The objective of this PhD research programme is to investigate the effectiveness of a finite volume method incorporating radial basis functions for simulating nonlinear transport processes. The finite volume method is the favoured numerical technique for solving the advection-diffusion equations that arise in transport simulation. The method transforms the original problem into a system of nonlinear, algebraic equations through the process of discretisation. The accuracy of this discretisation determines to a large extent the accuracy of the final solution. A new method of discretisation is presented that employs radial basis functions (rbfs) as a means of local interpolation. When combined with Gaussian quadrature integration methods, the resulting finite volume discretisation leads to accurate numerical solutions without the need for very fine meshes, and the additional overheads they entail. The resulting nonlinear, algebraic system is solved efficiently using a Jacobian-free Newton-Krylov method. By employing the new method as an extension of existing shape function-based approaches, the number of nonlinear iterations required to obtain convergence can be reduced. Furthermore, information obtained from these iterations can be used to increase the efficiency of subsequent rbf-based iterations, as well as to construct an effective parallel reconditioner to further reduce the number of nonlinear iterations required. Results are presented that demonstrate the improved accuracy offered by the new method when applied to several test problems. By successively refining the meshes, it is also possible to demonstrate the increased order of the new method, when compared to a traditional shape function basedmethod. Comparing the resources required for both methods reveals that the new approach can be many times more efficient at producing a solution of a given accuracy.

Thesis (M.S.) -- Mississippi State University. Department of Electrical and Computer Engineering. / Title from title screen. Includes bibliographical references.

Made available in DSpace on 2014-12-17T14:55:44Z (GMT). No. of bitstreams: 1 AllanDGA_DISSERT.pdf: 1992696 bytes, checksum: 87d8b1dbc6fe4df6df2f85f90481f9be (MD5) Previous issue date: 2010-01-13 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / This study aims to seek a more viable alternative for the calculation of differences in images of stereo vision, using a factor that reduces heel the amount of points that are considered on the captured image, and a network neural-based radial basis functions to interpolate the results. The objective to be achieved is to produce an approximate picture of disparities using algorithms with low computational cost, unlike the classical algorithms / O presente trabalho visa buscar uma alternativa mais vi?vel para o c?lculo das disparidades em imagens de vis?o est?reo, utilizando um fator de salto que reduz a quantidade de pontos que s?o considerados da imagem capturada, e uma rede neural baseada em fun??es de base radial para interpolar os resultados obtidos. O objetivo a ser alcan?ado ? produzir uma imagem de disparidades aproximada da real com algoritmos de baixo custo computacional, diferentemente dos algoritmos tradicionais

Made available in DSpace on 2014-12-17T14:55:44Z (GMT). No. of bitstreams: 1 JoilsonBAR_DISSERT.pdf: 5903616 bytes, checksum: bee0d51eb1c54833e1d9a19364c80c76 (MD5) Previous issue date: 2010-07-30 / The present work describes the use of a mathematical tool to solve problems arising from control theory, including the identification, analysis of the phase portrait and stability, as well as the temporal evolution of the plant s current induction motor. The system identification is an area of mathematical modeling that has as its objective the study of techniques which can determine a dynamic model in representing a real system. The tool used in the identification and analysis of nonlinear dynamical system is the Radial Basis Function (RBF). The process or plant that is used has a mathematical model unknown, but belongs to a particular class that contains an internal dynamics that can be modeled.Will be presented as contributions to the analysis of asymptotic stability of the RBF. The identification using radial basis function is demonstrated through computer simulations from a real data set obtained from the plant / O presente trabalho descreve a utiliza??o de uma ferramenta matem?tica na solu??o de problemas decorrentes da teoria de controle, incluindo a identifica??o, a an?lise do retrato de fase e a estabilidade, bem como a evolu??o temporal da planta de corrente do motor de indu??o. A identifica??o de sistemas ? uma ?rea da modelagem matem?tica que tem como objetivo o estudo de t?cnicas que possam determinar um modelo din?mico na representa??o de um sistema real. A ferramenta utilizada na identifica??o e an?lise do sistema din?mico n?o linear ser? as Fun??es de Base Radial (RBF). O processo ou a planta que ser? utilizada possui um modelo matem?tico desconhecido, mas pertence a uma determinada classe que cont?m uma din?mica interna que pode ser modelada. Ser? apresentada como contribui??es a an?lise da estabilidade assint?tica da RBF. A identifica??o utilizando Fun??es de Base Radial ? demonstrada atrav?s de simula??es computacionais a partir de um conjunto de dados reais obtidos da planta de corrente do motor de indu??o

