Electromagnetic analysis of periodic structures under a waveguide viewpoint

The fundamental objective of this Ph. D. dissertation is to demonstrate that, under particular circumstances which cover most of the structures with practical interest, periodic structures can be understood and analyzed by means of closed waveguide theories and techniques. To that aim, in the first place a transversely periodic cylindrical structure is considered and the wave equation, under a combination of perfectly conducting and periodic boundary conditions, is studied. This theoretical study runs parallel to the classic analysis of perfectly conducting closed waveguides. Under the light shed by the aforementioned study it is clear that, under certain very common periodicity conditions, transversely periodic cylindrical structures share a lot of properties with closed waveguides. Particularly, they can be characterized by a complete set of TEM, TE and TM modes. As a result, this Ph. D. dissertation introduces the transversely periodic waveguide concept. Once the analogies between the modes of a transversely periodic waveguide and the ones of a closed waveguide have been established, a generalization of a well-known closed waveguide characterization method, the generalized Transverse Resonance Technique, is developed for the obtention of transversely periodic modes. At this point, all the necessary elements for the consideration of discontinuities between two different transversely periodic waveguides are at our disposal. The analysis of this type of discontinuities will be carried out by means of another well known closed waveguide method, the Mode Matching technique. This Ph. D. dissertation contains a sufficient number of examples, including the analysis of a wire-medium slab, a cross-shaped patches periodic surface and a parallel plate waveguide with a textured surface, that demonstrate that the Transverse Resonance Technique - Mode Matching hybrid is highly precise, efficient and versatile. Thus, the initial statement: ”periodic structures can be understood and analyzed by means of closed waveguide theories and techniques”, will be corroborated. Finally, this Ph. D. dissertation contains an adaptation of the aforementioned generalized Transverse Resonance Technique by means of which the analysis of laterally open periodic waveguides, such as the well known Substrate Integrated Waveguides, can be carried out without any approximation. The analysis of this type of structures has suscitated a lot of interest in the recent past and the previous analysis techniques proposed always resorted to some kind of fictitious wall to close the structure. vii Resumen El principal objetivo de esta tesis doctoral es demostrar que, bajo ciertas circunstancias que se cumplen para la gran mayoría de estructuras con interés práctico, las estructuras periódicas se pueden analizar y entender con conceptos y técnicas propias de las guías de onda cerradas. Para ello, en un primer lugar se considera una estructura cilíndrical transversalmente periódica y se estudia la ecuación de onda bajo una combinación de condiciones de contorno periódicas y de conductor perfecto. Este estudio teórico y de caracter general, sigue el análisis clásico de las guías de onda cerradas por conductor eléctrico perfecto. A la luz de los resultados queda claro que, bajo ciertas condiciones de periodicidad (muy comunes en la práctica) las estructuras cilíndricas transversalmente periódicas guardan multitud de analogías con las guías de onda cerradas. En particular, pueden ser descritas mediante un conjunto completo de modos TEM, TE y TM. Por ello, ésta tesis introduce el concepto de guía de onda transversalmente periódica. Una vez establecidas las similitudes entre las soluciones de la ecuación de onda, bajo una combinación de condiciones de contorno periódicas y de conductor perfecto, y los modos de guías de onda cerradas, se lleva a cabo, con éxito, la adaptación de un conocido método de caracterización de guías de onda cerradas, la técnica de la Resonancia Transversal Generalizada, para la obtención de los modos de guías transversalmente periódicas. En este punto, se tienen todos los elementos necesarios para considerar discontinuidades entre guías de onda transversalmente periódicas. El analisis de este tipo de discontinuidades se llevará a cabo mediante otro conocido método de análisis de estructuras cerradas, el Ajuste Modal. Esta tesis muestra multitud de ejemplos, como por ejemplo el análisis de un wire-medium slab, una superficie de parches con forma de cruz o una guía de placas paralelas donde una de dichas placas tiene cierta textura, en los que se demuestra que el método híbrido formado por la Resonancia Transversal Generalizada y el Ajuste Modal, es tremendamente preciso, eficiente y versátil y confirmará la validez de el enunciado inicial: ”las estructuras periódicas se pueden analizar y entender con conceptos y técnicas propias de las guías de onda cerradas” Para terminar, esta tésis doctoral incluye también una modificación de la técnica de la Resonancia Transversal Generalizada mediante la cual es posible abordar el análisis de estructuras periódica abiertas en los laterales, como por ejemplo las famosas guías de onda integradas en sustrato, sin ninguna aproximación. El análisis de este tipo de estructuras ha despertado mucho interés en los últimos años y las técnicas de análisis propuestas hasta ix el momento acostumbran a recurrir a algún tipo de pared ficticia para simular el carácter abierto de la estructura.

