A semi-analytical solution for multilayer diffusion in a composite medium consisting of a large number of layers

Diffusion in a composite slab consisting of a large number of layers provides an ideal prototype problem for developing and analysing two-scale modelling approaches for heterogeneous media. Numerous analytical techniques have been proposed for solving the transient diffusion equation in a one-dimensional composite slab consisting of an arbitrary number of layers. Most of these approaches, however, require the solution of a complex transcendental equation arising from a matrix determinant for the eigenvalues that is difficult to solve numerically for a large number of layers. To overcome this issue, in this paper, we present a semi-analytical method based on the Laplace transform and an orthogonal eigenfunction expansion. The proposed approach uses eigenvalues local to each layer that can be obtained either explicitly, or by solving simple transcendental equations. The semi-analytical solution is applicable to both perfect and imperfect contact at the interfaces between adjacent layers and either Dirichlet, Neumann or Robin boundary conditions at the ends of the slab. The solution approach is verified for several test cases and is shown to work well for a large number of layers. The work is concluded with an application to macroscopic modelling where the solution of a fine-scale multilayered medium consisting of two hundred layers is compared against an “up-scaled” variant of the same problem involving only ten layers.

Flow of a micropolar fluid bounded by a stretching sheet

We consider boundary layer flow of a micropolar fluid driven by a porous stretching sheet. A similarity solution is defined, and numerical solutions using Runge-Kutta and quasilinearisation schemes are obtained. A perturbation analysis is also used to derive analytic solutions to first order in the perturbing parameter. The resulting closed form solutions involve relatively complex expressions, and the analysis is made more tractable by a combination of offline and online work using a computational algebra system (CAS). For this combined numerical and analytic approach, the perturbation analysis yields a number of benefits with regard to the numerical work. The existence of a closed form solution helps to discriminate between acceptable and spurious numerical solutions. Also, the expressions obtained from the perturbation work can provide an accurate description of the solution for ranges of parameters where the numerical approaches considered here prove computationally more difficult.

Constructive development of the solutions of linear equations in introductory ordinary differential equations

The solution of linear ordinary differential equations (ODEs) is commonly taught in first year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognising what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to tables of solutions, is an important skill for students to carry with them to advanced studies in mathematics. In this study we describe a teaching and learning strategy that replaces the traditional algorithmic, transmission presentation style for solving ODEs with a constructive, discovery based approach where students employ their existing skills as a framework for constructing the solutions of first and second order linear ODEs. We elaborate on how the strategy was implemented and discuss the resulting impact on a first year undergraduate class. Finally we propose further improvements to the strategy as well as suggesting other topics which could be taught in a similar manner.

A method of lines approach for modelling saltwater intrusion in coastal aquifiers

The equations governing saltwater intrusion in coastal aquifers are complex. Backward Euler time stepping approaches are often used to advance the solution to these equations in time, which typically requires that small time steps be taken in order to ensure that an accurate solution is obtained. We show that a method of lines approach incorporating variable order backward differentiation formulas can greatly improve the efficiency of the time stepping process.

Homegardening as a panacea : a case study of South Tarawa

The Republic of Kiribati is a small, highly infertile Pacific Island nation and is one of the most challenging locations to attempt to support dense urban populations. Kiribati, like other nations in the Pacific, faces an urban future where food insecurity, unemployment, waste management and malnutrition will become increasing issues. Homegardening is suggested as one way to address many of these problems. However, the most recent study on agriculture production in urban centres in Kiribati shows that, in general, intensive cultivation of homegardens is not a common practice. This disparity between theory and practice creates an opportunity to re-examine homegardening in Kiribati and, more broadly, in the Pacific. This paper examines the practice of homegardening in urban centres in Kiribati and explores reasons why change has or has not occurred through interviews with homegardeners and government/donor representives. Results show that homegardening has increased significantly in the past five years, largely because of the promotion of homegardens and organic composting systems by donor organisations. While findings further endorse homegardening as an excellent theoretical solution to many of the problems that confront urban settlements in Kiribati and the Pacific, it raises additional questions regarding the continuation of homegarden schemes beyond donor support programmes.

