Ce travail de thèse a pour objectif de fournir un outil de modélisation rapide de l'interaction d'une source électromagnétique 3D avec une pièce de géométrie ou de propriétés physiques complexes dans le domaine des basses fréquences (régime quasi-statique). La principale application est la simulation d'un procédé de Contrôle Non Destructif (CND) d'une pièce conductrice présentant une surface ou des propriétés physiques perturbées. La plateforme logicielle CIVA, comportant un module dédié à la simulation des procédés de CND par courants de Foucault intègre à l’heure actuelle des modèles semi-analytiques limités aux géométries canoniques : pièces planes, cylindriques. Afin de lever ce verrou, le formalisme des équations de Maxwell covariantes, déjà très utilisé dans le domaine optique pour la caractérisation des réseaux de diffraction (méthode des coordonnées curvilignes) est étendu au régime quasi-statique. L’utilisation d’un nouveau système de coordonnées curvilignes non-orthogonal associé à la géométrie de la pièce conduit à écrire très facilement et de manière analytique les conditions de passage aux interfaces de formes complexes. La résolution numérique des équations de Maxwell sous leur forme covariante est abordée par une approche modale qui repose sur le calcul préalable de solutions propres d’un système d’équations différentielles en absence de source. La représentation des composantes du champ électromagnétique à partir de deux fonctions de potentiels du second ordre (SOVP) ou potentiels de Hertz dans des systèmes de coordonnées canoniques est d’abord étendue au système de coordonnées curvilignes. On obtient alors les expansions modales des composantes covariantes et contra-variantes du champ électromagnétique. Les coefficients de ces expansions modales sont déterminés ensuite en introduisant le champ d’excitation et en imposant les conditions de passage adéquates entre les différents milieux. Cette approche est ensuite couplée d'une part à un algorithme récursif (les paramètres S) afin de prendre en compte la présence d'interfaces internes complexes dans la pièce, et d'autre part à une méthode numérique d'ordre élevé (Méthode pseudo-Spectrale) afin de prendre en compte de façon rigoureuse des variations des propriétés physiques (perméabilité magnétique et/ou conductivité électrique...) du matériau avec la profondeur. La validation de la méthode numérique proposée s’appuie sur des comparaisons avec des données simulées à l'aide d'un logiciel commercial de calcul par éléments finis et des données expérimentales obtenues au laboratoire. En outre, les codes développés ont été intégrés à une version de développement de la plateforme CIVA afin de répondre aux besoins des partenaires dans le cadre du projet européen SIMPOSIUM. / This PhD work concerns the development of fast numerical tools, dedicated to the computation of the electromagnetic interaction between a low frequency 3D current source and a complex conductor, presenting rough interfaces and/or conductivity (and/or permeability) variations. The main application is the simulation of the Eddy Current non-destructive testing process applied to complex specimens. Indeed, the existing semi-analytical models currently available in the CIVA simulation platform are limited to canonical geometries. The method we propose here is based on the covariant Maxwell equations, which allow us to consider the physical equations and relationships in a non-orthogonal coordinate system depending on the geometry of the specimen. Historically, this method (cf. C-method) has been developed in the framework of optical applications, particularly for the characterization of diffraction gratings. Here, we transpose this formalism into the quasi-static regime and we thus develop an innovative formulation of the Second Order Vector Potential formalism, widely used for the computation of the quasi-static fields in canonical geometries. Then, we determine numerically a set of modal solutions of the source-free Maxwell equations in the coordinate system introduced, and this allows us to represent the unknown fields as modal expansions in source-free domains. Then, the coefficients of these expansions are computed by introducing the source fields and enforcing the boundary conditions that the total fields must verify at the interfaces between media. In order to tackle the case of a layered conductor presenting rough interfaces, the generalized SOVP formalism is coupled with a recursive algorithm called the S-matrices. On the other hand, the application case of a complex shape specimen with depth-varying physical properties is treated by coupling the modal method we developed with a high-order numerical method: pseudo-spectral method. The validation of these codes is carried out numerically by comparison with a commercial finite element software in some particular configurations. Besides, the homogeneous case is also validated by comparison with experimental data.

