Neste trabalho é desenvolvida uma metodologia baseada na aplicação do método dos momentos conjugado à aproximação por diferenças finitas, para análise no domínio do tempo da distribuição de correntes transitórias em meios condutores. O procedimento computacional permite que sejam considerados sistemas de condutores longos, com seção transversal genérica, discretizados em elementos filiformes. Assim, a equação integral que descreve o problema é substituída por uma representação matricial. No caso de meios não homogêneos, o método dos elementos finitos é usado para a obtenção da matriz que permite a solução do problema. Como o método proposto baseia-se na solução passo a passo no tempo, é interessante, para maior eficiência computacional, que as matrizes tenham sua dimensão reduzida, o que pode ser realizado por meio da Transformada Discreta Wavelet, como investigado no trabalho. Também é feita uma análise da estabilidade do procedimento computacional, para verificação das condições de aplicabilidade do mesmo. A validação do procedimento desenvolvido é feita a partir da comparação dos resultados obtidos para problemas com solução já conhecida por meio de medições ou por outros métodos de resolução. São apresentados resultados da aplicação do método proposto a alguns casos de interesse prático, na área de Compatibilidade Eletromagnética, destacando-se a análise do efeito de blindagem e o estudo de crosstalk em um sistema multicondutor. O procedimento desenvolvido ainda pode ser aprimorado com a utilização de elementos com formas mais gerais do que a filiforme proposta, permitindo sua extensão a problemas tridimensionais. Também pode ser estudado o pré-condicionamento da matriz a ser reduzida pela aplicação da Transformada Wavelet, a fim de permitir um melhor desempenho do processo computacional. / In this work, a methodology is developed, based on the application of the moment method associated to a finite difference approximation, for time domain analysis of transient current distribution in conducting media. The computational procedure allows that long conductors with any cross section can be considered, approximated by filamentary elements. Thus, the integral equation that describes the problem is substituted by a matrix representation. In the case of non homogeneous regions, the finite element method is used to obtain the matrix that allows the solution of the problem. As the proposed method works in a time step by step scheme, it is interesting, for increased computational efficiency, to reduce the size of the involved matrices. This can be done by application of the Discrete Wavelet Transform, as investigated in this work. An analysis of the numerical stability of the computational procedure is also done, aiming to establish its applicability conditions. Validation of the numerical procedure developed is done by comparing the results obtained for problems whose solution is known by measurement or by another method. Some problems of practical interest in Electromagnetic Compatibility were solved and the results are presented, concerning particularly shielding effects and multi conductor crosstalk. The developed procedure can be improved employing elements with more general shapes, besides the filamentary ones, allowing the consideration of three dimensional systems. Also, pre conditioning of the matrices before application of the Wavelet Transform can be studied, for a better performance of the method.

On s'interesse aux problèmes elliptiques avec coéficients variables à travers des interfaces intérieures. La solution et ses dérivées normales peuvent subir des variations significatives à travers les frontières intérieures. On présente une méthode compacte aux différences finies sur des maillages adaptés de type octree conçues pour une résolution en parallèle. L'idée principale est de minimiser l'erreur de troncature sur la discretisation locale, en fonction de la configuration du maillage, en rapprochant une convergence à l'ordre deux. On montrera des cas 2D et 3D des résultat liés à des applications concrètes. / We consider problems governed by a linear elliptic equation with varying coéficients across internal interfaces. The solution and its normal derivative can undergo significant variations through these internal boundaries. We present a compact finite-difference scheme on a tree-based adaptive grid that can be efficiently solved using a natively parallel data structure. The main idea is to optimize the truncation error of the discretization scheme as a function of the local grid configuration to achieve second order accuracy. Numerical illustrations relevant for actual applications are presented in two and three-dimensional configurations.

