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This thesis presents research into the micromechanical modelling of composite materials using numerical techniques. Composite materials are generally examined from two points of view: macromechanics and micromechanics, owing to their inherent heterogeneous nature. In this research, the material behaviour is examined on a microscopic scale, as the properties of interest, i.e. strength and toughness, are dependent on local phenomena. In general, the strength and toughness of composite materials are not as well understood as the simpler elastic properties, because in many cases the modes of failure under a given system of external load are not predictable in advance. Previous research in this field has typically involved specially designed experiments, theoretical/statistical studies, or the use of numerical models. In this study, advanced implementations of numerical methods in continuum mechanics, i.e. the boundary element and the finite element methods are employed to gain a greater understanding of composite behaviour. The advantage of using numerical methods, as opposed to experimental studies, is that the geometric and material characteristics can be investigated parametrically, in addition to the reduced time and expense involved. However, to model the complete behaviour of real composites is still not possible, due to the degree of complexity and uncertainty involved in modelling the various mechanisms of damage and failure, etc. and also due to the immense computational cost. Therefore, simplified models must be employed which are limited by their assumptions. For the preliminary studies within this thesis, geometrically simplified models are presented to provide an understanding of the influence of embedding second phase inclusions on the local stress fields, and also to validate the numerical techniques with readily available analytical solutions. These models are then extended to accommodate additional phenomena, such as inclusion interaction, spatial inclusion arrangement, material formulation, i.e. consisting of two- and three-phases of various material properties. The influence of such factors on the local stress concentrations, which play an important role in determining the strength of the composite, is analysed through a series of parametric studies. The localised toughening of composites is also considered through novel investigations into the interaction between a propagating crack with inclusions and microcracks. Through the development of the numerical models a more realistic representation of composite behaviour is achieved, which in tum, provides an improved knowledge of the factors that control strength and toughness. Such information is invaluable to composite material designers, who presently rely heavily on experimental studies to develop composite materials.

This thesis is concerned with a mathematical study of several problems involving infiltration from irrigation channels into an unsaturated homogeneous soil. All the problems considered are two dimensional and are solved numerically by employing boundary integral equation techniques. In the first chapter I introduce some of the literature and ideas surrounding my thesis. Some background information is stated followed by an outline of the thesis and a list of author’s published works that support the material in the thesis. Full descriptions of the fundamental equations used throughout the thesis are provided in chapter 2. Chapter 3 contains the first problem considered in this thesis which is infiltration from various shapes of single and periodic irrigation channels. Specifically strip, semi-circular, rectangular and v shaped channels. The solutions are obtained using the boundary element technique. The solutions are then compared with the results obtained by Batu [14] for single and periodic strip sources. In chapter 4 a boundary integral equation method is adopted for the solution of flow from single and periodic semi-circular channels into a soil containing impermeable inclusions. The impermeable inclusions considered are of rectangular, circular and square shapes. The aim is to observe how the various shapes of inclusions can affect the direction of the flow particularly in the region adjacent to the zone where plant roots would be located. Chapter 5 solves the problem of infiltration from single and periodic semicircular irrigation channels into a soil containing impermeable layers. A modification is made to the boundary integral equation in order to include the impermeable layers with the integration over the layers involving Hadamard finite-part integrals. The objective of the work is to investigate how the number and the depth of the impermeable layers affects the flow. Chapter 6 employs a particular Green’s function in the boundary integral equation. The Green’s function is useful for flow from a single channel since it removes the need to evaluate the boundary integral along the soil surface outside the irrigation channel. A time dependent infiltration problem is considered in chapter 7. The Laplace transform is applied to the governing equations and the boundary integral equation technique is used to solve the resulting partial differential equation. The Laplace transform is then inverted numerically to obtain the time dependent values of the matric flux potential. / Thesis (Ph.D.) - University of Adelaide, School of Mathematical Sciences, 2008

