Cette thèse a pour objectif principal de développer un outil numérique capable de prédire la ductilité des matériaux polycristallins. Cet outil est basé sur le couplage de l’approche multi-échelles autocohérente à deux critères d’instabilités plastiques : la théorie de bifurcation et l’approche d’imperfection initiale. Le schéma autocohérent est utilisé pour déterminer le comportement d’un agrégat polycristallin (supposé représentatif du matériau étudié) à partir du comportement des monocristaux constitutifs. Le comportement à l’échelle monocristalline est formulé dans le cadre des grandes déformations élastoplastiques. Deux différentes versions du critère de Schmid sont successivement utilisées pour modéliser l’écoulement plastique monocristallin : la version classique et une version régularisée. Pour intégrer numériquement les équations constitutives à l’échelle monocristalline, deux algorithmes ont été développés : un algorithme de type évolutif et un algorithme de type retour radial. Les équations gouvernant le schéma autocohérent ont été revisitées. Pour résoudre ces équations, un nouvel algorithme numérique a été proposé, qui est montré être plus efficace que les algorithmes existants communément basés sur la méthode du point fixe. De plus, une approche numérique robuste a été développée, qui permet de coupler le modèle autocohérent à l’approche d’imperfection initiale. La performance ainsi que la robustesse des différents algorithmes et schémas numériques développés ont été mis en évidence à travers plusieurs résultats de simulation. L’effet de plusieurs paramètres et choix de modélisation sur la prédiction de formabilité des tôles métalliques a été extensivement analysé. / The main objective of this PhD thesis is to develop a numerical tool capable of predicting the ductility of polycrystalline materials. This tool is based on the coupling of the self-consistent multiscale approach with two plastic instability criteria: the bifurcation theory and the initial imperfection approach. The self-consistent scheme is used to derive the mechanical behavior of a polycrystalline aggregate (assumed to be representative of the studied material) from that of its microscopic constituents (the single crystals). The constitutive framework at the single crystal scale follows a finite strain rate-independent formulation. Two different versions of the Schmid law are successively used to model the plastic flow: the classical version and a regularized one. To solve the constitutive equations at the single crystal scale, two numerical algorithms have been developed: one is based on the usual return-mapping scheme and the other on the so-called ultimate scheme. The equations governing the self-consistent approach have been revisited. To solve these equations, a new numerical scheme has been developed, which is shown to be more efficient than the existing schemes commonly based on the fixed point method. Also, a robust numerical approach has been developed to couple the self-consistent model to the initial imperfection approach. The performance and the robustness of the different numerical schemes and algorithms developed have been highlighted through several simulation results. The impact of various parameters and modeling choices on the formability prediction of sheet metals has been extensively analyzed.

Este trabalho apresenta um estudo sobre o método da continuação aplicado ao problema de fluxo de potência. Definições e conceitos de estabilidade de tensão são descritos de forma a explicitar as diferenças e semelhanças existentes com relação ao estudo de máximo carregamento em sistemas de energia elétrica. Uma síntese da teoria da bifurcação aborda sua importância em estudos de colapso de tensão. É proposta uma técnica de controle do tamanho do passo para o método da continuação com o objetivo de determinar o ponto de máximo carregamento (PMC) sem a necessidade de especificar, por meio do usuário, um valor para o tamanho inicial do passo. Os resultados dos estudos realizados em sistemas testes do IEEE (14, 30, 57 e 118 barras) mostram a aplicação do método da continuação convencional e de sua associação à técnica de controle do tamanho do passo. / This work presents a research about the continuation method applied to the power flow problem. Voltage stability definitions and concepts are described in a way to highlight and point out the differences and the similarities among several methods used to determine the maximum loading of electrical power systems. A short description of the bifurcation theory is also presented in order to show its importance to the voltage collapse studies. A technique based on automatically controlling the step size is proposed as an innovation of the continuation method. The objective of this technique is to determine the maximum loading point without the traditional need of asking the user for the initial step size. The results compare the performance between the conventional and the new method. These methods are analyzed using IEEE test systems (14, 30, 57 and 118-bus).

