Rozprawy doktorskie na temat „Periodic and quasi-Periodic media”

L'objectif de la thèse est de développer des méthodes numériques originales pour la résolution de l'équation des ondes en régime harmonique dans des milieux quasi-périodiques, dans l'esprit des méthodes développées précédemment pour des milieux périodiques. L'idée est d'utiliser comme dans des travaux d'homogénéisation quasi-périodique le fait que l'étude d'une EDP elliptique avec des coefficients quasi-périodiques se ramène à l'étude d'une EDP elliptiquement dégénérée en dimension supérieure, mais dont les coefficients sont périodiques. Le caractère périodique permet d'utiliser des outils adaptés, mais le caractère non-elliptique rend toutefois l'analyse mathématique et numérique de l'EDP délicate. Une des applications étudiées dans ce manuscrit concerne des problèmes de transmission entre des demi-plans périodiques (typiquement des cristaux photoniques) quand (1) l'interface ne coupe pas les demi-plans périodiques dans une direction de périodicité, ou (2) quand les milieux périodiques n'ont pas des périodes commensurables le long de l'interface<br>The goal of this thesis is to develop efficient numerical methods for the solution of the time-harmonic wave equation in quasiperiodic media, in the spirit of methods previously developed for periodic media. The goal is to use as in quasiperiodic homogenization the idea that an elliptic PDE with quasiperiodic coefficients can be interpreted as the cut of a higher-dimensional PDE which is elliptically degenerate, but with periodic coefficients. The periodicity property allows to use adapted tools, but the non-elliptic aspect makes the mathematical and numerical analysis of the PDE delicate. One application concerns transmission problems between periodic half-spaces (typically photonic crystals) when (1) the interface does not cut the periodic half-spaces in a direction of periodicity, or (2) when the periodic media have noncommensurate periods along the interface

Phononic crystals are composite media, with two different elastic materials modulated in a periodic fashion. The two-dimensional system under study is comprised of an array of cylindrical tubes arranged in a square Bravais lattice. The propagation of a time-harmonic acoustic Bloch waves is investigated experimentally and the findings compared with those obtained from the finite element method. The band structure, that is a well-known characteristic of Bloch wave dispersion, is also determined. This demonstrates the existence of band gaps of forbidden transmission at certain frequency ranges. These arise due to the interference when the wavelength of the incident wave is comparable to the periodic spacing of the modulated media. A second phononic crystal with slotted steel tubes is also studied. The opening in the tube permitting air to flow to the internal cavity, and constituting a Helmholtz resonator. The band structure of such a structure possesses additional band gaps, which arise due to the excitation of the resonators. Furthermore, it has recently been shown that periodically distributed Helmholtz resonator structures, can exhibit negative refraction phenomena. Using the transmission line technique, as outlined by Y.Cheng et al. we demonstrate this behaviour for our experimental system. It is shown that, at certain frequency ranges, the locally resonant phononic crystal exhibits negative effective density and bulk modulus, the two conditions required to obtain a negative index of refraction.

La problématique récente de la conception de métamatériaux a renouvelé l'intérêt dans les théories de l'homogénéisation en régime dynamique. En particulier, la théorie de l'homogénéisation élastodynamique initiée par J.R. Willis a reçu une attention particulière suite à des travaux sur l'invisibilité élastique. La présente thèse reformule la théorie de Willis dans le cas des milieux périodiques, examine ses implications et évalue sa pertinence physique au sens de quelques ``conditions d'homogénéisabilité'' qui sont suggérées. En se basant sur les résultats de cette première partie, des développements asymptotiques approximatifs de la théorie de Willis sont explorés en relation avec les théories à gradient. Une condition nécessaire de convergence montre alors que toutes les branches optiques de la courbe de dispersion sont omises quand des développements asymptotiques de Taylor de basse fréquence et de longue longueur d'onde sont déployés. Enfin, une nouvelle théorie de l'homogénéisation est proposée. On montre qu'elle généralise la théorie de Willis et qu'elle l'améliore en moyenne fréquence de sorte qu'on retrouve certaines branches optiques omises auparavant. On montre également que le milieu homogène effectif défini par la nouvelle théorie est un milieu généralisé dont les champs satisfont une version élastodynamique généralisée du lemme de Hill-Mandel<br>The recent issue of metamaterials design has renewed the interest in homogenization theories under dynamic loadings. In particular, the elastodynamic homogenization theory initiated by J.R. Willis has gained special attention while studying elastic cloaking. The present thesis reformulates Willis theory for periodic media, investigates its outcome and assesses its physical suitability in the sense of a few suggested ``homogenizability conditions''. Based on the results of this first part, approximate asymptotic expansions of Willis theory are explored in connection with strain-gradient media. A necessary convergence condition then shows that all optical dispersion branches are lost when long-wavelength low-frequency Taylor asymptotic expansions are carried out. Finally, a new homogenization theory is proposed to generalize Willis theory and improve it at finite frequencies in such a way that selected optical branches, formerly lost, are recovered. It is also proven that the outcome of the new theory is an effective homogeneous generalized continuum satisfying a generalized elastodynamic version of Hill-Mandel lemma

Diffusion is the fundamental process behind many molecular phenomena such as the mixing of substances. Its physical basis is the random motion of particles in a fluid. In complex porous media, diffusion is restricted by interactions with internal structures. In this work, we present studies of restricted diffusion that aim to efficiently produce quantitative models for obtaining detailed information about the morphology of biological porous media from diffusion tensor imaging experiments. We achieved this by developing a Langevin dynamics algorithm to provide physically realistic modelling of water/barrier interactions and the Lattice-Path Count algorithm to enumerate all available particle trajectories to evaluate molecular transport properties. We also performed diffusion tensor imaging experiments of the fibre networks of tissue engineering scaffolds. The findings of this thesis provide further insight into the physics underlying restricted diffusion.

Using the bond fluctuation algorithm of Carmesin and Kremer (Carmesin and Kremer 1988), we investigate the static and dynamic properties of self-avoiding linear polymers embedded in static, two-dimensional (d=2), quasi-periodic arrays of obstacles with entropic traps. The phenomenon of polymer collapse, the closely related enrichment and depletion of polymer configurations, the conformational relaxation, and the diffusive behaviour are all investigated within the framework of the lattice Monte Carlo method. Several distinct dynamical regimes are encountered: the (obstacle-free) Rouse-like regime (obstacle sub-array concentration c=0), the reptation regime for chains in perfectly periodic obstacle sub-arrays (c=1), and, in the presence of disorder and entropic traps (0&lt;c&lt;1), the anomalous regimes where the scaling properties differ from those predicted by the Rouse and reptation theories. Prior to the onset of normal diffusion, even systems characterized by very slight disorder (i.e., the existence of random isolated void spaces) are shown to lead to long, transient, subdiffusive regimes where the mean square displacement of the centre of mass scales as RCM 2&sim;D*tbeta where 0.5&lt;beta&lt;1 is the anomalous diffusion exponent and D* is the anomalous diffusion coefficient. In such disordered systems, conformational relaxation is shown to be coupled with centre of mass subdiffusion, resulting in long, time-stretched, exponential relaxation of the Rouse coordinates, viz. exp.[-(t/tau) alpha]. The stretching exponents 0.5&lt;alpha&lt;1 are shown to be closely related to the anomalous diffusion exponents beta and where the alpha, for a given chain, are shown to decrease with increasing mode number and with strong disorder. The molecular size-dependence of the steady-state diffusion coefficient, as well as that of the conformational relaxation time, is shown to be greatest when the concentration of obstacles is large and when that of the voids is non-vanishing (c &lsim; 1). Thus, the dynamical scaling in entropic trapping systems is non-monotonic with respect to the concentration of obstacles. Polymer reptation dynamics thus appears to be intrinsically unstable with respect to static disordered systems of obstacles. Having demonstrated the coupling of centre of mass subdiffusion and conformational relaxation, we introduce a new relaxation length scale, lambda=(2dD*t alpha)1/2, that is more appropriate for characterizing disordered systems than is the ubiquitous radius of gyration used in both the Rouse and reptation theories. However, lambda could not be distinguished from the radius of gyration in terms of the molecular size scaling given the uncertainty in our data. Finally, having proposed a theoretical dynamic model of entropic trapping for dilute polymer solutions in embedded mesoscopic voids, we investigate the effect of polymer solution concentration on the dynamics for both monodisperse and polydisperse polymer solutions. New, unexplored dynamical behaviours are manifest as the conformational and translational entropies compete to minimize the system free energy.

