An Improved Adj oint Method for Design of Microwave Devices

with 3D Finite Elements

Hatem Akel

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Abstract

A numerical method is described for detPrmining the derivatives of the scattering

parameters of an arbitrary 3D microwave device with respect to changes in mode1

geometry. The new technique is based on the adjoint variable method as applied to h i t e

element analysis, but does not require the separate calculation of an adjoint solution. The

new technique is applied to the optixnization of the design of microwave components,

using a derivative-based, quasi-Newton methad.

The denvatives of the scattering parameters of three devices are computed and

compared to analytical or finite ciiffience values. The agreement is very good. The three

devices are: a length of short-circuited waveguide, an E-plane miter-bend, and a

waveguide transformer.

The short-circuited waveguide length is then optimized to achieve a specified phase of

the reflection coefficient. The miter-bend and the transformer are optimized for minimum

r e m loss. Al1 optimizations reached the (local) optimum solution after only 4-12

computations of the field.

Une méthode numérique pour calculer les dérivées de la matrice de dispersion d'un

circuit micro-onde 3D en fonction des paramètres géométriques est presentée.

Sirnilairement à la méthode des éléments finis la nouvelle technique fait appel à la

methode de la variable adjointe, mais ne requiert pas le calcul de la solution adjointe. La

nouvelle technique est utilisée conjointement avec une méthode quasi-newton basée sur

les dérivées pour concevoir des composantes micro-ondes optimales.

Les dérivées des paramètres de dispersion pour trois composantes sont ainsi calculées et

comparées, selon la disponibilité des résultats, à des valeurs analytiques ou à des résultats

de différences finies. Une très bonne corrélation entre les différents résultats a été

observée. Les trois composantes sont : un guide d'onde terminé par un court-circuit, un

coude à onglet dans le plan E et un transformateur guide d'onde.

Le guide d'onde court-circuité a été optimisé en fonction de la phase du coefficient de

réflexion. Les deux autres composantes ont, quand à elles' été optimisées pour minimiser

les pertes par réflexion. Toutes les simulations ont atteint la solution optimale (Iocale)

après seulement 4 à 12 itérations.

Table of Contents

Abstract ............................................................................................ " Resume .. .................................................................. iü ..........................

................................................................................. Table of Contents * ..................................................................................... List of Figures vi

CHAPTER 1 Introduction ..................................................................... 1

......................................... CHAPTER 2 Computiug the Scattering Parameters .................................................................................... 2.1 Introduction

.......................................................................... 2.2 Maxwell's Equation ............................. ....................................... 2.3 Boun&uy Conditions ...

.................................................. 2.3.1 Dirichlet Boundary Condition for E ................................................. 2.3.2 Ne~~~larm Boundary Condition for E

..................................................... 2.3.3 Absorbing Bouadary Conditions ............................................................. 2.3.4 Port Boundary Conditions

.................................................................. 2.4 Weak Fonn of the Problem ............................................................... 2.5 Finite Element Discretization

............................................................................. 2.6 Scatterhg Ma& ........................................................................... 2.7 Propagation Modes

.............................. ......*.........*............ 2.6.1 Rectangular Waveguide ... .................................................................... 2.6.2 Circular Waveguide

CHAP'IER 3 The Gradient of Scatterhg Parameters ........................................ 23 3.1 uitroduction ................................................................................... 23 3.2 The Adjoint Method .......................................................................... 23 3.3 The Nature of the Adjoint Method .................................... .. ................... 25 3.4 Evaluahg the Derivative of ............................................................ 27

.......................................... CHAPTER 4 Optimization of Microwave Devices ............................................................ ..................... 4.1 Introduction ...

..................................................... 4.2 Cost Function of Microwave Devices .......................................................... 4.3 Conditions for a Locd Minimum

........................................ 4.4 Multivariate Gradient Methods of Minimization ..................................................................... 4.5 Quasi-Newton Methods

.................................................................................... 4.6 Line Search .................................................. 4.6.1 Establishing Bounds for the Search

................................................................... 4.6.2 Locating a Minimum

CHAPTER 5 Cornputer Program Implementation ........................................... 40 5.1 Introduction .................................................................................... 40 5.2 Program Structures ........................................................................... 40

5.2.1 Quasi-Newton Optimizer ............................................................... ............................................................................... 5.2.2 Line Se&

5.2.3 Me& Generator .......................................................................... ..................................... 5.2.4 Nodal-to-Edge Converter ... ............