A model of teacher professional development based on the principles of lesson study

The researcher’s professional role as an Education Officer was the impetus for this study. Designing and implementing professional development activities is a significant component of the researcher’s position description and as a result of reflection and feedback from participants and colleagues, the creation of a more effective model of professional development became the focus for this study. Few studies have examined all three links between the purposes of professional development that is, increasing teacher knowledge, improving teacher practice, and improving student outcomes. This study is significant in that it investigates the nature of the growth of teachers who participated in a model of professional development which was based upon the principles of Lesson Study. The research provides qualitative and empirical data to establish some links between teacher knowledge, teacher practice, and student learning outcomes. Teacher knowledge in this study refers to mathematics content knowledge as well as pedagogical-content knowledge. The outcomes for students include achievement outcomes, attitudinal outcomes, and behavioural outcomes. As the study was conducted at one school-site, existence proof research was the focus of the methodology and data collection. Developing over the 2007 school year, with five teacher-participants and approximately 160 students from Year Levels 6 to 9, the Lesson Study-principled model of professional development provided the teacher-participants with on-site, on-going, and reflective learning based on their classroom environment. The focus area for the professional development was strategising the engagement with and solution of worded mathematics problems. A design experiment was used to develop the professional development as an intervention of prevailing teacher practice for which data were collected prior to and after the period of intervention. A model of teacher change was developed as an underpinning framework for the development of the study, and was useful in making decisions about data collection and analyses. Data sources consisted of questionnaires, pre-tests and post-tests, interviews, and researcher observations and field notes. The data clearly showed that: content knowledge and pedagogical-content knowledge were increased among the teacher-participants; teacher practice changed in a positive manner; and that a majority of students demonstrated improved learning outcomes. The positive changes to teacher practice are described in this study as the demonstrated use of mixed pedagogical practices rather than a polarisation to either traditional pedagogical practices or contemporary pedagogical practices. The improvement in student learning outcomes was most significant as improved achievement outcomes as indicated by the comparison of pre-test and post-test scores. The effectiveness of the Lesson Study-principled model of professional development used in this study was evaluated using Guskey’s (2005) Five Levels of Professional Development Evaluation.

A point interpolation method with locally smoothed strain field (PIM-LS2) for mechanics problems using triangular mesh

A point interpolation method with locally smoothed strain field (PIM-LS2) is developed for mechanics problems using a triangular background mesh. In the PIM-LS2, the strain within each sub-cell of a nodal domain is assumed to be the average strain over the adjacent sub-cells of the neighboring element sharing the same field node. We prove theoretically that the energy norm of the smoothed strain field in PIM-LS2 is equivalent to that of the compatible strain field, and then prove that the solution of the PIM- LS2 converges to the exact solution of the original strong form. Furthermore, the softening effects of PIM-LS2 to system and the effects of the number of sub-cells that participated in the smoothing operation on the convergence of PIM-LS2 are investigated. Intensive numerical studies verify the convergence, softening effects and bound properties of the PIM-LS2, and show that the very ‘‘tight’’ lower and upper bound solutions can be obtained using PIM-LS2.

Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.

An investigation of foot and ankle problems experienced by nurses

Relatively little information has been reported about foot and ankle problems experienced by nurses, despite anecdotal evidence which suggests they are common ailments. The purpose of this study was to improve knowledge about the prevalence of foot and ankle musculoskeletal disorders (MSDs) and to explore relationships between these MSDs and proposed risk factors. A review of the literature relating to work-related MSDs, MSDs in nursing, foot and lower-limb MSDs, screening for work-related MSDs, foot discomfort, footwear and the prevalence of foot problems in the community was undertaken. Based on the review, theoretical risk factors were proposed that pertained to the individual characteristics of the nurses, their work activity or their work environment. Three studies were then undertaken. A cross-sectional survey of 304 nurses, working in a large tertiary paediatric hospital, established the prevalence of foot and ankle MSDs. The survey collected information about self-reported risk factors of interest. The second study involved the clinical examination of a subgroup of 40 nurses, to examine changes in body discomfort, foot discomfort and postural sway over the course of a single work shift. Objective measurements of additional risk factors, such as individual foot posture (arch index) and the hardness of shoe midsoles, were performed. A final study was used to confirm the test-retest reliability of important aspects of the survey and key clinical measurements. Foot and ankle problems were the most common MSDs experienced by nurses in the preceding seven days (42.7% of nurses). They were the second most common MSDs to cause disability in the last 12 months (17.4% of nurses), and the third most common MSDs experienced by nurses in the last 12 months (54% of nurses). Substantial foot discomfort (Visual Analogue Scale (VAS) score of 50mm or more) was experienced by 48.5% of nurses at sometime in the last 12 months. Individual risk factors, such as obesity and the number of self-reported foot conditions (e.g., callouses, curled toes, flat feet) were strongly associated with the likelihood of experiencing foot problems in the last seven days or during the last 12 months. These risk factors showed consistent associations with disabling foot conditions and substantial foot discomfort. Some of these associations were dependent upon work-related risk factors, such as the location within the hospital and the average hours worked per week. Working in the intensive care unit was associated with higher odds of experiencing foot problems within the last seven days, foot problems in the last 12 months and foot problems that impaired activity in the last 12 months. Changes in foot discomfort experienced within a day, showed large individual variability. Fifteen of the forty nurses experienced moderate/substantial foot discomfort at the end of their shift (VAS 25+mm). Analysis of the association between risk factors and moderate/substantial foot discomfort revealed that foot discomfort was less likely for nurses who were older, had greater BMI or had lower foot arches, as indicated by higher arch index scores. The nurses’ postural sway decreased over the course of the work shift, suggesting improved body balance by the end of the day. These findings were unexpected. Further clinical studies examining individual nurses on several work shifts are needed to confirm these results, particularly due to the small sample size and the single measurement occasion. There are more than 280,000 nurses registered to practice in Australia. The nursing workforce is ageing and the prevalence of foot problems will increase. If the prevalence estimates from this study are extrapolated to the profession generally, more than 70,000 hospital nurses have experienced substantial foot discomfort and 25-30,000 hospital nurses have been limited in their activity due to foot problems during the last 12 months. Nurses with underlying foot conditions were more likely to report having foot problems at work. Strategies to prevent or manage foot conditions exist and they should be disseminated to nurses. Obesity is a significant risk factor for foot and ankle MSDs and these nurses may need particular assistance to manage foot problems. The risk of foot problems for particular groups of nurses, e.g. obese nurses, may vary depending upon the location within the hospital. Further research is needed to confirm the findings of this study. Similar studies should be conducted in other occupational groups that require workers to stand for prolonged periods.

