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The accuracy of moment method solutions to electromagnetic scattering problems has been studied by many researchers. Error bounds for the moment method have been obtained in terms of Sobolev norms of the current solution. Motivated by the historical origins of Sobolev spaces as energy spaces, it is shown that the Sobolev norm used in these bounds is equivalent to the forward scattering amplitude, for the case of 2D scattering from a PEC circular cylinder. A slightly weaker relationship is obtained for 3D scattering from a PEC sphere. These results provide a physical meaning for abstract solution error bounds in terms of the power radiated by the error in the current solution. It is further shown that bounds on the Sobolev norm of the current error imply a bound on the error in the computed backscattering amplitude. Since Sobolev-based error bounds do not provide the actual error in a solution nor identify its source, the error in typical moment method scattering solutions for smooth cylindrical geometries is analyzed. To quantify the impact of mesh element size, approximate integration of moment matrix elements, and geometrical discretization error on the accuracy of computed surface currents and scattering amplitudes, error estimates are derived analytically for the circular cylinder. These results for the circular cylinder are empirically compared to computed error values for other smooth scatterer geometries, with consistent results obtained. It is observed that moment method solutions to the magnetic field integral equation are often less accurate for a given grid than corresponding solutions to the electric field integral equation. Building from the error analysis, the cause of this observation is proposed to be the identity operator in the magnetic formulation. A regularization of the identity operator is then derived that increases the convergence rate of the discretized 2D magnetic field integral equation by three orders.

We investigate the spatial evolution of an intense laser pulse as it generates high-order harmonics in a long gas cell, filled with 80 torr of helium. A thin foil separates the gas-filled region of the cell from a subsequent evacuated region. The exit plane of the gas cell can be scanned along the laser axis so that the evolution of the laser throughout the focus can be observed (full scanning range of 9 cm). We constructed an apparatus that images the laser radial energy profile as it exits the cell. The high harmonics, odd orders ranging from 45 to 91, are observed at the same time that the laser spot is characterized. Re-absorption of the harmonics within the gas cell restricts the region of harmonic emission to the final centimeter (or less) of the cell. We present the first direct evidence (to our knowledge) of laser filamentation under conditions ideal for high-order harmonic generation. The 30 fs, 4 mJ, laser pulses were observed to undergo double focusing within the gas cell, with about 4 cm separating the two foci. The region with best harmonic emission occurs midway between the two foci. The radial profile of the laser focus, 150-200 microns in diameter, evolves from a Gaussian-like profile to a more square-top profile as it propagates over several centimeters. The filamentation phenomenon as well as the brightness of the harmonics improves when an aperture is partially closed on the laser beam before reaching the focusing mirror. A spectral sampling of the imaged laser focus revealed a 4 nm blue-shift associated with the generation of plasma in the gas cell. The blue-shifting occurs primarily in the center of the laser beam and less at the wider radii. The initial laser pulse had a spectrum centered at 800 nm with a 35 nm bandwidth. The energy associated with each of the observed 26 harmonic beams was found to be approximately 1 nJ, yielding a conversion efficiency of approximately 2e−7.

<p> High index contrast optical waveguides have recently attracted much attention as a promising platform for ultradense photonic integrated circuits. The vector nature and fine geometry of such waveguiding structures impose new challenges for numerical modeling. By introducing the high-order finite-difference method, highly accurate and efficient modeling techniques have been developed in this thesis for simulation and design of high index contrast waveguiding structures with compact size.</p> <p> High-order mode solving techniques are first presented for modal analyses. Their advantages in accuracy have been demonstrated for high index contrast optical waveguides and bent waveguides with small bending radius.</p> <p> Later, a class of high-order propagation algorithms, including the paraxial and wide-angle beam propagation methods, reflective operator method and bidirectional beam propagation method, have been developed for modeling longitudinally slow-varying structures, single waveguide discontinuity and piecewise z-invariant structures, respectively. All the proposed propagation algorithms have been shown to provide significant improvement in accuracy and efficiency in comparison with conventional methods, especially when simulating high index contrast structures with small feature size.</p> <p> Accurate modeling of evanescent waves is critical for the simulation of strongly reflecting structures with high longitudinal index contrast. Various rational approximations to square root operators used in the bidirectional beam propagation method have been comprehensively assessed. Useful guidelines for accurate modeling of evanescent and propagating modes are provided.</p> <p> Finally, the efficient high-order bidirectional beam propagation method is introduced for the design of Bragg gratings on high index contrast and plasmonic waveguides. Good performance is achieved.</p> / Thesis / Doctor of Philosophy (PhD)

