In this chapter we introduce the framework for two alternative methods, the finite volume (FV) and finite element (FE) methods, as they apply to time domain electromagnetic problems. These two methods are particularly useful in cases where the geometry of the problem involves complicated, curved surfaces, the scattering from which we would like to model with high accuracy. In such scenarios, the FDTD method's inherent “staircasing” of the grid cells means that those curved surfaces are not well described in the model.

In Section 15.2 we will introduce the finite volume method, and in Section 15.3 we will discuss the finite element method. We conclude the chapter with a brief introduction to the discontinuous Galerkin method, using features of both finite volume and finite element methods. Our intention here is not to provide a full treatment of these methods, but simply to introduce them to the student, in order to show their relationship to the FDTD method, and to clarify the conditions under which these alternative methods might be used. As such we provide only brief introductions to the FV and FE methods and refer the readers to other materials for a full treatment. Before describing these methods, we provide an introduction to advanced grid techniques, which will be required for both of these methods.

Irregular grids

The FDTD method as we have described it so far involves a regular, orthogonal grid of identical cells with dimensions Δx, Δy, and Δz, in Cartesian coordinates.

In this final chapter, we conclude the main part of this book with a selection of more advanced topics. Although primarily relating to the finite element method, hybridization with both the MoM and FDTD will be discussed, so as a final chapter it appropriately draws together these three apparently quite different methods.

The previous chapter concluded with one method for terminating open-region problems with radiation boundary conditions, specifically the use of an absorbing boundary condition. An alternative to the application of an approximate local ABC is to use an exact global RBC, the MoM; this leads to the hybrid FEM/MoM formulation. This approach has proven very powerful for specialized applications, and an application to radiation exposure assessment near a base-station antenna will be presented.

To this point in this book, all the finite element work has proceeded in the frequency domain; in this chapter, time domain finite element analysis (FETD) is discussed, and a connection made with the FDTD, which is explored in detail, revisiting the one-dimensional wave analysis problem introduced in Chapter 9 in the time domain. This connection permits the consideration of hybrid FETD/FDTD schemes.

We conclude the chapter with a discussion on two issues which impact on efficiency. Firstly, sparse matrix storage schemes are briefly outlined, and secondly, error estimation and the use of mesh adaptation based on this is discussed.

The coverage in this chapter is at a higher level than in much of the rest of this book.

Introduction

In the preceding chapter, an introduction to the finite element method was provided by way of a one-dimensional problem. In the course of that development, a number of core features of a typical finite element analysis and FEM code were presented, including the concepts of the variational boundary value problem (VBVP) — which is solved instead of the original differential equation, the importance of boundary conditions, assembly-by-elements, rates of convergence and higher-order elements. Whilst very useful indeed for didactic purposes, the one-dimensional introduction does not permit one to address a number of important issues, which can indeed be addressed in two dimensions. The most important of these is the necessity of a new type of element, originally known as an edge element, but now generally called a vector element, where the degrees of freedom no longer reside at element nodes, but rather along element edges (in their lowest-order form, as edge elements), on faces, and (in three dimensions) over the volume of the element.

However, before vector elements are addressed, there are still some very useful topics to discuss with scalar (nodal) elements in two dimensions, and the first part of this chapter will revisit some topics which were deferred from the previous one, as well as demonstrate an application of a two-dimensional solver to a quasi-static problem (the quasi-TEM analysis of a microstrip transmission line), where the electric fields can be adequately represented as the gradient of the scalar electric potential ø.

On graduating twenty years back, in 1984, my first job was as a research engineer working on computational electromagnetics (CEM) at the National Institute for Aeronautical Systems Technology (as it was then called) of the Council for Scientific and Industrial Research (CSIR) in Pretoria, South Africa. It was an exciting time to be working in this field. Although a number of methods had already been successfully introduced, including the three which will be discussed in detail in this book, major advances were being made in all of these methods, and the power of desktop computers was growing in leaps and bounds. No commercial programs (or codes, as they are generally called) were then available for RF problems, but some US government-sponsored codes, in particular the NEC-2 code, were becoming available for general use.

