Studies on Some Aspects of Light Beam Propagation Through Certain Nonlinear Media

This thesis deals with the study of light beam propagation through different nonlinear media. Analytical and numerical methods are used to show the formation of solitonS in these media. Basic experiments have also been performed to show the formation of a self-written waveguide in a photopolymer. The variational method is used for the analytical analysis throughout the thesis. Numerical method based on the finite-difference forms of the original partial differential equation is used for the numerical analysis.In Chapter 2, we have studied two kinds of solitons, the (2 + 1) D spatial solitons and the (3 + l)D spatio-temporal solitons in a cubic-quintic medium in the presence of multiphoton ionization.In Chapter 3, we have studied the evolution of light beam through a different kind of nonlinear media, the photorcfractive polymer. We study modulational instability and beam propagation through a photorefractive polymer in the presence of absorption losses. The one dimensional beam propagation through the nonlinear medium is studied using variational and numerical methods. Stable soliton propagation is observed both analytically and numerically.Chapter 4 deals with the study of modulational instability in a photorefractive crystal in the presence of wave mixing effects. Modulational instability in a photorefractive medium is studied in the presence of two wave mixing. We then propose and derive a model for forward four wave mixing in the photorefractive medium and investigate the modulational instability induced by four wave mixing effects. By using the standard linear stability analysis the instability gain is obtained.Chapter 5 deals with the study of self-written waveguides. Besides the usual analytical analysis, basic experiments were done showing the formation of self-written waveguide in a photopolymer system. The formation of a directional coupler in a photopolymer system is studied theoretically in Chapter 6. We propose and study, using the variational approximation as well as numerical simulation, the evolution of a probe beam through a directional coupler formed in a photopolymer system.

On the Synthesis and Multifunctional Properties of some Nanocrystalline Spinel Ferrites and Magnetic Nanocomposites

This thesis lays importance in the preparation and characterization of a few selected representatives of the ferrite family in the nanoregime. The candidates being manganese zinc ferrite and cobalt ferrite prepared by coprecipitation and sol-gel combustion techniques respectively. The thesis not only stresses importance on the preparation techniques and optimization of the reaction conditions, but emphasizes in investigating the various properties namely structural, magnetic and electrical. Passivated nickel nanocomposites are synthesized using polystyrene beads and adopting a novel route of ion exchange reduction. The structural and magnetic properties of these magnetic nanocomposites are correlated. The magnetocaloric effect (MCE) exhibited by these materials are also investigated with a view to finding out the potential of these materials as magnetic refrigerants. Calculations using numerical methods are employed to evaluate the entropy change on selected samples.

Wave Height Ano Speotral Transformation In The Shallow Waters Of Kerala Ooast And Their Preoiotion

There are basically two methods for prediction of shallow water waves, viz. the graphical method and the numerical method. The numerical methods are being widely used, now—a—days, because they are fast, accurate and are especially useful when the prediction over a large spatial frame is required. Practically little has been done on the development of numerical models for the prediction of height and spectral transformation of waves as applicable to our coasts. Synchronized deep and shallow water wave measurements which are essential for study of wave transformation are very much lacking for our coasts. Under these circumstances, a comprehensive study of the wave transformation in the shallow waters of our coast was felt very important and is undertaken in the present investigation.

Meshing highly regular structures: The case of super carbon nanotubes of arbitrary order

Mesh generation is an important step inmany numerical methods.We present the “HierarchicalGraphMeshing” (HGM)method as a novel approach to mesh generation, based on algebraic graph theory.The HGM method can be used to systematically construct configurations exhibiting multiple hierarchies and complex symmetry characteristics. The hierarchical description of structures provided by the HGM method can be exploited to increase the efficiency of multiscale and multigrid methods. In this paper, the HGMmethod is employed for the systematic construction of super carbon nanotubes of arbitrary order, which present a pertinent example of structurally and geometrically complex, yet highly regular, structures. The HGMalgorithm is computationally efficient and exhibits good scaling characteristics. In particular, it scales linearly for super carbon nanotube structures and is working much faster than geometry-based methods employing neighborhood search algorithms. Its modular character makes it conducive to automatization. For the generation of a mesh, the information about the geometry of the structure in a given configuration is added in a way that relates geometric symmetries to structural symmetries. The intrinsically hierarchic description of the resulting mesh greatly reduces the effort of determining mesh hierarchies for multigrid and multiscale applications and helps to exploit symmetry-related methods in the mechanical analysis of complex structures.

