Numerical Methods for Ports in Closed Waveguides

Christer Johansson

Stockholm 2003 Licentiate Thesis Stockholm University Department of Numerical Analysis and Computer Science Akademisk avhandling som med tillstånd av Kungl Tekniska Högskolan framlägges till offentlig granskning för avläggande av licentiatexamen tisdagen den 28 oktober 2003 kl 16.00 i sal D31, Kungl Tekniska Högskolan, Lindstedtsvägen 5, Stockholm.

ISBN 91-7283-598-2 TRITA-NA-0319 ISSN 0348-2952 ISRN KTH/NA/R-03/19-SE

c Christer Johansson, October 2003

Högskoletryckeriet, Stockholm 2003 Abstract

Waveguides are used to transmit electromagnetic signals. Their geometry is typi- cally long and slender their particular shape can be used in the design of computa- tional methods. Only special modes are transmitted and eigenvalue and eigenvector analysis becomes important. We develop a finite-element code for solving the electromagnetic field problem in closed waveguides filled with various materials. By discretizing the cross-section of the waveguide into a number of triangles, an eigenvalue problem is derived. A general program based on Arnoldi’s method and ARPACK has been written using node and edge elements to approximate the field. A serious problem in the FEM was the occurrence of spurious solution, that was due to improper modeling of the null space of the curl operator. Therefore edge elements has been chosen to remove non physical spurious solutions that arises. Numerical examples are given for homogeneous and inhomogeneous waveguides, in the homogeneous case the results are compared to analytical solutions and the right order of convergence is achieved. For the more complicated inhomogeneous waveguides with and without striplines, comparison has been done with results found in literature together with grid convergence studies. The code has been implemented to be used in an industrial environment, to- gether with full 3-D time and frequency domain solvers. The 2-D simulations has been used as input for full 3-D time domain simulations, and the results have been compared to what an analytical input would give.

ISBN 91-7283-598-2 • TRITA-NA-0319 • ISSN 0348-2952 • ISRN KTH/NA/R-03/19-SE

iii iv Acknowledgments

I would like to thank all the people involved in the GEMS project. Particularly I want to thank Erik Abenius and Anders Ålund for their help in debugging the code. I would also like to thank the industrial partners that give a broader perspective to my work. I would also want to thank my advisor Björn Enquist for finding time for sci- entific discussions. He also arranged visits for me to University of California, Los Angeles and Princeton University. This work was supported with computing re- sources at Paralelldatorcentrum (PDC), Royal Institute of Technology. Special thanks to Anders Höök for the geometrical setup of Clavin antenna, and Katarina Gustavsson for proofreading my thesis. Financial support has been provided by NADA, KTH, NUTEK (PSCI).

v vi Contents

1 Introduction 1

2 General Framework 5 2.1Waveguidemodeling...... 6 2.2Staticfield...... 8 2.3Full-Waveanalysis...... 10 2.3.1Homogeneousmedia...... 10 2.3.2 Inhomogeneousmedia...... 12 2.4Analyticalsolutions...... 12 2.4.1Rectangular...... 13 2.4.2 Circular...... 13 2.4.3 Coaxial...... 14 2.4.4 Transverseresonancetechnique...... 15

3 Variational Formulation 17 3.1Potential...... 17 3.2Vector...... 18

4 Domain Decomposition and Hybrid Methods 19

5 Finite Dimensional Approximation 21 5.1Potential...... 23 5.2Vector...... 23 5.3 Spurious modes ...... 25

6 Generalized Algebraic Eigenvalue Problem 27 6.1Arnoldi’smethod...... 27 6.2Reordering...... 31

vii viii Contents

7 Numerical Results for 3-D Waveguides 33 7.1Homogeneouswaveguide...... 33 7.1.1 Rectangularwaveguide...... 33 7.1.2 Circularwaveguide...... 41 7.1.3 Coaxialwaveguide...... 41 7.2Inhomogeneouswaveguide...... 42 7.3EnclosedStripline...... 46 7.3.1Homogeneous...... 46 7.3.2 Inhomogeneous...... 47

8 Direct Numerical Simulation of 3-D Waveguides 49 8.1Coaxialwaveguide...... 52 8.2Rectangularwaveguide...... 55 8.3Clavinantenna...... 58

9Conclusion 59

A Element Matrices 61 A.1Laplace...... 61 A.2Helmholtz...... 62

B Algorithm Description 63 B.1Laplace...... 63 B.2Helmholtz...... 64

CTheGemsProject 65 Chapter 1

Introduction

Waveguides are used for the transmission of electromagnetic signals. The electro- magnetic fields in these waveguides are given by the Maxwell equations,

∇·D = ρ ∇·B =0 ∂B = −∇ × E ∂t ∂D = ∇×H − J ∂t where, E(x, y, z, t) is the electric field, D(x, y, z, t) is the electric flux density, H(x, y, z, t) is the magnetic field, B(x, y, z, t) is the magnetic flux density, J(x, y, z, t) is the electric current density and ρ is the charge density. These equations describe general electromagnetic phenomena. The equations are linear but there are only in cases with special boundary interface conditions that analytical solutions can be obtained. Electromagnetic applications have become more complex and the need to predict their performance more important. Therefore numerical methods have become a useful tool based on faster computers and improved algorithms. One area of application is systems that rely on waveguides for the transmission of the signal, like antennas see figure 1.1. At high frequencies this is the only practical way of transmitting electromagnetic radiation. Waveguides can handle high power with low loss, but they are often bulky and expensive. In modern day an often used waveguide are planar transmission lines (striplines) used in integrated microwave circuits. These are very compact and with low cost. For many years, something called lumped element circuit [34], has been used for simulating transmission lines. This method makes the simplifying assumption that each element, such as resistors, inductors, capacitors, is an infinitesimal point in space, and that the wires connecting elements are perfect conductors. However,

1 2 Chapter 1. Introduction

Figure 1.1. Cross-section of a horn antenna. The horn is feed by an waveguide which in turn is feed by a coaxial line. The dashed line shows there one may want to use ports this method is not valid at microwave frequencies where the dimension of the de- vice is of the same order as the micro wavelength. These high frequencies (short wavelength) are difficult to analyze but very important in applications like radar and communication systems, since

• More bandwidth, more information can be transmitted

• In radar systems the effective reflection area is proportional to the targets electrical size

In resent years numerical methods has become a more useful tool when designing these waveguides. The simulation problem is a full 3-D problem, but by using separation of variables, the propagation direction is eliminated and the problem is reduced to a much smaller 2-D problem. This new problem is studied and used in the full 3-D simulation of for example antennas where the waveguide is the cable connecting to the antenna. With the introduction of more complex waveguides, it becomes important to be able to predict the wavenumber, or propagation constant for the medium [11]. One typically wants to evaluate propagation characteristics of waveguides with arbitrarily shaped cross section and the finite-element method (FEM) is often used. Other typical methods are methods of moments (MoM), spectral-domain and finite- difference methods (FD). One should note that FD may not work well on every type of waveguide because of geometric restrictions, but is generally faster and easier to implement. For complex geometries, with different materials one has to solve for all the field components to determine the electric and magnetic fields. Also, the normal components may suffer a discontinuity at the interfaces when we have domains with different materials. 3

We will focus on the FE approach in this thesis, since it allows complex geomet- ric structures containing inhomogeneous materials. By discretizing the cross-section of the waveguide into a number of elements, often triangles, and by using the vari- ational formulation, FEM can now be used to predict propagation characteristics of the waveguide. An often used basis in FEM is the nodal basis, but it is not straightforward since these elements cannot reproduce jump discontinuities easily. One way to solve this is to solve for the potentials, that are continuous. However the differentiation required to obtain the electric and magnetic fields suffers from loss of accuracy. Another a serious problem in the FEM is the occurrence of spurious solutions. This is due to improper modeling of the null space of the curl operator. An often used method to eliminate the spurious solutions is the use of a penalty function. The penalty function method does not remove the spurious modes but shift their cutoff frequency outside the frequency range of interest. Another approach to eliminate the spurious solutions, is to use tangential vector finite elements. In this case the following properties are imposed on the basis:

• solenodal on each element • nonzero component tangential to one edge only

With this approach, only the tangential component of the field is imposed across element boundaries. This leads to a more natural way of handling sharp points and edges. Imposing boundary conditions on these type of elements was introduced by, Nedelec [30]. In chapter 2 we will give a brief description of the governing equations and the waveguide modeling together with some analytical solutions for simple but important waveguides. In chapter 3 the equations are reformulated into the weak form to be used in the finite-element formulation. We will briefly discuss the concept of domain decomposition (DD) in chapter 4. In chapter 5 we employ the Galerkin’s procedure and a generalized eigenvalue prob- lem is derived where the eigenvalues correspond to the propagation constant for the waveguide. There are several methods to solve this problem, e.g. the inverse iter- ation method (IIM) as described in, [16],[3],[40]. This leads to the use of Lanczos method, [26] and therefore the free software ARPACK, [35] will be used a study of the convergence of the eigenvalue problem. This is presented in section 6.1. Furthermore a reordering the unknowns to reduce the bandwidth is applied, see section 6.2. Numerical result for a static field (TEM modes), which arise when we have two transmission lines with different potentials, and more general cases are presented and validated against analytical result. In chapter 7. For static fields ordinary Lagrangian nodal bases will be used to solve for the potential. For the transverse component, a linear vector element is used an Lagrangian elements of first order are used for the longitudinal component. We have also restricted ourself to two side wall boundary conditions, perfect electric and perfect magnetic walls. 4 Chapter 1. Introduction

There is one problem in the current implementation. If there are no propagating modes, the spurious solutions are mapped to the eigenvalue zero. All these zeros has to be filtered out to reach the first evanescent mode, this is not handled in the current program. In section 7.1we verify our implementation by looking at analytical grid convergence for homogeneous waveguides. In sections 7.2-7.3 the convergence for the inhomogeneous and enclosed microstrips are studied and the results are compared with other results found in the literature. In chapter 8, full 3-D calculations simulations are done with the full-TD code written in the GEMS project, see appendix C. In figure 1.2 we see an example of a direct waveguide simulation.