High harmonic generation (HHG) is a process that occurs when an intense laser interacts with a material and generates new frequencies of light. HHG has many practical applications, namely as a spectroscopy technique and source for high frequency light and attosecond pulses. While HHG has been done extensively in gases, HHG in solids is a relatively new field. Solids are appealing as an HHG medium as they require much simpler equipment and are subsequently much more compact, and thus may have a variety of applications previously inaccessible to gas-phase HHG. However, the generation mechanism of HHG in solids has not been fully characterized yet, as the processes behind HHG in gases and solids are not synonymous. Here, we study the influence of polarization, symmetry, and setup geometry on HHG in solids. We study the propagation effects in a transmission geometry setup and use Jones calculus to counteract the polarization change from propagation. We compare these results to a reflection geometry setup, which naturally does not have propagation effects, to determine the validity of the polarization correction technique. We also look at the electric field symmetry dependence on HHG through the manipulation of the laser electric field with a two-color interferometer. The impact of symmetry dependence and propagation effects both contribute to a better understanding of the HHG process in solids.

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No description available.

No description available.

Studying the effects of free stream turbulence on noise, vibration, and heat transfer on structures is very important in engineering applications. The problem of the interaction of large scale turbulence with a flat-plate cascade is a model of important problems in propulsion systems. Addressing the problem of large scale turbulence interacting with a flat plate cascade requires flow simulation over a large number of plates (6-12 plates) in order to be able to represent numerically integral length scales on the order of blade-to-blade spacing. Having such a large number of solid surfaces in the simulation requires very large computational grid points to resolve the boundary layers on the plates, and that is not possible with the current computing resources. In this thesis we develop a computational technique to predict the distortion of homogeneous isotropic turbulence as it passes through a cascade of thin flat plates. We use Large-Eddy Simulation (LES) to capture the spatial development of the incident turbulence and its interaction with the plates which are assumed to be inviscid walls. The LES is conducted for a linear cascade composed of six plates. Because suppression of the normal component of velocity is the main mechanism of distortion, we neglect the presence of mean shear in the boundary layers and wakes, and allow slip velocity on the plate surfaces. We enforce the zero normal velocity condition on the plates. This boundary condition treatment is motivated by rapid distortion theory (RDT) in which viscous effects are neglected, however, the present LES approach accounts for nonlinear and turbulence diffusion effects by a sub-grid scale model. We refer to this type of turbulence-blade interaction as shear-free interaction. To validate our calculations, we computed the unsteady loading and radiated acoustic pressure field from flat plates interacting with vortical structures. We consider two fundamental problems: (1) A linear cascade of flat plates excited by a vortical wave (gust) given by a 2D Fourier mode, and (2) The parallel interaction of a finite-core vortex with a single plate. We solve the nonlinear Euler equations by a high-order finite-differece method. We use nonreflecting boundary conditions at the inflow and outflow boundaries. For the gust problem, we found that the cascade response depends sensitively on the frequency of the convicted gust. The unsteady surface pressure distribution and radiated pressure field agree very well with predictions of the linear theory for the tested range of reduced frequency. We have also investigated the effects of the incident gust frequency on the undesirable wave reflection at the inflow and outflow boundaries. For the vortex-plate interaction problem, we investigate the effects of the internal structure of the vortex on the strength and directivity of radiated sound. Then we solved the turbulence cascade interaction problem. The normal Reynolds stresses and velocity spectra are analyzed ahead, within, and downstream of the cascade. Good agreement with predictions of rapid distortion theory in the region of its validity is obtained. Also, the normal Reynolds stress profiles are found to be in qualitative agreement with available experimental data. As such, this dissertation presents a viable computational alternative to rapid distortion theory (RDT) for the prediction of noise radiation due to the interaction of free stream turbulence with structures. / Ph. D.

