This thesis deals with different computational techniques related to some classes of nonlinear response regimes of engineering interest. The work is mainly divided into two parts. In the first part different numeric-analytic integration techniques for nonlinear oscillators are developed. In the second part, procedures for handling arbitrarily large rotations are addressed and a few novel developments are reported in the process. To begin the first part, we have proposed an explicit numeric-analytic technique, based on the Adomian decomposition method, for integrating strongly nonlinear oscillators. Numerical experiments suggest that this method, like most other numerical techniques, is versatile and can accurately solve strongly nonlinear and chaotic systems with relatively larger step-sizes. It is then demonstrated that the procedure may also be effectively employed for solving two-point boundary value problems with the help of a shooting algorithm. This has been followed up with the derivation and numerical exploration of variants of a recently developed numeric-analytic technique, the multi-step transversal linearization (MTrL), in the context of nonlinear oscillators of relevance in engineering dynamics. A considerable generalization and improvement over the original form of a MTrL strategy is achieved in this study. Finally, we have used the concept of MTrL method on the nonlinear variational (rate) equation corresponding to a nonlinear oscillator and thus derive another family of numeric-analytic techniques, presently referred to as the multi-step tangential linearization (MTnL). A comparison of relative errors through the MTrL and MTnL techniques consistently indicate a superior quality of approximation via the MTrL route. In the second part of the thesis, a scheme for numerical integration of rigid body rotation is proposed using only rudimentary tensor analysis. The equations of motion are rewritten in terms of rotation vectors lying in same tangent spaces, thereby facilitating vector space operations consistent with the underlying geometric structure of rotation. One of the most important findings of this part of the dissertation is that the existing constant-preserving algorithms are not necessarily accurate enough and may not be ideally applicable to cases wherein numerical accuracy is of primary importance. In contrast, the proposed rotation-algorithms, the higher order ones in particular, are significantly more accurate for conservative rotational systems for reasonably long time. Similar accuracy is expected for dissipative rotational systems as well. The operators relating rotation variables corresponding to different tangent spaces are also investigated and this should provide further insight into the understanding of rotation vector parametrization. A rotation update is next proposed in terms of rotation vectors. This update, employed along with interpolation of relative rotations, gives a strain-objective and path independent finite element implementation of a geometrically exact beam. The method has the computational advantage of requiring considerably less nodal variables due to the use of rotation vector parametrization. We have proposed a new isoparametric interpolation of nodal quaternions for computing the rotation field within an element. This should be a computationally efficient alternative to the interpolation of local rotations. It has been proved that the proposed interpolation of rotation leads to the objectivity of strain measures. Several numerical experiments are conducted to demonstrate the frame invariance, path-independence and other superior aspects of the present approach vis-`a-vis the existing methods based on the rotation vector parametrization. It is emphasized that, in order to develop an objective finite element formulation, the use of relative rotation is not mandatory and an interpolation of total rotation variables conforming with the rotation manifold should suffice.

