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In this thesis, we study the entropy stability of the compressible Navier-Stokes model along with a modification of the model. We use the discretization of the inviscid terms with the Ismail-Roe entropy conservative flux. Then, we study entropy stability of the augmentation of viscous, heat and mass diffusion finite difference approximations to the entropy conservative flux. Additionally, we look at different choices of the diffusion coefficient that arise from combining the viscous, heat and mass diffusion terms. Lastly, we present numerical results of the discretizations comparing the effects of the viscous terms on the oscillations near the shock and show that they preserve entropy stability.

Finite difference approximations are methods for solving differential equations by approximating derivatives. This work will begin with how to solve a partial differential equation (PDE) called the advection equation, ut + cux = 0. Both analytically, and approximately with three different finite difference methods for the spatial part of the equation: • Central in space, • First order upwind in space, • Beam-Warming in space, and forward Euler for the temporal part. We then use the theoretical approximations considered for the advection equation and apply it on Maxwell’s equations for electromagnetism in 1D. This is a system of advection equations that describes how electromagnetic waves propagate through a dielectric material. In the end of this work we will model this electromagnetic wave, or wave of light moving through materials with different refraction indexes.

<p> The modeling, design and simulation of optical waveguides and integrated optical triplexer filters are presented. The work includes two subjects. One is application of improved three-point fourth-order finite-difference method and the other is design of triplexer optical filter for fiber-to-the-home passive optical network.</p> <p> The improved three-point fourth-order finite-difference method utilizes special format of one dimensional Helmholtz Equation and adopts generalized Douglas scheme and boundary conditions matching at interface. The modal analysis of dielectric slab waveguides and metal slab waveguides that support Surface Plasmon Plaritons by using this improved fourth-order finite-difference method is compared by using traditional first-order central difference method. The application of using improved three-point fourth-order finite-difference method in modal analysis of optical fiber waveguide is also provided.</p> <p> The modeling, design and simulation of monolithically integrated triplexer optical filter based on silicon wire waveguide are presented in detail. The design of this device facilitates multi-mode interference device (MMI) and arrayed waveguide grating (AWG) device to function as coarse wavelength division multiplexing and dense wavelength division multiplexing respectively. The MMI is used to separate downstream signs for upstream signal and AWG is used to further separate two down-stream signals with different bandwidths required. This design is validated by simulation that shows excellent performance in terms of spectral response as well as insertion loss.</p> / Thesis / Master of Applied Science (MASc)

<p> This thesis contributes significantly towards the development of a robust algorithm for design sensitivity analysis and the optimization of microwave structures. Based on the frequency-domain finite-element method, the approach provides accurate sensitivity information using both 2-D and 3-D formulations. It also significantly accelerates the optimization process.</p> <p> The design sensitivity analysis method greatly influences the efficiency and accuracy of gradient-based optimization by providing the response gradient (response Jacobians) for the whole range of parameter values. However, common commercial electromagnetic simulators provide only specific engineering responses, such as Z- or S-parameters. No sensitivity information is made available for further exploration of the design-parameter space. It is common to compute the design sensitivities from the response themselves using finite-difference or higher-order approximations at the response level. Consequently, for each design parameter of interest, at least one additional full-wave analysis is performed. However, when the number of design parameters becomes large, the simulation time becomes prohibitive for electromagnetic design procedures.</p> <p> The self-adjoint sensitivity analysis (SASA) is so far the most efficient way to extract the sensitivity information for the network parameters with the finite-element method. As an improvement of the adjoint-variable method (AVM), it eliminates the additional (adjoint) system analyses. With one single full-wave analysis, the sensitivities with respect to all design parameters are computed. This significantly improves the efficiency of the sensitivity computations. Through our proposed method, the finite-difference frequency-domain self-adjoint sensitivity analysis (FDFD-SASA), the process is further improved by eliminating the need for exporting the system matrix, thus improving both compatibility and computation time. The only requirement for integrating the sensitivity solver with the commercial EM simulators is the ability to access the field solution at the user-defined grid points. The sensitivity information is obtained by simple manipulation of the field solution as a post-process and hence, it adds little or no overhead to the simulation time.</p> <p> We explore the feasibility of implementing our newly proposed method using field solutions from a frequency-domain commercial solver HFSS v 11. We confirm the accuracy of the FDFD-SASA for shape parameters of metallic structures. Both 2-D and 3-D examples are presented, where the reference results are provided through the traditional finite-difference approximations. Also, the efficiency of the FDFD-SASA is validated by a filter design example, exploiting gradient-based optimization algorithm.</p> / Thesis / Master of Applied Science (MASc)