Figure 1.2. A cylindrical slot antenna feed by rectangular waveguides (top left). The unstructured grid around the slot (bottom left). An instantaneous solution where part of the geometry is removed (right). In the vertical field plane we can also see the radiation through the slot. Chapter 2

General Framework

Electromagnetic phenomena are described by Maxwell’s equations,

∇·D = ρ (2.1a) ∇·B =0 (2.1b) ∂B = −∇ × E (2.1c) ∂t ∂D = ∇×H − J (2.1d) ∂t where, (x, y, z)=Ω⊂ R3, t>0, E(x, y, z, t) is the electric field, D(x, y, z, t) is the electric flux density, H(x, y, z, t) is the magnetic field, B(x, y, z, t) is the magnetic flux density, J(x, y, z, t) is the electric current density and ρ is the charge density, bold letters denote vector valued functions. Boundary conditions are given on the boundary of Ω. We consider linear, isotropic and non-dispersive materials with the property

B = µH and D = εE. If we also allow materials that have electric losses, the following relation between the magnetic flux density and the electric field holds

J = σE. For lossless materials, σ =0, this yields

∇·(εE)=ρ (2.2a) ∇·(µH)=0 (2.2b) ∂H µ = −∇ × E (2.2c) ∂t ∂E ε = ∇×H (2.2d) ∂t

5 6 Chapter 2. General Framework

3 Consider a source-free region, ρ =0,ofspaceΩ=ΩD × R ⊂ R containing an inhomogeneity characterized by relative permittivity, εr, and permeability, µr. Across the interface between two media where we have different materials, the conservation law, [24], imply that

n × (E2 − E1)=0 (2.3a) n × (H2 − H1)=0 (2.3b) n · (D2 − D1)=0 (2.3c) n · (B2 − B1)=0 (2.3d)

Here the superscript denotes the two different domains. ∂ By considering a single frequency represention, the time derivative ∂t may be replaced by jω,andweobtain,

∇×E = −jωµ0µrH (2.4a)

∇×H = jωε0εrE (2.4b)

∇·(ε0εrE)=0 (2.4c)

∇·(µ0µrH)=0 (2.4d) where E(x, y, z)=(Ex,Ey,Ez) and H(x, y, z)=(Hx,Hy,Hz). For simplicity we also define Et(x, y, z)=(Ex,Ey) and Ht(x, y, z)=(Hx,Hy) so the field can be written as, E =(Et,Ez) and H =(Ht,Hz). By combining (2.4a) and (2.4b) and eliminating either E or H, we obtain the vector Helmholtz equations 1 ∇× ∇×E − k2ε E =0 (2.5a) µ 0 r r 1 2 ∇× ∇×H − k0µrH =0 (2.5b) εr

2 2 where k0 = ω µ0ε0. The parameter k0 is known as the wavenumber in free space and µ0 and ε0 are the permeability and permittivity in vacuum, respectively. The boundary conditions that are considered perfect electric conductor, (pec) equa- tion (2.6a) or perfect magnetic conductor, (pmc), equation (2.6b).

n × E =0 (2.6a) n × H =0 (2.6b)

2.1 Waveguide modeling

Given the geometry, Ω=ΩD × [0,L], the waveguide is often aligned along one of the axes. We assume it to be the z-axis and that the propagation can be described 2.1. Waveguide modeling 7 as e±jβz. The propagation constant β depends on the material and the geometry of the waveguide and can be given an analytical expression for some important but simple cases. A propagating waveguide mode can now be expressed as,

+ −jβz + −jβz E(x, y, z)= C Et(x, y)e + C Ez(x, y)e + −jβz + −jβz H(x, y, z)=C Ht(x, y)e + C Hz(x, y)e in the positive z-direction and

− jβz − jβz E(x, y, z)= C Et(x, y)e − C Ez(x, y)e − jβz − jβz H(x, y, z)=−C Ht(x, y)e + C Hz(x, y)e in the negative z-direction, where C+ and C− are constants related to the ampli- tude. Next let the following equivalent voltage and current waves be introduced, [11]

V (x, y, z)=V +(x, y)e−jβz + V −(x, y)ejβz (2.9a) I(x, y, z)= I+(x, y)e−jβz − I−(x, y)ejβz (2.9b) where

+ + − − V = K1C ,V= K1C + + − − I = K2C ,I= K2C here, K1 and K2 are proportionality constants to be determined later. In order to conserve power, one have that

+ 2 1 + + ∗ |C | ∗ V |I | = E × H · zdˆ ΩD 2 2 ΩD or ∗ ∗ K1K2 = E × H · zdˆ ΩD. (2.11) ΩD ∗ should be fulfilled. By a normalization of E and H, the product K1K2 can be made equal to unity. This normalization will be used in the actual computations when comparing the modefield in different parts of the waveguide. In the computation this is used then certain waveguide parameters are to be determined. We start by looking at the N-port in figure 2.1. If a wave with + associated equivalent voltage V1 is incident on the junction at terminal P1,a + − reflected wave S11V1 = V1 will be produced in line one, where S11 is the reflection coefficient. Waves will also be transmitted out of the other lines. These waves will + − have amplitudes proportional to SnmVm = Vn , n =2, 3, ..., N where Snm is the transmission coefficient on line n from line m. If we have waves incident in all lines, 8 Chapter 2. General Framework

− − + + V1 ,I1 V1 ,I1 P1

Pn P2

Figure 2.1. N-port microwave circuit then the scattered wave in each line has contributions from all incident waves. This can be written as:  −     +  V1 S11 S12 ··· S1N V1  −     +   V2   S21 S22 ··· T2N   V2   .  =  . . . .   .  , (2.12)  .   . . .. .   .  − ··· + VN SN1 SN2 SNN VN or more compactly as V − = SV +,whereS is called the scattering matrix. For a simple two port system we have, − + V1 S11 S12 V1 − = + . (2.13) V2 S21 S22 V2

If we now send in port one, S11 can be determined by − V1 S11 = + V1 + + assuming that V2 is zero or its contribution will be registered. Since Vn is known, − we know what we send into the waveguide. Only Vn needs to be computed. We will discuss this in more details in connection to the numerical simulations, see chapter 8.

2.2 Static field

Static fields arise when there is a potential, this can happen in coaxial cables or enclosed striplines. These fields are referred to as TEM modes. In static fields 2.2. Static field 9 there is no time variance so by putting the time derivative in (2.2) equal to zero and assume ε and µ constant together with a source free region we get,

∇·E =0 (2.14a) ∇·H =0 (2.14b) ∇×E =0 (2.14c) ∇×H =0. (2.14d)

Observe that the static fields have zero curl and this means that E can be derived from a scalar potential. This will be used when solving for TEM modes that arise from static fields and have Hz = Hz =0. These modes exists for example in coaxial lines or between two infinite parallel plates where there exists a potential. For these 2 2 √ modes, the propagation constant is given by k = β where k = k0 µrεr is the wavenumber for the material, see [34]. For these fields a scalar potential function can be found from the following relation

E(x, y)=−∇tΦ(x, y), (2.15) ∇ ∂ ∂ where t =(∂x, ∂y ). By using (2.14a) we see that Φ(x, y) is a solution of the two-dimensional Laplace equation, ∇2E t (x, y)=0 with appropriate boundary conditions. The electric field is then given by

−jβz E(x, y, z)=−∇tΦ(x, y)e .

To derive an equation for the magnetic field H, let us rewrite equation (2.4a) using −jβz the assumption of z dependence, e .Then∇ = ∇t − jβzˆ and

−jβz −jβz ∇×E =(∇t − jβzˆ) × (Et + Ez)e = −jωµrµ0(Ht + Hz)e where zˆ is the unit vector in the z-direction. Expand this and using the fact that Ez and Hz are equal to zero, yields

∇t × Et − jβzˆ × Et = −jωµrµ0Ht. (2.16)

If equation (2.16) is divided into a transverse and an axial components we get

∇t × Et =0

−jβzˆ × Et = −jωµrµ0Ht.

By using the potential formulation for the electric field, the magnetic field is then given by ε H (x, y, z)=− zˆ ×∇ Φ(x, y)e−jβz. (2.18) t µ t 10 Chapter 2. General Framework

For these type of modes we want to solve

∇2Φ(x, y)= 0, in Ω (2.19a) Φ=Φ0, on Γ0 (2.19b) Φ= Φ1 on Γ1 (2.19c) where Γ0 ∪ Γ1 = ∂ΩD.

2.3 Full-wave analysis

We start by looking at the field components when we have two media that form an interface along the x-axis. Assume that the two dielectrics have different permit- tivity and permeability constants, ε1, µ1 and ε2, µ2. At the interface the tangential magnetic field must be continuous, (2.3b) so the following condition must hold,

1 2 Hx = Hx

Now, by using equation (2.4a), we obtain at the interface

1 1 1 2 (∇×E )x = (∇×E )x. µ1 µ2

This together with the continuity of the electric flux, equation (2.3c) and using the z-dependence, e±jβz, yields

1 2 − ∂E2 − ∂Ez − ∂Ez µ1ε1 µ1ε1 y ± µ1ε1 µ1ε1 2 µ2 µ1 = = jβ Ey . ∂y ∂y ε1 ∂z ε1

If µ1ε1 − µ1ε1 =0 theremustbesomecontributionfromEz to support nonzero 1 2 Ex, Ex at the interface, see equation (2.3a) with n =(0, 1, 0). Similar arguments can be used for the magnetic field to show that we must have a contribution from Hz. A mode in a homogeneous media with Ez =0is called a TE-mode. If Hz =0 themodeiscalledaTM-mode. This can be used to derive two dimensional scalar equation for the field in terms of Ez or Hz, see section 2.3.1. Then the transversal field is obtained by differentiating the solution. This approach is not valid in the inhomogeneous case and we will use a formulation that is valid for both cases.

2.3.1 Homogeneous media In the homogeneous case formulas can be derived that can be used in the compu- tations. By using the vector relation,

∇×(∇×F)=∇(∇·F) −∇2F, 2.3. Full-wave analysis 11 together with the condition of a divergence free vector field, equation (2.5a) can be rewritten as 2 2 ∇ E +(k0εrµr)E =0.

Now, using the assumption that we can write the electric field as, jβz Et(x, y, z)=Et(x, y)e and that Ez(x, y, z)=0for the TE mode, we obtain

∇2 2 − 2 t Et +(k0εrµr β )Et =0.

A similar derivation for the TM mode can be done using the H field instead of the E 2 2 − 2 field. Denote kc = k0εrµr β , and rearrange the terms as

2 2 − 2 β = k kc

2 2 where k = k0εrµr. From this we see that if the operating frequency is too low 2 then β is negative and we have no modes propagating, hence kc is called the cutoff wavenumber. This relation is used later on in the program when we want to calculate the propagation constant for several frequencies and have homogeneous material in the waveguide. Often an alternative expression for the tangential fields is derived for TE or TM modes. We will look at TE modes. For TM modes similar expression can be derived expanding equations (2.4) and using the fact that Ez =0, Hz =0 and that the z-dependence is given by e−jβz. The transverse field can now be expressed as, [34],

jβ ∂Hz Hx = − 2 (2.20a) kc ∂x jωµ0µr ∂Hz Ex = − 2 (2.20b) kc ∂y jβ ∂Hz Hy = 2 (2.20c) kc ∂y jωµ0µr ∂Hz Ey = 2 . (2.20d) kc ∂x

−jβz To determine the transverse field we need to find Hz(x, y, z)=hz(x, y)e from the Helmholtz equation, ∂ ∂ + + k2 h (x, y)=0. (2.21a) ∂x ∂x c z

This is a scalar equation and has the benefit that the problem size in the discrete case will be smaller. We will not use this equation for the special case of pure TE or TM modes, instead the more general version described earlier, see equation 2.5, will be used. 12 Chapter 2. General Framework

2.3.2 Inhomogeneous media For an inhomogeneous waveguide the separation into TE/TM no longer apply, and there is no simple relation between the propagation constant and the wavenumber. By splitting the electric field as,

E = E˜t +ˆzE˜z where zˆ is the unit vector in the z direction and using the assumption that the z -dependence is E˜(x, y, z)=E˜(x, y)ejβz, equation (2.5a) can be rewritten as, 1 1 2 2 ∇ × ∇ × E˜ − jβ∇ E˜ − β E˜ = k0ε E˜ (2.22a) t µ t t µ t z t r t r r 1 2 − ∇t · (∇tE˜z + jβE˜t) = k0εrE˜z. (2.22b) µr By introducing the scaling E −E E˜ = t , E˜ = z . t β z j in equations (2.22a) and (2.22b), collecting the terms and multiplying (2.22b) with β2 we get, 1 1 2 2 2 ∇t × ∇t × Et + β ∇tEz + β Et = k0εrEt (2.23a) µr µr 2 β 2 2 − (∇t · (∇tEz + Et)) = β k0εrEz (2.23b) µr The boundary conditions are rewritten as n × Et =0 on Γ1 Ez =0 (∇tEz + Et) · n =0 on Γ2 ∇t × Ez =0 where Γ1 and Γ2 are the electric and magnetic walls, respectively and Γ1 ∪ Γ2 = ∂ΩD. Here we have only considered the electric field since the magnetic field can be treated in the same way.

