Advanced electrode models and numerical modelling for high frequency Electrical Impedance Tomography systems

The thesis discusses various electrode models and finite element analysis methods for Electrical Impedance Tomography (EIT) systems. EIT is a technique for determining the distribution of the conductivity or admittivity in a volume by injecting electrical currents into the volume and measuring the corresponding potentials on the surface of the volume. Various electrode models were investigated for operating EIT systems at higher frequencies in the beta-dispersion band. Research has shown that EIT is potentially capable to distinguish malignant and benign tumours in this frequency band. My study concludes that instrumental effects of the electrodes and full Maxwell effects of EIT systems are the major issues, and they have to be addressed when the operating frequency increases. In the thesis, I proposed 1) an Instrumental Electrode Model (IEM) for the quasi-static EIT formula, based on the analysis of the hardware structures attached to electrodes; 2) a Complete Electrode Model based on Impedance Boundary Conditions (CEM-IBC) that introduces the contact impedances into the full Maxwell EIT formula; 3) a Transmission line Port Model (TPM) for electrode pairs with the instrumental effects, the contact impedance, and the full Maxwell effects considered for EIT systems. Circuit analysis, Partial Differential Equations (PDE) analysis, numerical analysis and finite element methods were used to develop the models. The results obtained by the proposed models are compared with widely used Commercial PDE solvers. This thesis addresses the two major problems (instrumental effects of the electrodes and full Maxwell effects of EIT systems) with the proposed advanced electrode models. Numerical experiments show that the proposed models are more accurate in the high frequency range of EIT systems. The proposed electrode models can be also applicable to inverse problems, and the results show promising. Simple hardware circuits for verifying the results experimentally have been also designed

Bayesian Methods for Gas-Phase Tomography

Gas-phase tomography refers to a set of techniques that determine the 2D or 3D distribution of a target species in a jet, plume, or flame using measurements of light, made around the boundary of a flow area. Reconstructed quantities may include the concentration of one or more species, temperature, pressure, and optical density, among others. Tomography is increasingly used to study fundamental aspects of turbulent combustion and monitor emissions for regulatory compliance. This thesis develops statistical methods to improve gas-phase tomography and reports two novel experimental applications. Tomography is an inverse problem, meaning that a forward model (calculating measurements of light for a known distribution of gas) is inverted to estimate the model parameters (transforming experimental data into a gas distribution). The measurement modality varies with the problem geometry and objective of the experiment. For instance, transmittance data from an array of laser beams that transect a jet may be inverted to recover 2D fields of concentration and temperature; and multiple high-resolution images of a flame, captured from different angles, are used to reconstruct wrinkling of the 3D reacting zone. Forward models for gas-phase tomography modalities share a common mathematical form, that of a Fredholm integral equation of the first-kind (IFK). The inversion of coupled IFKs is necessarily ill-posed, however, meaning that solutions are either unstable or non-unique. Measurements are thus insufficient in themselves to generate a realistic image of the gas and additional information must be incorporated into the reconstruction procedure. Statistical inversion is an approach to inverse problems in which the measurements, experimental parameters, and quantities of interest are treated as random variables, characterized by a probability distribution. These distributions reflect uncertainty about the target due to fluctuations in the flow field, noise in the data, errors in the forward model, and the ill-posed nature of reconstruction. The Bayesian framework for tomography features a likelihood probability density function (pdf), which describes the chance of observing a measurement for a given distribution of gas, and prior pdf, which assigns a relative plausibility to candidate distributions based on assumptions about the flow physics. Bayes’ equation updates information about the target in response to measurement data, combining the likelihood and prior functions to form a posterior pdf. The posterior is usually summarized by the maximum a posteriori (MAP) estimate, which is the most likely distribution of gas for a set of data, subject to the effects of noise, model errors, and prior information. The framework can be used to estimate credibility intervals for a reconstruction and the form of Bayes’ equation suggests procedures for improving gas tomography. The accuracy of reconstructions depends on the information content of the data, which is a function of the experimental design, as well as the specificity and validity of the prior. This thesis employs theoretical arguments and experimental measurements of scalar fluctuations to justify joint-normal likelihood and prior pdfs for gas-phase tomography. Three methods are introduced to improve each stage of the inverse problem: to develop priors, design optimal experiments, and select a discretization scheme. First, a self-similarity analysis of turbulent jets—common targets in gas tomography—is used to construct an advanced prior, informed by an estimate of the jet’s spatial covariance. Next, a Bayesian objective function is proposed to optimize beam positions in limited-data arrays, which are necessary in scenarios where optical access to the flow area is restricted. Finally, a Bayesian expression for model selection is derived from the joint-normal pdfs and employed to select a mathematical basis to reconstruct a flow. Extensive numerical evidence is presented to validate these methods. The dissertation continues with two novel experiments, conducted in a Bayesian way. Broadband absorption tomography is a new technique intended for quantitative emissions detection from spectrally-convolved absorption signals. Theoretical foundations for the diagnostic are developed and the results of a proof-of-concept emissions detection experiment are reported. Lastly, background-oriented schlieren (BOS) tomography is applied to combustion for the first time. BOS tomography employs measurements of beam steering to reconstruct a fluid’s optical density field, which can be used to infer temperature and density. The application of BOS tomography to flame imaging sets the stage for instantaneous 3D combustion thermometry. Numerical and experimental results reported in this thesis support a Bayesian approach to gas-phase tomography. Bayesian tomography makes the role of prior information explicit, which can be leveraged to optimize reconstructions and design better imaging systems in support of research on fluid flow and combustion dynamics