5.2.5 3D Finite Element Solver ............................................................... 5.2.6 Remodeb ...............................................................................

.................................. 5.3 Data Files .. ................................................ 5.3.1 Geo.dat .......................................... ..... ...............

..................................................................................... 53.2 2D.dat .................................................................................. 5.3.3 Input-dat

5.3.4 Ports.dat .................................................................................. .................................................................................. 5.3.5 Matsdat ................................................................................. 5.3.6 Param.dat ............................................................................... 5.3.7 Tracing.log

.................................................................................... 5.3.8 Tetsdat .................................................................................. 5.3.9 Edges.dat

5.3.10 Resultsdat ..............................................................................

............................................................................. CHAPTER 6 Results ................................................................................... 6.1 Introduction

.................................................. 6.2 Short-circuited Rectangular Waveguide 6.2.1 Verifjing the Gradient of LS,, ........................... .. .................................

............................................................................ 6.2.2 Optimizaiion ............................................................ 6.3 E-plane Miter Bend Waveguide

6.3.1 Venfying the Gradient ................................................................ ............................................................................ 6.3.2 Optimization

....................................... 6.4 Single Stage Waveguide Impedance Transformer 6.4.1 Verifjing the Gradient .................................................................

............................................................................ 6.4.2 Optimization ............................................................... .................... 6.5 Summary ..

CHAPTER 7 Conclusion ........................................................................ 66 7.1 Summary ....................................................................................... 66 7.2 Original Contributions ....................................................................... 67 7.3 Suggestions for Further Work .............................................................. 67

APPENDIX A: Matrix Assembly for 3D Edge Elements ................................... 69

APPENDIX B: Quasi-Minimal Residual Method ............................................ 72

APPENDIX C: Derivations ...................................................................... 74

References ........................... .. .. .. ....................................................... 83

Table of Figures

.......................................... Figure 2.1 : Closed and Open Suspended Strip Line ....................................................... Figure 2.2 : An N-port Microwave Device

Figure 2.3 : a is a Unit vector Outward h m the Device .................................... Figure 2.4 : 2D Elements ........................................................................ Figure 2.5 : 3D Elements ......................................................................... Figure 2.6 : (a) Rectangular Waveguide (b) Circular Waveguide .......................... Figure 5.1 : Representative Flow Chart of the Relation

.................................................. Between the Six Main Modules Figure 6.1 : Rectangular Waveguide Short-Circuited at a Distance L nom the Port ... Figure 6.2 : Emr in % of the Phase of SI* and its Gradient ................................ Figure 6.3 : Enor in the phase of Sir Versus Solution Number ............................ Figure 6.4 : Cost Function in the Vicinity of the Optimum Solution ......................

................................................................ Figure 6.5 : E-pIane Miter Bend Figure 6.6 : Displacement of the Mesh Nodes Located on the Surface

of the Inclineci Wall ............................................................... Figure 6.7 : % Error in the Gradient of ISfIl of the miter-bend example ................... Figure 6.8 : Miter-Bend at Two Different Positions ......................................... Figure 6.9 : Miter-Bend Position as a Function of the Solution Number ...................

.............................................. Figure 6.10 : Cost (dB) Versus Solution Number Figure 6 . 1 1 : Rehim Loss of the Optimum Miter-Bend ..................................... Figure 6.12 : Single Stage Rectangdar Transfomer .......................................... Figure 6.13 : % Error in the Gradient of lSfIl ................................................. Figure 6.14 : The Cost (dB) as a Function of the Solution Number ......................... Figure 6.15 : Retum Loss of the Initial and the Optimized Geometry ......................

CHAPTER 1

INTRODUCTION

Many modem communications and radar systems employ microwave devices, e.g.

antennas, filters, polarizers, phase shifters, circulators, and diplexers. Over the 1st

decade, the application of high fkquency devices has grown very rapidly, and this has

generated a tremendous demand for accurate and easy-to-use analysis and design

software.

The performance of any microwave device can be described in terms of a scattering

matrix. The scattering matrix is an N x N matrix relating the input and output waves at the

N ports of the device. Atthough the scattering parameters of some microwave devices can

be obtained as closed form expressions, which can be used to design and optimize them,

in general this is not possible. Some components can be modeled using a lumped-element

circuit. However many of these circuits turn out to be poor approximations, especially for

wide-band applications. The finite element method (FEM) is one of many field-based

numencal techniques which can analyze an arbitrarily-shaped device and obtain very

good results compared to measurements [Il. However, a drawback of FEM even with

advanced computers and improved algorithms, is still its execution time. This problem is

worse when FEM is used in an optïmization routine where rnany solutions are required.

