Recently the computational speed and the functions of the numerical methods are advancing rapidly. It is the future trend that using the computational fluid dynamics (CFD) to perform simulation for making up the experimental deficiency, reducing the risk, improving the quality of the product, and saving the cost of research and development. A two-dimensional simulation was employed to study the melt/air and melt/solid interface shapes of the miniature molten zone formed in the laser-heated pedestal growth (LHPG) system. Using non-orthogonal body-fitting grid system with control-volume finite difference method, the interface shape can be determined both efficiently and accurately. During stable growth, the dependence of the molten-zone length and shape on the heating CO2 laser is examined in detail under both the maximum and the minimum allowed powers with various growth speeds. The effect of gravity for the miniature molten zone is also simulated, which reveals the possibility for a horizontally oriented LHPG system. Such a horizontal system is good for the growth of long crystal fibers. After comparing with the shape of the molten zone in terms of the experiment and the analysis of the simulation shown as above. Heat transfer and fluid flow in the LHPG system are analyzed near the deformed interfaces. The global thermal distributions of the crystal fiber, the melt, and the source rod are described by temperature and its axial gradient within length of ~10 mm. As compared with the growth of bulk crystal of several centimeters in dimension, natural convection drops six orders in magnitude due to smaller melt volume; therefore, conduction rather than convection determines the temperature distribution in the molten zone. Moreover, thermocapillary convection rather than mass-transfer convection becomes dominant. The symmetry and mass flow rate of double eddy pattern are significantly influenced by the molten-zone shape due to the diameter reduction and the large surface-tension-temperature coefficient in the order of 10-4~10-3. According to the analysis shown as above, the results could be further extended for the analysis of the concentration profile and study of horizontal growth.

Disertacijoje išnagrinėta trečiosios eilės vienmatė pseudoparabolinė lygtis su dviejų tipų nelokaliosiomis sąlygomis. Šiems uždaviniams spręsti sudarytos skirtuminės schemos, kurių stabilumas tiriamas, taikant skirtuminių operatorių su nelokaliosiomis sąlygomis spektro struktūrą. Trečiosios eilės vienmatėms ir dvimatėms pseudoparabolinėms lygtims su integralinėmis sąlygomis sudarytos ir išnagrinėtos padidinto tikslumo skirtuminės schemos. Išnagrinėta dvimatė pseudoparabolinė lygtis su nelokaliosiomis integralinėmis sąlygomis viena koordinačių kryptimi. Tokiam uždaviniui spręsti pritaikytas ir išnagrinėtas lokaliai vienmatis metodas, ištirtos šio metodo stabilumo sąlygos. Taip pat išnagrinėtos: trisluoksnės skirtuminės schemos vienmatei pseudoparabolinei lygčiai su įvairiomis, taip pat ir nelokaliosiomis, sąlygomis; trisluoksnių išreikštinių skirtuminių schemų stabilumo sąlygos. / The thesis analyzes the third-order one-dimensional pseudoparabolic equations with two types of nonlocal conditions. The stability of difference schemes for this problem was studied using the analysis of the spectrum structure of a difference operator with nonlocal conditions. The analysis of the increased accuracy difference schemes for third-order one-dimensional and two-dimensional pseudoparabolic equations with integral conditions has been made. The thesis considers a two-dimensional pseudoparabolic equation with nonlocal integral conditions in one coordinate direction. This problem was solved by a locally one-dimensional method. The stability of a difference scheme has been investigated based on the spectrum structure. The doctoral disertation investigates three-layer difference schemes for one-dimensional pseudoparabolic equations with various, including nonlocal, conditions. Also, the conditions for the stability of three-layer explicit difference schemes have been explored.