Variable resolution method helps to study the local scale phenomenon of interest within the context of global scale atmosphere/ocean dynamics. Global spectral methods based on spherical harmonics as basis functions are known to resolve a given function defined on the sphere, in an uniform manner. Though known for its mathematical elegance and higher order accuracy, global spectral methods are considered to be restrictive for developing mesh-refinement strategies. The only mesh refinement strategy available until now is due to the pioneering work of F. Schmidt. Schmidt transformation can study only one region with higher resolution. The study of tropical dynamics is an interesting theme due to the presence of teleconnections between various phenomena, especially Indian Monsoon and the El-Nino. The Inter-Tropical Convergence Zone (ITCZ)is a continental scale phenamenon. It is in the ITCZ, many monsoon systems and tropical cyclones do occur. To study such phenomena under variable resolution method, high resolution is required in the entire tropical belt. Hitherto such a kind of mesh refinement strategies were not available in global spectral models. In this work, a new variable resolution method is developed that can help to study the tropical sub-scale phenomena with high resolution, in global spectral models. A new conformal coordinate transformation named ’High resolution Tropical Belt Transformation(HTBT)’ is developed to generate high resolution in the entire tropical belt. Mathematical demonstrations are given to show the existence of additional conformal transformations available on the sphere, indicating additional degrees of freedom available to create variable resolution global spectral method. Variable resolution global spectral method with high resolution over tropics is created through HTBT. The restriction imposed by Schmidt’s framework that the map-ping factor of the transformation need to have a finite-decomposition in the spectral space of the transformed domain is relaxed, by introduction of a new framework. The new framework uses transformed spherical harmonics Bnm as basis for spectral computations. With the use of FFT algorithm and Gaussian quadrature, the efficiency of the traditional spectral method is retained with the variable resolution global spectral method. The newly defined basis functions Bnm are the eigenvalues of the transformed Laplacian operator . This property of Bnm provide an elegant direct solver for the transformed Helmholtz operator on the sphere. The transformed Helmholtz equations are solved accurately with the variable resolution method. Advection experiments conducted with variable resolution spectral transport scheme on the HTBT variable grid produces near-dispersion free advection on the tropical belt. Transport across homogeneous resolution regions produce very less dispersion errors. Transport of a feature over the poles result in severe grid representation errors. It is shown that an increase in resolution around the poles greatly reduces this error. Transport of a feature from a point close to poles but not over it, does not produce such representation errors. Fourth-order Runge-Kutta scheme improves the accuracy of the transport scheme. The second order Magazenkov time-scheme proves to be better accurate than the leap-frog scheme with Asselin filter. The non-divergent barotropic vorticity equation is tested with two exact solutions namely Rochas solution and Rossby-Haurwitz wave solutions. Each of the solution tests certain unique and contrasting characteristic of the system. The numerical behaviour of the solutions show non-linear interactions in them. The singularity at the poles, arising due to the unbounded nature of the latitudinal derivative of the map factor of HTBT, triggers Gibbs phenomena for certain functions. However the recent advances in spectral methods, especially spectral viscosity method and Boyd-Vandeven filtering strategy provide ways to control the Gibbs oscillation and recover higher accuracy; make the variable resolution global spectral method viable for accurate meteorological computations.

