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.

No description available.

In this study, the effect of a passive trapped-vortex cell on lift to drag (L/D) ratio of an FFA-W3-301 airfoil is studied. The upper surface of the airfoil was modified to incorporate a cavity defined by seven parameters. The L/D ratio of the airfoil is modeled using a radial basis function metamodel. This model is used to find the optimal design parameter values that give the highest L/D. The numerical results indicate that the L/D ratio is most sensitive to the position on an airfoil’s upper surface at which the cavity starts, the position of the end point of the cavity, and the vertical distance of the cavity end point relative to the airfoil surface. The L/D ratio can be improved by locating the cavity start point at the point of separation for a particular angle of attack. The optimal cavity shape (o19_aXX) is also tested for a NACA0024 airfoil.

Numerical methods for solving partial differential equations are commonplace in the engineering community and their popularity can be attributed to the rapid performance improvement of modern workstations and desktop computers. The ubiquity of computer technology has allowed all areas of engineering to have access to detailed thermal, stress, and fluid flow analysis packages capable of performing complex studies of current and future designs. The rapid pace of computer development, however, has begun to outstrip efforts to reduce analysis overhead. As such, most commercially available software packages are now limited by the human effort required to prepare, develop, and initialize the necessary computational models. Primarily due to the mesh-based analysis methods utilized in these software packages, the dependence on model preparation greatly limits the accessibility of these analysis tools. In response, the so-called meshless or mesh-free methods have seen considerable interest as they promise to greatly reduce the necessary human interaction during model setup. However, despite the success of these methods in areas demanding high degrees of model adaptability (such as crack growth, multi-phase flow, and solid friction), meshless methods have yet to gain notoriety as a viable alternative to more traditional solution approaches in general solution domains. Although this may be due (at least in part) to the relative youth of the techniques, another potential cause is the lack of focus on developing robust methodologies. The failure to approach development from a practical perspective has prevented researchers from obtaining commercially relevant meshless methodologies which reach the full potential of the approach. The primary goal of this research is to present a novel meshless approach called MIMS (Model Integrated Meshless Solver) which establishes the method as a generalized solution technique capable of competing with more traditional PDE methodologies (such as the finite element and finite volume methods). This was accomplished by developing a robust meshless technique as well as a comprehensive model generation procedure. By closely integrating the model generation process into the overall solution methodology, the presented techniques are able to fully exploit the strengths of the meshless approach to achieve levels of automation, stability, and accuracy currently unseen in the area of engineering analysis. Specifically, MIMS implements a blended meshless solution approach which utilizes a variety of shape functions to obtain a stable and accurate iteration process. This solution approach is then integrated with a newly developed, highly adaptive model generation process which employs a quaternary triangular surface discretization for the boundary, a binary-subdivision discretization for the interior, and a unique shadow layer discretization for near-boundary regions. Together, these discretization techniques are able to achieve directionally independent, automatic refinement of the underlying model, allowing the method to generate accurate solutions without need for intermediate human involvement. In addition, by coupling the model generation with the solution process, the presented method is able to address the issue of ill-constructed geometric input (small features, poorly formed faces, etc.) to provide an intuitive, yet powerful approach to solving modern engineering analysis problems.