2.4 Analytical solutions

In many applications there are simple but important cases of waveguides, e.g. ho- mogeneous rectangular, circular and coaxial guides. In these cases an analytical solution can be found, see [34]. These analytical solutions will be used to validate the program. 2.4. Analytical solutions 13

2.4.1 Rectangular We will only derive the analytical fields for TE modes. Consider a homogeneous rectangular, waveguide with dimensions (x, y) ∈ [0,a] × [0,b] with a perfect electric boundary,

ex(x, 0) = 0, and ex(x, b)=0,

ey(0,y)=0, and ey(a, y)=0.

Equation (2.21a) is solved by separation of variables and the general solution can be written as

hz(x, y)=(A cos(kxx)+B sin(kxx))(C cos(kyy)+D sin(kyy))

2 2 2 where kc = kx + ky. Using the relations for the transverse fields (2.20) together with the boundary conditions give that, nπ mπ D =0,k= ,n=0, 1, 2..., B =0,k= ,m=0, 1, 2, ... y b x a and mπx nπy h (x, y)=C cos( )sin( ) z mn a b where Cmn is an arbitrary constant related to the amplitude. The transverse fields are then given by

jωµ0µr nπ mπx nπy ex = Cmn 2 cos( )sin( ) bkc a b jωµ0µr mπ mπx nπy ey = −Cmn 2 sin( )cos( ) akc a b jβmπ mπx nπy hx = Cmn 2 sin( )cos( ) akc a b jβnpi mπx nπy hy = Cmn 2 cos( )sin( ) bkc a b and the propagation constant is mπ 2 nπ 2 β = k2 − − . a b

2.4.2 Circular Next we consider a hollow tube with a circular cross section and with a radius a.We consider only the TE case with perfect electrical boundary. Cylindrical coordinates, (r, φ), is natural to use in this case and equation (2.21a) can be expressed as, ∂2 1 ∂ 1 ∂2 + + + k2 h (r, φ)=0. (2.26) ∂r2 r ∂r r2 ∂φ2 c z 14 Chapter 2. General Framework

This equation is solved, as before, using separation of variables, this yields the general solution,

hz(r, φ)=(A sin(nφ)+B cos(nφ))(CJn(kcr)+DYn(kcr) (2.27) where Jn(kcr) and Yn(kcr) are the Bessel function of first and second kind. Since Yn(kcr) becomes infinite at r =0the only physical solution is (D=0),

hz(r, φ)=(A sin(nφ)+B cos(nφ))Jn(kcr). Now we find that

jωµrµ  eφ(r, φ)= (A sin(nφ)+B cos(nφ))Jn(kcr) kc where A and B are arbitrary amplitude constants. Since we want Eφ to vanish at   r = a we require that Jn(kca)=0. Define the roots of Jn(x) as ρnm,thenkc must have the value ρ k = nm . cnm a The values of ρnm can be found in [1]. The propagation constant for the TEnm mode is ρ 2 β = k2 − k2 = k2 − nm . nm c a

2.4.3 Coaxial As we saw in section (2.14) a coaxial line with inner radius b and outer radius a can support TEM modes. Since we have a circular domain, equation (2.19) is expressed in cylindrical coordinates. The resulting equation is solved by separation of variables and the assumption that Φ0 =0and Φ1 =0 . The solution is

Φ1 ln(b/r) Φ(r, φ)= . ln(b/a) The field is obtained according to the relations given in (2.15) and (2.18). Coaxial waveguides can also support TE/TM modes. As in the circular case we arrive at the general solution (2.27), but since we have two disjunct boundaries the boundary conditions are eφ(r, φ)=0,r= a, b and the electrical field is given by

jωµrµ   eφ(r, φ)= (A sin(nφ)+B cos(nφ))(CJn(kcr)+DYn(kcr)). kc The boundary conditions now yield,   CJn(kca)+DYn(kca)=0   CJn(kcb)+DYn(kcb)=0 2.4. Analytical solutions 15 and we assume that we do not have the trivial solution C = D =0. This means that the determinant has to be equal to zero,     Jn(kca)Yn(kcb)=Jn(kcb)Yn(kca).

To determine the cutoff, kc this equation has to be solved numerically.

2.4.4 Transverse resonance technique We have a parallel-plate waveguide, aligned along the x-axis, partially filled with dielectric material (0 ≤ y ≤ a) with dielectric constant εr and the rest is filled with air (a ≤ y ≤ b). It is constant along the x- and z-directions. For this waveguide the propagation constant must satisfy, √ √ k0 = ω ε0µ0 <β< εrk0 = k. 2 2 − 2 2 − 2 So, kc = k0 β in the air region equals kck β in the dielectric region. The propagation constant must be the same in both regions, since the tangential electric and magnetic fields must match at the air-dielectric interface. Now assume that the cutoff constant for the dielectric part is kd and ka is the cutoff constant in air, then 2 − 2 2 − 2 ka k0 = kd k Since this equation must satisfy (2.21a) we get for the two regions d2e z + k2e =0 0≤ y ≤ a (2.28a) dy2 d z d2e z + k2e =0 a ≤ y ≤ b (2.28b) dy2 a z with boundary conditions

ez(y)=0,y=0,b

ez(y) continuous at y = a.

At the interface y = a, Hx must be continuous. This leads to

εr ∂ez 1 ∂ez 2 = 2 kd ∂y ka ∂y The solution, to the equations (2.28a) and (2.28b) are

ez(y)=C1 sin kdy 0 ≤ y ≤ a

ez(y)=C2 sin ka(b − y)0≤ y ≤ a where C1 and C2 are amplitude constants. Using the boundary conditions the equation becomes kd tan(lda)=−εrka tan ka(b − a) 16 Chapter 2. General Framework together with 2 − 2 − 2 kd ka =(εr 1)k0.

This determines an infinite number of solutions for the wavenumbers kd and ka. Note that if ka becomes imaginary we have to change the equation, see [11], [34], [10] for a deeper discussion Chapter 3

Variational Formulation

The most essential part of the finite element method is the requirement of an integral formulation of the physical problem. Two possible formulations can be used either variational principle obtained by expressing the physical problem as the extremum of a functional or an integral obtained from the differential system as a weak formulation, see [7].

3.1 Potential

To form the weak formulation of (2.19) we multiply with a test function Ψ from a suitable function space, [7]. This space will be defined below. − Ψ∇2ΦdΩ=0. Ω Integration by parts now yields ∂Φ ∇Ψ∇ΦdΩ − Ψ =0 Ω ∂ΩD ∂n or by splitting the boundary integral ∂Φ ∂Φ ∇Ψ∇ΦdΩ − Ψ − Ψ =0. Ω Γ0 ∂n Γ1 ∂n Since we are solving for the potential, Ψ,wecansetitequaltozeroonΓ0 without loss of generality. Define the function space Vg(x)(Ω) as 2 2 Vg(x)(Ω) = {Ψ: |Ψ| + |∇Ψ| dΩ < ∞ and Ψ=g(x) on ∂Ω}. Ω and define the function h(x) as, Ω1 on Γ1 h(x)= (3.1) 0 on Γ0

17 18 Chapter 3. Variational Formulation

Problem 1 Find Φ ∈ Vh(x)(Ω),sothat

∇Ψ∇ΦdΩ=0 ∀Ψ ∈ V0(Ω). Ω

3.2 Vector

Since the differential operator in (2.23) is symmetric but not positive-definite, we use the following weak form [23]. Define the function space V as

2 3 V = {[Bt,Bz]:ˆz ·∇t × Bt ∈ L2(Ω), ∇tBz ∈{L2(Ω)} , [Bt,Bz] ∈{L2(Ω)} , | | } Bt =0Γ1 ,Bz =0Γ1 with the norm ||v||2 ||v||2 ||∇ × v||2 V = 0 + 0

Where Lp(Ω) is the set of p-integrable functions in the domain Ω and ||·||s denotes the standard (Hs(Ω))3 Sobolev norm in space. Now multiply equation (2.23a) with the test function Bt and equation (2.23b) with the test function Bz.

2 1 β 2 Bt ·∇t × ∇t × Et + (Bt ·∇tEz + Bt · Et) dΩ=k0 εrBt · EtdΩ Ω µr µr Ω 2 1 2 2 −β Bz (∇t · (∇tEz + Et)) dΩ=β k0 εrBzEzdΩ Ω µr Ω integration by parts now yields,

Problem 2 Find a complex number β and a function [Et,Ez] ⊂ V ,sothat 1 2 [(∇t × Bt) · (∇t × Et)+β (Bt ·∇tEz + Bt · Et)]dΩ Ω µ r (3.3) 2 1 = k0 εrBt · EtdΩ − Bt · (n ×∇t × Et)ds Ω ∂Ω µr 2 1 β [∇tBz ·∇tEz + ∇Bz · Et]dΩ Ω µ r (3.4) 2 2 2 1 ∂Ez = β k0 εrBzEzdΩ+β Bz[ + n · Et]ds. Ω ∂Ω µr ∂n holds for all [Bt,Bz] ⊂ V . If we have pec (Γ1) on the boundary, then Bt =0and Bz =0or if we have p.m.c (Γ2) then the condition are n ×∇×Et =0and n · (∇Ez + Et)=0. Hence the boundary integrals in (3.3) and (3.4) can be neglected. Chapter 4

Domain Decomposition and Hybrid Methods

We will give an brief overview of these topics in this chapter. The main reasons one want to use domain decomposion are:

• Parallel computing and distribution of data. The decomposing of data struc- tures for computing with independent numerical methods on different pro- cessors, systems. The main reason for distribution of data is when the com- putational domain is very large. This often result in large systems of linear equations, coming from a finite element discretization. To solve these systems an Algebraic Multigrid Solver (AMG) is often used. Then, a parallel AMG a strategy called geometric division that use overlapping or non-overlapping data decomposition, can be used, see [19].

• If the domain can be divided into simpler domains that allows one to use faster and simpler methods, then one do domain decomposion based on geometric and computational complexity. A common method then using finite element is the finite element tearing and interconnecting method, FETI developed by Farhat and associates it is a non overlapping method. By solving in parallel each subproblem, the FETI approach uses Lagrange multipliers to enforce continuity on the finite element functions on the interfaces of the subdomains. If the number of subdomains become large then the calculation of the multipliers become time consuming and there is precondition methods in use. The condition that we have continuity across subdomains may not always be the case and one way to solve this is described in, [27]. There a overlapping method is used for Helmholt’s equation to allow discontinuities across the interfaces.