Two codes have been developed and implemented for use on massively parallelsuper computers to simulate two-dimensional and quasi-geostrophic turbulence.The codes have been found to scale well with increasing resolution and width ofthe simulations. This has allowed for the highest resolution simulations of twodimensionaland quasi-geostrophic turbulence so far reported in the literature.The direct numerical simulations have focused on the statistical characteristicsof turbulent cascades of energy and enstrophy, the role of coherent vorticesand departures from universal scaling laws, theoretized more than 40 yearsago. In particular, the investigations have concerned the enstrophy and energycascades in forced and decaying two-dimensional turbulence. Furthermore, theapplicability of Charney’s hypotheses on quasi-geostrophic turbulence has beentested. The results have shed light on the flow evolution at very large Reynoldsnumbers. The most important results are the robustness of the enstrophycascade in forced and decaying two-dimensional turbulence, the sensitivity toan infrared Reynolds number in the spectral scaling of the energy spectrumin the inverse energy cascade range, and the validation of Charney’s predictionson the dynamics of quasi-geostrophic turbulence. It has also been shownthat the scaling of the energy spectrum in the enstrophy cascade is insensitiveto intermittency in higher order statistics, but that corrections apply to the”universal” Batchelor-Kraichnan constant, as a consequence of large-scale dissipationanomalies following a classical remark by Landau (Landau & Lifshitz1987). Another finding is that the inverse energy cascade is maintained bynonlocal triad interactions, which is in contradiction with the classical localityassumption. / QC 20101029

Die Zeitverzögerung bei der Streuung in zwei Dimensionen ist eine Funktion von zwei unabhängigen Parametern. Wenn diese Funktion Sattelpunkte aufweist, so hat der entsprechende Funktionswert theoretisch ein unendlich großes Gewicht in der Wahrscheinlichkeitsverteilung der Zeitverzögerungen. Dieser Zusammenhang soll analytisch und numerisch nachgewiesen und detailliert beschrieben werden. Insbesondere soll die klassische und quantenmechanische Wahrscheinlichkeitsverteilung der Zeitverzögerung für ein Modellsystem aus mehreren nichtüberlappenden zentralsymmetrischen Potentialen berechnet werden. Erwartete Ergebnisse sind Aussagen über die Parameterwerte, bei denen der oben genannte Effekt zu beobachten ist sowie Näherungsformeln für die Verteilung der Zeitverzögerung in der Nähe der Singularitäten. Außerdem soll die quantenmechanisch zu erwartende Glättung der Verteilungsfunktion quantitativ beschrieben werden. / For scattering problems in two dimensions, time-delay is a function of two independent parameters. If this function features saddle points, the corresponding function value should theoretically have an infinite weight in the probability distribution of time-delays. This correlation shall be confirmed analytically and numerically and studied in-depth. In particular, the classical and quantum-mechanical probability distribution of time-delays shall be calculated for a model system consisting of multiple non-overlapping potentials with rotational symmetry. We expect to obtain information about the parameter values where the aforementioned effects can be observed, and analytical approximations for the time-delay distribution near the singularities. Furthermore, the smoothing of the distribution in the quantummechanical regime shall be quantified.

Made available in DSpace on 2016-06-02T20:18:17Z (GMT). No. of bitstreams: 1 4192.pdf: 6083601 bytes, checksum: a411c24424aa71f8ff5f0404ef48215b (MD5) Previous issue date: 2012-02-16 / Universidade Federal de Sao Carlos / The multifactorial nature of coronary artery disease (CAD) includes complications related to angina and acute myocardial infarction (AMI) and disorders involving sympathetic and parasympathetic cardiac autonomic modulation. The objective of this study was to evaluate the autonomic modulation of heart rate (HR) by linear and non-linear methods in healthy men and in patients with AMI and different percentages of coronary stenosis, as well as its relation with CAD risk factors. In order to evaluate heart rate variability (HRV), the HR and the RR intervals were recorded for 15 min in the supine position. Based on the results of this study, three manuscripts were written: The first manuscript