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Estimation of the Radar Cross Section of large inhomogeneous scattering objects such as composite aircrafts, ships and biological bodies at high frequencies has posed large computational challenge. The detection of scattering from wake vortex leading to detection and possible identification of low observable aircrafts also demand the development of computationally efficient and rigorous numerical techniques. Amongst the various methods deployed in Computational Electromagnetics, the Method of Moments predicts the electromagnetic characteristics accurately. Method of Moments is a rigorous method, combined with an array of modeling techniques such as triangular patch, cubical cell and tetrahedral modeling. Method of Moments has become an accurate technique for solving electromagnetic problems from complex shaped homogeneous and inhomogeneous objects. One of the drawbacks of Method of Moments is the fact that it results into a dense matrix, the inversion of which is a computationally complex both in terms of physical memory and compute power. This has been the prime reason for the Method of Moments hitherto remaining as a low frequency method. With recent advances in supercomputing, it is possible to extend the range of Method of Moments for Radar Cross Section computation of aircraft like structures and radiation characteristic of antennas mounted on complex shaped bodies at realistic frequencies of practical interest. This thesis is a contribution in this direction. The main focus of this thesis is development of parallel Method of Moments solvers, applied to solve real world electromagnetic wave scattering and radiation problems from inhomogeneous objects. While the methods developed in this thesis are applicable to a variety of problems in Computational Electromagnetics as shown by illustrative examples, in specific, it has been applied to compute the Radar Cross Section enhancement due to acoustic disturbances and flow inhomogeneities from the wake vortex of an aircraft, thus exploring the possibility of detecting stealth aircraft. Illustrative examples also include the analysis of antenna mounted on an aircraft. In this thesis, first the RWG basis functions have been used in Method of Moments procedure, for solving scattering problems from complex conducting structures such as aircraft and antenna(s) mounted on airborne vehicles, of electrically large size of about 45 and 0.76 million unknowns. Next, the solver using SWG basis functions with tetrahedral and pulse basis functions with cubical modeling have been developed to solve scattering from 3D inhomogeneous bodies. The developed codes are validated by computing the Radar Cross Section of spherical homogeneous and inhomogeneous layered scatterers, lossy dielectric cylinder with region wise inhomogeneity and high contrast dielectric objects. Aerodynamic flow solver ANSYS FLUENT, based on Finite Volume Method is used to solve inviscid compressible flow problem around the aircraft. The gradients of pressure/density are converted to dielectric constant variation in the wake region by using empirical relation and interpolation techniques. Then the Radar Cross Section is computed from the flow inhomogeneities in the vicinity of a model aircraft and beyond (wake zone) using the developed parallel Volume Surface Integral Equation using Method of Moments and investigated more rigorously. Radar Cross Section enhancement is demonstrated in the presence of the flow inhomogeneities and detectability is discussed. The Bragg scattering that occurs when electromagnetic and acoustic waves interact is also discussed and the results are interpreted in this light. The possibility of using the scattering from wake vortex to detect low visible aircraft is discussed. This thesis also explores the possibility of observing the Bragg scattering phenomenon from the acoustic disturbances, caused by the wake vortex. The latter sets the direction for use of radars for target identification and beyond target detection. The codes are parallelized using the ScaLAPACK and BiCG iterative method on shared and distributed memory machines, and tested on variety of High Performance Computing platforms such as Blue Gene/L (22.4TF), Tyrone cluster, CSIR-4PI HP Proliant 3000 BL460c (360TF) and CRAY XC40 machines. The parallelization speedup and efficiency of all the codes has also been shown.

