A Unified View of Computational Electromagnetics

Electromagnetic modeling is a critical analysis and design methodology in various fields such as radio‐frequency (RF) and microwave engineering, antennas, radar, remote sensing, medical diagnostics, bio‐electromagnetics, physics and radio astronomy, and electromagnetic compatibility (EMC) and interference. It formulates and solves electromagnetic field problems using mathematical, physical, or computational models that relate the electromagnetic functionality of devices to their geometrical and material parameters or predict how electromagnetic fields interact with physical objects and materials. This chapter presents the most common methods for modeling electromagnetic scenarios in the frequency, the spectral, and the time domain, as well as their hybrid combinations. They are introduced as projective approximations of field solutions using the method of weighted residuals (MWR). This unified perspective best illustrates their properties, differences, and their suitability for solving a particular electromagnetic problem, thus helping the reader to appreciate the connections, similarities, and differences among the various methods. It also reflects historical and methodological developments and aims to provide a foundational understanding of current electromagnetic modeling methods.

This article deals with a problem that describes the propagation of surface waves in a layer of an inhomogeneous fluid. The authors present a mathematical model that describes wave motions on the surface of an ideal exponentially stratified fluid. In the equations and boundary conditions, the transition to dimensionless variables and quantities has been completed. Next, a linear version of the problem follows, the solution of which is in the form of progressive waves of a steady-state form with unknown amplitude coefficients. This type of solution is substituted into the equations and boundary conditions of the linear problem, which makes it possible to reduce the determination of unknown quantities to the problem of solving a system of ordinary differential equations. Solving the system has allowed identifying two areas of physical parameters with different nature of wave motion. Expressions are obtained for the unknown components of the fluid velocity, pressure, free surface shape, and wave frequency. This article contains the analysis of the influence of various parameters of the problem on the wave motion: graphs of the dependence of the phase velocity of the wave on the stratification parameter are constructed for different layer depths and wavelengths. For a better understanding of the nature of wave motion, the expressions for the trajectories of liquid particles are determined. This has required writing the equations of motion of particles using the obtained expressions for the components of the velocity vector; these equations are solved with the method of asymptotic approximations. A graphical analysis of the effect of the stratification parameter value on the particle trajectory shape is carried out. The results have revealed that an increase in stratification leads to a compression of the trajectory in the vertical direction.

A large number of numerical methods have been proposed and continue to be developed for solving electromagnetic structure problems. This paper presents a new look at these numerical methods and a way of unifying them with the node-based method of weighted residuals (MWR). As a result, a unifying approach to constructing of numerical methods is shown. It can then be applied to develop new numerical methods catering to specific electromagnetic issues or problems. A new divergence-preserved meshless method is developed as an example.

Maxwell's equations are valid from the atomic to intergalactic length scales. Hence, features at various spatial or temporal scales may all be significant in electromagnetic (EM) problems of interest and may affect the results of multiscale modeling and simulation in computational electromagnetics (CEM). Solving electromagnetic problems that involve multiscale structures efficiently is challenging and is of growing interest in CEM. In recent decades, various approaches have been proposed for addressing the multiscale issue of electrically large problems. These methods can be roughly categorized into three types: the “geometry‐based,” “algebra‐based,” and “physics‐based” approaches. In this chapter, we introduce these three representative approaches for modeling multiscale problems in CEM: (1) the discontinuous Galerkin integral equation method (DGIE), which is a general domain decomposition framework for dealing with structures with mesh nonconformity; (2) the nested dissection accelerated H‐LU‐based fast direct solver for sparse matrices, which is used to construct effective preconditioners; and (3) the resistive sheet boundary condition‐based hybrid finite element‐boundary integral‐multilevel fast multipole algorithm (FE–BI–MLFMA), which can simulate electromagnetic field interactions with microwave‐absorbing honeycomb structures efficiently. After we present the formulation for each approach, we present examples to demonstrate its accuracy, capability, and performance.

Measured Equation of Invariance (MEI) is a new concept in computational electromagnetics. It has been demonstrated that the MEI is such an efficient boundary truncation technique that the meshes can be terminated very close to the object and still strictly preserves the sparsity of the FD equations. Therefore, the final system matrix encountered by MEI is a sparse matrix with size similar to that of integral equation methods. However, complicated Green's function and disagreeable Sommerfeld integrals make the traditional MEI very difficult, if not impossible, to be applied to analyze multilayer and multiconductor interconnects. In this paper, we propose the Geometry Independent MEI (GIMEI) which substantially improved the original MEI method. We use GIMEI for capacitance extraction of general two-dimension VLSI multilayer and multiconductor interconnect. Numerical results are in good agreement with various published data. We also include a simple three-dimensional example and compared GIMEI with FASTCAP from MIT. The accuracy is maintained while GIMEI care generally an order of magnitude faster than FASTCAP with much less memory usage.

