Using a finite element method, we numerically solve the time-dependent Ginzburg-Landau equations of superconductivity to explore vortex nucleation in type II superconductors. We consider a cylindrical geometry and simulate the transition from a superconducting state to a mixed state. Using saddle-node bifurcation theory we evaluate the superheating field for a cylinder. We explore how surface roughness and thermal fluctuations influence vortex nucleation. This allows us to simulate material inhomogeneities that may lead to instabilities in superconducting resonant frequency cavities used in particle accelerators.

We study a Ginzburg--Landau model for an inhomogeneous superconductor in the singular limit as the Ginzburg--Landau parameter tends to infinity. The inhomogeneity is represented by a potential term which vanishes when the order parameter equals a given smooth function, the pinning term, which is assumed to become negative in finitely many smooth subdomains, the ''normally included'' regions. For large exterior magnetic field, we study the Gamma-limit of this inhomogeneous Ginzburg-Landau functional. The vanishing of the given smooth function near the inner boundaries imply that the associated operators are strictly but not uniformly elliptic, leading to many questions to be resolved near the boundaries of the normal regions. The method we use is an extension of many techniques including the product estimate from Sandier-Serfaty, Jacobian estimates from Jerrard-Soner and an appropriate Hodge decomposition adapted to our problem. To resolve these problems, we first study the Gamma-limit in the simpler case when the pinning term is varying but bounded below by a positive constant. Second, we consider singular limits of the three-dimensional Ginzburg-Landau functional for a superconductor with thin-film geometry, in a constant external magnetic field. The superconducting domain is multiply connected and has a small characteristic thickness, and we consider the simultaneous limit as the thickness tends to zero and the Ginzburg-Landau parameter to infinity. We do this when the applied field is strong in its components tangential to the film domain. Finally, we study the Gamma-limit of the inhomogeneous superconducting Ginzburg-Landau model with the pinning term vanishing on the boundary of the normal regions. / Thesis / Doctor of Science (PhD)

<p>The goal of this thesis is to perform a simple theoretical analysis of the problem of two optical cavities coupled by a common mirror which is movable. The mirror position controls the electromagnetic mode structure of the double cavity. Modes can be transferred from one side to the other by moving the mirror, thereby allowing deterministic and on-demand transfer of photons between two cavities. By mapping the Maxwell wave equation onto the Schr\"{o}dinger wave equation, we are able to make use of the Landau-Zener result for the transition probability at an avoided crossing to obtain the conditions for adiabatic transfer.</p> / Master of Science (MS)

This work is concerned with the formulation and analysis of a simplified GinzburgLandau type model of superconductivity which is valid for large K and large magnetic field strengths. This model, referred to as the High kappa model, is derived via formal asymptotic expansion of the full, time-dependent Ginzburg-Landau equations. The model accounts for the effects of both applied magnetic fields and currents of constant magnitude. A notable feature of our model is that the systems for the leading order terms for the magnetic potential and the order parameter are decoupled. Finite element approximations of the High kappa model are introduced using standard Galerkin discretization in space and Backward-Euler and Crank-Nicolson discretization schemes in time. We establish existence and uniqueness results for the fully-discrete equations as well as optimal L2 and HI error estimates for the Backward-Euler-Galerkin and the Crank-Nicolson-Galerkin problems. Computational experiments are performed with several combinations of spatial and time discretizations of the High kappa model equations. Among other things our numerical approximations show good agreement for rates of convergence in space and time with the corresponding theoretical values. Finally, some well known steady-state and dynamic phenomena valid for type II superconductors are illustrated numerically. / Ph. D.

