Preface

When I was invited with multiple times by the representative of Wiley to write this book in 2015, I was very hesitant because I worried about whether or not there will be enough readers who want to read the book. This is because the method of moments (MoM) as the primary numerical method for solving electromagnetic (EM) integral equations has been so widely used and accepted, it is very difficult to attract readers' interest by presenting another similar numerical method without demonstrating its obvious advantages. Fortunately, the Nyström method as one of nontraditional numerical methods has received more and more attention and has been fast-increasingly applied to solve various EM problems since it was introduced to EM community in 1990. The Nyström method has shown certain merits which the MoM does not have and has become a strong competitor and alternative in many applications. Also, as a nontraditional and relatively-new approach, it is natural to be lately accepted since people need some time to recognize its value, but this should not become a reason to stop studying and promoting it. No new exploration and competition means no advance! The rule should be applicable to all situations, including the investigation on different numerical methods for solving EM problems. Therefore, we can confidently write this book now.

This book is a monograph instead of a textbook which is mainly used for research reference. Many authors of such type of books said that the reason why they wrote their books was that there were not related books on relevant topics or the existent books were too old, so they should present new books. This is of course a strong reason, but we think that more books on similar topics in some research areas are also needed and are very worthy to be written even though there have been other books which have not been very old, because a diversity of books should be presented to satisfy different readers' diverse needs. Also, different books can provide different views or emphases even on similar topics. More importantly, keeping a moderate competition can certainly improve the quality of books just like papers, tons of which can simultaneously address the same or similar topics.

The technical contents of books usually come from published papers which have gone through a peer review, but writing books is still necessary and is actually a very important thing even if readers can find similar contents in papers. This is because books and papers serve different groups of readers who could have very different needs. The papers mainly aim at those professional researchers who have enough background and knowledge on related topics and can easily understand what papers present. In the contrast, the books primarily serve the newbies or inexperienced researchers whose representatives could be graduate students who have not had enough background and knowledge on relevant topics and could feel hard to directly read papers. An investigation by our graduate students association shows that more than 80% graduate students strongly rely on reading books instead of papers in their research work. The books should re-narrate the sophisticated technical contents of papers by an easier-understanding style so that readers can more easily understand and accept them. This requires that the authors of books creatively compose, explain, clarify, or illustrate the technical contents of papers according to their understanding and experience, instead of just a simple and mechanical collection or repeat of papers' contents. Due to the significant difference of functions between the books and papers, we should not make the books become a collective copy of some papers as some books' authors did. Those authors often claimed that their books mainly wanted to provide a convenience for readers to read related papers. To avoid such kind of style, we have provided an extensive introduction to the background, basic principle or mechanism, implementation method, physical applications, etc. for the book's theme, i.e. the Nyström method, with a more easily-accepted manner. We believe that these non-paper contents are very necessary for inexperienced researchers and students although experienced researchers may not need them.

This book is a summary for the work that we have done on the Nyström method in the past sixteen years. Thanks to Professor Weng Cho Chew who initialized the study on the interesting method and gave a continuous and huge instruction and advice on later investigations, resulting in making a significant progress on this method. Compared with other similar books, we specially strengthen the introduction to the singularity treatment techniques which could be viewed as the life and core of the Nyström method and its great expansion on practical applications, including the incorporation with the multilevel fast multipole algorithm (MLFMA), solution of multiphysics problems, and solution of time-domain integral equations. These features have never or seldom been dealt with by other authors and could be the distinct highlights of this book.

Also, the Nyström method is an integral equation solver or belongs to an integral equation approach, so it is intimately related to integral equations and knowing the EM integral equations is a prerequisite for reading this book. We already wrote a book entitled "Integral Equation Methods for Electromagnetic and Elastic Waves", which was published by Morgan & Claypool, San Rafael, CA, in 2008. That book mainly focuses on addressing the mathematics and physics behind the EM integral equations although it also includes some numerical solution methods and numerical examples. As a comparison, this book can be thought of as an engineering book which emphasizes the implementations and applications of the Nyström method for solving EM integral equations for practical engineering problems. Therefore, it may be a good idea to jointly read or refer to these two books.

This book has ten chapters in total and is divided into two parts. The first part includes the first five chapters which mainly address the background of EM or computational EM (CEM), and the basic principle and necessary elements of implementation such as quadrature rules and singularity treatment in the Nyström method. The second part includes other five chapters which mainly deal with the applications and physical implementations of the Nyström method for solving various EM problems as we have emphasized in the above. The first chapter was written by Professor Chew and the other nine chapters were written by myself. The primary contents of each chapter are summarized as follows.

Chapter 1 can be thought of as Professor Chew's recent observation and thinking about the EM and CEM. He first recalls the history of classical EM theory and draws a picture describing its connection with other well-known physical theories like quantum electrodynamics and Yang-Mills theory to emphasize the mathematics and physics behind the EM theory. He then clarifies the relationship of the rising quantum optics and quantum EM to the classical EM which can be viewed as the descendent of the fluid physics. Immediately followed is his narration about the complete development of the Maxwell's equations and derivation of wave equations, which allows him to divide the wave into three regimes in terms of frequency, i.e. the circuit physics, wave physics, and ray physics, respectively, and the mechanism of plasmonic resonance is also explained by the way. With the physics, he then starts to address the solution methods for the wave problems. He first narrates the closed-form solutions and their asymptotic approximations at high frequencies. Next, he addresses the CEM or numerical methods and shows a procedure of converting an operator equation into a matrix equation which is the underpinning method behind the finite element method (FEM) and the MoM. After that, he further addresses fast algorithms and particularly introduces the MLFMA and domain decomposition method (DDM). Finally, he reviews some key components of high-frequency solutions, inverse problems, metamaterials, and small antennas. With those reviews, he predicts that EM will still remain an important area of study and a fundamental status impacting many other technologies, even if the onset of quantum mechanics could bring new possibilities to make a big change.

Chapter 2 also addresses the CEM as the background of applying the Nyström method, but it is in a more microscopic manner compared with the more macroscopic picture and wider range of Chapter 1. The CEM, which is the basis of solving EM problems by computer modeling and simulation, is first defined and categorized. The analytical solutions of spherical objects, which are Mie-series solutions and are usually used as exact solutions, are then presented. After that, a brief introduction to the three mainstream numerical methods, i.e. the finite-difference time-domain (FDTD) method, FEM, and MoM, is given. Finally, all EM integral equations for different cases are summarily presented and they serve the basis of applying the Nyström method to solve EM problems.

Chapter 3 presents a careful look on the theme of this book, i.e. the Nyström method. It introduces the history, basic principle, and implementation scheme for the Nyström method, but there is no demonstration and application which will be shown in later chapters. It also deals with the singularity treatment which is the key of the Nyström method and describes a higher-order scheme in principle. Finally,...