Wave propagation is a one of the most studied phenomenons in history due to the variety of applications such as quantum mechanics, electrodynamics and acoustics. In this thesis, the possibilities of improving numerical methods for solving the wave equation will be studied. More specifically, the dispersion relation will be used as a focal point. Generally there is a difference between the dispersion relation in the numerical solution and the analytic solution and the aim will be to decrease this difference and study the consequences. The numerical method that will be used and improved is the finite difference method (FDM). A dispersion relation for the numerical scheme will be derived including parameters from the spatial discretisation. These parameters will be optimised with the gradient descent method while retaining the second order accuracy of the derivative approximation. Performance is tested with numerical examples and the method of optimising for improved dispersion relation is proved to be successful. The optimised second order accurate schemes outperforms the standard second order accurate method in all simulated examples. When comparing the optimised stencil with the equally computationally expensive fourth order accurate method the optimised stencil performs better for sparse grids, especially when the spatial variation in the solution is high. For finer grids the fourth order accurate method quickly achieves smaller errors and is therefore preferable.

Modelling ionospheric total electron content (TEC) is an important area of interest for radio wave propagation, geodesy, surveying, the understanding of space weather dynamics and error correction in relation to Global Navigation Satellite Systems (GNNS) applications. With the utilisation of improved ionosonde technology coupled with the use of GNSS, the response of technological systems due to changes in the ionosphere during both quiet and disturbed conditions can be historically inferred. TEC values are usually derived from GNSS measurements using mathematically intensive algorithms. However, the techniques used to estimate these TEC values depend heavily on the availability of near-real time GNSS data, and therefore, are sometimes unable to generate complete datasets. This thesis investigated possibilities for the modelling of TEC values derived from the South African Global Positioning System (GPS)receiver network using linear regression methods and artificial neural networks (NNs). GPS TEC values were derived using the Adjusted Spherical Harmonic Analysis (ASHA) algorithm. Considering TEC and the factors that influence its variability as “dependent and independent variables” respectively, the capabilities of linear regression methods and NNs for TEC modelling were first investigated using a small dataset from two GPS receiver stations. NN and regression models were separately developed and used to reproduce TEC fluctuations at different stations not included in the models’ development. For this purpose, TEC was modelled as a function of diurnal variation, seasonal variation, solar and magnetic activities. Comparative analysis showed that NN models provide predictions of GPS TEC that were an improvement on those predicted by the regression models developed. A separate study to empirically investigate the effects of solar wind on GPS TEC was carried out. Quantitative results indicated that solar wind does not have a significant influence on TEC variability. The final TEC simulation model developed makes use of the NN technique to find the relationship between historical TEC data variations and factors that are known to influence TEC variability (such as solar and magnetic activities, diurnal and seasonal variations and the geographical locations of the respective GPS stations) for the purposes of regional TEC modelling and mapping. The NN technique in conjunction with interpolation and extrapolation methods makes it possible to construct ionospheric TEC maps and to analyse the spatial and temporal TEC behaviour over Southern Africa. For independent validation, modelled TEC values were compared to ionosonde TEC and the International Reference Ionosphere (IRI) generated TEC values during both quiet and disturbed conditions. This thesis provides a comprehensive guide on the development of TEC models for predicting ionospheric variability over the South African region, and forms a significant contribution to ionospheric modelling efforts in Africa.

Satellite systems are being planned for two-way communication with mobile vehicles using UHF and L-band frequencies. Of special concern in the system design are the characteristics of propagation in suburban and rural areas where fading occurs due to multipath effects and vegetative shadowing. A review of the literature was performed to study these propagation impairments. Available experimental data are examined, compared, and summarized. Propagation through vegetation is studied in order to compare reported modeling efforts and to determine the parameter dependences of path loss. A simple deterministic path model is then presented to estimate vegetative path loss. An overall statistical model is also proposed to describe the signal level fading statistics. The statistical model is compared to data, and the deterministic path model is used to determine the mean of signal level distribution functions in the presence of shadowing. / Master of Science

