This paper examines and develops matrix methods to approximate the eigenvalues of a fourth order Sturm-Liouville problem subjected to a kind of fixed boundary conditions, furthermore, it extends the matrix methods for a kind of general boundary conditions. The idea of the methods comes from finite difference and Numerov's method as well as boundary value methods for second order regular Sturm-Liouville problems. Moreover, the determination of the correction term formulas of the matrix methods are investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q = 0 are known in this case. Finally, some numerical examples are illustrated.

Partial differential equations (PDEs) are used to model various phenomena in nature and society, ranging from the motion of fluids and electromagnetic waves to the stock market and traffic jams. There are many methods for numerically approximating solutions to PDEs. Some of the most commonly used ones are the finite volume method, the finite element method, and the finite difference method. All methods have their strengths and weaknesses, and it is the problem at hand that determines which method that is suitable. In this thesis, we focus on the finite difference method which is conceptually easy to understand, has high-order accuracy, and can be efficiently implemented in computer software. We use the finite difference method on summation-by-parts (SBP) form, together with a weak implementation of the boundary conditions called the simultaneous approximation term (SAT). Together, SBP and SAT provide a technique for overcoming most of the drawbacks of the finite difference method. The SBP-SAT technique can be used to derive energy stable schemes for any linearly well-posed initial boundary value problem. The stability is not restricted by the order of accuracy, as long as the numerical scheme can be written in SBP form. The weak boundary conditions can be extended to interfaces which are used either in domain decomposition for geometric flexibility, or for coupling of different physics models. The contributions in this thesis are twofold. The first part, papers I-IV, develops stable boundary and interface procedures for computational fluid dynamics problems, in particular for problems related to the Navier-Stokes equations and conjugate heat transfer. The second part, papers V-VI, utilizes duality to construct numerical schemes which are not only energy stable, but also dual consistent. Dual consistency alone ensures superconvergence of linear integral functionals from the solutions of SBP-SAT discretizations. By simultaneously considering well-posedness of the primal and dual problems, new advanced boundary conditions can be derived. The new duality based boundary conditions are imposed by SATs, which by construction of the continuous boundary conditions ensure energy stability, dual consistency, and functional superconvergence of the SBP-SAT schemes.

Because current full‐core neutronic‐calculations use two‐group neutron diffusion and rely on homogenizing fuel assemblies, reconstructing pin powers from such a calculation is an elaborate and not very accurate process; one which becomes more difficult with increased core heterogeneity. A three‐dimensional Heterogeneous Finite Element Method (HFEM) is developed to address the limitations of current methods by offering fine‐group energy representation and fuel‐pin‐level spatial detail at modest computational cost. The calculational cost of the method is roughly equal to the calculational cost of the Finite Differences Method (FDM) using one mesh box per fuel assembly and a comparable number of energy groups. Pin‐level fluxes are directly obtained from the method’s results without the need for reconstruction schemes. / UOIT

A two-dimensional thin-layer Navier-Stokes cascade flow solver for turbomachinery is developed. A second-order finite-difference scheme and a second and fourth-difference dissipation scheme are used. Periodic and non-reflecting inlet and outlet boundary conditions are implemented into the approximate-factorization numerical method. Turbulence is modeled through the one-equation Spalart-Allmaras model. A two-dimensional turbomachinery cascade structured grid generator is developed to produce six-block H-type grids. The validity of this work is tested in various ways. A grid convergence study is performed showing the effect of grid density. The non-reflecting inlet and outlet boundary conditions are tested for boundary placement influence. Comparisons of the flow solver numerical results are performed against experimental results. A Mach number sweep and angle of attack sweep are performed on two similar transonic turbine cascades.

A two-dimensional thin-layer Navier-Stokes cascade flow solver for turbomachinery is developed. A second-order finite-difference scheme and a second and fourth-difference dissipation scheme are used. Periodic and non-reflecting inlet and outlet boundary conditions are implemented into the approximate-factorization numerical method. Turbulence is modeled through the one-equation Spalart-Allmaras model. A two-dimensional turbomachinery cascade structured grid generator is developed to produce six-block H-type grids. The validity of this work is tested in various ways. A grid convergence study is performed showing the effect of grid density. The non-reflecting inlet and outlet boundary conditions are tested for boundary placement influence. Comparisons of the flow solver numerical results are performed against experimental results. A Mach number sweep and angle of attack sweep are performed on two similar transonic turbine cascades.