Mechanical mass-spring networks have long acted to motivate, and gain qualitative intuition, in solid-state physics, continuous media containing periodic arrays of inclu- sions such as phononic crystals, and more recently in metamaterials. While in some cases an exact or approximate analogy between the continuous model and its discrete representation can be systematically drawn, more often such analogies are introduced heuristically to aid interpretation with the lumped parameters estimated and accepted as qualitative. This thesis builds towards making the analogy exact; we first look at the discrete masses and springs lattices and apply multiple-scales methods directly to Green’s function integrals to extract the behaviour near critical frequencies. The features we uncover, and the asymptotics, are generic for many lattice structures. We then identify and study a new class of materials, two- and three- dimensional phononic crystals formed by closely spaced rigid cylinders or interconnected perforated boxes, respectively, and show that such materials constitute a versatile and tuneable family of subwavelength metamaterials. Intuitively, the voids and narrow gaps that characterise the crystals form an interconnected network of Helmholtz-like resonators. We use this intuition to argue that these continuous phononic crystals are in fact asymptotically equivalent, at low frequencies, to discrete mass-spring networks whose lumped param- eters we derive explicitly. The crystals are tantamount to metamaterials as their entire acoustic branch is squeezed into a subwavelength regime where the ratio of wavelength to period scales like the ratio of period to gap width raised to the power 1/4 in two dimensions and 1/2 in three dimensions; at yet larger wavelengths we accordingly find a comparably large effective refractive index. The fully analytical dispersion relations predicted by the discrete models yield dispersion curves that agree with those from finite-element simulations of the continuous crystals.

Pulse tube cryocoolers (PTC) are a class of rugged and high-endurance refrigeration systems that operate without moving parts at their low temperature ends, and are capable of reaching temperatures down to and below 123 K. PTCs are particularly suitable for applications in space, guiding systems, cryosurgery, medicine preservation, superconducting electronics, magnetic resonance imaging, weather observation, and liquefaction of gases. Applications of these cryocoolers span across many industries including defense, aerospace, biomedical, energy, and high tech. Among the challenges facing the PTC research community is the improvement of system efficiency, which is a direct function of the regenerator component performance. A PTC implements the theory of oscillatory compression and expansion of the gas within a closed volume to achieve desired refrigeration. An important deficiency with respect to the state of art models dealing with PTCs is the limited understanding of the hydrodynamic and thermal transport parameters associated with periodic flow of a cryogenic fluid in micro-porous structures. In view of the above, the goals of this investigation include: 1) experimentally measuring and correlating the steady and periodic flow Darcy permeability and Forchheimer’s inertial hydrodynamic parameters for available rare-Earth ErPr regenerator filler; 2) employing a CFD-assisted methodology for the unambiguous quantification of the Darcy permeability and Forchheimer’s inertial hydrodynamic parameters, based on experimentally measured steady and periodic flow pressure drops in porous structures representing recently developed regenerator fillers; and 3) performing a direct numerical pore-level investigation for steady and periodic flows in a generic porous medium in order to elucidate the flow and transport processes, and quantify the solid-fluid hydrodynamic and heat transfer parameters. These hydrodynamic resistances parameters were found to be significantly different for steady and oscillatory flows.

The natural dynamics are not ideal or linear. To understand their complex behavior, we needs to study the nonlinear dynamics in more simple models. This thesis is consist of two main setups. Both setups are simplified models for the behavior occurs in the complex systems. We studied in both systems the same nonlinear dynamics such as higher-harmonics, sub-harmonics, solitary waves,...etc. In Chapter (2), the propagation of nonlinear waves in a lattice of repelling particles is studied theoretically and experimentally. A simple experimental setup is proposed, consisting in an array of coupled magnetic dipoles. By driving harmonically the lattice at one boundary, we excite propagating waves and demonstrate different regimes of mode conversion into higher harmonics, strongly in influenced by dispersion. The phenomenon of acoustic dilatation of the chain is also predicted and discussed. The results are compared with the theoretical predictions of FPU equation, describing a chain of masses connected by nonlinear quadratic springs. The results can be extrapolated to other systems described by this equation. We studied theoretically and experimentally the generation and propagation of kinks in the system. We excite pulses at one boundary of the system and demonstrate the existence of kinks, whose properties are in very good agreement with the theoretical predictions, that is the equation that approaches, under the conditions of our experiments, the one corresponding to full model describing a chain of masses connected by magnetic forces. The results can be extrapolated to other systems described by this equation. Also, In the case of a lattice of finite length, where standing waves are formed, we report the observation of subharmonics of the driving wave. In chapter (3), we studied the propagation of intense acoustic waves in a multilayer crystal. The medium consists in a structured fluid, formed by a periodic array of fluid layerswith alternating linear acoustic properties and quadratic nonlinearity coefficient. We presents the results for different mathematicalmodels (NonlinearWave Equation,Westervelt Equation and Constitutive equations). We show that the interplay between strong dispersion and nonlinearity leads to new scenarios of wave propagation. The classical waveform distortion process typical of intense acoustic waves in homogeneous media can be strongly altered when nonlinearly generated harmonics lie inside or close to band gaps. This allows the possibility of engineer a medium in order to get a particular waveform. Examples of this include the design of media with effective (e.g. cubic) nonlinearities, or extremely linear media. In chapter (4), the oscillatory behavior of a microbubble is investigated through an acousto-mechanical analogy based on a ring-shaped chain of coupled pendula. Observation of parametric vibration modes of the pendula ring excited at frequencies between 1 and 5 Hz is considered. Simulations have been carried out and show spatial mode, mixing and localization phenomena. The relevance of the analogy between a microbubble and the macroscopic acousto-mechanical setup is discussed and suggested as an alternative way to investigate the complexity of microbubble dynamics.<br>La dinámica natural no es ideal ni lineal. Para entender su comportamiento complejo, necesitamos estudiar la dinámica no lineal en modelos más simples. Esta tesis consta de dos configuraciones principales. Ambas configuraciones son modelos simplificados de el comportamiento que se produce en los sistemas complejos. Estudiamos en ambos sistemas la misma dinámica no lineal como son la generación de armónicos superiores, los sub-armónicos, las ondas solitarias, etc. En elCapítulo (2), se estudia, tanto teórica comoexperimentalmente, la propagación de ondas no lineales en sistemas periodicos de partículas acopladas mediante fuerzas repulsivas. Se propone una configuración experimental simple, que consiste en una matriz de dipolos magnéticos acoplados. Inyectando armónicamente la señal en un extremo, excitamos ondas de propagación y demostramos diferentes regímenes de conversión de modos en armónicos, fuertemente influenciados por la dispersión. También se predice y se discute el fenómeno de dilatación acústica de la cadena. Los resultados se comparan con las predicciones teóricas de la ecuación FPU, describiendo una cadena de masas conectadas por muelles cuadráticos no lineales. Los resultados pueden ser extrapolados a otros sistemas descritos por esta ecuación. Estudiamos también teórica y experimentalmente la generación y propagación de kinks. Excitamos pulsos en la frontera del sistema y demostramos la existencia de kinks cuyas propiedades están en muy buen acuerdo con las predicciones teóricas, es decir, con la ecuación que aproxima bajo las condiciones de nuestros experimentos la correspondiente al modelo completo que describe un cadena de masas conectadas por fuerzas magnéticas. Los resultados pueden ser extrapolados a otros sistemas descritos por esta ecuación. Además, en el caso de una red finita, donde se forman ondas estacionarias, describimos la observación de subarmónicos del armónico principal. En el capítulo (3), estudiamos la propagación de ondas acústicas intensas en un cristal multicapa. El medio consiste en un fluido estructurado, formado por un conjunto periódico de capas fluidas con propiedades acústicas lineales alternas y coeficiente de no linealidad cuadrática. Presentamos los resultados de diferentes modelos matemáticos (ecuación de ondas no lineal, ecuación de Westervelt y ecuaciones constitutivas). Mostramos que la interacción entre la fuerte dispersión y la no linealidad conduce a nuevos escenarios de propagaciónde ondas. El proceso de distorsión de la onda clásica, típico de las ondas acústicas intensas en medios homogéneos, puede ser alterado de forma importante cuando los armónicos generados no linealmente se encuentran dentro o cerca del gap. Esto permite la posibilidad de diseñar un medio con el fin de obtener una forma de onda en particular. Ejemplos de esto incluyen el diseño demedios con no linealidad efectiva (por ejemplo, cúbica) o medios extremadamente lineales. En el capítulo (4), el comportamiento oscilatorio de una microburbuja se investiga a través de una analogía acusto-mecánica basada en una cadena en forma de anillo de péndulos acoplados. Se estudian los modos de vibración paramétrica del anillo pendular excitado a frecuencias entre 1 y 5 Hz. Se han llevado a cabo simulaciones que muestran la presencia de modos espaciales, mixtos y fenómenos de localización. Se discute la relevancia de la analogía entre una microburbuja y la configuración macroscópica acústico-mecánica y se sugiere como una vía alternativa para investigar la complejidad de la dinàmica de microburbujas.<br>La dinàmica natural no és ideal ni tampoc lineal. Per entendre el seu comportament complex, es necessita estudiar la dinàmica no lineal dels models més simples. Aquesta tesi consisteix en l'estudi de dues configuracions principals, que són models simplificats del comportament que es produeix en els sistemes complexos. Estudiem en ambdós sistemes la mateixa dinàmica no lineal, com és la generació d'harmònics superiors, sub-harmònics, ones solitàries, etc. En el capítol (2), estudiem, tant teòrica com experimentalment, la propagació de les ones no lineals en sistemes periòdics de partícules acoblades mitjançant forces repulsives. Es proposa una configuració experimental simple, que consisteixen en una matriu de dipols magnètics acoblats. En conduint harmònicament la xarxa en un límit, excitemla propagació de les ones i demostrem diferents règims de conversió de modes en harmònics més alts, força influenciada per la dispersió. El fenomen de la dilatació acústica de la cadena també es prediu i es discuteix. Els resultats es comparen amb les prediccions teòriques que descriu una cadena de masses conectades per molls quadràtics no lineals. Els resultats es poden extrapolar a altres sistemes descrits per aquesta equació. Hem estudiat teòrica i experimentalment la generació i propagació de Kinks. Excitem polsos a la frontera del sistema i demostrem l'existència d'Kinks, les propietats desl quals estan en molt bon acord amb les prediccions teòriques, és a dir, de l'equació que aproxima sota les condicions dels nostres experiments la corresponent al model complet que descriu un cadena demasses connectades per forcesmagnètiques. Els resultats es poden extrapolar a altres sistemes descrits per aquesta equació. A més, en el cas d'una xarxa finita, on es formen ones estacionàries, descrivim l'observació de subarmónicos de l'harmònic principal. En el capítol (3), s'estudia la propagació d'ones acústiques intenses en un medi multicapa. El medi consisteix en un fluid estructurat, format per una matriu periòdica de capes de fluid amb l'alternança de propietats acústiques lineals i coeficient de no linealitat de segon grau. Es presenten els resultats per a diferents models matemàtics no lineals (equació d'ones no lineal, equació de Westervelt i les equacions constitutives). Es demostra que la interacciò entre la forta dispersió i no linealitat condueix a nous escenaris de propagació de l'ona. El procés de distorsió en formad'ona clàssica, típica de les ones acústiques intenses en medis homogenis, es pot alterar de manera significativa quan els harmònics generats de forma no lineal es troben dins o a prop del gap. Això obri la possibilitat de dissenyar unmedi per tal d'obtenir una forma d'ona particular. Exemples d'això inclouen el disseny delsmedis amb una no linealitat efectiva (per exemple cúbica), o medis extremadament lineals. En el capítol (4), el comportament oscilatori d'una micro-bombolla és investigat a través d' una analogia acústica-mecànica basada en una cadena en forma d'anell de pèndols acoblats. Es considera l'observació dels modes de vibració paramètriques de l'anell pendular excitat amb freqüències entre 1 i 5 Hz. S'han dut a terme simulacions que mostren la presència de moes espacilas, mixtes i fenòmens de localització. Es discuteix la relevància de l'analogia entre les microbambolles i la configuració macroscòpica acústica-mecànica i es suggereix una formaalternativa per a investigar la complexitat de la dinàmica demicrobombolles.<br>Mehrem Issa Mohamed Mehrem, A. (2017). Nonlinear acoustics in periodic media: from fundamental effects to applications [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/80289<br>TESIS

