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.

We discuss the numerical solution of large-scale sparse projected discrete-time periodic Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations in lifted form with low-rank right-hand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.:1 Introduction 2 Periodic descriptor systems 3 ADI method for causal lifted Lyapunov equations 4 Smith method for noncausal lifted Lyapunov equations 5 Application to model order reduction 6 Numerical results 7 Conclusions

Linear time-invariant models are widely used in the control community. They often serve as approximations of nonlinear systems. For control purposes linear approximations are often good enough since feedback control systems are inherently robust to model errors. In this thesis some of the possibilities for linear time-varying modeling are studied. In the thesis it is shown that the balanced truncation procedure can be applied to reduce the order of linear time-varying systems. Many of the attractive properties of balanced truncation for time-invariant systems can be generalized into the time-varying framework. For example, it is shown that a truncated input-output stable system will be input-output stable, and computable simple worst-case error bounds are derived. The method is illustrated with model reduction of a nonlinear diesel exhaust catalyst model. It is also shown that linear time-periodic models can be used for analysis of systems with power converters. Power converters produce harmonics in the power grids and give frequency coupling that cannot be modeled with standard time-invariant linear models. With time-periodic models we can visualize the coupling and also use all the available tools for linear time-varying systems, such as balanced truncation. The method is illustrated on inverter locomotives. / QC 20120208

The fuel consumption of a seagoing vessel is, through a combination of Box-Cox transforms and principal component analysis, reduced to a univariatefunction of the primary principle component with mean model error −3.2%and error standard deviation 10.3%. In the process, a Latin-hypercube-inspired space partitioning sampling technique is developed and successfully used to produce a representative sampleused in determining the regression coefficients. Finally, a formal optimisation problem for minimising the fuel use is described. The problem is derived from a parametrised expression for the fuel consumption, and has only 3, or 2 if simplified, free variables at each timestep. Some information has been redacted in order to comply with NDA restrictions. Most redactions are either names (of vessels or otherwise), units, andin some cases (especially on figures) quantities. / <p>Presentation was performed remotely using Zoom.</p>

Lors de l'optimisation géométrique de structures, un grand nombre d'évaluations de champs est nécessaire. L'idée, développée dans cette thèse, est la construction efficace et rapide d'approximations de ces champs à travers la Proper Generalized Decomposition (PGD), une méthode de réduction de modèle. Les résultats, calculés une fois pour toutes, sont stockés dans des abaques virtuels pour une utilisation ultérieure dans un processus d'optimisation. Le problème considéré est paramétrique et les paramètres définissent la géométrie. Ce type de problème est particulièrement adapté à la PGD. En effet, de nombreux travaux ont traité de la résolution de problèmes paramétriques et des premières études ont porté, en particulier, sur la prise en compte de paramètres géométriques. Toutefois, ce qui caractérise nos travaux est d'aller vers des outils aptes à traiter des situations significatives de la complexité des problèmes rencontrés au niveau industriel. En particulier, l'exploitation de codes éléments finis commerciaux est une contrainte majeure. Il a été décidé de développer des méthodes permettant de traiter des problèmes à paramètres géométriques par la PGD, et, en partenariat avec AIRBUS Defence & Space, d'appliquer ces techniques à un démonstrateur industriel présentant une géométrie complexe (type splines) et de fortes non-linéarités (géométriques, matériaux). Notre démarche a été implémentée dans un process industriel utilisant des codes éléments finis commerciaux. On propose aussi une nouvelle extension de la PGD aux paramètres discrets autorisant la prise en considération, dans une même résolution, de configurations de structures complètement différentes (cas de chargement, matériaux, etc.). / During shape optimisation of structures, numerous evaluations of fiels are necessary. The idea, developed in this PhD thesis, is the efficient construction of approximations of these fiels through the Proper Generalized Decomposition (PGD), a model reduction technique. The results, computed once and for all, are stored in virtual charts for a subsequent use into an optimisation process. Geometry variations correspond to a parametric problem, where the parameters is the geometry. This kind of problem is well suited for PGD. Many studies dealt with the resolution of parametric problems and recent works treated, particularly, the introduction of geometric parameters. However, our approach is to deal with configurations of the complexity of industrial problems. The use of commercial finite elements software is a crucial issue. It was decided, in partnership with AIRBUS Defence & Space, to develop techniques allowing the resolution of geometrically parametrised problems thanks to PGD and to apply them to an industrial demonstrator. The geometry considered is defined by splines and the behaviour of the structure is highly non-linear (geometric and material non-linearities). The approach was implemented into a genuine industrial design process using commercial finite elements software. The thesis proposed, also, a new extension of PGD to discrete parameters. It allows to take into account completely different configurations (loadings, materials, etc.) in the same resolution.

