249 research outputs found
The hydrodynamic approach for plasmonics in graphene
Dissertação de mestrado em PhysicsThe hydrodynamic model approach to plasmonics is based on the simultaneous solution of Euler’s equa tion, Poisson’s equation, and the continuity equation. The quantum mechanical effects enter the model
via the statistical pressure induced by the gas of electrons. It is also a form of including the effect of
nonlocality. The aim of this thesis is to study the dispersion relation of the surface plasmon-polaritons and
the plasmonic wakes created by an external potential, when a graphene sheet is in the vicinity of a metal.
It is known they disperse linearly with the wave vector, therefore are of acoustic nature. This problem has
been studied for normal plasmons, but the study for acoustic plasmons is missing. In the first part of
this thesis, the hydrodynamic model will be used to solve some electrostatic boundary-value problems in
planar geometry, that will give the linear dispersion of the SPPs in graphene near a semi-infinite and finite
nonlocal metal. The study for the metals, gold and titanium, showed that the nonlocal effects are more
visible in titanium, due to its intrinsic proprieties, such as plasmon frequency and background permittivity.
The dielectric separation between graphene and metal also enhances the nonlocal effects. The decrease
of the dielectric thickness increases the nonlocality. Regarding the finite metal, the results show that the
increase of the metal thickness results in a higher energy of the surface plasmon-polaritons in graphene.
In this case, the dispersion is also linear in the wavenumber . The second part encompasses the study of
the induced potential in graphene, due to an external charge moving parallel to graphene in the y-direction
at an height 0. When graphene is in the vicinity of a dielectric an oscillatory V-shaped pattern was per ceived, and the dependence of the angle on the Froude number (or dependence on the velocity of the
external charge) provided two different regions, a constant angle region for low Froude numbers where
the wake angle takes the value of 21◦
. This is similar to the Kelvin region, where the angle takes the
constant value of 19.47◦
. A transition for a Mach region occurs for a plasmonic Froude number of 2.2,
where the decrease of the angle happens for higher velocities following the law 1/. When a local metal
is added to the system, the oscillatory behavior vanishes and a more continuous V-shaped wake appears
in graphene. In this case, the angles follow a quadratic polynomial law, where these decrease with the
increasing Froude number. Studying the phase velocity and the dispersion for the classical water wakes
and the plasmonic wakes it is possible to see two limiting cases for = −1 and = 0, which correspond
to pure gravity waves in deep water and gravity waves in shallow water, respectively. In such manner, it is
possible to make an analogy between gravity waves and the plasmonic waves in graphene.A abordagem do modelo hidrodinâmico à plasmônica é baseada na solução simultânea da equação de
Euler, da equação de Poisson e da equação de continuidade. Os efeitos da mecânica quântica entram no
modelo através da pressão estatística induzida pelo gás de eletrões. É também uma forma de incluir o
efeito da não-localidade. O objetivo desta tese é estudar a relação de dispersão e as perturbações plasmô nicas criadas por um potencial externo, quando o grafeno está nas proximidades de um metal. É sabido
que os plasmões dispersam linearmente com o vetor de onda, portanto possuem uma natureza acústica.
Este problema foi estudado para plasmões normais, no entanto falta o estudo para plasmões acústicos.
Na primeira parte desta tese, o modelo hidrodinâmico será usado para resolver alguns problemas eletros táticos de valor de fronteira, que darão a dispersão linear dos SPPs quando o grafeno está próximo a um
metal não local semi-finito e finito. O estudo para os metais ouro e titânio mostraram que os efeitos não
locais são mais visíveis no titânio, devido às suas propriedades intrínsecas, como frequência plasmônica
e permissividade de fundo. A separação dielétrica entre o grafeno e o metal, também potencializa os
efeitos não locais. A diminuição da espessura do dielétrico, aumenta a não-localidade. Em relação ao
metal finito, os resultados mostram que o aumento da espessura do metal leva ao aumento da energia
dos plasmões-polaritões de superfície no grafeno. Neste caso, a dispersão também é linear no vetor de
onda . A segunda parte abrange o estudo do potencial induzido no grafeno, devido a uma carga externa
movendo-se paralelamente ao grafeno na direção dos y’s, a uma altura 0. Quando o grafeno está na
vizinhança de um dielétrico, foi visto um padrão oscilatório em forma de ”V”, e a dependência do ângulo
no número de Froude (ou dependência da velocidade da carga externa) mostra duas regiões diferentes,
a primeira de ângulo constante para números de Froude baixos onde o ângulo do cone é constante e
de valor 21◦
. Sendo semelhante à região de Kelvin, onde o ângulo constante toma o valor de 19.47◦
.