19 20 Chapter 4. Domain Decomposition and Hybrid Methods

• Precondition, then one subdivide the solution of a large system into smaller problems whose solution are then used as preconditioner to the large prob- lem. One example is the classical alternating Schwarz method, one solve the problem in each subdomain and using the result in the previous iteration as boundary condition on the interior boundaries, [6],[4]. • Heterogeneous division also called multiphysics, separation of the physical do- main into regions that can be modeled with different equations. By doing a subdomain splitting and using different type of differential equations in each computational domain. This type of decomposition is can be done in elas- ticity, [43], advection -diffusion problems, [32], [8] or in compressible viscous flows, [9] . This is the basis for hybrid methods and is used in the GEMS project, see appendix C and is used then we couple the 2-D calculations to the full 3-D Maxwell simulations. Another way of using heterogeneous division together with the above mentioned method of geometric and computational complex- ity. Is to assume that one has semi-infinite long structures so that one can reduce the problem into fewer dimensions, [39] and there solve the problem with a suitable method. We have concentrated on this approach in this thesis. Chapter 5

Finite Dimensional Approximation

The main part in the modal analysis is to derive an eigenvalue problem. This is done by using finite elements to approximate the solution. The solution represents the eigensolutions and eigenmodes for the waveguide at a certain frequency. In the present approach the domain is split into triangles where µr and εr are con- stant. The infinite dimensional space V is replaced by a finite-dimensional subspace V h ⊂ V .Let φ(x, y)=a + bx + cy, (5.1) then inside a triangular element, a function f can adequately be approximated linearly by φ(x, y). At the corners of the triangle, φ(x, y) can be expressed as

φ1 = a + bx1 + cy1 φ2 = a + bx2 + cy2 (5.2) φ3 = a + bx3 + cy3 where (xi,yi), i =1, 2, 3 are the corners in the triangle. By solving for a, b and c in (5.2) and insert the solution in equation (5.1) we get, 3 φ(x, y)= φiLi(x, y) i=1 where 1 L (x, y)= (a + b x + c y) i 2A i i i ai = xi+1yi+2 − xi+2yi+1

bi = yi+1 − yi−1

ci = xi−1 − xi+1.

21 22 Chapter 5. Finite Dimensional Approximation

The area of the triangle is represented by A and the indices i assumes the values 1,2 and 3 cyclically, so if i =3then i +1=1.TheLi are called simplex coordinates. This nodal basis can be extended to represent a vector function in 2-D,

Ψ(x, y)=ˆx(a + bx + cy)+ˆy(d + ex + fy) where xˆ and yˆ are the unit vectors in x-andy- direction, respectively. This function is complete to polynomial of order 1. Although the completeness, it appears to be inappropriate to use for discretizing Helmholtz equation. In 1980 Nedelec, [30], proposed some constraints to be used on the interpolation functions. For the linear functions in 2D the constraints are,

∂Ψ x =0 ∂x ∂Ψ y =0 ∂y ∂Ψ ∂Ψ x + y =0. ∂y ∂x

If these constraints are used, Ψ is reduced to

Ψ(x, y)=ˆx(a + cy)+ˆy(d − cx).

The last expression can also be written, using the simplex coordinates as

Bi = ωi(Li+1∇Li+2 − Li+2∇Li+1) where ωi denotes the length and direction of edge i and

1 ∇L (x, y)= (ˆxb +ˆyc ). i 2A i i

The function Bi has the following properties

ω ∇×B = ±zˆ i i A and

∇·Bi =0, where the Bi are called edge or Whitney elements. 5.2. Vector 23

Figure 5.1. The constant tangential/linear normal (CT/LN) curl-conforming vector basis functions Bi.

5.1 Potential

We can now define the finite-dimensional subspaces, h { h h i h | } V0 = B0 : B0 = e Li and B0 =0Γ1 i and h { h h i h | } Vh(x) = Bh(x) : Bh(x) = ezLi and Bh(x) = h(x) Γ1 i where h(x) is defined as in equation (3.1). The discrete problem, is formed as (A)ee = b (5.5)

Ωe where Ωe means summation over the contributions from each element and ∇ h ·∇ h (A)e = [ B0 Bh(x)]dΩ. Ωe The evaluation of the integral and the right hand side vector, b, is given in ap- pendix A.1.

5.2 Vector

The basis functions Bi, see figure 5.1, have constant curl and zero divergence within a cell. We can now define our finite-dimensional subspace V h h { h h h i h i h | h | } V = [Bt ,Bz ]:Bt = etBi,Bz = ezLi and Bt =0Γ1 ,Bz =0Γ1 i i 24 Chapter 5. Finite Dimensional Approximation

We will state a theorem from [29] that give estimates on the computed field in the general case,

Theorem 1 Suppose τh is a regular triangulation; then k+1 3 h 1. if u ∈ (H (Ω)) , uh ∈ V , then there exists a constant C independent of h such that k ||u − uh||V ≤ Ch ||u||k+1,

1,s 3 2. if τh is quasi-uniform and if u ∈ (W (Ω)) , s>2, there exist a constant C(s) independent of h and u such that

||u − uh||0 + h||∇ × (u − uh)||0 ≤ Ch||u||W 1,s(Ω

3. if u ∈ (H1(Ω))3 and ∇×u ∈ (Hk+1(Ω))3,then

k ||∇ × (u − uh)||0 ≤ Ch ||∇ × u||k+1 where W s,p(Ω) is the standard Sobolev space of functions with s derivatives in Lp(Ω). We can now form the generalized eigenvalue problem, tt tt tz (A )e 0 et − 2 (D )e (D )e et = β zt zz (5.6) 00 ez (D )e (D )e ez Ωe Ωe where 1 (Att ) = [ (∇ × Bh) · (∇ × Bh) − k2ε Bh · Bh]dΩ mn e µ t t t t 0 r t t Ωe r tt 1 h · h (Dmn)e = Bt Bt dΩ Ω µr 1 (Dzz ) = [ ∇ Bh ·∇Bh − k2ε BhBh]dΩ mn e µ t z t z 0 r z z Ωe r tz 1 ∇ h · h (Dmn)e = Bz Bt dΩ Ω µr zt 1 h ·∇ h (Dmn)e = Bt Bz dΩ. Ω µr The evaluation of the integrals for triangular cells are given in appendix A.2. After assembling the contributions from each element the system (5.6) can be written as, tt tt tz A 0 et − 2 D D et Ax = = β zt zz = λBx. (5.8) 00 ez D D ez This system can then be solved by calling suitable procedures from ARPACK, see section (6.1). The width of the waveguide is often a multiple of the height. The 5.3. Spurious modes 25 simplest case is the quadratic. For these waveguides there will be eigenfunctions that have common eigenvalues, often called degenerated modes and they are harder to find with good accuracy. This is explained below. Factorize B = LDLT and write the system, (5.8) as

Cy = D−1L−1AL−T y = λy (5.9) where y = LT x. (5.10) To get further we look at Block Diagonal Decomposion of a general matrix. Suppose   T11 T12 ··· T1q   0 T22 ··· T2q H   Q CQ = T =  . . . .  , (5.11)  . . .. .  00··· Tqq

n×n is a Schur decomposion of C ∈ C and assume that the Tii are square. If n×n σ(Tii)∩σ(Tjj)=∅ whenever i = j, then there exists a nonsingular matrix Y ∈ C such that −1 (QY ) AC(QY )=diag(T11, ..., Tqq). (5.12) This give us the possibility to write a matrix C on the form,

−1 ni×ni (X) AC(X)=diag(λ1I + N1, ..., λqI + Nq),Ni ∈ C , (5.13) where λ1, ..., λq are distinct, n1+...+nq = n and each Ni is strictly upper triangular. The following give estimates of the eigenvalues for a matrix C that has been perturbed with a matrix E.LetXH CX = D + U be a Schur decomposition of C ∈ Cn×n.Ifµ ∈ λ(C +E) and r is the smallest positive integer such that |U|r =0, then min |λ − µ|≤max(θ, θ1/r) (5.14) λ∈λ(A) where r−1 || || || ||k θ = E p U p. (5.15) k=0 From this we see that a defective eigenvalue is more sensitive to perturbations, for a deeper discussion of these statements see [18].

5.3 Spurious modes

We have two types of solutions to Helmholtz equation (2.5a), but we are only interested in the vector field that is divergence free. 26 Chapter 5. Finite Dimensional Approximation

Here we will briefly state some vector fields properties. It is important that the discrete versions will retain these properties. Every vector field can be written as the sum of two terms, see [24],

F = Fi + Fd where Fi is the irrotational part and has ∇×Fi =0, while Fd is the divergence free part and has ∇·Fd =0. The irrotational field Fi, is assumed to be non-zero and can be written as the gradient of a scalar potential

Fi = −∇φ.

2 Inserting this into equation (2.5a) we see that k0 =0for irrotational fields. 2 Now by taking the divergence equation (2.5a) we see that if k0 =0 then,

∇·E =0 hence it must be divergence free. By multiplying the last expression with a scalar potential that is zero on the boundary and integrate over the domain Ω one get E ·∇φdΩ=0. Ω

This orthogonal property between the two solutions of equation (2.5a) should be retained in the discrete Helmholtz equation. If we looking at the discrete version of Helmholtz, equation (5.8) and put the eigenvalue equal to zero, then we have Att 0 0 Ax = =0 00 v where v =0 . From this we see that the discrete irrotational fields form the null space of A and that there is exactly m such fields there m is the number of internal nodes. We also have that, Dtt Dtz u 0=xT Ay = λxT By = λ 0 v = λvDztu Dzt Dzz 0 so the discrete version of Helmholtz equation preserves the orthogonality between the irrotational solution, v and the divergence free solution, u.Them solutions with eigenvalue equal to zero are the so-called spurious modes. They are static solutions to (2.5a) with nonzero divergence and are not physically interesting. Chapter 6

Generalized Algebraic Eigenvalue Problem

6.1 Arnoldi’s method

For solving the generalized eigenvalue problem, tt tt tz A 0 et − 2 D D et Ax = = β zt zz = λBx 00 ez D D ez derived in chapter 5.2, a subspace iteration (SI) technique has been used in the FEM analysis. In 1998, J. Mielewski and M. Mrozowski [28] studied the performance of Krylov space based methods, such as the Arnoldi method. Their computations showed that the Arnoldi method with implicit restart is more efficient than the SI method when solving large nonsymetrical eigenproblems. The Implicitly Restarted Arnoldi Method (IRAM) is a robust and efficient method for solving large sparse generalized eigenvalue problems. It is an itera- tive method to compute a small set of eigenvalues with user specified features. In our case the wanted eigenvalues are positive and the propagating mode is the square root of the eigenvalue. A set of Schur basis vectors for the desired k dimensional eigenspace is computed which is numerically orthogonal to working precision. Arnoldi’s algorithm constructs an orthogonal basis of the Krylov subspace, Km, in exact arithmetic. The algorithm is as follows.

