Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2018. / Submitted by Raquel Viana (raquelviana@bce.unb.br) on 2018-07-12T19:56:22Z No. of bitstreams: 1 2018_HenriqueRennóZanata.pdf: 483481 bytes, checksum: ff042c7e9956f29cd0ab4c203aa6e7cb (MD5) / Approved for entry into archive by Raquel Viana (raquelviana@bce.unb.br) on 2018-07-14T20:16:45Z (GMT) No. of bitstreams: 1 2018_HenriqueRennóZanata.pdf: 483481 bytes, checksum: ff042c7e9956f29cd0ab4c203aa6e7cb (MD5) / Made available in DSpace on 2018-07-14T20:16:45Z (GMT). No. of bitstreams: 1 2018_HenriqueRennóZanata.pdf: 483481 bytes, checksum: ff042c7e9956f29cd0ab4c203aa6e7cb (MD5) Previous issue date: 2018-07-06 / Kirchhoff estacionárias em um domínio limitado Ω ⊂R^Ne uma classe de equações de Kirchhoff- Schrödinger estacionárias em R^2. A primeira envolve uma espécie de competição entre termos côncavo e convexo perto da origem e crescimento arbitrário no infinito. A segunda envolve crescimento exponencial crítico no sentido da desigualdade de Trudinger-Moser. / Ibno uthnids ewdo drko,m waein a Ωpp layn vda ari actiloanssa l omf estthaotidosn atroy sKtuirdcyh hao cffl-aSscsh oröf dsitnagtieorn aerqyu aKtiirocnhsh oinff .e qTuhaet ifoirnsst ionn ea involves a sort of competition between concave and convex terms near the origin and arbitrary growth at infinity. The second one involves critical exponential growth at infinity in the sense of Trudinger-Moser inequality.

La géologie, et plus particulièrement la géophysique, repose sur l’observation, directe et indirecte, de phénomènes se produisant en surface et dans les profondeurs de la Terre. Ces observations nous permettent d’étudier et définir la structure et les dynamiques globales de la Terre. L’étude des ondes sismiques générées par les tremblements de terre les plus puissants permet, par exemple, d’entrevoir la structure des hétérogénéités dans les premières centaines de kilomètres de la Terre. Dans cette thèse, nous nous intéressons au champ d’onde diffracté, qui est composé des arrivées tardives qui suivent les ondes incidentes. Par définition, les ondes diffractés contiennent de l’information liée aux hétérogénéités diffractantes, autrement dit les structures à petite échelle de la Terre, qu’elles rencontrent le long de leur trajet. De ce fait, il est possible d’étudier les variations rapides de vitesses sismiques grâce au champ d’onde diffracté, alors que ces informations seraient perdues dans les méthodes tomographiques à cause des facteurs de régularisation. Afin d’exploiter le champ d’onde diffracté, on a recours aux fonctions récepteurs (« receiver function » en anglais, RF) et à la migration sismique en profondeur de pré-empilage. Les procédures standard de migration sismiques sont de deux types principaux. Le premier type de procédures, dont l’exemple type est la migration en point de conversion communs (« common conversion point » en anglais, CCP) est rapide mais repose sur l’hypothèse fondamentale que les discontinuités que l’on cherche à imager sont horizontales. Le second type de procédures, pour lesquelles on peut citer la « reverse time migration » (RTM), ou la « generalized radon transform » (GRT), ne font pas d’hypothèse sur la structure du sous-sol, mais demandent une forte intensité des calculs et sont de fait souvent limités à des géométries bidimensionnelles. Au cours de ce manuscrit, nous développons une migration sismique de type Kirchhoff qui se base sur des calculs de temps de trajet sismique rapides en trois dimensions et quasiment aucune hypothèse sur la structure du milieu sous-jacent. Cet algorithme efficace nous permet de nous affranchir des traditionnelles limitations à des études 1D ou 2D. Notre principe d’imagerie prend en compte les ondes diffractées transmises et réfléchies, et se place dans la suite des travaux de Cheng et al. (2016). Nous adaptons la migration de type Kirchhoff élastique aux géométries de diffraction inhérentes à la sismologie passive et prenons en compte les multiples de surface. Les temps de trajet de toutes les ondes diffractées sont calculées grâce à la « fast marching method » (FMM). Les amplitudes et la polarité des signaux des RF sont corrigées à l’aide du calcul de figures de diffraction 3D. Pour extraire l’information des conversions transmises et réfléchies de façon cohérente, les résultats pour chaque mode de diffraction sont sommés de plusieurs façons (linéaire, à filtre de phase, et à filtre d’amplitude non linéaire). Afin de démontrer l’efficacité et la précision de notre méthode de migration, nous procédons à des tests synthétiques, aussi bien dans des situations réalistes