the DAE modeled power system, even in the presence of singular surfaces.

As equações polinomiais são estudadas desde a antiguidade e atualmente são utilizadas, por exemplo, para modelar problemas do cotidiano nas mais variadas áreas do conhecimento. As técnicas de solução de equações polinomiais nem sempre são triviais, principalmente quando envolvem equações de alta ordem e raízes complexas. O ensino desse tema no Ensino Básico é limitado a equações de segundo ou terceiro grau e coeficientes inteiros, o que restringe a aplicação em problemas mais realistas. Assim, o objetivo deste trabalho é trazer uma contribuição aos estudantes, aos professores do Ensino Básico e aos demais interessados, apresentando um material que aborde técnicas de resolução para equação polinomial de diversas naturezas. Iniciamos por uma revisão dos números complexos e dos polinômios, suas operações e propriedades. Embasamos o trabalho com teoremas e permeamos de exemplos com um crescente grau de dificuldade. Dividimos as técnicas de resolução em analíticas e numéricas. Entre as primeiras, tratamos das relações de Girard, das fórmulas resolventes e de alguns casos particulares de equações. Entre as técnicas numéricas, estudamos o método de Newton, o método das secantes e o método de Newton-Bairstow, este último para encontrar raízes complexas. / Polynomial equations have been studied since antiquity and are currently used, for example, to model everyday problems in the most varied areas of knowledge. The solution techniques of polynomial equations are not always trivial, especially when they involve high order equations and complex roots. The teaching of this subject in Basic Education is limited to second or third degree equations and integer coefficients, which restricts the application to more realistic problems. Thus, the objective of this work is to bring a contribution to students, teachers of Basic Education and other interested parties, presenting a material that treats of resolution techniques for polynomial equation of different natures. We begin with a review of complex numbers and polynomials, their operations and properties. We support the work with theorems and permeate examples with an increasing degree of difficulty. We divide the techniques of resolution into analytical and numerical. Among the first, we deal with Girards relations, the resolvent formulas, and some particular cases of equations. Among numerical techniques, we studied the Newton method, the secant method, and the Newton-Bairstow method, the last one to find complex roots.

Orientadora: Profª Dra. Ana Carolina Boero. / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional - PROFMAT, 2014. / This work is dedicated to the study of algebraic equations. A historical overview of the subject has been done, from ancient times to Galois. We have studied methods for solving algebraic equations of degree less than or equal to 4 and we present an example of an equation of 5th degree that is not solvable by radicals. We also propose an approach to the subject in high school. Finally, there is a chapter dedicated to teachers who wish to deepen their knowledge in the subject. In it, we show how some facts of calculus can be used to explain the nature of the roots of the equation x3 + px + q = 0 using the sign of its discriminant. / Este trabalho se dedica ao estudo das equações algébricas. Foi feito um apanhado histórico do assunto, desde os tempos antigos até Galois. Estudamos métodos para resolver equações algébricas de grau menor ou igual a 4 e exibimos um exemplo de equação de grau 5 que não é solúvel por radicais. Apresentamos, também, uma proposta de abordagem do tema no Ensino Médio. Encerramos este trabalho com um capítulo dedicado aos professores do Ensino Médio que desejam se aprofundar no assunto. Nele, mostramos como alguns fatos do cálculo podem ser usados para explicar a natureza das raízes da equação x3 + px + q = 0 a partir do sinal de seu discriminante.