Numerical modelling of industrial microwave heating

The numerical modelling of electromagnetic waves has been the focus of many research areas in the past. Some specific applications of electromagnetic wave scattering are in the fields of Microwave Heating and Radar Communication Systems. The equations that govern the fundamental behaviour of electromagnetic wave propagation in waveguides and cavities are the Maxwell's equations. In the literature, a number of methods have been employed to solve these equations. Of these methods, the classical Finite-Difference Time-Domain scheme, which uses a staggered time and space discretisation, is the most well known and widely used. However, it is complicated to implement this method on an irregular computational domain using an unstructured mesh. In this work, a coupled method is introduced for the solution of Maxwell's equations. It is proposed that the free-space component of the solution is computed in the time domain, whilst the load is resolved using the frequency dependent electric field Helmholtz equation. This methodology results in a timefrequency domain hybrid scheme. For the Helmholtz equation, boundary conditions are generated from the time dependent free-space solutions. The boundary information is mapped into the frequency domain using the Discrete Fourier Transform. The solution for the electric field components is obtained by solving a sparse-complex system of linear equations. The hybrid method has been tested for both waveguide and cavity configurations. Numerical tests performed on waveguides and cavities for inhomogeneous lossy materials highlight the accuracy and computational efficiency of the newly proposed hybrid computational electromagnetic strategy.

Stability and convergence of an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term

Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractional Fokker-Planck equations with a nonlinear source term (TSFFPE-NST), which involve the Caputo time fractional derivative (CTFD) of order α ∈ (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order μ ∈ (1, 2). Approximating the CTFD and RSFD using the L1-algorithm and shifted Grunwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.

Remediation strategies of shaft and common mode voltages in adjustable speed drive systems

