Made available in DSpace on 2014-10-09T12:31:02Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T13:56:48Z (GMT). No. of bitstreams: 1 01368.pdf: 3640499 bytes, checksum: ccba7944ce0eb5025a31bde960ef457a (MD5) / Dissertacao (Mestrado) / IPEN/D / Instituto de Pesquisas Energeticas e Nucleares - IPEN/CNEN-SP

Made available in DSpace on 2014-10-09T12:25:30Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:02:44Z (GMT). No. of bitstreams: 1 06213.pdf: 4263088 bytes, checksum: 543c6cb711764dac098c3b7d24f8c9cc (MD5) / Desenvolve-se aqui um novo método para calcular coeficientes de sensibilidade. Este novo método é uma combinação entre as duas metodologias usadas para calcular estes coeficientes, que são o método diferencial e o método da teoria da perturbação generalizada. O método consiste em fazer como parâmetro integral o fluxo médio em uma região arbitrária do sistema. Dessa forma, o coeficiente de sensibilidade passa a conter somente o termo correspondente ao fluxo de nêutrons. Para obtenção do novo coeficiente de sensibilidade é feito o cálculo do coeficiente de sensibilidade desse parâmetro integral com relação a σ através do método de perturbação e são obtidas as derivadas funcionais do parâmetro integral genérico com relação a σ e Φ utilizando o método diferencial. / Dissertação (Mestrado em Tecnologia Nuclear) / IPEN/D / Instituto de Pesquisas Energeticas e Nucleares - IPEN-CNEN/SP

Neste trabalho é desenvolvido um método, que utiliza funções de Green, para a solução analítica da equação de difusão de nêutrons em multigrupo, para um reator refletido, cujo fluxo tem dependência apenas radial e com distribuição de combustível não uniforme no cerne. As propriedades de moderação, difusão e absorção são consideradas diferentes no cerne e refletor. Uma distribuição de densidade de potência, que estabelece a condição de criticalidade do reator, é assumida a priori e determina a distribuição de combustível no cerne. Com auxílio das funções de Green e das condições de continuidade do fluxo e da densidade de corrente de nêutrons na interface cerne-refletor, a equação de difusão em multigrupo é transformada em um sistema de equações lineares, contendo como incógnitas os valores dos fluxos na interface entre as regiões. Resolvido esse sistema, obtém-se os valores dos fluxos na interface e, com eles, a distribuição de fluxo em cada região e para cada grupo. Como verificação do método proposto, é feita uma aplicação numérica, utilizando dois grupos de energia, para um reator TRIGA de 1MW. Nessa aplicação são calculadas, além das distribuições de fluxos para os dois grupos de energia, a distribuição de combustível no cerne, a massa crítica e a potência específica linear, para diferentes distribuições de densidade de potência. / In the present work a method is developed for applying Green\'s functions to obtain an analytical solution o£ the neutron diffusion equation to the case o£ a reflected reactor. The problem of a non-uniform fuel distribution in the core is treated. Multigroup theory is used and the neutron flux is assumed to have only radial dependence. Different values are employed to characterize the moderation, diffusion and absorption properties o£ the core and the reflector. A power density distribution which establishes the reactor critica1 condition \"a priori\" is assumed and is then used to calculate the fuel distribution. By using the Green\'s functions and the continuity relations (for neutron fluxes and neutron current densities) at the core-reflector interface, the multigroup diffusion equation is transformed into a system of linear equations. In this system o£ equations the unknowns are the neutron fluxes at the core- reflector interface. Once this system is solved and the interface fluxes are determined, it follows immediately that the neutron flux distribution in the core and in the reflector is determined. The method employed and proposed in the present study has been applied to the problem of calculating the neutron distribution in a 1MW TRIGA reactor, using two energy group. This numerical application, in addition to calculating the two-group flux distribution, the fuel distribution in the core, the critical mass and the linear specific power for different assumed power density distribution have been evaluated.