Cette thèse a pour sujet les effets que l’information a sur les incitations. Les trois articles fournissent et explorent des résultats lorsque l’information est la principale variable d’intérêt, est endogène, pas homogène entre les acteurs et évolue dans le temps d’une manière qui n’est pas nécessairement rationnelle. Le premier article étudie les problèmes de hold-up dans les hiérarchies verticales avec la sélection adverse montrant qu’alors que le pouvoir de négociation des travailleurs augmente, les distorsions provenant de l’asymétrie d’information disparaissent. En outre, il étudie l’effet de la scolarité et du degré d’hétérogénéité de la population de travailleurs sur la répartition du pouvoir de négociation dans les marchés réglementés. Le deuxième article assouplit l’hypothèse des croyances homogènes dans les relations principal-agent avec sélection adverse. Dans un apprentissage évolutif qui est imitatif, les principaux peuvent avoir des croyances différentes sur la répartition des types d’agents dans la population. La convergence à une croyance uniforme dépend de la taille relative de la polarisation dans les croyances. En outre, le modèle est une version d’un cobweb stable. Notre approche offre des explications pour l’alternance des périodes avec quantité oscillante et relativement stable. Le troisième article étudie la fac¸on dont le contenu informatif des politiques juridiques, comme la responsabilité stricte et négligence, en cas de soucis morales, influence la conception optimale des régimes de responsabilité. Plusieurs cas récents ont montré qu’un individu ayant causé un dommage s’expose non seulement a une sanction légale — par exemple, une amende — mais aussi a un boycott social, la désapprobation ou la stigmatisation. L’article montre que le choix d’une stratégie dépend de façon complexe de l’importance du dommage et du “coût moral”. / This thesis focuses on the effects that information has on incentives. The three papers provide and explore some results when the information is the main variable of interest, it is made endogenous, not homogeneous between actors and evolving over time in a way that is not necessarily rational. The first paper studies hold-up problems in vertical hierarchies with adverse selection showing that as the bargaining power of the worker increases, distortions coming from asymmetric information vanish. Moreover, it studies the effect of schooling and degree of heterogeneity in the workforce on the allocation of bargaining power in regulating markets. The second paper relaxes the common assumption of homogeneous beliefs in principal-agent relationships with adverse selection. In an evolutionary learning set-up, which is imitative, principals can have different beliefs about the distribution of agents’ types in the population. Convergence to a uniform belief depends on the relative size of the bias in beliefs. In addition, the set-up is a version of a stable cobweb model. Our approach offers explanations for alternating periods of oscillating and relatively steady quantity. The third paper studies how the informative content of legal policies as strict-liability and fault-based, in case of moral concerns, influences the optimal design of liability regimes. Many recent cases show that an individual found to have caused harm faces not only the possibility of a legal sanction — e.g., the damages he must pay — but also social boycott, disapproval or stigma. The paper shows that the choice of a policy depends in a complex way on the magnitude of the harm and the “moral cost”.Keywords: Bargaining, Adverse Selection, Hold-up, Evolutionary Game Theory, Heterogeneous Beliefs, Bifurcation Theory, Boycott, Law Enforcement, Strict Liability, Negligence.

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Cette thèse présente quelques contributions à l'étude qualitative de solutions d'équations aux dérivées partielles non linéaires dans des modèles issus de l'optique et de la mécanique des fluides. Nous nous intéressons plus précisément à l'existence de solutions et à leur stabilité temporelle. Le Chapitre 1 est consacré à l'équation de Lugiato-Lefever, qui est une variante de l'équation de Schrödinger non linéaire et qui a été dérivée dans plusieurs contextes en optique. En utilisant des outils de la théorie des bifurcations et des formes normales, nous procédons à une étude systématique des solutions stationnaires de cette équation, et prouvons l'existence de solutions périodiques et localisées. Dans le Chapitre 2, nous présentons un critère simple d'instabilité linéaire pour des ondes non linéaires. Nous appliquons ce résultat aux équations de Lugiato-Lefever, de Kadomtsev-Petviashvili-I et de Davey-Stewartson. Ces deux dernières équations sont des équations modèles dérivées en mécanique des fluides. Dans le Chapitre 3, nous montrons un critère d'instabilité linéaire pour des solutions périodiques de petite amplitude, par rapport à certaines perturbations quasipériodiques. Ce résultat est ensuite appliqué à l'équation de Lugiato-Lefever. / In this thesis we present several contributions to qualitative study of solutions of nonlinear partial differential equations in optics and fluid mechanics models. More precisely, we focus on the existence of solutions and their stability properties. In Chapter 1, we study the Lugiato-lefever equation, which is a variant of the nonlinear Schrödinger equation arising in sereval contexts in nonlinear optics. Using tools from bifurcation and normal forms theory, we perfom a systematic analysis of stationary solutions of this equation and prove the existence of periodic and localized solutions. In Chapter 2, we present a simple criterion for linear instability of nonlinear waves. We then apply this result to the Lugiato-Lefever equation, to the Kadomtsev-Petviashvili-I equation and the Davey-Stewartson equations. These last two equations are model equations arising in fluid mechanics. In Chapter 3, we prove a criterion for linear instability of periodic solutions with small amplitude, with respect to certain quasiperiodic perturbations. This result is then applied to the Lugiato-Lefever equation.