Escoamentos viscoelásticos não estacionários com superfícies livres são comuns em muitos processos industriais e diversas técnicas numéricas têm sido empregadas para reproduzir computacionalmente estes processos. A maioria dos modelos empregados utiliza equações diferenciais na definição do tensor de tensões. Porém, para alguns grupos de fluidos complexos, por exemplo, fluidos de Boger, os modelos integrais mostram-se mais capacitados em fornecer uma boa aproximação para os comportamentos não lineares desses fluidos. Este trabalho trata da solução numérica do modelo constitutivo integral KBKZ-PSM para escoamentos transientes bidimensionais com superfícies livres. O método numérico proposto é uma técnica numérica que utiliza diferenças finitas para simular escoamentos com superfícies livres na presença de paredes sólidas. As principais características do método numérico proposto são: solução das equações de conservação de quantidade de movimento e massa utilizando um método semi-implícito; a condição de contorno na superfície livre é acoplada à equação de Poisson, o que garante conservação de massa; a discretização do tempo t é realizada por uma nova técnica numérica; o tensor de Finger é calculado pelo método dos campos de deformação e avançado no tempo pelo método de Euler modificado. Essa nova técnica é verificada em escoamentos cisalhantes e elongacionais. Adicionalmente, uma solução analítica desenvolvida para escoamentos em canais bidimensionais é empregada para verificar e analisar a convergência do método proposto. Com relação a escoamentos com superfícies livres, a convergência é verificada por meio de refinamento de malha nas simulações de um jato incidindo sobre placa rígida e no problema do inchamento do extrudado. Finalmente, o método é aplicado para investigar os problemas jet buckling e inchamento do extrudado de fluidos KBKZ-PSM. / Unsteady viscoelastic free surface flows are common in many industrial processes and a variety of numerical techniques have been employed to simulate these flows. The majority of constitutive models employed are based on differential equations to define the extra stress tensor. However, for some complex fluids, for instance, Boger fluids, integral models are more adequate to approximate the nonlinear behaviour of these fluids. This work deals with the numerical solution of the integral constitutive model KBKZ-PSM for two-dimensional unsteady free surface flows. The proposed numerical method is a numerical technique that employs finite differences to simulate moving free surface flows that interact with solid walls. The main features of the method are: numerical solution of the momentum and mass equations by an implicit method; the pressure condition on the free surface is implicitly coupled with the Poisson equation for obtaining the pressure field from mass conservation; a novel scheme for defining the past times t is employed; the Finger tensor is calculated by the deformation fields method and is advanced in time by the modified Euler method. This new technique is verified by solving shear and uniaxial elongational flows. Moreover, an analytic solution for channel flows is obtained that is used in the verification and convergence analysis of the proposed methodology. For free surface flows, the assessment of convergence lies on the mesh refinement on the simulation of a jet impinging on a flat surface and the extrudade swell problem. Finally, the new method is applied to investigate the jet buckling phenomenon and extrudate swell of KBKZ-PSM fluids.