Public-Private-Partnership (P3) is gaining momentum as the delivery method for the development of public infrastructure. These projects, however, are exposed to economic risks. If the private parties are not comfortable with the level of the risks, they would not participate in the project and, as a result, the infrastructure will most likely not be realized. As an incentive for participation in the P3 project, private parties are sometimes offered guarantees against unfavorable economic risks. Therefore, the valuation of these guarantees is essential for deciding whether or not to participate in the project. While previous works focused on the valuation of guarantees, the incorporation of credit risk in the value of the P3 projects and the guarantees has been neglected. The effect of credit risk can be taken into account by using the rigorous Credit Value Adjustment method (CVA). CVA is a computationally demanding method that the valuation methods currently in the literature are not capable of handling. This research offers a novel approach for the valuation of guarantees and P3 projects which is computationally superior to the existing methods. Because of this computational efficiency, CVA can be implemented to account for credit risk. For the development of this method, a continuous stochastic differential equation (SDE) is derived from the forecasted curve of an economic risk. Using the SDE, the partial differential equation (PDE) governing the value of the guarantees will be derived. Then, the PDE will be solved using Finite Difference Method (FDM). A new feature for this method is that it obtains exercise strategies for the Australian guarantees. The present work extends the literature by providing a valuation method for the cases that multiple risks affect P3 projects. It also presents an approach for the valuation of the Asian style guarantee, a contract which reimburses the private party based on the average of risk factor. Finally, a hypothetical case study illustrates the implementation of the FDM-based valuation method and CVA to obtain the value of the P3 project and the guarantees adjusted for the counterparty credit risk.

Public-Private-Partnership (P3) is gaining momentum as the delivery method for the development of public infrastructure. These projects, however, are exposed to economic risks. If the private parties are not comfortable with the level of the risks, they would not participate in the project and, as a result, the infrastructure will most likely not be realized. As an incentive for participation in the P3 project, private parties are sometimes offered guarantees against unfavorable economic risks. Therefore, the valuation of these guarantees is essential for deciding whether or not to participate in the project. While previous works focused on the valuation of guarantees, the incorporation of credit risk in the value of the P3 projects and the guarantees has been neglected. The effect of credit risk can be taken into account by using the rigorous Credit Value Adjustment method (CVA). CVA is a computationally demanding method that the valuation methods currently in the literature are not capable of handling. This research offers a novel approach for the valuation of guarantees and P3 projects which is computationally superior to the existing methods. Because of this computational efficiency, CVA can be implemented to account for credit risk. For the development of this method, a continuous stochastic differential equation (SDE) is derived from the forecasted curve of an economic risk. Using the SDE, the partial differential equation (PDE) governing the value of the guarantees will be derived. Then, the PDE will be solved using Finite Difference Method (FDM). A new feature for this method is that it obtains exercise strategies for the Australian guarantees. The present work extends the literature by providing a valuation method for the cases that multiple risks affect P3 projects. It also presents an approach for the valuation of the Asian style guarantee, a contract which reimburses the private party based on the average of risk factor. Finally, a hypothetical case study illustrates the implementation of the FDM-based valuation method and CVA to obtain the value of the P3 project and the guarantees adjusted for the counterparty credit risk.

In order to handle turbulent flow problems, one equation turbulence models are implemented in to a previously developed explicit, Reynolds averaged Navier-Stokes solver. Discretization of Navier-Stokes solver is based on cell-vertex finite volume formulation combined with single step Lax-Wendroff numerical method which is second order accurate in space. Turbulent viscosity is calculated by using one equation Spalart-Allmaras and Baldwin-Barth turbulence transport equations. For the discretization of Spalart-Allmaras and Baldwin-Barth equations, both finite volume scheme which is used for Navier-Stokes equation in this work and explicit finite difference discretization method are used. In order to increase the convergence rate of the solver, local time stepping technique is applied. Stabilization of non-physical oscillations resulting from the numerical scheme is maintained by adding second and fourth order artificial smoothing terms. Three test cases are considered. In order to validate the accuracy of the Navier-Stokes solver, solver is tested over a laminar flat plate. The results are compared with analytical solutions. Later, in order to check the performance of the turbulence models, turbulent flow over flat plate and turbulent transonic flow over NACA-0012 airfoil are handled. For turbulent flow over flat plate obtained results are compared with analytical and empirical solutions, whereas for transonic turbulent flow obtained results are compared with numerical and experimental solutions.