Dans le contexte du stockage des déchets radioactifs en milieu poreux, nous considérons l’équation de convection-diffusion instationnaire et sa discrétisation par des méthodes numériques. La discontinuité des paramètres physiques et la variabilité des échelles d’espace et de temps conduisent à utiliser des discrétisations différentes en temps et en espace dans différentes régions du domaine. Nous choisissons dans cette thèse le schéma volumes finis en dualité discrète (DDFV) et le schéma de Galerkin Discontinu en temps couplés à une méthode de décomposition de domaine de Schwarz de type relaxation d’ondes optimisées (OSWR), ce qui permet de traiter des maillages espace-temps non conformes. La principale difficulté réside dans l’obtention d’une discrétisation amont du flux convectif qui reste locale à un sous-domaine et telle que le schéma monodomaine soit équivalent au schéma multidomaine. Ces difficultés sont appréhendées d’abord en une dimension d’espace où différentes discrétisations sont étudiées. Le schéma retenu introduit une inconnue hybride sur les interfaces entre cellules. L’idée du décentrage amont par rapport à cette inconnue hybride est reprise en dimension deux d’espace, et adaptée au schéma DDFV. Le caractère bien posé de ce schéma et d’un schéma multidomaine équivalent est montré. Ce dernier est résolu par un algorithme OSWR dont la convergence est prouvée. Les paramètres optimisés des conditions de Robin sont obtenus par l'étude du taux de convergence continu ou discret. Différents cas-tests, dont l’un est inspiré du stockage des déchets nucléaires, illustrent ces résultats. / In the context of nuclear waste repositories, we consider the numerical discretization of the non stationary convection diffusion equation. Discontinuous physical parameters and heterogeneous space and time scales lead us to use different space and time discretizations in different parts of the domain. In this work, we choose the discrete duality finite volume (DDFV) scheme and the discontinuous Galerkin scheme in time, coupled by an optimized Scwharz waveform relaxation (OSWR) domain decomposition method, because this allows the use of non-conforming space-time meshes. The main difficulty lies in finding an upwind discretization of the convective flux which remains local to a sub-domain and such that the multidomain scheme is equivalent to the monodomain one. These difficulties are first dealt with in the one-dimensional context, where different discretizations are studied. The chosen scheme introduces a hybrid unknown on the cell interfaces. The idea of upwinding with respect to this hybrid unknown is extended to the DDFV scheme in the two-dimensional setting. The well-posedness of the scheme and of an equivalent multidomain scheme is shown. The latter is solved by an OSWR algorithm, the convergence of which is proved. The optimized parameters in the Robin transmission conditions are obtained by studying the continuous or discrete convergence rates. Several test-cases, one of which inspired by nuclear waste repositories, illustrate these results.

The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H^1 seminorm that can be applied to anisotropic finite element meshes. A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably. Furthermore three modifications of these estimators are introduced and discussed. Numerical experiments for all estimators complement and confirm the theoretical results.

This thesis deals with problems arising in the study of nonlinear partial differential equations arising from many-body problems. It is divided into two parts: The first part concerns the derivation of a nonlinear diffusion equation from a microscopic stochastic process. We give a new method to show that in the hydrodynamic limit, the particle densities of a one-dimensional zero range process on a periodic lattice converge to the solution of a nonlinear diffusion equation. This method allows for the first time an explicit uniform-in-time bound on the rate of convergence in the hydrodynamic limit. We also discuss how to extend this method to the multi-dimensional case. Furthermore we present an argument, which seems to be new in the context of hydrodynamic limits, how to deduce the convergence of the microscopic entropy and Fisher information towards the corresponding macroscopic quantities from the validity of the hydrodynamic limit and the initial convergence of the entropy. The second part deals with problems arising in the analysis of nonlinear Schrödinger equations of Gross-Pitaevskii type. First, we consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in the literature. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case. Finally, a mathematical framework for optimal bilinear control of nonlinear Schrödinger equations arising in the description of Bose-Einstein condensates is presented. The obtained results generalize earlier efforts found in the literature in several aspects. In particular, the cost induced by the physical work load over the control process is taken into account rather then often used L^2- or H^1-norms for the cost of the control action. We prove well-posedness of the problem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Also a numerical solution method is proposed, which is based on a Newton type iteration, and used to solve several coherent quantum control problems.