Realización de un procedimiento para la medida del coeficiente de absorción sonora normal

El objetivo de este trabajo es la elaboración de un procedimiento para la medida del coeficiente de absorción sonora normal en un tubo de impedancia. Para ello se han estudiado los fundamentos básicos de la ecuación de ondas y sus soluciones. Se han considerado las soluciones pertinentes que describen el comportamiento de una onda sonora dentro de un tubo rígido. Se ha considerado también la teoría básica de funciones de transferencia. Estas teorías son claves a la hora de poder desarrollar el procedimiento de medida, ya que el coeficiente de absorción acústica se obtendrá con la ayuda de un tubo de impedancias que mide las funciones de transferencia entre dos posiciones de micrófonos incorporados en una de las caras del tubo. La utilización de esta técnica tiene como principal ventaja, la necesidad de poco espacio en un laboratorio y el empleo de muestras pequeñas de material. La implementación de los visto teóricamente a su aplicación práctica se ha hecho a través de un procedimiento de medida que sigue la Norma UNE-EN ISO 10534-2 (2002) “Determinación del coeficiente de absorción sonoro y la impedancia en tubos de impedancia Parte 2: método función de transferencia”. El valor del coeficiente de absorción se puede obtener a través de una instrumentación específica y un programa computador. Para poder validar los cálculos que realiza el programa utilizado, se ha realizado una batería de medidas del coeficiente de absorción a diferentes tipos de materiales acústicos, y los cálculos se han hecho por la vía del programa y por la vía de una hoja de cálculo. Como parte del procedimiento de medida se ha calculado la incertidumbre en las medidas. En definitiva se pretende contribuir con este trabajo a establecer un procedimiento de medida del comportamiento acústico de diversos materiales. SUMMARY. The aim of this work is the development of a procedure for measuring the sound absorption coefficient normal of an impedance tube. To this end we have studied the basics of the wave equation and its solutions. We have considered the relevant solutions that describe the behavior of a sound wave in a rigid tube. It has also considered the basic theory of transfer functions. These theories are key when we want to develop the measurement method, since the absorption coefficient is obtained with the aid of an impedance tube measuring transfer functions between two positions of microphones incorporated into one side of the tube. The use of this technique has the main advantage, the need of little space on a laboratory and use of small samples of material. The implementation of theoretically seen to his practical application has been made through a measurement procedure following the UNE-EN ISO 10534-2 (2002) "Determination of sound absorption coefficient and impedance in impedance tubes Part 2 : transfer function method ". The value of the absorption coefficient can be obtained through a specific instrumentation and computer software. In order to validate the calculations performed by the program used, there has been realized a series of measures of the absorption coefficient at different types of acoustical materials, and calculations were made by means of the program and by means of a spreadsheet. As part of the measurement procedure has been estimated uncertainty in the measurements. Ultimately it’s tried to contribute with this work to establish a procedure measuring the acoustic behavior of various materials.

