Numerical modeling and simulation play a central role in the solution of many scientific and engineering problems nowadays. Modeling and simulation require a deep understanding of the mathematical equations governing the physical phenomena being studied, as well as algorithms capable of solving these equations efficiently and accurately. The governing equations can be either posed in differential or integral form, or a combination of the thereof. Differential equations (DE) historically have been solved using finite difference and finite element methods (FDM/FEM). Integral equations (IEs) often are solved using boundary element methods (BEM), often called the method of moments (MoM) by electrical engineers. Although classically formulated DE and IE solvers are useful in the solution of many real-world problems, recently, several nontraditional discretization schemes (NDSs) for solving DEs and IEs pertinent to the analysis of device, network, and field phenomena have gained significant traction. Examples of NDSs include Nyström methods; meshless techniques including partition of unity and generalized FEM and BEM schemes, and as well as stochastic methods including random walk and walk-on-sphere techniques. Generally, NDSs aim to overcome certain drawbacks of traditional discretization methods. For example, Nyström-based IE methods often are easier to implement than their Galerkin counterparts, especially when used in conjunction with point-based fast multipole algorithms. Meshless DE and IE solvers significantly reduce the cost of developing computer-aided design models and offer substantial flexibility when analyzing problems involving deformable and moving objects and boundaries. Finally, random walk and walk-on-sphere methods provide for particularly simple, stable, and easy to parallelize avenues for solving certain DEs.

This special issue reports on recent advances in NDSs and their applications to challenging real-world problems involving the modeling and simulation for electronic networks, devices, and fields. After a rigorous process of peer review, 16 papers were selected from those submitted in response to the Call for Papers. Among them, five papers study the state-of-the-art meshless techniques, two papers discuss the development of new Nyström methods, four papers address the nontraditional finite-difference time-domain (FDTD) implementations, four papers deal with the nontraditional MoM discretizations, and one paper introduces a novel non-uniform grid interpolation method for fast evaluation of wave fields.

This special issue starts with the five papers on meshless methods. The first paper by Song and Chen studies the cutoff wavelength of elliptical waveguides by a regularized meshless method (RMM). It employs the RMM combined with the determinant rule to analyze the cutoff wavelength of elliptical waveguides with arbitrary eccentricity. The method is shown to yield well-conditioned systems and to be devoid of fictitious boundary effects. The next paper by Ala, Di Blasi, and Francomano presents a meshless particle method for solving the magnetoencephalography forward problem. The work considers quasi-stationary Maxwell's curl equations. The adopted meshless particle model is shown to provide results that are in excellent agreement with analytical data. The approach is validated via analysis of a model of a realistic human brain cortex and comparison with the results obtained using a BEM. In the subsequent paper, Razmjoo, Movahhedi, and Hakimi introduce an improved and truly meshless method on the basis of a new shape function and nodal integration. Specifically, the work develops an efficient and stable nodal integration technique on the basis of Taylor series expansions to avoid the use of background meshes that are inevitable in traditional meshless methods. Numerical experiments for 3D electrostatic and electromagnetic problems show that the improved method often is superior to traditional meshless methods. Gordon and Hutchcraft present a meshless method with radial basis functions and use it to determine the electromagnetic fields near material interfaces in two-dimensional and three-dimensional settings. Their method provides a convenient approach for implementing boundary conditions at material interfaces in a meshless context. The approach is verified by comparing its results with analytical solutions and good performance is observed. The final paper in this category is a review article by Kaufmann, Yu, Engström, Chen, and Fumeaux and provides a comprehensive overview of the state-of-the-art in meshless radial point interpolation methods (RPIM) for time-domain electromagnetics. Particularly, it summarizes the localized RPIM scheme and discusses the dependence of its interpolation accuracy on various parameters. Also, an unconditionally stable RPIM scheme is introduced, and its advantages for hybridization with the classical RPIM scheme are demonstrated. Finally, the capabilities of an adaptive time-domain refinement strategy derived from those used in a frequency-domain solver are illustrated.

