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.

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.

Cette thèse présente les travaux réalisés en vue de l’élaboration d’un modèle cardiaque associant croissance, mouvement et circulation sanguine pour permettre ensuite la construction d’un modèle patient à partir d’un modèle de population. Le premier axe de ce travail est la simulation de la croissance bi-ventriculaire. Un modèle existant de surface unique, calculé à l’aide de méthodes statistiques, a été généralisé à un modèle bi-ventriculaire puis appliqué à la tétralogie de Fallot (ToF). Le deuxième axe concerne la modélisation du mouvement cardiaque au niveau de la population. Un modèle d’ordre réduit basé sur un modèle Polyaffine et LogDemons a été proposé. Il simule la dynamique cardiaque avec peu de paramètres. Les paramètres de transformation sont analysés par des méthodes statistiques. Un modèle de mouvement moyen a été calculé pour représenter le mouvement standard de la population. Le troisième axe s'intéresse à la simulation de l’écoulement sanguin à l’échelle de la population. La complexité des simulations spécifiques à un patient a été réduite grâce à l’utilisation de méthodes d’analyse d’image, de dynamique des fluides numérique et de réduction d’ordre de modèle. La simulation du flux sanguin dans l’artère pulmonaire pour des patients ToF a permis de mieux comprendre l’impact du sang régurgité sur la pression et la vitesse. Étant donné nos contributions sur ces trois axes, nous sommes maintenant en bonne position pour élaborer le modèle couplé des contributions interdépendantes de la croissance, du mouvement et de l'écoulement sanguin. Ce modèle pourrait être utilisé afin d'aider la planification de la thérapie chez les patients atteints de maladies cardiaques. / This thesis presents work towards a coupled model of cardiac growth, motion, and blood flow to enable predictive patient-specific models to be built from a population-based model. The first axis of this work is to simulate bi-ventricular growth through aging. A previously proposed single surface model computed using statistical methods was extended to a bi-ventricular model and applied to Tetralogy of Fallot patients to model the complex evolution of the ventricles due to the pathology. The second axis concerns the development of a model to simulate cardiac motion at a population level. A reduced-order cardiac-specific motion model was proposed to simulate the motion dynamics with a small number of parameters using a Polyaffine and LogDemons based model. From the computed transformations, the parameters were analysed using statistical methods to obtain population-based measures of normality. A mean motion model was derived to represent the normal motion for a given population. The third axis is to develop a model of population-based flow dynamics. The complexity of patient-specific simulations was reduced by combining image analysis, computational fluid dynamics and model order reduction techniques. Blood flow through the pulmonary artery in Tetralogy of Fallot patients was simulated to better understand the impact of regurgitated blood on pressure and velocity. Given our contributions on these three axes, we are now in a good position to couple the models in order to capture the interrelated contributions of growth, motion and flow. Such a model could be used to aid in therapy planning and decision making for patients with heart disease.

L'étude numérique est un outil de recherche qui est devenu incontournable, en particulier pour la compréhension et le contrôle des systèmes complexes. La simulation des plasmas de fusion par confinement magnétique s'inscrit parfaitement dans cette démarche. Les larges rapports d'échelle en temps et espace, la nature chaotique des plasmas et les très fortes anisotropies imposent l'utilisation de méthodes numériques avancées. C'est dans ce cadre que les deux volets de ma thèse s'inscrivent.Le premier volet est l’originalité de ma thèse, la mise en place la méthode PoPe, une procédure générale de vérification de codes et réduction de modèles. Le principe de cette méthode est de déterminer les équations qui ont permis de générer un ensemble de données : si les données sont issues d'un code de simulation, retrouver ces équations et les comparer au modèle théoriquement implémenté est équivalent à vérifier le code. La précision de la procédure permet de caractériser l'erreur commise jusqu'à retrouver l'ordre des schémas numériques employés, même en régime chaotique.Le second volet de ma thèse se consacre à l’étude du transport turbulent qui détermine la performance des plasmas de fusion. L’étude du transport sous forme d’avalanches dans un modèle de bord fluide est entreprise en quantifiant l’impact du chaos sur l’auto-organisation. Pour un modèle cinétique restreint aux instabilités basse fréquence, la capacité de se bloquer dans deux régimes exclusifs, l’un isolant, l’autre conducteur, est étudiée. Ce modèle est amélioré pour permettre des relaxations entre ces deux états. Pour ces modèles fluide et cinétique, des modèles réduits obtenus avec la méthode PoPe sont proposés. / Numerical analysis is now a key component of research, especially for the understanding and the control of complex systems. Simulations of magnetic confinement plasmas fall within this approach. One of the difficulties of this field is the wide range of spatial scales, time scales, the chaotic nature of plasmas and the strong anisotropies require advanced numerical methods. Each of the two parts of my thesis takes place in this frame of numerical simulation and fusion plasmas.The first part of my thesis is dedicated to the method PoPe, a general method for code verification and model reduction. The principle of this method is to determine the equations which have generated a set of data. If the data was produced by a simulation tool, finding these equations and comparing them to the ones theoretically implemented is equivalent to verifying this simulation tool. The accuracy of this procedure allows to characterize the numerical error and to recover the order of each numerical scheme used.The second part of my thesis deals with the study of turbulent transport which determines the efficiency of fusion plasma. The chaotic avalanches of a fluid model are studied considering the impact of the chaos on the self-organization. For a kinetic model restricted to the low frequency instabilities, the ability to block itself in two regimes, one insulating and the other conducting, is studied. Upgrades of this model are undertaken in order to introduce the possibility of relaxations between the two previous states. For both the fluid and the kinetic model, reduce models are proposed thank to the PoPe method.