Análisis, modelado e igualación no lineal de arrays paramétricos acústicos

La presente Tesis analiza y desarrolla metodología específica que permite la caracterización de sistemas de transmisión acústicos basados en el fenómeno del array paramétrico. Este tipo de estructuras es considerado como uno de los sistemas más representativos de la acústica no lineal con amplias posibilidades tecnológicas. Los arrays paramétricos aprovechan la no linealidad del medio aéreo para obtener en recepción señales en el margen sónico a partir de señales ultrasónicas en emisión. Por desgracia, este procedimiento implica que la señal transmitida y la recibida guardan una relación compleja, que incluye una fuerte ecualización así como una distorsión apreciable por el oyente. Este hecho reduce claramente la posibilidad de obtener sistemas acústicos de gran fidelidad. Hasta ahora, los esfuerzos tecnológicos dirigidos al diseño de sistemas comerciales han tratado de paliar esta falta de fidelidad mediante técnicas de preprocesado fuertemente dependientes de los modelos físicos teóricos. Estos están basados en la ecuación de propagación de onda no lineal. En esta Tesis se propone un nuevo enfoque: la obtención de una representación completa del sistema mediante series de Volterra que permita inferir un sistema de compensación computacionalmente ligero y fiable. La dificultad que entraña la correcta extracción de esta representación obliga a desarrollar una metodología completa de identificación adaptada a este tipo de estructuras. Así, a la hora de aplicar métodos de identificación se hace indispensable la determinación de ciertas características iniciales que favorezcan la parametrización del sistema. En esta Tesis se propone una metodología propia que extrae estas condiciones iniciales. Con estos datos, nos encontramos en disposición de plantear un sistema completo de identificación no lineal basado en señales pseudoaleatorias, que aumenta la fiabilidad de la descripción del sistema, posibilitando tanto la inferencia de la estructura basada en bloques subyacente, como el diseño de mecanismos de compensación adecuados. A su vez, en este escenario concreto en el que intervienen procesos de modulación, factores como el punto de trabajo o las características físicas del transductor, hacen inviables los algoritmos de caracterización habituales. Incluyendo el método de identificación propuesto. Con el fin de eliminar esta problemática se propone una serie de nuevos algoritmos de corrección que permiten la aplicación de la caracterización. Las capacidades de estos nuevos algoritmos se pondrán a prueba sobre un prototipo físico, diseñado a tal efecto. Para ello, se propondrán la metodología y los mecanismos de instrumentación necesarios para llevar a cabo el diseño, la identificación del sistema y su posible corrección, todo ello mediante técnicas de procesado digital previas al sistema de transducción. Los algoritmos se evaluarán en términos de error de modelado a partir de la señal de salida del sistema real frente a la salida sintetizada a partir del modelo estimado. Esta estrategia asegura la posibilidad de aplicar técnicas de compensación ya que éstas son sensibles a errores de estima en módulo y fase. La calidad del sistema final se evaluará en términos de fase, coloración y distorsión no lineal mediante un test propuesto a lo largo de este discurso, como paso previo a una futura evaluación subjetiva. ABSTRACT This Thesis presents a specific methodology for the characterization of acoustic transmission systems based on the parametric array phenomenon. These structures are well-known representatives of the nonlinear acoustics field and display large technological opportunities. Parametric arrays exploit the nonlinear behavior of air to obtain sonic signals at the receptors’side, which were generated within the ultrasonic range. The underlying physical process redunds in a complex relationship between the transmitted and received signals. This includes both a strong equalization and an appreciable distortion for a human listener. High fidelity, acoustic equipment based on this phenomenon is therefore difficult to design. Until recently, efforts devoted to this enterprise have focused in fidelity enhancement based on physically-informed, pre-processing schemes. These derive directly from the nonlinear form of the wave equation. However, online limited enhancement has been achieved. In this Thesis we propose a novel approach: the evaluation of a complete representation of the system through its projection onto the Volterra series, which allows the posterior inference of a computationally light and reliable compensation scheme. The main difficulty in the derivation of such representation strives from the need of a complete identification methodology, suitable for this particular type of structures. As an example, whenever identification techniques are involved, we require preliminary estimates on certain parameters that contribute to the correct parameterization of the system. In this Thesis we propose a methodology to derive such initial values from simple measures. Once these information is made available, a complete identification scheme is required for nonlinear systems based on pseudorandom signals. These contribute to the robustness and fidelity of the resulting model, and facilitate both the inference of the underlying structure, which we subdivide into a simple block-oriented construction, and the design of the corresponding compensation structure. In a scenario such as this where frequency modulations occur, one must control exogenous factors such as devices’ operation point and the physical properties of the transducer. These may conflict with the principia behind the standard identification procedures, as it is the case. With this idea in mind, the Thesis includes a series of novel correction algorithms that facilitate the application of the characterization results onto the system compensation. The proposed algorithms are tested on a prototype that was designed and built for this purpose. The methodology and instrumentation required for its design, the identification of the overall acoustic system and its correction are all based on signal processing techniques, focusing on the system front-end, i.e. prior to transduction. Results are evaluated in terms of input-output modelling error, considering a synthetic construction of the system. This criterion ensures that compensation techniques may actually be introduced, since these are highly sensible to estimation errors both on the envelope and the phase of the signals involved. Finally, the quality of the overall system will be evaluated in terms of phase, spectral color and nonlinear distortion; by means of a test protocol specifically devised for this Thesis, as a prior step for a future, subjective quality evaluation.

