High power Gallium Nitride (GaN) based field effect transistors are used in many high power applications from RADARs to communications. These devices dissipate a large amount of power and sustain high electric fields during operation. High power dissipation occurs in the form of heat generation through Joule heating which also results in localized hot spot formation that induces thermal stresses. In addition, because GaN is strongly piezoelectric, high electric fields result in large inverse piezoelectric stresses. Combined with residual stresses due to growth conditions, these effects are believed to lead to device degradation and reliability issues. This work focuses on studying these effects in detail through modeling of Heterostructure Field Effect Transistors (HFETs) and metal oxide semiconductor hetero-structure field effect transistor (MOSHFETs) under various operational conditions. The goal is to develop a thorough understanding of device operation in order to better predict device failure and eventually aid in device design through modeling. The first portion of this work covers the development of a continuum scale model which couples temperature and thermal stress to find peak temperatures and stresses in the device. The second portion of this work focuses on development of a micro-scale model which captures phonon-interactions at the device scale and can resolve local perturbations in phonon population due to electron-phonon interactions combined with ballistic transport. This portion also includes development of phonon relaxation times for GaN. The model provides a framework to understand the ballistic diffusive phonon transport near the hotspot in GaN transistors which leads to thermally related degradation in these devices.

Linear discontinuous (LD) spatial discretization of the transport operator can generate negative angular flux solutions. In slab geometry, negativities are limited to optically thick cells. However, in multi-dimension problems, negativities can even occur in voids. Past attempts to eliminate the negativities associated with LD have focused on inherently positive solution shapes and ad-hoc fixups. We present a new, strictly non-negative finite element method that reduces to the LD method whenever the LD solution is everywhere positive. The new method assumes an angular flux distribution, e , that is a linear function in space, but with all negativities set-to- zero. Our new scheme always conserves the zeroth and linear spatial moments of the transport equation. For these reasons, we call our method the consistent set-to-zero (CSZ) scheme. CSZ can be thought of as a nonlinear modification of the LD scheme. When the LD solution is everywhere positive within a cell, psi csz = psi LD. If psi LD < 0 somewhere within a cell, psi csz is a linear function psi csz with all negativities set to zero. Applying CSZ to the transport moment equations creates a nonlinear system of equations which is solved to obtain a non-negative solution that preserves the moments of the transport equation. These properties make CSZ unique; it encompasses the desirable properties of both strictly positive nonlinear solution representations and ad-hoc fixups. Our test problems indicate that CSZ avoids the slow spatial convergence properties of past inherently positive solutions representations, is more accurate than ad-hoc fixups, and does not require significantly more computational work to solve a problem than using an ad-hoc fixup. Overall, CSZ is easy to implement and a valuable addition to existing transport codes, particularly for shielding applications. CSZ is presented here in slab and rect- angular geometries, but is readily extensible to three-dimensional Cartesian (brick) geometries. To be applicable to other simulations, particularly radiative transfer, additional research will need to be conducted, focusing on the diffusion limit in multi-dimension geometries and solution acceleration techniques.

Recent emphasis on the prediction of temperature and concentration fields in the atmosphere has led to the investigation of accurate solution methods of the time-dependent conservation equations for mass, momentum, energy and species. Atmospheric radiation is the key component of this system. Therefore, atmospheric radiation models were developed in isolation from the climate models. The time-dependent multi-dimensional governing equations of atmospheric models must be solved in conjunction with the radiative transfer equation for accurate modeling of the atmosphere. In order to achieve this objective, a 1-D Radiative-Convective Model for Earth-Atmosphere System (RCM4EAS) was developed for clear and cloudy sky atmospheres. The radiative component of the code is Santa Barbara DISORT (Discrete Ordinate Radiative Transfer) Atmospheric Radiative Transfer (SBDART) integrated with exponential sum-fitting method as the radiative property estimation technique. The accuracy of SBDART was tested by comparing its predictions of radiative fluxes with those of Line-by-Line Radiative Transfer Model (LBLRTM) for 1-D longwave (3.33-100 &micro / m) clear sky atmosphere and a good agreement was obtained. A parametric study aiming at finding the optimum parameters to be used as input in SBDART regarding the wavelength increment and order of approximation was also carried out. Variable wavelength and eight streams were selected as optimum parameters for the accuracy and computational efficiency. The code was then coupled with a 1-D Radiative-Convective Model (RCM) to obtain the time dependent code, (RCM4EAS), which was applied to the investigation of the sensitivity of climate to changes in the CO2 concentration for clear and cloudy sky conditions. CO2 sensitivity analyses revealed that doubling the CO2 concentration in the earth&rsquo / s atmosphere from its present value (387 ppm) results in an increase in equilibrium surface temperature of 4.2 K in the clear sky atmosphere as opposed to 2.1 K in cloudy sky atmosphere with typical cloud physical parameters. It is worth noting that times required to reach equilibrium surface temperatures are approximately 2000 and 6000 days for clear and cloudy sky atmospheres, respectively and these temperature increases are calculated assuming that all the other parameters except CO2 concentration remain unchanged within these time periods. Therefore, it should be noted that these temperature increases reflect only the effect of CO2 doubling and excludes the effect of other forcings which might positively or negatively affect these temperature increases. Overall evaluation of the performance of the code developed in this thesis study indicates that it can be used with confidence in 1-D radiative-convective modeling of the earth-atmosphere systems.