Observa-se uma transição para a região de Mach ocorre para um número de Froude plasmónico de 2.2,
onde a diminuição do ângulo ocorre para velocidades mais altas seguindo a lei 1/. Quando um me tal local é adicionado ao sistema, o comportamento oscilatório é quebrado e uma onda em forma de
”V”contínua aparece no grafeno. Neste caso, os ângulos seguem uma lei polinomial quadrática, onde os
ângulos diminuem com o aumento do número de Froude. Estudando a velocidade de fase e a dispersão
para as ondas clássicas na água e as ondas plasmônicas é possível ver dois casos limites para = −1
e = 0, que correspondem a ondas gravíticas puras em águas profundas e ondas gravíticas em águas
rasas, respectivamente. Desta forma, é possível fazer uma analogia entre as ondas gravitacionais e as
ondas plasmônicas no grafeno
Study of compression techniques for partial differential equation solvers
Partial Differential Equations (PDEs) are widely applied in many branches of science, and solving them efficiently, from a computational point of view, is one of the cornerstones of modern computational science. The finite element (FE) method is a popular numerical technique for calculating approximate solutions to PDEs. A not necessarily complex finite element analysis containing substructures can easily gen-erate enormous quantities of elements that hinder and slow down simulations. Therefore, compression methods are required to decrease the amount of computational effort while retaining the significant dynamics of the problem. In this study, it was decided to apply a purely algebraic approach. Various methods will be included and discussed, ranging from research-level techniques to other apparently unrelated fields like image compression, via the discrete Fourier transform (DFT) and the Wavelet transform or the Singular Value Decomposition (SVD)
Elastic shape analysis of geometric objects with complex structures and partial correspondences
In this dissertation, we address the development of elastic shape analysis frameworks for the registration, comparison and statistical shape analysis of geometric objects with complex topological structures and partial correspondences. In particular, we introduce a variational framework and several numerical algorithms for the estimation of geodesics and distances induced by higher-order elastic Sobolev metrics on the space of parametrized and unparametrized curves and surfaces. We extend our framework to the setting of shape graphs (i.e., geometric objects with branching structures where each branch is a curve) and surfaces with complex topological structures and partial correspondences. To do so, we leverage the flexibility of varifold fidelity metrics in order to augment our geometric objects with a spatially-varying weight function, which in turn enables us to indirectly model topological changes and handle partial matching constraints via the estimation of vanishing weights within the registration process. In the setting of shape graphs, we prove the existence of solutions to the relaxed registration problem with weights, which is the main theoretical contribution of this thesis. In the setting of surfaces, we leverage our surface matching algorithms to develop a comprehensive collection of numerical routines for the statistical shape analysis of sets of 3D surfaces, which includes algorithms to compute Karcher means, perform dimensionality reduction via multidimensional scaling and tangent principal component analysis, and estimate parallel transport across surfaces (possibly with partial matching constraints).
Moreover, we also address the development of numerical shape analysis pipelines for large-scale data-driven applications with geometric objects. Towards this end, we introduce a supervised deep learning framework to compute the square-root velocity (SRV) distance for curves. Our trained network provides fast and accurate estimates of the SRV distance between pairs of geometric curves, without the need to find optimal reparametrizations. As a proof of concept for the suitability of such approaches in practical contexts, we use it to perform optical character recognition (OCR), achieving comparable performance in terms of computational speed and accuracy to other existing OCR methods.
Lastly, we address the difficulty of extracting high quality shape structures from imaging data in the field of astronomy. To do so, we present a state-of-the-art expectation-maximization approach for the challenging task of multi-frame astronomical image deconvolution and super-resolution. We leverage our approach to obtain a high-fidelity reconstruction of the night sky, from which high quality shape data can be extracted using appropriate segmentation and photometric techniques
Physics-Aware Convolutional Neural Networks for Computational Fluid Dynamics
Determining the behavior of fluids is of interest in many fields. In this work, we focus on
incompressible, viscous, Newtonian fluids, which are well described by the incompressible
Navier-Stokes equations. A common approach to solve them approximately is to perform
Computational Fluid Dynamics (CFD) simulations. However, CFD simulations are very
expensive and must be repeated if the geometry changes even slightly.