This thesis has made two significant contributions to the field of reactive crystallisation. First, new data from batch cooling crystallisation and semi-batch reactive crystallisation experiments of mono-ammonium phosphate (MAP) were obtained to describe the key factors that influence crystal nucleation and growth rates, crystal size distribution (CSD), and crystal shape. The second contribution is the development of a numerical scheme for solving the population balance equations, which can be used to describe the evolution of CSD during the crystallisation process. This scheme combines the finite difference method with a wavelet method, and is the first reported application of this approach for crystallisation modelling and simulation. / Experiments into the batch cooling crystallisation of MAP were conducted both with and without seed crystals. The effects of key factors such as cooling rate, initial level of supersaturation and seeding technique, including seed concentration and seed size, on the real time supersaturation, final CSD, crystal yield and crystal shape were investigated. It was found that a seed concentration of 20-30% effectively suppressed nucleation. The growth and nucleation rate were estimated by using an isothermal seeded batch approach and their parameters were calculated by non-linear optimisation techniques. / The second series of experiments involved the semi-batch reactive crystallisation of MAP. Both single-feed and dual-feed systems were investigated. In the single-feed arrangement, an ammonia solution was fed into a charge of phosphoric acid. In the dual-feed system, phosphoric acid and ammonia solution were fed into a charge of saturated MAP solution. The molar ratio of the reactants, initial supersaturation, presence or absence of seed crystals, initial MAP concentration, reactants’ flow rate, feeding time and stirring speed were varied, and the effects upon the real time supersaturation, final CSD, crystal yield, crystal shape and solution temperature were measured. X-ray diffraction analysis showed that MAP can be produced in both the single-feed and dual-feed arrangements. For the single feed system, the N/P mole ratio controlled the degree of reaction and the CSD of the product. Di-ammonium phosphate (DAP) was not be observed in the single-feed system due to its high solubility. In the dual-feed system, a seeded solution with slow feed addition, moderate stirring speed and a low initial supersaturation provided the most favourable conditions for generating a desirable supersaturation profile, and thus obtaining a product with good CSD and crystal shape. / A comparative numerical study was undertaken in order to evaluate the existing numerical schemes for solving the population balance equations (PBE) that describe crystallisation. Several analytical solutions to the PBE were used to benchmark the following numerical schemes: Upwind Finite Difference, Biased Upwind Finite Difference, Orthogonal Collocation with Finite Elements, and Wavelet Orthogonal Collocation. The Wavelet Finite Difference (WFD) method has been applied here for the first time for solving PBE problems. The WFD scheme was adapted to solve the batch cooling and the semi-batch reactive crystallisation models, and the solutions were validated against experimental data that we obtained. / In summary, the experimental data provide an improved understanding of MAPreaction and crystallisation mechanisms. The adaptability of the WFD method has beendemonstrated validating the two crystallisation systems, and this should help extendthe application of wavelet-based solutions beyond crystallisation processes and intomore diverse areas of chemical engineering.

Submitted by Raniere Barreto (raniere.barros@ufvjm.edu.br) on 2018-05-07T19:57:19Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) victor_luiz_batista_aguiar.pdf: 2508647 bytes, checksum: 87d9596e1cf8e04b90b12aa373fa4748 (MD5) / Approved for entry into archive by Rodrigo Martins Cruz (rodrigo.cruz@ufvjm.edu.br) on 2018-05-08T12:30:45Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) victor_luiz_batista_aguiar.pdf: 2508647 bytes, checksum: 87d9596e1cf8e04b90b12aa373fa4748 (MD5) / Made available in DSpace on 2018-05-08T12:30:45Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) victor_luiz_batista_aguiar.pdf: 2508647 bytes, checksum: 87d9596e1cf8e04b90b12aa373fa4748 (MD5) Previous issue date: 2017 / A crescente demanda pela utiliza??o de recursos h?dricos em nossa sociedade faz com que seja necess?ria a amplia??o dos meios de capta??o de ?gua. As ?guas subterr?neas surgem como alternativa ? capta??o convencional, tendo em vista que o armazenamento da ?gua nas camadas inferiores do solo garanta a manuten??o da qualidade deste recurso, uma vez que passam por processos naturais de filtra??o. Nesse sentido, a caracteriza??o dos sistemas aqu?feros por meio de modelagem matem?tica surge