27 28 Chapter 6. Generalized Algebraic Eigenvalue Problem

Algorithm 1 Arnoldi algorithm 1: {Initialization part} 2: Choose start vector v1, ||v1|| =1 3: for j=1,..,k do 4: hi,j =(Avj ,vi), for i=1,..,j − j 5: wj := Avj i=1 hi,j vi 6: hj+1,j = ||wj ||2 7: if hj+1,j =0,thenstop 8: vj+1 = wj/hj+1,j 9: end for

There is a free version of IRAM, the serial version ARPACK and parallel version named PARPACK, see [35]. The ARPACK software does not require that the user supply the matrices, instead all that is needed is the action of these matrices, this is called reverse communication. The memory used is of the order N · O(Nk)+O(k2) where N is the dimension of the system and k the desired number of eigenvalues. This does not include the storage of the matrices. We will here state some important feature of ARPACK: • Fixed storage requirement normally, N · O(k)+O(k2). • Reverse communication. • Need only the action of the matrix A on a vector v, in our case we need the action B−1A on the vector v. • Multiple eigenvalues offer no theoretical difficulty. • Ability to return k eigenvalues which satisfy a user specified criterion such as largest real part, largest absolute value, largest algebraic value (symmetric case), etc. When using ARPACK several user specified parameters are required, there are two important and problem dependent parameters. One parameter is the numerical tolerance used for determining convergence. The stopping criteria is

||Axˆ − xθˆ || ≤ max(εmach||Hm||,tol·|θ|), where (ˆx, θ) is the Ritz pair approximating the wanted eigenvalue. Hm is an m × m upper Hessenberg matrix, that arise in the Arnoldi factorization. The other parameter is “ncv”, the number of Lanczos vectors to use. It must be at least k +1. If the eigenvalues are well separated then ncv≈2k is acceptable. Well separated is defined as,

|λi − λj | >τ|λN − λ1| for all j = i with τ>>εmach. In the computations we want to choose ncv as small as possible and the tolerance as high as possible. Figure 6.1and 6.2 show the relation between ncv and the numerical tolerance on a simple rectangular waveguide for the two different modes. 6.1. Arnoldi’s method 29

−5 x 10 100 basis vectors 1.1354

1.1354

1.1354

1.1354 rel 1.1354

1.1354

1.1354

1.1354 −15 −10 −5 0 10 10 10 10 tol

−5 x 10 50 basis vectors 1.1355

1.1355

rel 1.1354

1.1354

1.1353 −14 −12 −10 −8 −6 −4 −2 10 10 10 10 10 10 10 tol

Figure 6.1. The relative error for the first propagating mode versus the tolerance given in the ARPACK package, the top figure is with 100 basis vectors and the lower with 50 basis vectors. 30 Chapter 6. Generalized Algebraic Eigenvalue Problem

100 basis vectors 2

1.5

1 rel

0.5

0 −15 −10 −5 0 10 10 10 10 tol 50 basis vectors 2

1.5

1 rel

0.5

0 −15 −10 −5 0 10 10 10 10 tol

−4 Figure 6.2. The relative error, relmin is of order 10 , for the second propagating mode versus the tolerance given in the ARPACK package.

1800 100 basis vectors 50 basis vectors 1600

1400

1200

1000

#iterations 800

600

400

200

0 −15 −10 −5 0 10 10 10 10 tol

Figure 6.3. The number of Arnoldi iterations versus the tolerance given in the ARPACK package. 6.2. Reordering 31

As can be seen from the figures, convergence to the first propagating mode is relatively fast and does not depend much on the tolerance given in ARPACK, but higher order modes are more sensitive. The convergence is a little bit faster if one have a larger basis but storage and computation time increase. During practical simulations we found that about two to three times the wanted modes is often enough. In figure 6.3 we see that the number of iterations performed in ARPACK increase rapidly when the numerical tolerance is chosen very small.

6.2 Reordering

As pointed out earlier, a reordering procedure is used to reduce the bandwidth of the matrix in the linear system of equations. There are several methods that permutes the unknowns. The method we choose is a level set method. A level set is defined as the set of all unmarked neighbour of all the nodes at a previous level set. A simple strategy is to pick a node as initial level set, mark its neighbour to belong to the next level set. If unmarked, continue this until all nodes are marked. This is called Breadth First Search (BFS) and is a common method used in graph theory. In BFS, the nodes at each level is traversed in the order they are listed. In the Cuthill-McKee Ordering, (C-M), the nodes at each level are traversed from lowest to highest. For a thorough description on ordering methods and algorithms see e.g [36],[18] and [22]. The following algorithm is taken from [36].

Algorithm 2 Cuthill-McKee Ordering 1: {input: initial node i1; Output: permutation array iperm.} 2: Set levset:={i1};next = 2 3: Set marker(i1 =1);iperm(1)=i1 4: while next

We use a version of this algorithm that is called Reverse Cuthill-McKee Order- ing, (R-C-M), i.e the first element from C-M goes to the last in R-C-M and so forth. Practice has shown that this often yields a lower fill in, [36]. Figure 6.4 and 6.5 show the elements wof matrix B before and after the reordering. 32 Chapter 6. Generalized Algebraic Eigenvalue Problem

0

100

200

300

400

500

600

700

0 100 200 300 400 500 600 700 nz = 8075

Figure 6.4. Original right hand side matrix, B.

0

100

200

300

400

500

600

700

0 100 200 300 400 500 600 700 nz = 8075

Figure 6.5. The B matrix then Reverse Cuthill-McKee Ordering have been applied. Chapter 7

Numerical Results for 3-D Waveguides

A general-purpose computer program has been written to assemble and solve the generalized eigenvalue system (5.8). To lower the bandwidth of the right hand side matrix we use reverse Cuthill-McKee algorithm to reorder the unknowns. A description of the program is given in appendix B.2. In 2-D we have N ∝ h−2, where h is the length of the largest side of a triangle in the domain and N is the number of elements.

7.1 Homogeneous waveguide

We validate the code against three test cases, a rectangular, see figure7.1, a circular, see figure 7.8, and a coaxial waveguide. The results are then compared to analytical results given in [34].

7.1.1 Rectangular waveguide First, we compare our computed propagation constant against analytical results. This is shown in Figure 7.2. We see that there are good agreement. Second, we look at the convergence of the wavenumber. Figure 7.3 shows that we have convergence like O(h2). It also shows that with a more random distribution of elements, the convergence is not as “nice” as if the mesh is refined in a more regular way, but it still 2 | comp − analytic|∝ 2 follow the O(h ) line. Hence we have β1,2 β1,2 h . For the convergence test on the electric field we use the following set of parameters:

εr = µr =1,a=2b =0.1 and f =10GHz. The convergence for the electric field is checked in two different locations of the triangles. One at the midpoints along the triangle edges and one in barycenter

33 34 Chapter 7. Numerical Results for 3-D Waveguides

y

br

εr

µr

ar x

Figure 7.1. Geometry of the rectangular waveguide, br/ar =0.25, εr =1.0 and µr =1.0.

1

0.9

0.8

0.7

0.6 0 0.5 β/k 0.4

0.3

0.2

0.1

0 3 4 5 6 7 8 9 k0a

β Figure 7.2. The dispersion of the propagating constant ,fortwothelowest k0 modes. The circles are the analytic solution. 7.1. Homogeneous waveguide 35

0 10

−1 10 β2, unif. grid refinement

−2 10 β1, unif. grid refinement abs(error) −3 10 O(h2)

−4 10

β1, random distr. of el.

−5 10 −5 −4 −3 −2 −1 10 10 10h2 10 10

comp analytic Figure 7.3. Convergence study of |βi − βi |, i =1, 2 then k0a =16, a =2b and looking at the TE case. of the triangle. For the first test, a simple average of the next finer grid will be used for comparison. In the second test case, a reference solution will be computed on a fine grid. To be able to compare the fields calculated on different meshes, a normalization, E × H∗dΩ=1 Ω is used, where Ω is the cross section of the waveguide and H∗ is the complex conjugate of H.

Case 1 Convergence validation using analytical fields, 1 2 N / √1 √ | j j − ana j |2 | j j − ana j |2 εj = Exi (mi ) Exi (mi ) + Eyi (mi ) Eyi (mi ) A N 2 i=1 j where A is the waveguide area, mi the midpoint on edge i and mesh j where the tangential component of the field is computed. We now assume that we have convergence like | | p p εj = c1∆xj + c2∆yj and that ∆xj =∆yj = hj. Refine the mesh uniformly by dividing hj by two, then we obtain ε log( j )=p log(2) εj+1 where we have ignored the constants c1 and c2. 36 Chapter 7. Numerical Results for 3-D Waveguides

The analytical expressions are given in section 2.4. We see from table 7.1that

hvaluesused(10−2) log ( εj ) 2 εj+1 {5, 2.5, 1.25, 0.625, 0.3125, 0.15625 } 1.9505 1.9898 1.9976 1.9994 1.9999

Table 7.1. Convergence of the field, first mode and TE case, then measured as, (E · ti)(mi) there ti is the tangent on edge i and mi is the midpoint on edge i. second order of accuracy for the field is achieved.

Case 2 Define, j j j j avg E (m )+E (m ) E j = xi+1 i xi−1 i , xi 2 j j j j avg E (m )+E (m ) E j = yi+1 i yi−1 i . yi 2 avgj j where the fields used to calculate Exi are taken as in figure 7.4 and (mi ) is the midpoint on edge i,meshj. For simplicity we have only used edges aligned with the coordinate axes.

j+1 j+1 Exi−1 Exi+1

j Exi

Figure 7.4. The continuous line is the coarse mesh j and the doted the next finer mesh j +1.

Then convergence on the edges is measured on mesh j as,

1 2 N / 1 avg avg √ √ | j j − j+1 |2 | j j − j+1 |2 εj = Exi (mi ) Exi + Eyi (mi ) Eyi A N 2 i=1 7.1. Homogeneous waveguide 37

−8 10

E εj

−9 10 O(h) H εj error

−10 2 10 O(h )

−11 10 −2 −1 10 h 10

Figure 7.5. Convergence study of the electric and magnetic field then looking at thefirstmodeintheTEcasewitha =2b =0.2, µr = εr =1and f=10GHz. where we have evaluated the components on the edge midpoint and used the tan- gential component, see figure 7.4 for the setup.

hvaluesused(10−2) log ( εj ) 2 εi+1 {1.25, 0.625, 0.3125, 0.15625 } 2.0182 2.0045

Table 7.2. Convergence of the field, first mode and TE case, then measured as, (E · ti)(mi) there ti is the tangent on edge i and mi is the midpoint on edge i.

We see from table 7.2 that measuring the field in the midpoint on each edge a second order accuracy is achieved, as reported in [29].

Case 3 Now we look at the convergence in the triangle midpoints where it is measured as,

1 2 N / √1 √ | j x − ref x |2 | j x − ref x |2 εj = Exi ( p) Exi ( p) + Eyi ( p) Eyi ( p) A N 2 i=1 where (xp) is the midpoint in triangle i on mesh j. Figure 7.5 shows first order convergence of the electric and magnetic field. For large N the comparison to the reference solution will generate low accuracy since 38 Chapter 7. Numerical Results for 3-D Waveguides then there are roundoff errors from the reference solution. This is why we see a turn of the two lines in the graph.