The 1980s saw the final decade of the Cold War, which in some areas (such as Southern Africa) was far from cold. New military technologies, in particular stealth, were driving CEM to address progressively more electromagnetically complex problems. However, when the Cold War ended, far from CEM work coming to a halt, new commercial markets, such as the rapidly developing market in mobile telephony and personal communication systems, and the proliferation of electronic systems in motor vehicles, continued to drive the technology forward at breakneck speed throughout the 1990s. This was also due to the widespread availability of cheap and progressively more powerful personal computers as a crucial enabling technology.

A substantial number of Matlab files are available on the publisher's website www.cambridge.org/Davidson, supporting the theoretical material in this book. This appendix briefly describes the functionality of each main script. (Most have a number of functions associated with them.)

FDTD codes

• fdtd_1D_demo This implements the 1D FDTD theory described in Section 2.4, for a transmission line with a sinusoidal excitation.

• fdtd_1D_WB_demo This implements the 1D FDTD theory described in Section 2.5, for a transmission line with a wideband excitation.

• fdtd_2D_demo This implements the 2D FDTD theory described in Section 3.2, for TE scattering from a PEC cylinder.

• fdtd_2D_pml_demo This implements the PML described in Section 3.3.

• fdtd_3D_demo This implements the 3D FDTD theory described in Section 3.4, for eigenanalysis of a PEC cavity.

MoM codes

• static_mom This implements the 1D MOM theory described in Section 4.2, for a charged thin wire.

• thin_dipole This implements the 1D MOM theory described in Section 4.3, for radiation from a center-fed thin dipole.

• MoM_2D_TM This implements the MOM theory described in Section 4.6, for TM scattering from a PEC cylinder.

• MoM3D_demo This implements the MOM theory described in Section 6.3, for scattering from a flat plate using the mixed potential EFIE using RWG basis functions.

• MoM_Som This computes the input impedance of a thin printed dipole using the Sommerfeld MPIE MOM formulation of Chapter 7.

In this penultimate chapter, the preceding discussion of the finite element method is extended to three dimensions. We start by extending the eigenanalysis of the preceding chapter to three dimensions, using Whitney elements. Although a 3D FEM code is non-trivial to implement, simple problems can nonetheless be addressed, and the eigenanalysis of a PEC cavity is used to illustrate the development of a 3D FEM code. Higher-order vector elements have already been introduced in the preceding chapter for two-dimensional analysis; in this chapter, a more detailed theoretical treatment will be presented for the more general three-dimensional case, and an LT/QN element is applied to the cavity eigenanalysis problem. Following this, the stationary functional formulation for deterministic (driven) problems will be outlined. In this and the preceding chapter, eigenvalue problems have been used to illustrate the FEM; in this chapter, a deterministic three-dimensional problem will also be discussed, namely the analysis of waveguide obstacles. Finite element analysis is ideal for this problem, and good results have been obtained by a number of workers. Results for two waveguide problems computed using FEM codes incorporating higher-order elements will be shown. The chapter concludes with a discussion on the use of absorbing boundary conditions for open-region problems, and a first-order ABC is presented. Treatment of a more sophisticated scheme is deferred to the final chapter.

Introduction

In the previous chapter, the basic concepts of the finite difference time domain method were introduced via a one-dimensional example. We will briefly reprise the issues one must attend to when doing an FDTD simulation, as follows:

• An FDTD mesh (or grid) must be created for the problem. (This is trivial in 1D, requires a little thought in 2D, and becomes quite a major problem in 3D.)

• This mesh must be fine enough – i.e. Δs must be no more than perhaps one-tenth of the minimum wavelength (i.e. maximum frequency) of interest (Δs represents the spatial step size; quite often, Δx, Δy and Δz are chosen equal and Δs is used as shorthand for this).

• The time step Δt must satisfy the Courant limit (but be as close to this as possible to minimize dispersion).

• Boundary conditions (the source and load resistors in our 1D example) must be specified.

• An appropriate signal shape (e.g. differentiated Gaussian) with suitable time duration for the desired spectral content must be chosen. Also, in general, its spatial position must be specified. (In the transmission line example, it was fixed as the source voltage generator.)

In this chapter, we will study the FDTD method in two and three dimensions. Firstly, we will develop a 2D simulator for a problem of scattering in free space. Following this, a very important development, the perfectly matched layer absorbing boundary condition, will be discussed and implemented.