An asymptotic scheme is generated that captures the motion of waves within discrete, semi-discrete and continuous periodic media by creating continuum homogenised equations. Conventional homogenisation theory is a well-known classical method valid when the wavelength of any disturbance is long relative to the microstructure. Unfortunately many of the features of interest in real applications involve wave oscillations that are of high frequency and that have wavelength of the same, or similar, order to the microstructure; this requires a new version of homogenisation theory: High frequency homogenisation. This has already been introduced for periodic microstructured continua and extended to discrete systems. Herein we extend high frequency homogenisation further, to deal with localised defect states and non-orthogonal geometries for both discrete and continuous media. We also apply the asymptotic theory to new models, such as in-plane oscillations of the discrete vector system. In each of the studies presented herein, the homogenisation method is verified using numerical and/or analytical solutions.

A periodic structure consists of spatially repeating unit cells. From man-made multi-span bridges to naturally occurring atomic lattices, periodic structures are ubiquitous. The periodicity can be exploited to generate frequency bands within which elastic wave propagation is impeded. A limitation to the linear periodic structure is that the filtering properties depend only on the structural design and periodicity which implies that the dispersion characteristics are fixed unless the overall structure or the periodicity is altered. The current research focuses on wave propagation in nonlinear periodic structures to explore tunability in filtering properties such as bandgaps, cut-off frequencies and response directionality. The first part of the research documents amplitude-dependent dispersion properties of weakly nonlinear periodic media through a general perturbation approach. The perturbation approach allows closed-form estimation of the effects of weak nonlinearities on wave propagation. Variation in bandstructure and bandgaps lead to tunable filtering and directional behavior. The latter is due to anisotropy in nonlinear interaction that generates low response regions, or "dead zones," within the structure.The general perturbation approach developed has also been applied to evaluate dispersion in a complex nonlinear periodic structure which is discretized using Finite Elements. The second part of the research focuses on wave dispersion in strongly nonlinear periodic structures which includes pre-compressed granular media as an example. Plane wave dispersion is studied through the harmonic balance method and it is shown that the cut-off frequencies and bandgaps vary significantly with wave amplitude. Acoustic wave beaming phenomenon is also observed in pre-compressed two-dimensional hexagonally packed granular media. Numerical simulations of wave propagation in finite lattices also demonstrated amplitude-dependent bandstructures and directional behavior so far observed.

In the late eighties, a new class of materials, known as photonic crystals (PCs), emerged enabling the propagation and generation of light to be potentially manipulated with unprecedented control. PCs consist of a periodic modulation of dielectric constant in one, two, or three dimensions, which can result in the formation of directional or omni-directional photonic band gaps (PBGs), spectral regions where light propagation is forbidden, and more remarkably, novel dispersion characteristics. Since PC properties scale with the dimension of the wavelength of interest, significant technological constraints must be fully addressed to manufacture 3D PBG materials for optical or infrared applications such as displays, lightning, and communications. PCs enable the unraveling of unique optical phenomena such as PBGs, spontaneous emission rate manipulation, sub-wavelength focusing, and superprism effects. This research focuses on the feasibility to achieve omni-directional PBGs in synthetic opal-based 3D PCs through precise nanoscale control to the original dielectric architecture. In particular, the optical response to the conformal deposition of dielectric layers using atomic layer deposition (ALD) within the porous template is strongly emphasized. Geometrical models were developed to faithfully model the manipulation of the synthetic opal architecture by ALD and then used in electromagnetic algorithms to predict the resulting optical properties. From these results, this research presents and investigates a scheme used to greatly enhance and adjust the PBG width and position, as well as simultaneously reducing the dielectric contrast threshold at which the PBG forms. This Thesis demonstrates that the unique opal architectures offered by ALD not only supports the formation of larger PBGs with high index materials; but also enables the use of optically transparent materials with reduced refractive index. Additionally, slight alteration of these structures facilitates the incorporation of non-linear (NL) electro-optical (EO) material for dynamic tuning capabilities and potentially offers a pathway for fabricating multi-functional photonic devices. Finally, low-temperature ALD was investigated as a means to manipulate band gaps and dispersion effects in 2D PC silicon slab waveguides and 3D organic biologically-derived templates. The results indicate the unique ability of ALD to achieve composite structures with desirable (large PBGs) or novel (slow light) optical properties.

This thesis examines the properties of transmission and transport of light and charged particles in periodic or quasiperiodic systems of solid state and optics, especially the nonlinear and external field effects and the dynamic properties of these systems.

