The authors focus on the solution of Maxwell’s equations by means of the Finite Difference Method (FDM), the Finite Element Method (FEM), and the Method of Moments (MoM). Electro- and magnetostatic, quasistatic, and time-depended problems are treated, the last also in the time-harmonic formulation. Convergence, error and stability analysis are included. The implementation of the algorithms is demonstrated by a collection of MATLAB programs which can be downloaded from the URL
http://ct.am.chalmers.sc/edu/books/cem/.
Other well-known numerical methods, used in computational electromagnetics, are characterized and compared shortly in the last of 8 Chapters. The use of the mentioned methods requires also solutions of large scale systems of linear algebraic equations. Thus, short advices for this part of computational electromagnetics are given in two Appendices. In order to help the reader to check his knowledge and to understand the theory each of the Chapters 2-7 contains the parts Review Questions, Summary, Problems, and Computer Projects, the last with implementation tasks. The well-written monograph is devoted to students at the undergraduate level, but is also useful for practising engineers.
The introductory Chapter starts with Maxwell’s equations (Ampère’s law, Faraday’s law, Poisson’s equation and the condition of solenoidal magnetic flux density) including the constitutive relations with the restriction to linear, isotropic and nondispersive media. Boundary conditions for the interface of two media are formulated. The derivation of the curl-curl equation or so-called vector wave equation and its frequency domain form, the vector Helmholtz equation, is outlined.
Analytical solutions of Maxwell’s equations are only known for special cases. Real-world applications require generally numerical methods, which give never exact results. Thus, in Chapter 2 some numerical computational concepts, such as numerical error, convergence, and extrapolation, are introduced using an electrostatic example.
The approximation of Maxwell’s equation by appropriate finite difference methods is introduced in Chapter 3 and demonstrated again by an electrostatic example. The application of the finite difference approximation to complex exponentials which are typically for the solution of the corresponding partial differential equations is studied. The discussion of numerical dispersion, spurios modes and so-called staggered grids in the subsequent Chapters is prepared.
Chapter 4 treats eigenvalue problems for Maxwell’s equations. In this case natural oscillation frequencies of the system are of interest rather than the response to a source. Because Maxwell’s equations can be solved in the time domain two computational methods are discussed. It is showed that Maxwell’s equations written as single second-order curl-curl equation in the case of linear, dispersion-free medium are self-adjoined, that means, only real eigenvalues occur. The frequency-domain eigenvalue calculation is demonstrated for the 1D Helmholtz equation using a MATLAB-routine for the calculation of the corresponding algebraic eigenvalue problem. The time-domain eigenvalue calculation works with the advantage of explicit time-stepping, that means, no matrix inversion is needed. After computing the signals, the eigenfrequencies can be determined by the fast Fourier transform including Pade approximation.
The time step \(\Delta t\) of the explicit time-stepping algorithm is restricted by stability reasons, thus a corresponding Neumann stability analysis is presented which leads to the well-known CFL (Courant-Friedrichs-Levy) condition, discussed in this book in Chapter 5 which is about the Finite-Difference Time-Domain algorithm (FDTD). The FDTD method was originally proposed by K.S. Yee and is also known as Yee scheme. First, the advantages (explicit time-stepping, no matrix has to be stored) and the disadvantages (time-step limit, boundaries that are not aligned with the Cartesian grid, no local mesh refinement) of the FDTD as well as the fields of applications are presented. In the three-dimensional FDTD Maxwell’s equation in the differential form are approximated by six scalar equations, three for Ampère’s law and three for Faraday’s law on staggered grids. The FDTD method is also derived using the integral representation of Maxwell’s equation, and it is showed in this connection that the Yee scheme preserves the solenoidal magnetic flux density for all times. Other topics of Chapter 5 are a dispersion analysis of the three-dimensional FDTD, the treatment of open regions by absorbing boundary conditions in form of the perfectly matched layer, and the near-to-far-field-transformation of the radiation pattern of an antenna. The FDTD is applied in connection with a discrete Fourier transform to compute the resonant frequencies of a cubical cavity using MATLAB.
Chapter 6, the most comprehensive part of the book, treats the Finite Element Method. First, the reader is informed about the advantages of the FEM, the ability to deal with complex geometries using unstructered grids and mesh refinements, and the disadvantages, the need to solve linear systems of equations, i.e., more computer resources are required. The basic concepts of the FEM – the elements, the basic functions, the residual, the test functions, and the reduction of the residual – are introduced in general. Nodal basic functions, element matrices, assembling and unstructered meshes are treated in connection with the solution of the one- and two-dimensional Helmholtz equation using Galerkin’s method. Local mesh refinement is demonstrated by an example. For the mesh generation the authors refer to the literature. The so far discussed basic functions are only for scalar equations. The next topic of this Chapter are edge elements for the curl-curl equation. These curl-conforming elements are very well suited for approximating electromagnetic fields. The reader is confronted with edge elements in Cartesian grids, hexahedra and in triangles. The properties of the edge elements are discussed calculating the eigenmodes of a resonator. A MATLAB function is presented for the computation of the mass and stiffness matrix for triangular edge elements. Other topics of edge elements are a time-dependent problem and eddy current problems. The last section of this Chapter deals with variational methods, including the Rayleigh-Ritz and the Galerkin’s method.
The third computational technique, treated in this monography, the Method of Moments, is presented in Chapter 7. Both the electrostatics, previously formulated as Laplace’s and Poisson’s equation, and the full Maxwell equations are introduced as integral equations. As stated by the authors, the electromagnetic community refers to these integral formulation as the MoM. Other literature sources consider the original MoM as a subclass of the Boundary Element Method (BEM). First, the integral formulation of electrostatics with the concept of the Green’s function is introduced, and it is verified that the Poisson and Coulomb formulations are equivalent. The MoM works very well for open geometries whereas the FDM and FEM have difficulties with truncating the open regions. Solving the integral equations the FEM technology is used. A MATLAB implementation is presented for a capacitance problem in an unbounded two-dimensional region using uniform and adaptive grids. Secondly, the MoM is applied to scattering problems in the frequency domain. To find the appropriate Green’s function scalar and vector potentials and a gauge condition are introduced. The related Electric Field Integral Equation (EFIE) is derived. The mathematical difficulties (the singularity of the Green’s function) and the more physical difficulties (the presence of internal resonances) are outlined, and methods to overcome these problems are discussed. Thus, also the Magnetic Field Integral Equation (MFIE) and the Combined Field Integral Equation (CFIE) are shortly introduced. For the complete derivation of these equations the authors refer to the literature. The FEM and Galerkin’s method are used to solve the integral equations. The scattering on thin conducting wires by means of the EFIE is presented as an example. A simplified one-dimensional version of the EFIE, known as Hallén’s equation, is described, and a corresponding MATLAB implementation is given.