qu’artificiellement compliquées, en nous servant du logiciel Raysum. Les résultats de ces tests prouvent que cette méthode de migration permet d’obtenir une image fidèle du milieu imagé quasiment sans artéfacts. En intégrant les trois composantes des RF dans la migration, cette méthode de migration est capable d’exploiter l’information d’ondes arrivant avec n’importe quel angle d’incidence et n’importe quel azimut. Finalement, cette méthode de migration multi-mode 3D est appliquée à deux jeux de données de terrain issus de réseaux sismiques déployés au dessus de zones de subduction, en Grèce et en Alaska / In geology, and in particular in geophysics, direct and indirect observations of processes occurring both at the surface of the Earth and at depth are used to understand the structure and dynamics of the Earth. For instance, seismic waves generated by large earthquakes can be used to study the structure of heterogeneities in the first few hundred kilometers inside the Earth. In this work, we use the scattered wavefield, which corresponds to energy arriving after the incident wavefield, to image the Earth. By nature, the scattered waves are linked to the scattering heterogeneities encountered along their propagation path, i.e. the fine scale structure of the Earth. Hence, the scattered wavefield has the ability to highlight structures where rapid velocity variations would otherwise be smoothed out by tomographic regularization, such as the structure of subducting slabs. To extract the information from the scattered wavefield, we resort to receiver function (RF) analysis and pre-stack depth migration. Standard migration procedures either rely on the assumption that underlying discontinuities are horizontal, such as in Common Conversion Point stacking (CCP), or are computationally expensive and usually limited to 2D geometries, such as in Reverse Time Migration (RTM) or Generalized Radon Transform (GRT). Here, we develop a Kirchhoff-type teleseismic imaging method that uses fast 3D travel-time calculations with minimal assumptions about the underlying structure. This provides high computational efficiency without limiting the problem to 1D or 2D geometries. In our method, we apply elastic Kirchhoff migration to transmitted and reflected teleseismic waves (i.e., RF). The approach expands on the work of Cheng et al. (2016). The 3D elastic Kirchhoff migration is adapted to the passive seismology scattering geometry and to account for free surface multiples. We use an Eikonal solver based on the fast marching method (FMM) to compute travel times for all scattered phases. 3D scattering patterns are computed to correct the amplitudes and polarities of the three component input signals. We consider three different stacking methods (linear, phase weighted and 2 nd root) to enhance the structures that are most coherent across scattering modes. To showcase the efficiency and accuracy of our migration procedure, we test it by conducting a series of synthetic tests in both artificially challenging and realistic scenarios. Results from synthetic tests show that our imaging principle can recover scattering structures accurately with minimal artifacts. We show that integrating the three components of the RF into the imaging principle allows to coherently retrieve the scattering potential for arbitrarily dipping discontinuities from all back-azimuths, and are able to retrieve a typical 2.5D subduction zone structure. We apply this novel 3D multi-mode Kirchhoff migration method to two different subduction zones, in Western Greece and Southern Alaska

Cette thèse porte sur le traitement mathématique du problème de Hele-Shaw dans l’approximation de Stokes-Leibenson. À l’aide d’une transformation de type Helmholtz- Kirchhoff, nous explicitons une équation d’évolution du contour fluide valable pour tout type d’écoulement plan. Cette équation généralise des résultats précédents et permet alors d’établir un schéma numérique performant dit du quasi-contour, qui se réduit à un problème de Cauchy. Nous considérons ensuite l’étude du problème par transformations conformes menant à l’équation de Polubarinova-Galin. Dans le cas simple d’un contour représenté par un trinôme à coefficients réels, nous réussissons à expliciter la solution exacte du problème. Notons que les trajectoires des solutions exactes permettent de préciser la position de la frontière des domaines d’univalence décrits par les trinômes. Enfin, nous introduisons des paramètres de contrôle sous forme de coefficients d’un multipôle superposé à la source. Des conditions suffisantes de contrôlabilité sont