Mathematical models displaying travelling waves appear in a variety of domains. These waves are often faced with different kinds of perturbations. In some cases, these perturbations result in propagation failure, also known as wave-blocking. Wave-blocking has been studied in the case of several specific models, often with the help of numerical tools. In this thesis, we will display a technique that uses symmetry and a center manifold reduction to find a criterion which defines regions in parameter space where a wave will be blocked. We focus on waves with low velocity and small symmetry-breaking perturbations, which is where the blocking initiates; the organising center. The range of the tools used makes the technique easily generalizable to higher dimensions. In order to demonstrate this technique, we apply it to the bistable equation. This allows us to do calculations explicitly. As a result, we show that wave-blocking occurs inside a wedge originating from the organising center and derive an expression for this wedge to leading order. We verify our results with some numerical simulations.

El principal objetivo de esta tesis es el desarrollo de un Método Modal para resolver dos ecuaciones: la Ecuación de la Difusión de Neutrones y la de las Ordenadas Discretas del Transporte de Neutrones. Además, este método está basado en el Método de Volúmenes Finitos para discretizar las variables espaciales. La solución de estas ecuaciones proporciona el flujo de neutrones, que está relacionado con la potencia que se produce en los reactores nucleares, por lo que es un factor fundamental para los Análisis de Seguridad Nuclear. Por una parte, la utilización del Método Modal está justificada para realizar análisis de inestabilidades en reactores. Por otra parte, el uso del Método de Volúmenes Finitos está justificado por la utilización de este método para resolver las ecuaciones termohidráulicas, que están fuertemente acopladas con la generación de energía en el combustible nuclear. En primer lugar, esta tesis incluye la definición de estas ecuaciones y los principales métodos utilizados para resolverlas. Además, se introducen los principales esquemas y características del Método de Volúmenes Finitos. También se describen los principales métodos numéricos para el Método Modal, que incluye tanto la solución de problemas de autovalores como la solución de Ecuaciones Diferenciales Ordinarias dependientes del tiempo. A continuación, se desarrollan varios algoritmos del Método de Volúmenes Finitos para el Estado Estacionario de la Ecuación de la Difusión de Neutrones. Se consigue desarrollar una formulación multigrupo, que permite resolver el problema de autovalores para cualquier número de grupos de energía, incluyendo términos de upscattering y de fisión en varios grupos de energía. Además, se desarrollan los algoritmos para realizar la computación en paralelo. La solución anterior es la condición inicial para resolver la Ecuación de Difusión de Neutrones dependiente del tiempo. En esta tesis se utiliza un Método Modal, que transforma el Sistema de Ecuaciones Diferenciales Ordinarias en uno de mucho menor tamaño, que se resuelve con el Método de la Matriz Exponencial. Además, se ha desarrollado un método rápido para estimar el flujo adjunto a partir del directo, ya que se necesita en el Método Modal. Por otra parte, se ha desarrollado un algoritmo que resuelve el problema de autovalores de la Ecuación del Transporte de Neutrones. Este algoritmo es para la formulación de Ordenadas Discretas y el Método de Volúmenes Finitos. En concreto, se han aplicado dos tipos de cuadraturas para las Ordenadas Discretas y dos esquemas de interpolación para el Método de Volúmenes Finitos. Finalmente, se han aplicado estos métodos a diferentes tipos de reactores nucleares, incluyendo reactores comerciales. Se han evaluado los valores de la constante de multiplicación y de la potencia, ya que son las variables fundamentales en los Análisis de Seguridad Nuclear. Además, se ha realizado un análisis de sensibilidad de diferentes parámetros como la malla y métodos numéricos. En conclusión, se obtienen excelentes resultados, tanto en precisión como en coste computacional. / The main objective of this thesis is the development of a Modal Method to solve two equations: the Neutron Diffusion Equation and the Discrete Ordinates Neutron Transport Equation. Moreover, this method uses the Finite Volume Method to discretize the spatial variables. The solution of these equations gives the neutron flux, which is related to the power produced in nuclear reactors; thus, the neutron flux is a paramount variable in Nuclear Safety Analyses. On the one hand, the use of Modal Methods is justified because one uses them to perform instability analyses in nuclear reactors. On the other hand, it is worth using the Finite Volume Method because one uses it to solve thermalhydraulic equations, which are strongly coupled with the energy generation in the nuclear fuel. First, this thesis defines the equations mentioned above and the main methods to solve these equations. Furthermore, the thesis describes the major schemes and features of the Finite