[pt] Este texto pode ser dividido em duas partes: a primeira trata da modelagem de sistemas dinâmicos, passando da chamada formulação forte ao conceito de formulação variacional, sem antes deixar de apresentar ferramentas básicas do cálculo variacional e o Princípio de Hamilton. Os conceitos são exemplificados por duas estruturas que acompanham todo o texto: um cabo unidimensional e uma placa. Ainda na primeira parte, é apresentado o problema de autovalor de sistemas contínuos e são mostradas as propriedades dos operadores autoadjuntos. Ao longo desta etapa e no apêndice, soluções analíticas para o problema de autovalor são desenvolvidas. Por ser a obtenção das soluções analíticas dos problemas por demais engenhosas ou até mesmo impossíveis, um outro caminho é proposto: a aproximação de soluções, sendo este o tema da segunda parte deste texto. Ela é iniciada pela apresentação de métodos de discretização dos sistemas contínuos sem deixar de exemplificá-los. Os métodos são usados como ferramentas de aproximação dos modos de vibração. São abordados os Métodos de Ritz, de Galerkin e o da Colocação. As funções usadas no primeiro e no segundo são geradas pelo Método dos Elementos Finitos e as aproximações dos modos por este método são usadas na redução de sistemas, para então se obter a resposta dinâmica dado um carregamento. Toda a teoria é reforçada ao final com dois problemas práticos: um cabo durante uma operação de abastecimento de uma plataforma de petróleo e o outro de uma placa durante uma operação de jateamento. Por último, mas não menos importante, um capítulo é dedicado ao Método da Colocação, onde polinômios de ordem superior, os polinômios de Chebyshev, são usados para a aproximação com o uso de diferentes grades de interpolação, a grade de Chebyshev-Gauss e a grade de Gauss-Lobatto. / [en] This text can be divided into two parts: the first deals with modeling of dynamic systems, passing through the so-called strong formulation to the concept of variational formulation, considering the basic tools of variational calculus and the Hamilton Principle. The concepts are exemplified by two structures that follows the whole text: a unidimensional cable and a plate. Still in the first part, the eigenvalue problem of continuous systems is presented and the properties of self-adjoint operators are shown. Throughout this stage and at the appendix, analytical solutions to the eigenvalue problem are developed. As to get the problems analytical solutions may be too ingenious or even impossible, another way is proposed: the use of approximate solutions, which is the theme of the second part of this text. It starts by presenting methods of discretization of continuous systems, exemplifying them. The methods are used as tools for approximation of the vibration modes. The Ritz, Galerkin and Collocation methods are exposed. The functions used at the first and at the second are generated by the Finite Element Method and the modes approximated by this method are used to reduce the systems to then obtain the dynamic response to a given dynamic loading. The whole theory is reinforced with two practical problems at the end: one is about a cable during a supplying operation of an oil rig and the other is about a plate during a shot blastening operation. Last but not least, a chapter is devoted to the Collocation Method, where higher-order polynomials, the Chebyshev polynomials, are used to the approximation using different interpolation grids, the Chebyshev-Gauss and the Gauss-Lobatto grid.

We investigate the numerical solution of stable Sylvester equations via iterative schemes proposed for computing the sign function of a matrix. In particular, we discuss how the rational iterations for the matrix sign function can efficiently be adapted to the special structure implied by the Sylvester equation. For Sylvester equations with factored constant term as those arising in model reduction or image restoration, we derive an algorithm that computes the solution in factored form directly. We also suggest convergence criteria for the resulting iterations and compare the accuracy and performance of the resulting methods with existing Sylvester solvers. The algorithms proposed here are easy to parallelize. We report on the parallelization of those algorithms and demonstrate their high efficiency and scalability using experimental results obtained on a cluster of Intel Pentium Xeon processors.

