L’objectif est de construire un modèle d’écoulement qui se prête bien à des problèmes de contrôle, en associant un faible nombre de degrés de liberté à la possibilité de décrire la dynamique d’un écoulement relativement complexe. Dans ce travail nous considérons un écoulement bidimensionnel laminaire autour d’un cylindre carré. Des actionneurs placés sur le cylindre permettent un contrôle actif par sou?age et aspiration. Ce contrôle peut être dé?ni par rétroaction, exploitant des mesures de la vitesse dans le sillage du cylindre. Nous construisons un modèle d’ordre réduit (ROM) des équations de Navier-Stokes incompressibles, basé sur la technique de décomposition orthogonale aux valeurs propres (POD). Une façon classique de construire un tel modèle est de réaliser une projection Galerkin des équations sur le sous-espace réduit obtenu par POD. Un tel modèle peut cependant être peu précis, voire instable. Une technique de calibration est alors mise en place pour assurer la bonne représentativité dynamique du modèle. Nous dé?nissons ensuite une stratégie pour mettre à jour le modèle au cours d’un processus d’optimisation. La méthode est en?n appliquée pour réduire la di?érence entre l’écoulement contrôlé et la solution stationnaire instable à Re = 150. / The aim is to build a ?ow model adapted for control applications combining a low number of degrees of freedom with the possibility of describing relatively complex ?ows. In this work a two-dimensional laminar ?ow past a square cylinder is considered. Actuators placed on the cylinder enable active control by blowing and suction. Proportional feedback control can then be applied using velocity measurements taken in the cylinder wake. The proper orthogonal decom- position (POD) approach is used to build a low order model of the incompressible Navier-Stokes equations. A classical way of obtaining a Reduced-Order Model (ROM) is to perform a Galerkin projection of the equations onto the subspace spanned by the POD modes. Such a model can however be inaccurate, even unstable. A calibration technique is therefore applied, leading to a model that is accurate and robust to variations of the control parameters. A strategy is then de?ned to update the model within an optimisation loop. The method is tested at Re = 150 for reducing the di?erence between the actuated ?ow ?eld and the steady unstable solution.

La dynamique d'instabilité d'un écoulement laminaire décollé est étudiée expérimentalement et son contrôle par le biais de la simulation numérique. La configuration étudiée est une couche limite laminaire décollée au dessus d'une géométrie de type bosse.Pour une certaine gamme de paramètres, l'écoulement de recirculation en aval de la bosse est caractérisé par un battement basse fréquence. L'étude expérimentale de cette dynamique a permis de retrouver les différents régimes d'instabilité mis a jour par voie numérique. Ces résultats prouvent notamment que les instabilités basse fréquence, dont l'existence a été surtout mise en évidence dans des configurations d'écoulements compressibles, sont un phénomène générique pour des bulles de recirculations allongées. Le contrôle du battement basse fréquence est ensuite étudié par voie numérique suivant deux approches complémentaires. Un asservissement en boucle fermée de la dynamique de perturbation linéaire est tout d'abord proposé. Les modes d'instabilité linéaires sont utilisés afin de construire des modèles réduits de la dynamique de perturbation. Cette réduction de modèle donne lieu à des estimateurs de faible dimension capables d'estimer la dynamique et de la contrôler. Ainsi la dynamique d'instabilité linéaire peut être supprimée en couplant le système de Navier-Stokes linéarisé avec le contrôleur.Le contrôle de la dynamique non linéaire est ensuite étudié en utilisant une méthode d'optimisation Lagrangienne. Cette méthode permet de calculer les lois de contrôle à partir de la dynamique non linéaire des équations de Navier-Stokes. / The dynamics and control of a separated boundary-layer flow have been investigated. Separation is triggered by a bump mounted on a flat plate and the transition dynamics has been investigated experimentally. For a certain parameter range, the recirculation region is subject to self-sustained low-frequency oscillations, and results from the numerical simulation for the same geometry are recovered. These results show that low frequency oscillations, observed mainly in compressible flow regimes, are inherent to elongated recirculation bubbles.The control of this low-frequency instability has been investigated using modern control theory based on two complementary approaches. Feedback control of the linear perturbation dynamics is first considered. Global instability modes are used to build reduced-order estimators. This model reduction gives rise to low-dimensional compensators capable of controlling the unstable dynamics. Once coupled to the unstable linearised Navier-Stokes system, the compensator is seen to succesfully control the unstable dynamics. The control of the nonlinear dynamics is then investigated using adjoint-based optimisation procedures. This method is used to compute control laws based on a complete knowledge of the nonlinear dynamics. Although the low-frequency instability is clearly attenuated, it seems difficult to control the flow towards its steady state, using only a few blowing/suction actuators localized on the wall.

