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This thesis focuses on some mathematical aspects and a few recent applications of the polarization tensor (PT). Here, the main concern of the study is to characterize objects presented in electrical or electromagnetic fields by only using the PT. This is possible since the PT contains significant information about the object such as shape, orientation and material properties. Two main applications are considered in the study and they are electrosensing fish and metal detection. In each application, we present a mathematical formulation of the PT and briefly discuss its properties. The PT in the electrosensing fish is actually based on the first order generalized polarization tensor (GPT) while the GPT itself generalizes the classical PT called as the P\'{o}lya-Szeg\H{o} PT. In order to investigate the role of the PT in electrosensing fish, we propose two numerical methods to compute the first order PT. The first method is directly based on the quadrature method of numerical integration while the second method is an adaptation of some terminologies of the boundary element method (BEM). A code to use the first method is developed in \textit{Matlab} while a script in \textit{Python} is written as an interface for using the new developed code for BEM called as \textit{BEM++}. When comparing the two methods, our numerical results show that the first order PT is more accurate with faster convergence when computed by \textit{BEM++}. During this study, we also give a strategy to determine an ellipsoid from a given first order PT. This is because we would like to propose an experiment to test whether electrosensing fish can discriminate a pair of different objects but with the same first order PT such that the pair could be an ellipsoid and some other object. In addition, the first order PT (or the P\'{o}lya-Szeg\H{o} PT) with complex conductivity (or complex permittivity) which is similar to the PT for Maxwell's equations is also investigated. On the other hand, following recent mathematical foundation of the PT from the eddy current model, we use the new proposed explicit formula to compute the rank 2 PT for a few metallic targets relevance in metal detection. We show that the PT for the targets computed from the explicit formula agree to some degree of accuracy with the PT obtained from metal detectors during experimental works and simulations conducted by the engineers. This suggests to alternatively use the explicit formula which depends only on the geometry and material properties of the target as well as offering lower computational efforts than performing measurements with metal detectors to obtain the PT. By using the explicit formula of the rank 2 PT, we also numerically investigate some properties of the rank 2 PT where, the information obtained could be useful to improve metal detection and also in other potential applications of the eddy current. In this case, if the target is magnetic but non-conducting, the rank 2 PT of the target can also be computed by using the explicit formula of the first order PT.

Over the last several decades, the development of fast, high-order accurate, and robust integral equation methods for computational physics has gained increasing attention. Using integral equation formulation as a global statement in contrast to a local partial differential equation (PDE) formulation offers several unique advantages. For homogeneous PDEs, the boundary integral equation (BIE) formulation allows accurate handling of complex and moving geometries, and it only requires a mesh on the boundary, which is much easier to generate as a result of the dimension reduction. With the acceleration of fast algorithms like the Fast Multipole Method (FMM), the computational complexity can be reduced to O(N), where N is the number of degrees of freedom on the boundary. Using standard potential theory decomposition, inhomogeneous PDEs can be solved by evaluating a volume potential over the inhomogeneous source domain, followed by a solution of the homogeneous part. Despite the advantages of BIE methods in easy meshing, near-optimal efficiency, and well conditioning, the accurate evaluation of nearly singular integrals is a classical problem that needs to be addressed to enable simulations for practical applications. In the first half of this thesis, we develop a series of product integration schemes to solve this close evaluation problem. The use of differential forms provides a dimensional-agnostic way of integrating the nearly singular kernels against polynomial basis functions analytically. So the problem of singular integration gets reduced to a matter of source function approximation. In 2D, this procedure has been traditionally portrayed by building a connection to complex Cauchy integral, then supplemented by a complex monomial approximation. In $3$D, the closed differential form requirement leads to the design of a new function approximation scheme based on harmonic polynomials and quaternion algebra. Under a similar framework, we develop a high-order accurate product integration scheme for evaluating singular and nearly singular volume integral equations (VIE) in complex domains using regular Cartesian grids discretization. A high-order accurate source term approximation scheme matching smooth volume integrals on irregular cut cells is developed, which requires no function extension. BIE methods have been widely used for studying Stokes flows, incompressible flows at low Reynolds' number, in both biological systems and microfluidics. In the second half of this thesis, we employ the BIE methods to simulate and optimize Stokes fluid-structure interactions. In 2D, a hybrid computational method is presented for simulating cilia-generated fluid mixing as well as the cilia-particle hydrodynamics. The method is based on a BIE formulation for confining geometries and rigid particles, and the method of regularized Stokeslets for the cilia. In 3D, we use the time-independent envelop model for arbitrary axisymmetric microswimmers to minimize the power loss while maintaining a target swimming speed. This is a quadratic optimization problem in terms of the slip velocity due to the linearity of Stokes flow. Under specified reduced volume constraint, we find prolate spheroids to be the most efficient micro-swimmer among various families of shapes we considered. We then derive an adjoint-based formulation for computing power loss sensitivities in terms of a time-dependent slip profile by introducing an auxiliary time-periodic function, and find that the optimal swimmer displays one or multiple traveling waves, reminiscent of the typical metachronal waves observed in ciliated microswimmers.