Afin de tenir compte de l'amorçage et de la propagation des fissures en fatigue, il est nécessaire de connaître l'historique des différentes quantités mécaniques dans la zone d'amorçage. Cela nécessite une connaissance des états mécaniques cycliques stabilisés. D'un point de vue numérique, les simulations numériques d'agrégats polycristallin avec les méthodes de résolution classiques ne sont réalisées que pour quelques cycles. Ce travail présente le développement des méthodes accélérées de calcul pour réduire le temps de calcul de la méthode des Éléments Finis dans le cas des simulations d'agrégats polycristallins soumis à un chargement cyclique. La première idée est de maintenir la matrice de rigidité constante afin d'effectuer une seule factorisation. Un premier algorithme a été écrit dans ce sens avec une résolution incrémentale et non incrémentale. La deuxième proposition est d'utiliser une méthode de réduction dimensionnelle en espace/temps couplé à la méthode des éléments finis. La PGD a été choisie. Cette méthode permet de découpler les variables spatiales et temporelles et les champs de déplacement ne sont calculés que pour un certain nombre de modes. Deux possibilités d'utilisation de la méthode PGD dans le cadre de la plasticité ont été proposées. La troisième proposition consiste à utiliser la stratégie de réduction adaptative APR comme solveur afin de résoudre un modèle d'ordre réduit en termes de nombre de ddl. Une stratégie incrémentale d'amélioration de la qualité de la base pour un certain intervalle de temps choisi a été mis en place dans ce cadre. Quatre possibilités d'utilisation de la méthode APR ont été proposées. L'analyse des performances des différentes méthodes est effectuée tout d'abord sur un problème élasto-plastique classique tridimensionnel présentant un défaut sphérique, ensuite à l'échelle de la microstructure avec un calcul visco-élasto-plastique d'agrégats polycristallins tridimensionnels. Les analyses ont montré que les réponses mécaniques macroscopique et mésoscopique des méthodes de réduction de modèle sont très proches de la méthode incrémentale. Le temps des simulations a été réduit par les différentes méthodes. Les gains sont plus importants quand on augmente la taille des maillages et la non linéarité du problème. Toutefois, l'idée de garder la matrice de rigidité constante avec une résolution incrémentale a permis un gain de temps encore plus conséquent à l'échelle de la microstructure. / In order to take account of fatigue cracks initiation and growth, it is necessary to know the history of the various mechanical quantities in fatigue initiation site. This requires knowledge of the stabilized cyclic mechanical states. From a numerical approach, numerical simulation of polycrystalline aggregates with conventional resolution methods are only carried out for a few cycles. This work presents the development of accelerated numerical methods to reduce the computation time of the Finite Element method in the case of numerical simulation of polycrystalline aggregates under cyclic loading. The first idea is to keep a constant stiffness matrix during overall simulation in order to get just one single factorization to carry out. An algorithm has been proposed in this sense with an incremental and non incremental resolution. The second proposal is based on the use of a model reduction method coupled with the finite element method to solve space/time problem. The PGD has been selected. This method allows to decouple spatial and time coordinates and the displacement fields are computed for a certain number of modes. Two possibilities have been proposed to implement the PGD method in the context of plasticity. The third proposal is to use an a priori adaptative approach based on the use of APR strategy to solve a reduced order model in terms of number of degrees of freedom. An incremental adaptive strategie has been proposed in order to master the quality of the reduced-basis for a certain time steps. Four possibilities of using the APR method have been proposed. The applicability and the performance of the different methods have been first analyzed on a conventional three-dimensional elastoplastic problem with a spherical defect, then on the scale of the microstructure with numerical simulation of polycrystalline aggregates under cyclic elasto-visco-plastic loading. The analyzes have shown that the macroscopic and mesoscopic mechanical responses of the model reduction methods are very close to the incremental method. The simulation time has been reduced by the different methods. The time gains are more significant by increasing the size of the meshes and the non-linearity of the problem. Nevertheless, the idea of keeping a constant stiffness matrix with the incremental method has enabled more CPU time saving at the microstructural scale.