Volume Method. In addition, the author also introduces the major methods used in the Modal Method, which include the methods used to solve the eigenvalue problem, as well as those used to solve the time dependent Ordinary Differential Equations. Next, the author develops several algorithms of the Finite Volume Method applied to the Steady State Neutron Diffusion Equation. In addition, the thesis includes an improvement of the multigroup formulation, which solves problems involving upscattering and fission terms in several energy groups. Moreover, the author optimizes the algorithms to do calculations with parallel computing. The previous solution is used as initial condition to solve the time dependent Neutron Diffusion Equation. The author uses a Modal Method to do so, which transforms the Ordinary Differential Equations System into a smaller system that is solved by using the Exponential Matrix Method. Furthermore, the author developed a computationally efficient method to estimate the adjoint flux from the forward one, because the Modal Method uses the adjoint flux. Additionally, the thesis also presents an algorithm to solve the eigenvalue problem of the Neutron Transport Equation. This algorithm uses the Discrete Ordinates formulation and the Finite Volume Method. In particular, the author uses two types of quadratures for the Discrete Ordinates and two interpolation schemes for the Finite Volume Method. Finally, the author tested the developed methods in different types of nuclear reactors, including commercial ones. The author checks the accuracy of the values of the crucial variables in Nuclear Safety Analyses, which are the multiplication factor and the power distribution. Furthermore, the thesis includes a sensitivity analysis of several parameters, such as the mesh and numerical methods. In conclusion, excellent results are reported in both accuracy and computational cost. / El principal objectiu d'esta tesi és el desenvolupament d'un Mètode Modal per a resoldre dos equacions: l'Equació de Difusió de Neutrons i la de les Ordenades Discretes del Transport de Neutrons. A més a més, este mètode està basat en el Mètode de Volums Finits per a discretitzar les variables espacials. La solució d'estes equacions proporcionen el flux de neutrons, que està relacionat amb la potència que es produïx en els reactors nuclears; per tant, el flux de neutrons és un factor fonamental en els Anàlisis de Seguretat Nuclear. Per una banda, la utilització del Mètode Modal està justificada per a realitzar anàlisis d'inestabilitats en reactors. Per altra banda, l'ús del Mètode de Volums Finits està justificat per l'ús d'este mètode per a resoldre les equacions termohidràuliques, que estan fortament acoblades amb la generació d'energia en el combustible nuclear. En primer lloc, esta tesi inclou la definició d'estes equacions i els principals mètodes utilitzats per a resoldre-les. A més d'això, s'introduïxen els principals esquemes i característiques del Mètode de Volums Finits. Endemés, es descriuen els principals mètodes numèrics per al Mètode Modal, que inclou tant la solució del problema d'autovalors com la solució d'Equacions Diferencials Ordinàries dependents del temps. A continuació, es desenvolupa diversos algoritmes del Mètode de Volums Finits per a l'Estat Estacionari de l'Equació de Difusió de Neutrons. Es conseguix desenvolupar una formulació multigrup, que permetre resoldre el problema d'autovalors per a qualsevol nombre de grups d'energia, incloent termes d' upscattering i de fissió en diversos grups d'energia. A més a més, es desenvolupen els algoritmes per a realitzar la computació en paral·lel. La solució anterior és la condició inicial per a resoldre l'Equació de Difusió de Neutrons dependent del temps. En esta tesi s'utilitza un Mètode Modal, que transforma el Sistema d'Equacions Diferencials Ordinàries en un problema de menor tamany, que es resol amb el Mètode de la Matriu Exponencial. Endemés, s'ha desenvolupat un mètode ràpid per a estimar el flux adjunt a partir del directe, perquè es necessita en el Mètode Modal. Per altra banda, s'ha desenvolupat un algoritme que resol el problema d'autovalors de l'Equació de Transport de Neutrons. Este algoritme és per a la formulació d'Ordenades Discretes i el Mètode de Volums Finits. En concret, s'han aplicat dos tipos de quadratures per a les Ordenades Discretes i dos esquemes d'interpolació per al Mètode de Volums Finits. Finalment, s'han aplicat estos mètodes a diversos tipos de reactors nuclears, incloent reactors comercials. S'han avaluat els valor de la constat de multiplicació i de la potència, perquè són variables fonamentals en els Anàlisis de Seguretat Nuclear. Endemés, s'ha realitzat un anàlisi de sensibilitat de diversos paràmetres com la malla i mètodes numèrics. En conclusió, es conseguix obtenir excel·lents resultats, tant en precisió com en cost computacional. / Bernal García, Á. (2018). Development of a 3D Modal Neutron Code with the Finite Volume Method for the Diffusion and Discrete Ordinates Transport Equations. Application to Nuclear Safety Analyses [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/112422 / TESIS