Consequently, it is important to h d ways to reduce the number of so1utiom needed to

optimize a microwave device.

This dissertation descnbes a method of calculating the derivative (or sensitivity) of the

@ scattering parameters of a microwave device with respect to changes in its geometry. The

0 calculations are obtained directly fkom the hnite elernent solution of a boundary value problem for the electric field. These denvatives are then used in an optimization routine,

which aims to irnprove the design of the device.

Lee and Itoh [2] âemonstrated in their paper the adjoint method where, for an N-port

problem, 2N solutions are required to calculate unIimited numbers of ( k t ) derivatives of

al1 the scattering parameters. Garcia and Webb [3] presented fonnulae to calculate the

derivatives of the admittance parameters of 2D microwave models using only N

solutions. The derivatives of the scattering parameters are denved fkom the denvatives of

the admittance parameters. In this dissertation, new fomiulae are presented for the

derivatives of the scattering parameters directly, and for 3D problems, using only N

solutions [4]. These derivatives are then used in an optimization scheme.

Optimization techniques can be classined into two main categories: stochastic, and

detenninistic. Stochastic techniques search the space of the unknowns randomly, seekhg

a global minimum of the cost (objective) hction. Deterministic methods generdly

follow a trajectory in search space, but often only find the local minimum that is closest

to the starting point. There are two subclasses of deterministic methods: direct, and

gradient, depending on whether they exploit the derivative of the cost funetion. The

reader can refer to [SI for an exhaustive review of optimization techniques.

Stochastic methods are best used with simple problems which have analytical

solutions, or can be solved in very short of time, because they require very large numbers

of cost fünction evaluations. There are many techniques that can be classified as

stochastic, e.g. random search. evolutionary strategy, and genetic algorithms [6].

Rechenberg [7] was the b t to use the evolution technique in 1973. Amdt et al.[8-181 in

theu mode-matching papers used the evolution technique to optimize all kinds of

microwave components, e.g. filters, polarizers, diplexers, couplers, and orthornode

transducers. This was a successful approach because in mode m a t c b g the gradients

cannot be extracteci easily, but on the other hand the analysis time is very short. Zhang

and Weiland [19] combined the evolution method with detemiinistic gradient techniques

to optimize waveguide-to-microstrip transitions. Holland [20] was the h t to use genetic

algorithms in 1992, and since then many papers have appeared on using genetic

algorithms for low and high fiequency electromagnetic applications. John and Jansen

[21] used the genetic algorithm to design and optimize M(M1C) components, Jones and

Joines 1221 used the genetic technique to design Yagi-Uda antenua, and Altshuler and

Linden 1231 applied it to a loaded monopole.

Direct techniques are used when no gradient information is available, or when the

target function is discontinuous. Some direct methods rely on pattern hding approaches

such as the simplex method, Hooke and Jeeves method, and Rosenbrock pattern search.

Other direct methods seek conjugate directions, such as Powell's method. Okoshi et al.

(241 used Powell's method to design and optimize a hybrid ring directional coupler.

Miyoshi and Shinhama 1251 used Powell's method to design planar circulators, and Aloss

and Guglielmi [26] used it also to design and optimize microwave filters.

The direct methods are considered inefficient techniques, due to the fact that they do

not make use of gradient (denvative) information. Examples of gradient methods are:

steepest descent, Newton Raphson, quickprop, quasi-Newton, and gradient associated

conjugate direction (GACD). The earliest literature regarding the use of gradients goes

back to Hadamard [27] in 1910. Lee et ai. [28] used the steepest descent method to design

ridged waveguides, and Rakanos 1291 used the quickprop technique for low frequency

applications. Garcia and Webb [3] [q used quasi-Newton techniques to optimize

rectanguiar waveguide H-ben& and circulators. Suzuici and Hosono [30] used the quasi-

Newton method to optimize the cross section of a waveguide giving the minimum

conductor loss for the dominant mode, and Lee and Itoh [2] used the quasi-Newton

methods to optimize waveguide-to-microstrip transitions. Zhang [31] 1191 proposed the

GACD method which he claims behaves better than the quasi-Newton methods, for some

cases.

Chapter 2 introduces the basic electromagnetic formulae and boundary conditions

encountered in microwave problems. The weak form is given as an alternative to solving

the vector wave equation in ternis of the electric field. The finite element modeling is

then presenteâ, and the formula used to calculate the scattering parameters is introduced.