In a recent invited paper in the IEEE Antennas and Propagation Magazine, some of the challenging problems in computational electromagnetics were presented. One of the objectives of this note is to simply point out that challenging to one may be simple to another. This is demonstrated through an example cited in that article. The example chosen is a Vivaldi antenna array. What we discuss here also applies to the other examples presented in that article, but we have chosen the Vivaldi antenna array to help us make our point. It is shown in this short article that a higher-order basis using a surface integral equation a la a PMCHWT (Poggio-Miller-Chu-Harrington-Wu-Tsai) method-of-moments formulation may still be the best weapon that one have in today's arsenal to deal with challenging complex electromagnetic analysis problems. Here, we have used the commercially available code WIPL-D to carry out all the computations using laptop/desktop systems. The second objective of this paper is to present an out-of-core solver. The goal is to demonstrate that an out-of-core 32-bit-system-based solver can be as efficient as a 64-bit in-core solver. This is quite contrary to the popular belief that an out-of-core solver is generally much slower than an in-core solver. This can be significant, as the difference in the cost of a 32-bit system can be 1/30 of a 64-bit system of similar capabilities using current computer architectures. For the 32-bit system, we consider a Pentium 4 system, whereas for the 64-bit system, we consider an Itanium 2 system for comparison. The out-of-core solver can go beyond the 2 GB limitation for a 32-bit system and can be run on ordinary laptop/desktop; hence, we can simultaneously have a much lower hardware investment while better performance for a sophisticated and powerful electromagnetic solver. The system resources and the CPU times are also outlined.

The ultimate goal in computational electromagnetics is to develop numerical methods and algorithms that are effective, stable, and give accurate and error-controllable results over a wide range of frequencies, material parameters, geometries, and applications. This article discusses obstacles and challenges in achieving that goal with frequency-domain surface integral-equation methods.

Many problems in Computational Electromagnetics (CEM) and Computational Fluid Dynamics (CFD) require millions of grid cells to properly represent the physics. Advances in parallel computing, both in hardware and software, make it feasible to solve such large scale simulations in a timely manner. The objective of this paper is to describe some of the recent work at Rockwell in performing parallel computing for providing quick-turnaround solutions to problems in CFD, CEM and other computational disciplines.

Chloride penetration is one of the major problems that affect the durability of reinforced concrete structures, and it is of great importance to study the chloride penetration mechanisms, to develop proper models and complete related computation, to enhance the durability design of reinforced concrete structures, and to improve the system of service life prediction and assessments of concrete structures. In this paper, chloride penetration in unsaturated concrete was studied by considering the coupled effects of moisture seepage and chloride diffusion. With weighted residue method, the weak forms of seepage and diffusion control equations were established. Using radial integration method, function approximating with compacted support radial basis function, and central difference scheme, a meshless boundary element method was developed. System equations were solved, and a self-contained computing code was constructed with the developed meshless boundary element method. Numerical examples demonstrate that the effect of moisture seepage is significantly greater than that of chloride diffusion in unsaturated concrete, and its influence on diffusion process is lasted after concrete is saturated. The developed method could give some bases for studying chloride penetration in unsaturated concrete during dry-wet cycling process.

We study the threshold conditions for the natural modes of the microsize plasmonic laser shaped as an infinite flat graphene strip grating symmetrically embedded into the gain-material layer, in the H-polarization case. For this purpose, we solve the lasing eigenvalue problem (LEP), which is a classical source-free electromagnetic field boundary-value problem, adapted to the presence of the active region by the corresponding sign of the imaginary part of the refractive index. In such a way we look for the eigenpairs, i.e. the stimulated emission real-valued frequency and the threshold gain index, specific to each mode. We transform LEP to a hypersingular integral equation for the on-strip current density and discretize it by the regularizing Galerkin technique. This procedure leads to a determinantal equation with guaranteed convergence to the exact LEP eigenpairs and controlled accuracy of their computation. The numerical analysis allows us to study the threshold conditions for various lasing modes of the microsize laser, identify them and trace their change when varying the parameters of the lasing structure.This article is part of the theme issue 'Analytically grounded full-wave methods for advances in computational electromagnetics'.

