The development of high-order solution methods remain a very active field of research in Computational Fluid Dynamics (CFD). These types of schemes have the potential to reduce the computational cost necessary to compute solutions to a desired level of accuracy. The goal of this thesis has been to develop a high-order Central Essentially Non Oscillatory (CENO) finite volume scheme for multi-block unstructured meshes. In particular, solutions to the compressible, inviscid Euler equations are considered. The CENO method achieves a high-order spatial reconstruction based on the k-exact method, combined with hybrid switching to limited piecewise linear reconstruction in non-smooth regions to maintain monotonicity. Additionally, fourth-order Runge-Kutta time marching is applied. The solver described has been validated through a combination of high-order function reconstructions, and solutions to the Euler equations. Cases have been selected to demonstrate high-orders of convergence, the application of the hybrid switching method, and the multi-block techniques which has been implemented.

Large-eddy simulation (LES) is emerging as a promising computational tool for reacting flows. High-order schemes for LES are desirable to achieve improved solution accuracy with reduced computational cost. In this study, a parallel, block-based, three-dimensional high-order central essentially non-oscillatory (CENO) finite-volume scheme for LES of premixed turbulent combustion is developed for Cartesian mesh. This LES formulation makes use of the flame surface density (FSD) for subfilter-scale reaction rate modelling. An algebraic model is used to approximate the FSD. A detailed explanation of the governing equations for LES and the mathematical framework for CENO schemes are presented. The CENO reconstruction is validated and is also applied to three-dimensional Euler equations prior to its application to the equations governing LES of reacting flows.

The continued research and development of high-order methods in Computational Fluid Dynamics (CFD) is primarily motivated by their potential to significantly reduce the computational cost and memory usage required to obtain a solution to a desired level of accuracy. In this work, a high-order Central Essentially Non-Oscillatory (CENO) finite-volume scheme is developed for the Euler and Navier-Stokes equations in three dimensions. The proposed CENO scheme is based on a hybrid solution reconstruction procedure using a fixed central stencil. A solution smoothness indicator facilitates the hybrid switching between a high-order k-exact reconstruction technique, and a monotonicity preserving limited piecewise linear reconstruction algorithm. The resulting scheme is applied to the compressible forms of the Euler and Navier-Stokes equations in three dimensions. The latter of which includes the application of this high-order work to the Large Eddy Simulation (LES) of turbulent non-reacting flows.