établies et des résultats de contrôle optimal sont explicités pour les solutions binomiales et trinomiales. L’introduction de paramètres de contrôle permet de comprendre le lien qui relie les moments de Richardson à l’équation de Polubarinova-Galin. / This PhD thesis deals with the mathematical treatment of the Hele–Shaw problem in the Stokes–Leibenson approximation. By an Helmholtz–Kirchhoff transformation, we exhibited an evolutive equation of the fluid contour applicable to all type of planar fows. This equation generalizes previous results and also allows to state an efficient numerical scheme called quasi-contour’s, which is a simple Cauchy problem. We then consider the study of this problem using conformal transformations leading to the Polubarinova-Galin equation. In the simple case of a contour representing by a trinomial with real coefficients, we succeeded in exhibiting the exact solution of the problem. Notice that the trajectories of the exact solutions enable to precise the position of frontiers of univalent domains described by trinomials. Finally, we introduce control parameters under the form of coefficients of a multipole superposed to the source. Sufficient conditions of controllability are stated and results on optimal control established for the binomial and trinomial cases. Introduction of control parameters allows us to understand the link, which bound Richardson’s moments to the Polubarinova-Galin equation.

O índice de refração não-linear n2 é o parâmetro chave da não-linearidade de terceira ordem em materiais ópticos. Um método simples, direto e sensível para medição do índice de refração não-linear (n2) é a técnica Z-scan proposta Sheik-Bahae et al 1. Nesse trabalho, um estudo sistemático quanto a esta técnica foi realizado. Utilizando a integral de difração de Fresnel-Kirchhoff, foi possível analisar a técnica investigando a influência do raio de abertura da íris. Foram obtidas expressões analíticas inéditas quando consideramos baixas não-linearidades induzidas por amostras finas. Esses resultados mostraram que o valor de n2 obtido é aproximadamente 36% maior do que n2 obtido por Sheik-Bahae para abertura linear de S=0,5, por exemplo.Extensões da técnica foram feitas, onde simulações para testar a viabilidade de medidas Z-scan no campo próximo e com dois feixes foram realizadas, de onde também foi possível encontrar resultados analíticos inéditos. Esses resultados teóricos foram testados tanto para feixe único quanto duplo em alguns experimentos, sendo obtido um bom acordo. Por fim, um novo método para medição do índice de refração não-linear não-degenerado foi proposto, em que é possível obter n2 no comprimento de onda do feixe de prova, num experimento com dois feixes. Esse resultado é importante uma vez que, em muitos casos, não é possível a realização de uma medida Z-scan convencional, devido ao fato da amostra não absorver o comprimento de onda do feixe de prova. Um resultado de n2 para rubi em 633nm foi obtido em bom acordo com resultados prévios da literatura, em que foram utilizadas técnicas de maior complexidade experimental. / The nonlinear refractive index n2 is the key parameter of the third order nonlinearity of optical materials. A simple, direct and sensitive measurement of nonlinear refractive index n2 is so-called Z-scan technique proposed by Sheik Bahae et al 1. In this work, a systematic study on this technique was performed. Based on Fresnel-Kirchhoff diffraction integral formula, it was possible to analyze the technique in order to investigate the influence of the apertures radius. We obtained unpublished analytical expressions when we consider small nonlinearities induced by thin samples. These results showed that the value of n2 obtained is about 36% larger than n2 obtained by Sheik-Bahae when the linear opening is 50%, for example.Extensions about the technique were made. Simulations were performed in order to test the feasibility of near-field Z-scan and two-color Z-scan measurements. Again it was also possible to find unpublished analytical results. The theoretical results were tested in some experiments and good agreement was found. Finally, a new method for measuring the nondegenerate refractive index was proposed. So, there is the possibility to obtain n2 at the wavelength of the probe beam in a two-color Z-scan experiment. This result is important since in many cases is impossible to perform a conventional z-scan measurement, due to the fact that the sample does not absorb the wavelength of the probe beam. A result of n2 for ruby at 633nm was obtained in good agreement with previous studies.

