A new discrete ordinates method designed for use on modern, large memory, vector and/or parallel processing super computers has been developed. The method is similar to conventional SN techniques in that the medium is divided into spatial mesh cells and that discrete directions are used. However, in place of an approximate differencing scheme, a nearly exact matrix representation of the streaming operator is determined. Although extremely large, this matrix can be stored on today's computers for repeated use in the source iteration. Since the source iteration is cast in matrix form it benefits enormously from vector and/or parallel processing, if available. Several test results are presented demonstrating the reduction in numerical diffusion and elimination of ray effects.

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.

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.

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.

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.

Neste trabalho, o fenômeno de evaporação em gases rarefeitos e analisado, para o caso de uma espécie de gás bem como de misturas binárias. Evaporação fraca e forte são consideradas para escoamentos de gases em canal e semi-espaco. Também e investigado o fenômeno conhecido como reverso de temperatura, típico de gases em estado de rarefação. O método ADO, uma versão analítica do método de ordenadas discretas, é utilizado para construção de soluções em forma fechada para os diversos problemas e quantidades de interesse, como perfis de temperatura e fluxos de calor. Para o caso de um gás, uma solução unificada e desenvolvida para problemas formulados a partir dos modelos cinéticos, derivados da equação de Boltzmann, BGK, S, Gross- Jackson e MRS. No caso de mistura binária de gases, a formulação matemática e baseada no modelo McCormack. Particularmente, quando a evaporação forte e abordada, e aspectos não lineares devem ser incluídos, a versão não linear do modelo BGK e utilizada. Neste caso, a solução ADO do modelo linear e utilizada em um processo chamado de pós-processamento para inclusão dos termos não lineares do problema e reavaliação das quantidades de interesse, evidenciando melhoria dos resultados obtidos pela formulação linear. Uma serie de resultados numéricos são listados e é observada, de forma geral, excelente exatidão e eficiência computacional. / In this work, evaporation phenomena in rarefied gas flow, for one gas case and binary mixtures, are analyzed. Weak and strong evaporation are considered in channel and half-space problems. The reverse of temperature problem, typical in rarefied gas dynamics, is also investigated. The ADO method, an analytical version of the discrete ordinates method, is used to develop closed form solutions, to several problems and quantities of interest, as temperature profiles and heat flows. For the one gas case, an unified solution is developed for the BGK, S, Gross-Jackson and MRS models, derived from the Boltzmann equation. For binary mixtures, the mathematical formulation is based on the McCormack model. Particularly, when strong evaporation is investigated, and nonlinear aspects have to be included, the nonlinear BGK model is used. In this case, the ADO solution, provided by the linear model, is considered in a post-processing procedure which takes into account the nonlinear terms to evaluate the quantities of interest, and improved results are obtained, in comparison with the linear version. A series of numerical results are listed and, in general, an excellent accuracy and good computational efficiency are observed.

Neste trabalho, o fenômeno de evaporação em gases rarefeitos e analisado, para o caso de uma espécie de gás bem como de misturas binárias. Evaporação fraca e forte são consideradas para escoamentos de gases em canal e semi-espaco. Também e investigado o fenômeno conhecido como reverso de temperatura, típico de gases em estado de rarefação. O método ADO, uma versão analítica do método de ordenadas discretas, é utilizado para construção de soluções em forma fechada para os diversos problemas e quantidades de interesse, como perfis de temperatura e fluxos de calor. Para o caso de um gás, uma solução unificada e desenvolvida para problemas formulados a partir dos modelos cinéticos, derivados da equação de Boltzmann, BGK, S, Gross- Jackson e MRS. No caso de mistura binária de gases, a formulação matemática e baseada no modelo McCormack. Particularmente, quando a evaporação forte e abordada, e aspectos não lineares devem ser incluídos, a versão não linear do modelo BGK e utilizada. Neste caso, a solução ADO do modelo linear e utilizada em um processo chamado de pós-processamento para inclusão dos termos não lineares do problema e reavaliação das quantidades de interesse, evidenciando melhoria dos resultados obtidos pela formulação linear. Uma serie de resultados numéricos são listados e é observada, de forma geral, excelente exatidão e eficiência computacional. / In this work, evaporation phenomena in rarefied gas flow, for one gas case and binary mixtures, are analyzed. Weak and strong evaporation are considered in channel and half-space problems. The reverse of temperature problem, typical in rarefied gas dynamics, is also investigated. The ADO method, an analytical version of the discrete ordinates method, is used to develop closed form solutions, to several problems and quantities of interest, as temperature profiles and heat flows. For the one gas case, an unified solution is developed for the BGK, S, Gross-Jackson and MRS models, derived from the Boltzmann equation. For binary mixtures, the mathematical formulation is based on the McCormack model. Particularly, when strong evaporation is investigated, and nonlinear aspects have to be included, the nonlinear BGK model is used. In this case, the ADO solution, provided by the linear model, is considered in a post-processing procedure which takes into account the nonlinear terms to evaluate the quantities of interest, and improved results are obtained, in comparison with the linear version. A series of numerical results are listed and, in general, an excellent accuracy and good computational efficiency are observed.