Este trabalho apresenta um estudo sobre o método da continuação aplicado ao problema de fluxo de potência. Definições e conceitos de estabilidade de tensão são descritos de forma a explicitar as diferenças e semelhanças existentes com relação ao estudo de máximo carregamento em sistemas de energia elétrica. Uma síntese da teoria da bifurcação aborda sua importância em estudos de colapso de tensão. É proposta uma técnica de controle do tamanho do passo para o método da continuação com o objetivo de determinar o ponto de máximo carregamento (PMC) sem a necessidade de especificar, por meio do usuário, um valor para o tamanho inicial do passo. Os resultados dos estudos realizados em sistemas testes do IEEE (14, 30, 57 e 118 barras) mostram a aplicação do método da continuação convencional e de sua associação à técnica de controle do tamanho do passo. / This work presents a research about the continuation method applied to the power flow problem. Voltage stability definitions and concepts are described in a way to highlight and point out the differences and the similarities among several methods used to determine the maximum loading of electrical power systems. A short description of the bifurcation theory is also presented in order to show its importance to the voltage collapse studies. A technique based on automatically controlling the step size is proposed as an innovation of the continuation method. The objective of this technique is to determine the maximum loading point without the traditional need of asking the user for the initial step size. The results compare the performance between the conventional and the new method. These methods are analyzed using IEEE test systems (14, 30, 57 and 118-bus).

Prédire la statistique des grandes échelles des écoulements turbulents constitue un enjeu important. Pour l'équation d'Euler 2D et des modèles analogues d'écoulements géophysiques, une auto-organisation est observée (formation de cyclones/anticyclones, jets intenses). La mécanique statistique d'équilibre des écoulements bidimensionnels s'est avérée fondamentale et pertinente même en présence de forçage et dissipation, dans la limite inertielle. La thèse est motivée par le phénomène de transitions aléatoires entre deux topologies différentes, lié à une bistabilité. Il s'agit de prédire la multiplicité des équilibres d'un écoulement (quasi) bidimensionnel. On développe une classification des transitions de phase, pour des équilibres (statistiques et/ou dynamiques) d'un tel écoulement. Les diagrammes de phase font apparaître la présence générique de points critiques et tricritiques, et des domaines d'inéquivalence d'ensembles statistiques. Dans le cas d'une géométrie annulaire, on décrit les effets de la topographie et de la conservation de deux circulations. Des analogies avec la bistabilité du courant océanique Kuroshio sont proposées à partir de cette étude académique. Enfin, pour le système Euler 2D, on détaille un résultat de mécanique statistique dans l'ensemble énergie-enstrophie : la distribution microcanonique, construite à partir du théorème de Liouville en dimension finie, correspond à la maximisation d'une entropie de mélange de la vorticité. / A most challenging problem in turbulence is to predict the statistics of flows at the large scales. In the case of the 2D Euler equation and analogous models for geophysical flows, the flow is observed to self-organize: cyclones/anticyclones and intense jets form. Equilibrium statistical mechanics has proven to be fundamental and relevant even in the presence of forcing and dissipation, in the inertial limit. The thesis is motivated by the phenomenon of random transitions between two different topologies. This phenomenon implies bistability. The goal is to predict the multiplicity of equilibria for a (quasi) two-dimensional flow. We develop a classification of phase transitions for the (statistical and/or dynamical) equilibria of this flow. Phase diagrams show critical and tricritical points as well as domains of statistical ensemble inequivalence, all this generically. In the case of an annular geometry, the effects of topography and of conserving two circulations are described. Analogies between the bistability of the ocean current Kuroshio and this academic study are suggested. Lastly, for the 2D Euler system, a statistical-mechanical result in the energy-enstrophy ensemble is detailed: the microcanonical distribution, constructed from Liouville's theorem in finite dimension, corresponds to the maximization of a vorticity-mixing entropy.

The purpose of this thesis was to examine a partial differential equation to model the Tacoma Narrows bridge failure. This thesis will examine the equation developed by Lazer and McKenna to model a suspension bridge in no wind.