The theme of the research is on the use of the hypersingular boundary element method for the modelling of surface waves. Surface waves in solids are known to be partially reflected & transmitted and mode converted into body waves at stress discontinuities, which suggests that a formulation continuous in stress and strain might prove beneficial for modelling purposes. Such continuity can be achieved with a subparametric approach where the geometry is approximated using linear elements and the field variables, displacement and traction, are approximated using cubic Hermitian and linear shape functions respectively. The higher order polynomial for approximating displacement is intended to be a more accurate representation of the physics relating to surface wave phenomena, especially at corners, and thus, is expected to capture this behaviour with greater accuracy than the standard isoparametric approach. The subparametric approach affords itself to continuity in stress and strain by imposing a smoothness in the elements, which is not available to the isoparametric approach. As the attention is focused primarily on the modelling of surface waves on the boundary of a medium rather than the interior, the boundary element method lends itself appropriately to this end.A 2D semi analytical integration scheme is employed to evaluate the integrals appearing in the hypersingular boundary integral formulation. The integration scheme is designed to reduce the errors incurred when integrals with singular integrands are evaluated numerically. The scheme involves the application of Taylor expansions to formulate the integrals into two parts. One part is regular and is evaluated numerically and the other part is singular but sufficiently simple to be evaluated analytically. The scheme makes use of the aforementioned subparametric approach and is applied to linear elements for the use in steady state elastodynamic boundary element method problems. The steady state problem is used as it is a simplified problem and is sufficient to permit the investigation of surface vibration at a constant motion. The 2D semi analytical integration scheme presented can be naturally extended to 3D.A particular focus and novelty of the work is the application of different limiting approaches to determine the free terms common to boundary integral methods. The accurate numerical solution of hypersingular boundary integral equations necessitates the precise evaluation of free terms, which are required to counter discontinuous and often unbounded behaviour of hypersingular integrals at a boundary. The common approach for the evaluation of free terms involves integration over a portion of a circular/spherical shaped surface centred at a singularity and allowing the radius of the circle/sphere to tend to zero. This approach is revisited in order to ascertain whether incorrect results are possible as a consequence of shape dependency, which is a recognised issue for hypersingular integrals.Two alternative methods, which are shape invariant, are proposed and investigated for the determination of free terms. The first approach, the point limiting method, involves moving a singularity towards a shrinking integration domain at a faster rate than the domain shrinks. Issues surrounding the choice of approach, shrinkage rates and path dependency are examined. A related and second approach, the boundary limiting method, involves moving an invariant, but shrinking, boundary toward the singularity, again at a faster rate than the shrinkage of the domain. The latter method can be viewed as a vanishing exclusion zone approach but the actual boundary shape is used for the boundary of the exclusion zone. Both these methods are shown to provide consistent answers and can be shown to be directly related to the result obtained by moving a singularity towards a boundary, that is, by comparison with the direct method. Unlike the circular/spherical approach the two methods involve integration over the actual boundary shape and consequently shape dependency is not an issue. A particular highlight of the point limiting approach is the ability to obtain free terms in mixed formulation, which is not available to the circular/spherical approach.There are three numerical problems considered in this research. The first problem considers the longitudinal vibration of a square plate. This is a problem for which a known analytical solution exists and is used to verify the equation formulation and integration scheme adopted for the isoparametric and subparametric formulations. Both formulations are as accurate as each other and produce results that are in keeping with the analytical solution, thus instilling confidence in their predictions.The second problem considers the simulation of surface waves on a square plate. Various boundaries of a square plate have displacement conditions imposed on them as a result of surface wave propagation. The results indicate that the surface wave behaviour is not captured. However, the analytical solution does not make any consideration for the effects from corners; the analytical solution is for a Rayleigh wave propagating upon a planar surface. It does not take into account the wave phenomena encountered at corners. Therefore, these results cannot be used to validate the predictions obtained on the boundary of the problem considered. The purpose of this problem is to illustrate the impact of corners on the surface wave propagation. Sensitivity studies are conducted to illustrate the effect of corners on the computed solution at the boundary.The final problem considers the simulation of surface waves on a circular plate. Various portions of the boundary of the circular plate have displacement conditions imposed on them as a result of surface wave propagation on curved surfaces. The results indicate that the isoparametric and subparametric predictions are similar to one another. However, both displacement profiles predict the presence of other waves. Given the multi faceted nature of the mesh, the computed solution is picking up mode conversion and partial reflection & transmission of surface waves. In reality, this is not expected as the surface of the boundary is smooth. However, due to the discretisation there are many corners in this problem.