La prévention des risques industriels nécessite de simuler la dispersion turbulente de polluants. Cependant, les outils majoritairement utilisés à ce jour ne permettent pas de traiter les champs proches dans le cas de géométries complexes, et il est nécessaire d'utiliser les outils de CFD (“ Computational Fluid Dynamics ”) plus adaptés, mais plus coûteux. Afin de simuler les écoulements atmosphériques avec dispersion de polluants, les modèles CFD doivent modéliser correctement d'une part, les effets de flottabilité, et d'autre part les effets de la turbulence. Plusieurs approches existent, notamment dans la prise en compte des effets de flottabilité et la modélisation de la turbulence, et nécessitent des méthodes numériques adaptées aux spécificités mathématiques de chacune d'entre elles, ainsi que des schémas numériques précis pour ne pas polluer la modélisation. Une formulation d'ordre élevé en volumes finis, sur maillages non structurés, parallélisée, est proposée pour simuler les écoulements atmosphériques avec dispersion de polluants. L'utilisation de schémas d'ordre élevé doit permettre d'une part de réduire le nombre de cellules et diminuer les temps de simulation pour atteindre une précision donnée, et d'autre part de mieux contrôler la viscosité numérique des schémas en vue de simulations LES (Large Eddy Simulation), pour lesquelles la viscosité numérique des schémas peut masquer les effets de la modélisation. Deux schémas d'ordre élevé ont été étudiés et implémentés dans un solveur 3D Navier Stokes incompressible sur des maillages volumes finis non structurés. Nous avons développé un premier schéma d'ordre élevé, correspondant à un schéma Padé volumes finis, et nous avons étendu le schéma de reconstruction polynomiale de Carpentier (2000) aux écoulements incompressibles. Les propriétés numériques des différents schémas implémentés dans le même code de calcul sont étudiées sur différents cas tests bi-dimensionnels (calcul de flux convectifs et diffusifs sur une solution a-priori, convection d'une tâche gaussienne, décroissance d'un vortex de Taylor et cavité entraînée) et tri-dimensionnel (écoulement autour d'un obstacle cubique). Une attention particulière a été portée à l'étude de la précision et du traitement des conditions limites. L'implémentation proposée du schéma polynomial permet d'approcher, pour un maillage identique, les temps de simulation obtenus avec un schéma décentré classique d'ordre 2, mais avec une précision supérieure. Le schéma compact donne la meilleure précision. En utilisant une méthode de Jacobi sans calcul implicite de la matrice pour calculer le gradient, le temps de simulation devient intéressant uniquement lorsque la précision requise est importante. Une alternative est la résolution du système linéaire par une méthode multigrille algébrique. Cette méthode diminue considérablement le temps de calcul du gradient et le schéma Padé devient performant même pour des maillages grossiers. Enfin, pour réduire les temps de simulation, la parallélisation des schémas d'ordre élevé est réalisée par une décomposition en sous domaines. L'assemblage des flux s'effectue naturellement et différents solveurs proposés par les librairies PETSC et HYPRE (solveur multigrille algébrique et méthode de Krylov préconditionnée) permettent de résoudre les systèmes linéaires issus de notre problème. / The prevention of industrial risks requires simulating turbulent dispersion of pollutants. However, the tools mostly used so far do not allow near fields treated in the case of complex geometries, and it is necessary to utilize the tools of CFD (Computational Fluid Dynamics ") more suitable but more expensive. To simulate atmospheric flows with dispersion of pollutants, the CFD models must correctly model the one hand, the effects of buoyancy, and secondly the effects of turbulence. Several approaches exist, including taking into account the effects of buoyancy and turbulence modeling, and require numerical methods adapted to the specific mathematics of each, and accurate numerical schemes to avoid pollution modeling. A formulation of high order finite volume on unstructured meshes, parallelized, is proposed to simulate the atmospheric flows with dispersion of pollutants. The use of high order schemes allow one hand to reduce the number of cells and decrease the simulation time to achieve a given accuracy, and secondly to better control the viscosity numerical schemes for simulation LES (Large Eddy Simulation), for which the numerical viscosity patterns may mask the effects of modeling. Two high-order schemes have been studied and implemented in a 3D Navier Stokes solver on unstructured mesh finite volume. We developed the first high-order scheme, corresponding to a Padé finite volume scheme, and we have extended the scheme of reconstruction polynomial Carpentier (2000) for incompressible flows. The numerical properties of the various schemes implemented in the same computer code are studied different two-dimensional test cases (calculation of diffusive and convective flow on a solution a priori, a task Gaussian convection, decay of a vortex of Taylor and driven cavity) and tri-dimensional (flow past an obstacle cubic). Particular attention has been paid to the study of the accuracy and treatment of boundary conditions. The implementation of the polynomial allows to obtain quasi identical simulation time compared to a classical upwind scheme of order 2, but with higher accuracy. The compact layout gives the best accuracy. Using a Jacobi method without calculation implied matrix to calculate the gradient, the simulation time becomes interesting only when the required accuracy is important. An alternative is the resolution of linear system by an algebraic multigrid method. This method significantly reduces the computation time of the gradient and the Padé scheme is effective even for coarse meshes. Finally, to reduce simulation time, the parallelization schemes of high order is achieved by a decomposition into subdomains. The assembly flow occurs naturally and different solvers provided by PETSc libraries and HYORE (algebraic multigrid solver and preconditioned Krylov method) used to solve linear systems from our problem. The work was to identify and determine the parameters that lead to lowest time resolution simulation. Various tests of speed-up and scale-up were used to determine the most effective and optimal parameters for solving linear systems in parallel from our problem. The results of this work have been the subject of a communication in an international conference "Parallel CFD 2008" and an article submitted to "International Journal for Numerical Methods in Fluids" (Analysis of high-order finite volume schemes for the incompressible Navier Stokes equations)