This dissertation presents techniques for high-order simulation of electromagnetic fields, particularly for problems involving ships with ferromagnetic hulls and active corrosion-protection systems. A set of numerically constrained hexahedral basis functions for volume integral equation discretization is presented in a method-of-moments context. Test simulations demonstrate the accuracy achievable with these functions as well as the improvement brought about in system conditioning when compared to other basis sets. A general method for converting between a locally-corrected Nyström discretization of an integral equation and a method-of-moments discretization is presented next. Several problems involving conducting and magnetic-conducting materials are solved to verify the accuracy of the method and to illustrate both the reduction in number of unknowns and the effect of the numerically constrained bases on the conditioning of the converted matrix. Finally, a surface integral equation derived from Laplace’s equation is discretized using the locally-corrected Nyström method in order to calculate the electric fields created by impressed-current corrosion protection systems. An iterative technique is presented for handling nonlinear boundary conditions. In addition we examine different approaches for calculating the magnetic field radiated by the corrosion protection system. Numerical tests show the accuracy achievable by higher-order discretizations, validate the iterative technique presented. Various methods for magnetic field calculation are also applied to basic test cases.

The exponentially weighted moving average chart based on the sample generalized variance is studied under the independent multivariate normal model for the vector of quality measurements. The performance of the chart is based on an analysis of the chart's initial and steady-state run length distributions. The three methods that are commonly used to determinate run length distribution, simulation, the integral equation method, and the Markov chain approximation are discussed. The integral equation and Markov chain approaches are analytical methods that require a nu- merical method for determining the probability density and cumulative distribution functions describing the distribution of the sample generalized variance. Two meth- ods for determining numerically these functions are discussed. The equivalence of the integral equation and Markov chain methods is shown resulting in a new method for obtaining a Markov chain approximation of the chart. Some examples of the implementation of these methods are given using MATLAB.

