Computational Fluid Dynamics (CFD) has been pivotal in scientific computing, providing critical insights into complex fluid dynamics unattainable through traditional experimental methods. Despite its widespread use, the accuracy of CFD results remains contingent upon the underlying modeling and numerical errors. A key aspect of ensuring simulation reliability is the accurate quantification of discretization error (DE), which is the difference between the simulation solution and the exact solution in the physical world. This study addresses quantifying DE through Error Transport Equations (ETE), which are an additional set of equations capable of quantifying the local DE in a solution. Historically, Richardson extrapolation has been a mainstay for DE estimation due to its simplicity and effectiveness. However, the method's feasibility diminishes with increasing computational demands, particularly in large-scale and high-dimensional problems. The integration of ETE into existing CFD frameworks is facilitated by their compatibility with existing numerical codes, minimizing the need for extensive code modification. By incorporating techniques developed for managing discontinuities, the study broadens ETE applicability to a wider range of scientific computing applications, particularly those involving complex, unsteady flows. The culmination of this research is demonstrated on unsteady discontinuous problems, such as Sod's problem. / Master of Science / In the ever-evolving field of Computational Fluid Dynamics (CFD), the quest for accuracy is paramount. This thesis focuses on discretization error estimation within CFD simulations, specifically on the challenge of predicting fluid behavior in scenarios marked by sudden changes, such as shock waves. At the core of this work lies an error estimation tool known as Error Transport Equations (ETE) to improve the numerical accuracy of simulations involving unsteady flows and discontinuities. Traditionally, the accuracy of CFD simulations has been limited by discretization errors, generally the largest numerical error, which is the difference between the numerical solution and the exact solution. With ETE, this research identifies these errors to enhance the simulation's overall accuracy. The implications of ETE research are far-reaching. Improved error estimation and correction methods can lead to more reliable predictions in a wide range of applications, from aeronautical engineering, where the aerodynamics of aircraft is critical, to plasma science, with applications in fusion and deep space propulsion.

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This dissertation presents an analytical study of a class of problems involving discontinuities, also referred to as shocks, propagating through one dimensional flexible objects such as strings and rods. The study entails interrogation of the classical balance laws of momentum, angular momentum, and energy across propagating discontinuities. A major part of this dissertation also concerns itself with a non-classical entity called the ``material momentum''. The balance of material momentum is studied in a variational context, where both the local and singular forms of it are derived from an action principle. A distinguishing aspect of discontinuities propagating in continua is that, unlike in the bulk, the balance of momentum and angular momentum are not sufficient to describe their mechanics, even when the discontinuities are energy conserving. In this work, it is shown that the additional information required to close the system of equations at propagating discontinuities can be obtained from the singular form of energy balance across them. This entails splitting of the energy balance by its invariance properties, and identifying the non-invariant and invariant part of the source term with the power input and energy dissipation respectively at the shock. This approach is in contrast with other treatments of such problems in the literature, where additional non-classical concepts such as ``material momentum'' and ``configurational force'' have been invoked. To further our understanding of the connections between the classical and non-classical approaches to problems involving discontinuities, a detailed exposition of the concept of material momentum is presented. The balance and conservation laws associated with material momentum are derived from an action principle. It is shown that the conservation of material momentum is associated with the material symmetry of the continuum, and that the conditions for the conservation of physical and material momentum are independent of each other. A new classification of the deformed configurations of the planar Euler elastica based on conserved quantities associated with the spatial and material symmetry of the rod is proposed. The manifestation of the balance of material momentum in seemingly unrelated fields of research, such as fracture mechanics, ideal fluids, and the mechanics of rods with discontinuities, is also discussed. / Ph. D. / One dimensional flexible bodies such as strings and rods can exhibit fascinating and counterintuitive behavior when they interact with rigid obstacles. For instance, a chain falling on a rigid surface falls faster than it would have if it were falling freely. When one end of a long chain piled up in a container placed at an elevation is pulled across the rim and let go, the chain flows out of the container like a water fountain. Discontinuities in the cross-sectional properties of an elastic rod contained in a curved frictionless channel can result in the generation of forces that propel the rod along the channel. Such counterintuitive phenomena are a consequence of the physics taking place at the point of partial contact where the flexible body comes in contact with a rigid surface. The purpose of this dissertation is to study the mechanics of such points of discontinuity. Several such phenomena where effectively one dimensional bodies interact with rigid surfaces are all around us. A familiar example is the peeling of an adhesive tape, where the peeling front qualifies as a point of discontinuity propagating through the tape as the peeling progresses. A good understanding of the mechanics of the peeling front is crucial in estimating the strength of the adhesive. Another such example of practical importance is a mooring line being placed on the seabed. In such situations, the existence of a reaction force acting at the touchdown point depends on whether or not the cable develops a kink at that point. Similar questions of importance can be asked in the context of deployment and unspooling of space tethers. In this dissertation, an analytical study of the general physics of the phenomena described above is presented. Standard theoretical tools of classical physics are employed to understand the mechanics of points of partial contact between flexible and rigid bodies. The conditions under which a flexible body could experience sharp changes in its geometry (e.g. a kink) at such points are investigated. In addition to that, we explore the implications of a nonclassical law of physics called the balance of “material momentum” in the context of such problems.