Case 4 We now look at the computation of a few more modes, see table 7.3. We see that to compute the lowest modes more then 800 elements are needed to get good accuracy for the wavenumber. Since we are mostly interested in the lowest modes when studying wave propagation in waveguides, the restriction on the mesh is not the wavenumber but the required accuracy in the computed field.

Computed, N = #elements mode N=384 N=800 N=3000 N=7000 N=12600 analytical 1 0.98053 0.98053 0.98053 0.98053 0.98053 0.98053 2 0.91955 0.91972 0.91962 0.91965 0.91966 0.91967 3 0.80743 0.80842 0.80783 0.80800 0.80805 0.80810 4 0.63091 0.62054 0.62257 0.62028 0.61965 0.61900 5 0.61634 0.62054 0.61787 0.61860 0.61879 0.61900 6 0.59839 0.58811 0.59047 0.58827 0.58766 0.58702 7 0.48778 0.47860 0.48142 0.47953 0.47901 0.47847 8 0.19261 0.20077 0.19166 0.19070 0.19044 0.19019

β Table 7.3. Comparison of computed and exact values then k0a =16for the k0 rectangular waveguide.

An investigation on grid dependence is also performed on the rectangular waveguide where we have used a =2b, since then the propagation constant for the second and third mode field are the same. The results are shown in the figures 7.6 and 7.7 where we have plotted the field for the four first propagating modes. This is of greater interest than doing the full 3-D simulation, since we have to run both the second and third mode and look at the sum of transmitted power. 7.1. Homogeneous waveguide 39

First Mode Second Mode 0.03 0.03

0.02 0.02

0.01 0.01

0 0

−0.01 −0.01 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04

Third Mode Fourth Mode 0.03 0.03

0.02 0.02

0.01 0.01

0 0

−0.01 −0.01 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04

Figure 7.6. Four lowest mode fields the using a structured mesh 40 Chapter 7. Numerical Results for 3-D Waveguides

First Mode Second Mode

0.03 0.03

0.02 0.02

0.01 0.01

0 0

−0.01 −0.01 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04

Third Mode Fourth Mode

0.03 0.03

0.02 0.02

0.01 0.01

0 0

−0.01 −0.01 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04

Figure 7.7. Four lowest mode fields the using a unstructured mesh 7.1. Homogeneous waveguide 41

7.1.2 Circular waveguide

y

εr µr x r

Figure 7.8. Geometry of the circular waveguide, r =1, εr =1.0 and µr =1.0.

mode (TE) computed analytical 1 0.98324 0.98291 2 0.98310 0.98291 3 0.95319 0.95222 4 0.95313 0.95222 5 0.92488 0.92367 6 0.90959 0.90748 7 0.90931 0.90748 8 0.85140 0.84687

β Table 7.4. Comparison of computed and exact values then k0r =10and using k0 678 elements for the circular waveguide.

For the circular waveguide we compute the nine lowest modes and compare them to the analytical values. Here we use the fact that for the lowest modes we do not need a very fine triangulation. The geometry for the problem is given in Figure 7.8 and results are shown in table 7.4. As can be seen we have the same number of correct decimals for mode 1to mode 7.

7.1.3 Coaxial waveguide This test case is to show that we can solve for the TEM modes using the edge formulation. But since we do not force both Ez and Hz to be equal to zero at the same time this method is not recommended. The test is performed at a rather high frequency. For this waveguide, the cutoff frequency for the next mode is a little bit over 18GHz, so we have only a TEM mode propagating. For the calculations two different meshes are used, mesh1 with 1256 elements and mesh2 with 5024 elements. 42 Chapter 7. Numerical Results for 3-D Waveguides

The coaxial waveguide has an inner radius of 0.5 mm and an outer radius of 1.5 mm and the material parameters are µr =1.0 and εr =2.04.

mode freq. mesh computed value |βanalytic − βcomp| κ(B)∞ TEM 18GHz 1 538.8237793484 9.970335·10−10 5.56·1012 TEM 18GHz 2 538.8237796979 3.504337·10−7 2.04·1014

Table 7.5.

As can be seen from table 7.5 we obtain an increase in the condition number for the right hand side matrix B, see equation (5.8). This could be the explanation for the accuracy decreases when we do a grid refinement. Since the vector formulation involves solving a system with the conditioned matrix, the potential formulation is better suited for these problems. The TEM solver will be used when we have a homogeneous enclosed stripline, see section 7.3.1.

7.2 Inhomogeneous waveguide

As a first example of an inhomogeneous waveguide, we investigate two dielectric- loaded waveguides.

Case 1 The first test is a comparison against results reported in [42] and [38]. Figure 7.9 shows the setup of the problem, the waveguide is of size 2a × a bounded by a perfect conductor. In table 7.6 and figure 7.10 we compare our computed wavenumbers against those given in [42]. From table 7.6 a good agreement with the exact wavenumber is achieved except for mode three, also in figure 7.10 one see a larger value for mode three. We have no good explanation for this phenomena.

y

a

εr

µr

2a x a

Figure 7.9. Geometry of the dielectric waveguide in case 1. 7.2. Inhomogeneous waveguide 43

N = #elements mode N=800 N=3000 N=9600 matlab code exact [42] paper [42] 1 1.27527 1.27569 1.27573 1.27501 1.27576 1.27102 2 0.97276 0.97140 0.97132 0.97361 0.97154 0.94546 3 0.77595 0.77932 0.77932 0.66732 0.72865 0.67683 4 0.59429 0.59431 0.59398 0.59455 0.59390 0.55718

β Table 7.6. Comparison of computed and exact values then k0a =3for the k0 inhomogeneous waveguide then εr =2.25 and µr =1.0 in lower part and εr =1.0 and µr =1.0 in upper part..

1.5

1 0 β/k

0.5

0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 k0a

Figure 7.10. Dispersion characteristics for the inhomogeneous waveguide, the “dots” is the exact solution.

Case 2 Next, we proceed to estimate the convergence rate for the solution and we use the same method as described in section 7.1.1 for both the estimate on the edges and in the triangle midpoint. Second order of accuracy is obtained if the field is measured in the edge midpoints, see table 7.7. We measure the field in the triangle midpoint only first order of accuracy is achieved. This can be seen in figure 7.12. An example of the mode field and its magnitude are shown in figures 7.13 and 7.14. We see that the field magnitude is greater where we have dielectric material. 44 Chapter 7. Numerical Results for 3-D Waveguides

y

b

b/2 εr µr

2b x

Figure 7.11. Geometry of the dielectric waveguide in case 2.

log ( εj ) 2 εi+1 hvaluesused(10−2) mode 1 mode 2 {2.5, 1.25, 0.625, 0.3125, 0.15625} 1.9952 1.9868 2.0001 2.0142 2.0001 2.0056

Table 7.7. Convergence of the field in the dielectric case, then measured as, (E · ti)(mi) there ti is the tangent on edge i and mi is the midpoint on edge i and f =1.5GHz, 0.2=a =2b =4H, εr =2.0 and µr =1.0 in lower part and εr =1.0 and µr =1.0 in upper part.

1 10

0 10

E εj

−1 10

O(h)

−2 10

−3 10 O(h2)

−4 10 −2 −1 10 h 10

E Figure 7.12. Convergence study of the electric εj field in the triangle midpoint,f = 1.5GHz, 0.2=a =2b =4H, εr =2.0 and µr =1.0 in lower part and εr =1.0 and µr =1.0 in upper part. 7.2. Inhomogeneous waveguide 45

0.05

0.04

0.03

0.02

0.01

0

−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Figure 7.13. The mode field with f =3.2GHz, 0.08 = a =2b =4H, εr =4.0 and µr =1.0 in lower part and εr =1.0 and µr =1.0 in upper part.

0.2

0.05 0.18

0.16 0.04

0.14

0.03 0.12

0.02 0.1

0.08 0.01

0.06

0 0.04

−0.01 0.02 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Figure 7.14. The mode field magnitude with f =3.2GHz, 0.08 = a =2b =4H, εr =4.0 and µr =1.0 in lower part and εr =1.0 and µr =1.0 in upper part. 46 Chapter 7. Numerical Results for 3-D Waveguides

7.3 Enclosed Stripline 7.3.1 Homogeneous Here we study the convergence of the Laplace solver for TEM modes. The convergence is measured in,

1 2 N / √1 √ | j x − ref x |2 | j x − ref x |2 εj = Exi ( p) Exi ( p) + Eyi ( p) Eyi ( p) A N 2 i=1 where (xp) is the midpoint in triangle i on mesh j and A is the black area in figure 7.15. Figure 7.16 shows the convergence rate for the electric field. We have only first order convergence since the derivatives of the potential are evaluated using the basis functions. w b

H

0 a

Figure 7.15. The marked area is used for the convergence test.

−2 10

−3 10

O(h)

εE −4 j 10 O(h2 )

−5 10 −3 −2 −1 10 10O(h) 10

Figure 7.16. Convergence rate for the homogeneous stripline waveguide, 0.2=a = 2b =2w =4H, εr =1.0 and µr =1.0. 7.3. Enclosed Stripline 47

7.3.2 Inhomogeneous The setup for the computations on the closed microstrip is given in figure 7.17.

y

b w

H εr µr

a x

Figure 7.17. Geometry of Microstrip

The convergence is measured in, N √1 √ | j x − ref x |2 εj = Exi ( p) Exi ( p) A N 3 i=1 1/2 | j x − ref x |2 | j x − ref x |2 + Eyi ( p) Eyi ( p) + Ezi ( p) Ezi ( p) where (xp) is the midpoint in triangle i on mesh j and A is the black area in figure 7.15. This example is taken from [12] and the same example has also been solved by [21]. The structure of this problem is showed in figure 7.17. It consists of a rectangular metal waveguide with a microstrip of zero thickness in the center. The microstrip as well as the outer surface is a perfect conductor. Our results agree well with those from [12] as can be seen from figure 7.19. The reason that we do not β2 get the same for 2 =3is that [12] uses a method that generates singular matrices k0 2 β for certain values of 2 . Our results also agree well with those found in [21], the k0 difficulty with singular matrices has been removed due to the use of conforming elements. In figure 7.18 we see that we have first order convergence as it should be when evaluating the field in the triangle midpoints. 48 Chapter 7. Numerical Results for 3-D Waveguides

1 10

0 10

E εj −1 10

O(h) −2 10

εH −3 j 10 O(h2)

−4 10 −2 −1 10 O(h) 10

Figure 7.18. Convergence rate for the stripline waveguide, 0.2=a =2b =2w = 4H, εr =4.0 and µr =1.0 in lower part and εr =1.0 and µr =1.0 in upper part.

3.5

3.4

3.3

3.2

2 3.1 ) 0

3 β/k (

2.9

2.8

2.7

2.6

2.5 0 0.1 0.2 ωH0.3 2 0.4 0.5 0.6 ( c )

Figure 7.19. comparison with results from [12], (the dots). Then εr =4.0, µr = 1.0, a =2b =2w =4H. Chapter 8

Direct Numerical Simulation of 3-D Waveguides

Port excitation

Eigenmode ËcaØØeÖiÒg ÑaØÖiÜ

Synthesizer Maxwell solver Öeg Analyzer

E ÑÓde

solver eÜc

´Øµ

E E iÒ !