We consider Convolutional Neural Networks (CNNs) as surrogate models for CFD
simulations for various geometries. This can also be considered as operator learning.
Typically, these models are trained on images of high-fidelity simulation results. The
generation of this high-fidelity training data is expensive, and a fully data-driven approach
usually requires a large data set. Therefore, we are interested in training a CNN in the
absence of abundant training data. To this end, we leverage the underlying physics in
the form of the governing equations to construct physical constraints that we then use to
train a CNN.
We present results for various model problems, including two- and three-dimensional
flow in channels around obstacles of various sizes and in non-rectangular geometries,
especially arteries and aneurysms. We compare our novel physics-aware approach to the
state-of-the-art data-based approach and also to a combination of the two, a combined
or hybrid approach. In addition, we present results for an extension of our approach to
include variations in the boundary conditions
Self-similar blow-up solutions in the generalized Korteweg-de Vries equation: Spectral analysis, normal form and asymptotics
In the present work we revisit the problem of the generalized Korteweg-de
Vries equation parametrically, as a function of the relevant nonlinearity
exponent, to examine the emergence of blow-up solutions, as traveling waveforms
lose their stability past a critical point of the relevant parameter , here
at . We provide a {\it normal form} of the associated collapse dynamics
and illustrate how this captures the collapsing branch bifurcating from the
unstable traveling branch. We also systematically characterize the
linearization spectrum of not only the traveling states, but importantly of the
emergent collapsing waveforms in the so-called co-exploding frame where these
waveforms are identified as stationary states. This spectrum, in addition to
two positive real eigenvalues which are shown to be associated with the
symmetries of translation and scaling invariance of the original
(non-exploding) frame features complex patterns of negative eigenvalues that we
also fully characterize. We show that the phenomenology of the latter is
significantly affected by the boundary conditions and is far more complicated
than in the corresponding symmetric Laplacian case of the nonlinear
Schr{\"o}dinger problem that has recently been explored. In addition, we
explore the dynamics of the unstable solitary waves for in the
co-exploding frame.Comment: 33 pages, 16 figure
Analytical study of ABC-fractional pantograph implicit differential equation with respect to another function
This article aims to establish sufficient conditions for qualitative properties of the solutions for a new class of a pantograph implicit system in the framework of Atangana-Baleanu-Caputo () fractional derivatives with respect to another function under integral boundary conditions. The Schaefer and Banach fixed point theorems (FPTs) are utilized to investigate the existence and uniqueness results for this pantograph implicit system. Moreover, some stability types such as the Ulam-Hyers , generalized , Ulam-Hyers-Rassias and generalized are discussed. Finally, interpretation mathematical examples are given in order to guarantee the validity of the main findings. Moreover, the fractional operator used in this study is more generalized and supports our results to be more extensive and covers several new and existing problems in the literature
An overdetermined eigenvalue problem and the Critical Catenoid conjecture
We consider the eigenvalue problem in
and along , being the
complement of a disjoint and finite union of smooth and bounded simply
connected regions in the two-sphere . Imposing that is locally constant along and that has infinitely
many maximum points, we are able to classify positive solutions as the
rotationally symmetric ones. As a consequence, we obtain a characterization of
the critical catenoid as the only embedded free boundary minimal annulus in the
unit ball whose support function has infinitely many critical points
The existence, uniqueness, and stability analyses of the generalized Caputo-type fractional boundary value problems
In this article, we derive some novel results of the existence, uniqueness, and stability of the solution of generalized Caputo-type fractional boundary value problems (FBVPs). The Banach contraction principle, along with necessary features of fixed point theory, is used to establish our results. An example is illustrated to justify the validity of the theoretical observations
On ABC coupled Langevin fractional differential equations constrained by Perov's fixed point in generalized Banach spaces
Nonlinear differential equations are widely used in everyday scientific and engineering dynamics. Problems involving differential equations of fractional order with initial and phase changes are often employed. Using a novel norm that is comfortable for fractional and non-singular differential equations containing Atangana-Baleanu-Caputo fractional derivatives, we examined a new class of initial values issues in this study. The Perov fixed point theorems that are utilized in generalized Banach spaces form the foundation for the new findings. Examples of the numerical analysis are provided in order to safeguard and effectively present the key findings
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