The next two papers report on the development of new Nyström methods. The paper by Balaban, Smotrova, Shapoval, Bulygin, and Nosich discusses Nyström-type techniques for solving electromagnetic IEs with smooth and singular kernels. It considers electromagnetic scattering, absorption, and emission by 2D or 3D metallic and dielectric objects. For each structure, the problem is formulated by boundary IEs that are discretized using Nyström-type quadrature formulas adapted to the singular kernel and the edge behavior of the unknown function. The accuracy and convergence behavior of the scheme are verified through a host of numerical examples. Another paper by Tsalamengas describes a method to generate quadrature rules for a certain class of singular integrals with logarithmic, Cauchy, or Hadamard-type singularities that are called for in Nyström-type approaches. The proposed rules possess several characteristics: they are easily derived from first principles, cast in terms of an arbitrarily selected external variables, do not involve derivatives of the unknowns, and are highly accurate. Numerical examples for the Nyström solution of wave diffraction problems illustrate the simplicity, flexibility, and accuracy of the algorithm.

Thereafter, follow four papers that deal with nontraditional MoM discretizations. The paper by Ylä-Oijala, Kiminki, Cools, Andriulli, and Järvenpää presents mixed discretization schemes for solving electromagnetic surface IEs. In particular, the authors develop stable and well-defined discretization procedures by testing integral operators using either their range or the dual of their range. This requires the use of dual spaces. It is found that the proposed testing scheme agrees with the Galerkin scheme for first kind of IEs but requires a Petrov–Galerkin scheme with dual testing function for second kind of IEs. The paper by Deng, Sheng, and Song focuses on the analysis of scattering by complex targets using an efficient mixed-order curved patch MoM scheme. In this work, the numerical performances of different types of higher-order basis functions are compared and an improved Duffy method for the higher-order discretization of curved patches is derived. On the basis of the optimized triangular hierarchical higher-order basis function and the improved Duffy method, an efficient and accurate computation scheme is developed. Numerical experiments are performed to show the accuracy and efficiency of the proposed scheme. The paper by Niino and Nishimura develops Calderón preconditioning approaches for Poggio–Miller–Chang–Harrington–Wu–Tsai formulations for analyzing scattering from material objects. The work proposes five different formulations that use different basis functions to represent surface electric and magnetic currents and preconditioning schemes. Numerical experiments are presented to demonstrate the performance of the proposed preconditioning approaches. The paper by Hu, Li, Feng, and Nie reports a non-conformal geometry discretization scheme for hybrid volume and surface IEs. Grid-robust high-order vector basis functions are adopted for the surface discretization of conductors so as to achieve flexibility in the modeling of geometries and a reduction of the number of the unknowns. For the volume discretization of dielectrics, non-conformal volume basis functions are used. Furthermore, a fast-Fourier-Transform (FFT) algorithm is applied to expedite matrix–vector multiplications. Some numerical results are presented to demonstrate the accuracy and efficiency of the method.