AC motors are largely used in a wide range of modern systems, from household appliances to automated industry applications such as: ventilations systems, fans, pumps, conveyors and machine tool drives. Inverters are widely used in industrial and commercial applications due to the growing need for speed control in ASD systems. Fast switching transients and the common mode voltage, in interaction with parasitic capacitive couplings, may cause many unwanted problems in the ASD applications. These include shaft voltage and leakage currents. One of the inherent characteristics of Pulse Width Modulation (PWM) techniques is the generation of the common mode voltage, which is defined as the voltage between the electrical neutral of the inverter output and the ground. Shaft voltage can cause bearing currents when it exceeds the amount of breakdown voltage level of the thin lubricant film between the inner and outer rings of the bearing. This phenomenon is the main reason for early bearing failures. A rapid development in power switches technology has lead to a drastic decrement of switching rise and fall times. Because there is considerable capacitance between the stator windings and the frame, there can be a significant capacitive current (ground current escaping to earth through stray capacitors inside a motor) if the common mode voltage has high frequency components. This current leads to noises and Electromagnetic Interferences (EMI) issues in motor drive systems. These problems have been dealt with using a variety of methods which have been reported in the literature. However, cost and maintenance issues have prevented these methods from being widely accepted. Extra cost or rating of the inverter switches is usually the price to pay for such approaches. Thus, the determination of cost-effective techniques for shaft and common mode voltage reduction in ASD systems, with the focus on the first step of the design process, is the targeted scope of this thesis. An introduction to this research – including a description of the research problem, the literature review and an account of the research progress linking the research papers – is presented in Chapter 1. Electrical power generation from renewable energy sources, such as wind energy systems, has become a crucial issue because of environmental problems and a predicted future shortage of traditional energy sources. Thus, Chapter 2 focuses on the shaft voltage analysis of stator-fed induction generators (IG) and Doubly Fed Induction Generators DFIGs in wind turbine applications. This shaft voltage analysis includes: topologies, high frequency modelling, calculation and mitigation techniques. A back-to-back AC-DC-AC converter is investigated in terms of shaft voltage generation in a DFIG. Different topologies of LC filter placement are analysed in an effort to eliminate the shaft voltage. Different capacitive couplings exist in the motor/generator structure and any change in design parameters affects the capacitive couplings. Thus, an appropriate design for AC motors should lead to the smallest possible shaft voltage. Calculation of the shaft voltage based on different capacitive couplings, and an investigation of the effects of different design parameters are discussed in Chapter 3. This is achieved through 2-D and 3-D finite element simulation and experimental analysis. End-winding parameters of the motor are also effective factors in the calculation of the shaft voltage and have not been taken into account in previous reported studies. Calculation of the end-winding capacitances is rather complex because of the diversity of end winding shapes and the complexity of their geometry. A comprehensive analysis of these capacitances has been carried out with 3-D finite element simulations and experimental studies to determine their effective design parameters. These are documented in Chapter 4. Results of this analysis show that, by choosing appropriate design parameters, it is possible to decrease the shaft voltage and resultant bearing current in the primary stage of generator/motor design without using any additional active and passive filter-based techniques. The common mode voltage is defined by a switching pattern and, by using the appropriate pattern; the common mode voltage level can be controlled. Therefore, any PWM pattern which eliminates or minimizes the common mode voltage will be an effective shaft voltage reduction technique. Thus, common mode voltage reduction of a three-phase AC motor supplied with a single-phase diode rectifier is the focus of Chapter 5. The proposed strategy is mainly based on proper utilization of the zero vectors. Multilevel inverters are also used in ASD systems which have more voltage levels and switching states, and can provide more possibilities to reduce common mode voltage. A description of common mode voltage of multilevel inverters is investigated in Chapter 6. Chapter 7 investigates the elimination techniques of the shaft voltage in a DFIG based on the methods presented in the literature by the use of simulation results. However, it could be shown that every solution to reduce the shaft voltage in DFIG systems has its own characteristics, and these have to be taken into account in determining the most effective strategy. Calculation of the capacitive coupling and electric fields between the outer and inner races and the balls at different motor speeds in symmetrical and asymmetrical shaft and balls positions is discussed in Chapter 8. The analysis is carried out using finite element simulations to determine the conditions which will increase the probability of high rates of bearing failure due to current discharges through the balls and races.

Lie group symmetry analysis for granular media stress equations

The Airy stress function, although frequently employed in classical linear elasticity, does not receive similar usage for granular media problems. For plane strain quasi-static deformations of a cohesionless Coulomb–Mohr granular solid, a single nonlinear partial differential equation is formulated for the Airy stress function by combining the equilibrium equations with the yield condition. This has certain advantages from the usual approach, in which two stress invariants and a stress angle are introduced, and a system of two partial differential equations is needed to describe the flow. In the present study, the symmetry analysis of differential equations is utilised for our single partial differential equation, and by computing an optimal system of one-dimensional Lie algebras, a complete set of group-invariant solutions is derived. By this it is meant that any group-invariant solution of the governing partial differential equation (provided it can be derived via the classical symmetries method) may be obtained as a member of this set by a suitable group transformation. For general values of the parameters (angle of internal friction and gravity g) it is found there are three distinct classes of solutions which correspond to granular flows considered previously in the literature. For the two limiting cases of high angle of internal friction and zero gravity, the governing partial differential equation admit larger families of Lie point symmetries, and from these symmetries, further solutions are derived, many of which are new. Furthermore, the majority of these solutions are exact, which is rare for granular flow, especially in the case of gravity driven flows.

A geometrical analysis of trajectory design for underwater vehicles

Designing trajectories for a submerged rigid body motivates this paper. Two approaches are addressed: the time optimal approach and the motion planning ap- proach using concatenation of kinematic motions. We focus on the structure of singular extremals and their relation to the existence of rank-one kinematic reduc- tions; thereby linking the optimization problem to the inherent geometric frame- work. Using these kinematic reductions, we provide a solution to the motion plan- ning problem in the under-actuated scenario, or equivalently, in the case of actuator failures. We finish the paper comparing a time optimal trajectory to one formed by concatenation of pure motions.