In this document we review the status of existing techniques for nonlinear model order reduction by investigating how well these techniques perform for typical industrial needs. In particular the TPWL-method (Trajectory Piecewise Linear-method) and the POD-approach (Proper Orthogonal Decomposion) is taken under consideration. We address several questions that are (closely) related to both the theory and application of nonlinear model order reduction techniques. The goal of this document is to provide an overview of available methods together with a classification of nonlinear problems that in principle could be handled by these methods.

For many decades, turbulence has been the subject of extensive numerical research and experimental work. A bottleneck problem in turbulence research has been to detect and characterize the energetic, space and time-dependent structures and give a mathematical definition to each topology. This research presents a fundamental study of coherent structures, embedded in turbulent flows, by use of Proper Orthogonal Decomposition (POD). The target is to detect dominant features which contain the largest fraction of the total kinetic energy and hence contribute more to a turbulent flow. POD is proven to be a robust methodology in multivariate analysis of non-linear problems. This method also helps to obtain a low-dimensional approximation of a high-dimensional process, like a turbulent flow. This manuscript-based dissertation consists of five chapters. The first chapter starts with a brief introduction to turbulence, available simulation techniques, limitations and practical applications. Next, POD is introduced and the step-by-step approach is explained in detail. Three submitted manuscripts are presented in the subsequent chapters. Each chapter starts with introducing the study case and explaining the contribution of the study to the whole topic and also has its topic-relevant literature review. Each article consists of two parts: flow simulation and verification of the results at the onset, followed by POD analysis and reconstruction of the turbulent flow fields. For flow simulation, Large Eddy Simulation (LES) was performed to obtain databases for POD analysis. The simulations were validated by making comparison with available experimental and numerical studies. For each case, coherent topologies are characterized and the contribution of kinetic energy for each structure is determined and compared with previous literature. The first manuscript focused on investigating the large-scale dynamics in the wake of an infinite square cylinder. This case is the first step towards the targeting study case of this research, i.e. flow over rib roughened walls. The main purpose the first step is to establish a benchmark for comparison to the more complicated cases of a square cylinder with a nearby wall and flow over a rib-roughened surface. For POD analysis, the three-dimensional velocity field is obtained from LES of the flow around an infinite square cylinder at a Reynolds number of Re = 500. The POD algorithm is examined and the total energy of the flow is found to be well captured by only a small number of eigenmodes. From the energy spectrum, it is learned that each eigenmode represents a particular flow characteristic embedded in the turbulent wake, and eigenmodes with analogous characteristics can be bundled as pairs. Qualitative analysis of the dominant modes provided insight as to the spatial distribution of dominant structures in the turbulent wake. Another outcome of this chapter is to develop physical interpretations of the energetic structures by examining the temporal coefficients and tracking their life-cycle. It was observed that the paired temporal coefficients are approximately sinusoidal with similar order of magnitude and frequency and a phase shift. Lastly, it was observed that the turbulent flow field can be approximated by a linear combination of the mean flow and a finite number of spatial modes. The second manuscript analyses the influence of a solid wall on the wake dynamics of an infinite square cylinder. Different cases have been studied by changing the distance between the cylinder and the bottom wall. From the simulation results, it is learned that the value of drag and lift coefficients can be significantly affected by a nearby solid wall. From the energy decay spectrum it is observed that the energy decay rate varies for different gap ratios and accordingly a physical explanation is developed. Visualization of coherent structures for each case shows that for larger gaps, although the structures are distorted and inclined away from the wall, the travelling wave characteristic persists. Lastly, it is observed that as the gap ratio gets smaller, energetic structures originated by the wall begin to appear in the lower index modes. The last manuscript presents a numerical study of the structures in turbulent Couette flow with roughness on one wall, which as mentioned earlier, is the targeting study case of this research. Flow over both roughened and smooth surfaces was examined in a single study. Comparison was made with experiments and other numerical studies to verify the LES results. The mean velocity distribution across the channel shows that the rib roughness on the bottom wall has a strong effect on the velocity profile on the opposite wall. The energetic coherent dynamics of turbulent flow were investigated by the use of POD. The energy decay spectrum was analysed and the influence of a roughened wall and each roughness element on formation of those structures was investigated. Coherent POD modes on a spanwise sampling plane are detected. A secondary swirling motion is visualized, for the first two modes and counter-rotating cells are observed in the lower region of the channel above the rough wall for the higher modes. At the end, a quantitative analysis of the POD temporal coefficients was performed, which characterize the life-cycle of each coherent dynamic. A motivating outcome of this analysis is to decompose the time trace curves into quasi-periodic and fluctuations curves and to detect a linkage between these life cycles and physical meaning and location of each energetic pattern. At the end, in a closuring chapter, concluding remarks of this research work are presented in more detail and some potential extensions have been proposed for future researchers.

