Multi-scale problems in numerical electromagnetics are becoming increasingly common with the advent and widespread usage of compact mobile phones, body area networks, small and nano antennas, sensors, high-speed interconnects, integrated circuits and complex electronic packaging structures, to name just a few commercial applications. Numerical electromagnetic modeling and simulation of structures with multi-scale features is highly challenging due to the fact that electrically small as well as large features are simultaneously present in the model which demands for discretization of the computational domain such that the number of degrees of freedom is very large, thus, levying a heavy burden on computational resources. The multi-scale nature of a given problem also exacerbates the challenge of generating very fine meshes which do not introduce instabilities or ill-conditioned behaviors. In this work we introduce a hybrid technique, which combines frequency domain and time domain techniques in a manner such that the fine features (electrically small) of the object being modeled are handled by the Method of Moments (MoM) technique while the electrically large parts of the structure are dealt with by using the Finite-Difference Time-Domain (FDTD) technique in order to reduce the computational burden. Recently, structures with multi-scale features have been simulated by using the dipole moment (DM) approach combined with the FDTD technique to handle fine features in a multi-scale geometry. However, when the size of the scatterer becomes larger in terms of the wavelength and the quasi-static assumption becomes invalid, extensive modifications of the DM/FDTD hybrid approach are needed resulting in a high computational cost.The research proposes a novel hybrid FDTD technique, which combines the Method of Moments and the Finite-Difference Time-Domain techniques directly in the time domain circumventing the need to carry out frequency transform calculations as required in the DM approach when the object size is not small (size>/20). The proposed technique utilizes piecewise sinusoidal basis functions to represent the currents on arbitrarily shaped wires with fine features, and modified RWG basis function for surfaces. The fields scattered by the object with fine features in MoM region are computed in the time domain on a planar interface. The time domain fields obtained at the planar interface are then combined with the FDTD update equations. In contrast to the existing techniques used to handle this type of problems, the proposed technique is both efficient as well as stable.

The last several decades have experienced an extraordinarily focused effort on developing general-purpose numerical methods in computational electromagnetics (CEM) that can accurately model a wide variety of electromagnetic systems. In turn, this has led to a number of techniques, such as the Method of Moments (MoM), the Finite Element Method (FEM), and the Finite-Difference-Time-Domain (FDTD), each of which exhibits their own advantages and disadvantages. In particular, the FDTD has become a widely used tool for modeling electromagnetic systems, and since it solves Maxwell's equations directly--without having to derive Green's Functions or to solve a matrix equation or--it experiences little or no difficulties when handling complex inhomogeneous media. Furthermore, the FDTD has the additional advantage that it can be easily parallelized; and, hence, it can model large systems using supercomputing clusters. However, the FDTD method is not without its disadvantages when used on platforms with limited computational resources. For many problems, the domain size can be extremely large in terms of the operating wavelengths, whereas many of the objects have fine features (e.g., Body Area Networks). Since FDTD requires a meshing of the entire computational domain, presence of these fine features can significantly increase the computational burden; in fact, in many cases, it can render the problem either too time-consuming or altogether impractical to solve. This has served as the primary motivation in this thesis for developing multi-scale techniques that can circumvent many of the problems associated with CEM, and in particular with time domain methods, such as the FDTD. Numerous multi-scale problems that frequently arise in CEM have been investigated in this work. These include: 1) The coupling problem between two conformal antennas systems on complex platforms; 2) Rigorous modeling of Body Area Networks (BANs), and some approximate human phantom models for path loss characterization; 3) Efficient modeling of fine features in the FDTD method and the introduction of the dipole moment method for finite methods; and, 4) Time domain scattering by thin wire structures using a novel Time-Domain- Electric-Field-Integral-Equation (TD-EFIE) formulation. Furthermore, it is illustrated, via several examples, that each problem requires a unique approach. Finally, the results obtained by each technique have been compared with other existing numerical methods for the purpose of validation.

The 91st London Mathematical Society Durham Symposium took place from July 5th to 15th 2010, with more than 100 international participants attending. The Symposium focused on Numerical Analysis of Multiscale Problems and this book contains 10 invited articles from some of the meeting's key speakers, covering a range of topics of contemporary interest in this area. Articles cover the analysis of forward and inverse PDE problems in heterogeneous media, high-frequency wave propagation, atomistic-continuum modeling and high-dimensional problems arising in modeling uncertainty. Novel upscaling and preconditioning techniques, as well as applications to turbulent multi-phase flow, and to problems of current interest in materials science are all addressed. As such this book presents the current state-of-the-art in the numerical analysis of multiscale problems and will be of interest to both practitioners and mathematicians working in those fields.

The dimmed outlines of phenomenal things all into one another unless we put on the merge focusing-glass of theory, and screw it up some times to one pitch of definition and sometimes to another, so as to see down into different depths through the great millstone of the world James Clerk Maxwell (1831 - 1879) For a long time after the foundation of the modern theory of electromag netism by James Clerk Maxwell in the 19th century, the mathematical ap proach to electromagnetic field problems was for a long time dominated by the analytical investigation of Maxwell's equations. The rapid development of computing facilities during the last century has then necessitated appropriate numerical methods and algorithmic tools for the simulation of electromagnetic phenomena. During the last few decades, a new research area "Computational Electromagnetics" has emerged com prising the mathematical analysis, design, implementation, and application of numerical schemes to simulate all kinds of relevant electromagnetic pro cesses. This area is still rapidly evolving with a wide spectrum of challenging issues featuring, among others, such problems as the proper choice of spatial discretizations (finite differences, finite elements, finite volumes, boundary elements), fast solvers for the discretized equations (multilevel techniques, domain decomposition methods, multipole, panel clustering), and multiscale aspects in microelectronics and micromagnetics.

The book will cover the past, present and future developments of field theory and computational electromagnetics. The first two chapters will give an overview of the historical developments and the present the state-of-the-art in computational electromagnetics. These two chapters will set the stage for discussing recent progress, new developments, challenges, trends and major directions in computational electromagnetics with three main emphases: a. Modeling of ever larger structures with multi-scale dimensions and multi-level descriptions (behavioral, circuit, network and field levels) and transient behaviours b. Inclusions of physical effects other than electromagnetic: quantum effects, thermal effects, mechanical effects and nano scale features c. New developments in available computer hardware, programming paradigms (MPI, Open MP, CUDA and Open CL) and the associated new modeling approaches These are the current emerging topics in the area of computational electromagnetics and may provide readers a comprehensive overview of future trends and directions in the area. The book is written for students, research scientists, professors, design engineers and consultants who engaged in the fields of design, analysis and research of the emerging technologies related to computational electromagnetics, RF/microwave, optimization, new numerical methods, as well as accelerator simulator, dispersive materials, nano-antennas, nano-waveguide, nano-electronics, terahertz applications, bio-medical and material sciences. The book may also be used for those involved in commercializing electromagnetic and related emerging technologies, sensors and the semiconductor industry. The book can be used as a reference book for graduates and post graduates. It can also be used as a text book for workshops and continuing education for researchers and design engineers.

Multiscale problems naturally pose severe challenges for computational science and engineering. The smaller scales must be well resolved over the range of the larger scales. Challenging multiscale problems are very common and are found in e.g. materials science, fluid mechanics, electrical and mechanical engineering. Homogenization, subgrid modelling, heterogeneous multiscale methods, multigrid, multipole, and adaptive algorithms are examples of methods to tackle these problems. This volume is an overview of current mathematical and computational methods for problems with multiple scales with applications in chemistry, physics and engineering.