The algorithms in this thesis have been implemented as a computer program written in C++. The program is designed to run on a Sun Sparc-10 workstation and is capable of using either the time domain or frequency domain finite element methods that will be described in Chapter 2. C++ was chosen because it allows an object oriented approach to program design. The finite element method is particularly suitable for this approach.
Chapter
2 Finite Element Method:
Electromagnetic Field Calculation
2.1 Introduction
The finite element method is a powerful tool for solving partial differential equations over irregularly shaped regions [Silvester & Ferrari, 1990]. The solution domain is divided into a number of sub-domains or elements each having simple geometry. The dependent variable is then represented within that element by a simple function. An equivalent discretized model for each element is constructed and the contributions from all elements to the system are assembled. The overall solution is therefore approximated by a summation of simple functions. The use of elements having a variety of shapes and sizes allows for irregularly shaped boundaries to be modelled accurately. Each element is treated individually so inhomogeneous regions can be handled in a straight forward manner: each element is assigned a material property and the material is assumed to be homogeneous inside the element.
Considerable attention has been paid by workers in the field to the finite element analysis of high frequency electromagnetic systems. The work carried out on closed problems generally falls into three categories:
• Two and three dimensional analysis of waveguides with complex geometries.
• Three dimensional eigen-analysis to predict modes in waveguides and cavities.
• Prediction of power density in lossy dielectrics.
The first two categories constitute the majority of work that has been carried out to date, for a survey see Davis [1993] or Rahman et al. [1991]. The work generally deals with waveguide systems containing lossless and irregularly shaped dielectric insertions, one of the primary interests being to determine the propagation constant of the various modes. Calculations in cavities have been carried out, but again this has mainly been to find the resonant modes, eigenvalues, of lossless resonators. Microwave heating requires a knowledge of the total field pattern produced rather than simply the frequencies of the individual modes. However, many of the techniques that have been developed in other areas of computational electromagnetics can be applied directly to microwave heating problems. Webb [1981] outlines the basic method for calculating the magnetic field for a microwave system using elements with node based expansion functions in the presence of lossy dielectrics, however, computational restrictions confine his attention to waveguide problems. The problem of spurious modes is common to many finite element solutions of electromagnetic problems. Section 2.1.2 discusses the origin of these modes and the techniques that can be used to either suppress or to eliminate them.
One of the major difficulties encountered when studying large cavities is the sheer number of unknowns than must be used in order to accurately represent the field, pro- ducing a very large system of algebraic equations. In the conventional frequency domain approach their efficient solution is made virtually impossible for some multimode cavity problems because of ill-conditioning. This dissertation presents a time domain finite element method that overcomes this problem of ill-conditioning. To the author’s knowl- edge this is the first time that the time domain finite element method has been applied to microwave heating. For multimode resonant cavities this proves to be considerably more efficient as well as providing other benefits, such as the ability to solve for more than one frequency simultaneously. The two approaches, frequency domain and time domain are outlined in this chapter. Only a brief description of the theory for the frequency domain method is given here as there are numerous detailed descriptions elsewhere, see for example Silvester & Ferrari [1990]. The time domain method is discussed in greater depth since the particular application is new.
2.1 Introduction 25
2.1.1 Governing Equations
The time dependent Maxwell’s equations that govern the electromagnetic fields in the applicator are,
∇ ×H =
Ã
²∂
∂t+σe
!
E+Js, (2.1)
∇ ×E = −µ∂H
∂t , (2.2)
∇ ·D = ρc, (2.3)
∇ ·H = 0, (2.4)
whereJs is the current density due to sources in the domain, E and H are the electric and magnetic field strengths respectively andDis the electric flux density. These equa- tions are then supplemented by appropriate boundary conditions, the simplest essential boundary condition being,
n×E= 0 on Γ, (2.5)
corresponding to a perfect electrical conductor over the surface Γ.
By eliminating H from equations (2.1) and (2.2) the wave equation is obtained;
∇ × 1
µ∇ ×E+σe∂E
∂t +²∂2E
∂t2 =−∂Js
∂t , (2.6)
on which the time domain formulation is based.
If the fields are assumed to be time harmonic, such that E=E0eωt, then equations (2.1) and (2.2) become,
∇ ×H = ²∗ωE+Js, (2.7)
∇ ×E = µωH. (2.8)
where ²∗ = ²0(²0 −σe/²0ω). Elimination of H from (2.7) and (2.8) gives the vector Helmholtz equation;
∇ × 1
µ∇ ×E−ω2²∗E =−ωJs. (2.9) which is used for the frequency domain discretisation.
2.1.2 Spurious modes
Spurious modes are non-physical solutions that can occur in numerical calculations and corrupt the true result. Their presence in numerical solutions to electromagnetic prob- lems has been known for some time and has prompted the development of a multitude of different techniques for their elimination or suppression1. The reasons for the occur- rence of these spurious modes, which are characterized by a non-zero divergence, is now well understood [Wong & Cendes, 1988; Pinchuk et al., 1988; Lynch & Paulsen, 1991;
Schroeder & Wolff, 1994]. Their presence stems from the improper approximation of the null-space of the curl operator. The double curl equation (2.9) has an infinite number of solutions at zero frequency, that is ∇ ×E= 0 which have the form E=−∇φ, where φ is a scalar field.
When traditional elements with node based expansion functions are used to model the field, many of the −∇φ modes are very poorly approximated, consequently they no longer have zero frequency and become mixed with the true modes. They are then indistinguishable from the physical modes in eigenvalue computations and corrupt the solution in driven problems. For elements with node based expansion functions the most popular method of eliminating these modes is to add a penalty termp∇(∇ ·E) to the weak form (2.16), wherepis a parameter to be chosen [Haraet al., 1983]. This enforces the divergence condition in a least squares way, thereby reducing the number of zero frequency modes. Paulsen & Lynch [1991] analyse this method and conclude that the only correct choice for pis unity, which corresponds to solving the Helmholtz equation
∇2E+ω2µ²∗E = 0, (2.10)
in homogeneous regions where Js = 0. Since the divergence condition is then explicitly included the spurious modes are eliminated. A simple example given by Collin [1991]
shows that enforcing the divergence removes the spurious modes but the new functional gives a poorer approximation to the physical modes because of the extra constraint imposed. This is further demonstrated by results at the end of this chapter.
An alternative method of eliminating the problem of spurious modes is to model the null space of the curl operator more accurately. The−∇φ modes are then approxi-
1See for example Koshiba et al. [1987] and Davis [1993] for a summary of methods.