Este trabalho estuda a transferência de calor por condução considerando a condutividade térmica como uma função constante por partes da temperatura. Esta relação, embora fisicamente mais realista que supor a condutividade térmica constante, permite obter uma forma explícita bem simples para a inversa da Transformada de Kirchhoff (empregada para tratar a não linearidade do problema). Como exemplo, apresenta-se uma solução exata para um problema com simetria esférica. Em seguida, propôe-se uma formulação variacional (com unicidade demonstrada) que introduz um funcional cuja minimização é equivalente à solução do problema na forma forte. Finalmente compara-se uma solução exata obtida pela inversa da Transformada de Kirchhoff com a solução obtida via formulação variacional. / This work studies conduction heat transfer considering thermal conductivity as a piecewise constant function of temperature. This relationship, although physically more realistic than assuming constant thermal conductivity, provides a simple explicit form for the inverse of Kirchhoff transformation (employed to deal with the problem non-linearity). An exact solution for a problem with spherical symmetry is presented, as an example. In the sequence, a variational formulation (with demonstrated uniqueness) is proposed. This formulation introduces a functional whose minimization is equivalent to the solution of the problem in the strong form. Finally an exact solution obtained using the inverse of Kirchhoff transformation is compared with the solution obtained via variational formulation.

Este trabalho estuda a transferência de calor por condução considerando a condutividade térmica como uma função constante por partes da temperatura. Esta relação, embora fisicamente mais realista que supor a condutividade térmica constante, permite obter uma forma explícita bem simples para a inversa da Transformada de Kirchhoff (empregada para tratar a não linearidade do problema). Como exemplo, apresenta-se uma solução exata para um problema com simetria esférica. Em seguida, propôe-se uma formulação variacional (com unicidade demonstrada) que introduz um funcional cuja minimização é equivalente à solução do problema na forma forte. Finalmente compara-se uma solução exata obtida pela inversa da Transformada de Kirchhoff com a solução obtida via formulação variacional. / This work studies conduction heat transfer considering thermal conductivity as a piecewise constant function of temperature. This relationship, although physically more realistic than assuming constant thermal conductivity, provides a simple explicit form for the inverse of Kirchhoff transformation (employed to deal with the problem non-linearity). An exact solution for a problem with spherical symmetry is presented, as an example. In the sequence, a variational formulation (with demonstrated uniqueness) is proposed. This formulation introduces a functional whose minimization is equivalent to the solution of the problem in the strong form. Finally an exact solution obtained using the inverse of Kirchhoff transformation is compared with the solution obtained via variational formulation.

O índice de refração não-linear n2 é o parâmetro chave da não-linearidade de terceira ordem em materiais ópticos. Um método simples, direto e sensível para medição do índice de refração não-linear (n2) é a técnica Z-scan proposta Sheik-Bahae et al 1. Nesse trabalho, um estudo sistemático quanto a esta técnica foi realizado. Utilizando a integral de difração de Fresnel-Kirchhoff, foi possível analisar a técnica investigando a influência do raio de abertura da íris. Foram obtidas expressões analíticas inéditas quando consideramos baixas não-linearidades induzidas por amostras finas. Esses resultados mostraram que o valor de n2 obtido é aproximadamente 36% maior do que n2 obtido por Sheik-Bahae para abertura linear de S=0,5, por exemplo.Extensões da técnica foram feitas, onde simulações para testar a viabilidade de medidas Z-scan no campo próximo e com dois feixes foram realizadas, de onde também foi possível encontrar resultados analíticos inéditos. Esses resultados teóricos foram testados tanto para feixe único quanto duplo em alguns experimentos, sendo obtido um bom acordo. Por fim, um novo método para medição do índice de refração não-linear não-degenerado foi proposto, em que é possível obter n2 no comprimento de onda do feixe de prova, num experimento com dois feixes. Esse resultado é importante uma vez que, em muitos casos, não é possível a realização de uma medida Z-scan convencional, devido ao fato da amostra não absorver o comprimento de onda do feixe de prova. Um resultado de n2 para rubi em 633nm foi obtido em bom acordo com resultados prévios da literatura, em que foram utilizadas técnicas de maior complexidade experimental. / The nonlinear refractive index n2 is the key parameter of the third order nonlinearity of optical materials. A simple, direct and sensitive measurement of nonlinear refractive index n2 is so-called Z-scan technique proposed by Sheik Bahae et al 1. In this work, a systematic study on this technique was performed. Based on Fresnel-Kirchhoff diffraction integral formula, it was possible to analyze the technique in order to investigate the influence of the apertures radius. We obtained unpublished analytical expressions when we consider small nonlinearities induced by thin samples. These results showed that the value of n2 obtained is about 36% larger than n2 obtained by Sheik-Bahae when the linear opening is 50%, for example.Extensions about the technique were made. Simulations were performed in order to test the feasibility of near-field Z-scan and two-color Z-scan measurements. Again it was also possible to find unpublished analytical results. The theoretical results were tested in some experiments and good agreement was found. Finally, a new method for measuring the nondegenerate refractive index was proposed. So, there is the possibility to obtain n2 at the wavelength of the probe beam in a two-color Z-scan experiment. This result is important since in many cases is impossible to perform a conventional z-scan measurement, due to the fact that the sample does not absorb the wavelength of the probe beam. A result of n2 for ruby at 633nm was obtained in good agreement with previous studies.

