Dissertationen zum Thema „Numerical analysis“

Magnetic nanoparticles (NPs) with sizes ranging from 2 to 20 nm in diameter represent an important class of artificial nanostructured materials, since the NP size is comparable to the size of a magnetic domain. They have potential applications in data storage, catalysis, permanent magnetic nanocomposites, and biomedicine. To begin with, a brief overview on the background of Fe-based bimetallic NPs and their applications for data-storage and catalysis was presented in Chapter 1. In Chapter 2, L10-ordered FePt NPs with high coercivity were directly prepared from a novel bimetallic acetylenic alternating copolymer P3 by a one-step pyrolysis method without post-thermal annealing. The chemical ordering, morphology and magnetic properties were studied. Magnetic measurements showed that a record coercivity of 3.6 T (1 T = 10 kOe) was obtained in FePt NPs. By comparison of the resultant FePt NPs synthesized under Ar and Ar/H2, the characterization proved that the incorporation of H2 would affect the nucleation and promote the growth of FePt NPs. The L10 FePt NPs were also successfully patterned on Si substrate by nanoimprinting lihthography (NIL). The highly ordered ferromagnetic arrays on a desired substrate for bit-patterned media (BPM) were studied and promised bright prospects for the progress of data-storage. Furthermore, we also reported a new FePt-containing metallopolymer P4 as the single-source precursor for metal alloy NPs synthesis, where the metal fractions were on the side chain and the ratio could be easily controlled. This polymer was synthesized from random copolymer poly(styrene-4-ethynylstyrene) PES-PS and bimetallic precursor TPy-FePt ([Pt(4’-ferrocenyl-(N^N^N))Cl]Cl) by Sonogashira coupling reaction. After pyrolysis of P4, the stoichiometry of Fe and Pt atoms in the synthesized NPs (NPs) is nearly close to 1:1, which is more precise than using TPy-FePt as precursor. Polymer P4 was also more favorable for patterning by high throughout NIL as compared to TPy-FePt. Ferromagnetic nanolines, potentially as bit-patterned magnetic recording media, were successfully fabricated from P4 and fully characterized. In Chapter 3, a novel organometallic compound TPy-FePd-1 [4’-ferrocenyl-(N^N^N)PdOCOCH3] was synthesized and structurally characterized, whose crystal structure showed a coplanar Pd center and Pd-Pd distance (3.17 Å). Two metals Fe and Pd were evenly embedded in the molecular dimension and remained tightly coupled between each other benefiting to the metalmetal (Pd-Pd) and ligand ππ stacking interactions, all of which made it facilitate the nucleation without sintering during preparing the FePd NPs. Ferromagnetic FePd NPs of ca. 16.2 nm in diameter were synthesized by one-pot pyrolysis of the single-source precursor TPy-FePd-1 under getter gas with metal-ion reduction and minimal nanoparticle coalescence, which have a nearly equal atomic ratio (Fe/Pd = 49/51) and exhibited coercivity of 4.9 kOe at 300 K. By imprinting the mixed chloroform solution of TPy-FePd-1 and polystyrene (PS) on Si, reproducible patterning of nanochains was formed due to the excellent self-assembly properties and the incompatibility between TPy-FePd-1 and PS under the slow evaporation of the solvents. The FePd nanochains with average length of ca. 260 nm were evenly dispersed around the PS nanosphere by self-assembly of TPy-FePd-1. In addition, the orientation of the FePd nanochains could also be controlled by tuning the morphology of PS, and the length was shorter in confined space of PS. Orgnic skeleton in TPy-FePd-1 and PS were carbonized and removed by pyrolysis under Ar/H2 (5 wt%) and only magnetic FePd alloy nanochains with domain structure were left. Besides, a bimetallic complex TPy-FePd-2 was prepared and used as a single-source precursor to synthesize ferromagnetic FePd NPs by one-pot pyrolysis. The resultant FePd NPs have a mean size of 19.8 nm and show the coercivity of 1.02 kOe. In addition, the functional group (-NCMe) in TPy-FePd-2 was easily substituted by a pyridyl group. A random copolymer PS-P4VP was used to coordinate with TPy-FePd-2, and the as-synthesized polymer made the metal fraction disperse evenly along the flexible chain. Fabrication of FePd NPs from the polymers was also investigated, and the size could be easily controlled by tuning the metal fraction in polymer. FePd NPs with the mean size of 10.9, 14.2 and 17.9 nm were prepared from the metallopolymer with 5 wt%, 10 wt% and 20wt% of metal fractions, respectively. In Chapter 4, molybdenum disulfide (MoS2) monolayers decorated with ferromagnetic FeCo NPs on the edges were synthesized through a one-step pyrolysis of precursor molecules in an argon atmosphere. The FeCo precursor was spin coated on the MoS2 monolayer grown on Si/SiO2 substrate. Highly-ordered body-centered cubic (bcc) FeCo NPs were revealed under optimized pyrolysis conditions, possessing coercivity up to 1000 Oe at room temperature. The FeCo NPs were well-positioned along the edge sites of MoS2 monolayers. The vibration modes of Mo and S atoms were confined after FeCo NPs decoration, as characterized by Raman shift spectroscopy. These MoS2 monolayers decorated with ferromagnetic FeCo NPs can be used for novel catalytic materials with magnetic recycling capabilities. The sizes of NPs grown on MoS2 monolayers are more uniform than from other preparation routines. Finally, the optimized pyrolysis temperature and conditions provide receipts for decorating related noble catalytic materials. Finally, Chapters 5 and 6 present the concluding remarks and the experimental details of the work described in Chapters 2-4.