Landslides cause significant damage to property and human lives throughout the world. Rainfall is the most common triggering factor for the occurrence of landslides. This dissertation presents two novel methodologies for assessment of rainfall-triggered shallow landslide hazard. The first method focuses on using remotely sensed soil moisture and soil surface properties in developing a framework for real-time regional scale landslide hazard assessment while the second method is a deterministic approach to landslide hazard assessment of the specific sites identified during first assessment. In the latter approach, landslide inducing transient seepage in soil during rainfall and its effect on slope stability are modeled using numerical analysis. Traditionally, the prediction of rainfall-triggered landslides has been performed using pre-determined rainfall intensity-duration thresholds. However, it is the infiltration of rainwater into soil slopes which leads to an increase of porewater pressure and destruction of matric suction that causes a reduction in soil shear strength and slope instability. Hence, soil moisture, pore pressure and infiltration properties of soil must be direct inputs to reliable landslide hazard assessment methods. In-situ measurement of pore pressure for real-time landslide hazard assessment is an expensive endeavor and thus, the use of more practical remote sensing of soil moisture is constantly sought. In past studies, a statistical framework for regional scale landslide hazard assessment using remotely sensed soil moisture has not been developed. Thus, the first major objective of this study is to develop a framework for using downscaled remotely sensed soil moisture available on a daily basis to monitor locations that are highly susceptible to rainfall- triggered shallow landslides, using a well-structured assessment procedure. Downscaled soil moisture, the relevant geotechnical properties of saturated hydraulic conductivity and soil type, and the conditioning factors of elevation, slope, and distance to roads are used to develop an improved logistic regression model to predict the soil slide hazard of soil slopes using data from two geographically different regions. A soil moisture downscaling model with a proven superior prediction accuracy than the downscaling models that have been used in previous landslide studies is employed in this study. Furthermore, this model provides satisfactory classification accuracy and performs better than the alternative water drainage-based indices that are conventionally used to quantify the effect that elevated soil moisture has upon the soil sliding. Furthermore, the downscaling of soil moisture content is shown to improve the prediction accuracy. Finally, a technique that can determine the threshold probability for identifying locations with a high soil slide hazard is proposed. On the other hand, many deterministic methods based on analytical and numerical methodologies have been developed in the past to model the effects of infiltration and subsequent transient seepage during rainfall on the stability of natural and manmade slopes. However, the effects of continuous interplay between surface and subsurface water flows on slope stability is seldom considered in the above-mentioned numerical and analytical models. Furthermore, the existing seepage models are based on the Richards equation, which is derived using Darcy’s law, under a pseudo-steady state assumption. Thus, the inertial components of flow have not been incorporated typically in modeling the flow of water through the subsurface. Hence, the second objective of this study is to develop a numerical model which has the capability to model surface, subsurface and infiltration water flows based on a unified approach, employing fundamental fluid dynamics, to assess slope stability during rainfall-induced transient seepage conditions. The developed model is based on the Navier-Stokes equations, which possess the capability to model surface, subsurface and infiltration water flows in a unified manner. The extended Mohr-Coulomb criterion is used in evaluating the shear strength reduction due to infiltration. Finally, the effect of soil hydraulic conductivity on slope stability is examined. The interplay between surface and subsurface water flows is observed to have a significant impact on slope stability, especially at low hydraulic conductivity values. The developed numerical model facilitates site-specific calibration with respect to saturated hydraulic conductivity, remotely sensed soil moisture content and rainfall intensity to predict landslide inducing subsurface pore pressure variations in real time.

In mammalian hearts, initiation of the heartbeat occurs in a region of specialised pacemaker cells known as the sinoatrial node (SAN). The SAN is a highly complex spatially distributed structure which displays considerable cellular heterogeneity and is subject to complex electrotonic interactions with the surrounding atrial tissue. In this study, biophysically detailed ionic models of central and peripheral SAN pacemaker cells are described. These models are able to accurately reproduce experimental recordings of the membrane potential from central and peripheral SAN tissue. These models are used to investigate frequency synchronisation of electrically coupled cardiac pacemaker cells. Based on simulation results presented, it is proposed that cellular heterogeneity in the SAN plays an important role in achieving rhythm coordination and possibly contributes to the efficient activation of the surrounding atrial myocardium. This represents an important, previously unexplored, mechanism underlying pacemaker synchronisation and cardiac activation in vivo. A spatial-gradient model of action potential heterogeneity within the SAN is then formulated using a large-scale least squares optimisation technique. This model accurately reproduces the smooth spatial variation in action potential characteristics observed in the SAN. One and two dimensional models of the intact SAN are then formulated and three proposed models of SAN heterogeneity are investigated: 1) the discrete-region model, in which the SAN consists of a compact central region surrounded by a region of transitional pacemaker cells, 2) the gradient model, in which cells of the SAN exhibit a smooth variation in properties from the centre to the periphery of the SAN, and 3) the mosaic model, in which SAN and atrial cells are scattered throughout the SAN region with the proportion of atrial cells increasing towards the periphery. Simulation results suggest that the gradient model achieves frequency entrainment more easily than the other models of SAN heterogeneity. The gradient model also reproduces action potential waveshapes and a site of earliest activation consistent with experimental observations in the intact SAN. It is therefore proposed that the gradient model of SAN heterogeneity represents the most plausible model of SAN organisation.