A Conservative Front Tracking Algorithm

The discontinuities in the solutions of systems of conservation laws are widely considered as one of the difficulties in numerical simulation. A numerical method is proposed for solving these partial differential equations with discontinuities in the solution. The method is able to track these sharp discontinuities or interfaces while still fully maintain the conservation property. The motion of the front is obtained by solving a Riemann problem based on the state values at its both sides which are reconstructed by using weighted essentially non oscillatory (WENO) scheme. The propagation of the front is coupled with the evaluation of "dynamic" numerical fluxes. Some numerical tests in 1D and preliminary results in 2D are presented.

Linear algebra with applications

Lecture slides and notes for a PhD level course on linear algebra for electrical engineers and computer scientists. This course is given in in the framework of the School of Electronics and Computer Science Mathematics Training Courses https://secure.ecs.soton.ac.uk/notes/pg_maths/ (ECS password required)

Optimal management of natural resources. Accounting for heterogeneity

Dynamic optimization methods have become increasingly important over the last years in economics. Within the dynamic optimization techniques employed, optimal control has emerged as the most powerful tool for the theoretical economic analysis. However, there is the need to advance further and take account that many dynamic economic processes are, in addition, dependent on some other parameter different than time. One can think of relaxing the assumption of a representative (homogeneous) agent in macro- and micro-economic applications allowing for heterogeneity among the agents. For instance, the optimal adaptation and diffusion of a new technology over time, may depend on the age of the person that adopted the new technology. Therefore, the economic models must take account of heterogeneity conditions within the dynamic framework. This thesis intends to accomplish two goals. The first goal is to analyze and revise existing environmental policies that focus on defining the optimal management of natural resources over time, by taking account of the heterogeneity of environmental conditions. Thus, the thesis makes a policy orientated contribution in the field of environmental policy by defining the necessary changes to transform an environmental policy based on the assumption of homogeneity into an environmental policy which takes account of heterogeneity. As a result the newly defined environmental policy will be more efficient and likely also politically more acceptable since it is tailored more specifically to the heterogeneous environmental conditions. Additionally to its policy orientated contribution, this thesis aims making a methodological contribution by applying a new optimization technique for solving problems where the control variables depend on two or more arguments --- the so-called two-stage solution approach ---, and by applying a numerical method --- the Escalator Boxcar Train Method --- for solving distributed optimal control problems, i.e., problems where the state variables, in addition to the control variables, depend on two or more arguments. Chapter 2 presents a theoretical framework to determine optimal resource allocation over time for the production of a good by heterogeneous producers, who generate a stock externalit and derives government policies to modify the behavior of competitive producers in order to achieve optimality. Chapter 3 illustrates the method in a more specific context, and integrates the aspects of quality and time, presenting a theoretical model that allows to determine the socially optimal outcome over time and space for the problem of waterlogging in irrigated agricultural production. Chapter 4 of this thesis concentrates on forestry resources and analyses the optimal selective-logging regime of a size-distributed forest.

Analysis and Simulation of Transverse Random Fracture of Long Fibre Reinforced Composites

La present tesi proposa una metodología per a la simulació probabilística de la fallada de la matriu en materials compòsits reforçats amb fibres de carboni, basant-se en l'anàlisi de la distribució aleatòria de les fibres. En els primers capítols es revisa l'estat de l'art sobre modelització matemàtica de materials aleatoris, càlcul de propietats efectives i criteris de fallada transversal en materials compòsits. El primer pas en la metodologia proposada és la definició de la determinació del tamany mínim d'un Element de Volum Representatiu Estadístic (SRVE) . Aquesta determinació es du a terme analitzant el volum de fibra, les propietats elàstiques efectives, la condició de Hill, els estadístics de les components de tensió i defromació, la funció de densitat de probabilitat i les funcions estadístiques de distància entre fibres de models d'elements de la microestructura, de diferent tamany. Un cop s'ha determinat aquest tamany mínim, es comparen un model periòdic i un model aleatori, per constatar la magnitud de les diferències que s'hi observen. Es defineix, també, una metodologia per a l'anàlisi estadístic de la distribució de la fibra en el compòsit, a partir d'imatges digitals de la secció transversal. Aquest anàlisi s'aplica a quatre materials diferents. Finalment, es proposa un mètode computacional de dues escales per a simular la fallada transversal de làmines unidireccionals, que permet obtenir funcions de densitat de probabilitat per a les variables mecàniques. Es descriuen algunes aplicacions i possibilitats d'aquest mètode i es comparen els resultats obtinguts de la simulació amb valors experimentals.