This dissertation work investigates the numerical method for solving EMC problems involving electrically large and complex platform. Numerical simulation of the antenna couplings on the aircraft is of great interest in EMC community. Conventional CEM solvers suffer inefficiency and inflexibility in modelling this kind of problems. One of the challenges comes from the multi-scale physics in the geometry containing both electrically large platform and antennas with electrically small structures. Also, using conventional CEM solvers, the user may have the dilemma of choosing between the PDE based methods , like FEM, which are convenient and accurate in modelling complex materials but requires volume mesh, and IE based methods, which only require surface discretization but are not convenient in modelling antennas with complex materials.In this dissertation, a MSDDM has been proposed to model the problems with electrically large and complex structures. Using MSDDM, the problem of antenna coupling can be decomposed into antenna sub-domains and platform sub-domains, for which different CEM solvers can be applied. This gives an efficient way to precondition the global system. It hybridizes the strength of PDE based methods and IE methods. Also, the CEM solvers in MSDDM can be stored in out-of-core memory, which enables it to solve large problem within limited memory.MSDDM is applied to solve the EMC problem on a mock-up aircraft platform. It has been recognized that the decomposition scheme for MSDDM is inflexible in modelling electrically large objects, which need to be decomposed into smaller blocks for memory reason. In most cases, it will be the platform who has the highest memory cost. With MSDDM, the user needs to decompose the geometry and re-mesh it, which is undesirable. A new non-overlapping IEDDM has been proposed in this dissertation to solve this issue. IEDDM can decompose the surface of PEC into non-overlapping sub-domains. Hence in dealing with the electrically large object, the user can obtain the mesh by decompose the original surface mesh into open partitions and analytically refine each of them. This decomposition can be easily implemented in an automatic fashion.With MSDDM and IEDDM, a two-level domain decomposition method can be implemented for the EMC applications. The first level decomposition is implemented with MSDDM, in which the original problem is decomposed according to their geometrical properties. For the platform sub-domains with large electrical size, the IEDDM is applied and it automatically decomposes the surface into smaller sub-domains to fit into memory. This decomposition can be totally transparent to the user. With the method developed in this work, the antenna coupling problem on an mock-up aircraft platform of more than 700λ0 was successfully conducted on a workstation with 48GB memory.Mathematically, the IEDDM slightly enlarges the function space for conventional EFIE formulation, which supports the normal discontinuity at the boundary. The continuity is weakly enforced by testing the residual by Galerkin method. The validity of this formulation has been tested. And it paves the way to IEDG in which diverse basis functions can be hybridized into IE solver.

In this dissertation a fast integral equation method, termed IE-FFT, is developed, analyzed and applied to solve electromagnetic (EM) scattering problems. The methodology is developed for the Method of Moments (MoM) solution of the Electric Field Integral Equation (EFIE) on electrically large Perfect Electric Conducting (PEC) structures. Similar to other Fast Fourier Transform (FFT) based algorithms, IE-FFT uses a Cartesian grid to drastically decrease memory storage and speed up the matrix-vector multiplication, when used with iterative solvers. The IE-FFT algorithm employs two discretizations, one for the unknown current on an unstructured triangular mesh and the other on a uniform Cartesian grid for interpolating the Green’s function. The uniform interpolation of the Green’s function allows the fast computation of well-separated MoM interaction terms with the aid of a global FFT. Nevertheless, the coupling between near-interaction terms should be adequately corrected. The major contribution of this dissertation lays on the Lagrangian interpolation of the Green’s function. This not only allows simple and efficient algorithmic implementation, but also naturally suggests a rigorous error analysis and tight error bounds of the algorithm. The efficiency of the method is based on the Toeplitz structure of the interpolated Green’s function. Therefore, it is applicable on both asymptotically-smooth and oscillatory kernels arising in static and wave propagation problems, respectively. Through the electromagnetic wave scattering from a PEC sphere, the complexity of the IE-FFT algorithm is found to scale as O(N1.5)and O(N1.5log(N)) for memory and CPU time, respectively. More accurate integration of weakly singular integrals for EFIE formulation is proposed. Subsequently the proposed integration improves the robustness and the accuracy of the EFIE greatly. Furthermore, the IE-FFT algorithm can be directly applied to Finite Element Method-Boundary Element Method (FEM-BEM) for solving the boundary integral equation.