In this thesis we will focus on the linear elastodynamic properties of complex materials. In Chapter 2, we review the linear theory of elasticity. This chapter provides a brief overview of the basic laws of elasticity theory for di_erent coordinate systems, that will facilitate the further development of our research. Dispersion properties of 2D periodic systems for di_erent lattices are reported in Chapter 3. The governing equations of out of plane wave and examples of coordinate transformations in Cartesian and cylindrical are considered in Chapter 4. Also the scattering problems by square and cylindrical uncloaked and cloaked holes are investigated analytically and numerically. In Chapter 5 a periodic transformation approach has been applied to the problem of out of plane shear wave propagation in an isotropic linear elastic material. The Chapter is organized as follows. In Section 5.1 we present initial and transformed equations of motion and corresponding boundary conditions, describing the periodic locally radial geometric transformation. In Section 5.2 we report the comparative analysis of dispersion properties and briey describe the applied multipole expansion method. In particular, we focus our attention on classical, overlapping and unfolding transformations by also performing a low-frequency, long wavelength homogenisation. In Section 5.3 we show several application including a transmission problems in a continuum and in a waveguide, the detection of defect modes and the design of the transformation for the existence of Dirac points. In Chapter 6, the mathematical model of a curved beam that is connected to two semi-in_nite straight beams is developed. Dispersion properties of curved beams are derived, characterized by three di_erent propagating regimes. By implementing the Transfer matrix approach, the reection and transmission coe_cients that depend on the curvature, frequency and total angle of the curved beam are determined. By analysing the e_ect of the curvature, frequency and total angle on energy ux, separation between high frequency/low curvature regime, where the incident wave is practically totally transmitted, and low frequency/high curvature regime where, in addition to reection there is a strong coupling between longitudinal and exural waves, are de_ned. Finally, general conclusions are given in the last chapter.

The purpose of this thesis is the derivation of corrector estimates justifying the upscaling of systems of partial differential equations (PDEs) with coupled fluxes posed in media with microstructures (like porous media). Such models play an important role in the understanding of, for example, drug-delivery mechanisms, where the involved chemical species diffusing inside the domain are assumed to obey perhaps other transport mechanisms and certain non-dissipative nonlinear processes within the pore space and at the boundaries of the perforated media (e.g. interaction, chemical reaction, aggregation, deposition). In this thesis, our corrector estimates provide a quantitative analysis in terms of convergence rates in suitable norms, i.e. as the small homogenization parameter tends to zero, the differences between the micro- and macro-concentrations and between the corresponding micro- and macro-concentration gradients are controlled in terms of the small parameter. As preparation, we are first concerned with the weak solvability of the microscopic models as well as with the fundamental asymptotic homogenization procedures that are behind the derivation of the corresponding upscaled models. We report results on three connected mathematical problems: 1. Asymptotic analysis of microscopic semi-linear elliptic equations/systems. We explore the asymptotic analysis of a prototype model including the interplay between stationary diffusion and both surface and volume chemical reactions in porous media. Our interest lies in deriving homogenization limits (upscaling) for alike systems, and particularly, in justifying rigorously the obtained averaged descriptions. We prove the well-posedness of the microscopic problem ensuring also the positivity and boundedness of the involved concentrations. Then we use the structure of the two-scale expansions to derive corrector estimates delimitating quantitatively the convergence rate of the asymptotic approximates to the macroscopic limit concentrations and their gradients. High-order corrector estimates are also obtained. The semi-linear auxiliary problems are tackled by a fixed-point homogenization argument. Our techniques include also Moser-like iteration techniques, a variational formulation, two-scale asymptotic expansions as well as suitable energy estimates. 2. Corrector estimates for a Smoluchowski-Soret-Dufour model. We consider a thermodiffusion system, which is a coupled system of PDEs and ODEs that account for the heat-driven diffusion dynamics of hot colloids in periodic heterogeneous media. This model describes the joint evolution of temperature and colloidal concentrations in a saturated porous tissue where the Smoluchowski interactions for aggregation process and a linear deposition process take place. By a fixed-point argument, we prove the local existence and uniqueness results for the upscaled system. To obtain the corrector estimates, we exploit the concept of macroscopic reconstructions as well as suitable integral estimates to control boundary interactions. 3. Corrector estimates for a non-stationary Stokes-Nernst-Planck-Poisson system. We investigate a non-stationary Stokes-Nernst-Planck-Poisson system posed in a perforated domain as originally proposed by Knabner and his co-authors (see e.g. [98] and [99]). Starting off with the setting from [99], we complete the results by proving corrector estimates for the homogenization procedure. Main difficulties are connected to the choice of boundary conditions for the Poisson part of the system as well as with the scaling of the Stokes part of the system.

For the last ten years, DVD-R media have played an important role in the storage of large amounts of digital data throughout the world. During this time it was assumed that the DVD-R was as long-lasting and stable as its predecessor, the CD-R. Several reports have surfaced over the last few years questioning the DVD-R's ability to maintain many of its claims regarding archival quality life spans. These reports have shown a wide range of longevity between the different brands. While some DVD-Rs may last a while, others may result in an early and unexpected failure. Compounding this problem is the lack of information available for consumers to know the quality of the media they own. While the industry works on devising a standard for labeling the quality of future media, it is currently up to the consumer to pay close attention to their own DVD-R archives and work diligently to prevent data loss. This research shows that through accelerated aging and the use of logistic regression analysis on data collected through periodic monitoring of disc read-back errors it is possible to accurately predict unrecoverable failures in the test discs. This study analyzed various measurements of PIE errors, PIE8 Sum errors, POF errors and jitter data from three areas of the disc: the whole disc, the region of the disc where it first failed as well as the last half of the disc. From this data five unique predictive equations were produced, each with the ability to predict disc failure. In conclusion, the relative value of these equations for end-of-life predictions is discussed.

In this thesis, the propagation of transient waves in heterogeneous media and the tuning of periodic elastic materials are studied. The behaviour of time harmonic waves in complex media is a well understood phenomenon. The primary aim of this text is to gain a deeper understanding into the propagation of transient waves in periodic media. The secondary aim is to explore the time harmonic behaviour of two dimensional pre-stressed elastic media and investigate the plausibility of band gap tuning. We begin this text by investigating the reflection of pulses from a semi-infinite set of point masses (we call 'beads') on a string. The reflected pulse is formulated using Fourier transforms which involve the harmonic reflection coefficient. We find that the reflected amplitude of a harmonic wave depends on its frequency. We then ask whether it is possible to find an effective reflection coefficient by assuming the beaded portion of the string is given by some effective homogeneous medium. An effective reflection coefficient is found by assuming the homogeneous medium has the wavenumber given by the infinite beaded string. This effective reflection coefficient is compared to the exact reflection coefficient found using the Wiener-Hopf technique. The results from studying the reflection problem gave inspiration to chapter 4, which focuses on the time dependent forcing of an infinite beaded string that is initially at rest. We again use the Fourier transform to find a time dependent solution. The z-transform is then used, after sampling the solution at the bead positions. We impose a sinusoidal loading which is switched on at a specified time. In doing this we are able to explore how the system behaves differently when excited in a stop band, a pass band and at a frequency on the edge between the two. An exact solution for the infinite beaded string is found at any point in time by expanding the branch points of the solution as a series of poles. We compare this exact solution to the long time asymptotics. The energy input into the system is studied with the results from the exact solution and long time approximation showing agreement. Interesting behaviour is discovered on the two edges between stop and pass bands. In chapter 5 the effect of a nonlinear elastic pre-stress on the wave band structure of a two dimensional phononic crystal is investigated. In this chapter we restrict ourselves to incompressible materials with the strain energy functions used being the neo-Hookean, Mooney-Rivlin and Fung. The method of small-on-large is used to derive the equation for incremental elastic waves and then the plane wave expansion method is used to find the band structure. Finally, chapter 6 focuses on the same geometry with a compressible elastic material. The strain energy function used is the one suggested by Levinson and Burgess. We use the theory of small-on-large to derive the incremental equations for coupled small amplitude pressure and shear waves in this material. In both compressible and incompressible materials we show how it is possible to control the stop bands in a material by applying a large elastic pre-stress.