Closed-loop ground source heat pumps (GSHPs) are used to transfer thermal energy between the subsurface and conditioned spaces for heating and cooling applications. A basic GSHP is composed of a ground heat exchanger (GHX), which is a closed loop of pipe buried in the shallow subsurface circulating a heat exchange fluid, connected to a heat pump. These systems offer an energy efficient alternative to conventional heating and cooling systems; however, installation costs are higher due to the additional cost associated with the GHX. By further developing our understanding of how these ground loops interact with the subsurface, it may possible to design them more intelligently, efficiently, and economically. To gain insight into the physical processes occurring between the GHX and the subsurface and to identify efficiencies and inefficiencies in GSHP design and operation, two main research goals were defined: comprehensive monitoring of a fully functioning GSHP and intensive simulation of these systems using computer models. A 6-ton GSHP was installed at a residence in Elora, ON. An array of 64 temperature sensors was installed on and surrounding the GHX and power consumption and temperature sensors were installed on the system inside the residence. The data collected were used to help characterize and understand the function of the system, provide motivation for further investigations, and assess the impact of the time of use billing scheme on GSHP operation costs. To simulate GSHPs, two computer models were utilized. A 3D finite element model was employed to analyse the effects of pipe configuration and pipe spacing on system performance. A unique, transient 1D finite difference heat conduction model was developed to simulate a single pipe in a U-tube shape with inter-pipe interactions and was benchmarked against a tested analytical solution. The model was used to compare quasi-steady state and transient simulation of GSHPs, identify system performance efficiencies through pump schedule optimization, and investigate the effect of pipe length on system performance. A comprehensive comparison of steady state and pulsed simulation concludes that it is possible to simulate transient operation using a steady state assumption for some cases. Optimal pipe configurations are identified for a range of soil thermal properties. Optimized pump schedules are identified and analysed for a specific heat pump and fluid circulation pump. Finally, the effect of pipe spacing and length on system performance is characterized. It was found that there are few design inefficiencies that could be easily addressed to improve general design practice.

Photonic Crystal (PC) devices are the most exciting advancement in the field of photonics. The use of computational techniques has made considerable improvements in photonic crystals design. We present here an ultrahigh quality factor (Q) photonic crystal slab nanocavity formed by the local width modulation of a line defect. We show that only shifting two holes away from a line defect is enough to attain an ultrahigh Q value. We simulated this double heterostructure nano cavity by using Finite Difference Time Domain (FDTD) technique. We observed that photonic crystal cavities are very sensitive to the frequency, size and position of the source. So we must choose the right values for these parameters.

This research characterizes the thermal stress resulting from temperature gradients in hybrid solid oxide fuel cells that are processed using a novel oxide powder slurry technology developed at Georgia Tech. The hybrid solid oxide fuel cell is composed of metallic interconnect and ceramic electrolyte constituents with integral mechanical bonds formed during high temperature processing steps. A combined thermo-mechanical analysis approach must be implemented to evaluate a range of designs for power output and structural integrity. As an alternative to costly CFD analysis, approximate finite difference techniques that are more useful in preliminary design are developed to analyze the temperature distributions resulting from a range of fuel cell geometries and materials. The corresponding thermal stresses are then calculated from the temperature fields using ABAQUS. This model analyzes the manufacturing, start-up, and steady state operating conditions of the hybrid solid oxide fuel cell.

The operation speed of power delivery system of packages has been upgraded to GHz. The instant current will pass to the power plane of the mother board by way of the IC pins and result in electromagnetic wave propagation between the power plane and the ground plane, then to produce the programs of electromagnetic interference. In this thesis, we will analyze the EMI of power delivery system of packages by finite-difference time-domain in two dimensions structure in three sections. In firist section, to computing EMI in finite-difference time-domain in two dimensions structure. In second section, to analyze more complicated power delivery plane, ex: EBG, in finite-difference time-domain in two dimensions structure. In three section, to add property of capacitors on power plane to reduce EMI in two dimensions structure. Above all, we hope to built a fast computing method to compute EMI to solve the time-consuming problems of full-wave simulated software. And to supply the engineer to deal with the similar problems in packages efficiently.

In modern high-speed digital circuits, the space of the traditional single-layered or double-layered circuit board is not enough, therefore multi-layered circuit and stacked distribution technology are widely applied to many applications. The signal via is a vertical interconnection structure to communicate different signal layers, which will be seriously interfere with the simultaneous switching noise by via through the parallel plate cavity that consists of power and ground plane. It is an important issue to minimize the influence from noise. In multi-layered printed circuit boards, shorting vias are usually utilized to interconnect the planes with the same voltage level. The major theme of this thesis is the placement of shorting vias affecting plane cavity mode. And we propose a design rule of the shorting vias to significantly decrease the simultaneous switching noise and improve the power integrity of multi-layered circuit board.