This work develops a one dimensional, slab geometry, multigroup collision probability code named Oklo which solves both criticality calculations and fixed source problems. The code uses the classical collision probabilities approach where the first flight collision probabilities are calculated analytically for void, reflected and periodic boundary conditions. The code has been verified against analytical criticality benchmark test sets from Los Alamos National Laboratory, which have been used to verify MCNP amongst other codes. The results from the code show a good agreement with the benchmark test sets for the critical systems presented in this report. The results from the code also match the infinite multiplication factors k and average scalar flux ratios for infinite multiplicative systems from the benchmark test sets. The criticality results and the fixed source results from the Oklo code have been compared with criticality results and fixed source results from a discrete ordinates code and the results for both types of problems show a good agreement with the results from the discrete ordinates code as we increase the N for the discreet ordinates code. / Thesis (M.Sc. Engineering Sciences (Nuclear Engineering))--North-West University, Potchefstroom Campus, 2012.

This work develops a one dimensional, slab geometry, multigroup collision probability code named Oklo which solves both criticality calculations and fixed source problems. The code uses the classical collision probabilities approach where the first flight collision probabilities are calculated analytically for void, reflected and periodic boundary conditions. The code has been verified against analytical criticality benchmark test sets from Los Alamos National Laboratory, which have been used to verify MCNP amongst other codes. The results from the code show a good agreement with the benchmark test sets for the critical systems presented in this report. The results from the code also match the infinite multiplication factors k and average scalar flux ratios for infinite multiplicative systems from the benchmark test sets. The criticality results and the fixed source results from the Oklo code have been compared with criticality results and fixed source results from a discrete ordinates code and the results for both types of problems show a good agreement with the results from the discrete ordinates code as we increase the N for the discreet ordinates code. / Thesis (M.Sc. Engineering Sciences (Nuclear Engineering))--North-West University, Potchefstroom Campus, 2012.

Neste trabalho são apresentadas soluções de caráter analítico, em forma fechada, para o problema de transporte para fótons, com dependência espectral, considerando o núcleo de Klein-Nishina para espalhamento Compton, o qual tem particular aplicação no cálculo de doses em tratamentos de radioterapia. Foram propostas duas abordagens: no caso 1, a variável comprimento de onda e discretizada, sendo que o termo integral da equação, em termos de energia, e aproximado por uma quadratura. No caso 2, uma expansão em termos de funções conhecidas é proposta para solução, de forma que se obtém uma expressão em forma fechada, dependendo continuamente em λ. Em todas as situações o problema resultante em termos da dependência angular, foi resolvido pelo método analítico de ordenadas discretas (ADO). Simulações numéricas são obtidas para uma placa plana, com o cálculo do fluxo escalar, das doses e do fator de "buildup". / In this work, closed form solutions to the transport equation for photons are presented. The Klein-Nishina kernel for Compton scattering is considered, for a particular application in the radiotherapy doses planning. Two approaches are proposed: case 1, where the wavelength variable is discretized and the integral term of the equation is approximated by a quadrature scheme; case 2, where the solution is proposed as an expansion in terms of known functions. In the second case, in the final form of the solution, the dependence on the wavelength is continuous. For all cases, the resulting problem, which depends on the angular variable, is solved by the analytical discrete ordinates method (ADO method). Numerical simulations are performed, in a slab geometry, to generate results for the scalar flux, doses and buildup factor.

Neste trabalho são apresentadas soluções de caráter analítico, em forma fechada, para o problema de transporte para fótons, com dependência espectral, considerando o núcleo de Klein-Nishina para espalhamento Compton, o qual tem particular aplicação no cálculo de doses em tratamentos de radioterapia. Foram propostas duas abordagens: no caso 1, a variável comprimento de onda e discretizada, sendo que o termo integral da equação, em termos de energia, e aproximado por uma quadratura. No caso 2, uma expansão em termos de funções conhecidas é proposta para solução, de forma que se obtém uma expressão em forma fechada, dependendo continuamente em λ. Em todas as situações o problema resultante em termos da dependência angular, foi resolvido pelo método analítico de ordenadas discretas (ADO). Simulações numéricas são obtidas para uma placa plana, com o cálculo do fluxo escalar, das doses e do fator de "buildup". / In this work, closed form solutions to the transport equation for photons are presented. The Klein-Nishina kernel for Compton scattering is considered, for a particular application in the radiotherapy doses planning. Two approaches are proposed: case 1, where the wavelength variable is discretized and the integral term of the equation is approximated by a quadrature scheme; case 2, where the solution is proposed as an expansion in terms of known functions. In the second case, in the final form of the solution, the dependence on the wavelength is continuous. For all cases, the resulting problem, which depends on the angular variable, is solved by the analytical discrete ordinates method (ADO method). Numerical simulations are performed, in a slab geometry, to generate results for the scalar flux, doses and buildup factor.