Nuestro grupo ha estudiado la estructura de la cromatina de núcleos de eritrocitos de pollo (Bartolomé et al., 1994; Bartolomé et al., 1995; Bermúdez et al., 1998). La consecuencia de estos estudios ha sido la elaboración de un modelo para el plegamiento de la fibra de cromatina con una elevada concentración local del DNA (Daban y Bermúdez, 1998; Daban, 2000). Sin embargo, el nivel máximo de condensación en la cromatina, se encuentra en el interior de los cromosomas metafásicos. Aunque la bibliografía ha planteado diferentes modelos para el plegamiento de la cromatina en el interior de éstos, existe un conocimiento muy escaso acerca de la estructura molecular de la cromatina en los cromosomas condensados.Se ha realizado un estudio exhaustivo de microscopía electrónica de transmisión sobre la estructura de los cromosomas metafásicos de células HeLa. Se han estudiado un total de 4410 micrografías de cromosomas metafásicos, que en su mayor parte han sido tratados con diversos medios parcialmente desnaturalizantes, para poder analizar su estructura interna.Morfológicamente, los cromosomas estudiados en este trabajo pueden agruparse en tres tipos diferentes: compactos, granulados y fibrilados. La morfología más abundante es la compacta y se observa en presencia de cationes monovalentes y divalentes a concentración similar a la presente en la cromatina metafásica (Mg2+ 1.7-40 mM). Estos cromosomas tienen las cromátidas muy densas y en sus bordes se aprecian una serie de estructuras planas superpuestas. En condiciones de menor concentración de cationes (Mg2+£ 1.7 mM), la morfología dominante es la granular. Estos cromosomas están compuestos principalmente por gran cantidad de cuerpos circulares de 30-40 nm de diámetro. Únicamente en condiciones de fuerza iónica extremadamente baja podemos encontrar la morfología fibrilar, la cual se caracteriza por la abundancia de fibras de 30-40 nm.Los resultados obtenidos con cromosomas parcialmente desnaturalizados nos permiten concluir que existen tres elementos estructurales en el interior de los cromosomas metafásicos: la fibra, el gránulo y la placa.Las fibras gruesas con diámetros que oscilan entre los 100 y los 500 nm son el resultado de la deformación plástica de las cromátidas durante los diferentes procesos de preparación de las muestras. En función de las condiciones iónicas del medio las fibras gruesas muestran gránulos o placas en su interior. Las fibras delgadas están formadas por una sucesión de cuerpos de 30-40 nm de diámetro unidos irregularmente mediante interacciones cabeza-cola. Las fibras delgadas se observan dominantemente en condiciones de concentración salina extremadamente baja.Los gránulos son unos cuerpos circulares compactos de unos 30-40 nm de diámetro. Estos cuerpos compactos descritos previamente por nuestro grupo y se interpretaron como una forma de plegamiento solenoidal de la fibra de 30 nm (Daban y Bermúdez, 1998). Se encuentran presentes en todas las condiciones estudiadas en este trabajo, siendo especialmente abundantes en presencia de iones divalentes a concentración baja y en muestras tratadas con nucleasa micrococal. La placa es un elemento estructural característico de los cromosomas cuando éstos se encuentran en su forma más compacta, en presencia de concentraciones elevadas de cationes divalentes. Esta estructura no había sido descrita previamente por otros laboratorios. Es una estructura cromatínica de gran regularidad y con una superficie muy lisa. Hemos estimado la altura de estas placas a través de muestras sombreadas unidireccionalemente con platino. El promedio de los valores obtenidos es de 6.7 ± 1.4 nm.En conjunto los resultados obtenidos en esta tesis permiten sugerir que el componente principal de la cromatina en los cromosomas metafásicos es el gránulo de 30-40 nm. Dependiendo de las condiciones iónicas, este elemento estructural fundamental se agrega a través de uniones cabeza-cola para formar fibras (fuerza iónica muy baja), o bien se agrega mediante interacciones laterales para formar placas (condiciones salinas próximas a las de la cromatina metafásica). / Our group has studied the chromatin structure in the chicken erythrocyte nuclei (Bartolome et al., 1994; Bartolomé et al., 1995; Bermúdez et al., 1998). The consequences of this studies has been the elaboration of a folding model of the chromatin fiber with a high local concentration of DNA. However, the maximum level of chromatin condensation, is found in the metaphase chromosomes. Although the bibliography has proposed different models to explain the chromatin folding inside the chromosomes, there is a low knowledge about the molecular structure of chromatin in the condensed chromosomes. In this thesis, we have carried out an exhaustive electron microscopy study about the HeLa cells metaphase chromosomes. We have studied a large number of chromosome electron micrographs (4410). Chromosomes were partially denaturated under a wide variety of conditions in order to observe some chromatin structural element inside them.Our studies indicate that chromosomes can adopt three global structural forms in function of the ionic conditions: compact, granular and fibrillar.The compact form is the most frequent and we can observe it in the presence of monovalent and divalent cations in similar concentrations than the ones found in metaphase chromatin (Mg2+ 1.7-40 mM). These chromosomes have highly condensed chromatids and we can appreciate overlapped chromatin plates around the chromosomes edges. When the chromosomes are incubated with solutions containing lower cations concentration (Mg 2+£ 1.7 mM) they become granular. The granular structures seen inside these chromosomes show a diameter of about 35 nm. Fibrillar chromosomes are observed only at very low ionic strength. The fibers seen emanating from the chromatids have a diameter of 30-40 nm.Our results obtained from partially denaturated chromosomes show that there are three structural elements inside the metaphase chromosomes: the fiber, the 30-40 nm chromatin granule and the plate.The largest fibers with a diameter of 100-400 nm, presumably are produced by mechanical deformation of chromosomes during the preparation processes. Depending of the ionic conditions these fibrillar structures are composed by plates or granules. The thinnest fibers are formed by face to face association of the 30-40 nm chromatin granules. These kind of fibers are usually found only at very low ionic strength.The chromatin granules are compact bodies with 35 nm of diameter. These compact bodies were previously described in our laboratory and were modeled as compact solenoids of nucleosomes forming (Daban and Bermúdez, 1998). They are usually seen at low divalent cation concentrations and in chromosome samples treated with micrococal nuclease.The plate is the most frequent structural element when the chromosomes are in their compact form (high ionic strength, similar to physiological conditions). This element has not been described by any group. It is a chromatin element with a regular structure and very smooth surface. We have estimated the height of the steps between layers in unidirectional shadowing experiments. The value obtained is 6.7 ± 1.4 nm.Our results suggest that the fundamental component inside the metaphase is the 30-40 nm chromatin granules. Depending of the ionic conditions, this basic structural element forms fibers through face to face interactions (very low ionic strength) or form plates through side to side interactions (high ionic strength similar to metaphase chromatin).