The analysis of wave propagation in hyperelastic waveguides has significant applications in various branches of engineering like Non-Destructive Testing and Evaluation, impact analysis, material characterization and damage detection. Linear elastic models are typically used for wave analysis since they are sufficient for many applications. However, certain solids exhibit inherent nonlinear material properties that cannot be adequately described with linear models. In the presence of large deformations, geometric nonlinearity also needs to be incorporated in the analysis. These two forms of nonlinearity can have significant consequences on the propagation of stress waves in solids. A detailed analysis of nonlinear wave propagation in solids is thus necessary for a proper understanding of these phenomena. The current research focuses on the development of novel algorithms for nonlinear finite element analysis of stress wave propagation in hyperelastic waveguides. A full three-dimensional(3D) finite element analysis of stress wave propagation in waveguides is both computationally difficult and expensive, especially in the presence of nonlinearities. By definition, waveguides are solids with special geometric features that channel the propagation of stress waves along certain preferred directions. This suggests the use of kinematic waveguide models that take advantage of the special geometric features of the waveguide. The primary advantage of using waveguide models is the reduction of the problem dimension and hence the associated computational cost. Elementary waveguide models like the Euler-Bernoulli beam model, Kirchoff plate model etc., which are developed primarily within the context of linear elasticity, need to be modified appropriately in the presence of material/geometric nonlinearities and/or loads with high frequency content. This modification, besides being non-trivial, may be inadequate for studying nonlinear wave propagation and higher order waveguide models need to be developed. However, higher order models are difficult to formulate and typically have complex governing equations for the kinematic modes. This reflects in the relatively scarce research on the development of higher order waveguide models for studying nonlinear wave propagation. The formulation is difficult primarily because of the complexity of both the governing equations and their linearization, which is required as part of a nonlinear finite element analysis. One of the primary contributions of this thesis is the development and implementation of a general, flexible and efficient framework for automating the finite element analysis of higher order kinematic models for nonlinear waveguides. A hierarchic set of higher order waveguide models that are compatible with this formulation are proposed for this purpose. This hierarchic series of waveguide models are similar in form to the kinematic assumptions associated with standard waveguide models, but are different in the sense that no conditions related to the stress distribution specific to a waveguide are imposed since that is automatically handled by the proposed algorithm. The automation of the finite element analysis is accomplished with a dexterous combination of a nodal degrees-of-freedom based assembly algorithm, automatic differentiation and a novel procedure for numerically computing the finite element matrices directly from a given waveguide model. The algorithm, however, is quite general and is also developed for studying nonlinear plane stress configurations and inhomogeneous structures that require a coupling of continuum and waveguide elements. Significant features of the algorithm are the automatic numerical derivation of the finite element matrices for both linear and nonlinear problems, especially in the context of nonlinear plane stress and higher order waveguide models, without requiring an explicit derivation of their algebraic forms, automatic assembly of finite element matrices and the automatic handling of natural boundary conditions. Full geometric nonlinearity and the hyperelastic form of material nonlinearity are considered in this thesis. The procedures developed here are however quite general and can be extended for other types of material nonlinearities. Throughout this thesis, It is assumed that the solids under investigation are homogeneous and isotropic. The subject matter of the research is developed in four stages: First, a comparison of different finite element discretization schemes is carried out using a simple rod model to choose the most efficient computational scheme to study nonlinear wave propagation. As part of this, the frequency domain Fourier spectral finite element method is extended for a special class of weakly nonlinear problems. Based on this comparative study, the Legendre spectral element method is identified as the most efficient computational tool. The efficiency of the Legendre spectral element is also illustrated in the context of a nonlinear Timoshenko beam model. Since the spectral element method is a special case of the standard nonlinear finite element Method, differing primarily in the choice of the element basis functions and quadrature rules, the automation of the standard nonlinear finite element method is undertaken next. The automatic finite element formulation and assembly algorithm that constitutes the most significant contribution of this thesis is developed as an efficient numerical alternative to study the physics of wave propagation in nonlinear higher order structural models. The development of this algorithm and its extension to a general automatic framework for studying a large class of problems in nonlinear solid mechanics forms the second part of this research. Of special importance are the automatic handling of nonlinear plane stress configurations, hierarchic higher order waveguide models and the automatic coupling of continuum and higher order structural elements using specially designed transition elements that enable an efficient means to study waveguides with local inhomogeneities. In the third stage, the automatic algorithm is used to study wave propagation in hyperelastic waveguides using a few higher order 1D kinematic models. Two variants of a particular hyperelastic constitutive law – the6-constantMurnaghanmodel(for rock like solids) and the 9-constant Murnaghan model(for metallic solids) –are chosen for modeling the material nonlinearity in the analysis. Finally, the algorithm is extended to study energy-momentum conserving time integrators that are derived within a Hamiltonian framework, thus illustrating the extensibility of the algorithm for more complex finite element formulations. In short, the current research deals primarily with the identification and automation of finite element schemes that are most suited for studying wave propagation in hyper-elastic waveguides. Of special mention is the development of a novel unified computational framework that automates the finite element analysis of a large class of problems involving nonlinear plane stress/plane strain, higher order waveguide models and coupling of both continuum and waveguide elements. The thesis, which comprises of 10 chapters, provides a detailed account of various aspects of hyperelastic wave propagation, primarily for 1D waveguides.