As a result of the increase in accessibility of computational resources and the increase in the power of the computers during the last two decades, designers are able to create computer models to simulate the behavior of a complex products. To address global competitiveness, companies are forced to optimize their designs and products. Optimizing the design needs several runs of computationally expensive simulation models. Therefore, using metamodels as an efficient and sufficiently accurate approximate of the simulation model is necessary. Radial basis functions (RBF) is one of the several metamodeling methods that can be found in the literature. The established approach is to add a bias to RBF in order to obtain a robust performance. The a posteriori bias is considered to be unknown at the beginning and it is defined by imposing extra orthogonality constraints. In this thesis, a new approach in constructing RBF with the bias to be set a priori by using the normal equation is proposed. The performance of the suggested approach is compared to the classic RBF with a posteriori bias. Another comprehensive comparison study by including several modeling criteria, such as problem dimension, sampling technique and size of samples is conducted. The studies demonstrate that the suggested approach with a priori bias is in general as good as the performance of RBF with a posteriori bias. Using the a priori RBF, it is clear that the global response is modeled with the bias and that the details are captured with radial basis functions. Multi-objective optimization and the approaches used in solving such problems are briefly described in this thesis. One of the methods that proved to be efficient in solving multi-objective optimization problems (MOOP) is the strength Pareto evolutionary algorithm (SPEA2). Multi-objective optimization of a disc brake system of a heavy truck by using SPEA2 and RBF with a priori bias is performed. As a result, the possibility to reduce the weight of the system without extensive compromise in other objectives is found. Multi-objective optimization of material model parameters of an adhesive layer with the aim of improving the results of a previous study is implemented. The result of the original study is improved and a clear insight into the nature of the problem is revealed.

Philosophiae Doctor - PhD / Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided. / South Africa