presents the results of 10 men with AMI (57±9 years old) (2nd and 7th day after coronary event) and 11 healthy men (53±4 years old). The HRV analysis was carried out using linear methods in the time domain (TD=RMSSD and SDNN) and frequency domain (FD= low frequency (LF) and high frequency (HF) in normalized units (nu) and LF/HF) and using the non-linear methods approximate entropy (ApEn). A significant relationship between the linear and non-linear methods and the RMSSD, SDNN, LFun, HFun and LF/HF and ApEn indexes was observed. The linear and non-linear HRV indexes from the healthy group were higher than those of the AMI group on the 2nd and 7th days, which suggests that the analysis of HRV with linear methods in the TD and FD and the use of ApEn for linear analysis are in agreement, both for healthy subjects and patients after AMI. The second manuscript presents the results of 52 men (54±5 years old) divided into two groups with coronary obstruction CAD+ &#8805; 50% (n=18) and CAD- < 50% (n=17) and one control group (n=17). HRV analysis was carried out with Shannon entropy (SE) and symbolic analysis (0V and 2ULV). The patients with DAC+ presented lower SE (complexity), 2ULV (vagal predominance) and higher 0V (sympathetic predominance) than the DAC- and control groups, which indicates that cardiac autonomic disorder is related to the degree of coronary occlusion and to cardiac impairment. The third manuscript presents the results for risk factors, ACE I/D polymorphism and the indexes in the TD and FD of 151 patients with CAD (56±8 years old, DD=54, DI=70 and II=27). The results show that there was no relation between the ACE I/D polymorphism and HR, BP or HRV. However, the highest indexes of the HRV, which reflect vagal autonomic modulation, are related to a lower percentage of stenosis and the use of ACE inhibitors. / A doença arterial coronariana (DAC) é de natureza multifatorial sendo que as principais complicações estão relacionadas à angina e infarto agudo do miocárdio (IAM), apresentando disfunção da modulação autonômica cardíaca simpática e parassimpática. Assim, o objetivo foi avaliar a modulação autonômica da frequência cardíaca (FC), a partir de métodos lineares e não lineares, de homens saudáveis, de pacientes com IAM e com diferentes percentuais de estenose coronariana e sua relação com os fatores de risco para a DAC. Para a análise da variabilidade da FC (VFC) foi realizada a captação dos intervalos RR e da FC, durante 15 min na posição supina. A partir dos resultados do estudo foram elaborados três manuscritos. Primeiro manuscrito: Foram apresentados os resultados de 10 homens com IAM (57±9 anos) (avaliados no 2º e 7 º dia após evento coronariano) e 11 homens saudáveis (53±4 anos). A análise da VFC foi realizada utilizando-se dos métodos lineares no domínio do tempo (DT=RMSSD e RMSM) e da frequência (DF=baixa frequência (BF) e alta frequência (AF) em unidades normalizadas (un) e BF/AF) e pelo método não linear de entropia aproximada (EnAp). As análises da relação entre os métodos lineares e o não linear (índices RMSSD, RMSM, BFun, AFun e BF/AF com a EnAp) foi significativa. Os índices lineares e não linear da VFC do grupo saudável foram maiores em relação ao grupo IAM no 2º e no 7º dia. Os resultados mostram que os métodos lineares no DT e no DF e o não linear , são concordantes, para análise da VFC, tanto para voluntários saudáveis como para pacientes após o IAM. Segundo manuscrito: Foram apresentados os resultados de 52 homens (54±5 anos) divididos em três grupos, sendo dois grupos com obstrução coronariana DAC+ (&#8805; 50%; n=18) e DAC- (< 50%; n=17) e um grupo controle (n=17). A análise da VFC foi pela entropia de Shannon (ES) e análise simbólica (0V e 2ULV). Os pacientes com DAC+ apresentam menor ES (complexidade) e 2ULV (predominância vagal) e maior 0V (predominância simpática) quando comparado aos grupos DAC- e controle, o que indica que a disfunção autonômica cardíaca está relacionada ao grau de oclusão coronariana. Terceiro manuscrito: Foram apresentados os resultados da possível relação existente entre dos fatores de risco, do polimorfismo I/D do gene da ECA com os índices no DT e no DF de 151 pacientes com DAC (56±8 anos, DD=54, DI=70 e II=27). Os resultados mostram que não há relação entre o polimorfismo I/D do gene da ECA com a FC, PA e VFC. Já os maiores índices da VFC que refletem a modulação autonômica vagal estão relacionados ao menor percentual de estenose e ao uso de inibidores da ECA.