Neste trabalho são apresentadas soluções de caráter analítico, em forma fechada, para o problema de transporte para fótons, com dependência espectral, considerando o núcleo de Klein-Nishina para espalhamento Compton, o qual tem particular aplicação no cálculo de doses em tratamentos de radioterapia. Foram propostas duas abordagens: no caso 1, a variável comprimento de onda e discretizada, sendo que o termo integral da equação, em termos de energia, e aproximado por uma quadratura. No caso 2, uma expansão em termos de funções conhecidas é proposta para solução, de forma que se obtém uma expressão em forma fechada, dependendo continuamente em λ. Em todas as situações o problema resultante em termos da dependência angular, foi resolvido pelo método analítico de ordenadas discretas (ADO). Simulações numéricas são obtidas para uma placa plana, com o cálculo do fluxo escalar, das doses e do fator de "buildup". / In this work, closed form solutions to the transport equation for photons are presented. The Klein-Nishina kernel for Compton scattering is considered, for a particular application in the radiotherapy doses planning. Two approaches are proposed: case 1, where the wavelength variable is discretized and the integral term of the equation is approximated by a quadrature scheme; case 2, where the solution is proposed as an expansion in terms of known functions. In the second case, in the final form of the solution, the dependence on the wavelength is continuous. For all cases, the resulting problem, which depends on the angular variable, is solved by the analytical discrete ordinates method (ADO method). Numerical simulations are performed, in a slab geometry, to generate results for the scalar flux, doses and buildup factor.

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

Computational time to model radiative heat transfer in a cylindrical Pressurized Oxy-Coal (POC) combustor was reduced by incorporating the multi-dimensional characteristics of the combustion field. The Discrete Transfer Method (DTM) and the Discrete Ordinates Method (DOM) were modified to work with a computational mesh that transitions from 3D cells to axisymmetric and then 1D cells, also known as a dimensionally adaptive mesh. For the DTM, three methods were developed for selecting so-called transdimensional rays, the Single Unweighted Ray (SUR) technique, the Multiple Unweighted Ray (MUR) technique, and the Single Weighted Ray (SWR) technique. For the DOM, averaging methods for handling radiative intensity at dimensional boundaries were developed. Limitations of both solvers with adaptive meshes were identified by comparison with fully 3D results. For the DTM, the primary limit was numerical error associated with view factor calculations. For the DOM, treatment of dimensional boundaries led to step changes that created numerical oscillations, the severity of which was lessened by both increased angular resolution and increased optical thickness. Performance of dimensionally adaptive radiation calculations, uncoupled to any other physical calculation, was evaluated with a series of sensitivity studies including sensitivity to spatial and angular resolution, dimensional boundary placement, and reactor scaling. Runtime was most impacted by boundary layer placement. For the upstream case which had 3D cells over 40% of the reactor length, the speedup versus the fully 3D calculations were 743%, 18%, 220%, and 76% for the SUR, MUR, SWR, and DOM calculations, respectively. The downstream case which had 3D cells over the first 60% of the reactor length, had speedups of 209%, 3%, 109%, and 37%, respectively. For the DTM, accuracy was most sensitive to optical thickness, with the average percent difference in incident heat flux for SUR, MUR, and SWR calculations versus fully 3D calculations being 0.93%, 0.86%, and 1.18%, respectively, for a reactor half the size of the baseline case. The case with four times the reactor size had average percent differences of 0.28%, 0.41%, and 0.39% for the SUR, MUR, and SWR, respectively. Accuracy of the DOM was comparatively insensitive to the different changes studied. Performance of dimensionally adaptive radiation calculations coupled with thermochemistry was also investigated for both pilot and industrial scale systems. For pilot scale systems, flux and temperature differences from either solver were less than 5% and 6%, respectively, with speedups being between 200% - 600%. For industrial systems, temperature differences as high as 15% - 20% and flux differences as high as 50% - 75% were seen. In the case of the DTM, these differences between fully 3D and adaptive results come from a combination of high property gradients and comparatively few rays being drawn and could therefore be improved, at the cost of additional computation time, by using a more sophisticated ray selection method. For the DOM, these issues stem from poor performance of the 1D portion of the solver and could therefore be improved by using a more sophisticated equation to model the radiative transfer in the 1D region.