Desenvolupament de mètodes de predicció de soroll i anàlisi de l'impacte acústic produït pel trànsit viari i el ferrocarril en la ciutat de Girona

La feina feta en aquest treball de tesis s'ha desenvolupat a partir de tres objectius vertebradors, que fonamentalment són: ·El primer dels objectius d'aquest treball de tesis és presentar un recull dels factors que intervenen en l'acústica urbanística: soroll produït pels diferents vehicles, fórmules de predicció de soroll, geometria dels edificis... , tot estudiant els seus efectes en la ciutat de Girona. ·Un altre objectiu ha estat desenvolupar uns mètodes numèrics propis, contrastats experimentalment i extrapolables a qualsevol entorn urbanístic, que permetin predir els valors de les pertorbacions acústiques produïdes pels diferents vehicles en diferents situacions, entre els que es destaquen: -Fórmula de predicció del soroll en un entorn urbà i la seva aplicació a Girona. -Càlcul de l'increment de soroll en un carrer provocat per les reflexions de les ones sonores en les façanes dels edificis. -Estudi del nivell de soroll en la boca de la cavitat d'un túnel produït pel pas del ferrocarril. -Determinació del soroll provocat pel pas d'un tren sobre un viaducte. -Mètode de distribució i planificació del trànsit urbà per disminuir l'impacte acústic sobre la zona. ·El darrer objectiu consisteix en fer una descripció analítica de les principals fonts de soroll que afecten a la ciutat: el trànsit viari i el ferrocarril. Per realitzar aquests objectius s'ha disposat d'un banc de dades amb més de 2.000 mesures sonores de Girona (nivells equivalents de 10 minuts de durada). La metodologia seguida i els principis en que es fonamenta es detallen a l'inici de cada apartat. La finalitat de tots aquests estudis, no és altre que millorar el confort acústic, i la qualitat de vida, de les ciutats. Gairebé tots els grans nuclis de població del planeta es veuen afectats per una gravíssima problemàtica mediambiental, doncs a l'anomenada contaminació acústica cal afegir uns alts índexs de pol·lució atmosfèrica (altes concentracions de biòxid de carboni, generació d'illes de calor...). Aquesta situació, generalitzada arreu del planeta, ha propiciat l'aparició de mesures dràstiques consistents fonamentalment en restringir l'accés dels vehicles motoritzats als nuclis i zones centrals de les àrees urbanes. Precisament aquesta opció s'ha proposat per les zones interiors de Girona on l'elevada densitat de les edificacions deixa un escàs marge per plantejar la construcció de noves rutes o vies alternatives. Cal esmentar que tots els càlculs i teories que es desenvolupen en aquest treball de tesis reflecteixen la realitat acústica actual provocada pels diferents mitjans de transport. Molt possiblement, en un futur no massa llunyà, els nivells de soroll (dB) enregistrats en situacions de tràfic similar seran força menors. Són molts els factors que poden contribuir a aquesta disminució de la intensitat de les emissions sonores: reducció del fregament mecànic, augment del coeficient aerodinàmic, nous materials pels pneumàtics i l'asfalt ... Sense cap mena de dubte, però, una millora transcendental, i no només pel que fa al confort acústic sinó per l'ecosistema en general, seria potenciar la construcció de motors elèctrics o d'hidrogen. Aquests últims per exemple, a diferència dels motors de combustió, funcionen mitjançant piles de combustible que converteixen, amb molta netedat, el gas hidrogen en electricitat i possibiliten l'existència de vehicles no contaminants propulsats per motors elèctrics menys sorollosos. Així, al haver-hi menys fregament entre les parts mòbils del motor (no hi ha pistons ni cilindres) el soroll generat es reduiria considerablement.

A high-order arbitrarily unstructured finite-volume model of the global atmosphere: tests solving the shallow-water equations

Simulations of the global atmosphere for weather and climate forecasting require fast and accurate solutions and so operational models use high-order finite differences on regular structured grids. This precludes the use of local refinement; techniques allowing local refinement are either expensive (eg. high-order finite element techniques) or have reduced accuracy at changes in resolution (eg. unstructured finite-volume with linear differencing). We present solutions of the shallow-water equations for westerly flow over a mid-latitude mountain from a finite-volume model written using OpenFOAM. A second/third-order accurate differencing scheme is applied on arbitrarily unstructured meshes made up of various shapes and refinement patterns. The results are as accurate as equivalent resolution spectral methods. Using lower order differencing reduces accuracy at a refinement pattern which allows errors from refinement of the mountain to accumulate and reduces the global accuracy over a 15 day simulation. We have therefore introduced a scheme which fits a 2D cubic polynomial approximately on a stencil around each cell. Using this scheme means that refinement of the mountain improves the accuracy after a 15 day simulation. This is a more severe test of local mesh refinement for global simulations than has been presented but a realistic test if these techniques are to be used operationally. These efficient, high-order schemes may make it possible for local mesh refinement to be used by weather and climate forecast models.