To improve the performance of the three-way catalytic (TWC) converter, advanced control strategies and on-board diagnostics (OBD) systems are needed. Both rely on a relatively accurate but computationally efficient TWC converter model. This thesis aims to develop a control-oriented model that can be employed to develop the control strategies and OBD systems of the TWC converter. The thesis consists of two parts, i.e., the high-fidelity model development and the model reduction. Firstly, a high-fidelity model is built using the energy and mass conservation principles. In this model, a constant inlet simulation is used to validate the warming-up characteristics, and a driving cycle simulation is used to calibrate the reaction rate parameters. The results of the simulation show that the high-fidelity model has adequate accuracy. Secondly, a reduced-order model is developed based on phase and reaction simplifications of the high-fidelity model. The aim of the development of the reduced-order model is to propose a computationally efficient model for further development of control strategies and state estimators for OBD systems. The accuracy of the reduced-order model is then validated by means of simulations. / Thesis / Master of Applied Science (MASc)

In this thesis, I examine filtering based stabilization methods to design new regularized reduced order models (ROMs) for under-resolved simulations of unsteady, nonlinear, convection-dominated systems. The new ROMs proposed are variable delta filtering applied to the evolve-filter-relax ROM (V-EFR ROM), variable delta filtering applied to the Leray ROM, and approximate deconvolution Leray ROM (ADL-ROM). They are tested in the numerical setting of Burgers equation, a nonlinear, time dependent problem with one spatial dimension. Regularization is considered for the low viscosity, convection dominated setting. / Master of Science / Numerical solutions of partial differential equations may not be able to be efficiently computed in a way that fully captures the true behavior of the underlying model or differential equation, especially if significant changes in the solution to the differential equation occur over a very small spatial area. In this case, non-physical numerical artifacts may appear in the computed solution. We discuss methods of treating these calculations with a goal of improving the fidelity of numerical solutions with respect to the original model.

In the effort to reduce emission and fuel consumption in industrial gas turbines, lean premixed combustion is utilized but is susceptible to thermoacoustic instabilities. These instabilities occur due to an in-phase relationship between acoustic pressure and unsteady heat release in a combustor. Thermoacoustic instabilities have been shown to cause structural damage and limit operability of combustors. To mitigate these instabilities, a variety of active and passive methods can be employed. The addition of porous inert media (PIM) is a passive mitigation technique that has been shown to be effective at mitigating an instability. Practical industrial application of a mitigation strategy requires early-stage design considerations such as reduced order modeling, which is often used to study a systems' stability response to geometric changes and mitigation approaches. These reduced order models rely on flame transfer functions (FTF) which numerically represent the relationship between heat release and acoustic perturbations. The accurate quantification of heat release is critical in the study of these instabilities and is a necessary component to improve the reduced order model's predictive capability. Heat release quantification presents numerous challenges. Previous work resolving heat release has used optical diagnostics. For perfectly premixed, laminar flames, it has been shown there are proportional relationships between OH* or CH* chemiluminescence to heat release. This is an ideal case; in reality, practical burners produce turbulent and partially premixed flames. Due to the additional straining of the flame caused by turbulence, the heat release is no longer proportional to chemiluminescence quantities. Also, partially premixed systems have spatially varying equivalence ratios and heat release rates, meaning analysis reliant on perfectly premixed assumptions cannot be used and techniques that can handle spatial variations is needed. The objective of this thesis is to incorporate PIM effects into a reduced order model and resolve quantities vital to understand how PIM is mitigating thermoacoustic instabilities in a partially premixed, turbulent combustion environment. The initial work presented in this thesis is the development of a reduced order model for predicting mode shapes and system stability with and without PIM. This was the first time that a reduced order model was developed to study PIM effects on the thermoacoustic response. Model development used a linear FTF and can predict the system frequency and stability response. Through the frequency response, mode shapes can be constructed which show the axial variation in acoustic values, along with node and anti-node locations. Stability trends can be predicted, such as the independent effects of system parameter variation, to determine its stability response. The model was compared to canonical case studies as well as experimental data with reasonable agreement. With PIM addition, it was shown that a combustor would be under stable operation at more flow conditions than without PIM. The work also shows the stability sensitivity to different porous parameters and PIM locations within the combustor. The model has been used to aid in the design of other combustion systems developed at Virginia Tech's Advanced Propulsion and Power Laboratory. To better understand how PIM is affecting the system stability and demonstrate measurements for the improvement of a numerical FTF, experimental work to resolve the spatially varying equivalence ratio fluctuations was conducted in an atmospheric, swirl-stabilized combustor. The experimental studies worked to improve the fundamental understanding of PIM and its mitigation effects through spatially and temporally resolved equivalence ratios during a self-excited instability. The experimental combustor has an optically accessible flame region which allowed for high speed chemiluminescence to be captured during the instability. Equivalence ratio values were calculated from a relation involving OH*/CH* chemiluminescence ratio. The acoustic perturbations