Modelling relativistic solitary wave interactions in over-dense plasmas: a perturbed nonlinear Schröndinger equation framework

We investigate the dynamics of localized solutions of the relativistic cold-fluid plasma model in the small but finite amplitude limit, for slightly overcritical plasma density. Adopting a multiple scale analysis, we derive a perturbed nonlinear Schrödinger equation that describes the evolution of the envelope of circularly polarized electromagnetic field. Retaining terms up to fifth order in the small perturbation parameter, we derive a self-consistent framework for the description of the plasma response in the presence of localized electromagnetic field. The formalism is applied to standing electromagnetic soliton interactions and the results are validated by simulations of the full cold-fluid model. To lowest order, a cubic nonlinear Schrödinger equation with a focusing nonlinearity is recovered. Classical quasiparticle theory is used to obtain analytical estimates for the collision time and minimum distance of approach between solitons. For larger soliton amplitudes the inclusion of the fifth-order terms is essential for a qualitatively correct description of soliton interactions. The defocusing quintic nonlinearity leads to inelastic soliton collisions, while bound states of solitons do not persist under perturbations in the initial phase or amplitude

Hard transition to chaotic dynamics in Alfven wave fronts

The derivative nonlinear Schrodinger DNLS equation, describing propagation of circularly polarized Alfven waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear, cubic coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. In a reduced three-wave model equal dampings of daughter waves, three-dimensional flow for two wave amplitudes and one relative phase, no matter how small the growth rate of the unstable wave there exists a parametric domain with the flow exhibiting chaotic relaxation oscillations that are absent for zero growth rate. This hard transition in phase-space behavior occurs for left-hand LH polarized waves, paralleling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable, with damping less than about unstable wave frequency 2/4 x ion cyclotron frequency. The structural stability of the transition was explored by going into a fully 3-wave model different dampings of daughter waves,four-dimensional flow; both models differ in significant phase-space features but keep common features essential for the transition.

Damping models in the truncated derivative nonlinear Schrödinger equation

Four-dimensional flow in the phase space of three amplitudes of circularly polarized Alfven waves and one relative phase, resulting from a resonant three-wave truncation of the derivative nonlinear Schrödinger equation, has been analyzed; wave 1 is linearly unstable with growth rate , and waves 2 and 3 are stable with damping 2 and 3, respectively. The dependence of gross dynamical features on the damping model as characterized by the relation between damping and wave-vector ratios, 2 /3, k2 /k3, and the polarization of the waves, is discussed; two damping models, Landau k and resistive k2, are studied in depth. Very complex dynamics, such as multiple blue sky catastrophes and chaotic attractors arising from Feigenbaum sequences, and explosive bifurcations involving Intermittency-I chaos, are shown to be associated with the existence and loss of stability of certain fixed point P of the flow. Independently of the damping model, P may only exist as against flow contraction just requiring.In the case of right-hand RH polarization, point P may exist for all models other than Landau damping; for the resistive model, P may exist for RH polarization only if 2+3/2.

