Yes / The Poisson's equation is an essential entity of applied mathematics for modelling many phenomena of importance. They include the theory of gravitation, electromagnetism, fluid flows and geometric design. In this regard, finding efficient solution methods for the Poisson's equation is a significant problem that requires addressing. In this paper, we show how it is possible to generate approximate solutions of the Poisson's equation subject to various boundary conditions. We make use of the discrete finite difference operator, which, in many ways, is similar to the standard finite difference method for numerically solving partial differential equations. Our approach is based upon the Laplacian averaging operator which, as we show, can be elegantly applied over many folds in a computationally efficient manner to obtain a close approximation to the solution of the equation at hand. We compare our method by way of examples with the solutions arising from the analytic variants as well as the numerical variants of the Poisson's equation subject to a given set of boundary conditions. Thus, we show that our method, though simple to implement yet computationally very efficient, is powerful enough to generate approximate solutions of the Poisson's equation. / Supported by the European Union’s Horizon 2020 Programme H2020-MSCA-RISE-2017, under the project PDE-GIR with grant number 778035.

<p> This thesis contributes to the development of a novel electromagnetic (EM) time-domain computational approach, the self-adjoint variable method, for the scattering parameter (S-parameter) sensitivity analysis of high frequency problems. </p> <p> The design sensitivity analysis provides sensitivity information in the form of the response gradient (response Jacobian). For that, various techniques are used, ranging from finite-difference approximations to quadratic and spline interpolations. However, when the number of design parameters becomes large, the simulation time would become unaffordable, which is especially the case with EM simulations. The proposed self-adjoint sensitivity analysis (SASA) approach aims at providing sensitivity information efficiently without sacrificing the accuracy. Its efficiency lies in the fact that regardless of the number of design parameters, only one simulation of the original structure is required- the one used to compute the S-parameters. Thus, the sensitivity computation has negligible overhead. At the same time, it has second-order accuracy. </p> <p> Currently, commercial EM simulators provide only specific engineering responses, such as Z- or S-parameters. No sensitivity information is actually made available. With the SASA approach, the only requirement for the EM solver is the ability to access the field solution at the perturbation grid points. This feature is generally available with all time-domain EM simulators. The manipulation of the field solutions in this approach is simple and it adds practically negligible overhead to the -simulation time. </p> <p> We confirm the validity of this approach for both the shape and constitutive parameters of the design structures. 2-D examples including metallic and dielectric details are presented, using the field solutions from an in-house time-domain solver. We also explore the feasibility of implementing this approach with one of the commercial solvers, XFDTD v. 6.3. </p> <p> Suggestions for future research are provided. </P> / Thesis / Master of Applied Science (MASc)

The importance of precise values for the parameters used in heat and mass transfer models has been demonstrated by many research studies. Thermal conductivity values used in previous models are usually empirical and fluctuate. Theoretical analysis and estimations of wood thermal conductivities in the radial and tangential directions were conducted with the geometric models built up from the macro- and micro-structure observations. Theoretically, thermal conductivity in the radial direction is about 1.5 to 2.5 times of the tangential direction for softwood species with moisture content (MC) below Fiber Saturation Point (FSP). When MC is over the FSP, tangential radial thermal conductivity both increase dramatically and are linear function of MC. The two thermal conductivity values are close with a ratio of near one estimated by the model for MC above the FPS. In hardwood species, radial thermal conductivity estimated by the model is 1.5 times of the tangential thermal conductivity. Validation tests for model estimations of thermal conductivities in the radial and tangential directions for three wood species showed the reliability of the geometric models developed in this project. Correlations between the wood thermal conductivity and structure parameters, such as latewood percentage and cell wall percentage, were examined. Linear relationships for the thermal conductivity and average temperature in wood were established in both radial and tangential directions of three wood species. A two-dimensional transient heat conduction model was developed utilizing thermal conductivity values derived from geometric models. The anisotropic material property affect on heat transport in radial and tangential directions was discussed using an assumed situation. The simulation run showed slightly faster heat flow in the radial direction than in the tangential direction due to higher thermal conductivity in the radial direction. Validation tests on practical wood blocks showed the 2D model with the use of theoretical thermal conductivity values can predict good temperature distribution in wood during the heating process. However, in the practical wood samples with curved rings on the cross section, no significant difference was found in the two transverse directions. Mathematica software was introduced in this study for the intense and complicated math calculations and model programming. Mathematica was found to be a powerful technique for solving sophisticated math problems. It had abundant and flexible plotting options for providing optimized presentations for the results. These advantages make Mathematica popular for engineering modeling research. / Ph. D.

