The modeling of nuclear reactors involves the solution of a multi-physics problem with widely varying time and length scales. This translates mathematically to solving a system of coupled, non-linear, and stiff partial differential equations (PDEs). Multi-physics applications possess the added complexity that most of the solution fields participate in various physics components, potentially yielding spatial and/or temporal coupling errors. This dissertation deals with the verification aspects associated with such a multi-physics code, i.e., the substantiation that the mathematical description of the multi-physics equations are solved correctly (both in time and space). Conventional paradigms used in reactor analysis problems employed to couple various physics components are often non-iterative and can be inconsistent in their treatment of the non-linear terms. This leads to the usage of smaller time steps to maintain stability and accuracy requirements, thereby increasing the overall computational time for simulation. The inconsistencies of these weakly coupled solution methods can be overcome using tighter coupling strategies and yield a better approximation to the coupled non-linear operator, by resolving the dominant spatial and temporal scales involved in the multi-physics simulation. A multi-physics framework, KARMA (K(c)ode for Analysis of Reactor and other Multi-physics Applications), is presented. KARMA uses tight coupling strategies for various physical models based on a Matrix-free Nonlinear-Krylov (MFNK) framework in order to attain high-order spatio-temporal accuracy for all solution fields in amenable wall clock times, for various test problems. The framework also utilizes traditional loosely coupled methods as lower-order solvers, which serve as efficient preconditioners for the tightly coupled solution. Since the software platform employs both lower and higher-order coupling strategies, it can easily be used to test and evaluate different coupling strategies and numerical methods and to compare their efficiency for problems of interest. Multi-physics code verification efforts pertaining to reactor applications are described and associated numerical results obtained using the developed multi-physics framework are provided. The versatility of numerical methods used here for coupled problems and feasibility of general non-linear solvers with appropriate physics-based preconditioners in the KARMA framework offer significantly efficient techniques to solve multi-physics problems in reactor analysis.

Three compositional reservoir simulators have been developed in the Department of Petroleum and Geosystems Engineering at The University of Texas at Austin (UT-Austin): UTCOMP (miscible gas flooding simulator), UTCHEM (chemical flooding simulator), and GPAS (General Purpose Adaptive Simulator). UTCOMP and UTCHEM simulators have been used by various oil companies for solving a variety of field problems. The efficiency and accuracy of each simulator becomes critically important when they are used to solve field problems. In this study, two well-developed solver packages, SAMG and HYPRE, along with existing solvers were compared. Our numerical results showed that SAMG can be an excellent solver for the usage in the three simulators for solving problems with a high accuracy requirement and long simulation times, and BoomerAMG in HYPRE package can also be a good solver for application in the UTCHEM simulator. In order to investigate the flexibility and the efficiency of a partitioned coupling method, the second part of this thesis presents a new implementation using a partition method for a thermal module in an equation-of-state (EOS) compositional simulator, the General Purpose Adaptive Simulator (GPAS) developed at The University of Texas at Austin. The finite difference method (FDM) was used for the solution of governing partial differential equations. Specifically, the new coupled implementation was based on the Schur complement method. For the partition method, two suitable acceleration techniques were constructed. One technique was the optimized choice of preconditioner for the Schur complement; the other was the optimized selection of tolerances for the two solution steps. To validate the implementation, we present simulation examples of hot water injection in an oil reservoir. The numerical comparison between the new implementation and the traditional, fully implicit method showed that the partition method is not only more flexible, but also faster than the classical, fully implicit method for the same test problems without sacrificing accuracy. In conclusion, the new implementation of the partition method is a more flexible and more efficient method for coupling a new module into an existing simulator than the classical, fully implicit method.The third part of this thesis presents another type of coupling method, iterative coupling methods, which has been implemented into GPAS with thermal module, FICM (Fully, Iterative Coupling Method) and GICM (General, Iterative Coupling Method), LICM (Loose, Iterative Coupling Method). The results show that LICM is divergent, and GICM and FICM can work normally. GICM is the fastest among the compared methods, and FICM has a similar efficiency as CFIM (Classic Fully Implicit Method). Although GICM is the fastest method, GICM is less accurate than FICM for in the test cases carried out in this study. / text