An obvious reason for studying unsteady flows is the prediction of the effect of unsteady aerodynamic forces on a flight vehicle, since these effects tend to increase the likelihood of aeroelastic instabilities. This is a major concern in aerodynamic design of aircraft that operate in transonic regime, where the flows are characterised by the presence of adjacent regions of subsonic and supersonic flow, usually accompanied by weak shocks. It has been a common expectation that the numerical approach as an alternative to wind tunnel experiments would become more economical as computers became less expensive and more powerful. However even with all the expected future advances in computer technology, the cost of a numerical flutter analysis (computational aeroelasticity) for a transonic flight remains prohibitively high. Hence it is vitally important to develop an efficient, cheaper (in the sense of computational cost) and physically accurate flutter simulation tech nique which is capable of reproducing the data, which would otherwise be obtained from wind tunnel tests, at least to some acceptable engineering accuracy, and that it is essentially appropriate for industrial applications. This need motivated the present research work on exploring and developing efficient and physically accurate computational techniques for steady, unsteady and time-linearised calculations of transonic flows over an aircraft wing with moving shocks. This dissertation is subdivided into eight chapters, seven appendices and a bibliography listing all the reference materials used in the research work. The research work initially starts with a literature survey in unsteady transonic flow theory and calculations, in which emphasis is placed upon the developments in these areas in the last three decades. Chapter 3 presents the small disturbance theory for potential flows in the subsonic, transonic and supersonic regimes, including the required boundary conditions and shock jump conditions. The flow is assumed irrotational and inviscid, so that the equation of state, continuity equation and Bernoulli's equation formulated in Appendices A and B can be employed to formulate the governing fluid equation in terms of total velocity potential. Furthermore for transonic flow with free-stream Mach number close to unity, we show in Appendix C that the shocks that appear are weak enough to allow us to neglect the flow rotationality. The formulations are based on the main assumption that aerofoil slopes are everywhere small, and the flow quantities are small perturbations about their free-stream values. In Chapter 4, we developed an improved approximate factorisation algorithm that solves the two-dimensional steady subsonic small disturbance equation with nonreflecting far-field boundary conditions. The finite difference formulation for the improved algorithm is presented in Appendix D, with the description of the solver used for solving the system of difference equations described in Appendix E. The calculation of steady and unsteady nonlinear transonic flows over a realistic aerofoil are considered in Chapter 5. Numerical solution methods, based on the finite difference approach, for solving the two-dimensional steady and unsteady, general-frequency transonic small disturbance equations are presented, with the corresponding finite difference formulation described in Appendix F. The theories and solution methods for the time-linearised calculations, in the frequency and time domains, for the problem of unsteady transonic flow over a thin planar wing undergoing harmonic oscillation are presented in Chapters 6 and 7, respectively. The time-linearised calculations include the periodic shock motion via the shock jump correction procedure. This procedure corrects the solution values behind the shock, to accommodate the effect of shock motion, and consequently, the solution method will produce a more accurate time-linearised solution for supercritical flow. Appendix G presents the finite difference formulation of these time-linearised solution methods. The aim is to develop an efficient computational method for calculating oscillatory transonic aerodynamic quantities efficiently for use in flutter analyses of both two- and three-dimensional wings with lifting surfaces. Chapter 8 closes the dissertation with concluding remarks and future prospects on the current research work.