Cette thèse traite de questions mathématiques posées par des problèmes issus du micromagnétisme ; un thème central en est les champs de vecteur de rotationnel nul et de norme 1, qu'on voit naturellement apparaître comme configurations minimisant des énergies micromagnétiques.Le premier chapitre est motivé par la question suivante : peut-on, en dimension plus grande que deux, caractériser les champs de vecteur de rotationnel nul et de norme 1 par une formulation cinétique ?Une telle formulation a d'abord été introduite en dimension 2 dans l'article \cite{Jabin_Otto_Perthame_Line_energy_2002} de Jabin, Otto et Perthame où elle apparaît naturellement dans le cadre de la minimisation d'une énergie de type Ginzburg-Landau. Ignat et De Lellis ont ensuite montré dans \cite{DeLellis_Ignat_Regularizing_2014} qu'une telle formulation cinétique caractérise les champs de rotationnel nul et de norme 1 possédant une certaine régularité en dimension 2. Le premier chapitre de cette thèse est consacré à l'étude d'une formulation cinétique similaire en dimension quelconque ; le résultat principal en est qu'en dimension strictement plus grande que 2, cette fomulation cinétique ne caractérise non plus tous les champs de rotationnel nul et de norme 1, mais seulement les champs constants ou les vortex.La caractérsation cinétique des champs de vecteur de rotationnel nul et de norme 1 en dimension 2,prouvée par De Lellis et Ignat et que nous venons de mentionner reposait sur la notion d'entropie.Ayant obtenu une formulation cinétique en dimension quelconque, il était naturel de vouloir l'exploiter un tentant d'étendre également la notion d'entropie aux dimensions supérieures à 2. C'est ce à quoi est consacré le deuxième chapitre de cette thèse ; nous y définissons en particulier une notion d'entropie en dimension quelconque. Le point central en est la caractérisation de ces entropies par un système d'\équations aux dérivées partielles, et leur description complète en dimension 3, ainsi que la preuve pour ces entropies de propriétés tout à fait semblables à celles des entropies deux dimensionnelles.Le troisième chapitre de cette thèse, qui expose les résultats d'un travail en collaboration avec Antonin Monteil, s'intéresse à la minimisation d'\'energies de type Aviles-Giga de la forme $\mathcal_f(m)=\int_f(|m^+-m^-|)$ o\`u $m$ est un champ de rotationnel nul et de norme 1 et où $J(m)$ désigne les lignes de saut de $m$. Deux questions classiques se posent pour ce type d'énergie : la solution de viscosité de l'équation eikonale est-elle un minimiseur et l'énergie est-elle semi-continue inférieurement pour une certaine topologie. Le résutat principal de cette partie est un construction, qui nous permet en particulier de répondre par la négative à ces deux questions dans les cas où $f(t)= t^p$ avec $p \in ]0,1[$ en donnant une condition nécessaire sur $f$ pour que $\mathcal_f$ soit semi-continue inférieurement.Enfin, le dernier chapitre de cette thèse est consacré à l'étude d'une variante de l'énergie de Ginzburg-Landau introduite par Béthuel, Brezis et Helein où on a remplacé la condition de bord par une pénalisation dépendant d'un paramètre. Nous y décrivons le comportement asymptotique de l'énergie minimale qui, suivant la valeur de ce paramètre, soit se comporte comme l'énergie de Ginzburg-Landau classique en privilégiant une configuration vortex, soit privilégie au contraire une configuration singulière suivant une ligne. / This thesis is motivated by mathematical questions arising from micromagnetism. One would say that a central topic of this thesis is curl-free vector fields taking value into the sphere. Such fields naturally arise as minimizers of micromagnetic-type energies. The first part of this thesis is motivated by the following question : can we find a kinetic formulation caracterizing curl-free vector fields taking value into the sphere in dimension greater than 2 ? Such a formulation has been found in two dimension by Jabin, Otto and Perthame in \cite. De Lellis and Ignat used this formulation in \cite{DeLellis_Ignat_Regularizing_2014} to caracterize curl-free vector fields taking value into the sphere with a given regularity. The main result of this part is the generalization of their kinetic formulation in any dimension and the proof that if $d>2$, this formulation caracterizes only constant vector fields and vorteces, i. e. vector fields of the form $\pm \frac$. The second part of this thesis is devoted to a generalization of the notion of \textit, which plays a key role in the article of De Lellis and Ignat we talked about above. We give a definition of entropy in any dimension, and prove properties quite similar to those enjoyed by the classical two-dimensional entropy. The third part of this thesis, which is the result of a joint work with Antonin Monteil, is about the study of an Aviles-Giga type energy. The main point of this part is a necessary condition for such an energy to be lower semi continuous. We give in particular an example of energy of this type for which the viscosity solution of the eikonal equation is \textit a minimizer. The last part, finally is devoted to the study of a Ginzburg-Landau type energy where we replace the boundary condition of the classical Ginzburg-Landau energy introduced by Béthuel, Brezis and Helein by a penalization within the energy at the critical scaling depending on a parameter. The core result of this part is the description of the asymptotic of the minimal energy, which, depending on the parameter, favorizes vortices-like configuration like in the classical Ginzburg-Landau case, or configurations singular along a line.