The simple exclusion process is formally defined as follows : each particle performs a simple random walk on a set of sites and interacts with other particles by never moving on occupied sites. Despite its simplicity, this process has properties that are found in many more complex statistical mechanics models. It is the combination of the simplicity of the process and the importance of the observed phenomena that make it one of the reference models in out of equilibrium statistical mechanics. In this thesis, I'm interested in the case of the totally asymmetric exclusion process (particles jump only to the right) on N to study its behavior according to the mechanism of particle creation : particles are created at site 0 with arate depending on the current configuration. Once this mechanism is no longer a Poisson process, the associated exclusion process does not admit a product invariant measure. As a consequence, classical computation methods with theinfinitesimal generator are rarely successful. So I used mainly the methods of coupling and second class particles.In the first part of the thesis, I'm interested in the model introduced by Grosskinsky for which I get the following result : if the maximum rate of creation and the initial density of particles are smaller than 12 and if the creation mechanism is of integrable range, there is no phase transition which means that there is only one invariant measure. In the second part of the thesis, my goal was to construct a process with finite and non-integrable range that has a phase transition. For this, I was inspired by methods developed for the process of specification of Bramson and Kalikow.

We introduce the notion of coupling distances on the space of Lévy measures in order to quantify rates of convergence towards a limiting Lévy jump diffusion in terms of its characteristic triplet, in particular in terms of the tail of the Lévy measure. The main result yields an estimate of the Wasserstein-Kantorovich-Rubinstein distance on path space between two Lévy diffusions in terms of the couping distances. We want to apply this to obtain precise rates of convergence for Markov chain approximations and a statistical goodness-of-fit test for low-dimensional conceptual climate models with paleoclimatic data.

Le processus d’exclusion simple est défini formellement de la façon suivante : chaque particule effectue une marche aléatoire sur un ensemble de sites et interagit avec les autres particules en ne se déplaçant jamais sur un site occupé.Malgré sa simplicité, ce processus présente des propriétés que l’on retrouve dans beaucoup de modèles de mécanique statistique plus complexes. C’est la conjonction de la simplicité du processus et de l’intérêt des phénomènes observés quien fait l’un des modèles de référence en mécanique statistique hors équilibre. Dans cette thèse, je me suis intéressé au cas du processus d’exclusion totalement asymétrique (les particules sautent uniquement vers la droite) sur Nafin d’étudier son comportement en fonction du mécanisme de création de particules: on crée des particules au site 0 avec un taux dépendant de la configurationactuelle. Dès que ce mécanisme n’est plus un processus de Poisson, le processusd’exclusion associé n’admet plus de mesure invariante sous forme de produitce qui fait que les méthodes classiques de calcul sur le générateur infinitésimaln’aboutissent que rarement. Je me suis donc appuyé principalement sur les méthodesde couplage et de particules de deuxième classe.Dans la première partie de la thèse, je me suis intéressé au modèle introduitpar Grosskinsky pour lequel j’ai obtenu les résultats suivants : si le taux maximumde création et la densité initiale de particules sont plus petits que 12 et sile mécanisme de création est à portée intégrable, il n’y a pas de transition dephase c’est-à-dire qu’il n’y a qu’une seule mesure invariante.Dans la deuxième partie de la thèse, je me suis intéressé au problème inversedont le but est de construire un processus à portée finie mais non-intégrableayant une transition de phase. Pour cela, je me suis inspiré des méthodes développéespour le processus des spécifications de Bramson et Kalikow. / The simple exclusion process is formally defined as follows : each particle performs a simple random walk on a set of sites and interacts with other particles by never moving on occupied sites. Despite its simplicity, this process has properties that are found in many more complex statistical mechanics models. It is the combination of the simplicity of the process and the importance of the observed phenomena that make it one of the reference models in out of equilibrium statistical mechanics. In this thesis, I’m interested in the case of the totally asymmetric exclusion process (particles jump only to the right) on N to study its behavior according to the mechanism of particle creation : particles are created at site 0 with arate depending on the current configuration. Once this mechanism is no longer a Poisson process, the associated exclusion process does not admit a product invariant measure. As a consequence, classical computation methods with theinfinitesimal generator are rarely successful. So I used mainly the methods of coupling and second class particles.In the first part of the thesis, I’m interested in the model introduced by Grosskinsky for which I get the following result : if the maximum rate of creation and the initial density of particles are smaller than 12 and if the creation mechanism is of integrable range, there is no phase transition which means that there is only one invariant measure. In the second part of the thesis, my goal was to construct a process with finite and non-integrable range that has a phase transition. For this, I was inspired by methods developed for the process of specification of Bramson and Kalikow.