Este trabalho visa a avaliar os efeitos dinâmicos em vigas esbeltas semi-infinitas com contato unilateral em apoio elástico sob flexão composta, e sua possível aplicação a dutos flexíveis de extração de petróleo offshore em configuração de catenária (Steel Catenary Risers) provocados por esforços de movimento da plataforma. O estudo se desenvolve com formulação de vigas semi-infinitas em duas dimensões (2D) com suporte unilateral elástico, considerando-se inicialmente apenas efeito de flexão, à qual se aplica o método das múltiplas escalas (MME) para obtenção dos modos de vibração, e posteriormente o método das variedades invariantes (MVI) para recuperação das relações modais entre as coordenadas generalizadas e as variáveis modais. Trata-se, a seguir, da formulação do problema com consideração da tração, tanto estática como dinâmica, e a projeção da equação de movimento completa segundo os modos obtidos no problema da flexão simples, para obtenção do modelo de ordem reduzida com efeito de tração, que permitiu estudar este último qualitativa e quantitativamente. A formulação inicial teve como ponto de partida a utilização de mudanças de variáveis que permitiu a transformação do problema de condições móveis em condições fixas de contorno, propiciando caracterizar o movimento vertical da viga na solução analítica do problema de vibrações livres, até o limite possível, para depois utilizar técnicas de integração numérica na solução do problema de vibrações forçadas. A motivação para estudo de tais efeitos está ligada à importância indiscutível que tem para a indústria petrolífera e para a economia mundial, além do crescente papel deste segmento em termos nacionais. A descoberta do Campo de Tupi, na Bacia de Santos, que representa a maior reserva de óleo e gás do País, alavancou pesquisas para extração de petróleo em águas ultra-profundas (profundidades da ordem de 2.500m) (fonte: www.petrobras.com.br), além da necessidade já estabelecida de maiores pesquisas em águas profundas (profundidades que podem chegar a 2.000m). Considere-se também a motivação da própria Petrobrás, cujo intuito é ter capacidade de extrair 4,5 milhões de barris por dia até 2020. Tal capacidade de produção deve exigir 45 sistemas de produção, sendo que cada sistema de produção necessita de 4 a 5 barcos de apoio, totalizando assim 200 barcos operando em 2020 (Fonte: O Estado de São Paulo, Ano 131, Nº 42786, Coluna Celso Ming). Apesar da crescente pesquisa em fontes alternativas de energia, a demanda não só por óleo e gás, mas também pela utilização de polímeros e componentes plásticos, ainda sustentará a extração de petróleo por no mínimo algumas décadas. Além disso, técnicas melhores na fixação de carbono (técnicas naturais e artificiais para seqüestro de carbono), permitem sonhar que a utilização de petróleo, desde a sua queima até a produção de material sintético possa se tornar cada vez menos impactante para o meio ambiente. / This work aims at analyzing the dynamical effects in slender reticulated beams on elastic support with unilateral contact under combined axial force and bending, and the possibility of its association with steel catenary risers connected to offshore structures, caused by movement of the platform. The study is developed based on 2D semi-infinite beams on unilateral elastic support, initially considering only bending effects, to which the multiple scales method (MSM) is applied to obtain vibration modes, and thereafter the invariant manifold method (IMM) to obtain the modal relationships between the generalized coordinates and the modal variables. It follows the formulation of the problem considering tension effects, both from static and dynamical sources, and the projection of the full equation of motion with respect to the vibration modes from the first formulation considering only bending effects, which allows us to obtain the reduced order model with tension effects, and its application to steel catenary risers (SCR). The starting point for the mathematical formulation is a variable transformation, which allows us to change a free boundary into a fixed boundary one. It is then possible to obtain the free-vibration analytical solution, as far as possible, and then use numerical integration schemes to obtain the forced vibration time responses. As motivation, we can firstly mention the unquestionable role of the oil and gas industry for the world economy, besides its increasing market share within the national economy. The discovery of the Tupi Field, in the Santos Basin, which represents the biggest oil and gas reserve of the Country, gave impulse to investments and research in all fields associated to the oil and gas area, in ultra deep water (depth around 2.500m), besides the well known need of research in deep water (depth around 2.000m). In spite of the growing research on alternative sources of energy, the world still demands not only oil and gas, but also polymers and plastic components, and this will still be the scenario for the next decades. Besides that, the increasing of knowledge in carbon sequestration may keep up the use of oil and gas for some more decades, hopefully with less environment impact.