In northeastern Mississippi, land cover types vary from agriculture, forests, urban surfaces, pasture, to bodies of water. Substantial evidence exists supporting the contribution of land cover and land cover discontinuities, or physiographic transition zones, to cloud formation on synoptically benign days in many areas across the globe. However, research is lacking on the specific type of land cover and/or land cover discontinuities that convection favors in the warm season. The objective of this study was to develop a synoptically benign convective cloud climatology for northeastern Mississippi and compare this climatology to land cover to determine whether a relationship between land cover type and convective cloud enhancement exists. The study shows a statistically significant clustered pattern occurring in the study area. In addition, enhanced convective events appear to favor land use regions of evergreen needleleaf forest; dryland, cropland, and pasture; and savanna. The study indicates that these three land cover types occur significantly more frequently for the enhancement points than in the study area. The findings support the existence of a significant relationship between land cover and convective enhancement in northeastern Mississippi and provide opportunities for additional future research on relationships between land cover and convection to improve forecast applications and our knowledge of mesoscale circulations.

The Meshless Local Petrov-Galerkin (MLPG) method has been employed to analyze the following linear and nonlinear solid mechanics problems: free and forced vibrations of a segmented bar and a cracked beam, pull-in instability of an electrostatically actuated microbeam, and plane strain deformations of incompressible hyperelastic materials. The Moving Least Squares (MLS) approximation is used to generate basis functions for the trial solution, and for the test functions. Local symmetric weak formulations are derived, and the displacement boundary conditions are enforced by the method of Lagrange multipliers. Three different techniques are employed to enforce continuity conditions at the material interfaces: Lagrange multipliers, jump functions, and MLS basis functions with discontinuous derivatives. For the electromechanical problem, the pull-in voltage and the corresponding deflection are extracted by combining the MLPG method with the displacement iteration pull-in extraction algorithm. The analysis of large deformations of incompressible hyperelastic materials is performed by using a mixed pressure-displacement formulation. For every problem studied, computed results are found to compare well with those obtained either analytically or by the Finite Element Method (FEM). For the same accuracy, the MLPG method requires fewer nodes but more CPU time than the FEM. / Ph. D.

The Karhunen-Loeve Transform (KLT) is a popular transform used in multiple image processing scenarios. Sometimes, the application of the KLT is not carried out as a single transform over an entire image Rather, the image is divided into smaller spatial regions (segments), each of which is transformed by a smaller dimensional KLT. Such a situation may penalize the transform efficiency. An improvement for the segmented KLT, aiming at mitigating discontinuities arising on the edge of adjacent regions, is proposed in this paper. In the case of moderately varying image regions, discontinuities occur as the consequence of disregarded similarity between transform domains, as the order and sign of eigenvectors in the transform matrices are mismatched. In the proposed method, the KLT is adjusted to guarantee the best achievable similarity via the optimal assignment and sign correspondence for eigenvectors. Experimental results indicate that the proposed transform improves the similarity between transform domains, and reduces RMSE on the edge of adjacent regions. In consequence, images processed by the adjusted KLT present better cohesion and continuity between independently transformed regions.

Seismic transitions of the subsurface are typically considered as zero-order singularities (step functions). According to this model, the conventional deconvolution problem aims at recovering the seismic reflectivity as a sparse spike train. However, recent multiscale analysis on sedimentary records revealed the existence of accumulations of varying order singularities in the subsurface, which give rise to fractional-order discontinuities. This observation not only calls for a richer class of seismic reflection waveforms, but it also requires a different methodology to detect and characterize these reflection events. For instance, the assumptions underlying conventional deconvolution no longer hold. Because of the bandwidth limitation of seismic data, multiscale analysis methods based on the decay rate of wavelet coefficients may yield ambiguous results. We avoid this problem by formulating the estimation of the singularity orders by a parametric nonlinear inversion method.