The miscible displacement equations provide the mathematical model for simulating the displacement of a mixture of oil and miscible fluid in underground reservoirs during the Enhance Oil Recovery(EOR) process. In this thesis, I propose a stable numerical scheme combining a mixed finite element method and space-time discontinuous Galerkin method for solving miscible displacement equations under low regularity assumption. Convergence of the discrete solution is investigated using a compactness theorem for functions that are discontinuous in space and time. Numerical experiments illustrate that the rate of convergence is improved by using a high order time stepping method. For petroleum engineers, it is essential to compute finely detailed fluid profiles in order to design efficient recovery procedure thereby increase production in the EOR process. The method I propose takes advantage of both high order time approximation and discontinuous Galerkin method in space and is capable of providing accurate numerical solutions to assist in increasing the production rate of the miscible displacement oil recovery process.

The Eshelby-type problems of an arbitrary-shape polygonal or polyhedral inclusion embedded in an infinite homogeneous isotropic elastic material are analytically solved using a simplified strain gradient elasticity theory (SSGET) that contains a material length scale parameter. The Eshelby tensors for a plane strain inclusion with an arbitrary polygonal cross section and for an arbitrary-shape polyhedral inclusion are analytically derived in general forms in terms of three potential functions. These potential functions, as area integrals over the polygonal cross section and volume integrals over the polyhedral inclusion, are evaluated. For the polygonal inclusion problem, the three area integrals are first transformed to three line integrals using the Green's theorem, which are then evaluated analytically by direct integration. In the polyhedral inclusion case, each of the three volume integrals is first transformed to a surface integral by applying the divergence theorem, which is then transformed to a contour (line) integral based on Stokes' theorem and using an inverse approach. In addition, the Eshelby tensor for an anti-plane strain inclusion with an arbitrary polygonal cross section embedded in an infinite homogeneous isotropic elastic material is analytically solved. Each of the newly derived Eshelby tensors is separated into a classical part and a gradient part. The latter includes the material length scale parameter additionally, thereby enabling the interpretation of the inclusion size effect. For homogenization applications, the area or volume average of each newly derived Eshelby tensor over the polygonal cross section or the polyhedral inclusion domain is also provided in a general form. To illustrate the newly obtained Eshelby tensors and their area or volume averages, different types of polygonal and polyhedral inclusions are quantitatively studied by directly using the general formulas derived. The numerical results show that the components of the each SSGET-based Eshelby tensor for all inclusion shapes considered vary with both the position and the inclusion size. It is also observed that the components of each averaged Eshelby tensor based on the SSGET change with the inclusion size.

Classically, researchers in Computational Physics and specifically in Computational Electromagnetics have sought to find numerical solutions to complex physical problems. Several techniques have been developed to accomplish such tasks, each of which having advantages over their counterparts. Typically, each solution method has been developed separately despite having numerous commonalities with other methods. This fact motivates a unified software tool to house each solution method to avoid duplicating previous efforts. Subsequently, these solution methods can be used alone or in conjunction with one another in a straightforward manner. The aforementioned goals can be accomplished by using an Object Oriented software approach. Thus, the goal of the presented research was to incorporate a specific solution technique, an Integral Equation Nystrm method, into a general, Object Oriented software framework.