In recent years, the injection of nanoscopic particles of zero-valent iron (nZVI) proved to be an effective intervention to remediate contaminated aquifer systems. Thus, the fate and transport of nZVI, as well as those of contaminants and harmful particles, needs to be completely characterised in order to design successful interventions. The theoretical framework in which this research field is enclosed is the study of flow and transport phenomena in porous media. Under laminar flow regime, computational fluid-dynamic (CFD) simulations of flow field and transport of both reacting and non-reacting solutes and colloidal particles were performed. The aim was to explore the correlations that link the spatial arrangement of the porous medium grains to the morphology of the flow field and to the onset of anomalous transport. In particular, the following dissertation is devoted to the study of three different periodic pore structures, namely face-centered cubic (FCC), body-centered cubic (BCC) and sphere-in-cube (SIC) arrangements, as they excellently balance the complexity of real porous media systems with the ease of implementation and low computational costs. The nature of the present thesis is essentially multi-scale: performing pore-scale simulations it was possible to describe the intimate characteristics of periodic porous media, and couple them with the analysis of macro-scale effective parameters (dispersion above all) that are of interest in practical applications. The emphasis was put on the presence of recirculation zones, portions of the fluid region where the streaklines bend and curl. Recirculation zones are strongly related to the structure of the porous medium and their presence overwhelmingly emerges also at low Reynolds numbers. The role of recirculation zones is highlighted in the various aspects of transport and deposition phenomena, starting from their link to the onset of anomalous transport, to their influence on filtration performances of the investigated periodic packing structures. At the macro-scale level, for periodic porous media, the choice of a proper transport model was thus influenced by the presence (or not) of recirculation zones. Both the classical advection-dispersion model and the dual porosity model were considered, the latter to take into account recirculation zones and their effects. The onset of anomalous transport was considered also through the assumption of the model parameters as functions of both the pore-scale Péclet number and the space variable, thus allowing a proper description of the transition from anomalous to the Fickian regime. Moving within the framework of colloid filtration theory (CFT), deposition phenomena were investigated for both FCC and BCC arrangements, thus highlighting some interesting potentiality of the use of periodic porous media in filtration processes. The macro-scale analyses were supported by the application of the method of moments, that under the Fickianity assumption is a rather good alternative to derive information about model parameters.

Cette thèse vise à revisiter les schémas d'homogénéisation d'ordre supérieur vers des continuums d'ordre ou de gradient supérieurs, successivement pour les matériaux et composites architecturés périodiques et quasi-périodiques, en se basant sur des principes variationnels et une extension de la condition de macrohomogénéité de Hill. Les méthodes d'homogénéisation continue sont exposées dans la première partie pour les milieux micropolaires et micromorphes, suivies par une présentation de l’homogénéisation discrète, alternative de l’homogénéisation continue.Nous avons étendu ces développements théoriques à la situation des matériaux quasi-périodiques, de microstructure régulière, qui peut être transformée en une configuration périodique de référence. L'idée commune aux méthodes d'homogénéisation périodique proposées (de nature continue ou discrète) est de décomposer le déplacement microscopique en une partie homogène représentative de la cinématique du milieu continu effectif adopté, et une fluctuation évaluée à partir d'un principe variationnel. En substance, les développements théoriques permettent l'élaboration de continuums enrichis (milieux continus généralisés) de type micromorphe, et des variantes qui en découlent en utilisant des conditions de dégénérescence appropriées. Des applications numériques ont été réalisées pour des matériaux architecturés et des composites à renforts de type inclusion sujets à de tels effets d'ordre supérieur en raison de leur architecture interne. Sur le plan théorique, les développements réalisés remédient à de nombreuses limitations des schémas d'homogénéisation d'ordre supérieur existants.Dans la partie II, les propriétés mécaniques effectives classiques et d'ordre supérieur des matériaux architecturés ont été évaluées sur la base de schémas d'homogénéisation discrets. En suivant l'idée d'une approche phénoménologique, des modèles consistants de type couple de contraintes de réseaux de poutres répétitifs ont été élaborés. Des milieux de Cosserat enrichis ont été élaborés dans l'esprit de la micromécanique, en adoptant des modèles de poutre de Timoshenko à un niveau microscopique, et en appliquant une méthode de continualisation vers un milieu de substitution effectif de Cosserat. La méthode de continualisation proposée s'avère précise et efficace en termes de calcul par rapport aux schémas d'homogénéisation continus et aux simulations par éléments finis réalisés sur la microstructure initiale. Un résultat essentiel des analyses effectuées est la quantification des effets de bord.Le contexte théorique qui sous-tend l'homogénéisation asymptotique quasi-périodique dans le cadre de l'élasticité anisotrope linéarisée est abordé dans la troisième partie. Différentes méthodologies d'évaluation des propriétés effectives quasi-périodiques ont été élaborées, conduisant à l'émergence de milieux effectifs à gradient de déformation. Les transformations conformes définissent une classe spécifique de transformations géométriques permettant de concevoir des matériaux architecturés générant un gradient de porosité interne, ce qui en fait de bons candidats pour des biosubstituts en biomécanique osseuse<br>This thesis aims to revisit higher-order homogenization schemes towards higher-order or higher gradient continua, successively for periodic and quasi-periodic architected materials and composites, based on variational principles and an extension of Hill macrohomogeneity condition. Continuous homogenization methods are exposed in Part I for micropolar and micromorphic media, followed by an exposition of the alternative discrete homogenization method.We have extended these theoretical developments to the situation of quasi-periodic materials, which still have a regular microstructure. The common idea to the proposed periodic homogenization methods of continuous or discrete nature is to split the microscopic displacement into a homogeneous part representative of the kinematics of the adopted effective continuum and a fluctuation evaluated from a variational principle. In substance, the theoretical developments allow the elaboration of enriched continua (generalized continua) of micromorphic type and all sub continua obtained using suitable degeneration conditions. Numerical applications have been made for architected materials and inclusion-based composites prone to higher-order effects due to their inner architecture. On the theoretical framework, the performed developments remedy many existing limitations of existing higher-order homogenization schemes.In Part II, repetitive lattice materials' effective classical and higher-order mechanical properties have been evaluated based on discrete homogenization schemes. Following the idea of a phenomenological approach, consistent couple stress models of repetitive beam lattices have been elaborated. Enriched Cosserat media have been derived in the spirit of micromechanics, adopting Timoshenko beam models at a microlevel, and applying a continualization method towards a Cosserat effective substitution medium. The proposed continualization method proves to be accurate and computationally efficient compared to continuous homogenization schemes and fully resolved finite element simulations. One key outcome of the performed analyses is the quantification of edge effects in the response of lattice structures, relying on the surface formulation of the extended Hill macrohomogeneity condition.The theoretical background underlying quasi-periodic asymptotic homogenization in the framework of linearized anisotropic elasticity deserves the development of Part III. Different methodologies for evaluating the effective quasi-periodic properties have been elaborated, leading to the emergence of strain gradient effective media. Conformal transformations define a specific class of geometrical mappings, allowing for designing compatible architected materials with inner porosity gradient, making them suitable bone biomechanics candidates

Metamaterials offer the freedom to tune the rich electromagnetic coupling between the constituent meta-atoms to tailor their collective electromagnetic response. Therefore, a comprehensive understanding of the nature of electromagnetic interactions between meta-atoms is necessary for novel metamaterial design, which is provided in the first part of this thesis. The subsequent work in the thesis applies the understanding from the first part to design and demonstrate novel one-dimensional metamaterials that overcome the limitations of metamaterials proposed in literature or exhibit electromagnetic responses not previously observed. Split-ring Resonators (SRRs) are a fundamental building block of many electromagnetic metamaterials. In the first part of the work in this thesis, it is shown that bianisotropic SRRs (with magneto-electric cross-polarisation) when in close proximity to each other, exhibit a rich coupling that involves both electric and magnetic interactions. The strength and nature of the coupling between two identical SRRs are studied experimentally and computationally as a function of their separation and relative orientation. The electric and magnetic couplings are characterised and it is found that, when SRRs are close enough to be in each other's near-field, the electric and magnetic couplings may either reinforce each other or act in opposition. At larger separations retardation effects become important. The findings on the electromagnetic interactions between bianisotropic resonators are next applied to developing a one-dimensional ultra-wideband backward-wave metamaterial waveguide. The key concept on which the metamaterial waveguide is built is electro-inductive wave propagation, which has emerged as an attractive solution for designing backward-wave supporting metamaterials. Stacked metasurfaces etched with complementary SRRs (CSRRs) have also been shown to exhibit a broadband negative dispersion. It is demonstrated through experiment and numerical modeling, that the operational bandwidth of a CSRR metamaterial waveguide can be improved by restricting the cross-polarisation effects in the constituent meta-atoms. The metamaterial waveguide constructed using the modified non-bianisotropic CSRRs are found to have a fractional bandwidth of 56.3\% which, based on a thorough search of relevant literature, is the broadest reported value for an electro-inductive metamaterial. A traditional coupled-dipole toy-model is presented as a tool to understand the field interactions in CSRR based metamaterials, and to explain the origin of their negative dispersion response. This metamaterial waveguide should be of assistance in the design of broadband backward-wave metamaterial devices, with enhanced electro-inductive waveguiding effects. In the final part of the thesis, a one-dimensional metamaterial prototype that permits simultaneous forward- and backward-wave propagation is designed. Such a metamaterial waveguide could act as a microwave analogue of nanoparticle chains that support electromagnetic energy transfer with a positive or a negative dispersion due to the excitation of their longitudinal or transverse dipole modes. The symmetry of the designed hybrid meta-atom permits the co-existence of two non-interfering resonances closely separated in frequency. It is experimentally and computationally shown that the metamaterial waveguide supports simultaneous non-interacting forward- and backward-wave propagation in an overlapping frequency band. The proposed metamaterial design should be suitable for realising bidirectional wireless power transfer applications.