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Interconnect modeling plays an important role in design and verification of VLSI circuits and packages. For low frequency circuits, great advances for parasitic resistance and capacitance extraction have been achieved and wide varieties of techniques are available. However, for high frequency circuits and packages, parasitic inductance and impedance extraction still poses a tremendous challenge. Existing algorithms, such as FastImp and FastHenry developed by MIT, are slow and inherently unable to handle multiple dielectrics and magnetic materials. In this research, we solve three problems in interconnect modeling for high frequency circuits and packages. 1) Multiple dielectrics are common in integrated circuits and packages. We propose the first Boundary Element Method (BEM) algorithm for impedance extraction of interconnects with multiple dielectrics. The algorithm uses a novel equivalentcharge formulation to model the extraction problem with significantly fewer unknowns. Then fast matrix-vector multiplication and effective preconditioning techniques are applied to speed up the solution of linear systems. Experimental results show that the algorithm is significantly faster than existing methods with sufficient accuracy. 2) Magnetic materials are widely used in MEMS, RFID and MRAM. We present the first BEM algorithm to extract interconnect inductance with magnetic materials. The algorithm models magnetic characteristics by the Landau Lifshitz Gilbert equation and fictitious magnetic charges. The algorithm is accelerated by approximating magnetic charge effects and by modeling currents with solenoidal basis. The relative error of the algorithm with respect to the commercial tool is below 3%, while the speed is up to one magnitude faster. 3) Since traditional interconnect model includes mutual inductances between pairs of segments, the resulting circuit matrix is very dense. This has been the main bottleneck in the use of the interconnect model. Recently, K = L-1 is used. The RKC model is sparse and stable. We study the practical issues of the RKC model. We validate the RKC model and propose an efficient way to achieve high accuracy extraction by circuit simulations of practical examples.

This paper is to study the scattering problem and radiation problem between incident wave and a moored dual pontoon platform by using both a fully nonlinear numerical wave tank (NWT) and a physical tank. The nonlinear numerical wave tank is developed based on the velocity potential function and the boundary element method (BEM). In addition, a moored dual floating pontoon physical model is tested in an experimental wave tank to validate the numerical model for simulation of wave and structure interaction including mooring tension, structure translation and rotation. The phenomena of wave reflection and transmission due to a floating platform are also considered in the study. The experimential results indicate that the platform surge-RAO decays as the wave frequency increases. Similarly, the platform heave-RAO decays first until at the vicinity of the resonance frequency happening where the vertical amplitude rises up and then decays again. The tension-RAO has two resonance frequencies, the lower resonance is resulted by the surge montion, while the higer resonance is caused by the heave motion. Both wave reflection and transmission coefficients decrease near the heave resonance frequency. This indicates that the platform has the best performance in wave shelter effect at heave resonance to protect costal zone. In general, the comparisons of the numerical simulations and experimental results indicate the numerical horizontal motion have a good agreement, but for the vertical motion, the numerical predictions are larger than experiments especially near the heave resonance frequency. This may be due to the structure vertical velocity increases dramatically causing flow separation occurred below the structure sharp corner, thus the fluid viscous damping effect may play an important role in heave motion.