Multitud de problemas en ciencia e ingeniería se plantean como ecuaciones en derivadas parciales (EDPs). Si la frontera del recinto donde esas ecuaciones han de satisfacerse se desconoce a priori, se habla de "Problemas de frontera libre", propios de sistemas estacionarios no dependientes del tiempo, o bien de "Problemas de frontera móvil", asociados a problemas de evolución temporal, donde la frontera cambia con el tiempo. La solución a dichos problemas viene dada por la expresión de la(s) variable(s) dependiente(s) de la(s) EDP(s) junto con la función que determina la posición de la frontera. Dado que este tipo de problemas carece en la mayoría de los casos de solución analítica conocida, se hace preciso recurrir a métodos numéricos que permitan obtener una solución lo suficientemente aproximada, y que además mantenga propiedades cualitativas de la solución del modelo continuo de EDP(s). En este trabajo se ha abordado el estudio numérico de algunos problemas de frontera móvil provenientes de diversas disciplinas. La metodología aplicada consta de dos pasos sucesivos: aplicación de la transformación de Landau o "Front-fixing transformation" al modelo en EDP(s) con el fin de mantener inmóvil la frontera del dominio, y posterior discretización a través de un esquema en diferencias finitas. De ahí se obtienen esquemas numéricos que se implementan por medio de la herramienta MATLAB. Mediante un exhaustivo análisis numérico, se estudian propiedades del esquema y de la solución numérica (positividad, estabilidad, consistencia, monotonía, etc.). En el primer capítulo de este trabajo se revisa el estado del arte del campo objeto de estudio, se justifica la necesidad de disponer de métodos numéricos adaptados a este tipo de problemas y se describe brevemente la metodología empleada en nuestro enfoque. El Capítulo 2 se dedica a un problema perteneciente a la Biología Matemática y que consiste en determinar la evolución de la población de una especie invasora que se propaga en un hábitat. Este modelo consiste en una ecuación de difusión-reacción unida a una condición tipo Stefan. Los resultados del análisis numérico confirman la existencia de una dicotomía propagación-extinción en la evolución a largo plazo de la densidad de población de la especie invasora. En particular, se ha podido precisar el valor del coeficiente de la condición de Stefan que separa el comportamiento de propagación del de extinción. Los Capítulos 3 y 4 se centran en un problema de Química del Hormigón con interés en Ingeniería Civil: el proceso de carbonatación del hormigón, fenómeno evolutivo que lleva consigo la degradación progresiva de la estructura afectada y finalmente su ruina, si no se toman medidas preventivas. En el Capítulo 3 se considera un sistema de dos EDPs de tipo parabólico con dos incógnitas. Para su resolución, hay que considerar además las condiciones iniciales, las de contorno y las de tipo Stefan en la frontera. Los resultados numéricos confirman la tendencia de la ley de evolución de la frontera móvil hacia una función del tipo "raíz cuadrada del tiempo". En el Capítulo 4 se considera un modelo más general que el anterior, en el que intervienen seis especies químicas que se encuentran tanto en la zona carbonatada como en la no carbonatada. En el Capítulo 5 se aborda un problema de transmisión de calor que aparece en diversos procesos industriales; en este caso, en el enfriamiento durante la colada de metal fundido, donde la fase sólida avanza y la líquida se va extinguiendo. La frontera móvil (frente de solidificación) separa ambas fases, siendo su posición en cada instante la variable a determinar, junto con las temperaturas en cada fase. Después de la adecuada transformación y discretización, se implementa un esquema en diferencias finitas, subdividiendo el proceso en tres estadios temporales, a fin de tratar las singularidades asociadas a posicione<br>Many problems in science and engineering are formulated as partial differential equations (PDEs). If the boundary of the domain where these equations are to be solved is not known a priori, we face "Free-boundary problems", which are characteristic of non-time dependent stationary systems; besides, we have "Moving-boundary problems" in temporal evolution processes, where the border changes over time. The solution to these problems is given by the expression of the dependent variable(s) of PDE(s), together with the function that determines the position of the boundary. Since the analytical solution of this type of problems is lacked in most cases, it is necessary to resort to numerical methods that allow an accurate enough solution to be obtained, and which also maintain the qualitative properties of the solution(s) of the continuous model. This work approaches the numerical study of some moving-boundary problems that arise in different disciplines. The applied methodology consists of two successive steps: firstly, the so-called Landau transformation, or "Front-fixing transformation", which is used in the PDE(s) model to maintain the boundary of the domain immobile; later, we proceed to its discretization with a finite difference scheme. Different numerical schemes are obtained and implemented through the MATLAB computational tool. Properties of the scheme and the numerical solution (positivity, stability, consistency, monotonicity, etc.) are studied by an exhaustive numerical analysis. The first chapter of this work reports the state of the art of the field under study, justifies the need to adapt numerical methods to this type of problem, and briefly describes the methodology used in our approach. Chapter 2 presents a problem in Mathematical Biology that consists in determining over time the evolution of an invasive species population that spreads in a habitat. This problem is modelled by a diffusion-reaction equation linked to a Stefan-type condition. The results of the numerical analysis confirm the existence of a spreading-vanishing dichotomy