Recentes descobertas de campos de petróleo e gás, a centenas de quilômetros da costa Sudeste do Brasil, em lâmina d\'água acima de 2.000m, demandam avanços científicos e tecnológicos para sua exploração segura. Entre os desafios para os engenheiros, a análise dinâmica de risers offshore é de extrema relevância devido à propensão à fadiga do material estrutural. Os movimentos de unidades flutuantes, que ocorrem devido às ondas de gravidade, impõem um tipo particular de carregamento dinâmico para os risers. Por uma questão de simplicidade, apenas o movimento de heave é considerado, embora pitch and roll também possam desempenhar um papel relevante na análise. De fato, o efeito de heave provoca a modulação de tensãoamplitude ao longo da estrutura tubular e flexível, o que pode conduzir à ressonância paramétrica como consequência da conhecida instabilidade de Mathieu. O riser vertical será o foco desse trabalho. A teoria bidimensional de vigas de Bernoulli-Euler é utilizada para se obter uma equação diferencial não linear de movimento para o riser submetido a uma carga axial e peso submerso. Modelos matemáticos - conhecidos como modelos de ordem reduzida (MORs) - com poucos graus de liberdade são adotados, mas com capacidade adequada para representar a resposta estrutural tanto qualitativa como quantitativamente. Utilizando os modos não lineares como funções de projeção no procedimento de Galerkin não linear, a interpretação física vem da igualdade dos trabalhos virtuais tanto no modelo de alta hierarquia quanto do MOR, com a consequente introdução de vínculos rígidos nos modos excluídos da análise. Aqui, a relação não linear entre as amplitudes modais e modos de vibração/frequências são levados em conta. Os resultados obtidos estão de acordo com os resultados dos testes experimentais de um modelo em escala reduzida realizado sob coordenação do LIFE&MO (Laboratório de Interação Fluido-Estrutura e Mecânica Offshore), fornecendo uma calibração do coeficiente de arrasto equivalente, para se levar em conta o sistema dissipativo que inclui tanto o amortecimento hidrodinâmico e estrutural. As respostas também são comparadas com as obtidas através do estudo de um modelo de elementos finitos, elaborado com o auxílio de um software comercial, o OrcaFlex®, e com funções clássicas de projecção, em particular, a função trigonométrica. Em seguida, variando-se os parâmetros de controle, as respostas do estado estacionário pós-crítica são mapeadas e plotadas em um diagrama policromático. / Recent discoveries of oil and gas fields, hundreds of kilometers off the Southeast coast of Brazil, in water depths above 2,000m, demand scientific and technological advances to support their safe exploitation. Among the challenges posed to engineers, the dynamic analysis of offshore risers is of utmost relevance, due to fatigue of the structural material. The motions of floating units, which occur due to gravity waves, impose a particular type of dynamic loading to the risers. As a matter of simplicity, only heave is considered herewith, although pitch and roll can also play a relevant role in the analysis. In fact, the effect of heave causes tension-amplitude modulation to the long and flexible tubular structure, which may drive parametric resonance as a consequence of the well-known Mathieus instability. The vertical riser will be the focus of this work. Bernoulli-Euler two-dimensional beam theory is used to obtain a nonlinear differential equation of motion for the riser subjected to an axial thrust and submerged weight. Mathematical models known as reduced-order models (ROMs) with few degrees of freedom are used, but with adequate capacity to represent the structural response both qualitatively and quantitatively. Using non-linear modes as projection functions within the non-linear Galerkins procedure, the physical interpretation of which being the identification of virtual works in both the high-hierarchy model and the ROM, with consequent introduction of rigid constraints in the modes excluded from the analysis. Here the non-linear relationship between modal amplitudes and mode shapes/frequencies are taken into account. The results obtained here agree well with those of experimental tests with a small-scale model carried out under coordination LIFE&MO (Laboratory of Fluid-Structure Interaction & Offshore Mechanics), provided calibration of the equivalent drag coefficient is carried out, to account for the overall system dissipation including both hydrodynamic and structural damping. The responses are also compared with those given by finite element models studied with the help of commercial software, the OrcaFlex® and with classical projection functions, in particular the trigonometric function. Then varying the control parameters, the responses of post-critical steady state are mapped and plotted in diagram polychromatic.