This thesis concerns the wake control and flow dynamic analysis for a flow around a circular cylinder at different Reynolds numbers using reduced models. The wake control and dynamics in the reduced space were addressed using the sensitivity theory and the adjoint method. In the case of wake control, it was possible to predict the physical parameters of the active and passive controllers on the wake of the main cylinder. On the other hand, in the construction of the reduced space, a new shift mode calculated from a perturbation of the mean flow was proposed using the sensitivity to base flow modifications. The mathematical basis of the reduced space was constructed using a Fourier modal decomposition of the flow enriched by the shift mode. The proposed reduced space made possible the recomposition of the flow and the comparison with the physical parameters calculated in the physical space. Additionally, using the reduced space, it was possible to determine the transition dynamics between the equilibrium point of the Navier Stokes equation and the non-linear saturation state using the Landau coefficients obtained in the reduced model, opening the possibility of solving the flow around a 2D and 3D cylinder with low computational cost. / Esta tese trata sobre o controle de esteira assim como a análise dinâmica do escoamento em torno de um cilindro a diferentes números de Reynolds usando modelos reduzidos. O controle de esteira e a dinâmica no espaço reduzido foram abordados usando a teoria da sensibilidade e o método adjunto. No caso de controle de esteira, foi possível prever os parâmetros físicos dos controladores ativos e passivos no escoamento do cilindro principal. Por outro lado, na construção do espaço reduzido, foi proposto um novo modo de deslocamento (shift mode) calculado a partir de uma perturbação do campo médio usando a sensibilidade às modificações do campo base. A base matemática do espaço reduzido foi construída usando uma decomposição modal de Fourier do escoamento enriquecido pelo modo de deslocamento (shift mode). O espaço reduzido proposto possibilitou a recomposição do escoamento e a comparação com os parâmetros físicos calculados no espaço físico. Além disso, usando o espaço reduzido, foi possível determinar a dinâmica de transição entre o ponto de equilíbrio da equação de Navier Stokes e o estado de saturação não linear usando os coeficientes de Landau obtidos no modelo reduzido, abrindo a possibilidade de resolver o escoamento em torno de um cilindro 2D e 3D com baixo custo computacional

Submitted by Emanuel Varela Cardoso (emanuel.varela@ufcg.edu.br) on 2018-10-16T16:53:51Z No. of bitstreams: 1 FRANKBELSON DOS SANTOS AZEVEDO – DISSERTAÇÃO (PPGFísica) 2015.pdf: 1064778 bytes, checksum: c929ffea2d7e7c51359bf2dfc237fbf9 (MD5) / Made available in DSpace on 2018-10-16T16:53:51Z (GMT). No. of bitstreams: 1 FRANKBELSON DOS SANTOS AZEVEDO – DISSERTAÇÃO (PPGFísica) 2015.pdf: 1064778 bytes, checksum: c929ffea2d7e7c51359bf2dfc237fbf9 (MD5) Previous issue date: 2015-09-24 / Capes / Este trabalho tem por objetivo estudar a dinâmica quântica planar de partículas neutras de spin-1/2 na presença de campos elétricos. A priori uma partícula com carga nula não devia interagir com campos eletromagnéticos, mas ao admitir partículas possuindo momento de dipolo magnético, vemos que a interação é possível. Tal estudo acontece através da equação de Dirac com acoplamento não mínimo, onde o termo de interação leva em conta o momento de dipolo magnético, spin da partícula e campo eletromagnético. A partir dessa equação, derivamos e resolvemos duas equações diferenciais de primeira ordem, mostrando que as soluções estão atreladas ao spin. As soluções para o efeito Aharonov-Casher é discutida em detalhes pela primeira vez neste trabalho. Também derivamos uma equação diferencial de segunda ordem, a partir da qual obtivemos o níveis de energia para uma partícula movimentando-se em um caminho circular de raio constante. Além disso, usando o método de extensão auto-adjunta, encontramos funções de onda de estados ligados e níveis de energia para o espaço completo, incluindo a região r = 0. Os níveis de energia obtidos são análogos aos níveis de Landau, e mostram uma dependência com o parâmetro projeção de spin. Por fim, tomamos o limite não relativístico para o espaço completo. / This work aims to study the planar quantum dynamics of neutral particles of spin- 1/2 in the presence of electric fields. A priori a particle with null charge should not interact with electromagnetic fields, but to admit particles having magnetic dipole moment, we see that the interaction is possible. Such study happen through the Dirac equation with non-minimal coupling, where the interaction term takes into account the magnetic dipole moment, spin of the particle and electromagnetic field. From this equation, we derive and solve two differential equations of first order, showing that the solutions is linked to spin. The solutions to the Aharonov-Casher effect is discussed in detail for the first time in this study. We also derive a differential equation of second order, from which we obtained the energy levels for a particle moving in a constant radius circular path. In addition, using the self-adjoint extension method, we find wave functions of bound states and energy levels to the full space, including the region r = 0. The Energy levels obtained are analogous to Landau levels, and show a dependence on the spin projection parameter. Finally, we take the non-relativistic limit for the full space.