Reduced three-wave model to study the hard transition to chaotic dynamics in Alfven wave-fronts

The derivative nonlinear Schrödinger (DNLS) equation, describing propagation of circularly polarized Alfven waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear, cubic coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. In a reduced three-wave model (equal damping of daughter waves, three-dimensional flow for two wave amplitudes and one relative phase), no matter how small the growth rate of the unstable wave there exists a parametric domain with the flow exhibiting chaotic dynamics that is absent for zero growth-rate. This hard transition in phase-space behavior occurs for left-hand (LH) polarized waves, paralelling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable.

Fully 3-wave model to study the hard transition to chaotic dynamics in alfven wave-fronts

The derivative nonlinear Schrödinger (DNLS) equation, describing propagation of circularly polarized Alfven waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. No matter how small the growth rate of the unstable wave, the four-dimensional flow for the three wave amplitudes and a relative phase, with both resistive damping and linear Landau damping, exhibits chaotic relaxation oscillations that are absent for zero growth-rate. This hard transition in phase-space behavior occurs for left-hand (LH) polarized waves, paralleling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable. The parameter domain developing chaos is much broader than the corresponding domain in a reduced 3-wave model that assumes equal dampings of the daughter waves

Fast magnetosonic wave excitation by an array of wires with time-modulated currents

The excitation of Fast Magnetosonic (FMS)waves by a cylindrical array of parallel tethers carrying timemodulated current is discussed. The tethers would fly vertical in the equatorial plane, which is perpendicular to the geomagnetic field when its tilt is ignored, and would be stabilized by the gravity gradient. The tether array would radiate a single FMS wave. In the time-dependent background made of geomagnetic field plus radiated wave, plasma FMS perturbations are excited in the array vicinity through a parametric instability. The growth rate is estimated by truncating the evolution equation for FMS perturbations to the two azimuthal modes of lowest order. Design parameters such as tether length and number, required power and mass are discussed for Low Earth Orbit conditions. The array-attached wave structure would have the radiated wave controlled by the intensity and modulation frequency of the currents, making an active experiment on non-linear low frequency waves possible in real space plasma conditions.

Direct estimation wave setup as a medium level in swash

The extreme runup is a key parameter for a shore risk analysis in which the accurate and quantitative estimation of the upper limit reached by waves is essential. Runup can be better approximated by splitting the setup and swash semi-amplitude contributions. In an experimental study recording setup becomes difficult due to infragravity motions within the surf zone, hence, it would be desirable to measure the setup with available methodologies and devices. In this research, an analysis is made of evaluated the convenience of direct estimation setup as the medium level in the swash zone for experimental runup analysis through a physical model. A physical mobile bed model was setup in a wave flume at the Laboratory for Maritime Experimentation of CEDEX. The wave flume is 36 metres long, 6.5 metres wide and 1.3 metres high. The physical model was designed to cover a reasonable range of parameters, three different slopes (1/50, 1/30 and 1/20), two sand grain sizes (D50 = 0.12 mm and 0.70 mm) and a range for the Iribarren number in deep water (ξ0) from 0.1 to 0.6. Best formulations were chosen for estimating a theoretical setup in the physical model application. Once theoretical setup had been obtained, a comparison was made with an estimation of the setup directly as a medium level of the oscillation in swash usually considered in extreme runup analyses. A good correlation was noted between both theoretical and time-averaging setup and a relation is proposed. Extreme runup is analysed through the sum of setup and semi-amplitude of swash. An equation is proposed that could be applied in strong foreshore slope-dependent reflective beaches.

Long wave theory for experimental devices with compressed/expanded surfactant monolayers

Surfactant monolayers are of interest in a variety of phenomena, including thin film dynamics and the formation and dynamics of foams. Measurement of surface properties has received a continuous attention and requires good theoretical models to extract the relevant physico- chemical information from experimental data. A common experimental set up consists in a shallow liquid layer whose free surface is slowly com- pressed/expanded in periodic fashion by moving two slightly immersed solid barriers, which varies the free surface area and thus the surfactant concentration. The simplest theory ignores the fluid dynamics in the bulk fluid, assuming spatially uniform surfactant concentration, which requires quite small forcing frequencies and provides reversible dynamics in the compression/expansion cycles. Sometimes, it is not clear whether depar- ture from reversibility is due to non-equilibrium effects or to the ignored fluid dynamics. Here we present a long wave theory that takes the fluid dynamics and the symmetries of the problem into account. In particular, the validity of the spatially-uniform-surfactant-concentration assumption is established and a nonlinear diffusion equation is derived. This allows for calculating spatially nonuniform monolayer dynamics and uncovering the physical mechanisms involved in the surfactant behavior. Also, this analysis can be considered a good means for extracting more relevant information from each experimental run.

