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<p>In this thesis we study hybrid numerical methods to be usedin computational electromagnetics. We restrict the methods tospectral domain and scattering problems. The hybrids consist ofcombinations of Boundary Element Methods and Geometrical Theoryof Diffraction.</p><p>In the thesis three hybrid methods will be presented. Onemethod has been developped from a theoretical idea to anindustrial code. The two other methods will be presented mainlyfrom a theoretical perspective. We will also give shortintroductions to the Boundary Element Method and theGeometrical Theory of Diffraction from a theoretical andimplementational point of view.</p><p><b>Keywords:</b>Maxwell’s equations, Geometrical Theoryof Diffraction, Boundary Element Method, Hybrid methods,Electromagnetic Scattering</p>

Pitting corrosion is a form of highly localized corrosion that can lead to crack and failure of a structure. Study on pitting corrosion is necessary in order to predict and prevent the risk of failure of structure susceptible to corrosion. In this thesis, a combination of Cellular Automata (CA) and Boundary Element Method (BEM) was developed to simulate pitting corrosion growth under certain environment. It is assumed that pitting corrosion can be simplified to electrochemical corrosion cell. The distribution of potential around this corrosion cell can then be simulated by BEM. This distribution potential represents cathodic and anodic reactions around the corrosion cell. A CA model was developed that uses transition rules reflecting mechanism of pitting corrosion. The CA model has two types of cell states, one reflecting BEM simulation results and the other reflecting the status of corrosion cell (anode, cathode, and passive metal’s surface). For every CA iteration, the CA decides the state of the corrosion cells (the location and size of anode, cathode) while BEM simulate the level of electrochemical activity at discrete location on the surface (represented by potential distribution). In order to demonstrate the methodology, a simple case of rectangular corrosion cell with varied dimensions and under different polarization functions is considered. Results show certain shapes tend to grow at certain type environment and these pits are comparable to commonly observed pit shapes. In addition, stress analysis was carried out to investigate the severity of corrosion pits of varying shapes and sizes. Results show that certain pits induced highly varying stress concentration as it grows over time, while others have more steady increase of stress concentration.

In bubble dynamics, an underwater bubble may evolve from being singly-connected to being toroidal. Furthermore, two or more individual bubbles may merge to form a single large bubble. These dynamics involve significant topological changes such as merging and breaking, which may not be handled well by front-tracking boundary element methods. In the level set method, topological changes are handled naturally through a higher-dimensional level set function. This makes it an attractive method for bubble simulation. In this paper, we present a method that combines the level set method and the boundary element method for the simulation of bubble dynamics. We propose a formulation for the update of a potential function in the level set context. This potential function is non-physical off the bubble surface but consistent with the physics on the bubble surface. We consider only axisymmetric cavitation bubbles in this paper. Included in the paper are some preliminary results and findings. / Singapore-MIT Alliance (SMA)

In this paper a precorrected FFT-Fast Multipole Tree (pFFT-FMT) method for solving the potential flow around arbitrary three dimensional bodies is presented. The method takes advantage of the efficiency of the pFFT and FMT algorithms to facilitate more demanding computations such as automatic wake generation and hands-off steady and unsteady aerodynamic simulations. The velocity potential on the body surfaces and in the domain is determined using a pFFT Boundary Element Method (BEM) approach based on the Green’s Theorem Boundary Integral Equation. The vorticity trailing all lifting surfaces in the domain is represented using a Fast Multipole Tree, time advected, vortex participle method. Some simple steady state flow solutions are performed to demonstrate the basic capabilities of the solver. Although this paper focuses primarily on steady state solutions, it should be noted that this approach is designed to be a robust and efficient unsteady potential flow simulation tool, useful for rapid computational prototyping. / Singapore-MIT Alliance (SMA)

<p>Steady-state sound fields in enclosures, with specular and diffuse reflection boundaries, are modeled with a first-principle energy-intensity boundary element method using uncorrelated broadband directional sources. The specular reflection field is represented by a limited set of spherical harmonics that are orthogonal on the half-space. The amplitudes of these harmonics are determined by a Lagrange multiplier method to satisfy the energy conservation integral constraint. The computational problem is solved using an iterative relaxation method starting from the 3-D diffuse reflection solution. At each iteration, directivity harmonics are estimated by post-processing and the influence matrix is refined accordingly. For internal sources, simple first reflection images improve accuracy with virtually no penalty on computation time. Monotonic convergence occurs in relatively few relaxation steps. Extrapolating to an infinite number of boundary elements and iterations gives very accurate results. The method is very computationally efficient. Results are compared to exact benchmark solutions obtained from a frequency-by-frequency modal analysis, and a broadband image method, demonstrating high accuracy. The method of absorption scaling is verified for complicated 3-D cases, and showing that the spatial variation in rooms is largely determined by source position and the relative distribution of absorption, but not the overall absorption level.</p> / Dissertation