The following four papers address nontraditional FDTD schemes. In the paper by Zygiridis, a fourth-order FDTD method that leverages error-controlling concepts is presented. Unlike other approaches that optimize the FDTD method for a single frequency, the proposed method is not narrowband and outperforms a conventional fourth-order FDTD algorithm applied to wideband simulations. The merits of the scheme have been verified through theoretical studies, as well as harmonic and multi-frequency numerical simulations. The paper by Oh, Kim, and Yook investigates optimum scaling factors for one-dimensional FDTD schemes applied to the Maxwell–Boltzmann system. The authors use a new one-dimensional FDTD method to analyze electromagnetic phenomena in plasmas without resorting to an effective permittivity model. Two scaling factors that control the electron density and collision frequency of the plasma are considered and optimized to achieve minimum error in the frequency band of interest. The optimized scaling factors are applied to the conventional FDTD method and their effectiveness is verified by comparisons with analytic solutions or data obtained using conventional FDTD methods. The paper by Zhu, Chen, Zhong, and Liu proposes a hybrid finite element/finite difference method with an implicit-explicit time stepping scheme for Maxwell's equations. In their method, the finite-element time-domain (FETD) method uses an unconditionally stable Crank–Nicholson method and a triangular mesh, whereas the standard FDTD method employs a staggered Cartesian grid for spatial discretization and a leap-frog scheme for time stepping. In particular, the hybrid method takes advantage of the modeling flexibility of the FETD method for complex structures and the efficiency of the FDTD method for simple ones. The hybrid implicit-explicit time stepping scheme allows a time step increment as large as the stability limit for the FDTD method allows. Numerical examples validate the efficiency of the proposed method. Kantartzis contributes the last paper of this category, which deals with hybrid unconditionally stable high-order nonstandard schemes with optimal error-controllable spectral resolution for analysis of complex microwave problems. The key asset of his scheme is a novel high-order nonstandard approximator, whose tensorial properties preserve the hyperbolic character of Maxwell's equations. The resulting formulation remains completely explicit and generates effective dual meshes free of artificial vector parasites and spurious modes. Also, the schemes are hybridized with an alternating direction implicit FDTD method to handle abruptly varying media boundaries and intricate geometries, yielding a significant decrease in dispersion errors, even when time-steps are chosen beyond the stability limit. Diverse real-world setups and composite configurations are used to verify the efficiency and universality of the proposed methodology. Finally, this special issue is capped off by the paper by Costa and Boag, which introduces a Cartesian non-uniform grid interpolation method for fast field evaluation on elongated domains. The method employs the idea of phase and amplitude compensation, facilitating conversion of fields radiated by spatially confined sources into bandlimited functions of suitably chosen coordinates and thereby enabling the fields or potentials to be represented by their samples on sparse non-uniform Cartesian grids. Upon integrating this approach with a divide-and-conquer strategy, the algorithm attains linear computational complexity.

The publication of this Special Issue requires significant efforts, both on the part of the authors and the reviewers. I thank the authors for submitting their high-quality contributions for publication in the Special Issue and the reviewers for their tireless efforts aimed at improving the manuscripts. Also, I would like to express my gratitude to the editor-in-chief, Prof. Eric Michielssen, and his publishing assistant, Ms. Alice Wood, for their encouragement and invaluable advice during the preparation of this Special Issue. Finally, I hope that you will find this Special Issue to be interesting and informative and that it may become a well-referenced publication in this rapidly evolving area.


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  • Mei Song Tong received his BS and MS degree from Huazhong University of Science and Technology, Wuhan, China, in 1985 and 1988, respectively, and his PhD degree from Arizona State University, Tempe, Arizona, USA, in 2004, all in Electrical Engineering. He is currently a distinguished professor and the chair of the Department of Electronic Science and Technology, School of Electronics and Information Engineering, Tongji University, Shanghai, China. Before he joined Tongji University, he was a research scientist at the Center for Computational Electromagnetics and Electromagnetics Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA. His research interests include computational electromagnetics, antenna theory and design, simulation and design of RF/microwave circuits and devices, interconnect and packaging analysis, and inverse electromagnetic scattering for imaging. He has published more than 100 papers in refereed journals and conference proceedings and co-authored a book.

    Prof Tong is a fellow of Electromagnetics Academy, a Senior Member of Institute of Electrical and Electronic Engineers (IEEE), a full member (Commission B) of the US National Committee for the International Union of Radio Science, and a member of the Applied Computational Electromagnetics Society and the Sigma Xi Honor Society. He is serving as a technical reviewer for more than 20 international journals and conferences and serving as an associate editor or guest editor for several well-known international journals, including IEEE Transactions on Antennas and Propagation, IEEE Transactions on Components, Packaging and Manufacturing Technology, International Journal of Numerical Modeling: Electronic Networks, Devices and Fields, Progress in Electromagnetics Research, and Journal of Electromagnetic Waves and Applications. He also served as an editorial board member for Applications and Applied Mathematics, and as an associate editor or guest editor for Waves in Random and Complex Media. In addition, Prof Tong has served as a session organizer, session chair, and technical program committee member for many prestigious international conferences, including the annual IEEE International Symposium on Antennas and Propagation and USNC/URSI National Radio Science Meeting, and Progress in Electromagnetics Research Symposium.