in the long-term evolution of the population density of the invasive species. In particular, it is possible to determine the value of the coefficient of the Stefan condition that separates the propagation behaviour from extinction. Chapters 3 and 4 focus on a problem of Concrete Chemistry with an interest in Civil Engineering: the carbonation of concrete, an evolutionary phenomenon that leads to the progressive degradation of the affected structure and its eventual ruin if preventive measures are not taken. Chapter 3 considers a system of two parabolic type PDEs with two unknowns. For its resolution, the initial and boundary conditions have to be considered together with the Stefan conditions on the carbonation front. The numerical analysis results agree with those obtained in a previous theoretical study. The dynamics of the concentrations and the moving boundary confirm the long-term behaviour of the evolution law for the moving boundary as a "square root of time". Chapter 4 considers a more general model than the previous one, which includes six chemical species, defined in both the carbonated and non-carbonated zones, whose concentrations have to be found. Chapter 5 addresses a heat transfer problem that appears in various industrial processes; in this case, the solidification of metals in casting processes, where the solid phase advances and liquid reduces until it is depleted. The moving boundary (the solidification front) separates both phases. Its position in each instant is the variable to be determined together with the temperature profiles in both phases. After suitable transformation, discretization is carried out to obtain a finite difference scheme to be implemented. The process was subdivided into three temporal stages to deal with the singularities associated with the moving boundary position in the initialisation and depletion stages.<br>Multitud de problemes en ciència i enginyeria es plantegen com a equacions en derivades parcials (EDPs). Si la frontera del recinte on eixes equacions han de satisfer-se es desconeix a priori, es parla de "Problemas de frontera lliure", propis de sistemes estacionaris no dependents del temps, o bé de "Problemas de frontera mòbil", associats a problemes d'evolució temporal, on la frontera canvia amb el temps. Atés que este tipus de problemes manca en la majoria dels casos de solució analítica coneguda, es fa precís recórrer a mètodes numèrics que permeten obtindre una solució prou aproximada a l'exacta, i que a més mantinga propietats qualitatives de la solució del model continu d'EDP(s). En aquest treball s'ha abordat l'estudi numèric d'alguns problemes de frontera mòbil provinents de diverses disciplines. La metodologia aplicada consta de dos passos successius: en primer lloc, s'aplica l'anomenada transformació de Landau o "Front-fixing transformation" al model en EDP(s) a fi de mantindre immòbil la frontera del domini; posteriorment, es procedix a la seva discretització a través d'un esquema en diferències finites. D'ací s'obtenen esquemes numèrics que s'implementen per mitjà de la ferramenta informàtica MATLAB. Per mitjà d'una exhaustiva anàlisi numèrica, s'estudien propietats de l'esquema i de la solució numèrica (positivitat, estabilitat, consistència, monotonia, etc.). En el primer capítol d'aquest treball es revisa l'estat de l'art del camp objecte d'estudi, es justifica la necessitat de disposar de mètodes numèrics adaptats a aquest tipus de problemes i es descriu breument la metodologia emprada en el nostre enfocament. El Capítol 2 es dedica a un problema pertanyent a la Biologia Matemàtica i que consistix a determinar l'evolució en el temps de la distribució de la població d'una espècie invasora que es propaga en un hàbitat. Este model consistix en una equació de difusió-reacció unida a una condició tipus Stefan, que relaciona les funcions solució i frontera mòbil a determinar. Els resultats de l'anàlisi numèrica confirmen l'existència d'una dicotomia propagació-extinció en l'evolució a llarg termini de la densitat de població de l'espècie invasora. En particular, s'ha pogut precisar el valor del coeficient de la condició de Stefan que separa el comportament de propagació del d'extinció. Els Capítols 3 i 4 se centren en un problema de Química del Formigó amb interés en Enginyeria Civil: el procés de carbonatació del formigó, fenomen evolutiu que comporta la degradació progressiva de l'estructura afectada i finalment la seua ruïna, si no es prenen mesures preventives. En el Capítol 3 es considera un sistema de dos EDPs de tipus parabòlic amb dos incògnites. Per a la seua resolució, cal considerar a més, les condicions inicials, les de contorn i les de tipus Stefan en la frontera. Els resultats de l'anàlisi numèrica s'ajusten als obtinguts en un estudi teòric previ. S'han dut a terme experiments numèrics, comprovant la tendència de la llei d'evolució de la frontera mòbil cap a una funció del tipus "arrel quadrada del temps". En el Capítol 4 es considera un model més general, en el que intervenen sis espècies químiques les concentracions de les quals cal trobar, i que es troben tant en la zona carbonatada com en la no carbonatada. En el Capítol 5 s'aborda un problema de transmissió de calor que apareix en diversos processos industrials; en aquest cas, en el refredament durant la bugada de metall fos, on la fase sòlida avança i la líquida es va extingint. La frontera mòbil (front de solidificació) separa ambdues fases, sent la seua posició en cada instant la variable a determinar, junt amb les temperatures en cada una de les dos fases. Després de l'adequada transformació i discretització, s'implementa un esquema en diferències finites, subdividint el procés en tres estadis temporals, per tal de tractar les singularitats asso<br>Piqueras García, MÁ. (2018). Numerical Methods for Multidisciplinary Free Boundary Problems: Numerical Analysis and Computing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/107948<br>TESIS