were studied to show how the equivalence ratio fluctuations were being generated and coupling with the acoustic waves. The fluctuation in equivalence ratio showed about 65% variation around its mean value during the period of an instability cycle. When porous media was added to the system, the fluctuation in equivalence ratio was mitigated and a reduction in peak frequency (sound pressure level) SPL of 38 dB was observed. Changes in the spatial distribution of equivalence ratio with PIM addition were shown to produce a more stable operation. Effects such as locally richer burning and changes to recirculation zones promoted more stable operation with PIM addition. Testing with forced acoustic input was also conducted to quantify the flame response. The results demonstrated that a flame in a system with PIM responds differently than without PIM, highlighting the need to develop FTF for systems with PIM. This experimental study was the first to study equivalence ratio in a turbulent, partially premixed combustor using PIM as a mitigation technique. A final experimental investigation was conducted to resolve the spatially defined heat release and its fluctuation during a thermoacoustic instability period. This was the first time that heat release had been investigated in a partially premixed, thermoacoustically unstable system, using PIM as a migration method. Heat release was quantified using equivalence ratio, strain rate, OH* intensity, and a correction factor determined from a chemical kinetic solver. The heat release analysis built upon the equivalence ratio study with additional flow field analysis using PIV. The velocity vectors showed prominent corner and central recirculation zones in the no PIM case which have been shown to be feedback mechanisms that support instability formation. With PIM addition, these flow features were reduced and decoupled from the combustor inlet reactants. The velocity results were decomposed using a spectral proper orthogonal decomposition (SPOD) method. The energy breakdown showed how PIM reduced and distributed the energy in the flow structures, creating a more stable flow field. Heat release results with velocity information demonstrated the significant coupling mechanisms in the flow field that were mitigated with the PIM addition. The no PIM case showed high heat release areas being directly influenced by the incoming flow fluctuations. The feedback mechanisms, both mean flow and acoustic, have a direct path to the incoming flow to the combustor. In the PIM case, there is significant mixing and burning taking place in locations where it is less likely that feedback can reach the incoming flow to couple to form an instability. The methodology to quantify heat release provides a framework for quantifying a non-linear FTF with PIM. The development and testing to determine a non-linear FTF with PIM are reserved for future work and discussed in the final chapter. The methodologies and modeling conducted here provided insight and understanding to answer why PIM is effective at mitigating a thermoacoustic instability and how it can be studied using a reduced order numerical tool. / Doctor of Philosophy / In the effort to reduce emission and fuel consumption in industrial gas turbines, lean premixed combustion is utilized but is susceptible to thermoacoustic instabilities. These instabilities occur due to a relationship between acoustic pressure and unsteady heat release in a combustor. Thermoacoustic instabilities have been shown to cause structural damage and limit operability of combustors. To mitigate these instabilities, a variety of active and passive methods can be employed. The addition of porous inert media (PIM) is a passive mitigation technique that has been shown to be effective at mitigating an instability. Implementation of these mitigation strategies require early-stage design considerations such as reduced order modeling, which is often used to study a systems' stability response to geometric changes and mitigation approaches. These reduced order models rely on flame transfer functions (FTF) which numerically model the flame response. The accurate quantification of heat release is critical in the study of these instabilities and is a necessary component to improve the reduced order model's predicative capability. Heat release quantification presents numerous challenges. Previous work resolving heat release has used optical diagnostics with varying levels of success. For perfectly premixed, laminar flames, it has been shown there are proportional relationships between flame light emission and heat release. This is an ideal case; in reality, practical burners produce complex turbulent flames. Due to complex turbulent flame, the heat release is no longer proportional to the flame light emission quantities. Also, partially premixed systems have spatially variant flame quantities, meaning analyses reliant on perfectly premixed assumptions cannot be used and techniques that can handle spatial variations are required. The objective of this thesis is to incorporate PIM effects into a reduced order model and resolve quantities vital to understand how PIM is mitigating thermoacoustic instabilities in a partially premixed, turbulent combustion environment. The initial work presented in this thesis is the development of a reduced order model for predicting mode shapes and system stability with and without PIM. The model uses a simple relationship to model the flame response in an acoustic framework. To improve the model and understanding of PIM mitigation, experimental data such as the local heat release rates and equivalence ratios need to be quantified. An experimental technique was utilized on an optically accessible atmospheric, swirl-stabilized combustor, to resolve the spatially variant equivalence ratio and heat release rates. From these results, better understanding of how PIM is improving the stability in a combustion environment is shown. Quantities such as velocity, acoustic pressure, equivalence ratio, and heat release are all studied and used to explain the improved stability with PIM addition. The methodologies and modeling conducted here provided insight and understanding to answer why PIM is effective at mitigating a thermoacoustic instability and how it can be studied using a reduced order numerical tool. Furthermore, the present work provides a framework for quantifying spatially varying heat release measurements, which can be used to develop FTF for use with thermoacoustic modeling approaches.