The Burgers equation, and related partial differential equations (PDEs), can be numerically challenging for small values of the viscosity parameter. For example, these equations can develop discontinuous solutions (or solutions with large gradients) from smooth initial data. Aside from numerical stability issues, standard numerical methods can also give rise to spurious oscillations near these discontinuities. In this study, we consider an equivalent form of the Burgers equation given by Constantin and Iyer, whose solution can be written as the expected value of a stochastic differential equation. This equivalence is used to develop a numerical method for approximating solutions to Burgers equation. Our preliminary analysis of the algorithm reveals that it is a natural generalization of the method of characteristics and that it produces approximate solutions that actually improve as the viscosity parameter vanishes. We present three examples that compare our algorithm to a recently published reference method as well as the vanishing viscosity/entropy solution for decreasing values of the viscosity. / Master of Science / Burgers equation is a Partial Differential Equation (PDE) used to model how fluids evolve in time based on some initial condition and viscosity parameter. This viscosity parameter helps describe how the energy in a fluid dissipates. When studying partial differential equations, it is often hard to find a closed form solution to the problem, so we often approximate the solution with numerical methods. As our viscosity parameter approaches 0, many numerical methods develop problems and may no longer accurately compute the solution. Using random variables, we develop an approximation algorithm and test our numerical method on various types of initial conditions with small viscosity coefficients.

Blood perfusion is the local, non-directional blood flow through tissue. It is measured as the volumetric flow rate of blood through a given volume of tissue. One method that has been developed for measuring blood perfusion is a probe that measures the temperature response of the tissue when a thermal event is applied. From the temperature response, the blood perfusion and contact resistance can be estimated by comparing the experimental response to a predicted response, and employing Gaussian minimization techniques to estimate the blood perfusion and contact resistance. The objective of this research was to assess the measurement repeatability and sensitivity of the blood perfusion probe by testing the probe on phantom tissue, such that the effects of physiologic or pathologic conditions on the blood perfusion could be eliminated. Another objective was to conduct a preliminary in vivo study using rats for the purpose of establishing proper experimental protocols for future testing of the blood perfusion probe. A phantom tissue test stand comprised of porous material and water to simulate tissue and blood, respectively, was constructed for the phantom study. Inlet flow rates into the porous media ranging between 0 cc/min and 30 cc/min were tested. To test the measurement repeatability 7 flow rates (0, 5, 10, 15, 20, 25 and 30 cc/min) were tested on two different days. To test the measurement sensitivity of the probe, flow rates between 0 and 10 cc/min, and 15 and 20 cc/min were tested at intervals of 1 cc/min. From the phantom study it was concluded that the probe displayed good measurement repeatability, as the trend in perfusion as a function of inlet flow rates for both days was found to be the same. It was also found that the data collected using the probe yielded significantly different perfusion estimates for different flow rates, as statistical analyses show that the average perfusion differences between flow rates are truly independent within a 90% confidence interval, for flow differences above 4 cc/min. It was found that for flow rates below 4 cc/min the probe sensitivity was significantly reduced. For the in vivo study it was concluded that the probe can be used to obtain estimates of perfusion from rats. This preliminary study also served to establish proper experimental protocols for future tests. / Master of Science

We have created a three-dimensional, implicit finite difference model that can accurately calculate temperatures within the bulk of a sample during a friction stir fabrication process. The model was written in Wolfram Mathematica® 7 for Students, and allows for time-efficient calculation of thermal profiles. The non-dimensionality of the model allows for accurate refinement of the temporospatial mesh, and provides portability across material types. The model provides insight as to the mechanism of heat generation by qualifying the fraction of mechanical energy converted to thermal energy for different material types and sample geometries. Finally, our model gives an understanding of the effects of the heat transfer at the boundaries of the workpiece and suggests a backside heat loss localized at the center of the tool due to a decrease in thermal contact resistance. We have explored the effects of processing parameters on the performance of the friction stir fabrication process. The process has four stages; tool insertion, warm-up, bead formation, and steady-state translation. The tool insertion phase is characterized by a rapid increase in system horsepower requirements. During the warm-up phase, the mechanical energy of the rotating tip is converted to thermal energy. Once enough thermal energy has been transferred to the workpiece, the volume between the tip and the workpiece is filled by feedstock material. Finally, the tool is translated under relatively steady-state conditions. The success or failure of the process is dependent on adequate material delivery to the system. The horsepower requirements of the process depend on the material type and the rate of material delivery. We have explored the effect of processing parameters on the microstructure of the processed samples. Optical microscopy shows that the stratification of layers within the weld and the depth of the weld are both dependent on the processing parameters. EBSD analysis coupled with Vicker's microhardness measurements of the processed pieces show that the grain size within the weld nugget is constant over the range of processing parameters available to the system. Data also show that pressure and heat inherent in friction stir processing of strain-hardened Al5083 counteract strengthening of the temper of the alloy. / Ph. D.