The observation of less chaoticity and enhanced interannual periodicity of transient waves in the Martian atmosphere in comparison with that of the Earth suggests the hypothesis of a low-dimensional underlying atmospheric attractor. Grounded on this hypothesis, two questions can be asked: is there a small set of atmospheric modes, measured and classified by a suitable norm, capable of describing the atmosphere of Mars? If this set exists, are those atmospheric modes able to reproduce the dynamical behaviour of the atmosphere of Mars? The answer to these questions, constituting the central focus of this thesis, has led to the first application of POD-Galerkin methods to a state-of-the-art Mars general circulation model. The proper orthogonal decomposition (POD) as a method for extracting coherent structures, called empirical orthogonal functions (EOFs), provided a means to answer the first question in the positive. An important amount of atmospheric total energy (TE) was found to be concentrated in a few EOFs (e.g., 90% TE in 20 EOFs). The most energetic EOFs were identified with atmospheric motions such as thermal tides and transient waves. The Galerkin projection of the hydrostatic primitive equations onto the span of the EOFs provided a systematic method to establish physically plausible interactions between the most energetic EOFs. These interactions were complemented with closure schemes representing interactions with unresolved modes. This requirement proved to be essential in order to obtain bounded behaviour. In the diagnostic analysis, represented by the POD alone, increasing the number of EOFs directly leads to a better approximation of the atmospheric state. In contrast, the dynamic reconstruction of the atmospheric evolution does not depend only on the number of included EOFs. Other important factors to obtain realistic evolution are the inclusion of every mode involved in the description of a particular kind of motion (diurnal tide, semidiurnal tide or transients) and the retention of higher order modes that may interact strongly with the modes of interest. Once these conditions are satisfied the behaviour of the reduced models is greatly improved. Implications of these findings for future work are discussed.

Cette thèse traite de la réduction d’ordre de modèle du phénomène communément rencontré dans la modélisation de microsystèmes, à savoir, dans la littérature anglaise, le « squeeze-film damping ». Dans un premier chapitre sont présentées les différentes méthodes de réduction d’ordre de modèle. Dans le cas des systèmes linéaires, elles ont un cadre théorique bien établi. Ces méthodes peuvent être adaptées pour les systèmes non-linéaires. La validité des modèles réduits résultants sera alors réduite à un certain espace des phases, leur établissement faisant intervenir certaines trajectoires particulières servant d’apprentissage. On présente finalement la méthode des modes normaux non-linéaires dont les modèles résultants ne dépendent pas d’une trajectoire d’apprentissage. Au chapitre 2, on s’intéresse plus particulièrement au phénomène de « squeeze-film damping » régi par l’équation de Reynolds. Après avoir détaillé son établissement à partir de certaines hypothèses, on décrit les différentes méthodes de résolution de l’équation linéaire puis non-linéaire de la littérature. On compare ensuite les résultats d’un modèle de l’équation de Reynolds à des simulations éléments finis de l’équation de Navier-Stokes afin de valider les hypothèses faites pour la dérivation de l’équation de Reynolds. On propose ensuite une résolution originale par changement de variable. On étudie aussi plusieurs autres résolutions possibles ainsi que plusieurs bases de projection parmi celles décrites dans le premier chapitre. Le chapitre 3 est consacré à la modélisation du problème couplé que constitue le micro-interrupteur MEMS qui est un candidat au remplacement des interrupteurs à base de transistors dans les communications RF. Sa modélisation fait intervenir trois domaines, la mécanique, l’électrostatique, et la fluidique à travers l’équation de Reynolds. Après voir décrit les différents modèles de la littérature, on propose un modèle réduit couplé dont le modèle fluidique est basé sur le modèle établi au chapitre 2. Ce modèle est validé par rapport à des modèles différences finies et à des résultats expérimentaux de la littérature.Enfin le quatrième chapitre traite de la réduction du coût d’évaluation du modèle réduit couplé de micro-interrupteur du chapitre 3. La première méthode proposée consiste à trouver une fonction d’approximation de la projection de la force fluidique sur le premier mode mécanique, fonction des coordonnées modales mécaniques position et vitesse. Cette méthode ne se révèle valable que dans le cas incompressible. Dans le cas compressible, la résolution de l’équation de Reynolds restant obligatoire, on utilise la méthode de Rewienski et al. qui consiste à linéariser par morceaux les fonctions régissant la dynamique. Une autre méthode de linéarisation par morceaux, tirant parti d’une particularité du modèle du chapitre 2 et permettant de s’affranchir d’une trajectoire d’apprentissage, est également proposée. / This thesis deals with reduced-order modelling of squeeze-film damping, a fluidic phenomenon that is commonly encountered in MEMS. In the first chapter, reduced-order modelling methods are presented. For linear systems, well-established theories exist. They can be adapted to nonlinear systems. However, the resulting reduced-order models are valid in a certain region of the state-space only, depending on the training trajectory. The method of nonlinear normal modes, which does not depend on a training trajectory is also introduced. Chapter two is focused on the squeeze-film damping phenomenon governed by the Reynolds equation. We first establish the equation from appropriate hypotheses, and then present the different resolutions of its linear and nonlinear form found in literature. The results from a model based on the Reynolds equation are then compared to results from a finite element Navier-Stokes model, in order to validate the various hypotheses made. An original method of resolution based on a change of variable is then proposed. Several other method of resolution are studied as well as different projection bases amongst those presented in chapter one.Chapter three is dedicated to the modelling a micro-switch, a candidate to the replacement of switches based on transistors in RF communications. Its modelling implies the coupling of three domains: mechanics, electrostatics, and fluidics with Reynolds equation. Following a description of the models from literature, a coupled model is proposed, the fluidic model being the one presented in chapter two. This model is validated compared to finite difference models as well as experimental data from the literature.Finally, the fourth chapter aims at reducing the evaluation cost of the coupled micro-switch model established in chapter three. The first method consists in finding an approximation function of the projection of the fluidic force on the first linear mechanical mode as a function of the mechanical modal coordinates, position and speed. This method is applicable in the incompressible case only. In the compressible case, the Reynolds equation has to be solved. The method of Rewienski and al., which consists in piecewise-linearizing the functions governing the dynamics, is used. Another method based on a piecewise-linear approach, taking advantage of the particular structure of the model presented in chapter two, thus not depending on a training trajectory, is proposed.

