For gravity wave trains propagating over an arbitrary wavy bottom, a perturbation expansion is developed to the third-order by employing three small perturbation parameters. Both the resonant and non-resonant cases are treated and the singular behavior at resonance is treated separately. All the theoretical results are presented in explicit forms and easy to apply. The bottom effects of different mean water depths and different degrees of undulation, as well as the steepness of incident waves, are clearly described by the theoretical results. In general non-resonant cases, the surface fluctuations deduced from undulated bottom topography decrease as the relative water depth increases and vice versa. The theory can be applied to the cases for wave trains propagating over wavy bottom topography with any arbitrary incident angles which are closer to natural phenomenon in coastal zone. Not only the well-known Bragg resonance but also the higher-order Bragg resonances are included in resonant cases. Unlike previous studies that analyze specific bottom topographies based on prescribed resonant conditions, both Bragg and higher-order Bragg resonances are revealed through the perturbation procedure step by step. For the resonant wave field, the amplification with propagating distance and time is revealed with the aid of the growth of energy flux. This theory is successfully verified by reducing to simpler situations. Also, the analytical results for the special case of two-dimensional wavy bottom are compared with experimental data for validation.

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The solution of large and sparse eigenvalue problems arising from the discretization of the diffusion equation is considered. The multigroup diffusion equation is discretized by means of the Nodal expansion Method (NEM) [9, 10]. A new formulation of the higher order NEM variants revealing the true nature of the problem, that is, a generalized eigenvalue problem, is proposed. These generalized eigenvalue problems are solved using the Jacobi-Davidson (JD) method [26]. The most expensive part of the method consists of solving a linear system referred to as correction equation. It is solved using Krylov subspace methods in combination with aggregation-based Algebraic Multigrid (AMG) techniques. In that context, a particular aggregation technique used in combination with classical smoothers, referred to as oblique geometric coarsening, has been derived. Its particularity is that it aggregates unknowns that are not coupled, which has never been done to our knowledge. A modular code, combining JD with an AMG preconditioner, has been developed. The code comes with many options, that have been tested. In particular, the instability of the Rayleigh-Ritz [33] acceleration procedure in the non-symmetric case has been underlined. Our code has also been compared to an industrial code extracted from ARTEMIS.

Cylindrical dielectric waveguides such as the optical fiber and photonic crystal fiber are very important passive devices in optical communication systems. There are many kinds of commercial software and methods of simulation at present. In this thesis, we proposed the following four methods to analyze arbitrarily profiled cylindrical dielectric waveguides: The first two methods are modified from published work while the last two methods are entirely developed by ourselves. 1. Cylindrical ABCD matrix method: We take the four continuous electromagnetic field components as main variables and derive the exact four-by-four matrix (with Bessel functions) to relate the four field vector within each homogeneous layer. The electromagnetic field components of the inner and outer layer can propagate toward one of the selected interface of our choice by using the method of ABCD matrix. We can then solve for the £]-value of the waveguide mode with this nonlinear inhomogeneous matrix equation. 2. Runge-Kutta method: Runge-Kutta method is mostly used to solve the initial value problems of the differential equations. In this thesis, we introduce the Runge-Kutta method to solve the first-order four-by-four nonlinear differential equation of the electromagnetic field components and find the £]-value of the cylindrical dielectric waveguides in a similar way depicted in method one. 3. Coupled Ez and Hz method: It uses the axial electromagnetic filed components to solve cylindrical dielectric waveguides. The formulation is similar to cylindrical ABCD matrix method, but it requires less variables then cylindrical ABCD matrix method. The numerical solution obtained from this method is most stable, but it is more complicated to derive harder to write the program. 4. Simple basis expansion method: The simple trigonometric functions (sine or cosine) are chosen as the bases of the horizontal coupled magnetic field equation derived from the second-order differential equation of the transverse magnetic field components. We do not select the horizontal coupling electric field because the normal component of the electric field is discontinuous on the interface. But the normal and tangential components of the magnetic field are continuous across the interfaces. The modal solution problem is converted to a linear matrix eigenvalue-eigenvector equation which is solved by the standard linear algebra routines. We will compare these four numerical methods with one another. The characteristics and advantage as well as the disadvantage of each method will be studied and compared in detail.

Transition metal alloys are an important class of materials in heterogeneous catalysis due in no small part to the often greatly enhanced activity and selectivity they exhibit compared to their monometallic constituents. A host of experimental and theoretical studies have demonstrated that, in many cases, these synergistic effects can be attributed to atomic-scale features of the catalyst surface. Realizing the goal of designing -- rather than serendipitously discovering -- new alloy catalysts thus depends on our ability to predict their atomic configuration under technologically relevant conditions. This dissertation presents original research into the development and use of computational tools to accomplish this objective. These tools are all based on a similar strategy: For each of the alloy systems examined, cluster expansion (CE) Hamiltonians were constructed from the results of density functional theory (DFT) calculations, and then used in Metropolis Monte Carlo (MC) simulations to predict properties of interest. Following a detailed description of the DFT+CE+MC simulation scheme, results for the AuPd/Pd(111) and AuPt/Pt(111) surface alloys are presented. These two systems exhibit considerably different trends in their atomic arrangement, which are explicable in terms of their interatomic interactions. In AuPd, a preference for heteronuclear, Au-Pd interactions results in the preferential formation of Pd monomers and other small ensembles, while in AuPt, a preference for homonuclear interactions results in the opposite. AuPd/Pd(100) and AuPt/Pt(100) were similarly examined, revealing not only the effects of the same heteronuclear/homonuclear preferences in this facet, but also a propensity for the formation of second nearest-neighbor pairs of Pd monomers, in close agreement with experiment. Subsequent simulations of the AuPd/Pd(100) surface suggest the application of biaxial compressive strain as a means increasing the population of this catalytically important ensemble of atoms. A method to incorporate the effects of subsurface atomic configuration is also presented, using AuPd as an example. This method represents several improvements over others previously reported in the literature, especially in terms of its simplicity. Finally, we introduce the dimensionless scaled pair interaction, whereby the finite-temperature atomic configuration of any bimetallic surface alloy may be predicted from a small number of relatively inexpensive calculations. / text