Cette thèse s'intéresse à la question récurrente qu'est la résolution d'un problème pour un grand nombre de configurations différentes. Malgré l'augmentation constante de la puissance de calcul que l'on connait aujourd'hui, le traitement direct d'un tel problème reste souvent hors de portée. La technique qui est développée ici est basée sur l'utilisation de la Proper Generalized Decomposition (PGD) dans le cadre de la méthode LATIN. On étudie tout d’abord la capacité de cette technique de réduction de modèle à résoudre un problème paramétré pour un espace de conception donné. Lors du traitement d’un tel problème, on génère une base réduite que l’on peut réutiliser et éventuellement enrichir en traitant un par un les problèmes correspondants aux jeux de paramètres étudiés. Le but devient alors de développer une stratégie, inspirée par la méthode « Reduced Basis », afin d’explorer de façon rationnelle l’espace des paramètres. L’objectif étant de construire, avec le minimum de résolutions, une base réduite « complète » qui permet de résoudre tous les autres problèmes de l’espace de conception sans enrichir cette base. On commence dès lors par montrer l’existence d’une telle base complète en extrayant les informations les plus pertinentes des solutions PGD d’un problème pour tous les jeux de paramètres de l’espace de conception. On propose ensuite une stratégie rationnelle pour construire cette base complète sans la nécessité préalable de la résolution du problème pour tous les jeux de paramètres. Enfin, les performances de la méthode proposée sont illustrées sur plusieurs exemples, montrant des gains conséquents lorsque des études récurrentes doivent être menées. / This thesis deals with the recurring question of the resolution of a problem for many different configu- rations, which can lead to highly expensive computations when using a direct treatment. The technique which is presented here is based on the use of Proper Generalized Decomposition (PGD) in the framework of the LATIN method. The feasibility of this model reduction technique approach is studied to compute the solution of a parametrized problem for a given space of parameters. For that purpose, a Reduced-Order Basis is generated, reused and eventually enriched, by treating, one-by-one, all the various parameter sets. The aim is to develop a strategy, inspired by the Reduced Basis method, to explore rationally the space of parameters. Then, the objective is to build, with the minimum of resolutions, a ‘‘complete’’ basis that enables to solve all the other problems without enriching the basis. We first exemplify the existence of a such complete basis by extracting the most relevant information from the PGD solutions of the problem for all the sets in the space of parameters. Secondly, we propose a rational strategy to build this complete basis without preliminary solving the problem for all the sets of parameters. Finally, the capabilities of the proposed method are illustrated through a variety of examples, showing substantial gains when recurrent studies need to be carried out.

In this dissertation, I consider the problem of model reduction in both oscillatory and networked systems. Previously, the Manifold Boundary Approximation Method (MBAM) has been demonstrated as a data-driven tool for reducing the parametric complexity of so-called sloppy models. To be effective, MBAM requires the model manifold to have low curvature. I show that oscillatory models are characterized by model manifolds with high curvature in one or more directions. I propose methods for transforming the model manifolds of these models into ones with low curvature and demonstrate on a couple of test systems. I demonstrate MBAM as a tool for data-driven network reduction on a small model from power systems. I derive multiple effective networks for the model, each tailored to a specific choice of system observations. I find several important types of parameter reductions, including network reductions, which can be used in large power systems models. Finally, I consider the problem of piecemeal reduction of large systems. When a large system is split into pieces that are to be reduced separately using MBAM, there is no guarantee that the reduced pieces will be compatible for reassembly. I propose a strategy for reducing a system piecemeal while guaranteeing that the reduced pieces will be compatible. I demonstrate the reduction strategy on a small resistor network.

京都大学 / 新制・課程博士 / 博士(情報学) / 甲第23321号 / 情博第757号 / 新制||情||129(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 太田 快人, 教授 山下 信雄, 教授 大塚 敏之 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DGAM