Doctor of Philosophy / Department of Electrical and Computer Engineering / Kenneth H. Carpenter / A computer simulation of transient current density distributions in thin conductors was developed using a time-stepped implementation of the integral equation method on a finite element mesh. A study of current distributions in thin conductors was carried out using AC analysis methods. The study of the AC current density distributions was used to develop a circuit theory model for the thin conductor which was then used to determine the nature of its transient response. This model was used to support the design and evaluation of the transient current density solver. A circuit model for strip lines was made using the Partial Inductance Method to allow for simulations with the SPICE circuit solver. Magnetic probes were designed and tested that allow for physical measurements of voltages induced by the magnetic field generated by the current distributions in the strip line. A comparison of the measured voltages to simulated values from SPICE was done to validate the SPICE model. This model was used to validate the finite-integration model for the same strip line. Formulation of the transient current density distribution problem is accomplished by the superposition of a source current and an eddy current distribution on the same space. The mathematical derivation and implementation of the time-stepping algorithm to the finite element model is explicitly shown for a surface mesh with triangular elements. A C++ computer program was written to solve for the total current density in a thin conductor by implementing the time-stepping integral formulation. Evaluation of the finite element implementation was made regarding mesh size. Finite element meshes of increasing node density were simulated for the same structure until a smooth current density distribution profile was observed. The transient current density solver was validated by comparing simulations with AC conduction and transient response simulations of the SPICE model. Transient responses are compared for inputs at different frequencies and for varying time steps. This program is applied to thin conductors of irregular shape.

An efficient integral equation method based on a method of moment (MoM) discretization of the Mixed-Potential Integral Equation (MPIE) for the analysis of 2.5D or 3D planar microwave circuits is presented. The robust Discrete Complex Image Method (DCIM) is employed to approximate the Greens functions in layered media for horizontal and vertical sources of fields, where closed-form formulations of the z-integrations are derived in the spectral domain. Meanwhile, an efficient and accurate numerical integration technique based on the Khayat-Wilton transform is used to integrate functions with 1/R singularities and near singularities. The fast iterative solver - Quadrature Sampled Pre-Corrected Fast Fourier Transform (QSPCFFT) - is associated with the MoM formulation to analyze electrically large, dense and complex microwave circuits.

This dissertation develops surface integral equations using constraint-based Helmholtz decompositions for electromagnetic modeling. This new approach is applied to the electric field integral equation (EFIE), and it incorporates a Helmholtz decomposition (HD) of the current. For this reason, the new formulation is referred to as the EFIE-hd. The HD of the current is accomplished herein via appropriate surface integral constraints, and leads to a stable linear system. This strategy provides accurate solutions for the electric and magnetic fields at both high and low frequencies, it allows for the use of a locally corrected Nyström (LCN) discretization method for the resulting formulation, it is compatible with the local global solution framework, and it can be used with non-conformal meshes. To address large-scale and complex electromagnetic problems, an overlapped localizing local-global (OL-LOGOS) factorization is used to factorize the system matrix obtained from an LCN discretization of the augmented EFIE (AEFIE). The OL-LOGOS algorithm provides good asymptotic performance and error control when used with the AEFIE. This application is used to demonstrate the importance of using a well-conditioned formulation to obtain efficient performance from the factorization algorithm.

Novel integral-equation methods for efficiently solving electromagnetic problems that involve more than a single length scale of interest in complex backgrounds are presented. Such multi-scale electromagnetic problems arise because of the interplay of two distinct factors: the structure under study and the background medium. Both can contain material properties (wavelengths/skin depths) and geometrical features at different length scales, which gives rise to four types of multi-scale problems: (1) twoscale, (2) multi-scale structure, (3) multi-scale background, and (4) multi-scale-squared problems, where a single-scale structure resides in a different single-scale background, a multi-scale structure resides in a single-scale background, a single-scale structure resides in a multi-scale background, and a multi-scale structure resides in a multi-scale background, respectively. Electromagnetic problems can be further categorized in terms of the relative values of the length scales that characterize the structure and the background medium as (a) high-frequency, (b) low-frequency, and (c) mixed-frequency problems, where the wavelengths/skin depths in the background medium, the structure’s geometrical features or internal wavelengths/skin depths, and a combination of these three factors dictate the field variations on/in the structure, respectively. This dissertation presents several problems arising from geophysical exploration and microwave chemistry that demonstrate the different types of multi-scale problems encountered in electromagnetic analysis and the computational challenges they pose. It also presents novel frequency-domain integral-equation methods with proper Green function kernels for solving these multi-scale problems. These methods avoid meshing the background medium and finding fields in an extended computational domain outside the structure, thereby resolving important complications encountered in type 3 and 4 multi-scale problems that limit alternative methods. Nevertheless, they have been of limited practical use because of their high computational costs and because most of the existing ‘fast integral-equation algorithms’ are not applicable to complex Green function kernels. This dissertation introduces novel FFT, multigrid, and FFT-truncated multigrid algorithms that reduce the computational costs of frequency-domain integral-equation methods for complex backgrounds and enable the solution of unprecedented type 3 and 4 multi-scale problems. The proposed algorithms are formulated in detail, their computational costs are analyzed theoretically, and their features are demonstrated by solving benchmark and challenging multi-scale problems. / text