Recent advances in non-local continuum models, notably peridynamics, have spurred a paradigm shift in solid mechanics simulation by allowing accurate mathematical representation of singularities and discontinuities. This doctoral work attempts to extend the use of this theory to a community more familiar with local continuum models. In this communication, a coupling strategy - the morphing method -, which bridges local and non-local models, is presented. This thesis employs the morphing method to ease use of the non-local model to represent problems with failure-induced discontinuities. First, we give a quick review of strategies for the simulation of discrete degradation, and suggest a hybrid local/non-local alternative. Second, we present the technical concepts involved in the morphing method and evaluate the quality of the coupling. Third, we develop a numerical tool for the simulation of the hybrid model for fracture and damage and demonstrate its capabilities on numerical model examples

Cette thèse porte sur la mise en œuvre de procédés innovants d'extraction de composés naturels bioactifs de plantes. Nous avons considéré ici l'extraction d'un composé anti-oxydant, l'acide rosmarinique, à partir du végétal mélisse (Melissa officinalis L.), qui contient également d'autres composés d'intérêt (citral et caryophyllène). Différentes techniques d'extraction-purification, soit à haute pression (extraction au CO2 supercritique) ou à pression atmosphérique (extraction Soxhlet, extraction batch, nanofiltration etc.) ont été envisagées. L'objectif de ce travail a été d'étudier expérimentalement l'influence des paramètres opératoires (débit, composition et concentration du solvant, taille des particules, pression et température) sur la cinétique des processus afin de sélectionner les meilleures conditions pour chaque opération. Les résultats expérimentaux ont été ensuite comparés avec plusieurs modèles mathématiques décrivant les phénomènes de transfert de masse et l'écoulement au travers du milieu poreux constitué par la matière végétale broyée. Cette démarche, à partir de la détermination des paramètres physiques du modèle, a fourni les éléments pour une extrapolation potentielle à l'échelle industrielle. Du point de vue du procédé complet d'extraction-purification, l'originalité du travail a été de proposer plusieurs scénarii d'enchainement d'opérations, couplant en synergie des opérations conventionnelles à pression atmosphériques (macération, nano-filtration etc.) et des opérations de traitement au CO2 supercritique avec co-solvant. / This thesis deals with the extraction of natural bioactive compounds from plants (case study with Lemon Balm (Melissa officinalis L.)) by using different separation techniques at high (supercritical extraction) and atmospheric pressure (Soxhlet extraction, batch extraction, nanofiltration etc.). The influence of main operational parameters (solvent composition and flow rate, particle size of the raw material, pressure, temperature) on the process kinetics is studied experimentally with the aim to determine appropriate operational conditions for better extraction. The experimental results are confronted to a number of mathematical models in order to estimate the applicability of different theoretical concepts to the particular process and to select and apply appropriate models for determination of important parameters, characterizing the mass transfer process and necessary for scale-up and design purposes. Coupling between different separation methods is also considered and a number of integrated process schemes are proposed resulting in better yield of the targeted compounds.