como ferramenta de aux?lio na compreens?o do fluxo subterr?neo local. A presente pesquisa tem como principal objetivo analisar o comportamento hidrogeol?gico do munic?pio de Te?filo Otoni - MG, com base em simula??es oriundas de modelos conceituais, pelo m?todo das diferen?as finitas. Por meio de simula??o de propriedades influentes no transporte h?drico como taxa de recarga, vaz?o de bombeamento de po?os e condutividade hidr?ulica, foi poss?vel observar as varia??es nos valores m?ximos e m?nimos de carga hidr?ulica em pontos discretizados nos limites do modelo conceitual proposto, auxiliando assim no entendimento dos resultados obtidos. / Disserta??o (Mestrado Profissional) ? Programa de P?s-Gradua??o em Tecnologia, Sa?de e Sociedade, Universidade Federal dos Vales do Jequitinhonha e Mucuri, 2017. / The growing demand for the use of water resources in our society makes it necessary to expand the means of abstraction of water. Groundwater emerges as an alternative to conventional abstraction, owing to the storage of water in the lower layers of the soil ensures the maintenance of the quality of this resource, since they undergo natural filtration processes. In this sense, the characterization of the aquifer systems through mathematical modeling appears as a tool to aid in the understanding of the local underground flow. The present research has as main objective to analyze the hydrogeological behavior of Te?filo Otoni - MG, based on simulations derived from conceptual models, by the finite difference method. By means of simulation of influent properties in the water transport as recharge rate, well pumping rate and hydraulic conductivity it was possible to observe the variations in the maximum and minimum values of hydraulic head in discretized points within the limits of the proposed conceptual model, thus helping in the understanding of the results obtained.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Apresenta-se uma abordagemnumérica para ummodelo que descreve a formação de padrões por sputtering iônico na superfície de ummaterial. Esse processo é responsável pela formação de padrões inesperadamente organizados, como ondulações, nanopontos e filas hexagonais de nanoburacos. Uma análise numérica de padrões preexistentes é proposta para investigar a dinâmica na superfície, baseada em ummodelo resumido em uma equação anisotrópica amortecida de Kuramoto-Sivashinsky, em uma superfície bidimensional com condições de contorno periódicas. Apesar de determinística, seu caráter altamente não-linear fornece uma rica gama de resultados, sendo possível descrever acuradamente diferentes padrões. Umesquema semi implícito de diferenças finitas com fatoração no tempo é aplicado na discretização da equação governante. Simulações foram realizadas com coeficientes realísticos relacionados aos parâmetros físicos (anisotropias, orientação do feixe, difusão). A estabilidade do esquema numérico foi analisada por testes de passo de tempo e espaçamento de malha, enquanto a verificação do mesmo foi realizada pelo Método das Soluções Manufaturadas. Ondulações e padrões hexagonais foram obtidos a partir de condições iniciais monomodais para determinados valores do coeficiente de amortecimento, enquanto caos espaço-temporal apareceu para valores inferiores. Os efeitos anisotrópicos na formação de padrões foramestudados, variando o ângulo de incidência. / A numerical approach is presented for amodel describing the pattern formation by ion beam sputtering on a material surface. This process is responsible for the appearance of unexpectedly organized patterns, such as ripples, nanodots, and hexagonal arrays of nanoholes. A numerical analysis of preexisting patterns is proposed to investigate surface dynamics, based on a model resumed in an anisotropic damped Kuramoto-Sivashinsky equation, in a two dimensional surface with periodic boundary conditions. While deterministic, its highly nonlinear character gives a rich range of results, making it possible to describe accurately different patterns. A finite-difference semi-implicit time splitting scheme is employed on the discretization of the governing equation. Simulations were conducted with realistic coefficients related to physical parameters (anisotropies, beam orientation, diffusion). The stability of the numerical scheme is analyzed with time step and grid spacing tests for the pattern evolution, and the Method ofManufactured Solutions has been used to verify the scheme. Ripples and hexagonal patterns were obtained from amonomodal initial condition for certain values of the damping coefficient, while spatiotemporal chaos appeared for lower values. The anisotropy effects on pattern formation were studied, varying the angle of incidence.