Le sujet porte sur l'étude de systèmes SAS (Synthetic Aperture Sonar) multistatiques. Ces systèmes permettent d'obtenir des images de cibles mieux résolues qu'avec un sonar classique à partir d'ondes acoustiques. Le SAS est largement exploité en configuration monostatique mais il existe très peu d'études à ce jour en SAS multistatique. Le travail consiste donc à évaluer les performances en configuration bistatique et multistatique et à les comparer à celles connues en configuration monostatique. Une méthode de calcul utilisée en radar a donc été mise en oeuvre en sonar de façon à expliciter la résolution en configuration bistatique, ce qui est un résultat original de ce travail. L'algorithme classiquement utilisé pour reconstruire des images repose sur l'hypothèse que la cible est une somme de points brillants. Cette hypothèse n'est pas bien adaptée en acoustique sous-marine. Un nouvel algorithme a donc été développé dans le but de se rapprocher des phénomènes de diffraction présents à l'interface entre l'eau et la cible. Le modèle de champ diffracté est obtenu par la combinaison d'équations intégrales de frontière avec l'approximation de Kirchhoff. Une méthode de reconstruction d'images par transformée de Fourier 2D de ce modèle a été implémentée et testée sur des données simulées, puis sur des données obtenues lors d'essais en cuve. Le nouvel algorithme montre une meilleure précision de la reconstruction et la capacité de pouvoir extraire de l'information quantitative de la cible. L'intérêt des configurations multistatiques pour la reconnaissance de cibles a également été démontré dans ces travaux de thèse. / This study deals with multistatic SAS (Synthetic Aperture Sonar) systems. SAS are high resolution imaging systems compared to classical sonar ones. The SAS technique is largly exploited in the monostatic configuration but few studies already exist in multistatic SAS. Thus, the work consists in evaluating resolution and detection performances in bistatic and multistatic configurations. Then, the objective is to compare these performances to monostatic ones. A radar method has been adapted to sonar to compute bistatic performances and this is an original result of this work.The classical algorithm to reconstruct images from acoustical waves lies on the hypothesis that the target is a sum of point scatterers. This hypothesis is not really well adapted to underwater acoustics that is why a new algogorithm has been developped in this study. The new algorithm would be better adapted to scattering diffraction phenomena at the interface between water and target than the classical one. The scattered field model of the target is obtained by combinating boundary integral equations and the Kirchhoff Approximation. An imaging reconstruction method by 2D Fourier Transform of this model has been implemented and tested on numerical and experimental datas. The new algorithm allow a better reconstruction accurency and is able to give quantitative information on targets. The interest of multistatic configurations for target identification has also been demonstrated in this PhD work.