Engine mapping is the process of empirically modelling engine behaviour as a function of adjustable engine parameters, predicting the output of the engine. The aim is to calibrate the electronic engine controller to meet decreasing emission requirements and increasing fuel economy demands. Modern engines have an increasing number of control parameters that are having a dramatic impact on time and e ort required to obtain optimal engine calibrations. These are further complicated due to transient engine operating mode. A new model-based transient calibration method has been built on the application of hierarchical statistical modelling methods, and analysis of repeated experiments for the application of engine mapping. The methodology is based on two-stage regression approach, which organise the engine data for the mapping process in sweeps. The introduction of time-dependent covariates in the hierarchy of the modelling led to the development of a new approach for the problem of transient engine calibration. This new approach for transient engine modelling is analysed using a small designed data set for a throttle body inferred air ow phenomenon. The data collection for the model was performed on a transient engine test bed as a part of this work, with sophisticated software and hardware installed on it. Models and their associated experimental design protocols have been identi ed that permits the models capable of accurately predicting the desired response features over the whole region of operability. Further, during the course of the work, the utility of multi-layer perceptron (MLP) neural network based model for the multi-covariate case has been demonstrated. The MLP neural network performs slightly better than the radial basis function (RBF) model. The basis of this comparison is made on assessing relevant model selection criteria, as well as internal and external validation ts. Finally, the general ability of the model was demonstrated through the implementation of this methodology for use in the calibration process, for populating the electronic engine control module lookup tables.

L’animation faciale est l’un des points clés dans le réalisme des scènes 3D qui mettent en scène des personnages virtuels. Ceci s’explique principalement par les raisons suivantes : le visage et les nombreux muscles qui le composent permettent de générer une multitude d’expressions ; ensuite, notre faculté de perception nous permet de détecter et d’analyser ses mouvements les plus fins. La complexité de ce domaine se retrouve dans les approches existantes par le fait qu’il est très difficile de créer une animation de qualité sans un travail manuel long et fastidieux. Partant de ce constat, cette thèse a pour but de développer des techniques qui contribuent au processus de création d’animations faciales. Trois thèmes sont principalement abordés. Le premier concerne la paramétrisation du visage pour l’animation. La paramétrisation a pour but de définir des moyens de contrôle pour pouvoir déformer et animer le visage. Le second s’oriente sur l’animation, et plus particulièrement sur le transfert d’animation. Le but est de proposer une méthode qui permette d’animer le visage d’un personnage à partir de données variées. Ces données peuvent être issues d’un système de capture de mouvement, ou bien elles peuvent être obtenues à partir de l’animation d’un personnage virtuel qui existe déjà. Enfin, nous nous sommes concentrés sur les détails fins liés à l’animation comme les rides. Bien que ces rides soient fines et discrètes, ces déformations jouent un rôle important dans la perception et l’analyse des émotions. C’est pourquoi nous proposons une technique d’acquisition mono-caméra et une méthode à base de poses références pour synthétiser dynamiquement les détails fins d’animation sur le visage. L’objectif principal des méthodes proposées est d’offrir des solutions afin de faciliter et d’améliorer le processus de création d’animations faciales réalistes utilisées dans le cadre d’applications en temps réel. Nous nous sommes particulièrement concentrés sur la facilité d’utilisation et sur la contrainte du temps réel. De plus, nous offrons la possibilité à l’utilisateur ou au graphiste d’interagir afin de personnaliser sa création et/ou d’améliorer les résultats obtenus / Facial animation is one of the key points of the realism of 3D scenes featuring virtual humans. This is due to several reasons : face and the many muscles that compose it can generate a multitude of expressions ; then, our faculty of perception provides us a great ability to detect and analyze its smallest variations. This complexity is reflected in existing approaches by the fact that it is very difficult to create an animation without a long and a tedious manual work. Based on these observations, this thesis aims to develop techniques that contribute to the process of creating facial animation. Three main themes have been addressed. The first concerns the rigging issue of a virtual 3D face for animation. Rigging aims at defining control parameters in order to deform and animate the face. The second deals with the animation, especially on the animation retargeting issue. The goal is to propose a method to animate a character’s face from various data. These data can be obtained from a motion capture system or from an existing 3D facial animation. Finally, we focus on animation finescale details like wrinkles. Although these are thin and discreet, their deformations play an important part in the perception and analysis of emotions. Therefore we propose a monocular acquisition technique and a reference pose based method to synthetise dynamically animation fine details over the face. The purpose is to propose methods to facilitate and improve the process of creating realistic facial animations for interactive applications. We focused on ease to use in addition to the real-time aspect. Moreover, we offer the possibility to the user or graphist to interact in order to personalize its creation and/or improve the results