An optimal solution of vortex-induced vibrations of structures would be a time-domain numerical simulation that simultaneously solves the fluid flow and structural response. Yet, the requirements in terms of computing power remains a major obstacle for implementing such a simulation. On the other hand, lower- or reduced-order models provide an alternative for determining structural response to forcing by fluid flow. The objective of this thesis is to provide a consistent approach for the development of reduced-order models for the lift and drag on oscillating cylinders and the identification of their parameters. Amplitudes and phases of higher-order spectral moments of the lift and drag coefficients data are combined with approximate solutions of the representative models to determine their parameters. The results show that the amplitude and phase of the trispectrum could be used to model the lift on the oscillating cylinder under different excitation conditions. Moreover, the amplitude and phase of the cross-bispectrum could be used to establish the lift-drag relation for oscillating cylinders. A forced van der Pol equation is used to represent the lift on a transversely oscillating cylinder, and a parametrically excited van der Pol equation is used to model the lift coefficient on an inline oscillating cylinder. All cases of excitations lead to close values for the damping and nonlinear parameters in the van der Pol equation. Consequently, and as shown in this thesis, different excitation cases could be used to identify the parameters in the governing equations. Moreover, the results show that the drag coefficient could be derived from the lift coefficient through a square relation that takes into account the effects of the forced motions. / Ph. D.

Heat transfer systems contain many sources of nonlinearity including temperature dependent material properties, radiation boundary conditions, and internal source terms. Despite progress in numerical simulations, producing accurate models that can predict these complex behaviors are still encumbered by lengthy processing times. Accurate models can be produced quickly by utilizing projection Reduced Order Modeling (ROM) techniques. For discretized systems, the Singular Value Decomposition technique is the preferred approach but has had limited success on treating nonlinearities. In this research, the treatment of nonlinear temperature dependent material properties was incorporated into a ROM. Additional sources of nonlinearities such as radiation boundary conditions, temperature dependent source heating terms, and complex geometry were also integrated. From the results, low conductivity, highly nonlinear material properties were predicted by the ROM within 1% of full order models, and additional nonlinearities were predicted within 8%. A study was then done to identify initial snapshots for use in developing a ROM that can accurately predict results across a wide range of inputs. From this, a step function was identified as being the most accurate and computationally efficient. The ROM was further investigated by a discretization study to assess computational gains in both 1D and 3D models as a function of mesh density. The lower mesh densities in the 1D and 3D ROMs resulted in moderate computational times (up to 40 times faster). However, highly discretized systems such as 5000 nodes in 1D and 125000 nodes in 3D resulted in computational gains on the order of 2000 to 3000 times faster than the full order model. / Ph. D.