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Interaction of flow structures from a three-cylinder system is complex and important for fundamental and engineering applications. In this study, experiments using hotwire, 2D PIV, and Tomography are to be conducted to characterize the fluid flow at various Re number and rotation speeds. The Reynolds number considered based on the diameter of the single-cylinder ranges from 37 to 1700. The peaks in the frequency spectrum obtained from the hotwire study show a unique relation of Strouhal number as a function of static incident angle, RPM, and Reynolds number. From the 2D PIV and 3D tomography experiment, vorticity and velocity results characterize the interaction of wake flow from individual cylinders and as a function of the rotational speeds. Besides, the Standard deviation map shows the turbulence intensity variation at the various static and rotating conditions. The obtained results at static conditions are found to be consistent with the previous computational study.

This thesis deals with the formulation and application of reduced order models based on extraction of dominant structures from a system utilizing the method of proper orthogonal decomposition. Time evolution of computed modes is described by a system of ordinary differential equations, which is gained by means of Galerkin projection of these modes onto the Navier-Stokes equations. This methodology was applied on two test cases Kármán vortex street and vortex rope. In both cases, a CFD simulation of one refference point was carried out and by utilizing gained modes, the corresponding reduced order models were formulated. Their results were assessed by comparing to the refference simulation.

In applied mechanics it is always necessary to understand the fundamental properties of a system in order to generate an accurate numerical model or to predict future operating conditions. These fundamental properties include, but are not limited to, the material parameters of a specimen, the boundary conditions inside of a system, or essential dimensional characteristics that define the system or body. However in certain instances there may be little to no knowledge about the systems conditions or properties; as a result the problem cannot be modeled accurately using standard numerical methods. Consequently, it is critical to define an approach that is capable of identifying such characteristics of the problem at hand. In this thesis, an inverse approach is formulated using proper orthogonal decomposition (POD) with an accompanying radial basis function (RBF) network to estimate the current material parameters of a specimen with little prior knowledge of the system. Specifically conductive heat transfer and linear elasticity problems are developed in this thesis and modeled with a corresponding finite element (FEM) or boundary element (BEM) method. In order to create the truncated POD-RBF network to be utilized in the inverse approach, a series of direct FEM or BEM solutions are used to generate a statistical data set of temperatures or deformations in the system or body, each having a set of various material parameters. The data set is then transformed via POD to generate an orthonormal basis to accurately solve for the desired material characteristics using the Levenberg-Marquardt (LM) algorithm. For now, the LM algorithm can be simply defined as a direct relation to the minimization of the Euclidean norm of the objective Least Squares function(s). The trained POD-RBF inverse technique outlined in this thesis provides a flexible by which this inverse approach can be implemented into various fields of engineering and mechanics. More importantly this approach is designed to offer an inexpensive way to accurately estimate material characteristics or properties using nondestructive techniques. While the POD-RBF inverse approach outlined in this thesis focuses primarily in application to conduction heat transfer, elasticity, and fracture mechanics, this technique is designed to be directly applicable to other realistic conditions and/or industries.

To accurately model or predict future operating conditions of a system in engineering or applied mechanics, it is necessary to understand its fundamental principles. These may be the material parameters, defining dimensional characteristics, or the boundary conditions. However, there are instances when there is little to no prior knowledge of the system properties or conditions, and consequently, the problem cannot be modeled accurately. It is therefore critical to define a method that can identify the desired characteristics of the current system without accumulating extensive computation time. This thesis formulates an inverse approach using proper orthogonal decomposition (POD) with an accompanying radial basis function (RBF) interpolation network. This method is capable of predicting the desired characteristics of a specimen even with little prior knowledge of the system. This thesis first develops a conductive heat transfer problem, and by using the truncated POD – RBF interpolation network, temperature values are predicted given a varying Biot number. Then, a simple bifurcation problem is modeled and solved for velocity profiles while changing the mass flow rate. This bifurcation problem provides the data and foundation for future research into the left ventricular assist device (LVAD) and implementation of POD – RBF. The trained POD – RBF inverse approach defined in this thesis can be implemented in several applications of engineering and mechanics. It provides model reduction, error filtration, regularization and an improvement over previous analysis utilizing computational fluid dynamics (CFD).

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