´! µ ص ´ Port registration

Figure 8.1. Program execution then doing full 3-D simulation

For 3-D simulations, codes developed in the GEMS project will be used. A schematic picture is shown in figure 8.1. A more detailed description can be found in appendix C. We will perform computations on three different waveguides,

Case 1 TEM computations on a simple coaxial waveguide with εr =2, µr =1 inner radius of 1mm, outer radius of 2 mm and length 35 mm, see figure 8.2. The two ports are located 5 mm from the endpoints. The waveguide is terminated with a absorbing boundary condition, unsplit perfect matching layer (u-pml) cells, see [17]. Comparisons of the scattering matrix with analytical mode fields and with numerical computed mode fields are done.

Case 2 TE computation on a simple rectangular waveguide with three ports and filled with air, see figure 8.3. The dimensions of the waveguide is, width 50mm,

49 50 Chapter 8. Direct Numerical Simulation of 3-D Waveguides

Port 2

Port 1

Figure 8.2. The coaxial used for the TEM simulation, excitation at Port 1 hight 30mm and length 80mm. The waveguide is terminated with a layer of u- pml cells. Comparisons of the scattering matrix with analytical mode fields and numerical computed fields are done.

Mode 1Mode 2 u-pml u-pml

Port 1Port 2 Port 3

Figure 8.3. Excitation at Port 1 and Port 3 (TE10,TE20)

Case 3 For this problem we use a waveguide that is rectangular with a slot and two monopoles, see figure 8.4, close up of the element configuration is shown in figure 8.5. This type of configuration is called a Clavin antenna, the geometry and 51 parameter setup was supplied from, Ericsson Microwave Systems. A more detailed description of Clavin antennas can be found in [15]. The antenna have the following parameter setup. The operating frequency is 4.1GHz to 7.0GHz, the width is 40.386mm, the height 20.193mm and the midpoint of the slot is located 90mm from the ports, 7mm from the waveguide centerline and has the dimensions, (30mm×3mm). The two monopoles have a radius of 0.5mm and a length of 22.88mm and are located symmetric on the sides of the slot 10.37mm apart. The waveguide is terminated with a layer of u-pml cells.

Figure 8.4. The setup for the Clavin antenna

For case 1, a finite-element code will be used for the 3-D simulation. In case 2, a finite-difference code will be used and in case 3, we will use finite-difference together with finite-element for the simulation. By convention we use the normalization, E × H∗dΩ=1 Ω where Ω is the cross section of the waveguide and H∗ is the complex conjugate of H. Here we will briefly describe the excitation and registration done in the 3-D codes. The description are based on the one given in the GEMS document [41]. Excitation for homogeneous waveguides are done in the following way, we look only at TE modes. Let em denote the tangential, ezm the longitudinal electric mode field and βm the propagating constant for mode m. The mode field can be normalized so that it is constant for the whole frequency interval. So we need only one em and all βm. Then a pulse f(ω) is specified and the tangential part, Etm,in the port is excited with, ∞ −jβmz −jωt Etm(t, x, y, z)=em(x, y) f(ω)e e dω, −∞ since Ez =0for TE modes. For inhomogeneous waveguides we can not do the normalization of the mode field so that it is constant for the frequency interval. 52 Chapter 8. Direct Numerical Simulation of 3-D Waveguides

Figure 8.5. Close up on the element configuration around the slot for the Clavin antenna

This yields ∞ −jβmz −jωt Em(t, x, y, z)= f(ω)(em(ω,x,y)+ˆzezm(ω,x,y,z))e e dω. −∞ For registration in homogeneous ports, we use E · e − Ω( t m)dΩ Vm = e · e Ω( m m)dΩ and for inhomogeneous ports E · j − Ω( t m)dΩ Vm = e · j Ω( m m)dΩ where jm =ˆz × hm and hm is the tangential magnetic mode field.

8.1 Coaxial waveguide

We will run this test case using two different meshes, This give rise to the two 8.1. Coaxial waveguide 53

mesh in 3D mesh1 mesh2 #unk 107,478 325,122

meshed port in 2-D mesh1 mesh2 #unk 184 1108 different meshes for the waveguide port, The time to run the 2-D meshes is under one minute so that is not a big issue, but if we look at the time it takes for the 3-D problem, it takes.

runtime mesh1 mesh2 time[s] 4787 23368

That is the larger mesh takes about five times as long to run as the smaller mesh. From figures 8.7 and 8.8 we see that mesh1 is to coarse to give a good mode field for computing the scattering parameters, even when using analytical mode fields. The large values at the endpoints, the low and high frequencies seen in figures 8.7 and 8.8 are due to the low power sent into the waveguide, see figure 8.6.

0.1

0.09

0.08

0.07

0.06

0.05 "power"

0.04

0.03

0.02

0.01

0 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 10 f(Hz) x 10

Figure 8.6. Powersentfromport1 54 Chapter 8. Direct Numerical Simulation of 3-D Waveguides

S , [num] 20 11 S , [num] 21 S , [ana] 11 S , [ana] 10 21

0

−10

−20 dB

−30

−40

−50

−60 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 10 f(Hz) x 10

Figure 8.7. Scattering parameters for mesh1, [ana] = analytical mode field and [num] = numerical mode field

20 S , [num] 11 S , [num] 21 S , [ana] 11 10 S , [ana] 21

0

−10 dB

−20

−30

−40

−50 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 10 f(Hz) x 10

Figure 8.8. Scattering parameters for mesh2, [ana] = analytical mode field and [num] = numerical mode field 8.2. Rectangular waveguide 55

8.2 Rectangular waveguide

As stated before, in this test case the setup is a rectangular homogeneous waveguide with three ports, two that send and one that only measures. From port one we send mode one (TE10)andfromport3wesendmodetwo(TE20). In figure 8.9 we see how much of mode 1that is coming back through port 1and port 2. As can be seen, we are over the cutoff frequency, 4GHz for the first mode. Figure 8.10 shows how much of mode 1that goes from port 1to port 2. For mode 2 we see in figure 8.11 that we have a low reflection when over the cutoff and figure 8.12 shows how much is coming through port 1and port 2 from the sending port 3. From the figure we see that we have approximately the same scattering param- eters for the computed mode field as then we use analytical fields.

S mode no. 1 analytic 11

0 Port−nr−1 Port−nr−2

−20 dB −40

−60

2 3 4 5 6 7 8 9 10 11 f (GHz) S mode no. 1 modesolver 11

0 Port−nr−1 Port−nr−2

−20 dB −40

−60

2 3 4 5 6 7 8 9 10 11 f (GHz)

Figure 8.9. The above figure is then analytical mode solution is used and the lower then computed is used 56 Chapter 8. Direct Numerical Simulation of 3-D Waveguides

S mode no. 1. Port−nr−1 to Port−nr−2 analytic 21 10

0

−10

−20 dB

−30

−40

−50 2 3 4 5 6 7 8 9 10 11 f (GHz) S mode no. 1. Port−nr−1 to Port−nr−2 modesolver 21 10

0

−10

−20 dB

−30

−40

−50 2 3 4 5 6 7 8 9 10 11 f (GHz)

Figure 8.10. The above figure is then analytical mode solution is used and the lower then computed is used

S mode no. 2. Port−nr−3 analytic. 11 10

0

−10

−20 dB

−30

−40

−50 7 7.5 8 8.5 9 9.5 10 10.5 11 f (GHz) S mode no. 2. Port−nr−3 modesolver. 11 10

0

−10

−20 dB

−30

−40

−50 7 7.5 8 8.5 9 9.5 10 10.5 11 f (GHz)

Figure 8.11. The above figure is then analytical mode solution is used and the lower then computed is used 8.2. Rectangular waveguide 57

S mode no. 2. Port−nr−3 to Port−nr−1 and Port−nr−3 to Port−nr−2 analytic. 21 10

0

−10 Port−nr−3 to Port−nr−1 −20 dB Port−nr−3 to Port−nr−2 −30

−40

−50 7 7.5 8 8.5 9 9.5 10 10.5 11 f (GHz) S mode no. 2. Port−nr−3 to Port−nr−1 and Port−nr−3 to Port−nr−2 modesolver. 21 10

0

−10 Port−nr−3 to Port−nr−1 −20 dB Port−nr−3 to Port−nr−2 −30

−40

−50 7 7.5 8 8.5 9 9.5 10 10.5 11 f (GHz)

Figure 8.12. The above figure is then analytical mode solution is used and the lower then computed is used 58 Chapter 8. Direct Numerical Simulation of 3-D Waveguides

8.3 Clavin antenna

This is a test case from Ericsson Microwave Systems (EMW) for validation of the code. We use FD for the whole domain except around the slot where FEM are used. In figure 8.13, S11 show how much that is coming back to the sending port and S12 how much that is coming from port 1to port 2. The dip at 5.25Ghz is the radiation out from the slot.

S−parameters for the Clavin antenna 5 S 11 S 21 0

−5

−10

−15 dB −20

−25

−30

−35

−40 4 4.5 5 5.5 6 6.5 7 f (GHz)

Figure 8.13. Scattering parameters for the first TE mode in the Clavin antenna Chapter 9

Conclusion

A finite element system for waveguide analysis has been developed. We have ver- ified our implementation in different types of test cases, and got satisfactory re- sults. We achieved second order convergence for the edge element formulation in the homogeneous case, satisfying the theoretical results. In the other two cases have we verified our simulations with other results reported in the literature and archived good agreement. We have also showed that we have grid convergence for the inhomogeneous case both with and without a stripline. The program has then successfully been used in full 3-D time domain simulations and for these we have showed that we arrive approximately at the same results as when analytical modes are used. Future work includes, parallelizing the code and introduce an iterative solver to handle larger systems in the Arnoldi process. Higher order elements should be used to increase the accuracy as well as other types of boundary conditions to be able to compute on e.g. open microstrips. Filtering to remove the zero eigenvalues the searching for the eigenmodes so that modes under cutoff can easily be found. We would also like to be able to handle anisotropic materials in the waveguide.

59 60 Appendix A

Element Matrices

We have these definitions,

ai = xi+1yi+2 − xi+2yi+1

bi = yi+1 − yi−1

ci = xi−1 − xi+1 where A represents the area of the triangle, xi,yi are the corner coordinates for the triangle and the indexes i assumes the values 1,2 and 3 cyclically, so that if i =3 then i +1=1.

A.1 Laplace

The node element matrix can be expressed as,

1 (A ) = (b b + c c ) mn e 4A n m n m and the right hand side b vector will contain

−Ω1 b = (b b + c c )δ m 4A n m n m n where δn is equal to one if node n is on Ω1. Note that if we have a node on the boundary Ω1 then corresponding row and column is removed from the elemental matrix.

61 62 Appendix A. Element Matrices

A.2 Helmholtz

The edge element matrices can be expressed as,

tt ωmωn − − (Amn)e = 3 (bm+1cm+2 bm+2cm+1)(bn+1cn+2 bn+2cn+1) (A.2a) 4A µr 2 2 2 ωmωn ij −k ε α γ (b +3− b +3− + c +3− c +3− ) 0 r 2A ij mn m i n j m i n j i=1 j=1 2 2 tt 2 ωmωn ij (D ) = k0 α γ (b +3− b +3− + c +3− c +3− ) (A.2b) mn e 2Aµ ij mn m i n j m i n j r i=1 j=1 2 zz 1 − k0εrA (Dmn)e = (bnbm + cncm) σmn (A.2c) 4Aµr 12 tz ωm − − (Dmn)e = 2 (bnbm+2 + cncm+2 bnbm+1 cncm+1) (A.2d) 24A µr where 1 if i = j α = ij −1 otherwise 1 ij 12 if m + i = n + j γ = 1 mn otherwise 24 2 if m = n σ = mn 1 otherwise Appendix B

Algorithm Description

B.1 Laplace

This is a schematic description of the different steps performed in the TEM case.