Risers são tubos que transportam fluidos do fundo do mar até as plataformas flutuantes e vice-versa. Diversas configurações e materiais são utilizados, porém apenas os steel catenary risers (SCR) são estudados neste trabalho. Os risers são estruturas extremamente esbeltas e, por isso, grande parte de seu trecho suspenso tem comportamento de cabo. Apenas em duas regiões a rigidez flexional é relevante: no hang-off (topo) e na touch-down zone (TDZ), sendo esta última a região mais complexa para análise devido ao contato unilateral com o solo. Em função dos diversos carregamentos dinâmicos a que o riser é submetido, grandes variações na curvatura ocorrem na TDZ, além de impacto e atrito com o solo, que podem reduzir a vida útil da estrutura ou até mesmo por em risco a sua integridade. Por estas razões, este trabalho visa à elaboração de uma metodologia analítica para a construção de um modelo de ordem reduzida (MOR) capaz de analisar o comportamento dinâmico não linear da TDZ de um SCR. Como na TDZ a rigidez flexional predomina sobre a rigidez geométrica, o riser é modelado como uma viga semi-infinita, tendo uma parte suspensa e outra apoiada sobre solo hipoteticamente elástico com contato unilateral. Na extremidade suspensa são impostos deslocamentos verticais e trações dinâmicas que fazem com que a posição do touch-down point (TDP) também varie com o tempo. Trata-se, portanto, de um problema com condições de contorno móveis. A metodologia adotada para a resolução deste problema foi transformá-lo em um problema de condições de contorno fixas por meio de uma transformação de variáveis. Contudo, paga-se um preço por tal transformação, e fortes não linearidades surgem na equação diferencial de movimento, tornando-a extremamente complexa para uma resolução analítica direta. Para o problema de flexão simples, consegue-se obter os modos de vibração não lineares através do método das múltiplas escalas. De posse destes modos, utiliza-se o método de Galerkin não linear para projetar a equação completa em um modo escolhido, transformando o modelo contínuo em um modelo de ordem reduzida com apenas um grau de liberdade, cuja coordenada generalizada modal é o deslocamento horizontal do TDP. Obtida a equação do MOR, nota-se que existem coeficientes que variam com o tempo, como na clássica equação de Mathieu, indicando a possibilidade de ocorrer ressonância paramétrica. Neste tipo de ressonância, entre outras possibilidades, pode ocorrer que a frequência de excitação seja o dobro da frequência natural trata-se da ressonância paramétrica principal. A equação do MOR é integrada numericamente e suas respostas são comparadas com as respostas obtidas por modelos de elementos finitos elaborados em softwares comerciais, como o Abaqus e o Orcaflex. Por fim, discutem-se as potencialidades e limitações do MOR, sendo uma grande vantagem a possibilidade de processar diversos casos facilmente, variando os parâmetros que influem nas respostas. Com este mapeamento das respostas, é possível estimar as amplitudes dos estados estacionários pós-críticos. / Risers are pipes that convey fluids from the seabed up to the floating platforms and vice-versa. Many configurations and materials are used, but only steel catenary risers (SCR) are studied in this work. Risers are extremely slender structures, and for this reason, most of the suspended part has cable behavior. Only in two regions the bending stiffness is important: at the hang-off and at the touch-down zone (TDZ), which is the most complex region for analysis because of the unilateral contact with the seabed. Due to several dynamic loads that the riser is subjected to, great curvature variations occur at the TDZ, apart from impacts and friction with the soil, which can reduce the life time of the structure or even jeopardize its integrity. For these reasons, this work aims at the development of an analytical methodology for the construction of a reduced-order model (ROM) able to analyze the nonlinear dynamic behavior of the TDZ of a SCR. As at the TDZ the bending stiffness prevails over the geometric stiffness, the riser is modeled as a semi-infinite beam, having a suspended part and another one resting on the elastic soil with unilateral contact. At the end of the suspended part, vertical displacements and dynamic tensions are imposed, that cause the TDPs position to vary with time. It is, therefore, a problem with moving boundary conditions. The methodology adopted for solving this problem was to transform it into a problem with fixed boundary conditions via a variable transformation. However, a price is paid for such a transformation, and strong nonlinearities appear in the differential equation of motion, making it extremely complex to solve analytically. For the simple bending problem, nonlinear vibration modes are obtained via the method of multiple scales. In possession of these modes, the nonlinear Galerkin method is used to project the complete equation into a chosen mode, transforming the continuum model into a reduced-order model (ROM) with only one degree of freedom whose modal generalized coordinate is the horizontal displacement of the TDP. After obtaining the ROM, it is noticed that there are coefficients that vary with time, as in the classic Mathieu equation, indicating the possibility of parametric resonance. In this kind of resonance, among other possibilities, the excitation frequency may be twice the natural frequency it is the so-called principal parametric resonance. The ROMs equation is integrated numerically and the responses are compared to those given by finite-element models studied with the help of commercial softwares, like Abaqus and Orcaflex. Finally, the potentialities and limitations of the ROM are discussed. One of the advantages is the possibility of processing several cases easily, changing the parameters that affect the responses. With this response mapping, it is possible to estimate the post-critical steady-state amplitudes that take place.