Differential-algebraic equations (DAEs) consist of mixed systems of ordinary differential equations (ODEs) coupled with linear or nonlinear equations. Such systems may be viewed as ODEs with integral curves lying in a manifold. DAEs appear frequently in applications such as classical mechanics and electrical circuits. This thesis concentrates on systems of index 2, originally index 3, and mixed index 2 and 3. Fast and efficient numerical solvers for DAEs are highly desirable for finding solutions. We focus primarily on the class of Gauss-Lobatto SPARK methods. However, we also introduce an extension to methods proposed by Murua for solving index 2 systems to systems of mixed index 2 and 3. An analysis of these methods is also presented in this thesis. We examine the existence and uniqueness of the proposed numerical solutions, the influence of perturbations, and the local error and global convergence of the methods. When applied to index 2 DAEs, SPARK methods are shown to be equivalent to a class of collocation type methods. When applied to originally index 3 and mixed index 2 and 3 DAEs, they are equivalent to a class of discontinuous collocation methods. Using these equivalences, (s,s)--Gauss-Lobatto SPARK methods can be shown to be superconvergent of order 2s. Symplectic SPARK methods applied to Hamiltonian systems with holonomic constraints preserve well the total energy of the system. This follows from a backward error analysis approach. SPARK methods and our proposed EMPRK methods are shown to be Lagrange-d'Alembert integrators. This thesis also presents some numerical results for Gauss-Lobatto SPARK and EMPRK methods. A few problems from mechanics are considered.

Aquesta tesi doctoral proposa formulacions de Galerkin discontinu (DG) d'alt ordre per fluxos viscosos incompressibles. Es desenvolupa un nou mètode de DG amb penalti interior (IPM-DG), que condueix a una forma feble simètrica i coerciva pel terme de difusió, i que permet assolir una aproximació espacial d'alt ordre. Aquest mètode s'aplica per resoldre les equacions de Stokes i Navier-Stokes. L'espai d'aproximació de la velocitat es descompon dins de cada element en una part solenoidal i una altra irrotacional, de manera que es pot dividir la forma dèbil IPM-DG en dos problemes desacoblats. El primer permet el càlcul de les velocitats i de les pressions híbrides, mentre que el segon calcula les pressions en l'interior dels elements. Aquest desacoblament permet una reducció important del número de graus de llibertat tant per velocitat com per pressió. S'introdueix també un paràmetre extra de penalti resultant en una formulació DG alternativa per calcular les velocitats solenoidales, on les pressions no apareixen. Les pressions es poden calcular com un post-procés de la solució de les velocitats. Es contemplen altres formulacions DG, com per exemple el mètode Compact Discontinuous Galerkin, i es comparen al mètode IPM-DG. Es proposen mètodes implícits de Runge-Kutta d'alt ordre per problemes transitoris incompressibles, permetent obtenir esquemes incondicionalment estables i amb alt ordre de precisió temporal. Les equacions de Navier-Stokes incompressibles transitòries s'interpreten com un sistema de Equacions Algebraiques Diferencials, és a dir, un sistema d'equacions diferencials ordinàries corresponent a la equació de conservació del moment, més les restriccions algebraiques corresponent a la condició d'incompressibilitat. Mitjançant exemples numèrics es mostra l'aplicabilitat de les metodologies proposades i es comparen la seva eficiència i precisió. / This PhD thesis proposes divergence-free Discontinuous Galerkin formulations providing high orders of accuracy for incompressible viscous flows. A new Interior Penalty Discontinuous Galerkin (IPM-DG) formulation is developed, leading to a symmetric and coercive bilinear weak form for the diffusion term, and achieving high-order spatial approximations. It is applied to the solution of the Stokes and Navier-Stokes equations. The velocity approximation space is decomposed in every element into a solenoidal part and an irrotational part. This allows to split the IPM weak form in two uncoupled problems. The first one solves for velocity and hybrid pressure, and the second one allows the evaluation of pressures in the interior of the elements. This results in an important reduction of the total number of degrees of freedom for both velocity and pressure. The introduction of an extra penalty parameter leads to an alternative DG formulation for the computation of solenoidal velocities with no presence of pressure terms. Pressure can then be computed as a post-process of the velocity solution. Other DG formulations, such as the Compact Discontinuous Galerkin method, are contemplated and compared to IPM-DG. High-order Implicit Runge-Kutta methods are then proposed to solve transient incompressible problems, allowing to obtain unconditionally stable schemes with high orders of accuracy in time. For this purpose, the unsteady incompressible Navier-Stokes equations are interpreted as a system of Differential Algebraic Equations, that is, a system of ordinary differential equations corresponding to the conservation of momentum equation, plus algebraic constraints corresponding to the incompressibility condition. Numerical examples demonstrate the applicability of the proposed methodologies and compare their efficiency and accuracy.

