Finite-difference models have been used for nearly 40 years to solve electromagnetic problems of heterogeneous nature. Further, these techniques are well known for being computationally expensive, as well as subject to various numerical artifacts. However, little is yet understood about the errors arising in the simulation of wideband sources with the finitedifference time-domain (FDTD) method. Within this context, the focus of this thesis is on two different problems. On the one hand, the speed and accuracy of current FDTD implementations is analysed and increased. On the other hand, the distortion of numerical pulses is characterised and mitigation techniques proposed. In addition, recent developments in general-purpose computing on graphics processing units (GPGPU) have unveiled new methods for the efficient implementation of FDTD algorithms. Therefore, this thesis proposes specific GPU-based guidelines for the implementation of the standard FDTD. Then, metaheuristics are used for the calibration of a FDTD-based narrowband simulator. Regarding the simulation of wideband sources, this thesis uses first Lagrange multipliers to characterise the extrema of the numerical group velocity. Then, the spread of numerical Gaussian pulses is characterised analytically in terms of the FDTD grid parameters. The usefulness of the proposed solutions to the previously described problems is illustrated in this thesis using coverage and wideband predictions in large-scale scenarios. In particular, the indoor-to-outdoor radio channel in residential areas is studied. Furthermore, coverage and wideband measurements have also been used to validate the predictions. As a result of all the above, this thesis introduces first an efficient and accurate FDTD simulator. Then, it characterises analytically the propagation of numerical pulses. Finally, the narrowband and wideband indoorto-outdoor channels are modeled using the developed techniques.

We study noise from wind turbines while taking irregular terrain and non-constant atmosphere into consideration. We will show that simulating the distribution of 3D acoustic waves can be done by using only low frequencies, thus reducing the computational complexity significantly.

This work presents an investigation into the optical scattering of heterogeneous cells with an application to two-photon imaging, optical scattering measurements and STED imaging. Using the finite difference time-domain (FDTD) method, the full-wave scattering by many cells containing multiple organelles with varying indices of refraction is computed. These simulations were previously limited to single cells for reasons of computational cost. A superposition approximation that uses the coherent linear superposition of FDTD-determined farfield scattering patterns of small numbers of cells to estimate the scattering from a larger tissue was developed and investigated. It was found that for the approximation to be accurate, the scattering sub-problems must at minimum extend along the incident field propagation axis for the full depth of the tissue, preserving the scattering that takes place in the direction of propagation. The FDTD method was used to study the scattering effects of multiple inhomogeneous cells on the propagation of a focused Gaussian beam with an application to two-photon imaging. It was found that scattering is mostly responsible for the reduction in two-photon fluorescence signal as depth is increased. It was also determined that for the chosen beam parameters and the cell and organelle configurations used, the nuclei are the dominant scatterers. FDTD was also utilized in an investigation of cellular scattering effects on the propagation of a common depletion beam used in STED microscopy and how scattering impacts the image obtained with a STED microscope. An axial doughnut beam was formulated and implemented in FDTD simulations, along with a corresponding focused Gaussian beam to simulate a fluorescence excitation beam. It was determined that the depletion beam will maintain a well-defined axial null in spite of scattering, although scattering will reduce the resulting fluorescence signal with focal depth. / text

A 3D velocity model and 3D wave propagation code have been employed to simulate long-period ground motions in the upper Mississippi embayment. This region is exposed to seismic hazard in the form of large earthquakes in the New Madrid seismic zone and observational data are sparse, making simulation a valuable tool for predicting the effects of large events. These simulations were undertaken in order to estimate ground-motion characteristics and to investigate the influence of the 3D embayment structure and finite-fault mechanics. There are three primary fault zones in the New Madrid seismic zone, each of which was likely associated with one of the three main shocks of the 1811-1812 earthquake sequence. For this study, three simulations have been conducted on each major segment, evaluating the effects of different epicentral locations and rupture directions on ground motions. The full wave field up to a frequency of 0.5 Hz was computed on a 200 × 200 × 50 km3 volume, and up to a frequency of 1.0 Hz on a 100 × 100 × 50 km3 volume, using a staggered-grid finitedifference code. Peak horizontal velocity, bracketed durations, and pseudospectral accelerations were calculated at the free surface. Animations showing the evolution of peak horizontal velocity through time at the free surface were also generated. The New Madrid seismic zone simulations indicate that for the considered bandwidth, finite-fault mechanics such as fault proximity, directivity effect, and slip distribution exert the most control on ground motions. The 3D geologic structure of the upper Mississippi embayment also influences ground motion, with indications that the bedrock surface acts as a wave guide, trapping waves in shallow, low-velocity parts of the embayment.