In Chapter 3, the adjoint method is presenteâ, and its main f o d a e are derived. Then

it is shown how the gradients of the scattering parameters can be obtained with no

separate adjoint solution. Finally, we present the details of calculating the gradients for

tetrahedrai edge elements.

Chapter 4 presents a general cost fùnction and its gradient with respect to a

geometrical parameter. Then, a brief description of the mathematical background of some

gradient methods of optimization are reviewed. The chosen gradient method (BFGS) is

presented in detail.

The computer program implementation is outlined in Chapter 5. The different data

files used in the program are àiscussed bnefly.

The results are given in Cbapter 6. In the first example the accuracy of the calcuiations

of the gradient is venfied, by cornparison with the analytical solutions. In the second

example, a onedimensional optimization at a single Erequency is presented. In the last

example, a mdtidimensional optimization over a range of fiequencies is considered.

The dissertation is concluded in chapter 7. Some recommendations for fùture work in

the microwave domain are given.

CHAPTER 2

Computing the Scattering Parameters

2.1 Introduction

The main objective of this chap ter is to summarize the application of the finite element

method to finding the network parameters of microwave devices. Finite element methods,

or other numencd analysis techniques, are used because most real problems do not have

analytical or closed form solutions. The numerical analysis of any microwave device c m

be formulated into a set of linear equations. This set of linear equations can then be

solved directly or iteratively to get the field distribution, and consequently, the scattering

parameters. This chapter presents the electromagnetic fomulae, and how they are

translated into a set of linear equations, which provide the means to calculate the field

distribution. We also present the necessary fomulae used to compute the scattenng

parameters from the field solution.

In section 2.2 we give Maxwell's equations, and in section 2.3 the most cornmonly

used boundary conditions are described. Using Maxwell's Equations and the boundary

conditions the weak fom of the mathematical problem is constructed in section 2.4. In

section 2.5 the finite eiement modeling is descnbed. In section 2.6 the scattering matrix

and the fomulae used to calculate it are presented, and in section 2.7 we discuss some

aspects of the theory of modes in guided structures.

2.2 Y axwell's Equations

Electromagnetic phenornena are govemed by a set of differential equations known as

Maxwell's Equations. For a wave travelling in a fiee space or in a guided device, with

Linear, isotropie, sourceless, and homogeneous or hhomogeneous media, the electncal

(E) and magnetic CR) field must satisfy the following four relationships [32]:

dH VxE=-p- (2. la) at

where o, p and E' are the conductivity, pemeability and pemiitivity of the media of

concem, respectively. When a tirne hamonic electromagnetic wave is assurned, complex

phasors can be introduced, i.e.,

where o is the angular fkequency of the wave.

Maxwell's equations can then be rewritten as:

P x E = -jco,uH

where B is the complex permitivity:

With some mathematical manipulation, one c m combine the four formulae of (2.3) to

obtain one formula with E or H as the unknown. The new fonnula is called the Curl-Curl

Equation:

where k is the fiee space wave number:

c is the speed of light in fiee space. pr and gr are the relative pemeability and

permitivity of the media. A similar formula For the H-field could be derïved. However, in

this thesis, only the E-field formula (2.5) is considered.

2.3 Boundary Conditions

The finite element method can only analyze fmite volume models. Consequently,

whether we are solving a guided structure, or an antenna problem, the device must be

enclosed by a boundary (Figure 2.1).

Four types of boundary condition are commonly encountered in guided microwave and

antema problems: Dirichlet, Neumann, Absorbing, and Port.

2.3.1 Dirichlet Boundary Condition for E

This condition constrains the tangentid component of the E-field over the surface:

E x n = E , (2-7)

where n is the unit vector normal to the surface. If Eo is equal to zero, we obtain the

homogeneous Dirichlet boundaiy conditions. This case is imposed on surfaces which are

touching perfectly conducting materials. The Dirichlet Boundary Conditions can also be

applied on some syrnmetrïcal planes where the tangential components of the E-field, due

to symmetry, must be equal to zero.

Fictit ious Boundary \

Figure 2.1 : (a) Closed and (b) Open Suspended Stnp Line. In (b) a fictitious bounâary was introduced to

tntncate the infinite Space,

2.3.2 Neumann Boundary Condition for E

This condition constrains the tangential component of the curl of E over the surface:

V x E x n = O (2.8)

which dso Unplies that:

H x n = O

i.e., the tangentid part of the magnetic field vanishes. This condition is satisfied on some

symmetrical surfaces; Le. surfaces that split a microwave device into two parts, that are

geometrically and magnetically symmetric.