Ce travail présente une caractérisation expérimentale du comportement mécanique et de la rupture par localisation de la déformation dans un grès des Vosges. L'évolution temporelle de la localisation a été caractérisée par des mesures de champs. Une nouvelle cellule triaxiale vraie a été développée au Laboratoire 3SR (Grenoble), qui permet une visualisation des échantillons sous chargement pour réaliser de la corrélation d'image numérique (CIN). Les essais ont été réalisés par compression en déformation plane (confinement de 20 à 50 MPa). La transition d'une déformation diffuse à localisée a été finement étudiée. Une analyse comparative a été ensuite effectuée entre les mesures de champs et la microstructure à l'échelle des grains observée par microscope (MEB). Enfin, une étude théorique basée sur une analyse en bifurcation a été menée pour comparer observations des bandes de cisaillement et prédiction sur la localisation de la déformation. / This work aims an experimental characterization of the mechanical behaviour and failure by strain localization on a Vosges sandstone. The time evolution of strain localization has been characterized by full-field measurements. A new true-triaxial apparatus has been developed at Laboratoire 3SR (Grenoble), which enables the observation of the specimens during mechanical loading for application of digital image correlation (DIC). Tests have been performed in plane strain compression (confining pressure from 20 to 50 MPa). The transition from diffuse to localised deformation regimes has been extensively studied. Then, a comparative analysis has been done between the strain fields (DIC) and microscope (SEM) observations to determine how closely the DIC fields are related to deformation mechanisms detected at the grain scale. Finally, a theoretical bifurcation analysis is presented to compare the experimental observations of shear bands with strain localization prediction.

Indiana University-Purdue University Indianapolis (IUPUI) / Physical and biological phenomena that involve oscillations on multiple time scales attract attention of mathematicians because resulting equations include a small parameter that allows for decomposing a three- or higher-dimensional dynamical system into fast/slow subsystems of lower dimensionality and analyzing them independently using geometric singular perturbation theory and other techniques. However, in most life sciences applications observed dynamics is extremely complex, no small parameter exists and this approach fails. Nevertheless, it is still desirable to gain insight into behavior of these mathematical models using the only viable alternative – ad hoc computational analysis. Current dissertation is devoted to this latter approach. Neural networks in the region of the brain called basal ganglia (BG) are capable of producing rich activity patterns. For example, burst firing, i.e. a train of action potentials followed by a period of quiescence in neurons of the subthalamic nucleus (STN) in BG was shown to be related to involuntary shaking of limbs in Parkinson’s disease called tremor. The origin of tremor remains unknown; however, a few hypotheses of tremor-generation were proposed recently. The first project of this dissertation examines the BG-thalamo-cortical loop hypothesis for tremor generation by building physiologically-relevant mathematical model of tremor-related circuits with negative delayed feedback. The dynamics of the model is explored under variation of connection strength and delay parameters in the feedback loop using computational methods and data analysis techniques. The model is shown to qualitatively reproduce the transition from irregular physiological activity to pathological synchronous dynamics with varying parameters that are affected in Parkinson’s disease. Thus, the proposed model provides an explanation for the basal ganglia-thalamo-cortical loop mechanism of tremor generation. Besides tremor-related bursting activity BG structures in Parkinson’s disease also show increased synchronized activity in the beta-band (10-30Hz) that ultimately causes other parkinsonian symptoms like slowness of movement, rigidity etc. Suppression of excessively synchronous beta-band oscillatory activity is believed to suppress hypokinetic motor symptoms in Parkinson’s disease. Recently, a lot of interest has been devoted to desynchronizing delayed feedback deep brain stimulation (DBS). This type of synchrony control was shown to destabilize synchronized state in networks of simple model oscillators as well as in networks of coupled model neurons. However, the dynamics of the neural activity in Parkinson’s disease exhibits complex intermittent synchronous patterns, far from the idealized synchronized dynamics used to study the delayed feedback stimulation. The second project of this dissertation explores the action of delayed feedback stimulation on partially synchronous oscillatory dynamics, similar to what one observes experimentally in parkinsonian patients. We employ a computational model of the basal ganglia networks which reproduces the fine temporal structure of the synchronous dynamics observed experimentally. Modeling results suggest that delayed feedback DBS in Parkinson’s disease may boost rather than suppresses synchronization and is therefore unlikely to be clinically successful. Single neuron dynamics may also have important physiological meaning. For instance, bistability – coexistence of two stable solutions observed experimentally in many neurons is thought to be involved in some short-term memory tasks. Bistability that occurs at the depolarization block, i.e. a silent depolarized state a neuron enters with excessive excitatory input was proposed to play a role in improving robustness of oscillations in pacemaker-type neurons. The third project of this dissertation studies what parameters control bistability at the depolarization block in the three-dimensional conductance-based neuronal model by comparing the reduced dopaminergic neuron model to the Hodgkin-Huxley model of the squid giant axon. Bifurcation analysis and parameter variations revealed that bistability is mainly characterized by the inactivation of the Na+ current, while the activation characteristics of the Na+ and the delayed rectifier K+ currents do not account for the difference in bistability in the two models.