All procedures in the literature for decomposing symmetric second rank (stress) tensor and symmetric fourth rank (elastic constant) tensor are elaborated and compared which have many engineering and scientific applications for anisotropic materials. The decomposition methods for symmetric second rank tensors are orthonormal tensor basis method, complex variable representation and spectral method. For symmetric fourth rank (elastic constant) tensor, there are four mainly decomposition methods namely as, orthonormal tensor basis, irreducible, harmonic decomposition and spectral. Those are applied to anisotropic materials possessing various symmetry classes which are isotropic, cubic, transversely isotropic, tetragonal, trigonal and orthorhombic. For isotropic materials, an expression for the elastic constant tensor different than the traditionally known form is given. Some misprints found in the literature are corrected. For comparison purposes, numerical examples of each decomposition process are presented for the materials possessing different symmetry classes. Some applications of these decomposition methods are given. Besides, norm and norm ratio concepts are introduced to measure and compare the anisotropy degree for various materials with the same or di&curren / erent symmetries. For these materials,norm and norm ratios are calculated. It is suggested that the norm of a tensor may be used as a criterion for comparing the overall e&curren / ect of the properties of anisotropic materials and the norm ratios may be used as a criterion to represent the anisotropy degree of the properties of materials. Finally, comparison of all methods are done in order to determine similarities and differences between them. As a result of this comparison process, it is proposed that the spectral method is a non-linear decomposition method which yields non-linear orthogonal decomposed parts. For symmetric second rank and fourth rank tensors, this case is a significant innovation in decomposition procedures in the literature.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-11T21:36:37Z No. of bitstreams: 1 HALLYSON GUSTAVO GUEDES DE MORAIS LIMA - DISSERTAÇÃO PPGMAT 2007..pdf: 2962280 bytes, checksum: 027c0c4dc68684f41c7b168cacb0b228 (MD5) / Made available in DSpace on 2018-07-11T21:36:37Z (GMT). No. of bitstreams: 1 HALLYSON GUSTAVO GUEDES DE MORAIS LIMA - DISSERTAÇÃO PPGMAT 2007..pdf: 2962280 bytes, checksum: 027c0c4dc68684f41c7b168cacb0b228 (MD5) Previous issue date: 2007-03 / CNPq / Neste trabalho deduzimos o sistema de Equações de Águas Rasas na forma Lagrangeana e obtemos a sua solução analítica. Aplicamos o Método Espectral na análise numérica deste sistema e mostramos que a propagação de ondas de águas rasas não depende do meio em que ela se propaga. / In this work we deduce the system of Shallow Water Equations in the Lagrangian form and we obtain its analytical solution. We have applied the spectral method in the numerical analysis of this system and we have shown that the propagation of the shallow water waves doesn't depend on the medium in which it spreads.

In this thesis we consider the problem of clustering the data lying in a union of subspaces using spectral methods. Though the data generated may have high dimensionality, in many of the applications, such as motion segmentation and illumination invariant face clustering, the data resides in a union of subspaces having small dimensions. Furthermore, for a number of classification and inference problems, it is often useful to identify these subspaces and work with data in this smaller dimensional manifold. If the observations in each cluster were to be distributed around a centric, applying spectral clustering on an a nifty matrix built using distance based similarity measures between the data points have been used successfully to solve the problem. But it has been observed that using such pair-wise distance based measure between the data points to construct a similarity matrix is not sufficient to solve the subspace clustering problem. Hence, a major challenge is to end a similarity measure that can capture the information of the subspace the data lies in. This is the motivation to develop methods that use an affinity tensor by calculating similarity between multiple data points. One can then use spectral methods on these tensors to solve the subspace clustering problem. In order to keep the algorithm computationally feasible, one can employ column sampling strategies. However, the computational costs for performing the tensor factorization increases very quickly with increase in sampling rate. Fortunately, the advances in GPU computing has made it possible to perform many linear algebra operations several order of magnitudes faster than traditional CPU and multicourse computing. In this work, we develop parallel algorithms for subspace clustering on a GPU com-putting environment. We show that this gives us a significant speedup over the implementations on the CPU, which allows us to sample a larger fraction of the tensor and thereby achieve better accuracies. We empirically analyze the performance of these algorithms on a number of synthetically generated subspaces con gyrations. We ally demonstrate the effectiveness of these algorithms on the motion segmentation, handwritten digit clustering and illumination invariant face clustering and show that the performance of these algorithms are comparable with the state of the art approaches.

Nous nous intéressons, dans cette thèse, à l'étude des modèles de mélange et des modèles linéaires généralisés, avec une application aux données de co-infection entre les arbovirus et les parasites du paludisme. Après une première partie consacrée à l'étude de la co-infection par un modèle logistique multinomial, nous proposons dans une deuxième partie l'étude des mélanges de modèles linéaires généralisés. La méthode proposée pour estimer les paramètres du mélange est une combinaison d'une méthode des moments et d'une méthode spectrale. Nous proposons à la fin une dernière partie consacrée aux mélanges de valeurs extrêmes en présence de censure. La méthode d'estimation proposée dans cette partie se fait en deux étapes basées sur la maximisation d'une vraisemblance. / We are interested, in this thesis, to the study of mixture models and generalized linear models, with an application to co-infection data between arboviruses and malaria parasites. After a first part dedicated to the study of co-infection using a multinomial logistic model, we propose in a second part to study the mixtures of generalized linear models. The proposed method to estimate the parameters of the mixture is a combination of a moment method and a spectral method. Finally, we propose a final section for studing extreme value mixtures under random censoring. The estimation method proposed in this section is done in two steps based on the maximization of a likelihood.