PurposeTo describe and extend existing concepts of discrete electromagnetism in a unified formalism; to give examples for the usefulness of the presented ideas for our theoretical work, especially with regard to energy.Design/methodology/approachAfter a concise introduction to the mathematical concepts of discrete electromagnetism, we introduce continuous de Rham currents and give their discrete counterpart. We define operators acting upon discrete currents, and apply the theory to electromagnetism.Findingsde Rham current theory yields a mathematical framework for the discussion of discrete electromagnetic problems: The focus is on energy‐balance equations; a discrete Lagrangian can be defined for various modeling problems; the Galerkin approach fits nicely into the proposed formalism; boundary terms in discrete formulations are an implicit feature to the theory.Research limitations/implicationsIn this paper, we use the interpolation of discrete fields by Whitney forms on a simplicial cell complex. The resulting discrete formulation is identical to a Galerkin finite‐element method. Other numerical techniques that do not resort to Whitney‐form interpolation can equally be discussed in de Rham‐current terminology.Originality/valueRather than a novel numerical technique, the paper presents a unified mathematical framework for the discussion of different practical approaches. We advocate a canonical treatment of energy‐related quantities and of boundary terms in discrete formulations.

Electromagnetic (EM) problems with conducting and penetrable media are formulated by volume-surface integral equations (VSIEs) which allow inhomogeneous and anisotropic materials. Traditionally, the VSIEs are solved by the method of moments (MoM) with Rao-Wilton-Glisson (RWG) and Schaubert-Wilton-Glisson (SWG) basis functions. We propose a point-matching scheme to discretize the VSIEs. The point-matching scheme does not rely on any basis and testing functions and allows the use of nonconforming meshes. Also, the integral kernels are free of material parameters by selecting current densities as unknowns. To accelerate the solution for electrically large problems, we incorporate the scheme with the multilevel fast multipole algorithm (MLFMA) and a typical numerical example is presented to validate its effectiveness.

Recently, the authors have developed a number of new hierarchical and singular basis functions for field representation in two- and three-dimensional cells. Scalar and vector basis functions of both interpolatory and hierarchical kind together with some singular functions of the additive kind are discussed in detail, with several results, in a recent SciTech-IET book (R.D. Graglia and A.F. Peterson: Higher-order Techniques in Computational Electromagnetics), as well as in several other publications (for example, Graglia et al., IEEE TAP, vol. 61, pp. 3674–3692, Jul. 2013; IEEE TAP, vol. 62, pp 3632–3644, Jul. 2014; IEEE AWPL, vol. 13, pp 1701–1704, 2014, and in the special issue on "Finite Elements for Microwave Engineering," Electromagnetics, vol. 34, pp. 171–198, 2014).

The Neumann expansion of Bessel functions (of integer order) of a function \(g:\mathbb {C}\rightarrow \mathbb {C}\) corresponds to representing g as a linear combination of basis functions φ0, φ1, …, i.e., \(g(s)=\sum _{\ell = 0}^\infty w_\ell \varphi _\ell (s)\), where φi(s) = Ji(s), i = 0, …, are the Bessel functions. In this work, we study an expansion for a more general class of basis functions. More precisely, we assume that the basis functions satisfy an infinite dimensional linear ordinary differential equation associated with a Hessenberg matrix, motivated by the fact that these basis functions occur in certain iterative methods. A procedure to compute the basis functions as well as the coefficients is proposed. Theoretical properties of the expansion are studied. We illustrate that non-standard basis functions can give faster convergence than the Bessel functions.

In earlier work, we proposed to approximate the constrained linear quadratic regulator problem by representing input and state trajectories as linear combinations of basis functions. As a result, a relatively low-dimensional optimization problem is obtained, while at the same time an infinite prediction horizon can be retained, providing benefits regarding computation and closed-loop stability when applied to model predictive control. In the following, we will extend the approach and present an optimization strategy to design basis functions that are tailored to the specific problem instance. We will show on a simulation example that the same approximation quality of the underlying problem can be achieved with fewer basis functions, which reduces the problem dimension and the computational load even further. The optimization of the basis functions can be done offline. We also discuss the constraint-free case and show that, by choosing the basis functions appropriately, the solution of the underlying linear quadratic regulator problem is recovered by the approximation.