This thesis focuses on the impact of spatially heterogeneous environments on the spatio-temporal behavior of planktonic systems. Specific emphasis placed is on the influence of spatial variations in the strength of random or chaotic movements (diffusion) of the organisms. Interaction between different species is described by ordinary differential equations. In order to describe movements in space, reaction–diffusion or advection–reaction–diffusion systems are studied. Examples are given for different approaches of diffusive motion as well as for the possible effects on the localized biological system. The results are discussed based on their biological and physical meanings. In doing so, different mechanisms are shown which are able to explain events of fast plankton growth near turbulent flows. In general, it is shown that local variation in the strength of vertical mixing can have global effects on the biological system, such as changing the stability of dynamical solutions and generating new spatiotemporal behavior. The thesis consists of five chapters. Three of them have been published in international peer-reviewed scientific journals. Chapter 1. Introduction: This chapter gives a general introduction to the history of plankton modeling and introduces basic ideas and concepts which are used in the following chapters. Chapter 2. Fokker-Planck law of diffusion: The influence of spatially in- homogeneous diffusion on several common ecological problems is analyzed. Dif- fusion is modeled with Fick’s law and the Fokker–Planck law of diffusion. A discussion is given about the differences between the two formalisms and when to use the one or the other. To do this, the discussion starts with a pure diffusion equation, then it turns to a reaction–diffusion system with one logistically growing component which invades the spatial domain. This chapter also provides a look at systems of two reacting components, namely a trimolecular oscillating chemical model system and an excitable predator–prey model. Contrary to Fickian diffusion, spatial inhomogeneities promote spatial and spatiotemporal pattern formation in the case of Fokker–Planck diffusion. A slightly modified version of this chapter has been published in the Journal of Mathematical Biology (Bengfort et al., 2016). Chapter 3. Plankton blooms and patchiness: Microscopic turbulent motions of water have been shown to influence the dynamics of microscopic species. Therefore, the number, stability, and excitability of stationary states in a predator– prey model of plankton species can change when the strength of turbulent motions varies. In a spatial system these microscopic turbulent motions are naturally of different strength and form a heterogeneous physical environment. Spatially neighboring plankton communities with different physical conditions can impact each other due to diffusive coupling. It is shown that local variations in the physical conditions can influence the global system in the form of propagating pulses of high population densities. For this, three local predator–prey models with different local responses to variation in the physical environment are considered. The degree of spatial heterogeneity can, depending on the model, promote or reduce the number of propagating pulses, which can be interpreted as patchy plankton distributions and recurrent blooms. This chapter has been published in the Journal Ecological Complexity (Bengfort et al., 2014). Chapter 4. Advection–reaction–diffusion model: Here, some of the models introduced in chapter 1 and 2 are modified to perform two dimensional spatial simulations including advection, reaction and diffusion. These models include assumptions about turbulent flows introduced in chapter 1. Chapter 5. Competition: Some plankton species, such as cyanobacteria, have an advantage in competition for light compared to other species because of their buoyancy. This advantage can be diminished by vertical mixing in the surround- ing water column. A non–spatial model, based on ordinary differential equations, which accounts for this effect is introduced. The main aim is to show that vertical mixing influences the outcome of competition between different species. Hystersis is possible for a certain range of parameters. Introducing a grazing predator, the system exhibits different dynamics depending on the strength of mixing. In a diffusively coupled horizontal spatial model, local vertical mixing can also have a global effect on the biological system, for instance, destabilization of a locally stable solution, or the generation of new spatiotemporal behavior. This chapter has been published in the Journal Ecological Modelling (Bengfort and Malchow, 2016).

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