Differential interconnects are extensively used in high-speed digital circuits at fast data rates and in environments of high noise like backplanes. For such applications they are preferred over single-ended lines owing to their ability to reject common-mode noise. Differential schemes like Low Voltage Differential Signaling (LVDS) are used for wireless base stations and ATM switches in telecommunication applications, flat panel displays and servers and for system-level clock distribution. LVDS applications use data rates from 100 Mbps to about 1.5 Gbps and are expected to be highly immune to noise. However, noise will also be injected into differential signals at these high data rates, if there are irregularities in the interconnect setup. These anomalies may be via transitions from differential lines through power planes in power distribution systems, via stubs, asymmetric lengths of differential lines, different transition points for each of the differential vias etc. The differential setup is expected to be immune to such imbalances; however, investigation of these discontinuities indicate that sufficient signal energy can be leaked to power distribution networks (PDN) of packages and boards. The effect of this energy loss was examined in time-domain and was found to cause signal integrity effects like jitter. Irregular differential structures were compared with the equivalent single-ended configuration and symmetrical perfect differential lines. This thesis work quantifies signal to power coupling caused by irregular differential structures in the presence of PDN planes in frequency domain. Presence of noise in differential signaling is verified through a set of test vehicles. The jitter induced as a result of signal to power coupling from differential lines was also investigated.

Made available in DSpace on 2014-06-11T19:28:33Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-08-19Bitstream added on 2014-06-13T19:58:06Z : No. of bitstreams: 1 claro_gks_me_bauru.pdf: 1638801 bytes, checksum: 64af55b7016b44956f54e04a50b76b3c (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Este trabalho tem o propósito de contribuir para a generalização tridimensional de problemas de análise numérica da propagação de fissura, mediante a formulação de elementos finitos com descontinuidade incorporada. Em problemas planos as descontinuidades correspondem a linhas que podem ser elaboradas de uma forma relativamente simples, através da construção sequencial de segmentos retos, orientados conforme a direção de falha no interior de cada elemento finito do sólido. Na análise tridimensional a construção do caminho de descontinuidade é mais complexa, pois devem ser construídas superficiais planas no interior de cada elemento e essas superfícies planas devem ser contínuas entre os elementos. É apresentada, nesse trabalho, uma técnica alternativa de construção do caminho de descontinuidade em análises tridimensionais baseado na solução de um problema análogo ao problema de condução de calor, estabelecido a partir de orientações locais de falha, baseado no estado de tensão do problema mecânico. A solução do problema equivalente é obtida utilizando a mesma malha e interpolações do problema mecânico. Para minimizar o esforço computacional, é proposta uma estratégia na qual a análise para mapear o caminho da descontinuidade é restrita ao domínio formado por alguns elementos próximos à superfície de fissura, que se desenvolve ao longo do processo de carregamento. Para validar a metodologia proposta foram realizadas análises tridimensionais de problemas básicos de fratura experimentais e seus resultados foram contrastados com os resultados encontrados na bibliografia. Realizou-se também a comparação do tempo de processamento entre o algoritmo proposto e o algaritmo global para as mesmas análises mencionadas acima. Como resultado, constatou-se que o algoritmo proposto conseguiu descrever satisfatoriamente as trajetórias de descontinuidade, consumindo menor tempo de processamento / This work contributes to the generalization to 3D problems of numerical analysis of crack propagation, through finite elements formulation with embedded discontinuity. In plane problems the discontinuities correspond to lines that can be tracked in a relatively simple way, by sequentially constructing straight segments, following the crack orientation inside each solid finite elements. In tree-dimensional analysis the tracking scheme is more complex since planar surfaces must be constructed inside each element and these planar surfaces must be continuous between elements. It is show in this work, an alternative version of discontinuity path construction technique in three-dimensional analysis based on the solution of an analogous heat conduction problem, established from the local failure orientation based on the stress state of the mechanical problem. The solution of the equivalent problem is obtained using the same mesh and interpolations of the mechanical problem. To minimize computational effort, a strategy is proposed in which the analysis to track the discontinuity path is restricted to the domain formed by few elements near the crack surface front, which develops along the loading process. To validate the poposed methodology three-dimensional analysis of experimental fracture test were performed and the results were contrasted with those obtained from the literature. The comparison between the process time of the proposed algorithm and the global algorithm was performed too. It was found that the proposed algorithm was able to describe the discontinuity path satisfactorily with reduced computational time