The Thermoelectric Perfusion Probe is a completely electronic system that cyclically heats and cools tissue to measure blood perfusion. The probe produces the thermal event with a thermoelectric cooler and then measures the resulting heat flux and temperatures: the arterial temperature and the sensor temperature (the temperature between the heat flux gage and the skin). The Thermoelectric Perfusion Probe was validated and calibrated on a phantom tissue test stand, a system that simulates perfusion with known, controlled flow. With the new pressed sensor technology, a thermocouple sealed to a heat flux gage, the sensor temperature and the heat flux are simultaneously recorded. The pressed sensor tests validated the program used to predict perfusion for the Thermoelectric Perfusion Probe. This perfusion estimation program can determine the tissues perfusion regardless of how the thermal event is created (i.e. convective cooling, convective heating, conductive heating). Based on experimentation, the Thermoelectric Perfusion Probe displays good repeatability and sensitivity for continuously measuring perfusion. The sensitivity and repeatability of the Thermoelectric Perfusion Probe was proven when the perfusion estimates were compared to the perfusion estimates predicted by the Convective Perfusion Probe, a previously validated perfusion probe, and the CFD Flow Model, a computational model of the phantom tissue test stand. / Master of Science

Size reduction is a key issue in the development of contemporary integrated photonics. This thesis is mainly devoted to study some integrated photonic components in sub-wavelength or nanometric scales, both theoretically and experimentally. The possible approaches to reduce the sizes or to increase the functionalities of photonic components are discussed, including waveguides and devices based on silicon nanowires, photonic crystals, surface plasmons, and some near-field plasmonic components. First, some numerical methods, including the finite-difference time-domain method and the full-vectorial finite-difference mode solver, are introduced. The finite-difference time-domain method can be used to investigate the interaction of light fields with virtually arbitrary structures. The full-vectorial finite-difference mode solver is mainly used for calculating the eigenmodes of a waveguide structure. The fabrication and characterization technologies for nano-photonic components are reviewed. The fabrications are mainly based on semiconductor cleanroom facilities, which include thin film deposition, electron beam lithography, and etching. The characterization setups with the end-fire coupling and the vertical grating coupling are also described. Silicon nanowire waveguides and related devices are studied. Arrayed waveguide gratings with 11nm and 1.6nm channel spacing are fabricated and characterized. The dimension of these arrayed waveguide gratings is around 100 μm, which is 1--2 order of magnitude smaller than conventional silica based arrayed waveguide gratings. A compact polarization beam splitter employing positive/negative refraction based on a photonic crystal of silicon pillars is designed and demonstrated. Extinction ratio of ~15dB is achieved experimentally in a wide wavelength range. Surface plasmon waveguides and devices are analyzed theoretically. With surface plasmons the light field can be confined in a sub-wavelength dimension. Some related photonic devices, e.g., directional couplers and ring resonators, are studied. We also show that some ideas and principles of microwave devices, e.g., a branch-line coupler, can be borrowed for building corresponding surface plasmon based devices. Near-field plasmonic components, including near-field scanning optical microscope probes and left handed material slab lenses, are also analyzed. Some novel designs are introduced to enhance the corresponding systems. / QC 20100908

The oil and gas industry defines ultra-deep-water regions as areas in which water depths are greater than 1500 m. It is now well established that there are hydrocarbons in these regions. The reservoirs in these areas are generally located below basalt rocks or below salts. The focus of this thesis is to understand reflections, refractions, diffractions and scattering for acoustic lenses located below basalt rocks. The results of this study can potentially be used to understand the effect of tube waves on borehole seismic data in ultra-deep waters. Finite-difference modeling technique was used for this study. Finite-difference modeling allowed us to model refractions, reflections, diffractions and scattering; actually all events in surface seismic data, as well as borehole seismic data can be modeled. However, because of limited computational resources, this study will be based on a 2D finite difference instead of a 3D finite difference. This limitation implies that laterally infinite lenses were used to describe cylindrical boreholes. The four main characteristics of the geological constructs used here in simulating the ultra-deep-water regions were the size of the water column, the topography of the sea floor, the interfaces of basalt layers with the surroundings rocks, and the structure of heterogeneities inside the basalt layers. The average wavelength of wave propagation below the basalt layer is 125 m, which is very large compared to the size of a typical borehole (0.1 m). A lens with a thickness of 2.5 m, which corresponds to a dimension 50 times smaller than the average wavelength, sub-basalt was constructed. Also included were some lateral extensions in the construction of the lens to simulate wash-out zones. This study investigates the wave propagation below the basalt rocks and the effect of tube waves on borehole seismic data below the basalt layer by using these lenses instead of a cylindrical borehole. As the borehole geometry is different from that of the lens, the results are considered preliminary. Results suggest that tube waves are negligible in ultra-deep waters below basalt rocks because the wavelength of the seismic waves is large in comparison to the wash-out zone (192 times larger).

Ultra-fast laser heating of nano-films is investigated using 3-D Dual Phase Lag heat transport equation with laser heating at different locations on the metal film. The energy absorption rate, which is used to model femtosecond laser heating, is modified to accommodate for three-dimensional laser heating. A numerical solution based on an explicit finite-difference method is employed to solve the DPL equation. The stability criterion for selecting a time step size is obtained using von Neumann eigenmode analysis, and grid function convergence tests are performed. DPL results are compared with classical diffusion and hyperbolic heat conduction models and significant differences among these three approaches are demonstrated. We also develop an implicit finite-difference scheme of Crank-Nicolson type for solving 1-D and 3-D DPL equations. The proposed numerical technique solves one equation unlike other techniques available in the literature, which split the DPL equation into a system of two equations and then apply discretization. Stability analysis is performed using a von Neumann stability analysis. In 3-D, the discretized equation is solved using delta-form Douglas and Gunn time splitting. The performance of the proposed numerical technique is compared with the numerical techniques available in the literature.