Handbook of Mathematical Geosciences

This Open Access handbook published at the IAMG's 50th anniversary, presents a compilation of invited path-breaking research contributions by award-winning geoscientists who have been instrumental in shaping the IAMG. It contains 45 chapters that are categorized broadly into five parts (i) theory, (ii) general applications, (iii) exploration and resource estimation, (iv) reviews, and (v) reminiscences covering related topics like mathematical geosciences, mathematical morphology, geostatistics, fractals and multifractals, spatial statistics, multipoint geostatistics, compositional data analysis, informatics, geocomputation, numerical methods, and chaos theory in the geosciences

An integrated approach to span design in open stope mining

In order to develop an appropriate mine design, a thorough understanding of the rock mass conditions and its potential response to mining is required. Rock mass characterisation is a key component in developing models of the rock mass and its engineering behaviour, and relies on disparate data collected by exploration geologists, mine geologists, rock mechanics engineers and technicians, in a variety of formats. Optimal rock mass model development requires the effective integration of all data sources, which currently requires considerable effort in collecting, managing, collating, validating and analysing this data.The importance of understanding the spatial variability of rock mass conditions has been highlighted as a major issue. The traditional approach of using simplistic models of “average” rock mass conditions can lead to sub-optimal designs, which may result in unplanned additional costs or economic implications of dilution and ore loss. The design of stope and pillars should be optimised for the prevailing rock mass conditions in the various regions of the mine.Some of the existing design tools used for open stope design have shown poor reliability in their performance predictions. Though some may have been originally developed to assist in initial stope size selection (i.e. pre-feasibility and feasibility levels), they are potentially being inappropriately relied upon for detailed design. Consideration of large scale structures on stability and their influence on local rock mass conditions are also important aspects of open stope design that are commonly over-looked. There is a need to select design methodologies that are optimised for the stage of project development. It is also important to emphasise the iterative, evolutionary and interdisciplinary nature of open stope design.This thesis proposes a framework that attempts to integrate different rock mass characterisation models, numerical modelling and stope performance data to assist in improving the overall excavation design process. The key philosophy behind design optimisation is the continual reduction in uncertainty in collected data, analysis and design methods used with a view to improving the overall reliability of the design. A stope span design optimisation approach is proposed which attempts to ensure that the appropriate methodologies in data collection, data analysis, rock mass model formulation and stope design are utilised at relevant project stages in order to minimise uncertainty and maximise design reliability. The design optimisation approach recognises that the appropriateness of a particular design methodology is highly dependant on the availability of an appropriate rock mass model, which is in turn dependant on the availability of quality rock mass data. With respect to the design of spans in open stope mining, the key aims of the proposed integrated approach are to; • Assess the suitability of data for analysis • If data is unsuitable, assess the most appropriate data collection strategy • Assess the most appropriate approach to rock mass modelling • Assess the most appropriate design methodologies • Assess the reliability of the design criteria and quantify the potential economic impact of the design on the projectOptimisation of the design process also requires integration of state-of-the-art techniques in data collection, analysis, modelling and engineering analysis and design at the appropriate stage of project development. During development of this thesis a number of improvements have been proposed in key areas in the rock engineering design process which can be incorporated into the integrated approach, including; • A rock mass data model has been developed that assists in facilitating the ongoing rock mass characterisation process. The data model is capable of integrating rock mass data from various sources, which promotes sharing of data and avoids duplication of data collection efforts. The data model is able to query rock mass data, define relationships between data types, apply bias corrections, and perform basic analysis for use in subsequent detailed analysis and rock mass modelling. • An implicit based approach to spatial rock mass and deterministic discontinuity modelling can be employed to improve understanding of the spatial variability of rock mass parameters, inter-relationships between rock mass characteristics on their role in design. For example, understanding the influence of large-scale structures on rock mass characteristics and excavation performance. • Improved scale independent geometrical assessments of stope performance have been proposed that maximise the use of stope performance data. • An integrated back analysis framework has been presented that is able to account for structural complexity, scale and features that cannot be directly incorporated into linear elastic numerical modelling codes. • With regard to linear elastic back analyses, an number of improvements have been proposed, as well as a suggested method to assess appropriateness of continuum models based on discontinuity intensity and critical span