The north-east Atlantic margin (NEAM) is important for hydrocarbon exploration because of the growing evidence of hydrocarbon reserves in the region. However, seismic exploration of the sub-surface is hampered by large deposits of flood basalts, which cover possible hydrocarbon-bearing reservoirs underneath. There are several hypotheses as to why imaging beneath basalt is a problem. These include: the high impedance contrast between the basalt and the layers above; the thin-layering of the basalt due to the many flows which make up a basalt succession; and the rough interfaces on the top-basalt interface caused by weathering and emplacement mechanisms. I perform forward modelling to assess the relative importance of these factors for imaging of sub-basalt reflections. The boundary element method (BEM) is used for the rough-interface modelling. The method was selected because only the interfaces between layers need to be discretized, in contrast to grid methods such as finite difference for which the whole model needs to be discretized, and so should lead to fast generation of shot gathers for models which have only a few homogeneous layers. I have had to develop criteria for accurate modelling with the boundary element method and have considered the following: source near an interface, two interfaces close together, removal of model edge effects and precise modelling of a transparent interface. I have improved efficiency of my code by: resampling the model so that fewer discretization elements are required at low frequencies, and suppressing wrap-around so that the time window length can be reduced. I introduce a new scheme which combines domain decomposition and a far-field approximation to improve the efficiency of the boundary element code further. I compare performance with a standard finite difference code. I show that the BEM is well suited to seismic modelling in an exploration environment when there are only a few layers in the model and when a seismic profile containing many shot gathers for one model is required. For many other cases the finite difference code is still the best option. The input models for the forward modelling are based on real seismic data which were acquired in the Faeroe-Shetland Channel in 2001. The modelling shows that roughness on the surface of the basalt has little effect on the imaging in this particular area of the NEAM. The thin layers in the basalt act as a low-pass filter to the seismic wave. For the real-data acquisition, even the topbasalt reflection is a low frequency event. This is most likely to be due to high attenuation in the layers above the basalt. I show that sea-surface multiple energy is considerable and that it could mask possible sub-basalt events on a seismic shot gather, but any shallow sub-basalt events should still be visible even with the presence of multiple energy. This leaves the possibility that there is only one major stratigraphic unit between the base of the basalt and the crystalline basement. The implication of the forward modelling and real data analysis for acquisition is that the acquisition parameters must emphasize the low frequencies, since the high frequencies are attenuated before they even reach the top-basalt interface. The implication for processing is that multiple removal is of prime importance.

This thesis presents numerical and experimental investigations into Electrochemical Machining (ECM). The aim is to develop a computer program to predict the shape of a workpiece machined by the ECM process. The program is able to simulate various applications of EC machining which are drilling, milling, turning and shaped tube electrochemical drilling (STED). The program has been developed in a MATLAB environment. In this present work, EC-drilling, EC-milling and EC-turning are analysed as three-dimensional problems whereas STED is simulated in two-dimensions. Experiments have been carried out to verify the accuracy of the predicted results in the cases of EC-milling and EC-turning. The ECM modeller is based on the boundary element method (BEM) and uses Laplace's equation to determine the current distribution at nodes on the workpiece surface. In 3D, the surfaces of the tool and the workpiece are discretised into continuous linear triangular element types whereas in 2D, the boundaries of the tool and workpiece are discretised into linear elements. The ECM modeller is completely self-contained, i.e. it does not rely on any other commercial package. The program contains modules to automatically discretize the surfaces/boundaries of the tool and workpiece. Since the simulation of the ECM process is a temporal problem, several time steps are required to obtain the final workpiece shape. At the end of each time step, the shape of the workpiece is calculated using Faraday's laws. However, the workpiece's shape changes with progressing time steps causing the elements to become stretched and distorted. Mesh refinement techniques are built in the ECM modeller, and these subdivide the mesh automatically when necessary.The effect of time step on the predicted 3D shape of a hole in EC-drilling is investigated. The effect of discontinuity in the slope between neighbouring elements is also studied. Results obtained from the ECM modeller are compared with 2D analytical results to verify the accuracy that can be obtained from the ECM modeller. Milling features ranging from a simple slot to a pocket with a complex protrusion were machined in order to determine the feasibility of the EC milling process. These features were machined on a 3-axes CNC machine converted to permit EC milling. The effect of tool geometry, tool feed rate, applied voltage and step-over distances on the dimensions, shape and surface finish of the machined features were investigated. A pocket with a human shape protrusion was machined using two different types of tool paths, namely contour-parallel and zig-zag. Both types resulted in the base surface of the pocket being concave and the final dimensions of the pockets are compared with the design drawing to determine the effect of tool path type on the accuracy of machining. The ECM modeller was used to simulate the machining of a thin-walled turned component. The machining parameters, i.e. initial gap, rotational speed, and applied voltage, were specified by the collaborating company. Since only a small amount of material had to be removed from the thin-walled component, the tool was held stationary i.e. a feed in the radial or longitudinal direction was not required. By taking advantage of the axi-symmetric nature of a turned component, only a sector of the component was analysed thereby reducing the computing time considerably. The accuracy of the modeller was verified by comparing the predicted time to machine the thin-walled component with the actual machining time. The initial investigations in STED were both experimental and numerical in nature and they studied the effect of applied voltage, tool feed rate and electrolyte pressure on the dimensions of the holes. Later investigations were numerical and an iterative methodology has been developed to calculate a set of feed rates which could machine a specified turbulator shape.

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