A Otimização Multidisciplinar em Projeto (em inglês, Multidisciplinary Design Optimization - MDO) é uma ferramenta de projeto importante e versátil e seu uso está se expandindo em diversos campos da engenharia. O foco desta metodologia é unir disciplinas envolvidas no projeto para que trabalhem suas variáveis concomitantemente em um ambiente de otimização, para obter soluções melhores. É possível utilizar MDO em qualquer fase do projeto, seja a fase conceitual, preliminar ou detalhada, desde que os modelos numéricos sejam ajustados às necessidades de cada uma delas. Este trabalho descreve o desenvolvimento de um código de MDO para o projeto conceitual de asas flexíveis de aeronaves, com restrição quanto ao fenômeno denominado flutter. Como uma ferramenta para o projetista na fase conceitual, os modelos numéricos devem ser razoavelmente precisos e rápidos. O intuito deste estudo é analisar o uso de metamodelos para a previsão do flutter de asas de aeronaves no código de MDO, ao invés de um modelo convencional, o que pode alterar significativamente o custo computacional da otimização. Para este fim são avaliados três técnicas diferentes de metamodelagem, que foram escolhidas por representarem duas classes básicas de metamodelos, a classe de métodos de interpolação e a de métodos de aproximação. Para representá-las foram escolhidos o método de interpolação por funções de base radial e o método de redes neurais artificiais, respectivamente. O terceiro método, que é considerado um método híbrido dos dois anteriores, é chamado de redes neurais por funções de bases radiais e é uma tentativa de acoplar as características de ambos em um único metamodelo. Os metamodelos são preparados utilizando um código para solução aeroelástica baseado no método dos elementos finitos acoplado com um modelo aerodinâmico linear de faixas. São apresentados resultados de desempenho dos três metamodelos, de onde se pode notar que a rede neural artificial é a mais adequada para previsão de flutter. O processo de MDO é realizado com o uso de um algoritmo genético multi-objetivo baseado em não-dominância, cujos objetivos são a maximização da velocidade crítica de flutter e a minimização da massa estrutural. Dois estudos de caso são apresentados para avaliar o desempenho do código de MDO, revelando que o processo global de otimização realiza de fato a busca pela fronteira de Pareto. / The Multidisciplinary Design Optimization, MDO, is an important and versatile design tool and its use is spreading out in several fields of engineering. The focus of this methodology is to put together disciplines involved with the design to work all their variables concomitantly, at an optimization environment to obtain better solutions. It is possible to use MDO in any stage of the design process, that is in the conceptual, preliminary or detailed design, as long as the numerical models are fitted to the needs of each of these stages. This work describes the development of a MDO code for the conceptual design of flexible aircraft wings, with restrictions regarding the phenomenon called flutter. As a tool for the designer at the conceptual stage, the numerical models must be fairly accurate and fast. The aim of this study is to analyze the use of metamodels for the flutter prediction of aircraft wings in the MDO code, instead of a conventional model itself, what may affect significantly the computational cost of the optimization. For this purpose, three different metamodeling techniques have been evaluated, representing two basic metamodel classes, that are, the interpolation and the approximation class. These classes are represented by the radial basis function interpolation method and the artificial neural networks method, respectively. The third method, which is considered as a hybrid of the other two, is called radial basis function neural networks and is an attempt of coupling the features of both in single code. Metamodels are prepared using an aeroelastic code based on finite element model coupled with linear aerodynamics. Results of the three metamodels performance are presented, from where one can note that the artificial neural network is best suited for flutter prediction. The MDO process is achieved using a non-dominance based multi-objective genetic algorithm, whose objectives are the maximization of critical flutter speed and minimization of structural mass. Two case studies are presented to evaluate the performance of the MDO code, revealing that overall optimization process actually performs the search for the Pareto frontier.