Doctor of Philosophy / Department of Mechanical and Nuclear Engineering / Liang-Wu Cai / X. J. Xin / Thin walled structures such as plates and shells have application in many fields of engineering because these structures are light weight and can support large loads when designed suitably. In real world, loads may cause these structures to vibrate which can be undesirable causing fatigue and failure of the structure. Such undesirable vibrations need to be reduced or eliminated. In this work, analytical studies of flexural wave propagation for idealized geometries are conducted and finite element method (FEM) is used to explore the effects of composite panel designs of finite size for the reduction of vibration transmission. In the analytical studies, the influence of the material properties on the reflection and transmission characteristics are explored for an infinite bi-material plate, and infinite plate with a strip inhomogeneity. In the analytical study of an infinite thin plate with a solid circular inclusion, the far and near field scattering characteristics are explored for different frequencies and material properties. All the analytical studies presented here and reported in the literature consider infinite plates to characterize the flexural wave propagation. Obtaining closed form solutions to characterize the flexural wave propagation in a finite plate with inclusions is mathematically difficult process. So, FEM is used to explore the composite panel designs. The understanding gained about the material properties influence on the flexural wave propagation from analytical studies helped with the choice of materials for FEM simulations. The concept of phononic crystals is applied to define the design variations that are effective in suppressing vibration transmission. Various design configurations are explored to study the effects of various parameters like scatterer’s material properties, geometry and spatial pattern. Based on the knowledge gained through a systematic parametric study, a final design of the composite sandwich panel is proposed with an optimum set of parameters to achieve the best vibration reduction. This is the first study focused on reducing vibration and wave transmission in composite rotorcraft fuselage panels incorporating the concept of phononic crystals. The optimum sandwich panel design achieved 98% vibration transmission reduction at the frequency of interest of 3000 Hz.