Indentation tests are widely used with simultaneous measurements of indentation depth and force especially for determining material properties. In this study<br>numerical and experimental investigation of the force-indentation measurements is presented. For indentation tests on anisotropic metals, a novel indenter which is not self similar is used with three transducers to measure the displacements. It is seen that in order to have high repeatability and accuracy at the tests, workpiece and indenter parameters have crucial importance. These parameters in the indentations are analyzed by finite element methods. Ideal dimensions of the workpiece are determined. It is shown that plane strain conditions can only be achieved by embedded indentations. Effect of surface quality and clamping on repeatability are investigated. It is shown that surface treatments have significant effects on the results. Also it is seen that clamping increases the repeatability drastically. Moreover, indentation tests are conducted to verify the results of numerical simulations. Effect of anisotropy on the force-displacement curves is clearly observed.

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO<br>Uma das atividades relacionadas à recuperação de barragens envolve o alteamento de barragens existentes, normalmente com o objetivo de aumentar a capacidade de armazenamento dos reservatórios, melhorar o fator de segurança dos taludes ou a proteção da estrutura contra possíveis cheias. O alteamento pode estar previsto no projeto original da barragem, mas na maioria dos casos trata-se de um novo estudo, com a barragem em operação, devendo-se verificar as novas condições de fluxo, efeitos na estabilidade de taludes e na resposta da barragem a carregamentos estáticos e sísmicos, estes principalmente em regiões de alta sismicidade, como no sul do Peru, onde se enontram a barragem de terra de Viña Blanca, aqui considerada. Nesta dissertação o método dos elementos finitos e o método de equilíbrio limite, isolada ou conjuntamente, são empregados para análise estática e dinâmica destas barragens de terra considerando diversas opções de alteamento, como a construção de muros parapeito, muros de gabião, solo compactado, solo reforçado com geotêxteis e solo reforçado com revestimento de concreto. De estudos de perigo sísmico efetuados no local das barragens, selecionou-se o valor de aceleração horizontal máximo para ser utilizado nos registros de aceleração ocorridos nos terremotos de Lima (1974) e de Moquegua (2001). As análises numéricas efetuadas mostram que as opções de alteamento consideradas não alteram significativamente as condições de segurança das barragens existentes, tanto do ponto de vista hidráulico como da estabilidade de taludes e resposta dinâmica durante a incidência de terremotos.<br>One of the activities related to dam constructions involves the raising of the existent structure, normally done with the objective of increasing the water storage capacity of the reservoirs, improving the safety factor of the embankment slopes or to ensure a better protection against possible water flooding. The raising of an earth dam can be predicted in the original dam plan, but in mostly situations consists of a new design, with the dam fully operational, where the effects of a new dam height and reservoir level should be assessed with respect to flow conditions, stability of the embankment soil slopes and the response of the revised structure under static and seismic loads, mainly in highly seismic regions, as in the South of Peru where the earth dam of Viña Blanca, herein studied, was constructed. In this dissertation, the finite element method and the limit equilibrium method were used for the static and dynamic analyses of these earth dams, in their original geometry as well after dam raising with reinforced soil, compacted soil and concrete or gabion structures, among other options. From studies of seismic risk analyses carried out at the dam sites, the value of maximum horizontal acceleration equal to 0.4g was chosen to be used as the peak acceleration in the Lima (1974) and Moquegua (2001) acceleration time histories. The numerical results indicate that all dam raising options investigated in this work do not affect the safety conditions of the dams significantly, either under the point of view of the hydraulic behavior as well as soil slope stabilities or the dynamic response of the earth dams to seismic loads.

The research uses CFD to investigate the internal flow of two hypersonic engine inlets: the Hypersonic Research Engine (HRE), a dual-mode ramjet/scramjet, and the Sustained Hypersonic Flight Experiment (SHyFE), a ramjet developed by QinetiQ. Various interactions are considered, namely shock-expansion, shock-shock and shock-boundary layer interactions. To isolate the different interactions, both inviscid and viscous turbulent computations are considered. For the HRE, axisymmetric computations are performed at Mach numbers of 5, 6 and 7, consistent with ground testing conditions used by NASA. The HRE was designed to cruise at a range of Mach numbers; for a given set of freestream flow conditions, dramatically different internal flow characteristics have been found depending on whether the engine arrived at the flow conditions through either acceleration or deceleration. CFD surface data and throat profiles have been compared to, and agree well with, experimental data obtained by NASA. Two flow conditions are investigated for the SHyFE inlet. Firstly, the self-starting characteristics of the SHyFE intake are examined, where the effect of increased internal compression is considered. The findings show undesirable wave interactions, which lead to flow non-uniformities, and decreased shock stabilization properties have adverse effects on the performance of the engine. Secondly, the effect of freestream incidence on the inlet is examined. The SHyFE engine is designed to cruise at a mean incidence of between 2° and 3°, however, it is conceivable that the engine will, at times, operate at 5°. Fully three dimensional computations are performed at an angle of attack of 5° where the resulting flows show that Mach reflections on the inner surface of the cowl can lead to shock-detachment, as well as showing that shock-boundary layer interactions on the centrebody can cause centrebody flow separation which can unstart the engine.

Part one of the thesis contains analysis of the methods of computation in navigation. We start with loxodromic navigation and, although this subject is well documented, we make a positive attempt to analyse the subject matter using the methods of differential geometry. We then turn to the problem of shortest path curves and set out an alternative method of solving the problem of navigating along the arc of a great circle on the surface of a sphere which can be generalised to other surfaces. In particular, a contribution made by the thesis is an analysis of the problem of navigating along shortest path geodesic arcs on the surface of a spheroid which introduces an algebraic representation of the geodesic curve by solving Clairaut' s equation using a cylindrical transformation. We are therefore able to compute the the coordinates of the positions of points along the path of the geodesic and the length of arc along the path of the. geodesic curve can then be computed step by step between these points by a numerical method - the Direct Cubic Spline method which was first introduced by this author in the Bachelor of Philosophy thesis in 1982 and is developed further in part 2 of this thesis. We apply this method also to the special problem of computing the distance along the shortest path between nearly antipodean points on the surface of a spheroid. We analyse the problem of computing an observer's position on the surface of the Earth using astronomical observations and show how a position locus is distorted when it is transferred over the surface. We offer a method of computing the observer's position by a series of observations of a single astronomical body taken over a comparitively short period of time and which does not necessarily include an observation at the tide of meridian passage of the body. In part two of the thesis we discuss the Direct Spline approximation to integrals and give some error bounds. The Direct Cubic Spline is a step by step method of fitting a cubic spline to the integral of a function directly which computes the value of the integral of the function step by step between the data points which need not be evenly spaced. We extend the idea to splines of higher order and give the formula from which they may be obtained but we show that, except for a particular special form of the direct quartic spline. the higher order direct splines do not yield algorithm for computing integrals which are as efficient as the Direct Cubic Spline.