Die Zeitverzögerung bei der Streuung in zwei Dimensionen ist eine Funktion von zwei unabhängigen Parametern. Wenn diese Funktion Sattelpunkte aufweist, so hat der entsprechende Funktionswert theoretisch ein unendlich großes Gewicht in der Wahrscheinlichkeitsverteilung der Zeitverzögerungen. Dieser Zusammenhang soll analytisch und numerisch nachgewiesen und detailliert beschrieben werden. Insbesondere soll die klassische und quantenmechanische Wahrscheinlichkeitsverteilung der Zeitverzögerung für ein Modellsystem aus mehreren nichtüberlappenden zentralsymmetrischen Potentialen berechnet werden. Erwartete Ergebnisse sind Aussagen über die Parameterwerte, bei denen der oben genannte Effekt zu beobachten ist sowie Näherungsformeln für die Verteilung der Zeitverzögerung in der Nähe der Singularitäten. Außerdem soll die quantenmechanisch zu erwartende Glättung der Verteilungsfunktion quantitativ beschrieben werden.:1 Einleitung 2 Zeitverzögerung in klassischen Streusystemen 2.1 Definition durch die Wirkung 2.2 Geometrisch motivierte Definitionen 2.2.1 Eigentliche Zeitverzögerung 2.2.2 Definition über retardierten Ort 2.2.3 Definition über Aufenthaltszeit 2.2.4 Numerische Bestimmung der Zeitverzögerung 2.3 Zeitverzögerungsfunktion und -verteilung 2.4 Rechenregeln 2.4.1 Koordinatensystemwechsel 2.4.2 Verkettung 3 Klassische Modellsysteme 3.1 Harte Scheibe 3.2 Verschobene harte Scheibe 3.2.1 Verhalten in der Umgebung von stationären Punkten 3.3 Weiches Scheibenpaar 3.3.1 Sattelpunkte 3.3.2 Extrempunkte 3.3.3 Zusammenfassung 4 Quantenmechanische Zeitverzögerung 4.1 Quantisierung der klassischen Definition 4.1.1 Definition über Aufenthaltszeit 4.1.2 Wigner-Smith-Matrix 4.1.3 Numerische Umsetzung 4.2 Einheitenlose Formulierung 4.3 Gegenüberstellung von Zeitentwicklungsmethoden 4.4 Split-Operator-Methode 4.4.1 Parameterwahl 4.4.2 Zur Abschätzung des systematischen Fehlers 4.5 Unterdrückung der periodischen Randbedingung 4.6 Harte Potentiale 5 Quantenmechanische Modellsysteme 5.1 Stationäre Punkte 5.2 Unschärfeeffekte 5.3 Numerische Ungenauigkeiten 5.3.1 Skalierungsverhalten der numerischen Methoden 5.4 Zusammenfassung der Ergebnisse 6 Zusammenfassung und Ausblick Anhang A Verhalten der Verteilung einer Funktion in der Nähe stationärer Punkte A.1 Umgebung eines Sattelpunktes A.2 Umgebung eines Extremums B Zeitverzögerung für das weiche Scheibenpaar / For scattering problems in two dimensions, time-delay is a function of two independent parameters. If this function features saddle points, the corresponding function value should theoretically have an infinite weight in the probability distribution of time-delays. This correlation shall be confirmed analytically and numerically and studied in-depth. In particular, the classical and quantum-mechanical probability distribution of time-delays shall be calculated for a model system consisting of multiple non-overlapping potentials with rotational symmetry. We expect to obtain information about the parameter values where the aforementioned effects can be observed, and analytical approximations for the time-delay distribution near the singularities. Furthermore, the smoothing of the distribution in the quantummechanical regime shall be quantified.:1 Einleitung 2 Zeitverzögerung in klassischen Streusystemen 2.1 Definition durch die Wirkung 2.2 Geometrisch motivierte Definitionen 2.2.1 Eigentliche Zeitverzögerung 2.2.2 Definition über retardierten Ort 2.2.3 Definition über Aufenthaltszeit 2.2.4 Numerische Bestimmung der Zeitverzögerung 2.3 Zeitverzögerungsfunktion und -verteilung 2.4 Rechenregeln 2.4.1 Koordinatensystemwechsel 2.4.2 Verkettung 3 Klassische Modellsysteme 3.1 Harte Scheibe 3.2 Verschobene harte Scheibe 3.2.1 Verhalten in der Umgebung von stationären Punkten 3.3 Weiches Scheibenpaar 3.3.1 Sattelpunkte 3.3.2 Extrempunkte 3.3.3 Zusammenfassung 4 Quantenmechanische Zeitverzögerung 4.1 Quantisierung der klassischen Definition 4.1.1 Definition über Aufenthaltszeit 4.1.2 Wigner-Smith-Matrix 4.1.3 Numerische Umsetzung 4.2 Einheitenlose Formulierung 4.3 Gegenüberstellung von Zeitentwicklungsmethoden 4.4 Split-Operator-Methode 4.4.1 Parameterwahl 4.4.2 Zur Abschätzung des systematischen Fehlers 4.5 Unterdrückung der periodischen Randbedingung 4.6 Harte Potentiale 5 Quantenmechanische Modellsysteme 5.1 Stationäre Punkte 5.2 Unschärfeeffekte 5.3 Numerische Ungenauigkeiten 5.3.1 Skalierungsverhalten der numerischen Methoden 5.4 Zusammenfassung der Ergebnisse 6 Zusammenfassung und Ausblick Anhang A Verhalten der Verteilung einer Funktion in der Nähe stationärer Punkte A.1 Umgebung eines Sattelpunktes A.2 Umgebung eines Extremums B Zeitverzögerung für das weiche Scheibenpaar

