Fluid dynamics models provide a powerful deterministic technique to approximate stochasticity in a variety of application areas. In this paper, we study two classes of fluid models, investigate their relationship as well as some of their applications. This analysis allows us to provide analytical models of travel times as they arise in dynamically evolving environments, such as transportation networks as well as supply chains. In particular, using the laws of hydrodynamic theory, we first propose and examine a general second order fluid model. We consider a first-order approximation of this model and show how it is helpful in analyzing the dynamic traffic equilibrium problem. Furthermore, we present an alternate class of fluid models that are traditionally used in the context of dynamic traffic assignment. By interpreting travel times as price/inventory-sojourn-time relationships, we are also able to connect this approach with a tractable fluid model in the context of dynamic pricing and inventory management. Finally, we investigate the relationship between these two classes of fluid models. / Singapore-MIT Alliance (SMA)

VehProLib is one of many libraries being developed for the object oriented multi-domain language Modelica. The layout and the current status of the library are shown. The aims of the library are to provide the user with a number of different components with different levels of complexity. The components included range from mean value engine components to in-cylinder models. An efficient way to handle parameters using records is provided. Different bus systems are implemented and discussed. Furthermore are replaceable fluid models introduced in the library. It will be shown that Modelica is a very efficient way to create an advanced modelling library.

A Generalized Cambridge Ring is a queueing system that can be used as an approximate model of some material handling systems used in modern factories. It consists of one or more vehicles that carry cargo from origins to destinations around a loop, with queues forming when cargo temporarily exceeds the capacity of the system. For some Generalized Cambridge Rings that satisfy the usual traffic conditions for stability, it is demonstrated that some nonidling scheduling polices are unstable. A good scheduling policy will increase the efficiency of these systems by reducing waiting times and by therefore also reducing work in process (WIP). Simple heuristic policies are developed which provide substantial improvements over the commonly used first-in-first-out (FIFO) policy. Variances are incorporated into previously developed fluid models that used only means to produce a more accurate partially discrete fluid mean-variance model, which is used to further reduce waiting times. Optimal policies are obtained for some simple special cases, and simulations are used to compare policies in more general cases. The methods developed may be applicable to other queueing systems. / text

Cette thèse est dédiée à la description et à l'analyse mathématique de phénomènes d'hétérogénéités et de congestion dans les modèles de la mécanique des fluides.On montre un lien rigoureux entre des modèles de congestion douce de type Navier-Stokes compressible qui intègrent des forces de répulsion à très courte portée entre composants élémentaires; et des modèles de congestion dure de type compressible/incompressible décrivant les transitions entre zones libres et zones congestionnées.On s'intéresse ensuite à la modélisation macroscopique de mélanges formés par des particules solides immergées dans un fluide.On apporte dans ce cadre une première réponse mathématique à la question de la transition entre les régimes de suspensions dictés par les interactions hydrodynamiques et les régimes granulaires dictés par les contacts entre les particules solides.On met par cette démarche en évidence le rôle crucial joué par les effets de mémoire dans le régime granulaire.Cette approche permet également un nouveau point de vue pour l'étude mathématique des fluides avec viscosité dépendant de la pression.On s'intéresse enfin à la modélisation microscopique et macroscopique du trafic routier.Des schémas numériques originaux sont proposés afin de reproduire des phénomènes de persistance d'embouteillages. / This thesis is dedicated to the description and the mathematical analysis of heterogeneities and congestion phenomena in fluid mechanics models.A rigorous link between soft congestion models, based on the compressible Navier--Stokes equations which take into account short--range repulsive forces between elementary components; and hard congestion models which describe the transitions between free/compressible zones and congested/incompressible zones.We are interested then in the macroscopic modelling of mixtures composed solid particles immersed in a fluid.We provide a first mathematical answer to the question of the transition between the suspension regime dictated by hydrodynamical interactions and the granular regime dictated by the contacts between the solid particles.The method highlights the crucial role played by the memory effects in the granular regime.This approach enables also a new point of view concerning fluids with pressure-dependent viscosities.We finally deal with the microscopic and the macroscopic modelling of vehicular traffic.Original numerical schemes are proposed to robustly reproduce persistent traffic jams.