A numerical investigation of dielectric barrier discharge aimed to simulate the electro hydro dynamic interaction is presented. A discharge drift diffusive model according to the Townsend avalanche is described and used to duplicate the plasma kinetics of a DBD actuator. The discharge characteristics dependence upon dielectric material and applied voltage are simulated and the EHD force field according to a simplified approach is presented and discussed. The coupling of DBD results with a fluid dynamic code is also shown. Finally, a new non invasive diagnostic technique for EHD interaction based on Schlieren imaging is computationally validated.

A numerical investigation of dielectric barrier discharge aimed to simulate the electro hydro dynamic interaction is presented. A discharge drift diffusive model according to the Townsend avalanche is described and used to duplicate the plasma kinetics of a DBD actuator. The discharge characteristics dependence upon dielectric material and applied voltage are simulated and the EHD force field according to a simplified approach is presented and discussed. The coupling of DBD results with a fluid dynamic code is also shown. Finally, a new non invasive diagnostic technique for EHD interaction based on Schlieren imaging is computationally validated.

Some real-world applications involve situations where different physical phenomena acting on very different time scales occur simultaneously. The partial differential equations (PDEs) governing such situations are categorized as "stiff" PDEs. Stiffness is a challenging property of differential equations (DEs) that prevents conventional explicit numerical integrators from handling a problem efficiently. For such cases, stability (rather than accuracy) requirements dictate the choice of time step size to be very small. Considerable effort in coping with stiffness has gone into developing time-discretization methods to overcome many of the constraints of the conventional methods. Recently, there has been a renewed interest in exponential integrators that have emerged as a viable alternative for dealing effectively with stiffness of DEs. Our attention has been focused on the explicit Exponential Time Differencing (ETD) integrators that are designed to solve stiff semi-linear problems. Semi-linear PDEs can be split into a linear part, which contains the stiffest part of the dynamics of the problem, and a nonlinear part, which varies more slowly than the linear part. The ETD methods solve the linear part exactly, and then explicitly approximate the remaining part by polynomial approximations. The first aspect of this project involves an analytical examination of the methods' stability properties in order to present the advantage of these methods in overcoming the stability constraints. Furthermore, we discuss the numerical difficulties in approximating the ETD coefficients, which are functions of the linear term of the PDE. We address ourselves to describing various algorithms for approximating the coefficients, analyze their performance and their computational cost, and weigh their advantages for an efficient implementation of the ETD methods. The second aspect is to perform a variety of numerical experiments to evaluate the usefulness of the ETD methods, compared to other competing stiff integrators, for integrating real application problems. The problems considered include the Kuramoto-Sivashinsky equation, the nonlinear Schrödinger equation and the nonlinear Thin Film equation, all in one space dimension. The main properties tested are accuracy, start-up overhead cost and overall computation cost, since these parameters play key roles in the overall efficiency of the methods.

A range of numerical linear algebra problems that arise in finite element-based structural analysis are considered. These problems were encountered when implementing the finite element method in the software package Oasys GSA. We present novel solutions to these problems in the form of a new method for error detection, algorithms with superior numerical effeciency and algorithms with scalable performance on parallel computers. The solutions and their corresponding software implementations have been integrated into GSA's program code and we present results that demonstrate the use of these implementations by engineers to solve real-world structural analysis problems.

Graph analysis uses graph data collected on a physical, biological, or social phenomena to shed light on the underlying dynamics and behavior of the agents in that system. Many fields contribute to this topic including graph theory, algorithms, statistics, machine learning, and linear algebra. This dissertation advances a novel framework for dynamic graph analysis that combines numerical, statistical, and streaming algorithms to provide deep understanding into evolving networks. For example, one can be interested in the changing influence structure over time. These disparate techniques each contribute a fragment to understanding the graph; however, their combination allows us to understand dynamic behavior and graph structure. Spectral partitioning methods rely on eigenvectors for solving data analysis problems such as clustering. Eigenvectors of large sparse systems must be approximated with iterative methods. This dissertation analyzes how data analysis accuracy depends on the numerical accuracy of the eigensolver. This leads to new bounds on the residual tolerance necessary to guarantee correct partitioning. We present a novel stopping criterion for spectral partitioning guaranteed to satisfy the Cheeger inequality along with an empirical study of the performance on real world networks such as web, social, and e-commerce networks. This work bridges the gap between numerical analysis and computational data analysis.

