We study a nonlinear kinetic model describing the interactions of particles in a granular medium, i.e. inelastic systems where kinetic energy is not conserved due to internal friction. Examples of particles that fall into this category are sand, ground coffee and many others. Originally studied by Benedetto, Caglioti and Pulvirenti in the one-dimensional setting (RAIRO Model. Math. Anal. Numer, 31(5): 615-641, (1997)) the original model contained inconsistencies later accounted for and corrected by invoking a mollifier (Modelisation Mathematique et Analyse Numerique, M2AN, Vol. 33, No 2, pp. 439–441 (1999)). This thesis approximates the generalized model presented by Agueh (Arch. Rational Mech., Anal. 221, pp. 917-959 (2016)) with the added assumption of a spatial mollifier present in the kinetic equation. In dimension d ≥ 1 this model reads as ∂tf + v · ∇xf = divv(f([ηα∇W] ∗(x,v) f)) where f is a non-negative particle density function, W is a radially symmetric class C2 velocity interaction potential, and and ηα is a mollifier. A physical interpretation of this approximation is that the particles are spheres of radius α > 0 as opposed to the original assumption of being point-masses. Properties lost by this approximation and macroscopic quantities that remain conserved are discussed in greater detail and contrasted. The main result of this thesis is a proof of the weak global existence and uniqueness. An argument utilizing the tools of Optimal Transport allows simple construction of a weak solution to the kinetic model by transporting an initial measure under the characteristic flow curves. Concluding regularity arguments and restrictions on the velocity interaction potential ascertain that global classical solutions are obtained. / Graduate

Esta tesis está dedicada al desarrollo, aplicación y test de métodos para la simulación numérica de problemas procedentes de la física y de la ingeniería electrónica. La principal herramienta aplicada a lo largo de todo el trabajo es la ecuación de Vlasov (transporte) en la forma de la Boltzmann Transport Equation (BTE) para la descripción del transporte de partículas cargadas en plasmas y dispositivos electrónicos: las cargas se mueven bajo el efecto de un campo de fuerza y sufren scattering debido a otras cargas o fonones (pseudo-partículas que describen de manera efectiva las vibraciones de los iones del retículo cristalino).La BTE ha de ser acoplada con una ecuación o sistema de ecuaciones para calcular el campo de fuerza: para estructuras simples se usa la ecuación de Poisson; para plasmas, donde los efectos magnéticos no se pueden despreciar debido a las altas velocidades de las partículas, se usa la fuerza de Lorentz, por lo cual se han de resolver las ecuaciones de Maxwell; en nanoestructuras, por ejemplo transistores con dimensiones confinadas, la ecuación de Poisson necesita ser acoplada con la ecuación de Schrödinger para la descripción de las dimensiones cuánticas y para la descomposición en sub-bandas, o niveles de energía.Las colisiones son el scattering que las cargas padecen debido a las interacciones con otras cargas o con el retículo cristalino fijo, representado en forma de fonones. En la tesis se emplean diversos operadores de scattering: los más simples son operadores lineales de relajación; se estudia un modelo para la simulación de semiconductores donde se tienen en cuenta colisiones con fonones acústicos, en aproximación elástica, y fonones ópticos.Tras la introducción, en el primer capítulo se desarrollan los métodos numéricos más importantes: primero un método de interpolación no oscilante (PWENO), necesario para evitar las oscilaciones producidas por la reconstrucción por polinomios de Lagrange, que incrementa la variación total cuando aparecen choques: las oscilaciones en el espacio de fases son características del problema, pero si el método añade oscilaciones espúreas (es decir, debidas al método en sí), entonces el resultado numérico no tiene sentido, o simplemente explota. El segundo método numérico fundamental es la técnica de splitting: cuando se resuelve un problema complicado, si se puede dividir en sub-problemas y resolverlos por separado, entonces se puede reconstruir una aproximación para el problema completo; esta técnica se usa para el time splitting (separación de la parte de transporte y de colisión) y el splitting dimensional (dividir el espacio de fases en posición y velocidad). La tercera herramienta fundamental es un sólver para advección lineal: se usan dos métodos, uno basado en trazar hacia atrás las características a nivel puntual