Pour concevoir les futurs réacteurs à fusion nucléaire, une bonne compréhension des mécanismes régissant l'intéraction plasma-paroi est requise. En particulier, il est nécessaire d'estimer quantitativement les flux de chaleurs impactant les matériaux et la contamination du coeur par les impuretés provenant du mur. Dans ce contexte, le code fluide SolEdge2D a été développé pour simuler le transport dans le plasma de bord. L'interaction plasma-paroi est prise en compte grâce à une méthode de pénalisation innovante et originale. Cette méthode permet en particulier de modéliser la géométrie complexe des éléments face au plasma avec une grande flexibilité. En parallèle, une étude plus théorique sur les propriétés du transport dans les milieux faiblement collisionels a été conduite avec les physiciens du groupe CSDC de l'université de Florence. Une généralisation de la loi de Fourier prenant en compte les corrélation spatio-temporelle à longue distance à été obtenue par l'analyse de modèles stochastiques 1D. Cette loi retrouve en particulier la transition entre un régime diffusif à forte collisionalté et un régime balistique à faible collisionalité. / In the scope of designing future nuclear fusion reactors, a clear understanding of the plasma-wall interaction is mandatory. Indeed, a predictive estimation of heat flux impacting the surface and the subsequent emission of impurities from the wall is necessary to ensure material integrity and energy confinement performances. In that perspective, the fluid code SolEdge2D has been developed to simulate plasma transport in the tokamak edge plasma. The plasma-wall interaction is modeled using an innovative penalization technique. This method enables in particular to take complex plasma facing components geometry into account. In parallel to this numerical effort, a theoretical work has been achieved to find appropriate corrections to fluid closures when collisionality drops. The study of stochastic 1D models has been realized in collaboration with physicists from the CSDC group in Florence. A generalized Fourier law taking long range spatio-temporal correlations has been found to properly account for ballistic transport in the low collisional regime. This formulation is expected to be used to model parallel heat flux or turbulent cross-field transport in tokamak plasmas.

This dissertation addresses some of the most important issues in delay-sensitive communication over wireless systems and networks. Traditionally, the design of communication networks adopts a layered framework where each layer serves as a “black box” abstraction for higher layers. However, in the context of wireless networks with delay-sensitive applications such as Voice over Internet Protocol (VoIP), on-line gaming, and video conferencing, this layered architecture does not offer a complete picture. For example, an information theoretic perspective on the physical layer typically ignores the bursty nature of practical sources and often overlooks the role of delay in service quality. The purpose of this dissertation is to take on a cross-disciplinary approach to derive new fundamental limits on the performance, in terms of capacity and delay, of wireless systems and to apply these limits to the design of practical wireless systems that support delay-sensitive applications. To realize this goal, we consider a number of objectives. 1. Develop an integrated methodology for the analysis of wireless systems that support delay-sensitive applications based, in part, on large deviation theory. 2. Use this methodology to identify fundamental performance limits and to design systems which allocate resources efficiently under stringent service requirements. 3. Analyze the performance of wireless communication networks that takes advantage of novel paradigms such as user cooperation, and multi-antenna systems. Based on the proposed framework, we find that delay constraints significantly influence how system resources should be allocated. Channel correlation has a major impact on the performance of wireless communication systems. Sophisticated power control based on the joint space of channel and buffer states are essential for delaysensitive communications.

Geodynamic modelling, via computer simulations, offers an easily controllable method for investigating the behaviour of an Earth system and providing feedback to conceptual models of geological evolution. However, most available computer codes have been developed for engineering or hydrological applications, where strains are small and post-failure deformation is not studied. Such codes cannot simultaneously model large deformation and porous fluid flow. To remedy this situation in the face of tectonic modelling, a numerical approach was developed to incorporate porous fluid flow into an existing high-deformation code called Ellipsis. The resulting software, with these twin capabilities, simulates the evolution of highly deformed tectonic regimes where fluid flow is important, such as in mineral provinces. A realistic description of deformation depends on the accurate characterisation of material properties and the laws governing material behaviour. Aside from the development of appropriate physics, it can be a difficult task to find a set of model parameters, including material properties and initial geometries, that can reproduce some conceptual target. In this context, an interactive system for the rapid exploration of model parameter space, and for the evaluation of all model results, replaces the traditional but time-consuming approach of finding a result via trial and error. The visualisation of all solutions in such a search of parameter space, through simple graphical tools, adds a new degree of understanding to the effects of variations in the parameters, the importance of each parameter in controlling a solution, and the degree of coverage of the parameter space. Two final applications of the software code and interactive parameter search illustrate the power of numerical modelling within the feedback loop to field observations. In the first example, vertical rheological contrasts between the upper and lower crust, most easily related to thermal profiles and mineralogy, exert a greater control over the mode of crustal extension than any other parameters. A weak lower crust promotes large fault spacing with high displacements, often overriding initial close fault spacing, to lead eventually to metamorphic core complex formation. In the second case, specifically tied to the history of compressional orogenies in northern Nevada, exploration of model parameters shows that the natural reactivation of early normal faults in the Proterozoic basement, regardless of basement topography or rheological contrasts, would explain the subsequent elevation and gravitationally-induced thrusting of sedimentary layers over the Carlin gold trend, providing pathways and ponding sites for mineral-bearing fluids.