Algorithm 3 Laplace Solver 1: Read in the geometry 2: Assemble the main matrix A and r.h.s vector b 3: Reduce the bandwidth of the matrix A with reverse Cuthill-McKee reordering solve A with routines from LaPack 4: calculate the electric and magnetic field for all frequencies

63 64 Appendix B. Algorithm Description

B.2 Helmholtz

This is a schematic description of the different steps performed in the program then solving for the propagation constant β.

Algorithm 4 Helmholtz solver 1: Read in the geometry 2: Assemble the submatrices 3: for all frequencies do 4: Assemble the main matrices A and B 5: Reduce the bandwidth of the matrix B with reverse Cuthill-McKee reordering LU factorize B with routines from LaPack 6: while not converged do 7: call arpack routine 8: perform Ax → b 9: solve By = b, with LaPack routines 10: end while 11: calculate the electric and magnetic field 12: end for Appendix C

The Gems Project

The research in this thesis has been conducted in the GEMS project at PSCI[33] Formally, PSCI is a center of excellence funded by an industrial consortium, VIN- NOVA, KTH, and Uppsala University. The industrial partners involved include large Swedish companies as well as smaller enterprises. GEMS is an abbreviation for General Electromagnetic Solvers. The project was one of the larger projects at PSCI and The partners in the project are Royal Institute of Technology (KTH), Uppsala University (UU), Chalmers University of Technology (CTH), The Institute for Applied Mathematics (ITM), The Defense Re- search Establishment (FOA), Ericsson Microwave Systems (EMW), Saab Ericsson space (SES) and Ericsson Saab Avionics (ESB). The main objective was to develop two hybrid codes one in time domain (TD) [2], [25], [13] and one in frequency domain (FD) [14], [31], [5], [20], [37] The TD hybrid codes are mainly funded by the National Aeronautical Research Program (NFFP). A small part is funded by PSCI. The FD codes are funded by PSCI. The TD code is a multi-block and out of core solver based on Finite-Differences (FDTD) on structured grids, explicit Finite Volumes (FVTD) and implicit Finite Elements (FETD) on unstructured grids. The code uses unstructed grid to resolve small geometrical details and curved boundaries but go back to structured grids for the rest of the computational domain. The FD hybrid code is based on the Method of Moments (MoM), Physical Optics (PO) and Geometrical Theory of Diffraction (GTD). Since the MoM has a com- putational complexity that increases fast with higher frequencies, it is hybridized with PO and GTD for medium and high frequency problems. In microwave systems it is common to use waveguides as transmission media for instance to feed antennas. Under certain conditions the waveguide problem can be solved separately and the solution be used as data in the full simulation. For this reason software was developed to be used by both types of codes. The code is

65 66 Appendix C. The Gems Project based on Finite Elements in the frequency domain ant interacts with the TD and TD codes as described in figure C.1 A 3-D hybrid grid of the waveguide geometry is created. Virtual, plane sur- faces or ports which are orthogonal to the direction of propagation are used for excitation and registration of waveguide modes. Then the frequency-domain, 2-D finite-element solver is used to calculate the mode field and propagation constant in the ports. The solution is transformed to time-domain and used as excitation in the full simulation by the Huygens surface technique. For the frequency domain there is no need to transform the solution, instead the magnetic currents are given in each port and used for excitation. In the present time the GEMS project has ended but have given rise to the continuation projects, GEMS2 and GEMS3, see [33] for more information.

Figure C.1. Interaction between the codes in the GEMS project Bibliography

[1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions.Dover, 1972.

[2] Ulf Andersson. Time-Domain Methods for Maxwell Equations.PhDthesis, Royal Institute of Technology, 2001. Dissertation.

[3] Javier Arroyo and Juan Zapata. Subspace iteration search method for gener- alized eigenvalue problems with sparse complex unsymetric matices in finite- element analysis of wavegudes. IEEE Trans. Microwave Theoryand Tech- niques, Vol. 46(8), 1998.

[4] Petter Björestad Barry Smith and William Gropp. Domain Decomposion, Par- allel Multilevel methods for Elliptic Partial Differential Equations. Cambridge University Press, 1996.

[5] F. Bergholm, S. Hagdahl, and S. Sefi. A modular approach to gtd in the context of solving large hybrid problems. In AP2000 Millennium Conference on Antennas and Propagation, 2000. Davoz, Switzerland.

[6] S. C. Brenner. An additive schwarz preconditioner for the feti method. Num. Math, Vol. 94:1–31, 2003.

[7] Susanne C. Brenner and L. Ridgeway Scott. The Mathematical Theoryof Finite Element Methods. Springer, 1994.

[8] Andrea Toselli. Caroline Lasser. An overlapping domain decomposition pre- conditioner for a class of discontinuous galerkin approximations of advection- diffusion problems. Comp. of Math., Vol. 72:1215–1238, 2003.

[9] Cristian A Coclici and Wolfgang L. Wendland. Analysis of a heterogeneous domain decomposition for compresible viscous flow(seen only summary). Math. Models Method Appl. Sci, no. 4:565–599, 2001.

[10] Robert E. Collin. Field Theoryof Guided Waves . Oxford, University Press, 1991.

67 68 Bibliography

[11] Robert E. Collin. Foundation for Microwave Enginering. McGraw-Hill, 1992. [12] P. Daly. Hybrid-Mode analysis of microstrip by finite-element methods. IEEE Trans. Microwave Theoryand Techniques , Vol. 19(1), 1971. [13] Fredrik Edelvik. Hybrid Solvers for the Maxwell Equations in Time-Domain. PhD thesis, Upsala University, 2002. Dissertation. [14] J. Edlund. A parallel, iterative method of moments and physical optics hybrid solver for arbitrarysurfaces . PhD thesis, Upsala University, 2001. Dissertation. [15] Robert S. Elliott. On the mutual admittance between clavin elements. IEEE Trans. on Antennas and Propagation, Vol. 28(6), 1980. [16] Hugo J. W. M. Hoekstra Jorn H. Povlsen Frank Wijnands, Thomas Rasmussen and Rene M. de Ridder. Efficient semivectorial mode solvers. IEEE Journal of quantum electronics, Vol. 33(3), 1997. [17] S. D. Gedney. An anisotropic pml absorbing media for fdtd simulation of fields in lossy and despersive media. Electromagnetics, Vol. 16(4):399–415, 1996. [18] Gene H Golub and Charles F. Van Loan. Matrix Computations, third Edition. Johns Hopkins, 1996. [19] Gundolf Haase. A parallel amg for overlapping and non-overlapping doamin decomposition. Electronic Trans. on Numerical Analysis, Vol. 10:41–55, 2000. [20] S. Hagdahl and B. Engquist. Numerical techniques for mixed high-low fre- quency em-problems. In Proceedings RVK 02, June 2002. [21] Mitsuo Hano. Finite-Element analysis of dielectric-loaded waveguides. IEEE Trans. Microwave Theoryand Techniques , Vol. 32(10), 1984. [22]A.M.ErismanI.S.DuffandJ.K.Reid.Direct Methods for Sparse Matrices. Clarendon Press Oxford, 1992. [23] D-K. Sun. Z. J. Cendes J-F. Lee. Full-Wave analysis of dielectric waveguides using tangential vector finite elements. IEEE Trans. Microwave Theoryand Techniques, Vol. 39(8), 1991. [24] J.D. Jackson. Classical Electrodynamics, second edition. John Wiley & Sons, 1975. [25] Gunnar Ledfelt. Hybrid Time-Domain Methods and Wire Models fo Compu- tational Electromagnetics. PhD thesis, Royal Institute of Technology, 2001. Dissertation. [26] Pao-Lo Liu and B. J. Li. Full-vectorial mode analysis of rib waveguide by iterative lanczos. IEEE Journal of quantum electronics, Vol. 29(12), 1993. Bibliography 69

[27] O. B. Widlund M. A. Casarin. Overlapping schwarz methods for helmholt’s equation. Eleventh Int. Conf. on Domain Decomposition Methods, pages 178– 189, 1999.

[28] Jacek Mielewski and Michal Mrozowski. Application of the arnoldi method in fem analysis of waveguides. IEEE Microwave and guided wave letters,Vol 8(1), 1998.

[29] P. Monk. An analysis of nedelec’s method for spatial discretization of maxwell’s equations. Journal of Computational and Applied Mathematics, 47:101–121, 1993.

[30] J. C. Nedelec. Mixed finite elements in r3. Num. Math, Vol. 35:315–341, 1980.

[31] M. Nilsson. Iterative solution of Maxwell’s equations in frequencydomain .PhD thesis, Upsala University, 2002. Dissertation.

[32] J.-I. Lions P. Gervasio and A. Quarteroni. Heterogeneous coupling by virtual control methods. Num. Math, Vol. 90:241–264, 2001.

[33] The Parallel and PSCI Scientific Computing Institute. http://www.psci.kth.se. 2003.

[34] David M. Pozar. Microwave Engineering. John Wiley & Sons, 1998.

[35] Danny Sorensen Rich Lehoucq, Kristi Maschhoff and Chao Yang. http://www.caam.rice.edu/software/ARPACK/. Rice University, 2001.

[36] Yousef Saad. Iterative Methods for Sparse Linear Systems ,Second Edition. Yousef Saad, 2000.

[37] Sandy Sefi. RayTracing Tools for High FrequencyElectromagnetics Simula- tion. PhD thesis, Royal Institute of Technology, 2003. Licentiate Thesis.

[38] Jan A. Svedin. A numerically efficient finite-element formulation for the general waveguide problem without spurious modes. IEEE Trans. Microwave Theory and Techniques, Vol. 37(11), 1989.

[39] T. Kano T. Kako. Numerical simulation of wave propagation phenomena in vocal tract and domain decomposition method. Eleventh Int. Conf. on Domain Decomposition Methods, pages 268–273, 1999.

[40] Jilin Tan and Guangwen. A new edge element analysis of dispersive waveguid- ing structuers. IEEE Trans. Microwave Theoryand Techniques , Vol. 43(11), 1995.

[41] Lars Pettersson Torleif Martin. Excitering och registrering av vågledare i fem. GEMS/FOA, 00-08-14, Utgåva 2.1, 2000. 70 Bibliography

[42] Masanori Matsuhara Tuptim Angkaew and Nobukai Kumagai. Finite-Element analysis of Waveguides Modes: A novel approach that eliminates spurious modes. IEEE Trans. Microwave Theoryand Techniques , Vol. 35(2), 1987. [43] I. I. Dyyak Y. H. Savula and V. V. Krevs. Heterogeneous mathematical models in numerical analysis of structures. Comp. and Math. with applications,Vol. 42:1201–1216, 2001.