y otro basado en reconstruir valores integrales en segmentos en lugar de puntos; el primero controla mejor las oscilaciones, el segundo fuerza la conservación de masa.En el capítulo 2 estos métodos se aplican a algunos tests conocidos para averiguar su solidez.En el capítulo 3 estos métodos se aplican a la simulación de un diodo, y los resultados se comparan con resultados anteriores obtenidos por esquemas Runge-Kutta basados en diferencias finitas para aproximar las derivadas parciales.El capítulo 4 está dedicado a la construcción y simulación de modelos intermedios entre una ecuación cinética, con operador de colisión de tipo relajación, y su aproximación más grosera, ésta última siendo la ecuación del calor. Para obtener modelos intermedios, se busca un cierre de las ecuaciones de los momentos de orden cero y uno. Se proponen esquemas "asymptotic-preserving" para la ecuación cinética, que evitan la stiffness de la parte de advección a través de una descomposición de la función de distribución en su media más fluctuaciones. En cuanto a las clausuras de las ecuaciones de los momentos, se proponen esquemas de relajación para aislar las no-linealidades. Estos métodos son aplicados a un test conocido, el Su-Olson test.El último capítulo está dedicado a la simulación de un MOSFET (Metal Oxide Semiconductor Field Effect Transistor) 2D de dimensión nanométrica en el que los electrones se comportan como partículas en una dimensión y como ondas en las dimensiones confinadas. La descomposición en sub-bandas se realiza a través de una ecuación de Schrödinger 1D en estado estacionario. Las dimensiones, así como las sub-bandas, están acopladas por la ecuación de Poisson en la expresión de la densidad, y por el operador de colisión. Se propone un sólver microscópico para estados transitorios, basado en técnicas de splitting para las BTEs (una para cada nivel de energía), métodos de características para el transporte y una iteración de tipo Newton para resolver el problema acoplado Schrödinger-Poisson para el cálculo del campo de fuerza. / This thesis is dedicated to the development, application and test of numerical methods for the numerical simulation of problems arising from physics and electronic engineering. The main tool which is used all along the work is the Vlasov (transport) equation in the form of the Boltzmann Transport Equation (BTE) for the description of the transport and collisions of charged particles in plasmas and electronic devices: charge carriers are driven by a force field and scattered by other carriers or phonons (pseudo-particles giving an effective representation of the oscillating field produced by the vibrating ions).The BTE must be coupled to an equation or a system of equations for the computation of the force field: for simple structures the Poisson equation is used; for plasmas, where the magnetic phenomena cannot be neglected due to the high velocities of the particles, the Lorentz force is used, so the Maxwell equations have to be solved; for nanostructures, e.g. transistors with confined dimensions, the Poisson equation needs coupling with Schrödinger equation for the description of the quantum dimensions and the decomposition into subbands, or energy levels.Collisions mean the scattering the carriers suffer due to the interactions with other carriers or the fixed lattice, in form of phonons. All along the thesis several scattering operator are used: the simplest ones are linear relaxation-time operators; a model for the simulation of a semiconductor is studied in which collisions are taken into account with acoustic phonons, in the elastic approximation, and optical phonons.After the introduction, in the first chapter the most important numerical methods are developed: first of all a pointwise non-oscillatory interpolation method (PWENO) needed to avoid the simple Lagrange polynomial reconstruction, which increases the total variation when shocks appear: oscillations are part of the physics of the problem, but if the method adds spurious, non-physical oscillations, then the numerical result is meaningless, or it simply blows up. The second fundamental numerical method is the splitting technique: when solving a complicated problem, if we are able to subdivide it into sub-problem and solve them for separate, then we can reconstruct an approximation for the complete problem; this technique is used for both time splitting (separate transport from collisions) and dimensional splitting (split the phase space into either dimensions). The third fundamental instrument is the solver for linear advections: two methods are used, one based on pointwise following backwards the characteristics and another one based on reconstructing integral values along segments instead of point values; the first one controls better oscillations, the second one forces mass conservation.These methods are applied in chapter 2 to some well-known benchmark tests to control their robustness.In chapter 3 these methods are applied to the simulation of a diode, and the results compared to previous results obtained by Runge-Kutta schemes based on finite differences schemes for the approximation of the partial derivatives.Chapter 4 is dedicated to the construction and simulation of intermediate models between a kinetic equation, with relaxation-time collision operator, and its coarsest approximation, this one being the heat equations. In order to obtain intermediate models, the moment equations are closed at zeroth and first order. Asymptotic-preserving schemes are proposed for the kinetic equation, which avoid the stiffness of the advection part by decomposing the distribution function into its average plus fluctuations. As for the moment closures, relaxation schemes are proposed in order to confine the non-linearities in the right hand side. These methods are then applied to a known benchmark, the Su-Olson test.The last chapter is dedicated to the simulation of a nanoscaled 2D MOSFET (Metal Oxide Field Effect Transistor) in which electrons behave as particles in one dimension and as waves in the confined dimensions. The subband decomposition is realized through a stationary-state 1D Schrödinger equation. The dimensions as well as the subbands are coupled by the Poisson equation in the expression of the density and by the collision operator. A transient-state microscopic solver is proposed, based on splitting techniques for the BTE's (one for each energy level), characteristics methods for the transport and a Newton iteration for the solution of the coupled Schrödinger-Poisson system for computing the force field.

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Made available in DSpace on 2014-10-09T12:34:20Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:10:33Z (GMT). No. of bitstreams: 0 / Em testes físicos de reatores que são realizados na criticalização, após uma troca de combustível, por exemplo, como ocorre nos reatores PWR, é muito importante monitorar continuamente a reatividade durante o processo de criticalização. Medir a reatividade utilizando o método da Cinética Inversa é um processo bastante utilizado durante a operação de um reator nuclear, e é possível determinar a reatividade em tempo real baseado nas equações de cinética pontual. Essa técnica é aplicada com sucesso a altas potências, ou em núcleos que trabalham sem a existência de uma fonte externa, já que nesse caso o Termo Fonte na equação de cinética pontual pode ser desprezado. Para operações a baixas potências, a contribuição da fonte de nêutrons precisa ser levada em consideração, e isso implica em conhecer um valor proporcional à intensidade da fonte e, portanto esse valor precisa ser determinado. O Método dos Mínimos Quadrados em Cinética Inversa (Least Square Inverse Kinetics Method LSIKM) foi testado com modelo teórico e aplicado em experimentos no Reator Nuclear IPEN/MB-01 para a determinação do Termo Fonte com a utilização de um reatímetro que ignora o Termo Fonte em primeira instância. Após a determinação da Fonte S de forma consistente, seu valor foi inserido ao algoritmo de Cinética Inversa, e utilizando dados de detectores durante a criticalização, o reatímetro com Termo Fonte foi usado para medir a reatividade no domínio subcrítico, nos passos de criticalização, após experimento. / Dissertação (Mestrado) / IPEN/D / Instituto de Pesquisas Energeticas e Nucleares - IPEN-CNEN/SP

L'objet de cette thèse est la construction de schémas numériques pour les équations cinétiques dans différents régimes de diffusion anormale. Comme le modèle devient raide en s'approchant du modèle asymptotique, les méthodes numériques standard deviennent coûteuses dans ce régime. Les schémas Asymptotic Preserving ont été introduits pour pallier à cette difficulté. Ils sont en effet stables le long de la transition du régime mésoscopique au régime microscopique. Dans le premier chapitre, nous considérons le cas d'une distribution d'équilibre qui est une fonction à queue lourde et dont le moment d'ordre 2 est infini. Le poids important des grandes vitesses de l'équilibre fait tomber la limite de diffusion usuelle en défaut, et on montre que le modèle asymptotique est une