Cette thèse porte sur la modélisation de nano-fils ferromagnétiques. La première par-tie est consacrée à la dérivation par processus asymptotique d'un modèle uni-dimen-sionnel de nano-fil ferromagnétique fini, courbé, torsadé et de section elliptique non constante, soumis à un courant électrique. Nous utilisons ensuite le modèle asympto-tique de jonction de fils pour considérer deux cas :- celui d'un fil infini présentant un coude dans la deuxième partie.-celui un fil rectiligne infini sur lequel on branche perpendiculairement un fil fini dans la troisième partie.Dans chacun des cas précédents, on explicite toutes les solutions stationnaires. Nous étudions ensuite la stabilité de ces solutions, en concluant que le coude et la jonction sont des points attracteurs du mur. Dans la dernière partie, nous introduisons une mé-thode numérique de type différences finis d'ordre 2 en espace adaptée à la simulation des systèmes de réseaux de nano-fils. Après avoir établi numériquement l'ordre de convergence de la méthode, nous validons le schéma en simulant soit des phénomènes décrits dans la littérature, soit des propriétés décrites de manières théoriques dans les parties précédents.Ainsi, nous calculons d'abord le seuil de Walker pour un fil rectiligne. De plus, nous vé-rifions que la configuration du mur est stable dans un fil pincé même en présence d'un petit champ appliqué dans la direction du fil. Par la suite nous vérifions les résultats de stabilité pour les cas d'un fil coudé de longueur finie et d'un jonction de trois fils finis. Enfin, nous étudions la propagation de plusieurs murs dans un réseau de fils sous forme d'un peigne en injectant un courant électrique. Dans cette partie toutes les simulations numériques sont faites en Python avec quelques visualisations en Matlab. / This thesis focuses on the modeling of ferromagnetic nanowires. In the first part, we derive a one-dimensional asymptotic model for the dynamics of the magnetic moment in a twisted ferromagnetic nanowire with variable elliptical cross-section, curvature and torsion, subjected to an electric current. Then, we use the new one-dimensional model to consider two cases: - the case of an infinite ferromagnetic nanowire having a bend in the second part.- the second case is when we connect perpendicularly a finite straight wire on a straight infinite horizontal wire in the third part.In both cases, we prove the existence of static solutions. We study the stability of these solutions, we conclude that the bend and the junction attract the wall profiles. In the last part, we introduce a finite difference of order 2 in space adapted to the si-mulation of nanowire network systems. After having numerically established the order of convergence of the method,we validate the scheme by simulating either phenomena described in the literature, or properties described in theoretical ways in the previous parts.We calculate the Walker field limit, for a straight wire. In addition, we verify that the wall configuration is stable in a pinched wire even in the presence of a small field ap-plied in the direction of the wire. Then we check the stability results for the case of a finite bent wire and a junction of three finite wires. Finally, we study the propagation of several walls in a network of wires in the form of a comb by injecting an electric current. In this part all the numerical simulations are made in Python with some visua-lizations in Matlab.