In this research, a focus is on the development of simplified models to represent the behavior of electric machinery within the time-domain models of power systems. Toward this goal, a generator model is considered in which the states include the machine’s active and reactive power. In the case of the induction machine, rotor slip is utilized as a state and the steady-state equivalent circuit of the machine is used to calculate active and reactive power. The power network model is then configured to accept the generator and induction machine active and reactive power as inputs and provide machine terminal voltage amplitude and angle as outputs. The potential offered by these models is that the number of dynamic states is greatly reduced compared to traditional machine models. This can lead to increased simulation speed, which has potential benefits in model-based control. A potential disadvantage is that the relationship between the reactive power and terminal voltage requires the solution of nonlinear equations, which can lead to challenges when attempting to predict system dynamics in real-time optimal control. In addition, the accuracy of the generator model is greatly reduced with variations in rotor speed. Evaluation of the models is performed by comparing their predictions to those of traditional machine models in which stator dynamics are included and neglected.

We investigate model reduction of large-scale linear time-invariant systems in generalized state-space form. We consider sparse state matrix pencils, including pencils with banded structure. The balancing-based methods employed here are composed of well-known linear algebra operations and have been recently shown to be applicable to large models by exploiting the structure of the matrices defining the dynamics of the system. In this paper we propose a modification of the LR-ADI iteration to solve large-scale generalized Lyapunov equations together with a practical convergence criterion, and several other implementation refinements. Using kernels from several serial and parallel linear algebra libraries, we have developed a parallel package for model reduction, SpaRed, extending the applicability of balanced truncation to sparse systems with up to $O(10^5)$ states. Experiments on an SMP parallel architecture consisting of Intel Itanium 2 processors illustrate the numerical performance of this approach and the potential of the parallel algorithms for model reduction of large-scale sparse systems.

Model reduction is a common theme within the simulation, control and optimization of complex dynamical systems. For instance, in control problems for partial differential equations, the associated large-scale systems have to be solved very often. To attack these problems in reasonable time it is absolutely necessary to reduce the dimension of the underlying system. We focus on model reduction by balanced truncation where a system theoretical background provides some desirable properties of the reduced-order system. The major computational task in balanced truncation is the solution of large-scale Lyapunov equations, thus the method is of limited use for really large-scale applications. We develop an effective implementation of balancing-related model reduction methods in exploiting the structure of the underlying problem. This is done by a data-sparse approximation of the large-scale state matrix A using the hierarchical matrix format. Furthermore, we integrate the corresponding formatted arithmetic in the sign function method for computing approximate solution factors of the Lyapunov equations. This approach is well-suited for a class of practical relevant problems and allows the application of balanced truncation and related methods to systems coming from 2D and 3D FEM and BEM discretizations.

We provide a unifying projection-based framework for structure-preserving interpolatory model reduction of parameterized linear dynamical systems, i.e., systems having a structured dependence on parameters that we wish to retain in the reduced-order model. The parameter dependence may be linear or nonlinear and is retained in the reduced-order model. Moreover, we are able to give conditions under which the gradient and Hessian of the system response with respect to the system parameters is matched in the reduced-order model. We provide a systematic approach built on established interpolatory $\mathcal{H}_2$ optimal model reduction methods that will produce parameterized reduced-order models having high fidelity throughout a parameter range of interest. For single input/single output systems with parameters in the input/output maps, we provide reduced-order models that are \emph{optimal} with respect to an $\mathcal{H}_2\otimes\mathcal{L}_2$ joint error measure. The capabilities of these approaches are illustrated by several numerical examples from technical applications.