Les aubes des turbines à haute pression des réacteurs d'avion subissent des chargements complexes dans un environnement réactif. Prédire leur durée de vie peut nécessiter une approche en tolérance aux dommages, basée sur la prédiction de la propagation d'une fissure supposée. Mais cette approche est confrontée au comportement non linéaire sous des chargements à amplitudes variables et au coût énorme des calculs elasto-plastiques des structures 3D complexes sur des millions des cycles. Dans ce cadre, un modèle incrémental de fissuration a été proposé. Ce modèle est basé sur la plasticité comme mécanisme principal de propagation de fissure par fatigue pure. Cette modélisation passe par une réduction de modèle de type POD. La plasticité en pointe de la fissure est alors modélisée par un nombre réduit de variables non locales et des variables internes. Un ensemble d'hypothèses doit être respecté pour garantir la validité de cette modélisation. Pour décliner ce modèle dans le cas d'un matériau anisotrope représentatif du comportement des monocristaux, une première étude a été faite sur le cas d'une élasticité cubique avec de la plasticité de Von-Mises. Une stratégie a été proposée pour identifier un modèle matériau basé sur les facteurs d'intensité non locaux. Cette stratégie comporte une détermination de la fonction critère basée sur les solutions élastiques en anisotrope. L'étude des directions d'écoulement plastique avec les variables non locales montre une forte dépendance à l'anisotropie élastique du modèle même avec une plasticité associée de Von-Mises. La stratégie comporte également une identification des variables internes.Dans la deuxième partie, le problème d'une fissure avec un modèle de plasticité cristalline a été traité. L'activation de différents systèmes de glissement a été alors prise en compte dans la modélisation. Finalement, différentes méthodologies ont été explorées en vue de transposer le modèle local de plasticité cristalline à l'échelle non locale de la région en pointe de la fissure. / The fatigue life prediction of high pressure turbine blades may require a damage tolerance approach based on the study of possible crack propagation. The nonlinear behavior of the material under complex nonproportional loadings and the high cost of running long and expensive elastic-plastic FE computations on complex 3D structures over millions of cycles are some major issues that may encounter this type of approach.Within this context, an incremental model was proposed based on plasticity as a main mechanism for fatigue crack growth.A model reduction strategy using the Proper Orthogonal Decomposition (POD) was used to reduce the cost of FEA. Based on a set of hypotheses, the number of the degrees of freedom of the problem is reduced drastically. The plasticity at the crack tip is finally described by a set of empirical equations of few nonlocal variables and some internal variables.In order to apply this modeling strategy to the case of anisotropic materials that represent the behavior of single crystals, a first study was done with cubic elasticity and a Von-Mises plasticity. Elastic and plastic reference fields, required to reduce the model, were determined. Then, a material model of the near crack tip region was proposed based on nonlocal intensity factors. A yield criterion function was proposed based on Hoenig's asymptotic solutions for anisotropic materials. The study of plastic flow directions with the nonlocal variables of the model shows a strong dependency on the cubic elasticity. A strategy to identify internal variables is proposed as well. In the second part, a crystal plasticity model was implemented. The activation of different slip systems was taken into account in the model reduction strategy. A kinematic basis was constructed for each slip system. Finally, a strategy was proposed to transpose the local crystal plasticity model to the nonlocal scale of the crack.