Based on a full-wave formulation, a vector boundary element method (VBEM) is proposed to model the photonic crystal fibers (PCFs) (microstructured fibers). The accuracy and efficiency of the approach are confirmed by comparing the results calculated with those in previous literatures. With employing the VBEM, the guiding characteristics, including the effective indexes, vector mode patterns, and the polarization properties of the PCFs are investigated. There polarization characteristics of the PCFs with elliptical air holes (EPCFs) and the one ring air-hole EPCF embedded in the step-index core are studied and discussed. In addition, based on the VBEM formulations, a novel and efficient numerical approach to calculate the dispersion parameters of the PCFs is also proposed. The effect of the PCF geometrical structure on the group velocity dispersion property is reviewed, and then the one-ring defect and two-ring defect PCFs are studied and designed for the ultra-flattened dispersion applications. As an example, a four-ring (two-ring defect) PCF with flattened dispersion of ¡Ó0.25 ps/km/nm from 1.295£gm to 1.725£gm wavelength is numerically demonstrated.

In this paper we develop a computational procedure to investigate linear fracture of two-dimensional problems in isotropic linearly elastic media. A symmetric Galerkin boundary element method (SGBEM), based on a weakly singular, weak-form traction integral equation, is adopted to model these fractures. In particular we consider multiple interacting cracks in an unbounded domain subject to internal pressure and remote stress. The growth of the cracks is driven by either linearly dependent injection pressures or volumes in each crack. A variety of crack geometries are investigated. / text

A novel numerical method solves the VIScous Vorticity Equation (VISVE) in 3D in order to model the Leading Edge Vortex (LEV) of propellers is proposed and implemented in this dissertation. The spatial concentration of the vorticity is exploited in the method, which is designed to be spatially compact and numerically efficient, in the meantime, capable of modeling complicated vorticity/solid boundary interaction in 2D and 3D. The numerical model can work as a viscous correction on top of the traditional Boundary Element Method (BEM) results. The proposed method is first applied in the case of a 2D hydrofoil at high angle of attack. The results are correlated with those from Navier-Stokes (N-S) simulation. The method is then used to model the LEV and tip vortex of a 3D swept wing. The results of the 3D simulation show great similarity to those from N-S. In the end, the method is applied in the case of propellers at low advance ratios. All the essential flow characteristics (LEV and tip vortex) are predicted. The objective of this dissertation is not developing a mathematically equivalent numerical method to the full-blown Reynolds-Averaged Navier-Stokes (RANS) solver, but inventing an accurate and computationally efficient tool to model the effects of the LEV on the propeller performance for engineering's purpose. / text

The effect of coupled thermo-poroelastic behavior on hydraulic fracture propagation is of much interest in geothermal- and petroleum-related geomechanics problems such as wellbore stability and hydraulic fracturing as pore pressure and temperature variations can significantly induce rock deformation, fracture initiation, and propagation. In this dissertation, a two-dimensional (2D) boundary element method (BEM) was developed to simulate the fully coupled thermo-poroelastic fracture propagation process. The influence of pore pressure and temperature changes on the fracture propagation length and path, as well as on stress and pore pressure distribution near wellbores and fractures, was considered in isotropic and homogeneous rock formations. The BEM used in this work consists of the displacement discontinuity (DD) method and the fictitious stress (FS) method. Also, a combined FS-DD numerical model was implemented for the hydraulically or thermally-induced fractures in the vicinity of a wellbore. The linear elastic fracture mechanics (LEFM) theory was adopted to numerically model within the framework of poroelasticity and thermo-poroelasticity theory. For high accuracy of crack tip modeling, a special displacement discontinuity tip element was developed and extended to capture the pore pressure and temperature influence at the tip. For poroelastic fracture propagation, a steadily propagating crack driven by fluid pressure was modeled to find the effect of pore pressure on crack path under the two limiting poroelastic conditions (undrained and drained). The results indicate that the pore pressure diffusion has no influence on the crack growth under the undrained condition because the crack propagation velocity is too fast for the diffusion effect to take place. On the other hand, its influence on the crack path under the drained condition with its low propagation velocity has significance because it induces a change in principal stress direction, resulting in an alteration of fracture orientation. For the thermal fracturing, when the rock around a wellbore and a main fracture is cooled by injecting cold water in a hot reservoir, the rapid decrease in temperature gives rise to thermal stress, which causes a crack to initiate and propagate into the rock matrix. The single and multiple fracture propagation caused by transient cooling in both thermoelastic and poro-thermoelastic rock were numerically modeled. The results of this study indicate that the thermal stresses induced by cooling may exceed the in-situ stress in the reservoir, creating secondary fractures perpendicular to main fracture. Furthermore, the faster cooling rate produces longer crack extension of the secondary thermal fractures. This implies that the faster cooling induces a higher tensile stress zone around the fracture, which tends to produce larger driving forces to make the secondary fractures penetrate deeper into the geothermal reservoir.