[pt] A formulação híbrida dos elementos finitos, proposta por Pian, com base no princípio variacional de Hellinger-Reissner, mostrou-se uma ótima alternativa para a construção de elementos finitos eficientes que atendessem a condições tanto de compatibilidade como de equilíbrio. O potencial de Hellinger-Reissner consiste na aproximação de dois campos: um campo tensões que satisfaz, a priori, as equações diferenciais homogêneas de equilíbrio do problema, e um campo de deslocamentos que atende a compatibilidade ao longo do contorno. O conjunto de funções não-singulares que satisfazem as equações governantes de um problema é conhecido como soluções fundamentais ou soluções de Trefftz, e é a base para a interpolação do campo de tensões no método híbrido de elementos finitos. O presente trabalho apresenta uma metodologia geral para a formulação de uma família de elementos finitos híbridos poligonais de membrana para problemas de elasticidade bidimensional, assim como elementos finitos híbridos simples e eficientes a para análise numérica de problemas de placa de Kirchhoff e Mindlin-Reissner. Algumas contribuições conceituais são introduzidas nas soluções fundamentais para a correta concepção dos elementos híbridos em problemas de placa espessa. O desempenho dos elementos é avaliado através de alguns exemplos numéricos, os quais os resultados são confrontados com os de outros elementos encontrados na literatura. / [en] The hybrid finite element formulation, proposed by Pian, on the basis of the Hellinger-Reissner variational principle, has proved to be a good alternative for the development of efficient finite elements that best attend compatibility and equilibrium conditions. The Hellinger-Reissner potential assumes two trial fields: a stress field that satisfies the equilibrium homogenous differential equation in the domain and a displacement field that attends the compatibility along the boundary. The set of nonsingular functions that satisfy the governing equations of the problem is known as Trefftz or fundamental solutions. This work presents a general methodology for the formulation of a family of polygonal hybrid elements for plane strain problems, as well as simple and efficient plate elements for the numerical evaluation of Kirchhoff and Mindlin-Reissner plate problems. Conceptual approaches are introduced for the correct use of fundamental solutions in the plate elements formulation. The performance of the proposed hybrid elements is assessed by means of several numerical examples from the literature.

Le développement de techniques de focalisation et d'imagerie à haute résolution pour les sources acoustiques et vibratoires à basse fréquence est l'un des enjeux de la recherche actuelle en acoustique, notamment pour exciter localement et analyser des structures vibroacoustiques complexes tout en conservant des propriétés de haute résolution. Ces propriétés sont nécessaires lorsque la taille des objets étudiés est plus petite que la longueur d'onde mise en jeu. Nous désirons une méthode flexible, rapide, précise, non invasive, et unifiée d'excitation et d'analyse. Celle-ci doit être applicable tant dans le domaine des vibrations dans les structures que dans le domaine des ondes acoustiques tridimensionnelles. Pour cela, nous nous basons sur la technique du puits à retournement temporel, qui n'a, à ce jour, été mise en oeuvre que pour la focalisation d'ondes de Lamb dans une cavité ergodique ou avec des ondes électromagnétiques. Aucune technique d'imagerie n'a, avant cette thèse, été dérivée du puits à retournement temporel. La méthode du puits à retournement temporel est adaptée pour la focalisation à basse fréquence. Elle permet d'exciter localement une structure avec une grande intensité, et possède des capacités de super-résolution. Malgré tout, nous démontrons que cette méthode est difficilement applicable en situation pratique, puisqu'elle fait perdre le caractère non invasif nécessaire à la plupart des applications. En revanche, nous présentons dans ce manuscrit une technique nouvelle d'imagerie de sources vibratoires et acoustiques, basée sur le puits à retournement temporel. Cette technique non invasive d'imagerie, utilisant des dispositifs de mesure similaires aux techniques de formations de voies ou d'holographie en champ proche, permet d'obtenir une image des sources vibratoires ou acoustiques à très haute résolution de manière rapide. L'approche de cette nouvelle méthode d'imagerie est décrite. Des applications à l'imagerie de sources d'impact sur une plaque encastrée, ainsi qu'à l'imagerie de sources acoustiques en champ libre et en milieu sous-marin profond sont proposées. Une application à l'imagerie de sources acoustiques à basse fréquence sur une guitare est développée. Ces résultats représentent les premières applications de l'imagerie par puits à retournement temporel numérique. Les limites, la théorie, et la mise en oeuvre de cette technique d'imagerie à haute résolution sont étudiées et détaillées. II est démontré que cet outil possède des performances et des limites similaires à l'holographie en champ proche, tout en dépassant les capacités à basse fréquence des techniques classiques de localisation limitées en résolution couramment utilisées, comme le beamforming ou le retournement temporel.