This text describes different aspects of the design of a Doctor-on-a-Chip device. Doctor-on-a-Chip is a DNA analysis system integrated on a single chip, which should provide all of the advantages that stem from the system integration, such as small sample volume, fast and accurate analysis, and low cost. The text describes all of the steps of the on-chip sample analysis, including DNA extraction from the sample, purification, PCR amplification, novel dielectrophoretic sorting of the DNA molecules, and finally detection. The overview is given of the technologies which are available to make the integration on a single chip possible. The microfluidic technologies that are used to manipulate the sample and other chemical reagents are already known and in this text they are analyzed in terms of their feasibility in the on-chip system integration. These microfluidic technologies include, but are not limited to, microvalves, micromixers, micropumps, and chambers for PCR amplification. The novelty in the DNA analysis brought by Doctor-on-a-Chip is the way in which the different DNA molecules in the sample (for example, human and virus DNA) are sorted into different populations. This is done by means of dielectrophoresis – the force experienced by dielectric particles (such as DNA molecules) when subject to a non-uniform electric field. Different DNA molecules within a sample experience different dielectrophoretic forces within the same electric field, which makes their separation, and therefore detection, possible. In this text, the emphasis is put on numerical modelling of the dielectrophoretic effect on biological particles. The importance of numerical modelling lies in the fact that with the accurate model it is easier to design systems of microelectrodes for dielectrophoretic separation, and tune their sub-micrometre features to achieve the maximum separation efficacy. The numerical model described in this text is also experimentally verified with the novel microelectrodes design for dielectrophoretic separation, which is successfully used to separate the mixture of different particles in the micron and sub-micron range.

Master of Science / Department of Mathematics / Nathan Albin / This work utilizes finite differences to approximate the first derivative of non-periodic smooth functions. Math literature indicates that stabilizing Partial Differential Equation solvers based on high order finite difference approximations of spatial derivatives of a non-periodic function becomes problematic near a boundary. Hagstrom and Hagstrom have discovered a method of introducing additional grid points near a boundary, which has proven to be effective in stabilizing Partial Differential Equation solvers. Hagstrom and Hagstrom demonstrated their method for the case of the one-dimensional advection equation using spatial derivative approximations of even orders up to twenty-second order. In this dissertation, we explore the efficacy of the Hagstrom and Hagstrom method for the same Partial Differential Equation with spatial derivative approximations of odd orders and orders higher than twenty-two and report the number and locations of additional grid points required for stability in each case.

Through perturbation analysis, a study of the role of Brinkman viscosity in the propagation of finite amplitude harmonic waves is carried out. Interplay between various parameters, namely, frequency, Reynolds number and beta are investigated. For systems with physically realizable Reynolds numbers, departure from the Darcy Jordan model (DJM) is noted for high frequency signals. Low and high frequency limiting cases are discussed, and the physical parameters defining the acoustic propagation are obtained. Through numerical analyses, the roles of Brinkman viscosity, the Darcy coefficient, and the coefficient of nonlinearity on the evolution of finite amplitude harmonic waves is stud- ied. An investigation of acoustic blow-ups is conducted, showing that an increase in the magnitude of the nonlinear term gives rise to blow-ups, while an increase in the strength of the Darcy and/or Brinkman terms mitigate them. Finally, an analytical study via a regular perturbation expansion is given to support the numerical results. In order to gain insight into the formation and evolution of nonlinear standing waves un- der the Brinkman model, a numerical analysis is conducted on the weakly nonlinear model based on Brinkman’s equation. We develop a finite difference scheme and conduct a param- eter study. An examination of the Brinkman, Darcy, and nonlinear terms is carried out in the context of their roles on shock formation. Finally, we compare our findings to those of previous results found in similar nonlinear equations in other fields. So as to better understand the behavior of finite-amplitude harmonic waves under a Brinkman-based poroacoustic model, approximations and transformations are used to recast the Brinkman equation into the damped Burger’s equation. An examination is carried out for the two special solutions of the damped Burger’s equation: the approximate solution to the damped Burger’s equation and the boundary value problem given an initial sinusoidal pulse. The effects of the Darcy coefficient, Reynolds number, and nonlinear coefficient on these solutions are investigated.