Risers são tubos que transportam fluidos do fundo do mar até as plataformas flutuantes e vice-versa. Diversas configurações e materiais são utilizados, porém apenas os steel catenary risers (SCR) são estudados neste trabalho. Os risers são estruturas extremamente esbeltas e, por isso, grande parte de seu trecho suspenso tem comportamento de cabo. Apenas em duas regiões a rigidez flexional é relevante: no hang-off (topo) e na touch-down zone (TDZ), sendo esta última a região mais complexa para análise devido ao contato unilateral com o solo. Em função dos diversos carregamentos dinâmicos a que o riser é submetido, grandes variações na curvatura ocorrem na TDZ, além de impacto e atrito com o solo, que podem reduzir a vida útil da estrutura ou até mesmo por em risco a sua integridade. Por estas razões, este trabalho visa à elaboração de uma metodologia analítica para a construção de um modelo de ordem reduzida (MOR) capaz de analisar o comportamento dinâmico não linear da TDZ de um SCR. Como na TDZ a rigidez flexional predomina sobre a rigidez geométrica, o riser é modelado como uma viga semi-infinita, tendo uma parte suspensa e outra apoiada sobre solo hipoteticamente elástico com contato unilateral. Na extremidade suspensa são impostos deslocamentos verticais e trações dinâmicas que fazem com que a posição do touch-down point (TDP) também varie com o tempo. Trata-se, portanto, de um problema com condições de contorno móveis. A metodologia adotada para a resolução deste problema foi transformá-lo em um problema de condições de contorno fixas por meio de uma transformação de variáveis. Contudo, paga-se um preço por tal transformação, e fortes não linearidades surgem na equação diferencial de movimento, tornando-a extremamente complexa para uma resolução analítica direta. Para o problema de flexão simples, consegue-se obter os modos de vibração não lineares através do método das múltiplas escalas. De posse destes modos, utiliza-se o método de Galerkin não linear para projetar a equação completa em um modo escolhido, transformando o modelo contínuo em um modelo de ordem reduzida com apenas um grau de liberdade, cuja coordenada generalizada modal é o deslocamento horizontal do TDP. Obtida a equação do MOR, nota-se que existem coeficientes que variam com o tempo, como na clássica equação de Mathieu, indicando a possibilidade de ocorrer ressonância paramétrica. Neste tipo de ressonância, entre outras possibilidades, pode ocorrer que a frequência de excitação seja o dobro da frequência natural trata-se da ressonância paramétrica principal. A equação do MOR é integrada numericamente e suas respostas são comparadas com as respostas obtidas por modelos de elementos finitos elaborados em softwares comerciais, como o Abaqus e o Orcaflex. Por fim, discutem-se as potencialidades e limitações do MOR, sendo uma grande vantagem a possibilidade de processar diversos casos facilmente, variando os parâmetros que influem nas respostas. Com este mapeamento das respostas, é possível estimar as amplitudes dos estados estacionários pós-críticos. / Risers are pipes that convey fluids from the seabed up to the floating platforms and vice-versa. Many configurations and materials are used, but only steel catenary risers (SCR) are studied in this work. Risers are extremely slender structures, and for this reason, most of the suspended part has cable behavior. Only in two regions the bending stiffness is important: at the hang-off and at the touch-down zone (TDZ), which is the most complex region for analysis because of the unilateral contact with the seabed. Due to several dynamic loads that the riser is subjected to, great curvature variations occur at the TDZ, apart from impacts and friction with the soil, which can reduce the life time of the structure or even jeopardize its integrity. For these reasons, this work aims at the development of an analytical methodology for the construction of a reduced-order model (ROM) able to analyze the nonlinear dynamic behavior of the TDZ of a SCR. As at the TDZ the bending stiffness prevails over the geometric stiffness, the riser is modeled as a semi-infinite beam, having a suspended part and another one resting on the elastic soil with unilateral contact. At the end of the suspended part, vertical displacements and dynamic tensions are imposed, that cause the TDPs position to vary with time. It is, therefore, a problem with moving boundary conditions. The methodology adopted for solving this problem was to transform it into a problem with fixed boundary conditions via a variable transformation. However, a price is paid for such a transformation, and strong nonlinearities appear in the differential equation of motion, making it extremely complex to solve analytically. For the simple bending problem, nonlinear vibration modes are obtained via the method of multiple scales. In possession of these modes, the nonlinear Galerkin method is used to project the complete equation into a chosen mode, transforming the continuum model into a reduced-order model (ROM) with only one degree of freedom whose modal generalized coordinate is the horizontal displacement of the TDP. After obtaining the ROM, it is noticed that there are coefficients that vary with time, as in the classic Mathieu equation, indicating the possibility of parametric resonance. In this kind of resonance, among other possibilities, the excitation frequency may be twice the natural frequency it is the so-called principal parametric resonance. The ROMs equation is integrated numerically and the responses are compared to those given by finite-element models studied with the help of commercial softwares, like Abaqus and Orcaflex. Finally, the potentialities and limitations of the ROM are discussed. One of the advantages is the possibility of processing several cases easily, changing the parameters that affect the responses. With this response mapping, it is possible to estimate the post-critical steady-state amplitudes that take place.