équation de diffusion fractionnaire. En nous basant sur une analyse asymptotique formelle de la convergence vers le modèle limite, nous construisons trois schémas AP pour le problème. La discrétisation en vitesse est discutée afin de prendre en compte correctement les grandes vitesses, et nous montrons que le troisième schéma est en outre uniformément précis au cours de la transition vers le régime microscopique. Dans le chapitre 2, nous étendons ces résultats au cas d'une fréquence de collision dégénérée en 0 qui mène aussi à une équation de diffusion fractionnaire. Nous adaptons ensuite ces méthodes numériques au cas d'une limite de diffusion normale avec scaling en temps anormal dans l'équation cinétique dans le chapitre 3. Dans ce cadre, la lenteur de la convergence vers le modèle asymptotique rend nécessaire une adaptation de l'approche AP des chapitres précédents. Enfin, le chapitre 4 présente un schéma AP pour l'équation cinétique dans le cas heavy-tail du chapitre 1 lorsque l'opérateur de collision est non-local. / In this thesis, we construct numerical schemes for kinetic equations in some anomalous diffusion regimes. As the model becomes stiff when reaching the asymptotic model, the standard numerical methods become costly in this regime. Asymptotic Preserving (AP) schemes have been designed to overcome this difficulty. Indeed, they are uniformly stable along the transition from the mesoscopic regime to the microscopic one. In the first chapter, we study the case of a heavy-tailed equilibrium distribution, with infinite second order moment. The importance of the high velocities in the equilibrium makes the classical diffusion limit fail, and one can prove that the asymptotic model is a fractional diffusion equation. We construct three AP schemes for this problem, based on a formal asymptotic analysis of the convergence towards the limit model. The discretization of the velocities is then discussed to take into account the high velocities. Moreover, we prove that the third scheme enjoys the stronger property of being uniformly accurate along the convergence towards the microscopic regime. In chapter 2, we extend these results to the case of a degenerated collision frequency, also leading to a fractional diffusion limit. In chapter 3, these methods are then adapted to the case of a classical diffusion limit with anomalous time scale in the kinetic equation. In this case, an adaptation of the AP approach of the previous chapter is needed, because of the slow convergence rate of the kinetic equation towards the limit model. Eventually, a AP scheme for kinetic equation with heavy-tailed equilibria and non local collision operator is presented in chapter 4.

Made available in DSpace on 2014-10-09T12:41:26Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:03:23Z (GMT). No. of bitstreams: 0 / Dissertação (Mestrado) / IPEN/D / Instituto de Pesquisas Energeticas e Nucleares - IPEN-CNEN/SP

Made available in DSpace on 2014-10-09T12:24:18Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:04:29Z (GMT). No. of bitstreams: 1 00613.pdf: 1081974 bytes, checksum: e5db2bcc8582f1c49a35565db5d7f94f (MD5) / Dissertacao (Mestrado) / IEA/D / Escola Politecnica, Universidade de Sao Paulo - POLI/USP

Made available in DSpace on 2014-10-09T12:41:26Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:03:23Z (GMT). No. of bitstreams: 0 / Este trabalho apresenta o estudo do efeito do refletor sobre o tempo de vida neutrônico do Reator IPEN/MB-01. O método empregado requer uma abordagem que leve em conta tanto o núcleo quanto o refletor, de modo que as equações de cinética pontual, as quais constituem a base teórica de todo desenvolvimento matemático, contemplem ambas as regiões do reator. A partir dessas equações, conhecidas como equações de cinética pontual do modelo duas regiões, são obtidas as expressões teóricas para as APSDs (Auto Power Spectral Densities), as quais são utilizadas para o ajuste por mínimos quadrados aos dados das APSDs experimentais obtidas em vários estados subcríticos. O tempo de geração de nêutrons prontos, o tempo de vida dos nêutrons no refletor e a fração desses nêutrons que retornam ao núcleo, são obtidos como parâmetros do ajuste. / Dissertação (Mestrado) / IPEN/D / Instituto de Pesquisas Energeticas e Nucleares - IPEN-CNEN/SP