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Apresenta-se uma abordagemnumérica para ummodelo que descreve a formação de padrões por sputtering iônico na superfície de ummaterial. Esse processo é responsável pela formação de padrões inesperadamente organizados, como ondulações, nanopontos e filas hexagonais de nanoburacos. Uma análise numérica de padrões preexistentes é proposta para investigar a dinâmica na superfície, baseada em ummodelo resumido em uma equação anisotrópica amortecida de Kuramoto-Sivashinsky, em uma superfície bidimensional com condições de contorno periódicas. Apesar de determinística, seu caráter altamente não-linear fornece uma rica gama de resultados, sendo possível descrever acuradamente diferentes padrões. Umesquema semi implícito de diferenças finitas com fatoração no tempo é aplicado na discretização da equação governante. Simulações foram realizadas com coeficientes realísticos relacionados aos parâmetros físicos (anisotropias, orientação do feixe, difusão). A estabilidade do esquema numérico foi analisada por testes de passo de tempo e espaçamento de malha, enquanto a verificação do mesmo foi realizada pelo Método das Soluções Manufaturadas. Ondulações e padrões hexagonais foram obtidos a partir de condições iniciais monomodais para determinados valores do coeficiente de amortecimento, enquanto caos espaço-temporal apareceu para valores inferiores. Os efeitos anisotrópicos na formação de padrões foramestudados, variando o ângulo de incidência. / A numerical approach is presented for amodel describing the pattern formation by ion beam sputtering on a material surface. This process is responsible for the appearance of unexpectedly organized patterns, such as ripples, nanodots, and hexagonal arrays of nanoholes. A numerical analysis of preexisting patterns is proposed to investigate surface dynamics, based on a model resumed in an anisotropic damped Kuramoto-Sivashinsky equation, in a two dimensional surface with periodic boundary conditions. While deterministic, its highly nonlinear character gives a rich range of results, making it possible to describe accurately different patterns. A finite-difference semi-implicit time splitting scheme is employed on the discretization of the governing equation. Simulations were conducted with realistic coefficients related to physical parameters (anisotropies, beam orientation, diffusion). The stability of the numerical scheme is analyzed with time step and grid spacing tests for the pattern evolution, and the Method ofManufactured Solutions has been used to verify the scheme. Ripples and hexagonal patterns were obtained from amonomodal initial condition for certain values of the damping coefficient, while spatiotemporal chaos appeared for lower values. The anisotropy effects on pattern formation were studied, varying the angle of incidence.

Para uma análise mais exata do sistema estrutural, é necessário um estudo do comportamento interativo entre as diversas partes que o compõe, entre eles, destaca-se a interação entre os elementos de fundação e o maciço de solos. Neste trabalho foram desenvolvidas formulações numéricas para a análise da interação estaca-solo via acoplamento entre diferentes métodos numéricos: método dos elementos de contorno, método dos elementos finitos e método das diferenças finitas. As estacas podem estar submetidas a carregamentos horizontais, verticais e momentos aplicados em seu topo. Nestas formulações foram utilizadas, além das equações integrais de deslocamentos, as equações de suas derivadas, levando a um grau maior de singularidade, porém permitindo a adoção de aproximações mais refinadas para os deslocamentos e tensões ao longo da estaca. Todos os deslocamentos e suas derivadas referentes à estaca foram compatibilizados com os correspondentes do solo. Desenvolvidas as formulações, feito o devido acoplamento entre eles, foram analisados exemplos, que foram comparados com os resultados obtidos por outros pesquisadores, demonstrando sua validade. / For a more accurate analysis of the structural system, it is necessary to study the interactive behavior between the various parts that compose it, among them, there is the interaction between the foundation elements and massive soil. In this work, numerical formulations were developed for the analysis of pile-soil interaction by coupling between different numerical methods: the boundary element method, finite element method and finite difference method. Piles can be subjected to horizontal loads, vertical and moments applied on its top. In these formulations were used in addition to the displacement integral equations, the equations of their derivatives, leading to a higher degree of uniqueness but allowing the adoption of more sophisticated approaches to displacements and contact tractions along the pile. All displacements and their derivatives relating to the pile were matched with the corresponding soil. Developed formulations made due coupling between them were analyzed examples, which were compared with results obtained by other authors, demonstrating its validity.