We study the regularization problem for linear, constant coefficient descriptor systems $E x^. = AX + Bu, y_1 = Cx, y_2=\Gamma x^.$ by proportional and derivative mixed output feedback. Necessary and sufficient conditions are given, which guarantee that there exist output feedbacks such that the closed-loop system is regular, has index at most one and $E +BG\Gamma$ has a desired rank, i.e. there is a desired number of differential and algebraic equations. To resolve the freedom in the choice of the feedback matrices we then discuss how to obtain the desired regularizing feedback of minimum norm and show that this approach leads to useful results in the sense of robustness only if the rank of E is decreased. Numerical procedures are derived to construct the desired feedbacks gains. These numerical procedures are based on orthogonal matrix transformations which can be implemented in a numerically stable way.

No description available.

Nesta pesquisa são analisados os mecanismos que levam um sistema elétrico de potência (SEP) à instabilidade proveniente de uma perturbação de grande porte e as formas de se avaliar diretamente a margem de estabilidade desse sistema quando o mesmo é modelado preservando a estrutura da rede de transmissão. O sistema foi matematicamente modelado por um conjunto de equações algébrico-diferenciais (EAD), que permite modelagens mais compreensivas da carga e da rede e possibilita melhor avaliação da estabilidade de um sistema quando comparado com o modelo de equações diferenciais ordinárias (EDO) utilizado tradicionalmente para o estudo de estabilidade transitória. A avaliação direta da margem de estabilidade do sistema de potência modelado por conjuntos de EAD foi realizada usando métodos diretos de análise com base no conceito de ponto de equilíbrio instável de controle (PEIC). Tais métodos permitem a obtenção da margem deforma local e rápida, sem requerer a integração numérica de equações diferenciais do modelo pós-falta. No entanto, existem alguns problemas abertos para se alcançar a completa fundamentação do método PEIC para modelos de EAD. Assim, neste estudo dá-se um passo nesta direção, mostrando que as definições existentes de PEIC e de outros pontos de interesse podem ser falhas, principalmente quando a trajetória de falta do sistema alcança superfícies singulares. Neste sentido, são propostos a correção destas definições e um método direto de detecção do PEIC. O método proposto é adequado para análise direta de estabilidade angular e de tensão de curto-termos devido a grandes perturbações e capaz de fornecer corretos tempos críticos de abertura e a identificação dos mecanismos de instabilização do sistema de EAD, mesmo quando as trajetórias do sistema alcançam superfícies singulares. / This thesis addresses to the mechanisms that lead an electric power system to instability due to large disturbances and to the methods to assess directly the stability margin when the system is modeled preserving the network structure. The system is modeled by a set of differential-algebraic equations (DAE) that permits more comprehensive models for the load and network and provides a better stability margin assessment when compared to the model of ordinary differential equations (ODE) traditionally used for transient stability analysis. The direct assessment of the stability margin was realized using direct methods based on the controlling unstable equilibrium point (CUEP) concept and permits to assess the margin in a local and fast manner, without requires the time integration of the post-fault system differential equations. Nevertheless, some open problems remain to be solved in order to provide a complete foundation of the CUEP method for DAE power system models. In this research a further step is given in this direction, showing that the existent definitions for the CUEP and other interest points may fail, mainly when the fault-on trajectory reaches singular surfaces. In this sense, it is proposed the correction of these definitions and a new CUEP method that is adequate to the angular and voltage short-term direct stability assessment due to large disturbances; capable to provide precise critical clearing times and the identification of the instability mechanisms for the DAE modeled power system, even in the presence of singular surfaces.