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<div>The Boltzmann equation, an integro-differential equation for the molecular distribution function in the physical and velocity phase space, governs the fluid flow behavior at a wide range of physical conditions, including compressible, turbulent, as well as flows involving further physics such as non-equilibrium internal energy exchange and chemical reactions. Despite its wide applicability, deterministic solutions of the Boltzmann equation present a huge computational challenge, and often the collision operator is simplified for practical reasons, hereby, referred to as linear kinetic models. These models utilize the moment of the underlying probability distribution to mimic some properties of the original collision operator. But, just because we know the moments of a distribution, doesn't mean we know the actual distribution. The approximation of reality can never supersede the reality itself. Because, all the facts (moments) about the world (distribution) cannot explain the world. The premise lies not in the fact that a certain flow behavior can be correctly predicted; but rather that the Boltzmann equation can reveal and explain previously unsuspected aspects of reality.</div><div><br></div><div>Therefore, in this work, we introduce accurate, efficient, and robust numerical schemes for solving the multi-species Boltzmann equation which can model general repulsive interactions. These schemes are high order spatially and temporally accurate, spectrally accurate in molecular velocity space, exhibit nearly linear parallel efficiency on the current generation of processors; and can model a wide-range of rarefied flows including flows involving momentum, heat, and diffusive transport. The single-species variant formed the basis of author's Masters' thesis.</div><div><br></div><div>While the first part of the dissertation is targeted towards multi-species flows that exhibit rich non-equilibrium phenomenon; the second part focuses on single-species flows that do not depart significantly from equilibrium. This is the case, for example, in micro-nozzles, where a portion of flow can be highly rarefied, whereas others can be in near-continuum regime. However, when the flow is in near-continuum regime, the traditional deterministic numerical schemes for kinetic equations encounter two difficulties: a) since the near-continuum is essentially an effect of large number of particles in an infinitesimal volume, the average time between successive collisions decrease, and therefore the discrete simulation timestep has to be made smaller; b) since the number of molecular collisions increase, the flow acquires steady state slowly, and therefore one needs to carry out time integration for large number of time steps. Numerically, the underlying issue stems from stiffness of the discretized ordinary differential equation system. This situation is analogous to low Reynolds number scenario in traditional compressible Navier-Stokes simulations. To circumvent these issues, we introduce a class of high order spatially and temporally accurate implicit-explicit schemes for single-species Boltzmann equation and related kinetic models with the following properties: a) since the Navier-Stokes can be derived from the asymptotics of the Boltzmann equation (using Chapman-Enskog expansion~\cite{cercignani2000rarefied}) in the limit of vanishing rarefaction, these schemes preserve the transition from Boltzmann to Navier-Stokes; b) the timestep is independent of the rarefaction and therefore the scheme can handle both rarefied and near-continuum flows or combinations thereof; c) these schemes do not require iterations and therefore are easy to scale to large problem sizes beyond thousands of processors (because parallel efficiency of Krylov space iterative solvers deteriorate rapidly with increase in processor count); d) with use of high order multi-stage time-splitting, the time integration over sufficiently long number of timesteps can be carried out more accurately. The extension of the present methodology to the multi-species case can be considered in the future. </div><div><br></div><div>A series of numerical tests are performed to illustrate the efficiency and accuracy of the proposed methods. Various benchmarks highlighting different scattering models, different mass ratios, momentum transport, heat transfer, and diffusive transport have been studied. The results are directly compared with the direct simulation Monte Carlo (DSMC) method. As an engineering use-case, the developed methodology is applied for the study of thermal processes in micro-systems, such as heat transfer in electronic-chips; and primarily, the ingeniously Purdue-developed, Microscale In-Plane Knudsen Radiometric Actuator (MIKRA) sensor, which can be used for flow actuation and measurement.</div>