Parent-child social games, such as peek-a-boo and patty-cake, are a key element of an infant's earliest social interactions. The analysis of children's behaviors in social games based on video recordings provides a means for psychologists to study their social and cognitive development. However, the current practice in the use of video for behavioral research is extremely labor-intensive, involving many hours spent extracting and coding relevant video clips from a large corpus. From the standpoint of computer vision, such real-world video collections pose significant challenges in the automatic analysis of behavior, such as cluttered backgrounds, the effect of varying camera angles, clothing, subject appearance and lighting. These observations motivate my thesis work - automatic retrieval of social games from unstructured videos. The goal of this work is both to help accelerate the research progress in behavioral science and to take the initial steps towards the analysis of natural human interactions in natural settings. Social games are characterized by repetitions of turn-taking interactions between the parent and the child, with variations that are recognizable by both of them. I developed a computational model for social games that exploits the temporal structure over a long time-scale window as quasi-periodic patterns in a time series. I presented an unsupervised algorithm that mines the quasi-periodic patterns from videos. The algorithm consists of two functional modules: converting image sequences into discrete symbolic sequences and mining quasi-periodic patterns from the symbolic sequences. When this technique is applied to video of social games, the extracted quasi-periodic patterns often correspond to meaningful stages of the games. The retrieval performance on unstructured, lab-recorded videos and real-world family movies is promising. Building on this work, I developed a new feature extraction algorithm for social game categorization. Given a quasi-periodic pattern representation, my method automatically selects the most relevant space-time interest points to construct the feature representation. Our experiments demonstrate very promising classification performance on social games collected from YouTube. In addition, the method can also be used to categorize TV videos of sports rallies, demonstrating the generality of this approach. In order to support and encourage more research on human behavior analysis in realistic contexts, a video database of realistic child play in natural settings has been collected and is published on our project website (http://www.cc.gatech.edu/cpl/projects/socialgames), along with annotations. The unsupervised quasi-periodic pattern mining method represents a substantial generalization of conventional periodic motion analysis. Its generality is evaluated by retrieving motions of a range of quasi-periodicity from unstructured videos. The performance was compared with that of a periodic motion detection method based on motion self-similarity. Our method demonstrates superior retrieval performance with a 100% precision when the recall is up to 92.04%, with much fewer parameters than that of the other method.

Engaging in the music set of one’s favorite artist or DJ is oftentimes leading to the result of a powerful and euphoric felt experience, a sensation partly also induced from dancing in beat to the music. In an attempt to simulate a similar dance experience, a user-study was designed in order to investigate when a user is let to dance in rhythm to a music playback and in addition, in control of a music playback tempo through the induced dance movements. A proof-of-concept prototype was built and tested in an initial study, followed by a main study where the prototype had been modified and 12 participants participated. A questionnaire was given containing various question statements to be rated through a Likert-scale regarding their subjective experience. Open-ended questions were also included to collect their own opinions. From the results, an enhanced engagement and enjoyment of the music could be identified when being able to manipulate the tempo.<br>Att engagera sig i ett musikset av ens favoritartist eller DJ leder ofta till resultatet av en kraftfull och euforisk känsloupplevelse, en känsla delvis framkallat av att man dansar i takt med musiken. I ett försök att simulera en liknande dansupplevelse undersöker denna användarstudie när en användare dansar i rytm till musik och dessutom är i kontroll av tempot genom de skapade dansrörelserna. En proof-of-concept prototyp konstruerades och testades i en första studie, följt av en huvudstudie där prototypen hade modifierats och 12 deltagare deltog. Ett frågeformulär gavs med olika frågor som skulle bedömas via en Likert-skala, med avseende på deras subjektiva erfarenhet. Öppna frågor ingick också för att samla deras egna åsikter. Från resultaten kunde ett ökat engagemang och en förhöjd njutning av musiken identifieras när man kunde manipulera tempot.

Understanding the phenomenon of wave scattering by random media is a ubiquitous problem that has instigated extensive research in the field. This thesis focuses on wave scattering by quasiperiodic media as an alternative approach to provide insight into the effects of structural aperiodicity on the propagation of the waves. Quasiperiodic structures are aperiodic yet ordered so have attributes that make them beneficial to explore. Quasiperiodic lattices are also used to model the atomic structures of quasicrystals; materials that have been found to have a multitude of applications due to their unusual characteristics. The research in this thesis is motivated by both the mathematical and physical benefits of quasiperiodic structures and aims to bring together the two important and distinct fields of research: waves in heterogeneous media and quasiperiodic lattices. A review of the past literature in the area has highlighted research that would be beneficial to the applied mathematics community. Thus, particular attention is paid towards developing rigorous mathematical algorithms for the construction of several quasiperiodic lattices of interest and further investigation is made into the development of periodic structures that can be used to model quasiperiodic media. By employing established methods in multiple scattering new techniques are developed to predict and approximate wave propagation through finite and infinite arrays of isotropic scatterers with quasiperiodic distributions. Recursive formulae are derived that can be used to calculate rapidly the propagation through one- and two-dimensional arrays with a one-dimensional Fibonacci chain distribution. These formulae are applied, in addition to existing tools for two-dimensional multiple scattering, to form comparisons between the propagation in one- and two-dimensional quasiperiodic structures and their periodic approximations. The quasiperiodic distributions under consideration are governed by the Fibonacci, the square Fibonacci and the Penrose lattices. Finally, novel formulae are derived that allow the calculation of Bloch-type waves, and their properties, in infinite periodic structures that can approximate the properties of waves in large, or infinite, quasiperiodic media.

The study of periodic and quasi-periodic orbits in nearly integrable Hamiltonian systems is a long standing and challenging problem, that dates back to Poincaré. Quoting Poincaré, they represent "the only opening through which we can try to enter a place which, up to now, was deemed inaccessible". The aim of this thesis is to find effective and constructive algorithms for constructing both periodic and quasi-periodic solutions via a modification of the normal form methods related to Kolmogorov's theorem. The thesis is divided in two parts. The first part concerns the classical problem of the continuation of periodic orbits surviving to the breaking of invariant maximal or lower dimensional completely resonant tori in nearly integrable Hamiltonian systems: we here propose a new scheme which allows to deal with the problem of degeneracy at any order of perturbation. The second part regards the development of a variation of the Kolmogorov's normalization algorithm, by avoiding the so-called translation step at the price of fixing only the final frequency, while the initial one can only be determined a posteriori.

In this thesis, a numerical design approach has been proposed and developed based on the transmission matrix method in order to characterize periodic and quasi-periodic photonic structures in silicon-on-insulator. The approach and its performance have been extensively tested with specific structures in 2D and its validity has been verified in 3D.

Les équations aux dérivées partielles (EDP) permettent d’aborder d’un point de vue mathématique des phénomènes observés dans tous les domaines des sciences. Certaines EDP non-linéaires modélisent des problèmes de mécanique statistique, mécanique des fluides, théories de la gravitation ou des mathématiques financières.L’objectif de ce travail de thèse est l’étude de certains problèmes d’ EDP non-linéaires et hamiltoniennes et la recherche des leurs solutions périodiques et quasi-périodiques<br>The aim of this thesis is the research of periodic and quasi-periodic solutions for some non-linear hamiltonian PDEs

For several decades, quasi-periodic pulsations (QPP) in flares have been a signature feature of solar dynamics. In the last fifteen years, the advent of new observational instruments has led to a much-improved scope for studying and understanding such phenomena. These events are particularly relevant to the field of coronal seismology, where impulsive events are used as diagnostic tools to estimate the physical parameters of the solar atmosphere remotely. In this thesis we investigate quasi-periodic pulsations in flares from both a numerical and observational perpective, mostly in terms of magnetohydrodynamic (MHD) waves. It has long been suggested that MHD modes may be the cause of QPP in flares as they are capable of modulating a wide range of observable quantities. We study one such mode in detail: the sausage mode. For a model including significant gas pressure, the characteristic period, the ratio of the mode harmonics and the behaviour of the wavenumber cutoff are all considered. Although the period and wavenumber are only marginally affected by this gas pressure, the density contrast ratio and length are important factors. An observational study of a flaring QPP event was undertaken, where new techniques were developed in an attempt to successfully diagnose the QPP mechanism. Cross-correlation mapping was applied to spatially resolved radio data, showing how the strength and phase relationship of a flaring oscillation can be mapped in space. Using this information, we were able to exclude many mechanisms as possible drivers for this event, and suggest that an MHD sausage mode is the likely candidate. A second flaring QPP event was considered, based on the possibility of multiple harmonic oscillations. A sequential spectral peak filtering method was used to demonstrate the presence of multiple significant periods in the flare. Analysis of the harmonic ratios indicated that an MHD wave such as a kink mode was the probable cause. Finally we explore the potential of a new technique in the context of the solar corona, the combination of empirical mode decomposition (EMD) and the Hilbert spectrum. It was established that, under certain circumstances, this method compared favourably with existing analysis techniques such as the Morlet wavelet, and may lead to significant future observational results.