<p>Boundary-contact problems (BCPs) are studied within the frames of</p><p>classical mathematical theory of elasticity and plasticity</p><p>elaborated by Landau, Kupradze, Timoshenko, Goodier, Fichera and</p><p>many others on the basis of analysis of two- and three-dimensional</p><p>boundary value problems for linear partial differential equations.</p><p>A great attention is traditionally paid both to theoretical</p><p>investigations using variational methods and boundary singular</p><p>integral equations (Muskhelishvili) and construction of solutions</p><p>in the form that admit efficient numerical evaluation (Kupradze).</p><p>A special family of BCPs considered by Shtaerman, Vorovich,</p><p>Alblas, Nowell, and others arises within the frames of the models</p><p>of squeezing thin multilayer elastic sheets. We show that</p><p>mathematical models based on the analysis of BCPs can be also</p><p>applied to modeling of the clich\'{e}-surface printing contacts</p><p>and paper surface compressibility in flexographic printing.</p><p>The main result of this work is formulation and complete</p><p>investigation of BCPs in layered structures, which includes both</p><p>the theoretical (statement of the problems, solvability and</p><p>uniqueness) and applied parts (approximate and numerical</p><p>solutions, codes, simulation).</p><p>We elaborate a mathematical model of squeezing a thin elastic</p><p>sheet placed on a stiff base without friction by weak loads</p><p>through several openings on one of its boundary surfaces. We</p><p>formulate and consider the corresponding BCPs in two- and</p><p>three-dimensional bands, prove the existence and uniqueness of</p><p>solutions, and investigate their smoothness including the behavior</p><p>at infinity and in the vicinity of critical points. The BCP in a</p><p>two-dimensional band is reduced to a Fredholm integral equation</p><p>(IE) with a logarithmic singularity of the kernel. The theory of</p><p>logarithmic IEs developed in the study includes the analysis of</p><p>solvability and development of solution techniques when the set of</p><p>integration consists of several intervals. The IE associated with</p><p>the BCP is solved by three methods based on the use of</p><p>Fourier-Chebyshev series, matrix-algebraic determination of the</p><p>entries in the resulting infinite system matrix, and</p><p>semi-inversion. An asymptotic theory for the BCP is developed and</p><p>the solutions are obtained as asymptotic series in powers of the</p><p>characteristic small parameter.</p><p>We propose and justify a technique for the solution of BCPs and</p><p>boundary value problems with boundary conditions of mixed type</p><p>called the approximate decomposition method (ADM). The main idea</p><p>of ADM is simplifying general BCPs and reducing them to a chain</p><p>of auxiliary problems for 'shifted' Laplacian in long rectangles</p><p>or parallelepipeds and then to a sequence of iterative problems</p><p>such that each of them can be solved (explicitly) by the Fourier</p><p>method. The solution to the initial BCP is then obtained as a</p><p>limit using a contraction operator, which constitutes in</p><p>particular an independent proof of the BCP unique solvability.</p><p>We elaborate a numerical method and algorithms based on the</p><p>approximate decomposition and the computer codes and perform</p><p>comprehensive numerical analysis of the BCPs including the</p><p>simulation for problems of practical interest. A variety of</p><p>computational results are presented and discussed which form the</p><p>basis for further applications for the modeling and simulation of</p><p>printing-plate contact systems and other structures of</p><p>flexographic printing. A comparison with finite-element solution</p><p>is performed.</p>