D'un point de vue industriel, la mise en place de nouvelles architectures de systèmes mécaniques nécessite un long processus de conception permettant de définir et d'anticiper le comportement. Dans le cas particulier des systèmes aéronautiques tels que les moteurs d'avions, un certain nombre de pièces sont particulièrement sensibles car elles doivent répondre à des impératifs stricts en termes d'encombrement, de performance et de tenue mécanique. Dans ce contexte, la prévision du comportement vibratoire revêt une importance particulière puisqu'elle permet d'évaluer le niveau des sollicitations cycliques appliquées sur le système et guide ainsi la détection en amont d'éventuels problèmes de fatigue des matériaux. La plupart du temps, des modèles numériques sont utilisés pour représenter les structures, et le comportement est simulé en résolvant un ensemble d'équations. Pour atteindre un niveau de détail répondant au besoin industriel, ces modèles peuvent être particulièrement gros, et la résolution des équations associées demande des ressources et des temps de calcul considérables. De plus, pour rendre compte au mieux des comportements observés expérimentalement, il est souvent nécessaire de prendre en compte des phénomènes non-linéaires, ce qui augmente encore la difficulté. Les travaux présentés dans ce manuscrit concernent cette problématique du comportement vibratoire des structures non-linéaires et s'orientent autour de deux axes : la réduction de modèle et le calcul des solutions multiples. L'objectif du premier axe est de contribuer à la construction de modèles numériques non linéaires réduits utilisables en conception de systèmes industriels et de proposer des outils d'exploitation et d'interprétation de ces modèles. En particulier, on considère le cas des méthodes de projection de Galerkin et on montre qu'elles sont à même de construire des modèles réduits réalistes. Des méthodes complémentaires de réduction de modèles sont également présentées dans le cas particulier de la recherche de solutions par la méthode de la balance harmonique (HBM) : on s'intéressera en particulier à des méthodes de sélection d'harmoniques. Après avoir comparé les différentes méthodes proposées sur un exemple simple de poutre non-linéaire, elles sont appliquées à un modèle de structure industrielle représentant une aube d'hélice d'open rotor. Le second axe de ces travaux concerne le calcul de solutions multiples pour les systèmes dynamiques non-linéaires. Une particularité de ces systèmes est en effet de présenter plusieurs configurations stables pour un état de sollicitation donné. Il s'agira ici de proposer des méthodes de calcul permettant de dresser la liste exhaustive des solutions possibles. Le travail présenté se concentre sur la recherche de solutions périodiques par la méthode de la balance harmonique pour des systèmes possédant des non-linéarités polynomiales. Ces restrictions conduisent à la résolution de systèmes polynomiaux pour lesquels il existe des méthodes permettant de calculer l'ensemble des solutions. En particulier, on propose l'utilisation originale de méthodes basées sur le calcul de bases de Groebner pour la résolution de systèmes polynomiaux issus de la mécanique. Les différentes méthodes présentées sont illustrées et comparées sur des exemples simples. Les résultats montrent que même pour des systèmes simples, le comportement dynamique peut être très complexe. / In an industrial context, the design of new mechanical systems requires long design processes in order to define and to anticipate the behavior of all the constitutive parts. In the particular case of aeronautical structures such as plane engines, design is especially critical since they have to meet various and strict needs (life duration, performances . . .). Then, anticipating vibratory behavior is very important as this provides information about cyclic solicitations and fatigue. Most often, numerical models are used to mimic the structure and mechanical behavior is simulated by solving a set of differential equations. In the case of industrial structures, such models can be quite large and their resolution very time-consuming. Moreover, in order to model experimental behavior realistically, it is often necessary to take nonlinear phenomena into account and thus increase the required computational effort. The work presented in this PhD deals with the study of mechanical nonlinear systems. It focuses on two principal directions : model reduction and multiple solutions computation. The goal of the first direction is to contribute to the building of numerical reduced order models usable in industrial context and to propose tools to exploit