Type-I intermittency with discontinuous reinjection probability density in a truncation model of the derivative nonlinear Schrödinger equation

In previous papers, the type-I intermittent phenomenon with continuous reinjection probability density (RPD) has been extensively studied. However, in this paper type-I intermittency considering discontinuous RPD function in one-dimensional maps is analyzed. To carry out the present study the analytic approximation presented by del Río and Elaskar (Int. J. Bifurc. Chaos 20:1185-1191, 2010) and Elaskar et al. (Physica A. 390:2759-2768, 2011) is extended to consider discontinuous RPD functions. The results of this analysis show that the characteristic relation only depends on the position of the lower bound of reinjection (LBR), therefore for the LBR below the tangent point the relation {Mathematical expression}, where {Mathematical expression} is the control parameter, remains robust regardless the form of the RPD, although the average of the laminar phases {Mathematical expression} can change. Finally, the study of discontinuous RPD for type-I intermittency which occurs in a three-wave truncation model for the derivative nonlinear Schrodinger equation is presented. In all tests the theoretical results properly verify the numerical data

A time-domain finite element method for seakeeping and wave resistance problems

El objetivo de la tesis es la investigación de algoritmos numéricos para el desarrollo de herramientas numéricas para la simulación de problemas tanto de comportamiento en la mar como de resistencia al avance de buques y estructuras flotantes. La primera herramienta desarrollada resuelve el problema de difracción y radiación de olas. Se basan en el método de los elementos finitos (MEF) para la resolución de la ecuación de Laplace, así como en esquemas basados en MEF, integración a lo largo de líneas de corriente, y en diferencias finitas desarrollados para la condición de superficie libre. Se han desarrollado herramientas numéricas para la resolución de la dinámica de sólido rígido en sistemas multicuerpos con ligaduras. Estas herramientas han sido integradas junto con la herramienta de resolución de olas difractadas y radiadas para la resolución de problemas de interacción de cuerpos con olas. También se han diseñado algoritmos de acoplamientos con otras herramientas numéricas para la resolución de problemas multifísica. En particular, se han realizado acoplamientos con una herramienta numérica basada de cálculo de estructuras con MEF para problemas de interacción fluido-estructura, otra de cálculo de líneas de fondeo, y con una herramienta numérica de cálculo de flujos en tanques internos para problemas acoplados de comportamiento en la mar con “sloshing”. Se han realizado simulaciones numéricas para la validación y verificación de los algoritmos desarrollados, así como para el análisis de diferentes casos de estudio con aplicaciones diversas en los campos de la ingeniería naval, oceánica, y energías renovables marinas. ABSTRACT The objective of this thesis is the research on numerical algorithms to develop numerical tools to simulate seakeeping problems as well as wave resistance problems of ships and floating structures. The first tool developed is a wave diffraction-radiation solver. It is based on the finite element method (FEM) in order to solve the Laplace equation, as well as numerical schemes based on FEM, streamline integration, and finite difference method tailored for solving the free surface boundary condition. It has been developed numerical tools to solve solid body dynamics of multibody systems with body links across them. This tool has been integrated with the wave diffraction-radiation solver to solve wave-body interaction problems. Also it has been tailored coupling algorithms with other numerical tools in order to solve multi-physics problems. In particular, it has been performed coupling with a MEF structural solver to solve fluid-structure interaction problems, with a mooring solver, and with a solver capable of simulating internal flows in tanks to solve couple seakeeping-sloshing problems. Numerical simulations have been carried out to validate and verify the developed algorithms, as well as to analyze case studies in the areas of marine engineering, offshore engineering, and offshore renewable energy.