<p>Regardless of their origin or pathology, many, if not all, diseases have long been regarded as complex. Yet, despite the progression in the understanding of complexity and the development of systems biology, the majority of biomedical research has been derived from qualitative principles. In comparison to the ethical, temporal and logistical limitations of human experimentation, <em>in vivo</em> animal models have served to provide a more advantageous means to elucidate the underlying disease mechanisms. However, given the additional limitations presented by such models, <em>in silico </em>models have emerged as an effective complement, and, in some cases, a replacement for <em>in vivo</em> experimentation. The <em>in silico </em>models presented in this thesis were developed using mathematical and computational methods to investigate the evolution of two complex, diverse diseases from a systems biology perspective: allergic asthma and cancer.</p> <p>We generated two novel <em>in silico</em> models of allergic asthma aimed at clarifying some dynamic aspects of allergic responses. Experimentally, we utilized an <em>in vivo</em> murine model of chronic exposure to the most pervasive aeroallergen worldwide, house dust mite (HDM), for up to 20 weeks, equivalent to at least 20 human years. Using a range of HDM concentrations, experimental data were collected to study local and systemic effects. The first model applied empirical mathematical techniques to establish equations for airway inflammation and HDM-specific immunoglobulins using an iterative approach of experimentation and validation. Using the equations generated, we showed that the model was able to accurately predict and simulate data. The model also demonstrated the non-linear relationship between HDM exposure and both airway inflammation and allergic sensitization and identified system thresholds.</p> <p>The second model used mechanistic mathematical techniques to investigate the trafficking of eosinophils as they migrated from bone marrow to the blood and, ultimately, to the lungs. Making use of a limited data set, the model determined the effect of individual processes on the system. We identified eosinophil production, survival and death as having the greatest impacts, while migration played a relatively minor role. Furthermore, the model was used to simulate knockout models and the use of antibodies <em>in silico</em>.</p> <p>In the context of cancer growth and metastasis, we developed a theoretical model demonstrating the spatio-temporal development of a tumour in two-dimensions. The model was encoded to create a computer graphic simulation program, which simulated the effects of various parameters on the size and shape of a tumour. Through simulations, we demonstrated the importance of the diffusion process in cancer growth and metastasis.</p> <p>Ultimately, we believe the greatest benefit of each <em>in silico</em> model is the ability to provide an understanding of each respective disease recognized as dynamic and formally complex, but predominantly studied in reductionist, static or un-integrated approaches.</p> / Doctor of Philosophy (Medical Science)

<p>Efficient global dynamic optimization (GDO) using spatial branch-and-bound (SBB) requires the ability to construct tight bounds for the dynamic model. This thesis works toward efficient GDO by developing effective convex relaxation techniques for models with ordinary differential equations (ODEs). In particular, a novel algorithm, based upon a verified interval ODE method and the McCormick relaxation technique, is developed for constructing convex and concave relaxations of solutions of nonlinear parametric ODEs. In addition to better convergence properties, the relaxations so obtained are guaranteed to be no looser than their underlying interval bounds, and are typically tighter in practice. Moreover, they are rigorous in the sense of accounting for truncation errors. Nonetheless, the tightness of the relaxations is affected by the overestimation from the dependency problem of interval arithmetic that is not addressed systematically in the underlying interval ODE method. To handle this issue, the relaxation algorithm is extended to a Taylor