Research on interfacial phenomena has a long history, which has attracted tremendous interest in recent years. One of the most successful tools is the phase-field approach. As phase-field models usually involve very complex dynamics and it is nontrivial to obtain analytical solutions, numerical methods have played an important role in various simulations. This thesis is mainly devoted to developing accurate, efficient and robust numerical methods and the related numerical analysis for three representative phase-field models, namely the Allen-Cahn equation, the Cahn-Hilliard equation and the thin film models. The first part of this thesis is mainly concentrated on the stability analysis for these three models, with particular attention to the Allen-Chan equation. We have established three stability criterion, i.e., nonlinear energy stability, L∞-stability and L2-stability. As shared by most phase-field models, one of the intrinsic properties of the Allen- Cahn and the Cahn-Hilliard equations is that they satisfy a nonlinear stability re- lationship, usually expressed as a time-decreasing free energy functional. We have studied several stabilized temporal discretization for both the Allen-Cahn and the Cahn-Hilliard equations so that the relevant nonlinear energy stability can be pre- served. The corresponding temporal discretization schemes are linear and are of second-order accuracy. We also apply multi-step implicit-explicit methods to ap- proximate the Allen-Cahn equation. We demonstrate that by suitably choosing the parameters in multi-step implicit-explicit methods the nonlinear energy stability can be preserved. Apart from studying the energy stability for the Allen-Cahn equation, we also establish the numerical maximum principle for some fully discretized schemes. We further extend our analysis technique to the fractional-in-space Allen-Cahn equation. A more general Allen-Cahn-type equation with a nonlinear degenerate mobility and a logarithmic free energy is also considered. The third stability under investigation is the L2-stability. We prove that the con- tinuum Allen-Cahn equation satisfies a uniform Lp-stability. Furthermore, we show that both semi-discretized Fourier Galerkin and Fourier collocation methods can in- herit this stability for p = 2, i.e., L2-stability. Based on the established L2-stability, we accomplish the spectral convergence estimate for the Fourier Galerkin methods. We adopt the second-order Strang splitting schemes in the temporal direction with Fourier collocation methods to demonstrate the uniform L2-stability in the fully dis- cretized scheme. Another contribution of this thesis is to propose a p-adaptive spectral deferred correction methods for the long time simulations for all three models. We develop a high-order accurate and energy stable scheme to simulate the phase-field models by combining the semi-implicit spectral deferred correction method and first-order stabilized semi-implicit schemes. It is found that the accuracy improvement may affect the overall energy stability. To compromise the accuracy and stability, a local p- adaptive strategy is proposed to enhance the accuracy by sacrificing some local energy stability in an acceptable level. Numerical results demonstrate the high effectiveness of our proposed numerical strategy. Keywords: Phase-field models, Allen-Cahn equations, Cahn-Hilliard equations, thin film models, nonlinear energy stability, maximum principle, L2-stability, adaptive simulations, stabilized semi-implicit schemes, finite difference, Fourier spectral meth- ods, spectral deferred correction methods, convex splitting

Thesis (M.S.)--West Virginia University, 2005.<br>Title from document title page. Document formatted into pages; contains x, 150 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 72-73).

In this thesis, bifurcation phenomena in dynamical systems with cosymmetry and Hamiltonian structure were investigated using numerical methods. Several numerical continuation methods and test functions for detecting bifurcations were presented. The numerical results for various examples are given using a numerical bifurcation analysis toolbox.

Indentation is a practical and easy method, therefore, is a preferred method of material characterization. Main aim of this thesis study is to determine anisotropic properties of metals by indentation tests. The basic property of the indenter used in the finite element analyses and experiments is that it is specific to this process. Thesis includes studies on optimization of the indenter geometry, analyses of effects of friction coefficient, multiple indentations, tilting of the indenter and clamping of the specimen on force-displacements curves during indentation by finite element analyses. This study also includes finite element analyses of compression tests where these experiments have been necessary to prove anisotropic behavior of the specimen material. In addition to compression, tension tests are done to have a reference for indentation tests. On the other hand, the upper bound method which is an analytical solution is applied on the assumption of plane strain indentation.

Nowadays, field development strategy has become increasingly dependent on the results of reservoir simulation models. Reservoir studies demand fast and efficient results to make investment decisions that require a reasonable trade off between accuracy and simulation time. One of the suitable options to fulfill this requirement is streamline reservoir simulation technology, which has become very popular in the last few years. Streamline (SL) simulation provides an attractive alternative to conventional reservoir simulation because SL offers high computational efficiency and minimizes numerical diffusion and grid orientation effects. However, streamline methods have weaknesses incorporating complex physical processes and can also suffer numerical accuracy problems. The main objective of this research is to evaluate the numerical accuracy of the latest SL technology, and examine the influence of different factors that may impact the solution of SL simulation models. An extensive number of numerical experiments based on sensitivity analysis were performed to determine the effects of various influential elements on the stability and results of the solution. Those experiments were applied to various models to identify the impact of factors such as mobility ratios, mapping of saturation methods, number of streamlines, time step sizes, and gravity effects. This study provides a detailed investigation of some fundamental issues that are currently unresolved in streamline simulation.