Most realistic financial derivatives models are too complex to allow explicit analytic solutions. The computational techniques used to implement those models fall into two broad categories: finite difference methods for the solution of partial differential equations (PDEs) and Monte Carlo simulation. Accordingly, the course consists of two sections. The first half of the course focuses on finite difference methods. The following topics are discussed; Parabolic PDEs, Black-Scholes PDE for European and American options; binomial and trinomial trees; explicit, implicit and Crank- Nicholson finite difference methods; far boundary conditions, convergence, stability, variance bias; early exercise and free boundary conditions; parabolic PDEs arising from fixed income derivatives; implied trees for exotic derivatives, adapted trees for interest rate derivatives. The second half of the course focuses on Monte Carlo. The following topics are discussed; Random number generation and testing; evaluation of expected payoff by Monte Carlo simulation; variance reduction techniques�antithetic variables, importance sampling, martingale control variables; stratification, low-discrepancy sequences and quasi-Monte Carlo methods; efficient evaluation of sensitivity measures; methods suitable for multifactor and term-structure dependent models. Computational Methods of Financial Mathematics is taught by Marcel Blais, a professor at Worcester Polytechnic Institute.

Em muitas simulações eletromagnéticas utilizando o método FDTD, é desejado que os campos radiados pelas estruturas em análise sejam transmitidos para fora do domínio computacional. Infelizmente isto não é possível de ser realizado através do método FDTD em sua forma original. Para resolver este problema, deve-se implementar, nas superfícies limítrofes dos domínios computacionais, condições especiais denominadas na literatura técnica de Condições de Fronteiras de Absorção, ou, em inglês, \"Absorbing Boundary Conditions\" (ABC´s). Essas Condições de Fronteiras de Absorção impedem que os campos radiados sejam refletidos nas superfícies limítrofes dos domínios computacionais, retornando para o interior do domínio e interferindo no resultado final das simulações. Não existe uma técnica de absorção ideal, ou seja, que elimine totalmente a reflexão. As técnicas atualmente existentes possuem vantagens e desvantagens, podendo ser mais ou menos eficientes, o que faz com que esse tema ainda seja motivo de extensivos estudos. O objetivo deste trabalho consiste no estudo, implementação e comparação de Condições de Fronteiras de Absorção e na indicação de uma possível melhoria nessa área. São realizadas simulações em domínios bidimensionais e tridimensionais para se determinar vantagens e desvantagens de cada técnica de absorção. A análise dos resultados das simulações está focalizada no grau de atenuação que as ABCs possuem e na carga computacional despendidas por elas. Este trabalho é concluído com simulações empregando as condições de fronteiras analisadas para três estruturas clássicas. As vantagens e desvantagens de cada ABC são apresentadas e uma melhoria proposta na técnica de Auto Teleportação de Campos, ou, em inglês, \"Self Teleportation of Fields\" é validada. As estruturas analisadas são uma microlinha de transmissão, um filtro planar e um cilindro metálico iluminado por uma onda plana uniforme. / In many electromagnetic computational simulations using the FDTD method, it is desired that the electromagnetic fields radiated by the structures under analysis can be transmitted outwards the computational domain. Unfortunately, this is impossible to be done by the FDTD method in its original form. To mitigate this problem, one must apply special conditions to the computational domain boundaries, known in the technical literature as Absorbing Boundary Conditions (ABCs) These Absorbing Boundaries Conditions prevent the radiated fields to be reflected by boundaries back into the computational domain. Without them, these fields would interfere with the final simulation results. However, there is no ideal technique that completely eliminates the reflections. The existing techniques have advantages and disadvantages, which make them more or less efficient, still making this subject a theme of extensive studies. This work is aimed at studying, implementing and comparing these Absorbing Boundary Conditions and at indicating a possible improvement in this field. Simulations in bi-dimensional and three-dimensional domains were made to evaluate advantages and disadvantages of each absorption technique. The analysis of the simulation results was focused in the attenuation degree of the ABCs and their computational burden. The work is concluded with simulations using the analyzed ABCs for three classic structures. The advantages and disadvantages of each ABC are presented and a proposed improvement on the \"Self Teleportation of Fields\" technique is validated. The analyzed structures are a microstrip line, a planar filter and a metallic cylinder illuminated by a uniform plane wave.

This work outlines the design and implementation of an algorithm to simulate two-polarisation bowed string motion, for the purpose of realistic sound synthesis. The algorithm is based on a physical model of a linear string, coupled with a bow, stopping fi ngers, and a rigid, distributed fingerboard. In one polarisation, the normal interaction forces are based on a nonlinear impact model. In the other polarisation, the tangential forces between the string and the bow, fingers, and fingerboard are based on a force-velocity friction curve model, also nonlinear. The linear string model includes accurate time-domain reproduction of frequency-dependent decay times. The equations of motion for the full system are discretised with an energy-balanced finite difference scheme, and integrated in the discrete time domain. Control parameters are dynamically updated, allowing for the simulation of a wide range of bowed string gestures. The playability range of the proposed algorithm is explored, and example synthesised gestures are demonstrated.