Cette thèse traite de la modélisation et de la simulation des écoulements diphasiques transitoires eau-vapeur. Dans de nombreuses installations industrielles, des écoulements monophasiques d'eau liquide sont susceptibles de devenir diphasiques lors de certaines situations transitoires. La modélisation de ces écoulements peut s'avérer délicate car deux phénomènes physiques interagissant fortement entre eux, le changement de phase et la propagation d'ondes de pression, sont alors à prendre en compte. Une approche bifluide statistique, ne supposant aucun équilibre entre les phases, est utilisée afin de modéliser de tels écoulements. Les modèles obtenus sont de type convection-source et s'apparentent au modèle de Baer-Nunziato. Différentes lois de fermeture pour ces modèles sont comparées sur des cas expérimentaux de transitoires eau-vapeur tels que les coups de bélier et la dépressurisation d'une tuyauterie d'eau liquide suite à une rupture.La simulation numérique des différents modèles est effectuée grâce à une méthode à pas fractionnaires. Un nouveau schéma de convection, robuste et efficace, capable de gérer toute equation d'état est utilisé dans la première étape de cette méthode. La seconde étape est dédiée au traitement des termes sources et requiert différents schémas implicites. Une grande attention est accordée à la vérification de tous les schémas numériques utilisés grâce à des études de convergence. Une nouvelle modélisation du transfert de masse est également proposée, sur la base de travaux dédiés à la brusque dépressurisation d'eau liquide en approche homogène. La validation du modèle est effectuée grâce de nombreuses comparaisons calcul-expérience. / This thesis deals with the modelling and the computation of steam-water two-phase flows. Liquid water flows are involved in many industrial facilities and a second phase may appear in some transients situations. Thus, pressure wave propagation and mass transfer are physical phenomena that need to be properly included in the modelling of such two-phase flows. A statistical two-fluid approach is used, leading to models similar to the Baer-Nunziato one. They include both convective and source terms without any assumption on the equilibrium between phases. Different closure laws for such models are compared on steam-water transient experiments including water-hammers and fast depressurization of liquid water. The computation of the different models is based on a fractional step method. A new convective scheme, able to deal with any Equation Of State, is used in the first step of the method. When compared with other schemes, it appears to be accurate, efficient and very robust. The second step of the method is dedicated to the treatment of source terms and requires several implicit schemes. Particular attention is paid to the verification of every scheme involved in the method. Convergence studies are carried out on test-cases with analytical solutions to that purpose. Based on existing work on fast depressurization of liquid water in an homogeneous approach, a new formulation of the mass transfer is proposed. Many comparisons between computational and experimental results are detailled in order to validate the models.