In this dissertation, we first provide a short introduction to qualitative homogenization of elliptic equations and systems. We collect relevant and known results regarding elliptic equations and systems with rapidly oscillating, periodic coefficients, which is the classical setting in homogenization of elliptic equations and systems. We extend several classical results to the so called case of perforated domains and consider materials reinforced with soft inclusions. We establish quantitative H1-convergence rates in both settings, and as a result deduce large-scale Lipschitz estimates and Liouville-type estimates for solutions to elliptic systems with rapidly oscillating periodic bounded and measurable coefficients. Finally, we connect these large-scale estimates with local regulartity results at the microscopic-level to achieve interior Lipschitz regularity at every scale.

Cette thèse traite de systèmes embarqués contrôlés par un ensemble de processus périodiques non synchronisés. Chaque processus est activé quasi-périodiquement, c'est-à-dire périodiquement avec une gigue bornée. Les délais de communication sont également bornés. De tels systèmes réactifs, appelés 'quasi-périodiques', apparaissent dès que l'on branche ensemble deux processus périodiques. Dans la littérature, ils sont parfois qualifiés de systèmes distribués temps-réels synchrones. Nous nous intéressons aux techniques de conception et d'analyse de ces systèmes qui n'imposent pas de synchronisation globale. Les langages synchrones ont été introduits pour faciliter la conception des systèmes réactifs. Ils offrent un cadre privilégié pour programmer, analyser, et vérifier des systèmes quasi-périodiques. En s'appuyant sur une approche synchrone, les contributions de cette thèse s'organisent selon trois thématiques: vérification,implémentation, et simulation des systèmes quasi périodiques.Vérification: 'L'abstraction quasi-synchrone' est une abstraction discrète proposée par Paul Caspi pour vérifier des propriétés de sûreté des systèmes quasi-périodiques. Nous démontrons que cette abstraction est en général incorrecte et nous donnons des conditions nécessaires et suffisantes sur le graphe de communication et les caractéristiques temps-réel de l'architecture pour assurer sa correction. Ces résultats sont ensuite généralisés aux systèmes multi-périodiques.Implémentation: Les 'LTTAs' sont des protocoles conçus pour assurer l'exécution correcte d'une application sur un système quasi-périodique. Nous proposons d'étudier les LTTA dans un cadre synchrone unifié qui englobe l'application et les contrôleurs introduits par les protocoles. Cette approche nous permet de simplifier les protocoles existants, de proposer des versions optimisées, et de donner de nouvelles preuves de correction. Nous présentons également dans le même cadre un protocole fondé sur une synchronisation d'horloge pour comparer les performances des deux approches.Simulation: Un système quasi-périodique est un exemple de modèle faisant intervenir des caractéristiques temps-réels et des tolérances. Pour ce type de modèle non déterministe, nous proposons une 'simulation symbolique', inspirée des techniques de vérification des automates temporisés. Nous montrons comment compiler un modèle mêlant des composantes temps-réel non déterministes et des contrôleurs discrets en un programme discret qui manipule des ensembles de valeurs. Chaque trace du programme résultant capture un ensemble d'exécutions possibles du programme source<br>In this thesis we study embedded controllers implemented as sets of unsynchronized periodic processes. Each process activates quasi-periodically, that is, periodically with bounded jitter, and communicates with bounded transmission delays. Such reactive systems,termed 'quasi-periodic', exist as soon as two periodic processes areconnected together. In the distributed systems literature they arealso known as synchronous real-time models. We focus on techniquesfor the design and analysis of such systems without imposing a globa lclock synchronization. Synchronous languages were introduced as domain specific languages for the design of reactive systems. They offer an ideal framework to program, analyze, and verify quasi-periodic systems. Based on a synchronous approach, this thesis makes contributions to the treatment of quasi-periodic systems along three themes: verification,implementation, and simulation.Verification: The 'quasi-synchronous abstraction' is a discrete abstraction proposed by Paul Caspi for model checking safety properties of quasi-periodic systems. We show that this abstractionis not sound in general and give necessary and sufficient conditionson both the static communication graph of the application and the real-time characteristics of the architecture to recover soundness. We then generalize these results to multirate systems.Implementation: 'Loosely time-triggered architectures' are protocols designed to ensure the correct execution of an application running on a quasi-periodic system. We propose a unified framework that encompasses both the application and the protocol controllers. This framework allows us to simplify existing protocols, propose optimized versions, and give new correctness proofs. We instantiate our framework with a protocol based on clock synchronization to compare the performance of the two approaches.Simulation: Quasi-periodic systems are but one example of timed systems involving real-time characteristics and tolerances. For such nondeterministic models, we propose a 'symbolic simulation' scheme inspired by model checking techniques for timed automata. We show how to compile a model mixing nondeterministic continuous-time and discrete-time dynamics into a discrete program manipulating sets of possible values. Each trace of the resulting program captures a set of possible executions of the source program

Numerical analysis of periodic structures in layered media is usually accomplished by using Method of Moments which requires the formation of the impedance matrix of the structure. The construction of this impedance matrix requires the evaluation of the periodic Green&rsquo<br>s function in layered media which is expressed as an infinite series in terms of the spectral domain Green&rsquo<br>s function. The slow converging nature of this series make these kinds of analysis computationally expensive. Although some papers have proposed methods to accelerate the computation of these series successfully for a single frequency point, it is still very computation intensive to obtain the frequency response of the structure over a band of frequencies. In this thesis, Discrete Complex Image Method (DCIM) is utilized for the efficient computation of the periodic Green&rsquo<br>s function. First, the spectral domain Green&rsquo<br>s function in layered media is approximated by complex exponentials through the use of DCIM. During the application of the DCIM, three-level approximation scheme is employed to improve accuracy. Then, Ewald&rsquo<br>s transformation is applied to accelerate the computation of the infinite series involved in the periodic Green&rsquo<br>s functions. The accuracy and the efficiency of the method is demonstrated through numerical examples.

The aim of this diploma work is the deposition and characterization of ScN/CrN superlattices with both periodic and quasi-periodic structures. ScN as semiconductor material with (2eV) band gap energy was selected due to its thermal and mechanical stability and its hardness. High resistivity against oxidation and high wear resistance were the reasons for choosing CrN as another candidate for the superlattices. The Rudin Shapiro structure was selected as quasi-periodic structure because of its more random structure. In this research both periodic and Rudin Shapiro as quasi-periodic structures have been deposited and investigated. The best optimized temperature for the deposition was 835°C and the selected periodic thickness was 6 nm for periodic structure with each ScN and CrN layers having each a thickness of 3 nm. The material ratio of Rudin Shapiro superlattices was kept the same as periodic samples. Evaluation of quasi-periodic and periodic superlattices was performed by X-ray diffraction measurements. Five peaks were recorded for superlattices measurement of periodic structure by diffraction. The envelope of the diffraction pattern represents two separated peaks in investigation of Rudin Shapiro investigation. The results of the X-ay measurements showed low quality of the superlattices for both deposited structures which suggest the need for further optimization of the deposition process or the use of other materials of superlattices.