In this dissertation, three different methods for solving the Boltzmann neutron transport equation (and its low-order approximations) are developed in general geometry and implemented in 1D slab geometry. The first method is for solving the fine-group diffusion equation by estimating the in-scattering and fission source terms with consistent coarse-group diffusion solutions iteratively. This is achieved by extending the subgroup decomposition method initially developed in neutron transport theory to diffusion theory. Additionally, a new stabilizing scheme for on-the-fly cross section re-condensation based on local fixed source calculations is developed in the subgroup decomposition framework. The method is derived in general geometry and tested in 1D benchmark problems characteristic of Boiling Water Reactors (BWR) and Gas Cooled Reactor (GCR). It is shown that the method reproduces the standard fine-group results with 3-4 times faster computational speed in the BWR test problem and 1.5 to 6 times faster computational speed in the GCR core. The second method is a hybrid diffusion transport method for accelerating multi-group eigenvalue transport problems. This method extends the subgroup decomposition method to efficiently couple a coarse-group high-order diffusion method with a set of fixed-source transport decomposition sweeps to obtain the fine-group transport solution. The advantages of this new high-order diffusion theory are its consistent transport closure, straight forward implementation and numerical stability. The method is analyzed for 1D BWR and High Temperature Test Reactor (HTTR) benchmark problems. It is shown that the method reproduces the fine-group transport solution with high accuracy while increasing the computationally efficiency up to 16 times in the BWR core and up to 3.3 times in the HTTR core compared to direct fine-group transport calculations. The third method is a new spatial homogenization method in transport theory that reproduces the heterogeneous solution by using conventional flux weighted homogenized cross sections. By introducing an additional source term via an “auxiliary cross section” the resulting homogeneous transport equation becomes consistent with the heterogeneous equation, enabling easy implementation into existing solution methods/codes. This new method utilizes on-the-fly re-homogenization, performed at the assembly level, to correct for core environment effects on the homogenized cross sections. The method is derived in general geometry and continuous energy, and implemented and tested in fine-group 1D slab geometries typical of BWR and GCR cores. The test problems include two single assembly and 4 core configurations. It is believed that the coupling of the two new methods, namely the hybrid method for treating the energy variable and the new spatial homogenization method in transport theory set the stage, as future work, for the development of a robust and practical method for highly efficient and accurate whole core transport calculations.

Sensitivity analysis based on a chemical transport model (CTM) serves as an important approach towards better understanding the relationship between trace contaminant levels in the atmosphere and emissions, chemical and physical processes. Previous studies on ozone control identified the high-order Decoupled Direct Method (HDDM) as an efficient tool to conduct sensitivity analysis. Given the growing recognition of the adverse health effects of fine particulate matter (i.e., particles with an aerodynamic diameter less than 2.5 micrometers (PM2.5)), this dissertation presents the development of a HDDM sensitivity technique for particulate matter and its implementation it in a widely used CTM, CMAQ. Compared to previous studies, two new features of the implementation are 1) including sensitivities of aerosol water content and activity coefficients, and 2) tracking the chemical regimes of the embedded thermodynamic model. The new features provide more accurate sensitivities especially for nitrate and ammonium. Results compare well with brute force sensitivities and are shown to be more stable and computationally efficient. Next, this dissertation explores the applications of HDDM. Source apportionment analysis for the Houston region in September 2006 indicates that nonlinear responses accounted for 3.5% to 33.7% of daily average PM2.5, and that PM2.5 formed rapidly during night especially in the presence of abundant ozone and under stagnant conditions. Uncertainty analysis based on the HDDM found that on average, uncertainties in the emissions rates led to 36% uncertainty in simulated daily average PM2.5 and could explain much, but not all, of the difference between simulated and observed PM2.5 concentrations at two observations sites. HDDM is then applied to assess the impact of flare VOC emissions with temporally variable combustion efficiency. Detailed study of flare emissions using the 2006 Texas special inventory indicates that daily maximum 8-hour ozone at a monitoring site can increase by 2.9 ppb when combustion is significantly decreased. The last application in this dissertation integrates the reduced form model into an electricity generation planning model, and enables representation of geospatial dependence of air quality-related health costs in the optimization process to seek the least cost planning for power generation. The integrated model can provide useful advice on selecting fuel types and locations for power plants.

The issue of local scale and smoothness presents a crucial and daunting challenge for numerical simulation methods in fluid dynamics. Yet in the interests of both accuracy and economy, how can one devise a general technique that efficiently resolves flow features of consequence and discriminates against others which are either ``negligible' or amenable to ``universal' modeling? This is particularly difficult because geometries of engineering interest are complex and multi-dimensional, precluding a priori knowledge of the flowfield. To address this challenge, the current work employs wavelet theory for the local scale decomposition of functions, which provides a natural mechanism for the adaptive compression of data. The resulting technique is known as the Multi-Resolution Discontinuous Galerkin (MRDG) method. This research successfully demonstrates that the multi-resolution framework and the discontinuous Galerkin method are well-suited for a new approach to accuracy and cost as demonstrated by the relative ease of their integration in spatial dimension greater than one. Some specific steps achieved include the implementation of suitable data encoding and compression algorithms, construction of multi-wavelet expansion bases in one and two dimensions, and derivation of the multi-resolution derivative operator that includes an upwind-type correction to the central scheme. Solutions with the MRDG method are observed to adapt to and track both smooth and discontinuous flow features in an entirely solution-driven manner without the need for a priori user knowledge of those flow features. Run-time efficiency and local adaptation characteristics are explored via a series of classic test problems.