Au cours de cette thèse, un modèle potentiel résolvant les équations d’Euler-Zakharov a été développé dans le but de simuler la propagation de vagues et d’états de mer irréguliers et multi-directionnels, du large jusqu’à la côte, sur des bathymétries variables. L’objectif est de représenter les effets non-linéaires et dispersifs le plus précisément possible pour des domainescôtiers bidimensionnels (dans le plan horizontal) de l’ordre de quelques kilomètres.La version 1DH initiale du modèle, résolvant le problème aux limites de Laplace à l’aide de schémas aux différences finies d’ordre élevé dans la direction horizontale combinés à une approche spectrale sur la verticale, a été améliorée et validée. L’implémentation de conditions aux limites de type Dirichlet et Neumann pour générer des vagues dans le domaine a été étudiée en détail. Dans la pratique, une zone de relaxation a été utilisée en complément deces conditions pour améliorer la stabilité du modèle.L’expression analytique de la relation de dispersion a été établie dans le cas d’un fond plat. Son analyse a montré que la représentation des effets dispersifs s’améliorait significativement avec l’augmentation de la résolution sur la direction verticale (i.e. avec le degré maximal de la basede polynômes de Tchebyshev utilisée pour projeter le potentiel des vitesses sur la verticale).Une étude de convergence menée pour des ondes solitaires modérément à fortement non-linéaires a confirmé la convergence exponentielle avec la résolution verticale grâce à l’approche spectrale, ainsi que les convergences algébriques en temps et en espace sur l’horizontale avec des ordres d’environ 4 (ou plus) en accord avec les schémas numériques utilisés.La comparaison des résultats du modèle à plusieurs jeux de données expérimentales a démontré les capacités du modèle à représenter les effets non-linéaires induits par les variations de bathymétrie, notamment les transferts d’énergie entre les composantes harmoniques, ainsi que la représentation précise des propriétés dispersives. Une formulation visco-potentielle a également été implémentée afin de prendre en compte les effets visqueux induits par la dissipation interne et le frottement sur le fond. Cette formulation a été validée dans le cas d’une faible viscosité avec un fond plat ou présentant une faible pente.Dans le but de représenter des champs de vagues 2DH, le modèle a été étendu en utilisant une discrétisation non-structurée (par nuage de points) du plan horizontal. Les dérivées horizontales ont été estimées à l’aide de la méthode RBF-FD (Radial Basis Function-Finite Difference), en conservant l’approche spectrale sur la verticale. Une étude numérique de sensibilité a été menée afin d’évaluer la robustesse de la méthode RBF-FD, en comparant différents types de RBFs, avec ou sans paramètre de forme et l’ajout éventuel d’un polynôme. La version 2DH du modèle a été utilisée pour simuler deux expériences en bassin, validant ainsi l’approche choisie et démontrant son applicabilité pour simuler la propagation 3D des vagues faisant intervenir des effets non-linéaires. Dans le but de réduire le temps de calcul et de pouvoir appliquer le code à des simulations sur de grands domaines, le code a été modifié pour utiliser le solveur linéaire direct en mode parallèle / In this work, a potential flow model based on the Euler-Zakharov equations was developed with the objective of simulating the propagation of irregular and multidirectional sea states from deep water conditions to the coast over variable bathymetry. A highly accurate representation of nonlinear and dispersive effects for bidimensional (2DH) nearshore and coastal domains on the order of kilometers is targeted.The preexisting 1DH version of the model, resolving the Laplace Boundary Value problem using a combination of high-order finite difference schemes in the horizontal direction and a spectral approach in the vertical direction, was improved and validated. The generation of incident waves through the implementation of specific Dirichlet and Neumann boundary conditions was studied in detail. In practice, these conditions were used in combination witha relaxation zone to improve the stability of the model.The linear dispersion relation of the model was derived analytically for the flat bottom case. Its analysis showed that the accuracy of the representation of dispersive effects improves significantly by increasing the vertical resolution (i.e. the maximum degree of the Chebyshev polynomial basis used to project the potential in the vertical). A convergence study conducted for moderate to highly nonlinear solitary waves confirmed the exponential convergence in the vertical dimension owing to the spectral approach, and the algebraic convergence in time and in space (horizontal dimension) with orders of about 4 (or higher) in agreement with the numerical schemes used.The capability of the model to represent nonlinear effects induced by variable bathymetry, such as the transfer of energy between harmonic components, as well as the accurate representation of dispersive properties, were demonstrated with comparisons to several experimental data sets. A visco-potential flow formulation was also implemented to take into account viscous effects induced by bulk viscosity and bottom friction. This formulation was validated inthe limit of small viscosity for mild slope bathymetries.To represent 2DH wave fields in complex nearshore domains, the model was extended using an unstructured discretization (scattered nodes) in the horizontal plane. The horizontal derivatives were estimated using the RBF-FD (Radial Basis Function - Finite Difference) method, while the spectral approach in the vertical remained unchanged. A series of sensitivity tests were conducted to evaluate numerically the robustness of the RBF-FD method, including a comparison of a variety of RBFs with or without shape factors and augmented polynomials. The 2DH version of the model was used to simulate two wave basin experiments, validating the approach and demonstrating the applicability of this method for 3D wave propagation, including nonlinear effects. As an initial attempt to improve the computational efficiency ofthe model for running simulations of large spatial domains, the code was adapted to use a parallelized direct linear solver

This thesis deals with neural modeling of electromagnetic field inside small aircrafts, witch can contain composite materials in their construction. Introduction to neural networks and its application in EMC of small airplanes is discussed in the first part of the text. In the second part of this thesis we design a simple EM model of small airplane. The airplane is simulated by two parallel dielectric layers (the left-hand side wall and the right hand side wall of the airplane). The layers are put into a rectangular metallic waveguide terminated by the absorber in order to simulate the illumination of the airplane by the external wave (both of the harmonic nature and pulse one). Numerical analyses are performed to search the relations between the distribution of an electromagnetic field inside the aircraft and electric parameters of model walls. The results of numerical analyses are used to train two types of neural network. In this way we can obtain accurate continuous model of electromagnetic field inside the aircraft. For the comparison with neural networks a multi-dimensional cubic spline interpolation is provided also. Neural classifiers are also investigated. We use them for classification of imaginary composite materials in terms of EMC. The nearest neighbour algorithm is applied as a classic approach to problem of classification.

Philosophiae Doctor - PhD / Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston's volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.

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