The thesis is focused on the investigation of Rayleigh-Bénard problem in an extended setting approximating the conditions in the Earth's mantle. The aim is to evaluate the influence of depth- and temperature- dependent material parameters, dissipation, adiabatic heating/cooling and heat sources on the qualitative characteristics of thermal convection. We identify the critical values of dimensionless parameters that determine the onset of convection and characterize the dominating convection patterns in marginally supercritical states. These issues are addressed by the application of linear stability analysis and weakly non-linear analysis. It has been found that the character of convection differ substantially from the standard case of Rayleigh-Bénard convection. Powered by TCPDF (www.tcpdf.org)

For some alloys martensitic transformation is responsible for the so-called shape memory effect and pseudoelasticity. These properties are used in a wide range of industry applications. Each of these materials is transformed to the shape it was manufactured in when heated to its critical temperature (austenite phase) no matter how seriously it was deformed at lower temperatures (martensite phase). Looking at the microstructure, one can observe significant change of crystalographic lattice depending on temperature and deformation. This the- sis focuses on modelling the evolution of microstructure during deformation for materials in the martensite phase. In this case, the creation of multiple variants of martensite is observed, divided by interfaces where a part of energy is stored. This behaviour can be described by the phase-field model. The numerical im- plementation of this model using the standard finite element method requires large computational costs. The aim of this thesis is to implement this model in MATLAB using a spectral method based on the fast Fourier transform, which is suitable for solving problems on a periodic domain. It is interesting to com- pare the computation using spectral method on a conventional PC with the computation written in FEniCS computed on a cluster. However, the...

The thesis deals with detection of T-wave alternans. The presence of T-wave in surface ECG is recognized as a marker of electrical instability of heart in stage his repolarization, arise increased risk of emergence ventricular fibrillation and sudden cardiac death. The goal of our project is familiarize with methods of detection T-wave alternans. In particular spectral method and spectral method which was realized in variant for running reading values in time (“sliding window”). To suggest a QRS complex detector, localize the T-wave and to make TWA detection using spectral method and modified spectral method. This project is to be made in C language in appropriate user interface.

Large amplitude internal solitary waves in the coastal ocean are commonly modelled with the Korteweg-de Vries (KdV) equation or a closely related evolution equation. The characteristic feature of these models is the solitary wave solution, and it is well documented that these provide the basic paradigm for the interpretation of oceanic observations. However, often internal waves in the ocean survive for several inertial periods, and in that case, the KdV equation is supplemented with a linear non-local term representing the effects of background rotation, commonly called the Ostrovsky equation. This equation does not support solitary wave solutions, and instead a solitary-like initial condition collapses due to radiation of inertia-gravity waves, with instead the long-time outcome typically being an unsteady nonlinear wave packet. The KdV equation and the Ostrovsky equation are formulated on the assumption that only a single vertical mode is used. In this thesis we consider the situation when two vertical modes are used, due to a near-resonance between their respective linear long wave phase speeds. This phenomenon can be described by a pair of coupled Ostrovsky equations, which is derived asymptotically from the full set of Euler equations and solved numerically using a pseudo-spectral method. The derivation of a system of coupled Ostrovsky equations is an important extension of coupled KdV equations on the one hand, and a single Ostrovsky equation on the other hand. The analytic structure and dynamical behaviour of the system have been elucidated in two main cases. The first case is when there is no background shear flow, while the second case is when the background state contains current shear, and both cases lead to new solution types with rich dynamical behaviour. We demonstrate that solitary-like initial conditions typically collapse into two unsteady nonlinear wave packets, propagating with distinct speeds corresponding to the extremum value in the group velocities. However, a background shear flow allows for several types of dynamical behaviour, supporting both unsteady and steady nonlinear wave packets, propagating with the speeds which can be predicted from the linear dispersion relation. In addition, in some cases secondary wave packets are formed associated with certain resonances which also can be identified from the linear dispersion relation. Finally, as a by-product of this study it was shown that a background shear flow can lead to the anomalous version of the single Ostrovsky equation, which supports a steady wave packet.