Preface

Computational electromagnetics (CEM) research aims at the modeling and simulation of scientific and engineering problems based on the solution of Maxwell's equations or their variations through the development of numerical algorithms and computer programs for the evaluation, prediction, and optimization purposes. Since their early developments in 1960s, CEM methods have been used in widespread areas, such as radar scattering evaluation, antennas and array design, microwave circuit analysis, electronics and nanodevice development, magnetic and electric machine modeling, high-power microwave system simulation, bioelectromagnetic effect and biomedical device modeling, and electromagnetic imaging and inversion, just to name a few.

Through the past 70 some years, CEM methods have evolved into two major categories, asymptotic and full-wave methods. The asymptotic methods are based on an optical description of electromagnetic waves at high frequencies when the problem is electrically large. This category includes the basic geometrical optics (GO) and physical optics (PO) methods, which are later extended with the theory of diffraction to obtain the methods based on the geometrical theory of diffraction (GTD) and the physical theory of diffraction (PTD). Later developments include the unified theory of diffraction (UTD) and the shooting and bouncing ray (SBR) methods. These asymptotic methods, in general, are very efficient in the simulation of electromagnetic problems due to their simplified description of electromagnetic waves and are accurate only in an asymptotic sense, which means their accuracy is good only when the operating frequency is very high, or the problem size is very large compared to the wavelength. As a result, the asymptotic methods are usually regarded as "high-frequency methods."

Another type of methods, known as the full-wave methods, are based on the rigorous numerical solution of Maxwell's equations, the Helmholtz equation, or various integral equations, in either the frequency or the time domain. Two types of numerical methods have been developed based on the mathematical nature of the governing equations, including the ones that solve partial differential equations (PDEs) and those that solve integral equations (IEs). Well-known PDE solvers are the frequency-domain finite difference method (FDM), finite-element method (FEM), and their time-domain counterparts, the finite-difference time-domain (FDTD) method, and the finite-element time-domain (FETD) method. On the IE side, the method of moments (MoM) has been widely used to solve surface integral equations (SIEs), volume integral equations (VIEs), and their combination, the volume-surface integral equations (VSIEs). The FDM and FDTD are very popular in electromagnetic and optical modeling and simulations due to their simplicity in formulation and implementation. But they generally suffer from the low geometrical modeling accuracy due to the use of structured meshes, the low spatial and temporal interpolation accuracy due to the use of finite differencing that is usually second-order accurate, and a large number of time steps due to the use of conditionally stable time integration schemes. Compared to FDM and FDTD, the FEM and FETD are very flexible and accurate in describing complex geometrical structures with unstructured conformal meshes, convenient in achieving higher-order accuracy with high-order interpolatory or hierarchical basis functions, and efficient in time integration with unconditionally stabile time marching schemes in the temporal discretization. However, although FEM and FETD convert PDEs into sparse matrix equations that have a linear storage complexity, the dimension of the matrix equations is usually very large since the simulation domains need to be discretized into volumetric meshes and consequently, the numerical solution of the matrix equations can be very time consuming. Based on the solution of IEs, the MoM only requires the discretization of either the surface or the volume of the objects without the need of modeling their surrounding background or the truncation boundary. As a result, the overall dimension of the matrix equations is much smaller compared to that of the same problem modeled by FEM. Unfortunately, due to the use of Green's function that depicts the global coupling of fields between every two points in the objects, the resulting system matrix is a fully populated dense matrix that requires an storage with N being the number of degrees of freedoms (DoFs) and either or solution cost with a direct or an iterative solver, respectively. This greatly limits the size of the problem that a full-wave method can handle. Since 1990s, various fast algorithms have been developed to reduce the storage and computational complexities, and domain decomposition methods based on the philosophy of "divide and conquer" have been developed. The further application of large-scale parallel computation boosted the modeling and simulation capabilities of the modern CEM methods significantly. Many EM problems that cannot be tackled by the full-wave methods in the past can now be solved with high accuracy and good efficiency.

Full-wave simulations can be performed in both the frequency and the time domains. The numerical solutions obtained from these two types of methods are related by Fourier/inverse Fourier transform. When performing wideband or transient simulations, the frequency- and time-domain methods are equivalent. On the one hand, frequency-domain simulations can be performed on multiple frequencies of interest and their solutions can be inverse Fourier transformed to the time domain to construct time-domain results. On the other hand, time-domain simulations can be performed using a transient excitation to calculate the time-domain solutions, which can then be Fourier transformed to the frequency domain and sampled at the frequencies of interest. Apparently, in solving ultra-wideband problems, the time-domain simulation will outperform its frequency-domain counterpart since the excitation in these applications only lasts for a very short period but will lead to a very large number of sampling frequencies and a very long simulation time when employing frequency-domain solvers. There are, in fact, many other scenarios where time-domain simulations are not only preferred but, many times, required. Typical examples include the simulation of time-modulated materials, nonlinear problems, and multiphysics problems.

In problems involving time-varying/time-modulated materials, such as those encountered in the research of metamaterials and metasurfaces, many material parameters or properties can be tuned using electrical/magnetic/optical/thermal modulation. For instance, an important application of tenability is to break the time invariance, thus achieving the magnetless Lorentz nonreciprocity, which may lead to many exotic phenomena. To simulate such problems, the only approach viable is the time-domain method. In multiphysics problems, multiple physical phenomena, such as EM, thermal, mechanical, and even chemical phenomena, are coupled together in a sense that the variation of the physical quantities in one physical process affects those in other physical processes and vice versa. To simulate such problems, the time-domain methods are also necessary because many of these multiphysics problems involve transient processes, and the mutual couplings usually take place in the temporal domain. More importantly, many multiphysics problems involve the nonlinear coupling of physics, which is naturally suitable and more convenient to simulate in the time domain than in the frequency domain that usually requires a time-harmonic assumption that is not always satisfied.

In this edited book, many important aspects and latest developments of time-domain methods have been covered and the interesting progress of time-domain simulation applications has been reported. These methods and applications have been divided into the following six themes.

1. Time-domain methods for analyzing nonlinear phenomena
2. Time-domain methods for multiphysics and multiscale modeling
3. Time-domain integral equation methods for scattering analysis
4. Applications of deep learning in time-domain methods
5. Parallel computation schemes for time-domain methods
6. Multidisciplinary explorations of time-domain methods

Specifically, Chapters 1-3 focus on the recent developments of time-domain methods for nonlinear phenomena. In Chapter 1, nonlinear circuit elements are integrated into the FDTD modeling of circuit problems, where three categories of methods are discussed in detail, including the use of specialized updating equations, the development of co-simulation approaches, and the employment of data-based models. It is shown that the FDTD technique is extremely well suited to the simulation of nonlinear devices due to its time-domain nature. Chapter 2 employs Maxwell-hydrodynamic model and the finite difference time domain perturbation method to analyze the electromagnetic response of the nonlinear metasurfaces. Several optical components are designed using the proposed algorithm. In Chapter 3, the FETD modeling of...