Computational electromagnetics is an active research area on the development and implementation of numerical methods and techniques for rigorous solutions of physical problems in the entire spectrum of electromagnetic waves from radio frequencies to gamma rays. While a set of Maxwell's equations are sufficient to model most of the electromagnetic scenarios, analytical solutions are available only for a few canonical objects, making computational techniques inevitable for the analysis of real -life structures. This book and its chapters promise to provide an extensive insight into new trends in computational electromagnetics, while the aim has never been to cover each and every new effort in the area. In fact, considering the volume of publications in the form of conference proceedings, journal papers, and books, it would be impossible to cover all new trends in computational electromagnetics within a single book. Instead of mentioning about everything, the authors of this book demonstrate what they are really doing and what they will be doing in the near future, as the experts of the topics.

Surface integral equations (SIEs) are widely used to simulate and analyze electromagnetic scattering and radiation from arbitrary -shaped conducting and piecewise homogeneous penetrable structures. These methods are based on the surface equivalence principle, where the original boundary value problem for (time -harmonic) Maxwell's equations is reformulated and expressed in terms of surface -integral operators and equivalent sources. The attractive feature of this procedure is that it essentially decreases the dimensionality of the problem by one. Another great advantage of SIEbased methods is that, in unbounded regions, radiation conditions are automatically satisfied, and thus, absorbing boundary conditions or mesh truncation techniques is not needed. These nice features, however, do not come without a cost. The linear system obtained after a discretization process involves a fully populated matrix that is expensive to solve and requiring advanced fast solution strategies as the problem size increases. Special integration routines are needed to evaluate singular integrals efficiently and accurately. The underlying integral operators, equations, and the corresponding discretized linear systems may suffer from low -frequency and dense-discretization breakdowns, low -frequency cancellation, or other types of inaccuracies, as well as instabilities and ill conditioning due to resonances and extreme material parameter

TDIE implementations have come a long way since their initial unstable first steps and are well on their way to becoming a fourth set of canonical methods in the computational electromagnetics toolbox. Five basic methods for temporal discretization have shown promise for the stable and accurate temporal discretization of TDIEs, and several fast methods have been concocted to improve their performances. While new applications of TDIEs are likely to continue pouring in, multiphysics and electromagnetic physics appear to be the short-term trajectory of these newest computational electromagnetics methods. Despite their unimpressive origins, TDIEs are finally poised to become a very new trend in computational electromagnetics.

This chapter addresses both the "old" body of knowledge and the "new" trends of research and practice in FEM as applied to electromagnetics. It presents the general mathematical background and numerical components of FEM and discusses FEM formulations, discretizations, and solution procedures, mostly in the context of the higher order FEM computation. This includes the generation of curvilinear elements for higher order modeling of geometry, implementation of polynomial vector basis functions for higher order modeling of fields within the elements, and Galerkin testing method for discretizing the wave equations. The chapter focuses on the higher order FEM as the most general and versatile approach, where the low -order modeling is naturally included in the higher order FEM paradigm.

This chapter reviews recent advances in computational electromagnetics regarding simple techniques for the systematic construction of higher order vector bases used by advanced numerical codes. Higher order functions are used in numerical solutions of differential and integrodifferential equations by the application of the finite element method (FEM) and the method of moments (MoM). First, we consider divergence-conforming and curl-conforming polynomial vector bases and then introduce substitutive and additive vector bases that are able to model field singularities in the vicinity of edges or vertices. The advantages offered by the use of these higher order models are illustrated by numerical results. Mathematical aspects and numerical techniques presented in this chapter are dealt with in detail in [1], except for the most recent developments concerning singular vector basis functions and their numerical implementation. For background information and further details, the interested reader may refer to [1] and references therein.

Rigorous solutions of large-scale radiation and scattering problems are permanently present among the goals of the scientific community dedicated to computational electromagnetics. Research aimed at solving complex electromagnetic problems that can involve large numbers of unknowns plays a relevant role in the development of many real-life applications. In this context, the fast multipole method (FMM) and the multilevel fast multipole algorithm (MLFMA) have been extensively used for accelerating iterative solutions of dense matrix systems resulting from the application of the method of moments (MoM) to problems formulated with surface integral equations (SIEs). The purpose of using these acceleration techniques is to extend the applicability of MoM, whose matrix storage requirement is O(N² ), while the number of operations is O(N³ ) for direct solutions or O(N² ) for iterative solutions, to larger problems. FMM and MLFMA reduce computational costs to O(N^1.5 ) and 0(N log N), respectively.

In this chapter, we discuss trends and problems in the design of preconditioned Krylov methods for large-scale problems, particularly when they are formulated with surface integral equations such that dense and large matrices arise. We cover various numerical linear algebra aspects, such as the choice of iterative methods, characteristics and performances of fast integral-equation solvers for the required matrix-vector products, and the design of algebraic preconditioners based on multilevel incomplete LU factorization, sparse approximate inverses, inner-outer methods, and spectral approaches, particularly when they are combined with fast solvers. As shown via examples, the developed numerical linear algebra tools can enable efficient solutions of large electromagnetic problems on moderate number of cores and processors.

In this chapter, efficient collocation methods for EM analysis are reviewed. Traditional SC methods leveraging tensor-product, sparse grid, and Stroud cubature rules are described first. These methods are rather straightforward to implement and suitable for EM problems involving smoothly varying QoI. Then, the ME-PC method for efficiently constructing a surrogate model of a rapidly varying QoI is presented. Also detailed is the iterative HDMR technique for EM problems involving large numbers of random variables. Finally, an approximation technique based on the spectral quantic TT (QTT) (SQTT) for constructing a surrogate model in a high-dimensional random domain is briefly reviewed, before the chapter is concluded by numerical examples demonstrating applications of cutting-edge UQ methods to various EM problems.