This dissertation studies the model reduction and distributed control problems for interconnected systems, i.e., systems that consist of multiple interacting agents/subsystems. The study of the analysis and synthesis problems for interconnected systems is motivated by the multiple applications that can benefit from the design and implementation of distributed controllers. These applications include automated highway systems and formation flight of unmanned aircraft systems. The systems of interest are modeled using arbitrary directed graphs, where the subsystems correspond to the nodes, and the interconnections between the subsystems are described using the directed edges. In addition to the states of the subsystems, the adopted frameworks also model the interconnections between the subsystems as spatial states. Each agent/subsystem is assumed to have its own actuating and sensing capabilities. These capabilities are leveraged in order to design a controller subsystem for each plant subsystem. In the distributed control paradigm, the controller subsystems interact over the same interconnection structure as the plant subsystems. The models assumed for the subsystems are linear time-varying or linear parameter-varying. Linear time-varying models are useful for describing nonlinear equations that are linearized about prespecified trajectories, and linear parameter-varying models allow for capturing the nonlinearities of the agents, while still being amenable to control using linear techniques. It is clear from the above description that the size of the model for an interconnected system increases with the number of subsystems and the complexity of the interconnection structure. This motivates the development of model reduction techniques to rigorously reduce the size of the given model. In particular, this dissertation presents structure-preserving techniques for model reduction, i.e., techniques that guarantee that the interpretation of each state is retained in the reduced order system. Namely, the sought reduced order system is an interconnected system formed by reduced order subsystems that are interconnected over the same interconnection structure as that of the full order system. Model reduction is important for reducing the computational complexity of the system analysis and control synthesis problems. In this dissertation, interior point methods are extensively used for solving the semidefinite programming problems that arise in analysis and synthesis. / Ph. D. / The work in this dissertation is motivated by the numerous applications in which multiple agents interact and cooperate to perform a coordinated task. Examples of such applications include automated highway systems and formation flight of unmanned aircraft systems. For instance, one can think of the hazardous conditions created by a fire in a building and the benefits of using multiple interacting multirotors to deal with this emergency situation and reduce the risks on humans. This dissertation develops mathematical tools for studying and dealing with these complex systems. Namely, it is shown how controllers can be designed to ensure that such systems perform in the desired way, and how the models that describe the systems of interest can be systematically simplified to facilitate performing the tasks of mathematical analysis and control design.

A grinding mill circuit forms a crucial part in the energy-intensive comminution process of extracting valuable metals and minerals from mined ore. The ability to control the grinding mill circuit is of primary importance to achieve the desired product specification with regards to quality and production rate. In order to achieve control objectives an accurate dynamic model of the milling circuit is required. Phenomenological models are preferred over linear-time-invariant models since the latter cannot describe the non-linear behaviour of the process. However, the available phenomenological models of grinding mill circuits are usually complex, use large parameter sets and are mostly aimed towards steady-state design of grinding mill circuits. This study investigates simplified non-linear dynamic models of grinding mill circuits suitable for process controller design. In the first part of this study, the number of size classes in a cumulative rates model of a grinding mill circuit is reduced to determine the minimum number required to provide a reasonably accurate model of the circuit for process control. Each reduced size class set is used to create a non-linear cumulative rates model which is linearized to design a linear model predictive controller. The accuracy of a model is determined by the ability of the corresponding model predictive controller to control important process variables in the grinding mill circuit as represented by the full non-linear cumulative rates model. The second part of the study validates a simple and novel non-linear model of a run-of-mine grinding mill circuit developed for process control and estimation purposes. This model is named the Hulbert-model and makes use of the minimum number of states and parameters necessary to produce responses that are qualitatively accurate. It consists of separate feeder, mill, sump and hydrocyclone modules that can be connected to model different circuit configurations. The model uses five states: rocks, solids, fines, water and steel balls. Rocks are defined as too large to be discharged from the mill, whereas solids, defined as particles small enough to leave the mill, consist of out-of-specification coarse ore and in-specification fine ore fractions. The model incorporates a unique prediction of the rheology of the slurry within the mill. A new hydrocyclone model is also presented. The Hulbert-model parameters are fitted to an existing plant’s sampling campaign data and a step-wise procedure is given to fit the model to steady-state data. Simulation test results of the model are compared to sampling campaign data of the same plant at different steady-state conditions. / Dissertation (MEng)--University of Pretoria, 2012. / Electrical, Electronic and Computer Engineering / unrestricted

