Neste trabalho, uma análise quantitativa e qualitativa para a estabilização dinâmica de um pêndulo invertido com uma força externa senoidal aplicada no ponto de suspensão é feita. Inicialmente, a perturbação externa é composta de um único cosseno, então uma generalização é feita, usando uma soma de N cossenos com diferentes amplitudes e frequências. Aproximações são testadas e o tempo durante o qual o pêndulo invertido permanece estável é explorado quando N é grande, a fim de recuperar o padrão do caso onde N = 1. O caso específico de oscilações periódicas e quase periódicas, quando N = 2, é analisado e diagramas de estabilidade considerando diferentes frequências e amplitudes são estudados. Depois, um ruído Gaussiano additivo é adicionado ao sistema para que a degradação dos diagramas de estabilidade gerados por variâncias diferentes possam ser estudados. Todos os pontos deste trabalho são corroborados por simulações, as quais integram numericamente as equações de movimento do sistema através do método de Runge-Kutta de quarta ordem. Os algoritmos e detalhes extras dos métodos de integração usados são explorados numa publicação deste trabalho, a qual está apresentada, nesta dissertação, como um apêndice. / In this work a quantitative and qualitative analysis of the dynamical stabilization of an inverted pendulum with a sinusoidal external perturbation applied at the suspension point is made. Initially, the external perturbation is composed of a single cosine, then a generalization is made using a sum of N cosines with different amplitudes and frequencies. Approximations are tested, and the time for which the inverted pendulum remains stable is explored when N is large, in order to recover the pattern of the case when N = 1. The specific case of periodic and almost periodic oscillations, when N = 2, is analysed and stability diagrams considering different frequencies and amplitudes are studied. Later, an additive Gaussian noise is added to the system so the degradation of the stability diagrams generated by different variances can be studied. All points of this work are corroborated by simulations, which numerically integrate the system’s equation of motion through a fourth order Runge-Kutta method. Algorithms and extra details on the integration methods used are explored in a publication of this work, which is presented in this thesis as an appendix.

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.

The focus of this study is the prediction of trajectories of solid particles injected into either a cylindrically- shaped solid rocket motor (SRM) or a bidirectional vortex chamber (BV). The Lagrangian particle trajectory is assumed to be governed by drag, virtual mass, Magnus, Saffman lift, and gravity forces in a Stokes flow regime. For the conditions in a solid rocket motor, it is determined that either the drag or gravity forces will dominate depending on whether the sidewall injection velocity is high (drag) or low (gravity). Using a one-way coupling paradigm in a solid rocket motor, the effects of particle size, sidewall injection velocity, and particle-to-gas density ratio are examined. The particle size and sidewall injection velocity are found to have a greater impact on particle trajectories than the density ratio. Similarly, for conditions associated with a bidirectional vortex engine, it is determined that the drag force dominates. Using a one-way particle tracking Lagrangian model, the effects of particle size, geometric inlet parameter, particle-to-gas density ratio, and initial particle velocity are examined. All but the initial particle velocity are found to have a significant impact on particle trajectories. The proposed models can assist in reducing slag retention and identifying fuel injection configurations that will ensure proper confinement of combusting droplets to the inner vortex in solid rocket motors and bidirectional vortex engines, respectively.

Delay differential equations (DDEs) are a class of differential equations that have received considerable recent attention and been shown to model many real life problems, traditionally formulated as systems of ordinary differential equations (ODEs), more naturally and more accurately. Ideally a DDE modeling package should provide facilities for approximating the solution, performing a sensitivity analysis and estimating unknown parameters. In this thesis we propose new techniques for efficient simulation, accurate sensitivity analysis and reliable parameter estimation of DDEs. We propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a general linear method (GLM) and can provide dense output. This is done by treating a general DDE as a special example of a discontinuous IVP. We identify a precise process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes. We introduce an equation governing the dynamics of sensitivities for the most general system of parametric DDEs. Then, having a similar view as the simulation (DDEs as discontinuous ODEs), we introduce a formula for finding the size of jumps that appear at discontinuity points when the sensitivity equations are integrated. This leads to an algorithm which can compute sensitivities for various kind of parameters very accurately. We also develop an algorithm for reliable parameter identification of DDEs. We propose a method for adding extra constraints to the optimization problem, changing a possibly non-smooth optimization to a smooth problem. These constraints are effectively handled using information from the simulator and the sensitivity analyzer. Finally, we discuss the structure of our evolving modeling package DDEM. We present a process that has been used for incorporating existing codes to reduce the implementation time. We discuss the object-oriented paradigm as a way of having a manageable design with reusable and customizable components. The package is programmed in C++ and provides a user-friendly calling sequences. The numerical results are very encouraging and show the effectiveness of the techniques.