Model order reduction is necessary even in a time where the parallel processing is usual in almost any personal computer. The recent Model Reduction Methods are useful tools nowadays on reducing the problem processing. This work intends to describe a combination between POD (Proper Orthogonal Decomposition) and Ritz vectors that achieve an efficient Galerkin projection that changes during the processing, comparing the development of the error and the convergence rate between the full space and the projection space, in addition to check the stability of the projection space, leading to an adaptive model order reduction for nonlinear dynamical problems more efficient. This model reduction is supported by a secant formulation, which is updated by BFGS (Broyden - Fletcher - Goldfarb - Shanno) method to accelerate convergence of the model, and a tangent formulation to correct the projection space. Furthermore, this research shows that this method permits a correction of the reduced model at low cost, especially when the classical POD is no more efficient to represent accurately the solution. / A Redução de ordem de modelo é necessária, mesmo em uma época onde o processamento paralelo é usado em praticamente qualquer computador pessoal. Os recentes métodos de redução de modelo são ferramentas úteis nos dias de hoje para a redução de processamento de um problema. Este trabalho pretende descrever uma combinação entre POD (Proper Orthogonal Decomposition) e vetores de Ritz para uma projecção de Galerkin eficiente que sofre alterações durante o processamento, comparando o desenvolvimento do erro e a taxa de convergência entre o espaço total e o espaço de projeção, além da verificação de estabilidade do espaço de projeção, levando a uma redução de ordem do modelo adaptativo mais eficiente para problemas dinâmicos não-lineares. Esta redução de modelo é assistida por uma formulação secante, que é atualizado pela formula de BFGS (Broyden - Fletcher- Goldfarb - Shanno) com o intuito de acelerar a convergência do modelo, e uma formulação tangente para a correção do espaço de projeção. Além disso, esta pesquisa mostra que este método permite a correção do modelo reduzido com baixo custo, especialmente quando o clássico POD não é mais eficiente para representar com precisão a solução.

When simulating mechanical systems the flexibility of the components often has to be taken into account. This is particularly important for simulations when high detailed information is demanded, e.g. to calculate stresses. To this end the Finite Element Method (FEM) is often used. However the models can become very large, containing millions of degrees of freedom. Solving large linear systems are computationally demanding. Therefore ways of reducing the problem is often sought. These reduction does, however, remove much of the details that was to be investigated. In this thesis this problem is addressed by creating a reduction scheme, using Proper Orthogonal Decomposition (POD), that significantly reduces a problem but still captures much of the details. A novel method for enriching regular POD-based model reduction methods with hierarchically determined enrichment POD-modes is developed. The method is proposed and validated in a FEM application towards dynamical simulation. The enriched method is compared against a regular POD reduction technique. An numerical study is made of a model example of linear elasticity in a gearwheel. The numerical study suggests that the error of displacements is around ten times smaller, on average, when using the enriched basis compared to a reference basis of equal dimensionality consisting of only regular POD modes. Also it is shown that local quantities as the von Mises stress in a gearwheel tooth is preserved much better using the enriched basis. An a posteriori error estimate is proposed and proved for the static case, showing that the error is bound. / När man simulerar mekaniska system så måste man ofta ta hänsyn till de ingående komponenternas flexibilitet. Detta är särskilt viktigt då man gör simuleringar med krav på hög detaljkännedom, såsom mätningar av spänningar i kugghjul etc. Till detta ändamål används ofta en Finit Element Metod (FEM). Dock kan modellerna ofta bli väldigt stora, med över en miljon frihetsgrader. Att lösa linjära system av den storleken är beräkningsmässigt krävande. Därför är det naturligt att försöka reducera problemen. Reduktion innebär dock att information försvinner, i synnerhet de detaljer som skulle beräknas. I detta examensarbete så behandlas problemet genom att skapa en ny metod för reducering av stora finita element modeller. Metoden bygger på tidigare kunskap om Proper Orthogonal Decomposition (POD) som ett sätt att reducera modeller. Den nya metoden reducerar finita ellement modeller samtidigt som den bibehåller hög detalj. En ny metod utvecklas för att berika en vanlig POD-baserad modellreduktion med hjälp av hieraktiskt bestämda berikningsmoder. Metoden beskrivs och testas i en dynamisk FEM-applikation av elasticitet i ett kugghjul i 2 dimensioner. Metoden för berikning jämförs numeriskt med en metod som använder vanlig POD-reduktion. Körningar visar att felet i den berikade metoden är omkring 10 gånger mindre, i genomsnitt, jämfört med en vanlig metod. Det visas också att spänningar bevaras på ett mycket bra sätt med den nya berikningsmetoden. Dessutom så formuleras och bevisas ett a posteriori estimat för statiska lastfall, vilket innebär att felet i metoden är bundet.