model ODE method, which can provide generally tighter enclosures with better convergence properties than the interval ODE method. This way, an improved version of the algorithm is achieved where the relaxations are generally tighter than those computed with the interval ODE method, and offer better convergence. Moreover, they are guaranteed to be no looser than the interval bounds obtained from Taylor models, and are usually tighter in practice. However, the nonlinearity and (potentially) nonsmoothness of the relaxations impedes their fast and reliable solution. Therefore, the algorithm is finally modified by incorporating polyhedral relaxations in order to generate relatively tight and computationally cheap linear relaxations for the dynamic model. The resulting relaxation algorithm along with a SBB procedure is implemented in the MC++ software package. GDO utilizing the proposed relaxation algorithm is demonstrated to have significantly reduced computational expense, up to orders of magnitude, compared to existing GDO methods.</p> / Doctor of Philosophy (PhD)

La gyrocinétique est un modèle clef pour la microturbulence en physique des plasmas. Elle présente encore plusieurs difficultés, qui pourraient invalider ses équations. Ce rapport de thèse clarifie trois d'entre elles. Tout d'abord, une de des coordonnées causait des soucis, d'un point de vue tant physique que mathématique ; une coordonnée adéquate est introduite, qui dissipe les difficultés et explique les structures intrinsèques sous-jacentes. Ensuite, des relations de récurrence explicites sont obtenues pour tous les ordres du développement perturbatif. Enfin, en utilisant la structure hamiltonienne de la dynamique, le couplage plasma-champ électromagnétique est implémenté d'un façon plus adaptée, avec d'importantes conséquences sur les équations gyrocinétiques.Plusieurs autres résultats sont obtenus, e.g. sur l'origine de l'invariant adiabatique centre-guide, sur une transformation centre-guide minimale très efficace, ou sur un modèle hamiltonien intermédiaire entre Vlasov-Maxwell et la gyrocinétique, dont les caractéristiques de Vlasov contiennent à la fois la dynamique lente centre-guide et la dynamique rapide du gyro-angle. Diverses méthodes de réduction sont utilisées, développées ou introduites, e.g. une transformée de Lie du mouvement, un relèvement transférant les réductions de la dynamique des particules à la dynamique des champs, ou une troncature reliée à la fois à la théorie des contraintes de Dirac et à une projection sur une sous-algèbre. Outre la gyrocinétique, cela clarifie d'autres réductions hamiltoniennes en plasmas, e.g. pour une dynamique incompressible ou électrostatique, pour la MHD, ou pour des fermetures fluides avec tenseur de pression. / Gyrokinetics is a key model for plasma micro-turbulence. It still suffers from several issues, which could imply to reconsider the equations. This thesis dissertation clarifies three of them. First, one of the coordinates caused questions, both from a physical and from a mathematical point of view; a suitable constrained coordinate is introduced, which removes the issues from the theory and explains the intrinsic structures underlying the questions. Second, explicit induction relations are obtained to go arbitrary order in the perturbative expansion. Third, using the Hamiltonian structure of the dynamics, the coupling between the plasma and the electromagnetic field is implemented in a more appropriate way, with strong consequences on the gyrokinetic equations. Several other results are obtained, for instance about the origin of the guiding-center adiabatic invariant, about a very efficient minimal guiding-center transformation, or about an intermediate Hamiltonian model between Vlasov-Maxwell and gyrokinetics, where the characteristics include both the slow guiding-center dynamics and the fast gyro-angle dynamics. In addition, various reduction methods are used, introduced or developed, e.g. a Lie-transform of the equations of motion, a litfing method to transfer particle reductions to the corresponding Hamiltonian field dynamics, or a truncation method related both to Dirac's theory of constraints and to projections onto Lie-subalgebras. Besides gyrokinetics, this is useful to clarify other Hamiltonian reductions in plasma physics, e.g. for incompressible or electrostatic dynamics, for magnetohydrodynamics, or for fluid closures including moments of order two.