L'allègement des structures est un enjeu économique important dans les domaines d'activités industrielles telles que l'automobile, l'aéronautique ou le naval, qui intègrent peu à peu les matériaux composites dans la fabrication des structures. Cet allègement s'accompagne d'un raidissement de la matière qui implique des problèmes de vibrations et d'isolation acoustique. Plusieurs méthodes de traitement existent pour diminuer les vibrations ou le bruit rayonné d'une paroi, mais ces méthodes ont l'inconvénient d'augmenter significativement la masse de la paroi. Afin de répondre à cette problématique, il est proposé dans cette thèse de modifier le comportement dynamique des structures à partir d'un réseau périodique de patchs piézoélectriques shuntés avec un circuit électrique dont il est possible de modifier l'impédance. En contrôlant ainsi le comportement dynamique des patchs piézoélectriques, il est possible de contrôler le comportement vibratoire de la structure et donc de traiter les problèmes de transmissions solidiennes ou de transmissions aériennes.La méthode de shunt choisie est la méthode dite de shunt à capacité négative qui permet de modifier la rigidité d'une structure. Cette méthode dite semi-passive présente plusieurs avantages : la mise en œuvre est simple, il est possible d'intégrer les patchs directement à l'intérieur de la paroi, elle consomme une faible quantité d'énergie électrique et sa mise en application est peu onéreuse<br>Making the structure lighter is an important economic stake in the field of industrial activities such as automotive, aeronautic or naval, which gradually integrate composite materials in the manufacturing of structures. This reduction of the mass goes along with a stiffening of the matter implying acoustics and vibrations issues. Several methods exist to reduce vibrations or acoustic radiations of structures, but these methods increase the mass. In order to answer the problematic, we propose to change the dynamic behavior of structures with a periodic lattice of piezoelectric patches shunted with an electrical circuit whose the impedance can be controlled. Therefore, the control of the coupled behavior of the piezoelectric patches allows the control of vibrational wave's diffusion inside the structure and so to treat the structure-borne vibrations and airborne acoustics emission. The shunt method chosen is negative capacitance shunt which allows to modify the rigidity of a structure. This semi-passive method has several advantages: the implementation is simple, it is possible to integrate the patches directly inside the wall, it consumes a low amonte of electrical energy and its implementation is inexpensive

Quasi-periodic pulsations (QPPs) are a phenomenon commonly observed in solar flares, and are also occasionally observed in stellar flares. They are time variations in the intensity of the flare emission that repeat with approximately constant timescales, or timescales that increase or decrease monotonically in the special case of non-stationary QPPs. There are two main reasons for the interest in QPPs. First is the potential for the diagnosis of plasma parameters in the corona, such as the magnetic field strength and plasma density, which are otherwise difficult to observe directly. If the mechanism causing the QPPs can be inferred, then they would join MHD oscillations of coronal loops as a coronal seismology tool (e.g. Nakariakov & Ofman 2001). Secondly, since QPPs have been found to be a common phenomenon in flares, flares cannot be fully understood without knowing the origin of QPPs. This thesis presents statistical studies of QPPs in both solar and stellar flares, with the aim of learning more about the nature of this phenomenon. The robust detection of QPPs in solar and stellar flares has been the topic of recent debate. In light of this, this thesis shows how a statistical method described by Vaughan (2005), originally developed to test for the presence of periodic variations of the X-ray emission from Seyfert galaxies, can be adapted to aid with the search for QPPs in are time series data. The method identifies statistically significant periodic signals in power spectra, and properly accounts for red noise as well as the uncertainties associated with the data. The method has been further developed to be used with rebinned power spectra, allowing QPPs whose signal is spread over more than one frequency bin to be detected. An advantage of this method is that there is no need to detrend the data prior to creating the power spectrum. Examples are given where the method has been applied to synthetic data, as well as real flare data from the Nobeyama Radioheliograph (NoRH). These show that, despite the transient nature of QPPs, peaks corresponding to the QPPs can be detected at a significant level in the power spectrum without any processing of the original time series data, providing the background trends are not too steep. The properties of a set of solar flares originating from a single active region (AR) that exhibit QPPs were investigated. In particular, any indication of QPP periods relating to AR properties was searched for, as might be expected if the characteristic timescale of the pulsations corresponds to a characteristic length scale of the flaring structure. The three AR properties used for this study were the area, bipole separation distance, and average magnetic field strength, which were all measured at the photosphere using SDO/HMI magnetogram data. The AR studied, best known as NOAA 12192, was unusually long-lived and persisted for over three Carrington rotations. During this time a total of 181 flares were observed by GOES. Data from the GOES, SDO/EVE, Fermi, Vernov and NoRH observatories were used to determine if QPPs were present in the flares. For the soft X-ray GOES and EVE data, the time derivative of the signal was used so that any variability in the impulsive phase of the flare was emphasised. Power spectra of the time series data, without any form of detrending, were inspected and flares with a peak above the 95% confidence level in the power spectrum were labelled as having candidate QPPs. The confidence levels were determined taking full account of data uncertainties and the possible presence of red noise. A total of 37 flares (20% of the sample) showed good evidence of having stationary or weakly non-stationary QPPs, and some of the pulsations can be seen in data from multiple instruments and in different wavebands. Because of the conservative detection method used, this may be a lower bound for the true number of flares with QPPs. The fact that a substantial fraction of the flare sample showed evidence of QPPs, using a strict detection method with minimal processing of the data, demonstrates that these QPPs are a real phenomenon that cannot be explained by the presence of red noise or the superposition of multiple unrelated flares. No correlations were found between the QPP periods and the AR area, bipole separation distance, or average magnetic field strength. This lack of correlation with the AR properties implies that the small-scale structure of the AR (which was not accounted for in this study) is important and/or that different QPP mechanisms act in different cases. Flares that are orders of magnitude larger than the most energetic solar flares have been observed on Sun-like stars, raising the question of whether the same physical processes are responsible for both solar and stellar flares, and hence whether the Sun is capable producing a devastating superflare. A study of QPPs in the decline phase of white-light stellar flares observed by Kepler was embarked upon. Out of the 1439 flares on 216 different stars detected in the short-cadence data using an automated search, 56 flares were found to have QPP-like signatures in the light curve, of which 11 had stable decaying oscillations. No correlation was found between the QPP period and the stellar temperature, radius, rotation period, or surface gravity, suggesting that the QPPs are independent of global stellar parameters. Hence they are likely to be the result of processes occurring in their local environment. There was also no significant correlation between the QPP period and flare energy, while there was evidence that the period scales with the QPP decay time for the Gaussian damping scenario, but not to a significant degree for the exponentially damped case. This same scaling has been observed for MHD oscillations on the Sun, suggesting that they could be the cause of the QPPs in those flares. Scaling laws of the flare energy were also investigated, supporting previous reports of a strong correlation between the flare energy and stellar temperature/radius. Additional analysis was performed on one flare with a rare multi-period QPP pattern. Two periodic signals were identified using the wavelet and autocorrelation techniques. The presence of multiple periods is an indication that the QPPs might have been caused by magnetohydrodynamic oscillations, and suggests that the physical processes operating during stellar flares could be the same as those in solar flares.

Dans ce travail, nous étudions les conditions d'existence de phénomènes quasi périodiques et les implications de leur présence dans plusieurs systèmes dynamiques. Nous nous concentrerons principalement sur la dynamique hamiltonienne où nous fournissons des critères pour la persistance des mouvements quasi-périodiques associés à des tores résonants invariants (chapitre 2), explorons dans quelle mesure l'existence de mouvements quasi-périodiques caractérise le système (chapitre 3) et donnons des exemples des points fixes elliptiques instables avec deux degrés de liberté (chapitre 4). Pour les homéomorphismes critiques du cercle (chapitre 5), nous établissons une relation entre la dimension de Hausdorff de leurs mesures invariantes et leur nombre de rotation. Dans la dernière partie de cette thèse (chapitre 6) nous prouvons, à l'aide des approximations périodiques localisées, l'existence de transformations lisses à entropie nulle mélangeants sur T3 dont le produit avec elles-mêmes est lâchement Bernoulli<br>In this work we study conditions for the existence of quasi-periodic phenomena and the implications of their presence in several dynamical systems. Our main focus will be Hamiltonian dynamics where we provide criteria for the persistence of quasi-periodic motions associated to invariant resonant tori (Chapter 2), explore to what extent the existence of quasi-periodic motions characterize the system (Chapter 3) and give examples of unstable elliptic equilibria with two degrees of freedom (Chapter 4). For critical circle homeomorphisms (Chapter 5) we establish a relation between the Hausdorff dimension of their invariant measures and their rotation number. In the last part of this thesis (Chapter 6) we prove, by means of localized periodic approximations, the existence of mixing zero entropy smooth transformations on T3 whose product with themselves is loosely Bernoulli

Thesis (Ph. D.)--Mechanical Engineering, Georgia Institute of Technology, 2008.<br>Jeremy P. Harvey, Committee Member ; Carl S. Kirkconnell, Committee Member ; Kurt D. Pennell, Committee Member ; S. Mostafa Ghiaasiaan, Committee Chair ; Prateen V. Desai, Committee Member ; Sheldon M. Jeter, Committee Member.