Hybrid neural-microelectronic systems, systems composed of biological neural networks and neuronal models, have great potential for the treatment of neural injury and disease. The utility of such systems will be ultimately determined by the ability of the engineered component to correctly replicate the function of biological neural networks. These models can take the form of mechanistic models, which reproduce neural function by describing the physiologic mechanisms that produce neural activity, and empirical models, which reproduce neural function through more simplified mathematical expressions. We present our research into the role of model complexity in creating robust and flexible behaviors in hybrid systems. Beginning with a complex mechanistic model of a leech heartbeat interneuron, we create a series of three systematically reduced models that incorporate both mechanistic and empirical components. We then evaluate the robustness of these models to parameter variation, and assess the flexibility of the models activities. The modeling studies are validated by incorporating both mechanistic and semi-empirical models in hybrid systems with a living leech heartbeat interneuron. Our results indicate that model complexity serves to increase both the robustness of the system and the ability of the system to produce flexible outputs.

Mathematical models that are built in order to simulate the behavior of structures, most often, tend to respond differently than the actual structures in their initial state. In order to use the mathematical models and their computational outputs instead of testing the real structure under every possible case, it is mandatory to have a mathematical model that reflects the characteristics of the actual structure in the best possible way. In this thesis, the so called model updating techniques used for updating the mathematical models in order to make them respond in the way the actual structures do are investigated. Case studies using computationally generated test data are performed using the direct and indirect modal updating techniques with the software developed for each method investigated. After investigating the direct and indirect modal updating techniques, two of them, one using frequency response functions and the other using modal sensitivities, are determined to be the most suitable ones for aircraft structures. A generic software is developed for the technique using modal sensitivities. A modal test is carried out on a scaled aircraft model. The test data is used for updating of the finite element model of the scaled aircraft using the modal sensitivities and the usability of the method is thus evaluated. The finite element model of a real aircraft using the modal test data is also updated using the modal sensitivities. A new error localization technique and a model updating routine are also proposed in this thesis. This modal updating routine is used with several case studies using computationally generated test data and it is concluded that it is capable of updating the mathematical models even with incomplete measured data.

The full model of a double-wishbone suspension has more than 30 differential-algebraic equations which takes a remarkably long time to simulate. By contrast, the look-up table for the same suspension is simulated much faster, but may not be very accurate. Therefore, developing reduced models that approximate complex systems is necessary because model reduction decreases the simulation time in comparison with the original model, enables real time applications, and produces acceptable accuracy. In this research, we focus on model reduction techniques for vehicle systems such as suspensions and how they are approximated by models having lower degrees of freedom. First, some existing model reduction techniques, such as irreducible realization procedures, balanced truncation, and activity-based reduction, are implemented to some vehicle suspensions. Based on the application of these techniques, their disadvantages are revealed. Then, two methods of model reduction for multi-body systems are proposed. The first proposed method is 2-norm power-based model reduction (2NPR) that combines 2-norm of power and genetic algorithms to derive reduced models having lower degrees of freedom and fewer number of components. In the 2NPR, some components such as mass, damper, and spring are removed from the original system. Afterward, the values of the remaining components are adjusted by the genetic algorithms. The most important advantage of the 2NPR is keeping the topology of multi-body systems which is useful for design purposes. The second method uses proper orthogonal decomposition. First, the equations of motion for a multi-body system are converted to explicit second-order differential equations. Second, the projection matrix is obtained from simulation or experimental data by proper orthogonal decomposition. Finally, the equations of motion are transferred to a lower-dimensional state coordinate system. The implementation of the 2NPR to two double-wishbone suspensions and the comparison with other techniques such as balanced truncation and activity-based model reduction also demonstrate the efficiency of the new reduction technique.

This thesis investigates the frequency weighted balanced model reduction problem for linear time invariant systems. Both continuous and discrete time systems are considered, in one and two-dimensions. First the frequency weighted balanced model reduction problem is formulated, then a novel frequency weighted, balanced, model reduction method for continuous time systems is proposed. This method is based on the retention of frequency weighted Hankel singular values of the original system, and yields stable reduced order models even when two sided weightings are employed. An alternative frequency weighted balanced model reduction technique (applicable for controller reduction applications) is then developed. This is based on a parametrized combination of the frequency weighted partial fraction expansion technique with balanced truncation and the singular perturbation approximation techniques. This method yields stable models even when two sided weightings are employed. An a priori error bound for the model reduction method is derived. Lower frequency response errors and error bounds are obtained using free parameters and equivalent anti-stable weightings. Based on the same idea, a novel parameterized frequency weighted optimal Hankel norm model reduction method with a tighter a priori error bound is proposed. The proposed methods are extended to include discrete time systems. A frequency interval Gramians based stability preserving model reduction scheme with error bounds is also presented. In this case, frequency weights are not explicitly predefined. Discrete time system related results are also included. Several frequency weighted model reduction results for two-dimensional (2-D) systems are also proposed. The advantages of these schemes include error bounds, guaranteed stability and applicability to general stable (non-separable denominator) weighting functions. Finally, a novel 2-D identification based frequency weighted model reduction scheme is outlined. Numerically robust algorithms based on square root and balancing free techniques are proposed for frequency weighted balanced truncation techniques. Several practical examples are included to illustrate the effectiveness of the algorithms.