<p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">This thesis consists of two different parts, related to two different fields in mathematical physics: wave propagation and kinetic theory of gases. Various mathematical and computational problems for equations from these areas are treated.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">The first part is devoted to high order finite difference methods for the Helmholtz equation and the wave equation. Compact schemes with high order accuracy are obtained from an investigation of the function derivatives in the truncation error. With the help of the equation itself, it is possible to transfer high order derivatives to lower order or to transfer time derivatives to space derivatives. For the Helmholtz equation, a compact scheme based on this principle is compared to standard schemes and to deferred correction schemes, and the characteristics of the errors for the different methods are demonstrated and discussed. For the wave equation, a finite difference scheme with fourth order accuracy in both space and time is constructed and applied to a problem in discontinuous media.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">The second part addresses some problems related to kinetic equations. A direct simulation Monte-Carlo method is constructed for the Landau-Fokker-Planck equation, and numerical tests are performed to verify the accuracy of the algorithm. A formal derivation of the method from the Boltzmann equation with grazing collisions is performed. The linear and linearized Boltzmann collision operators for the hard sphere molecular model are studied using exact reduction of integral equations to ordinary differential equations. It is demonstrated how the eigenvalues of the operators are found from these equations, and numerical values are computed. A proof of existence of non-zero discrete eigenvalues is given. The ordinary diffential equations are also used for investigation of the Chapman-Enskog distribution function with respect to its asymptotic behavior.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p>

<p>This thesis consists of four papers on the following topics in analysis and probability: analysis on Wiener space, asymptotic properties of orthogonal polynomials, and convergence rates in the central limit theorem. The first paper gives lower bounds on the constants in the Meyer inequality from the Malliavin calculus. It is shown that both constants grow at least like <i>(p-1)</i>^-1 or like <i>p</i> when <i>p</i> approaches 1 or ∞ respectively. This agrees with known upper bounds. In the second paper, an extremal problem on Wiener chaos motivates an investigation of the <i>L</i>^p-norms of Hermite polynomials. This is followed up by similar computations for Charlier polynomials in the third paper. In both cases, the <i>L</i>^p-norms present a peculiar behaviour with certain threshold values of p, where the growth rate and the dominating intervals undergo a rapid change. The fourth paper analyzes a connection between probability and numerical analysis. More precisely, known estimates on the convergence rate of finite difference equations are "translated" into results on convergence rates of certain functionals in the central limit theorem. These are also extended, using interpolation of Banach spaces as a main tool. Besov spaces play a central role in the emerging results.</p>

<p>Laser beam control is important in many applications. Phase modulating spatial light modulators (SLMs) can be used to electronically alter the phase distribution of an optical wave-front and thus change the direction and shape of a laser beam. Physical constraints set limitations to the SLM and an ideal phase distribution can usually not be realised. In order to understand how such components can be used for non-mechanical beam control three nematic liquid crystal (NLC) SLMs have been thoroughly characterised and modelled.</p><p>The pixel structure and phase quantisation give a discrepancy between ideal and realised phase distributions. The impact on beam steering capability was examined by measurements and simulations of the intensity distribution in the far-field.</p><p>In two of the studied SLMs the pixel period was shorter than the thickness of the LC layer giving the optical phase shift. This results in a so-called “fringing field”, which was shown to degrade the phase modulation and couple light between polarisation modes. The deformation of the LC was simulated and a finite-difference time-domain (FDTD) algorithm was used to calculate how polarised light propagates through the optically anisotropic SLM.</p><p>Non-mechanical beam steering and tracking in an optical free-space communication link were demonstrated. Continual optimisation of the steering angle was achieved by feedback from a video camera.</p><p>The optical properties of the SLM in the time period right after a voltage update were studied. It was shown how light is redistributed between orders during the switching from one blazed grating to another. By appropriate choice of the blazed gratings the effects on the diffraction efficiency can be minimised.</p><p>The detailed knowledge of the SLM structure and its response to electronic control makes it possible to predict and optimise the device performance in future systems.</p>

A hybrid FEM/FDM algorithm for particle migration of quasi-steady flow in concentrated suspension materials is proposed in this study. This hybrid FEM/FDM algorithm in which the planar variables, such as pressure field, are described in terms of finite element method, and gapwise variables of temperature, density concentration and time derivatives are expressed by finite difference method. The particle concentration inhomogeneities can be predicted, which is ignored by the existing injection molding simulation packages. Simulation results indicated that powder concentration variation could be significant in practical processing in PIM. / Singapore-MIT Alliance (SMA)