The Finite-Difference Time-Domain (FDTD) method has become a very powerful tool for the analysis of propagating electromagnetic waves. It involves the discretization of Maxwell's equations in both time and space that leads to a numerical solution of the wave propagation problem in the time domain. The technique's main benefits are that it permits the description of wave propagation in non-uniform media, it can easily accommodate a wide range of boundary conditions, and it can be used to model nonlinear effects as well as the wave behaviour near localized structures or material defects. In this study, we extend this technique to mechanical wave propagation in piezoelectric crystals. It is observed to give large reflection artefacts generated by the computational boundaries which interfere with the desired wave propagation. To solve this problem, the renowned absorbing boundary condition called perfectly matched layer (PML) is used. PML was first introduced in 1994 for electromagnetic wave propagation. Our research has further developed this idea for acoustic wave propagation in piezoelectric crystals. The need to improve the large reflection artefacts by introducing a finite thickness PML has reduced acoustic wave reflection occurring due to practical errors to less than 0.5 %. However, it is found that PML can generate numerical instabilities in the calculation of acoustic fields in piezoelectric crystals. Theses observations are also discussed in this report. / Thesis / Master of Applied Science (MASc)

No / The use of the hybrid method of moments/finite difference time domain technique can be effective for solution of electromagnetic problems which are intractable for a single numerical method. Using this method, a study of the effects of human proximity on the polarisation purity of different types of circularly-polarised handset antennas for personal satellite communications was undertaken. Associated with this, assessments of the specific absorption rate in the head were made. The method gave stable results, in accordance with physical expectations; good agreement with the pure method of moments was shown in simplified cases omitting the head

No / A heterogeneous hybrid computational electromagnetics method is presented, which enables different parts of an antenna simulation problem to be treated by different methods, thus enabling the most appropriate method to be used for each part. The method uses a standard frequency-domain moment-method program and a finite-difference time-domain program to compute the fields in two regions. The two regions are interfaced by surfaces on which effective sources are defined by application of the Equivalence Principle. An extension to this permits conduction currents to cross the boundary between the different computational domains. Several validation cases are examined and the results compared with available data. The method is particularly suitable for simulation of the behavior of an antenna that is partially buried, or closely coupled with lossy dielectric volumes such as soil, building structures or the human body.

This study explores the application of GPU-based algorithms in radiowave propagation modeling, specifically through the scope of solving parabolic wave equations. Radiowave propagation models are crucial in the field of wireless communications, where they help predict how radio waves travel through different environments, which is vital for planning and optimization. The research specifically examines the implementation of two numerical methods: the Split Step Method and the Finite Difference Method. Both methods are adapted to utilize the parallel processing capabilities of modern GPUs, harnessing a parallel computing framework known as CUDA to achieve considerable speed enhancements compared to traditional CPU-based methods.Our findings reveal that the Split Step method generally achieves higher speedup factors, especially in scenarios involving large system sizes and high-frequency simulations, making it particularly effective for expansive and complex models. In contrast, the Finite Difference Method shows more consistent speedup across various domain sizes and frequencies, suggesting its robustness across a diverse range of simulation conditions. Both methods maintained high accuracy levels, with differences in computed norms remaining low when comparing GPU implementations against their CPU counterparts.

Increasing interest, from the U.S. Army Corps of Engineers (USACE) and other agencies, in using deep-mixing methods (DMM) to improve the stability of levees constructed on soft ground is driven by the need to reduce levee footprints and environmental impacts and to allow for more rapid construction. Suitable methods for analysis and design of these systems are needed to ensure that the DMM technology is properly applied. DMM shear walls oriented perpendicular to the levee alignment are an effective arrangement for supporting unbalanced lateral loads. Shear walls constructed by overlapping individual DMM columns installed with single-axis or multiple axis equipment include vertical joints caused by the reduced width of the wall at the overlap between adjacent columns. These joints can be made weaker by misalignment during construction, which reduces the efficiency of the overlap. Depending on the prevalence and strength of these joints, complex failure mechanisms, such as racking due to slipping along vertical joints between adjacent installations in the shear walls, can occur. Ordinary limit equilibrium analyses only account for a composite shearing failure mode; whereas, numerical stress-strain analyses can account for other failure modes. Five case studies provided by the USACE were analyzed to evaluate the behavior of levee and floodwall systems founded on soft ground stabilized with DMM shear walls. These identified and illustrated potential failure mechanisms of these types of systems. Two-dimensional numerical stability and settlement analyses were performed for the case studies using the FLAC computer program. The key findings and conclusions for the individual case studies were assessed and integrated into general conclusions about design of deep-mixing support for levees and floodwalls. One of the significant findings from this research was to identify the potential for a partial depth racking failure, which can control design when the DMM shear walls are socketted into a relatively strong bearing layer. The potential for partial depth racking failure is not discussed in the literature and represents a new failure mode identified by this research. This discovery also highlights the importance of adapting suitable methods for analysis and design of these systems to address all potential failure modes. / Ph. D.