Tomography in soft X-ray microscopy is an emerging technique for obtaining quantitative 3D structural information about cells. One of its strengths, compared with other techniques, is that it can image intact cells in their near-native state at a few 10 nm’s resolution, without staining. However, the methods for reconstructing 3D-data rely on algorithms that assume projection data, which the images are generally not due to the imaging systems’ limited depth of focus. To bring out the full potential of tomography in soft X-ray microscopy an improved understanding of the image formation is desired. This Thesis reviews zone plate-based X-ray microscopy for biological imaging and the theory necessary for a numerical implementation of a 3D image formation model. Furthermore, a novel reconstruction approach is proposed that improves the overall resolution in a reconstruction of a tomographically imaged object. This is demonstrated by simulations and experiments. Finally, this Thesis covers work on the Stockholm X-ray microscope, including an upgrade of the X-ray source yielding unprecedented brightness for a compact system. With this upgrade it was possible to do high-quality imaging of cells in their near-native state with only 10 second exposures. / Tomografi i mjukröntgenmikroskopi är en ny teknik för att få ut kvantitativ strukturell 3D information om celler. Dess styrka jämfört med andra tekniker är att den kan avbilda intakta celler i deras nära naturliga tillstånd med ett par 10 nm upplösning, utan omfattande preparering. Dock är metoderna för att rekonstruera 3D-data beroende av algoritmer som antar projektionsdata, vilket bilderna i allmänhet inte är på grund av avbildningsystemens begränsade skärpedjup. För att få ut den fulla potentialen av tomografi i röntgenmikroskopi behövs en ökad förståelse för avbildningsprocessen. Denna avhandling behandlar zonplatte-baserad röntgenmikroskopi för biologisk avbildning och den nödvändiga teorin för en numerisk implementering av en avbildningsmodell i 3D. En ny rekonstruktionsmetod föreslås som förbättrar upplösningen i rekonstruktionen för ett tomografiskt avbildat objekt. Detta visas i simuleringar och experiment. Slutligen omfattar denna avhandling arbete på Stockholms mjukröntgenmikroskop, inklusive en uppgradering av röntgenkällan som ger oöverträffad ljusstyrka för ett kompakt system. Denna uppgradering möjliggör högkvalitativ avbildning av celler i deras nästan naturliga tillstånd med endast 10 sekunders exponering. / <p>QC 20160324</p>

Surface mount devices (SMDs), such as flip chip packages and ball grid array (BGA) packages are gaining in popularity in microelectronics industry because they provide high density inputs/outputs, better electrical and thermal performance. However, these solder bump interconnections in SMDs are sandwiched between the silicon die and the substrate, which makes them challenging to be inspected. Current non-destructive solder bump inspection techniques like electrical testing, X-ray and acoustic microscopy have some application gaps. New solder bump inspection technique is urgently needed to fill these gaps. Previous work has shown the potential of using a non-contact, non-destructive laser ultrasonics and laser interferometer based inspection system for assessing solder bump qualities. The system uses a pulsed Nd:YAG laser to induce ultrasound in the chip packages and a laser interferometer to measure the transient out-of-plane displacement on the package surface. The quality of the solder bumps can be evaluated by analyzing the out-of-plane displacement. However, there are still some gaps that need to be addressed before the system is ready on the shelf. This dissertation focuses on addressing some of these existing issues. The research work consists of the following: 1) a control interface was developed to integrate all the different modules to achieve automation. 2) a new signal-processing method for analyzing the transient out-of-plane displacement signals without requiring a known-good reference chip was developed. 3) the application scope of the system was expanded to inspect the second level solder bumps in BGA packages. Two types of process-induced defects including poor-wetting and solder bump voids were investigated. Meanwhile, solder bump fatigue caused by cyclic mechanical bending and thermal cycle was also studied using this system. 4) a finite element analysis was performed to study the thermo-mechanical reliability of solder bumps in PBGA package under cyclic thermal loads. The successful completion of the research objectives has led to a laser ultrasound solder bump inspection system prototype with more user-friendliness, higher throughputs, better repeatability and more flexibility, which accelerate the commercialization the system.

When sound waves propagate in a duct in the presence of turbulent flow, turbulent mixing can cause attenuation of the sound waves extra to that caused by the viscothermal effects. Experiments show that compared to the viscothermal effects, this turbulent absorption becomes the dominant contribution to the sound attenuation at sufficiently low frequencies. The mechanism of this turbulent absorption is attributed to the turbulent stress and the turbulent heat transfer acting on the coherent perturbations (including the sound waves) near the duct wall, i.e. sound-turbulence interaction. The purpose of the current investigation is to understand the mechanism of the sound-turbulence interaction in low-Mach-number internal flows by theoretical modeling and numerical simulations. The turbulence absorption can be modeled through perturbation turbulent Reynolds stresses and perturbation turbulent heat flux in the linearized perturbation equations. In this thesis, the linearized perturbation equations are reviewed, and different models for the turbulent absorption of the sound waves are investigated. A new non–equilibrium model for the perturbation turbulent Reynolds stress is also proposed. The proposed model is validated by comparing with experimental data from the literature, and with the data from Direct Numerical Simulations (DNS) of pulsating turbulent channel flow. Good agreement is observed. / <p>QC 20150526</p>