an interpret them. Particularly, Galerkin projection methods are investigated in the context of nonlinear systems reduction, showing that those methods are, under certain conditions, able to give a reliable picture of full system behavior. In the case of the harmonic balance method, complementary methods are also proposed to reduce the size of the algebraic equations system by using harmonic selection techniques. The presented methods are firstly illustrated and compared on a simple nonlinear beam example ; they are then applied to an industrial model of open rotor blade. The second direction of this work deals with the computation of multiple solutions arising in nonlinear dynamical systems. Indeed, it has been shown that such systems can present different stable configurations for a given solicitation. The objective here is to provide tools for computing such multiple solutions. We only consider the case of periodic solutions for systems with polynomial nonlinearities, treated with harmonic balance method. These hypotheses enable one to search for multiple states as solutions of polynomial algebraic systems of equations, for which some methods exist to compute the entire set of solutions. In particular, we propose to use methods relying on Groebner basis computation, in order to compute the whole set of solutions. The proposed methods are illustrated and compared on simple examples, showing that even such simple systems can present very complex dynamical behavior.

京都大学 / 新制・課程博士 / 博士(工学) / 甲第23511号 / 工博第4923号 / 新制||工||1769(附属図書館) / 京都大学大学院工学研究科電気工学専攻 / (主査)教授 松尾 哲司, 教授 雨宮 尚之, 准教授 久門 尚史 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

This dissertation contains work that addresses both theoretical and numerical aspects of reduced order models (ROMs). In an under-resolved regime, the classical Galerkin reduced order model (G-ROM) fails to yield accurate approximations. Thus, we propose a new ROM, the data-driven variational multiscale ROM (DD-VMS-ROM) built by adding a closure term to the G-ROM, aiming to increase the numerical accuracy of the ROM approximation without decreasing the computational efficiency. The closure term is constructed based on the variational multiscale framework. To model the closure term, we use data-driven modeling. In other words, by using the available data, we find ROM operators that approximate the closure term. To present the closure term's effect on the ROMs, we numerically compare the DD-VMS-ROM with other standard ROMs. In numerical experiments, we show that the DD-VMS-ROM is significantly more accurate than the standard ROMs. Furthermore, to understand the closure term's physical role, we present a theoretical and numerical investigation of the closure term's role in long-time integration. We theoretically prove and numerically show that there is energy exchange from the most energetic modes to the least energetic modes in closure terms in a long time averaging. One of the promising contributions of this dissertation is providing the numerical analysis of the data-driven closure model, which has not been studied before. At both the theoretical and the numerical levels, we investigate what conditions guarantee that the small difference between the data-driven closure model and the full order model (FOM) closure term implies that the approximated solution is close to the FOM solution. In other words, we perform theoretical and numerical investigations to show that the data-driven model is verifiable. Apart from studying the ROM closure problem, we also investigate the setting in which the G-ROM converges optimality. We explore the ROM error bounds' optimality by considering the difference quotients (DQs). We theoretically prove and numerically illustrate that both the ROM projection error and the ROM error are suboptimal without the DQs, and optimal if the DQs are used. / Doctor of Philosophy / In many realistic applications, obtaining an accurate approximation to a given problem can require a tremendous number of degrees of freedom. Solving these large systems of equations can take days or even weeks on standard computational platforms. Thus, lower-dimensional models, i.e., reduced order models (ROMs), are often used instead. The ROMs are computationally efficient and accurate when the underlying system has dominant and recurrent spatial structures. Our contribution to reduced order modeling is adding a data-driven correction term, which carries important information and yields better ROM approximations. This dissertation's theoretical and numerical results show that the new ROM equipped with a closure term yields more accurate approximations than the standard ROM.