In this thesis, a digitalization method with nite word-length (resolu- tion N ) of a given stable analog controller guaranteeing the minimum dierence in terms of frequency responses is treated. The challenge has consisted in nding a relevant frequency responses mismatch met- ric and in relating it to the word niteness issue. The analog con- troller is represented in modal state-space form and digitalized with a stability-maintaining approximation (ramp invariance) for dierent sampling periods. It results in digital controllers with block diago- nal transition matrices whose coecients (poles) are inside the unit circle. The format is chosen to match the poles dynamical range. The matrix is then coded and the mismatch measure allows for the selection of the "best" poles coded controllers. The remaining ma- trices are then scaled and coded for these selected controllers. The measure is computed for each of them. The procedure nally gives the "optimal" coded controller. This algorithm is shown to perform well and better that a simple rounding after the analog controller discretization phase. iii

In a first part, we are interested in the behavior of a system of Stochastic PDEs with two time-scales- more precisely, we focus on the approximation of the slow component thanks to an efficient numerical scheme. We first prove an averaging principle, which states that the slow component converges to the solution of the so-called averaged equation. We then show that a numerical scheme of Euler type provides a good approximation of an unknown coefficient appearing in the averaged equation. Finally, we build and we analyze a discretization scheme based on the previous results, according to the HMM methodology (Heterogeneous Multiscale Method). We precise the orders of convergence with respect to the time-scale parameter and to the parameters of the numerical discretization- we study the convergence in a strong sense - approximation of the trajectories - and in a weak sense - approximation of the laws. In a second part, we study a method for approximating solutions of parabolic PDEs, which combines a semi-lagrangian approach and a Monte-Carlo discretization. We first show in a simplified situation that the variance depends on the discretization steps. We then provide numerical simulations of solutions, in order to show some possible applications of such a method.

A convertible bond is a nancial instrument which has both an equity part and a xed-income part. The pricing of nancial securities has for quite obvious reasons become extensively studied in the past decades. In this paper we study the Black-Scholes model, based on the equity value, where the equity is modelled by geometric brownian motion. We introduce the pricing of nancial securities in general by Partial Differential Equation (PDE) approach. We continue by studying the convertible bond with a call feature, which is a derivative of the stock price. Our model leads to a free boundary problem together with a parabolic partial differential equation. We also give some analytical results on uniqueness and monotonicity of the solutions. This paper ends with a numerical study of the solutions for different bond features.

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO<br>COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR<br>PROGRAMA DE EXCELENCIA ACADEMICA<br>Na atualidade, devido à necessidade de realizar obras de engenharia de grande porte em ambiente urbano, o engenheiro defronta-se muitas vezes com o desafio de executar escavações profundas, as quais devem ser projetadas para serem estáveis e limitar deformações a níveis aceitáveis. Uma escavação profunda estável é aquela cujas paredes não colapsam e o seu fundo não experimenta levantamento descontrolado. Deformações do solo podem afetar construções vizinhas, vias urbanas e outras instalações públicas, com consequências que dependem tanto da magnitude quanto do padrão do movimento do solo ao redor da escavação. A previsão do comportamento de uma escavação profunda envolve análises tanto de estabilidade quanto de deformação. Análises de estabilidade podem em geral ser feitas através de métodos de equilíbrio limite, mas as análises de deformações, por outro lado, são mais difíceis de serem previstas, necessitando do auxilio de métodos numéricos. Nesta dissertação, o comportamento de escavações profundas é investigado numericamente pelo método de elementos finitos, com especial atenção à ocorrência de deformações, para cuja previsão requer-se a utilização de modelos constitutivos que representem muitos dos aspectos de comportamento de solos reais.<br>At the present, due to the need for engineering works of large scale in urban environment, the engineer is confronted often with the challenge of performing deep excavations, which should be designed to be stable and with acceptable levels of deformations. A stable deep excavation is that, whose walls do not collapse and its base does not experience uncontrolled heave. Soil deformations can affect neighboring buildings, urban roads and other public facilities, with consequences that depend on both the magnitude and the pattern of the movement of the soil around the excavation. The prediction of the behavior of a deep excavation therefore involves stability analysis as well as deformation analysis. Stability analysis can generally be made through limit equilibrium methods, but deformation analyses are more difficult to predict, requiring the assistance of numerical methods. In this dissertation the behavior of deep excavations is numerically investigated by the finite element method, with special attention to the occurrence of deformations, for this prediction requires the use of constitutive models that represent many aspects of behavior of real soils.

Thesis (MEng) -- Stellenbosch University, 2014<br>ENGLISH ABSTRACT: The electrodynamics of arbitrary, point-to-plane electrospraying geometries, were investigated in this research both analytically and numerically. Electrospraying is the process during which particles of sizes in the nanometre range are simultaneously generated and charged by means of an applied electrostatic field. A high electrostatic potential is applied to a conductive capillary needle, which overcomes the force exerted by the liquid surface tension. One of the primary limitations of this process are corona discharges. The effect of corona discharges have not been studied quantitatively, even though it is frequently reported in the electrospraying literature. The main objective of this research was to understand the corona discharge thresholds associated with electrospraying. Previously, only one theoretical, and two empirical investigations studied this phenomenon, over a time period of approximately forty years. It was clear that by better understanding these thresholds, electrospraying could be applied much more effectively. A corona discharge threshold model is proposed, using either a numerical or analytical model for the calculation of polarization fields. When compared with the experimental results of other researchers, both these two models have average relative percentage errors of approximately 15%. These are the first models proposed in the literature for the calculation of electrospraying corona thresholds. A new method to determine surface tension using electrospraying is described theoretically. In addition to this method, the calculation of corona discharge thresholds have various applications. For example, the dynamics of electrostatic ion thrusters are much better described, powder production by means of electrospraying can be optimised, and pattern generation using pulsed electrospraying cone-jets can be optimised as well.<br>AFRIKAANSE OPSOMMING: Die elektrodinamika van arbitrêre, punt-tot-vlak elektrosproei geometrieë was beide analities en numeries ondersoek in hierdie tesis. Dit is die proses waartydens nanodeeltjies gelyktydig gegenereer en elektrostaties gelaai word. Deur 'n hoë elektriese potensiaal aan te lê tot 'n geleidende kapillêr, is dit moontlik om die krag van die oppervlakte spanning te oorkom. Een van die primêre beperkings van elektrosproei is corona ontladings. Die effek van corona ontladings was nog nie kwantitatief bestudeer nie, selfs al word dit dikwels rapporteer in die elektrosproei literatuur. Die primêre doel van hierdie navorsing was om die corona ontlading drempels te verstaan wat geassosieer word met elektrosproei. In die verlede was daar nog net een teoretiese, en twee empiriese ondersoeke gewees wat hierdie verskynsel bestuur het, oor 'n tydperk van ongeveer veertig jaar. Dit was duidelik dat deur 'n beter begrip te hê van hierdie elektrosproei drempels, kan hierdie proses baie meer doeltreffend toegepas word. In hierdie tesis word 'n corona ontlading drempel model voorgestel, wat gebruik maak van 'n analitiese of numeriese model om die polarisasie velde te bereken. Wanneer vergelyk met die resultate van ander navorsers, het beide die modelle 'n gemiddelde relatiewe persentasie fout van ongeveer 15%. Hierdie is die eerste modelle wat voorgestel word vir die berekening van corona ontlading drempels. Deur gebruik te maak van elektrosproei, word 'n nuwe metode ook voorgestel om die oppervlakte spanning te bereken. In byvoeging tot hierdie, het die berekening van corona ontlading drempels vele ander toepassings. As 'n voorbeeld, die dinamika van elektrostatiese ioon stuwers word beter beskryf, en poeier produksie deur middel van elektrosproei kan optimeer word.