The purpose of this study is to examine the propagation of waves of finite deformation in rubbers through experiments and analysis. First, attention is focused on the propagation of one-dimensional dispersive waves in strips of latex and nitrile rubber. Tensile wave propagation experiments were conducted at high strain-rates by holding one end fixed and displacing the other end at a constant velocity. A high-speed video camera was used to monitor the motion and to determine the evolution of strain and particle velocity in rubber strips. Analysis of the response through the theory of finite wave propagation indicated a need for an appropriate constitutive model for rubber; by quantitative matching between the experimental observations and analytical predictions, an appropriate instantaneous elastic response for the rubbers was obtained. This matching process suggested that a simple power-law constitutive model was capable of representing the high strain-rate response for both rubbers used. Next, the propagation of one-dimensional shock waves in strips of latex and nitrile rubber is examined. Shock waves have been generated under tensile impact in pre-stretched rubber strips; analysis of the response yields the tensile shock adiabat for rubbers. The propagation of shocks is analyzed by developing an analogy with the theory of detonation. Attention is then focused on the propagation of unloading waves of finite deformation in a rubber specimen analytically and experimentally. A rubber strip stretched to many times its initial length is released at one end and the resulting unloading is examined. Dispersive waves as well as shock waves are observed in these experiments. Quantitative discrepancies between the analytical model and experimental observations are again used to motivate a power-law model. Hysteresis in the response is attributed to strain-induced crystallization and melting phase transitions in natural latex rubber, and to nonequilibrium microstructural deformation in nitrile rubber. Finally, a Kolsky experiment is conducted and analyzed under the framework of dispersive loading and unloading waves utilized in the previous experiments. In this experiment, a phase boundary is introduced separating low and high strain phases of the rubber and is demonstrated to persist as a stationary boundary in latex rubber. / text

Characterizing natural and man-made fracture networks is fundamental to predicting the storage capacity and pathways for flow of both carbonate and shale reservoirs. The goal of this study is to determine the seismic response specifically to networks of fractures clustered closely together through the analysis of seismic wavefield scatter, directional phase velocities, and amplitude attenuation. To achieve this goal, finite element modeling techniques are implemented to allow for the meshing of discontinuous fracture interfaces and, therefore, provide the most accurate calculation of seismic events from these irregular surfaces. The work presented here focuses on the center layer of an isotropic model that is populated with two main phases of fracture network alteration: a single large-scale cluster and multiple smaller-scale clusters. Phase 1 first confirms that the seismic response of a single idealized vertically fractured cluster is distinct crosscutting energy within a seismogram. Further investigation shows that, as fracture spacing within the cluster decreases, the depth at which crosscutting energy appears exponentially increases, placing it well below the true location of the cluster. This relationship holds until 28% of the fractures are moved from their uniformly spaced locations to random locations within the cluster. The vertical thickness of the cluster has little effect on the location or strength or the crosscutting signature. Phase 2 shows that, although clusters of more randomly spaced fractures mask crosscutting energy, a marked decrease in amplitude coinciding with a bend in the wavefront produces a heterogeneous anisotropic seismic response. This amplitude decay and heterogeneous anisotropy is visible until cluster spacing drops below one half of the wavelength or the ratio of fractured material to matrix material within a cluster drops below 37%. Therefore, the location of an individual fracture cluster can be determined from the location of amplitude decay, heterogeneous anisotropy, and crosscutting energy. Furthermore, the density of the cluster can be determined from the degree of amplitude decay, the angle of heterogeneous anisotropy, and the depth of cross-cutting energy. These relationships, constrained by limits on their detectability, can aid fracture network interpretation of real seismic data. / text