The boundary element method (BEM) has become a powerful method for the numerical solution of boundary-value problems (BVPs), due to its ability (at least for problems with constant coefficients) of reducing a BVP for a linear partial differential equation (PDE) defined in a domain to an integral equation defined on the boundary, leading to a simplified discretisation process with boundary elements only. On the other hand, the coefficients in the mathematical model of a physical problem typically correspond to the material parameters of the problem. In many physical problems, the governing equation is likely to involve variable coefficients. The application of the BEM to these equations is hampered by the difficulty of finding a fundamental solution. The first part of this thesis will focus on the derivation of the boundary integral equation (BIE) for the Laplace equation, and numerical results are presented for some examples using constant elements. Then, the formulations of the boundary-domain integral or integro-differential equation (BDIE or BDIDE) for heat conduction problems with variable coefficients are presented using a parametrix (Levi function), which is usually available. The second part of this thesis deals with the extension of the BDIE and BDIDE formulations to the treatment of the two-dimensional Helmholtz equation with variable coefficients. Four possible cases are investigated, first of all when both material parameters and wave number are constant, in which case the zero-order Bessel function of the second kind is used as fundamental solution. Moreover, when the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or a BDIDE. Finally, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. In the third part, the radial integration method (RIM) is introduced and discussed in detail. Modifications are introduced to the RIM, particularly the fact that the radial integral is calculated by using a pure boundary-only integral which relaxes the “star-shaped” requirement of the RIM. Then, the RIM is used to convert the domain integrals appearing in both BDIE and BDIDE for heat conduction and Helmholtz equations to equivalent boundary integrals. For domain integrals consisting of known functions the transformation is straightforward, while for domain integrals that include unknown variables the transformation is accomplished with the use of augmented radial basis functions (RBFs). The most attractive feature of the method is that the transformations are very simple and have similar forms for both 2D and 3D problems. Finally, the application of the RIM is discussed for the diffusion equation, in which the parabolic PDE is initially reformulated as a BDIE or a BDIDE and the RIM is used to convert the resulting domain integrals to equivalent boundary integrals. Three cases have been investigated, for homogenous, non-homogeneous and variable coefficient diffusion problems.

A numerical implementation of the direct Boundary-Domain Integral Equation (BDIE)/ Boundary-Domain Integro-Differential Equations (BDIDEs) and Localized Boundary-Domain Integral Equation (LBDIE)/Localized Boundary-Domain Integro-Differential Equations (LBDIDEs) related to the Neumann and Dirichlet boundary value problem for a scalar elliptic PDE with variable coefficient is discussed in this thesis. The BDIE and LBDIE related to Neumann problem are reduced to a uniquely solvable one by adding an appropriate perturbation operator. The mesh-based discretisation of the BDIE/BDIDEs and LBDIE/LBDIDEs with quadrilateral domain elements leads to systems of linear algebraic equations (discretised BDIE/BDIDEs/LBDIE/BDIDEs). Then the systems obtained from BDIE/BDIDE (discretised BDIE/BDIDE) are solved by the LU decomposition method and Neumann iterations. Convergence of the iterative method is analyzed in relation with the eigen-values of the corresponding discrete BDIE/BDIDE operators obtained numerically. The systems obtained from LBDIE/LBDIDE (discretised LBDIE/LBDIDE) are solved by the LU decomposition method as the Neumann iteration method diverges.

For years, researchers have studied the movement of cells and mathematicians have attempted to model the movement of the cell using various methods. This work is an extension of the work done by Zheltukhin and Lui (2011), Mathematical Biosciences 229:30-40, who simulated the stress and displacement of a one-dimensional cell using a model based on viscoelastic theory. The report is divided into three main parts. The first part considers viscoelastic models with a first-order constitutive equation and uses the standard linear model as an example. The second part extends the results of the first to models with second-order constitutive equations. In this part, the two examples studied are Burger model and a Kelvin-Voigt element connected with a dashpot in series. In the third part, the effects of substrate with variable stiffness are explored. Here, the effective adhesion coefficient is changed from a constant to a spatially-dependent function. Numerical results are generated using two different functions for the adhesion coefficient. Results of this thesis show that stress on the cell varies greatly across each part of the cell depending on the constitute equation we use, while the position and velocity of the cell remain essentially unchanged from a large-scale point of view.