Short-distance digital communication links, between chips on a circuit board, or between different circuit boards for example, have traditionally been built by using electrical interconnects - metallic tracks and wires. Recent technological advances have resulted in improvements in the speed of information processing, but have left electrical interconnects intact, thus creating a serious communication problem. Free-space optical interconnects, made up of arrays of vertical-cavity surface-emitting lasers, microlenses, and photodetectors, could be used to solve this problem. If free-space optical interconnects are to successfully replace electrical interconnects, they have to be able to support large rates of information transfer with high channel densities. The biggest obstacle in the way of reaching these requirements is laser beam diffraction. There are three approaches commonly used to model the effects of laser beam diffraction in optical interconnects: one could pursue the path of solving the diffraction integral directly, one could apply stronger approximations with some loss of accuracy of the results, or one could cleverly reinterpret the diffraction problem altogether. None of the representatives of the three categories of existing solutions qualified for our purposes. The main contribution of this dissertation consist of, first, formulating the mode expansion method, and, second, showing that it outperforms all other methods previously used for modelling diffraction in optical interconnects. The mode expansion method allows us to obtain the optical field produced by the diffraction of arbitrary laser beams at empty apertures, phase-shifting optical elements, or any combinations thereof, regardless of the size, shape, position, or any other parameters either of the incident optical field or the observation plane. The mode expansion method enables us to perform all this without any reference or use of the traditional Huygens-Kirchhoff-Fresnel diffraction integrals. When using the mode expansion method, one replaces the incident optical field and the diffracting optical element by an effective beam, possibly containing higher-order transverse modes, so that the ultimate effects of diffraction are equivalently expressed through the complex-valued modal weights. By using the mode expansion method, one represents both the incident and the resultant optical fields in terms of an orthogonal set of functions, and finds the unknown parameters from the condition that the two fields have to be matched at each surface on their propagation paths. Even though essentially a numerical process, the mode expansion method can produce very accurate effective representations of the diffraction fields quickly and efficiently, usually by using no more than about a dozen expanding modes. The second tier of contributions contained in this dissertation is on the subject of the analysis and design of microchannel free-space optical interconnects. In addition to the proper characterisation of the design model, we have formulated several optical interconnect performance parameters, most notably the signal-to-noise ratio, optical carrier-to-noise ratio, and the space-bandwidth product, in a thorough and insightful way that has not been published previously. The proper calculation of those performance parameters, made possible by the mode expansion method, was then performed by using experimentally-measured fields of the incident vertical-cavity surface-emitting laser beams. After illustrating the importance of the proper way of modelling diffraction in optical interconnects, we have shown how to improve the optical interconnect performance by changing either the interconnect optical design, or by careful selection of the design parameter values. We have also suggested a change from the usual 'square' to a novel 'hexagonal' packing of the optical interconnect channels, in order to alleviate the negative diffraction effects. Finally, the optical interconnect tolerance to lateral misalignment, in the presence of multimodal incident laser beams was studied for the first time, and it was shown to be acceptable only as long as most of the incident optical power is emitted in the fundamental Gaussian mode.

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It is well known that a finite-size scatterer has a set of natural resonances, which are uniquely determined by the physical properties of the scatterer. This is also the case for a breast tumor which can be regarded as a dielectric scatterer. Since the scatterer is naturally "tuned" at the resonances, it is expected that an increased electromagnetic coupling would take place at the resonance frequencies compared to other frequencies. For a breast tumor, this would mean a higher power absorption, indicating a faster temperature increase resulting in more efficient hyperthermia. In this dissertation, an adaptive microwave concept is demonstrated for breast cancer applications. The general approach is to detect and identify the tumor-specific resonance, determine the electrical location of the tumor, and apply the focused microwave hyperthermia using the identified resonance and the electrical location. The natural resonances vary depending on the tumor size, shape, and breast tissue configuration. Therefore, an adaptive tuning of the microwave source to tumor-specific resonance frequencies could improve the overall efficiency of hyperthermia treatment by allowing for a faster and more effective heating to achieve a desired therapeutic temperature level. Applying the singularity expansion method (SEM), both the resonances and the electrical location can be obtained from the poles and residues, respectively. This SEM-based approach is computationally inexpensive and can easily be implemented as a combination processing into emerging UWB microwave systems. Alternatively, a relatively simple microwave system based on this concept can potentially be used in conjunction with existing mammography. / Ph. D.