To meet the requirements for the next generation of space missions, a paradigm shift is required from current structures that are static, heavy and stiff, to innovative structures that are adaptive, lightweight, versatile, and intelligent. This work proposes the use of a novel morphing structure, the thermally actuated anisogrid morphing boom, to meet the design requirements by making the primary structure actively adapt to the on-orbit environment. The proposed concept achieves the morphing capability by applying local and global thermal gradients and using the resulting thermal strains to introduce a 6 Degree of Freedom (DOF) morphing control. To address the key technical challenges associated with implementing this concept, the work is broken into four sections. First, the capability to develop and reduce large dynamic models using the Data Based Loewner-SVD method is demonstrated. This reduction method provides the computationally efficient dynamic models required for evaluation of the concept and the assessment of a vast number of loading cases. Secondly, a sensitivity analysis based parameter ranking methodology is developed to define parameter importance. A five parameter model correlation effort is used to demonstrate the ability to simplify complex coupled problems. By reducing the parameters to only the most critical, the resulting morphing optimization computation and engineering time is greatly reduced. The third piece builds the foundation for the thermal morphing anisogrid structure by describing the concept, defining the modeling assumptions, evaluating the design space, and building the performance metrics. The final piece takes the parameter ranking methodology, developed in part two, and the modeling capability of part three, and performs a trust-region optimization to define optimal morphing geometric configuration. The resulting geometry, optimized for minimum morphing capability, is evaluated to determine the morphing workspace, the frequency response capability, and the minimum and maximum morphing capability in 6 DOF. This work has demonstrated the potential and provided the technical tools required to model and optimize this novel smart structural concept for a variety of applications. / Ph. D.

Cette thèse est dédiée au développement de méthodes numériques pour la poromécanique, et à leur application dans un contexte de modélisation cardiaque.Elle est motivée par la prise en compte, dans les modèles de coeur humain, du réseau coronarien qui perfuse le myocarde, afin de mieux décrire les maladies vasculaires coronariennes.Nous appuyant sur des travaux existants, nous proposons un modèle de coeur perfusé, ainsi qu'une réduction 0D permettant de reproduire, à moindre coût de calcul, un cycle cardiaque réaliste avec masse et pression de perfusion. Le modèle mis au point nous permet de reproduire des phénomènes physiologiques auparavant inaccessibles dans les modèles, et d'une grande importance pour des applications cliniques, tels que la vasodilatation et les pathologies coronariennes.L'intégration d'un compartiment poreux pour représenter le myocarde perfusé dans les modèles 3D représente un défi technique d'un autre ordre. Nous inspirant des schémas en temps de type splitting établis en interaction fluide-structure pour modéliser les vaisseaux sanguins, nous proposons une discrétisation semi-implicite d'une formulation générale de poromécanique, satisfaisant un bilan d'énergie au niveau discret. Afin d'illustrer et valider notre démarche, l'environnement de calcul élément finis FreeFem++ nous permet de reproduire des cas tests classiques de gonflement et de drainage de milieux poreux en 2D, puis de vérifier le bilan énergétique discret.Enfin, motivés par l'étude de la discrétisation spatiale de notre problème, nous établissons dans un cadre linéaire un résultat de convergence totale du schéma sous conditions. Cela nous permet de proposer une méthode d'implémentation facile à mettre en oeuvre et présentant de bons résultats de stabilité. FreeFem++ nous permet à nouveau de valider nos résultats en illustrant les pathologies numériques associées à l'incompressibilité, et leur traitement efficace par les stratégies proposées, dans le cadre linéaire puis dans une situation plus générale. / This thesis is dedicated to the development of numerical methods for poromechanics, and to their application in a cardiac modeling context. It is motivated by the introduction into existing cardiac models of the coronary network that perfuses the myocardium, to better describe coronary vascular diseases.Drawing our inspiration from existing works, we propose a perfused heart model, and a 0D reduction allowing the cost-effective reproduction of a realistic cardiac cycle with perfusion mass and pressure. The model derived illustrates physiological phenomena inaccessible in former models, and with great clinical application potential, such as vasodilatation and coronary diseases.The integration of a porous compartment to represent the perfused myocardium within 3D models is more challenging. Relying on splitting time schemes established for fluid-structure interaction to model blood vessels, we propose a semi-implicit discretization of a general poromechanics formulation, satisfying a discrete energy balance. In order to illustrate and validate our approach, we reproduce in the finite element software FreeFem++ classical swelling and drainage 2D test cases, and we monitor the discrete energy balance.Finally, motivated by the study of spatial discretization aspects of our problem, we establish in a linear framework a conditional total convergence result. This enables us to propose a computational method easy to implement and presenting good stability results. FreeFem++ enables us again to validate our results, illustrating numerical pathologies associated with incompressibility, and their efficient treatment with the proposed strategies, first in a linear framework and then in a more general situation.