A Markov process is defined by its transition matrix. A skip-free Markov process is a stochastic system defined by a level that can only change by one unit either upwards or downwards. A regular perturbation is defined as a modification of one or more parameters that is small enough not to change qualitatively the model.<p>This thesis focuses on a category of methods, called matrix analytic methods, that has gained much interest because of good computational properties for the analysis of a large family of stochastic processes. Those methods are used in this work in order i) to analyze the effect of regular perturbations of the transition matrix on the stationary distribution of skip-free Markov processes; ii) to determine transient distributions of skip-free Markov processes by performing regular perturbations.<p>In the class of skip-free Markov processes, we focus in particular on quasi-birth-and-death (QBD) processes and Markov modulated fluid models.<p><p>We first determine the first order derivative of the stationary distribution - a key vector in Markov models - of a QBD for which we slightly perturb the transition matrix. This leads us to the study of Poisson equations that we analyze for finite and infinite QBDs. The infinite case has to be treated with more caution therefore, we first analyze it using probabilistic arguments based on a decomposition through first passage times to lower levels. Then, we use general algebraic arguments and use the repetitive block structure of the transition matrix to obtain all the solutions of the equation. The solutions of the Poisson equation need a generalized inverse called the deviation matrix. We develop a recursive formula for the computation of this matrix for the finite case and we derive an explicit expression for the elements of this matrix for the infinite case.<p><p>Then, we analyze the first order derivative of the stationary distribution of a Markov modulated fluid model. This leads to the analysis of the matrix of first return times to the initial level, a charactersitic matrix of Markov modulated fluid models.<p><p>Finally, we study the cumulative distribution function of the level in finite time and joint distribution functions (such as the level at a given finite time and the maximum level reached over a finite time interval). We show that our technique gives good approximations and allow to compute efficiently those distribution functions.<p><p><p>----------<p><p><p><p><p><p>Un processus markovien est défini par sa matrice de transition. Un processus markovien sans sauts est un processus stochastique de Markov défini par un niveau qui ne peut changer que d'une unité à la fois, soit vers le haut, soit vers le bas. Une perturbation régulière est une modification suffisamment petite d'un ou plusieurs paramètres qui ne modifie pas qualitativement le modèle.<p><p>Dans ce travail, nous utilisons des méthodes matricielles pour i) analyser l'effet de perturbations régulières de la matrice de transition sur le processus markoviens sans sauts; ii) déterminer des lois de probabilités en temps fini de processus markoviens sans sauts en réalisant des perturbations régulières. <p>Dans la famille des processus markoviens sans sauts, nous nous concentrons en particulier sur les processus quasi-birth-and-death (QBD) et sur les files fluides markoviennes. <p><p><p><p>Nous nous intéressons d'abord à la dérivée de premier ordre de la distribution stationnaire – vecteur clé des modèles markoviens – d'un QBD dont on modifie légèrement la matrice de transition. Celle-ci nous amène à devoir résoudre les équations de Poisson, que nous étudions pour les processus QBD finis et infinis. Le cas infini étant plus délicat, nous l'analysons en premier lieu par des arguments probabilistes en nous basant sur une décomposition par des temps de premier passage. En second lieu, nous faisons appel à un théorème général d'algèbre linéaire et utilisons la structure répétitive de la matrice de transition pour obtenir toutes les solutions à l’équation. Les solutions de l'équation de Poisson font appel à un inverse généralisé, appelé la matrice de déviation. Nous développons ensuite une formule récursive pour le calcul de cette matrice dans le cas fini et nous dérivons une expression explicite des éléments de cette dernière dans le cas infini.<p>Ensuite, nous analysons la dérivée de premier ordre de la distribution stationnaire d'une file fluide markovienne perturbée. Celle-ci nous amène à développer l'analyse de la matrice des temps de premier retour au niveau initial – matrice caractéristique des files fluides markoviennes. <p>Enfin, dans les files fluides markoviennes, nous étudions la fonction de répartition en temps fini du niveau et des fonctions de répartitions jointes (telles que le niveau à un instant donné et le niveau maximum atteint pendant un intervalle de temps donné). Nous montrerons que cette technique permet de trouver des bonnes approximations et de calculer efficacement ces fonctions de répartitions. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Les travaux de thèse sont axés sur les méthodes numériques pour les écoulements diphasiques, compressibles, à faible vitesse, avec apparition soudaine de forts gradients de pression. La vitesse matérielle de chacune des phases étant très petite devant la célérité des ondes acoustiques, le régime d'écoulement est dit à faible nombre de Mach. Dans ce travail, la loi d'état de la phase considérée contient toujours une information mesurant sa plus ou moins grande compressibilité. Ainsi, la faible compressibilité de l'eau peut produire un régime d'écoulement où des sauts de pression importants apparaissent même si le nombre de Mach est très faible. La première partie de la thèse s'est focalisée sur un modèle diphasique dit homogène-équilibré. Les deux phases de l'écoulement ont alors la même vitesse, pression, température et même potentiel chimique. Un premier travail a été la construction de solveurs de Riemann approchés dits tout-nombre-de-Mach. En l'absence de transitoire rapide, ces solveurs basent leur contrainte de pas de temps sur la vitesse des ondes matérielles lentes et sont donc précis pour suivre ces dernières. En revanche, lorsqu'une onde de choc rapide traverse l'écoulement, ces solveurs s'adaptent automatiquement afin de la capturer. La seconde partie de la thèse s'est focalisée sur la prise en compte du couplage convection-source dans le cadre des modèles en approche bi-fluides avec effets de relaxation pression-vitesse. Dans ces modèles, les deux phases de l'écoulement possèdent leur propre jeu de variables. Dans ce travail, un schéma implicite à mailles décalées, basé sur l'influence des termes sources dans des problèmes de Riemann linéaires, a été proposé / The present work focuses on numerical methods for low-material velocity compressible two-phase flows with high pressure jumps. In this context, the material velocity of both phases is small compared with the celerity of the acoustic waves. The flow is said to be a low-Mach number flow. In this work, the equation of state of the considered phase always contains information relative to its compressibility. For example, the low-compressibility of liquid water may lead to fast transients in which high pressure jumps are produced even if the flow Mach number is low. The first part of this work has leaned on two-phase homogeneous-equilibrium models. Thus, both phases have the same velocity, pressure, temperature and the same chemical potential. The construction of what is called an all-Mach-number approximate Riemann solver has been conducted. When no fast transients come through the flow, the above solvers enable computations with CFL conditions based on low-material velocities. As a result, they remain accurate to follow slow material interfaces, or subsonic contact discontinuities. However, when fast shock waves propagate, these solvers automatically adapt in order to capture them. The second part of the thesis has been dedicated to the design of numerical methods enhancing the coupling between convection and relaxation for two-fluid models containing pressure-velocity relaxation effects. In such models, both phases have their own set of variables. A time-implicit staggered scheme, based on the influence of relaxation source terms on linear Riemann problems has been proposed.