Model order reduction is necessary even in a time where the parallel processing is usual in almost any personal computer. The recent Model Reduction Methods are useful tools nowadays on reducing the problem processing. This work intends to describe a combination between POD (Proper Orthogonal Decomposition) and Ritz vectors that achieve an efficient Galerkin projection that changes during the processing, comparing the development of the error and the convergence rate between the full space and the projection space, in addition to check the stability of the projection space, leading to an adaptive model order reduction for nonlinear dynamical problems more efficient. This model reduction is supported by a secant formulation, which is updated by BFGS (Broyden - Fletcher - Goldfarb - Shanno) method to accelerate convergence of the model, and a tangent formulation to correct the projection space. Furthermore, this research shows that this method permits a correction of the reduced model at low cost, especially when the classical POD is no more efficient to represent accurately the solution. / A Redução de ordem de modelo é necessária, mesmo em uma época onde o processamento paralelo é usado em praticamente qualquer computador pessoal. Os recentes métodos de redução de modelo são ferramentas úteis nos dias de hoje para a redução de processamento de um problema. Este trabalho pretende descrever uma combinação entre POD (Proper Orthogonal Decomposition) e vetores de Ritz para uma projecção de Galerkin eficiente que sofre alterações durante o processamento, comparando o desenvolvimento do erro e a taxa de convergência entre o espaço total e o espaço de projeção, além da verificação de estabilidade do espaço de projeção, levando a uma redução de ordem do modelo adaptativo mais eficiente para problemas dinâmicos não-lineares. Esta redução de modelo é assistida por uma formulação secante, que é atualizado pela formula de BFGS (Broyden - Fletcher- Goldfarb - Shanno) com o intuito de acelerar a convergência do modelo, e uma formulação tangente para a correção do espaço de projeção. Além disso, esta pesquisa mostra que este método permite a correção do modelo reduzido com baixo custo, especialmente quando o clássico POD não é mais eficiente para representar com precisão a solução.

Cette thèse présente un algorithme automatique pour la construction d’un modèle NURBS volumique à partir d’un modèle représenté par ses bords (maillages ou splines). Ce type de modèle est indispensable dans le cadre de l’analyse isogéométrique utilisant les NURBS comme fonctions de forme. Le point d’entrée de l’algorithme est une triangulation du bord du modèle. Après deux étapes de décomposition, le modèle est approché par un polycube. Ensuite un paramétrage surfacique entre le bord du modèle et celui du polycube est établi en calculant un paramétrage global aligné à un champ de direction interpolant les directions de courbure principales du modèle. Finalement, le paramétrage volumique est obtenu en se basant sur ce paramétrage surfacique. Dans le contexte des études paramétriques basées sur des paramètres de formes géométriques, cette méthode peut être appliquée aux techniques de réduction de modèles pour obtenir la même représentation pour des objets ayant différentes géométries mais la même topologie. / This thesis presents an effective method to automatically construct trivariate tensor-product spline models of complicated geometry and arbitrary topology. Our method takes as input a solid model defined by its triangulated boundary. Using cuboid decomposition, an initial polycube approximating the input boundary mesh is built. This polycube serves as the parametric domain of the tensor-product spline representation required for isogeometric analysis. The polycube's nodes and arcs decompose the input model locally into quadrangular patches, and globally into hexahedral domains. Using aligned global parameterization, the nodes are re-positioned and the arcs are re-routed across the surface in a way to achieve low overall patch distortion, and alignment to principal curvature directions and sharp features. The optimization process is based on one of the main contributions of this thesis: a novel way to design cross fields with topological (i.e., imposed singularities) and geometrical (i.e., imposed directions) constraints by solving only sparse linear systems. Based on the optimized polycube and parameterization, compatible B-spline boundary surfaces are reconstructed. Finally, the interior volumetric parameterization is computed using Coon's interpolation and the B-spline surfaces as boundary conditions. This method can be applied to reduced order modeling for parametric studies based on geometrical parameters. For models with the same topology but different geometries, this method allows to have the same representation: i.e., meshes (or parameterizations) with the same topology.