Since the publication of the first edition of this book about eight years ago, much progress has been made in the development of the finite element method for the analysis of electromagnetics problems, especially in the following five areas. The first is the development of higher-order vector finite elements, which make it possible to obtain highly accurate and efficient solutions of vector wave equations. The second is the development of perfectly matched layers as an absorbing boundary condition. Although the perfectly matched layers were intended primarily for the time-domain finite-difference method, they have also found applications in the finite element simulations. The third is perhaps the development of hybrid techniques that combine the finite element and asymptotic methods for the analysis of large, complex problems that were unsolvable in the past. The fourth is the further development of the finite element--boundary integral methods that incorporate fast integral solvers, such as the fast multipole method, to reduce the computational complexity associated with the boundary integral part. The last, but not the least, is the development of the finite element method in time domain for transient analysis. As a result of all these efforts, the finite element method has gained more popularity in the computational electromagnetics community and has become one of the preeminent simulation techniques for electromagnetics problems.

In this second edition, we have updated the subject matter and introduced new advances in the finite element technology. In Chapter 1, which presents the basic electromagnetic equations and concepts, we have added a brief review of vector analysis because of its importance in the finite element formulation of electromagnetics problems. We have also added sections on the field--source relations, Huygens's principles, and definitions of radar cross section, since they are used frequently in the subsequent chapters.

Chapter 2 introduces the basic concepts of the finite element method after a brief review of classical methods for boundary-value problems. Minor changes have been made to improve clarity. The next three chapters develop the finite element method in one, two, and three dimensions and its application to electromagnetics problems. We have added sections on isoparametric elements that can provide a superior geometrical modeling, in addition to accurate representation of unknown functions to be computed, and sections on dispersion analysis to illustrate the convergence of higher-order finite elements.

Chapter 6 discusses various variational principles to establish the variational expression for a given electromagnetic boundary-value problem. We have added one section to present the most general variational principle and illustrate its application to electromagnetics problems involving anisotropic media. This topic is useful since anisotropic media have been used widely in electronic and electro-optical devices.

Chapter 7, which describes the finite element analysis of eigenvalue problems, remains unchanged except for some minor modifications to update the topic.

Chapter 8 introduces vector finite elements for the modeling of electromagnetic vector wave equations. Since the first edition of this book presented only the lowest-order vector elements in two and three dimensions, major revisions have been made to cover the developments of higher-order vector elements with examples to demonstrate their superior performance. The higher-order vector elements are expected to significantly impact the application of the finite element method to electromagnetics problems.

Chapter 9 is a new chapter, which is devoted to the important topic of absorbing boundary conditions. This topic was addressed briefly in Appendix C of the first edition of this book. It is now fully expanded to cover two-dimensional scalar and three-dimensional vector absorbing boundary conditions, adaptive absorbing boundary conditions, fictitious absorbers, and perfectly matched layers. The adaptive absorbing boundary condition is a relatively new approach that can systematically improve the accuracy of the solution obtained using an absorbing boundary condition. The concept of perfectly matched layers was proposed only a few years ago; hence, it is an entirely new topic. Efforts have been made to present it to suit the finite element applications since it has been used mostly for the time-domain finite-difference simulations. A section has been included for the finite element analysis of scattering and radiation by complex body-of-revolution structures using perfectly matched layers.

Chapter 10 addresses the development of a hybrid technique that combines the finite element and boundary integral methods for open-region scattering and radiation problems. Efforts have been made to improve the treatment of the interior resonance problem in both two and three dimensions. A section has been added to present a highly effective preconditioner to accelerate the iterative solution of the finite element--boundary integral method, with numerical examples to demonstrate its great potential. Coupled with fast integral solvers, the hybrid finite element--boundary integral method is promising for dealing with large-scale problems involving complex structures and inhomogeneous materials.

Chapter 11 describes the use of eigenfunction expansions for the finite element analysis of open-region problems. New material has been added to present eigenfunction expansions on elliptical boundaries that can significantly reduce the size of the computational domain in comparison to circular ones. New examples have also been included to demonstrate the capability of the method to simulate microwave devices such as circulators, filters, and junctions.

Chapter 12 is another new chapter, which describes the development of the finite element method for the time-domain analysis of transient electromagnetics problems. Time-domain simulations are important because of their ability to model nonlinear materials. This chapter covers basic time-marching schemes and their stability analysis. It also discusses the modeling of dispersive media, the formulation of orthogonal vector basis functions, and application of the time-domain finite element method to open-region scattering and radiation problems with the aid of absorbing boundary conditions, perfectly matched layers, and boundary integral equations.

Chapter 13 presents solution methods and algorithms for linear algebraic equations arising from the finite element discretization, which include the banded matrix method, the profile storage method, and the conjugate and biconjugate gradient methods. In this new edition, we have added other useful direct and iterative solvers and discussed preconditioning techniques to speed up iteration convergence and reordering schemes for the bandwidth reduction. A new section has also been added to describe the asymptotic waveform evaluation method for fast frequency-sweep analysis.

Chapter 14 is also a new chapter and was suggested by a reviewer. It presents another very powerful computational method in electromanetics---the method of moments---and its fast solvers. The inclusion of this chapter is justifiable because the method of moments is very closely related to the finite element method, at least in terms of basic principles. Moreover, good understanding of the method of moments can help a great deal in the development of the hybrid finite element--boundary integral method presented in Chapter 10. The fast solvers, including the FFT-based method, adaptive integral method, and fast multipole method, can all be incorporated into the hybrid finite element--boundary integral method to reduce the computational complexity associated with the boundary integral part and thus further expand the capability of the hybrid method.

Appendix A has been slightly expanded to list some formulae and integral theorems for vector analysis on surfaces, which are useful for the manipulation of absorbing boundary conditions, boundary integrals, and higher-order vector basis functions on curved surfaces. Appendix B remains unchanged.

The remaining appendices (C--E) are all new. To be specific, Appendix C presents the definition and derivation of Green's functions and their applications in electromagnetics. We have included this because Green's functions are the basis for boundary integral equations and near-to-far field calculations. Appendix D describes a numerical procedure to evaluate singular integrals, which is essential for the method of moments and the finite element--boundary integral method. Finally, Appendix E lists the definitions and useful properties of some special functions used in this book.

As a result of the revision, more than one-third of this second edition is new. The book is still intended as a textbook for use in computational electromagnetics courses at the graduate level and a research reference for scientists and electrical engineers. It is detailed enough for self study as well. We have included some exercises to supplement and reinforce the concepts and ideas presented, and also to facilitate its use as a textbook. For teaching purposes, I have developed a set of PowerPoint viewgraphs about the finite element method and will be happy to make them available to those teaching the method for electromagnetic analysis.