Delay differential equations (DDEs) are a class of differential equations that have received considerable recent attention and been shown to model many real life problems, traditionally formulated as systems of ordinary differential equations (ODEs), more naturally and more accurately. Ideally a DDE modeling package should provide facilities for approximating the solution, performing a sensitivity analysis and estimating unknown parameters. In this thesis we propose new techniques for efficient simulation, accurate sensitivity analysis and reliable parameter estimation of DDEs. We propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a general linear method (GLM) and can provide dense output. This is done by treating a general DDE as a special example of a discontinuous IVP. We identify a precise process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes. We introduce an equation governing the dynamics of sensitivities for the most general system of parametric DDEs. Then, having a similar view as the simulation (DDEs as discontinuous ODEs), we introduce a formula for finding the size of jumps that appear at discontinuity points when the sensitivity equations are integrated. This leads to an algorithm which can compute sensitivities for various kind of parameters very accurately. We also develop an algorithm for reliable parameter identification of DDEs. We propose a method for adding extra constraints to the optimization problem, changing a possibly non-smooth optimization to a smooth problem. These constraints are effectively handled using information from the simulator and the sensitivity analyzer. Finally, we discuss the structure of our evolving modeling package DDEM. We present a process that has been used for incorporating existing codes to reduce the implementation time. We discuss the object-oriented paradigm as a way of having a manageable design with reusable and customizable components. The package is programmed in C++ and provides a user-friendly calling sequences. The numerical results are very encouraging and show the effectiveness of the techniques.

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.

Modeling and prediction of gene-expression patterns has an important place in computational biology and bioinformatics. The measure of gene expression is determined from the genomic analysis at the mRNA level by means of microarray technologies. Thus, mRNA analysis informs us not only about genetic viewpoints of an organism but also about the dynamic changes in environment of that organism. Different mathematical methods have been developed for analyzing experimental data. In this study, we discuss the modeling approaches and the reasons why we concentrate on models derived from differential equations and improve the pioneering works in this field by including affine terms on the right-hand side of the nonlinear differential equations and by using Runge- Kutta instead of Euler discretization, especially, with Heun&rsquo / s method. Herewith, for stability analysis we apply modified Brayton and Tong algorithm to time-discrete dynamics in an extended space.

Neste trabalho, uma análise quantitativa e qualitativa para a estabilização dinâmica de um pêndulo invertido com uma força externa senoidal aplicada no ponto de suspensão é feita. Inicialmente, a perturbação externa é composta de um único cosseno, então uma generalização é feita, usando uma soma de N cossenos com diferentes amplitudes e frequências. Aproximações são testadas e o tempo durante o qual o pêndulo invertido permanece estável é explorado quando N é grande, a fim de recuperar o padrão do caso onde N = 1. O caso específico de oscilações periódicas e quase periódicas, quando N = 2, é analisado e diagramas de estabilidade considerando diferentes frequências e amplitudes são estudados. Depois, um ruído Gaussiano additivo é adicionado ao sistema para que a degradação dos diagramas de estabilidade gerados por variâncias diferentes possam ser estudados. Todos os pontos deste trabalho são corroborados por simulações, as quais integram numericamente as equações de movimento do sistema através do método de Runge-Kutta de quarta ordem. Os algoritmos e detalhes extras dos métodos de integração usados são explorados numa publicação deste trabalho, a qual está apresentada, nesta dissertação, como um apêndice. / In this work a quantitative and qualitative analysis of the dynamical stabilization of an inverted pendulum with a sinusoidal external perturbation applied at the suspension point is made. Initially, the external perturbation is composed of a single cosine, then a generalization is made using a sum of N cosines with different amplitudes and frequencies. Approximations are tested, and the time for which the inverted pendulum remains stable is explored when N is large, in order to recover the pattern of the case when N = 1. The specific case of periodic and almost periodic oscillations, when N = 2, is analysed and stability diagrams considering different frequencies and amplitudes are studied. Later, an additive Gaussian noise is added to the system so the degradation of the stability diagrams generated by different variances can be studied. All points of this work are corroborated by simulations, which numerically integrate the system’s equation of motion through a fourth order Runge-Kutta method. Algorithms and extra details on the integration methods used are explored in a publication of this work, which is presented in this thesis as an appendix.