Dans cette thèse de doctorat nous présentons une étude théorique de deux types d'instabilitésferroélectriques: celles apparaissant dans des géométries confinés et celles induites par le magnétismedans dans composés massifs de structure perovskite. Dans une première partie nous abordons leproblème des instabilités ferroélectriques apparaissant dans des nanotubes et des nanocoquillesoù nous développons un modèle théorique phénoménologique approprié à ces structures. Nousétudions comment l'émergence de la polarisation est affectée par (i) l'épaisseur des nanostructures,(ii) par la réponse diélectrique des matériaux environant la couche ferroélectrique et (iii) les conditionsaux interfaces. Nous observons un effet de taille finie topologique qui peut promouvoirune compétition inhabituelle entre deux types de distribution de la polarization, irrotationel eten vortex, dans la limite des très petites épaisseurs. Dans une deuxième partie nous utilisons descalculs ab-initio à base de la théorie de la fonctionnelle de la densité pour étudier les instabilitésferroélectriques des perovskites manganites à base de terres rares (RMnO3). A partir de ces calculsnous prédisons qu'il est possible d'induire une transition de phase sous pression dans EuMnO3 lefaisant transiter d'un ordre antiferromagnétique de type A isolant vers un ordre ferromagnétiquemétallique sous pression. Ce type de transition n'avait jamais été reporté précédemment dans lesmatériaux RMnO3. Nous étendons ensuite cette analyse à l'étude des effets de strain épitaxial dansles films minces de TbMnO3 et EuMnO3. Nos résultats montrent que le diagramme de phase souscontrainte d'épitaxie est bien plus riche que celui sous pression hydrostatique. Nous trouvons queles types antiferromagnétiques E-AFM et E*-AFM sont stabilisés dans le cas de TbMnO3, où letype E*-AFM est une phase métallique polaire. Dans le cas de EuMnO3, nous trouvons une phaseantiferromagnétique de type E qui n'a pas été observée sous pression hydrostatique. / In this thesis, we present a theoretical study of two types of ferroic instabilities: the ferroelectric instability in novel confined geometries and magnetic instabilities controlled by the distortion of the underlying crystal lattice. On the one hand, we consider in detail the ferroelectric instability, specifically, in the nanotubes and the spherical nanoshells and develop a phenomenological theory for describing such an instability. We determine how the emergence of polarization is affected bythe thickness of the nanoparticle, the dielectric properties of the surrounding media and the interfacial boundary conditions. We finnd an intriguing topological finite-size effect that can promote an unexpected competition between two different types of distribution of polarization - irrotational and vortex-like - in the ultra-thin limit. One the other hand, we employ a different formalism to investigate the structural, electronic and magnetic properties of the rare-earth manganites. Specifically,we conduct a theoretical investigation from first-principles calculations. First, we predict a pressure-induced A-AFM insulator to FM metal transition on EuMnO3 under hydrostatic pressure, that is unprecedented in the multiferroic rare-earth manganites RMnO3. This investigation is extended to the study to the epitaxial strain effects on both EuMnO3 and TbMnO3 thin films. We show that epitaxial strain generates a much richer phase diagram compared to hydrostatic pressure. We predict novel magnetically-induced insulator { metal and polar { non-polar transitions. More specifically, we find that both the multiferroic E-AFM order and the polar metallic E*-AFM state are stabilized in TbMnO3 by means of epitaxial strain. In the contrast, we find a novel epitaxial-strain-induced multiferroic E-AFM state in EuMnO3 that cannot be obtained by means of just hydrostatic pressure.

Made available in DSpace on 2016-06-02T20:28:26Z (GMT). No. of bitstreams: 1 DissGFM.pdf: 388964 bytes, checksum: 0930871c8482b09826976215c487e16c (MD5) Previous issue date: 2004-04-23 / Financiadora de Estudos e Projetos / In this work we study the problem of existence of non-constant stable equilibria to some parabolic systems. Specifically, the Ginzburg-Landau system, the Landau-Lifshitz system and systems with skew-gradient structure. In all cases, we note that the geometry of the domain has a fundamental role in the problem above: if the domain has a smooth boundary and is convex, then there are no non-constant stable equilibrium solutions, that is, every non-constant equilibrium is unstable. / Neste trabalho estudamos o problema da existência de equilíbrios estáveis não-constantes de alguns sistemas parabólicos, sendo eles o sistema de Ginzburg-Landau, o sistema de Landau-Lifshitz e sistemas de reação-difusão com estrutura anti-gradiente. Em todos os casos, evidencia-se que a geometria do domínio tem um papel fundamental para uma resposta ao problema: se o domínio tem fronteira suave e é convexo, então não existem soluções de equilíbrio não-constantes estáveis, ou seja, todo equilíbrio não-constante é instável.