Statistical interpolation has been proven to be a legitimate and efficient approach for data processing in the field of geodetic and geophysical sciences. Pursuing the minimization of the mean squared prediction error, the technique, known as Kriging or least-squares collocation, is able to densify, respectively filter a spatially and/or temporally referenced dataset, provided that its associated covariance model is given or estimated in advance. The involvement of the covariance matrix which to some extent reflects the physical behavior of the underlying process may, however, potentially lead to an ill-conditioned situation when the data are observed at a relatively high sampling rate. A new perspective, interpreting the Kriging equation in the continuous sense, is therefore proposed in this research so that, instead of matrix terms, a convolution equation is set up for the Green’s function where the covariance function is preserved in its analytic form. Two methods to approximate the solution of such a convolution equation are employed: One transforms the unknown Green’s function into a series consisting of a linear combination of (partial) derivatives of the covariance function so that the approximation of the Green’s function can be determined through a term-by-term approach; the other one manipulates the convolution equation in the spectral domain where the inversion can be treated within the space of real number. The proposed approach has been applied to various covariance models, especially several more recently established spatial-temporal models which have attracted increasing interests for geophysical applications. Examples from geodetic science include the cases of data fusion and terrain profile monitoring; although based on simulated data, the demonstration of this innovative approach shows great potential.

This thesis describes the theoretical and practical aspects of a new technique for quantitative, non-invasive and non-destructive imaging of the near-surface moisture content distribution of composite materials. The technique relies on the alteration by a nearby composite material, of the propagation velocity of an electromagnetic pulse along a parallel transmission line, through distortion of the evanescent field. A set of measurements taken at different relative positions of the transmission line and composite material are, in conjunction with a forward model describing propagation velocity on the line, inverted to provide the image of moisture content distribution. Development of the technique, called 'Time Domain Reflectometry Imaging' (TDRI), involved four steps: 1. Instrumentation to obtain a set of measurements of propagation times; 2. A forward model; 3. An inverse procedure; and 4. Conversion of a calculated permittivity distribution to a moisture distribution. Critical to the success of the inverse method is a set of measurements of propagation velocity that provide pico-second propagation time accuracy, and are sufficiently linearly independent to enable discrimination of the permittivity of each discretised cell within the composite material. Using commercial time domain reflectometry (TDR) instruments, a switched reference measurement, waveform subtraction and intersecting waveform tangents, sufficient timing accuracy has been achieved. The forward model was developed using the moment method. The advantage of such an integral equation method is that recalculation is not required when changing the impressed field. Hence for a particular model of the composite material's moisture distribution, just one execution of the forward model provides predicted propagation velocities for all positions of the transmission line. A new pseudo 3-D variant of the volume integral equation approach was developed to suit the 2-D transmission line, and resulted in a 100 fold reduction in memory use, and a greater than 10 fold reduction in execution time. The forward solution uses the telegrapher's equation to predict propagation velocity from an arbitrary permittivity distribution surrounding the line. Inversion of the measured data was accelerated by the use of three novel tactics: a rapid electric field surrogate for the Jacobian; a dynamic method of determining the conjugate gradient weighting factor; and a new blocking technique that accelerated the convergence of buried cells that have only a small influence on propagation velocity. The final TDRI step is a numerical model to translate both the a priori moisture distribution data to a permittivity distribution, and conversely the solution permittivity distribution to moisture content. A dielectric model based on an earlier model of Looyenga was adapted to include both the different characteristic of tightly held water, and the Debye relaxation of free water. The intention was a model with applicability to a range of composite materials. It was tested with data for soil, bentonite clay and wood, and except for one free parameter, model parameters were set by measurable physical properties of the host material.

Industrial processes are subject to changes that can adversely affect product quality. A change in the process that increases the variability of the output of the process causes the output to be less uniform and increases the probability that individual items will not meet specifications. Statistical control charts for monitoring process variance can be used to detect an increase in the variability of the output of a process so that the situation can be repaired and product uniformity restored. Control charts that increase the sampling rate when there is evidence the variance has changed gather information more quickly and detect changes in the variance more quickly (on average) than fixed sampling rate procedures. Several variable sampling rate procedures for detecting increases in the process variance will be developed and compared with fixed sampling rate methods. A control chart for the variance is usually used with a separate control chart for the mean so that changes in the average level of the process and the variability of the process can both be detected. A simple method for applying variable sampling rate techniques to dual monitoring of mean and variance will be developed. This control chart procedure increases the sampling rate when there is evidence the mean or variance has changed so that changes in either parameter that will negatively impact product quality will be detected quickly.