Em problemas de viscoelasticidade computacional, a discretização espacial para a solução global das equações de equilíbrio é acoplada à discretização temporal para a solução de um problema de valor inicial local do fluxo viscoelástico. É demonstrado que este acoplamento espacial-temporal (ou global-local) éconsistente se o tensor de deformação total, agindo como elemento acoplador, tem uma aproximação de ordem p ao longo do tempo igual à ordem de convergência do método de integração de Runge-Kutta (RK). Para a interpolação da deformação foram utilizados polinômios baseados em soluções obtidas nos tempos tn+1, tn, . . ., tn+2−p, p ≥ 2, fornecendo dados consistentes de deformação nos estágios do RK. Em uma situação onde tal regra para a interpolação da deformação não é satisfeita, a integração no tempo apresentará, consequentemente, redução de ordem, baixa precisão e, por conseguinte, eficiência inferior. Em termos gerais, o propósito é generalizar esta condição de consistência proposta pela literatura, formalizando-a matematicamente e o demonstrando através da utilização de métodos de Runge-Kutta diagonalmente implícitos (DIRK) até ordem p = 4, aplicados a modelos viscoelásticos não-lineares sujeitos a deformações finitas. Através de exemplos numéricos, os algoritmos de integração temporal adaptados apresentaram ordem de convergência nominal e, portanto, comprovam a validade da formalização do conceito de interpolação consistente da deformação. Comparado com o método de integração de Euler implícito, é demonstrado que os métodos DIRK aqui aplicados apresentam um ganho considerável em eficiência, comprovado através dos fatores de aceleração atingidos. / In computational viscoelasticity, spatial discretization for the solution of the weak form of the balance of linear momentum is coupled to the temporal discretization for solving a local initial value problem (IVP) of the viscoelastic flow. It is shown that this spatial- temporal (or global-local) coupling is consistent if the total strain tensor, acting as the coupling agent, exhibits the same approximation of order p in time as the convergence order of the Runge-Kutta (RK) integration algorithm. To this end we construct interpolation polynomials based on data at tn+1, tn, . . ., tn+2−p, p ≥ 2, which provide consistent strain data at the RK stages. If this novel rule for strain interpolation is not satisfied, time integration shows order reduction, poor accuracy and therefore less efficiency. Generally, the objective is to propose a generalization of this consistency idea proposed in the literature, formalizing it mathematically and testing it using diagonally implicit Runge-Kutta methods (DIRK) up to order p = 4 applied to a nonlinear viscoelasticity model subjected to finite strain. In a set of numerical examples, the adapted time integrators obtain full convergence order and thus approve the novel concept of consistency. Substantially high speed-up factors confirm the improvement in the efficiency compared with Backward Euler algorithm.

Neste trabalho, uma análise quantitativa e qualitativa para a estabilização dinâmica de um pêndulo invertido com uma força externa senoidal aplicada no ponto de suspensão é feita. Inicialmente, a perturbação externa é composta de um único cosseno, então uma generalização é feita, usando uma soma de N cossenos com diferentes amplitudes e frequências. Aproximações são testadas e o tempo durante o qual o pêndulo invertido permanece estável é explorado quando N é grande, a fim de recuperar o padrão do caso onde N = 1. O caso específico de oscilações periódicas e quase periódicas, quando N = 2, é analisado e diagramas de estabilidade considerando diferentes frequências e amplitudes são estudados. Depois, um ruído Gaussiano additivo é adicionado ao sistema para que a degradação dos diagramas de estabilidade gerados por variâncias diferentes possam ser estudados. Todos os pontos deste trabalho são corroborados por simulações, as quais integram numericamente as equações de movimento do sistema através do método de Runge-Kutta de quarta ordem. Os algoritmos e detalhes extras dos métodos de integração usados são explorados numa publicação deste trabalho, a qual está apresentada, nesta dissertação, como um apêndice. / In this work a quantitative and qualitative analysis of the dynamical stabilization of an inverted pendulum with a sinusoidal external perturbation applied at the suspension point is made. Initially, the external perturbation is composed of a single cosine, then a generalization is made using a sum of N cosines with different amplitudes and frequencies. Approximations are tested, and the time for which the inverted pendulum remains stable is explored when N is large, in order to recover the pattern of the case when N = 1. The specific case of periodic and almost periodic oscillations, when N = 2, is analysed and stability diagrams considering different frequencies and amplitudes are studied. Later, an additive Gaussian noise is added to the system so the degradation of the stability diagrams generated by different variances can be studied. All points of this work are corroborated by simulations, which numerically integrate the system’s equation of motion through a fourth order Runge-Kutta method. Algorithms and extra details on the integration methods used are explored in a publication of this work, which is presented in this thesis as an appendix.