碩士<br>義守大學<br>電機工程學系<br>89<br>In this thesis we developed numerical software to analyze the characteristics of the optical fiber. Our results are consistent with those of published papers. We also calculated dispersions of many optical fiber profiles including dispersion-shifted fiber and dispersion-flattened fiber. Furthermore, we discussed the relationship between parameters of fibers and dispersions. Finally, we analyzed the properties of the acoustics-induced fiber Bragg grating reflector by using the phase-matching conditions. Both the contra- and co- directional coupling mechanisms are applied to illustrate the transmission and reflection spectrum of the device. We also discussed the relationship between device parameters and the device.

In the study of partial differential equations (PDEs) one rarely finds an analytical solution. But a numerical solution can be found using different methods such as finite difference, finite element, etc. The main issue using such numerical methods is whether the numerical solution will converge to the “real" analytical solution and if so how fast will it converge as we shrink the discretization parameter. In the first part of this thesis discrete versions of well known inequalities from analysis are used in proving the convergence of certain numerical methods for the one dimensional Poisson equation with Dirichlet boundary conditions and with Neumann boundary conditions. A matrix is monotone if its inverse exists and is non-negative. In the second part of the thesis we will show that finite difference discretization of two PDEs result in monotone matrices. The monotonicity property will be used to demonstrate stability of certain methods for the Poisson and Biharmonic equations. Convergence of all schemes is also shown.<br>October 2016

Reflux condensation occurs in a vertical tube when there is an upward core flow of vapour (or gas-vapour mixture) and a downward flow of the liquid film. The understanding of this condensation configuration is crucial in the design of reflux condensers and in loss-of-coolant safety analyses in nuclear power plant steam generators. A range of modelling approaches exists for co-current film condensation from gas-vapour mixtures in parallel-plate channels and tubes. These methods are based on marching from the inlet down the tube and do not apply to the reflux condensation. In this research, however, a two-dimensional two-phase model was developed that solves the steady, full elliptic governing equations in both the film and the gas-vapour core flow on a non-orthogonal mesh that dynamically adapts to the phase interface. Gas-vapour shear and heat and mass transfer at the interface were accounted for fundamentally. This modelling is a big step ahead of current capabilities by removing the limitations of previous reflux condensation models which inherently cannot account for the detailed local balances of shear, mass, and heat transfer at the phase interface. The model was developed and applied for co-current and counter-current flows in vertical parallel-plate channels, followed by vertical tubes. In each stage, the model results were compared against the available experimental and numerical data for validation purposes. A wide range of boundary conditions and geometries have been studied to examine the details of co-current and counter-current condensation phenomena. Velocity, temperature, pressure, and gas mass fraction profiles along with the axial variation of various parameters such as local Nusselt number, film thickness, interface and centre-line temperature and gas mass fraction are presented in parametric studies.<br>February 2017

碩士<br>國立屏東科技大學<br>車輛工程系碩士班<br>92<br>This thesis plans to use a code that combines fluid and acoustic simulation. The parameters in the near field of sound are computed by FLUENT that is a CFD solver with various turbulent models and the far field sound intensity, that are governed by the Ffowcs-Williams and Hawkins equation (FW-H). There are three steps in the task. The first stage is to demonstrate the numerical package to predict aero-noise of the incompressible flow, through simple 3D geometry sample models are used for validation. The second stage is to reform the numerical package accuracy and efficient, and apply to side view mirror and road vehicle. The present simulation results are found to be in good agreement with the published experimental results. At engineering environment, DES model is an acceptable technique except LES model predicting aero-noise. Aero-noise of rough side view mirror is louder than smooth one.The results indicate that the installation of an appropriately angled tail fin reduces the aerodynamic lift coefficient, and hence improves the vertical stability of the vehicle under high speed driving conditions. Furthermore, it is shown that the use of an end plate reduces the noise profile behind the car. The present steady also proposed an optimal tail fin / end plate configuration, which possesses enhanced aerodynamic characteristics and aero-acoustic properties, and hence improves driver safety and comfort.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1986.<br>Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 226-231).