2.3.3 Absorbing Boundary Conditions

Absorbing boundary conditions are used with open boundary problem, e.g. antenna

problems, to tmncate the infinite space. Many versions of the Absorbing Boundary

Condition can be found in the literature [1][33-401. The theory discussed in this thesis is

applicable to both closed and open problems. However, in this research, only closed

structures are investigated, so no absorbing boundary condition is implemented.

2.3.4 Port Boundary Conditions

Consider an N-port microwave device (Figure 2.2), where each port rnay or may not

support a propagating mode. Let the transverse part of the incident wave of mode rn at

port i be:

Einc '.

Figure 2.2: An N-port microwave device.

(2.1 Oa)

(2.1 Ob)

where a: is the wave amplitude of the incident mode, and en, and hm represent the

transverse electric and magnetic field distribution across the port for mode m,

respectively. e, and hm are related as follows:

where n is a unit normal vector outward fiom the device, Le., opposite to the direction of

propagation (Figure 2.3). 2, is the wave impedance:

where y, is the propagation constant of mode m. e, and hm are nomalized such that

when a:' is equal to 1, mode rn will be then canying a unit power, i.e. e , and hm satisfy

the relationship:

Also, fkom mode orthogonality,

The outward wave of mode m at port i can also be written in terms of e, and hm :

where bm, is the wave amplitude of the outward mode m.

Hence, the total E- and H-fields due to mode rn are:

0 E=V,e ,

Where

v:' md 1: are called the generalized voltage and current of mode m at port i,

respectively. From mode orthogonality, the voltage can be calculated fkom the total E-

field using the following projection [41]:

where Si is the cross section area of port i.

Let port i be excited by unit power mode m only, i.e. an is equal to 1, with al1 other

ports perfectly matched, i.e. no modes are coming into the device through these ports. Let

the elecaic field solution witbin the device under these conditions be denoted EL'. Then,

the port boundary condition for port j can be shown to be equal to:

n

where V~)(E:)~S the voltage of mode n at port j, due to exciting port i with unit power

mode rn, and 6, is the Kronecker delta.

The wave equation (2.5) and the above boundary conditions fùlly represent the field in

fiee space and inside any microwave-guided device. Solving them together analytically or

numerically gives the field distribution (E- or H-field).

2.4 Weak Form of the Problem

Given the boundary value problem defined by the differential equation (2.5), and

subject to the boundary conditions descnbed by formulae (2.7-8, and-18), the solution

E: to the boundary value problem may be obtained by solving:

for al1 weight hc t ions w, where a is the bilinear form

where q, is the intrinsic impedance of the free space and it is approximately equal to 377

Formula (2.19a) is called the weak form or weighted-residual equation. The same

result can be obtained if a variational formulation is used.

2.5 Finite Element Discretization

In the finite element method, the region of concem, is discretized into a set of small

subregions. In 2D, the subregions could be triangles, rectangles, or 2D cwilinear shaped

elements, as shown in Figure 2.4. In 3D, the subregions could be tetrahedral, hexahedral

or 3D curvilinear shaped elements, Figure 2.5.

In the 3D finite element rnethod, tetrahedrals are the most commonly used elements,

due to their capability to approximate any cornplicated 3D structure and the possibility of

automatic generation of tetrahedral meshes. in the 3D solver developed for this thesis,

tetrahedral elements were used.

By discretizing a 3D problem into a set of M tetrahedra, the volume integral of

formula (2.19b) can then be decomposed into M integrals:

where V, is the volume of element m, and

Cal

Figure 2.4: 2D Elements. (a) Triangle, (b) Rectangle, and (c ) Curvilinear Element

6, is the simplex coordinate of a 3D tetrahedral element.

The field inside any element can be written as [42] :

Cbl

Figure 2.5: 3D Finite Elements. (a) Tetrahedral, (b) Hexahedral, and (c) 3D Curvilinear Element

N, is the 3D basis function associated with the rth unknown. The unknowns in a 3D

FEM nodal-solver are the components o f the E-field at each node, while in an edge-solver

they are the E-field along each edge in the mesh. In this thesis an edge solver was

developed.

Let g in formula (2.20) be replaced by formula (2.22) for al1 elements, and replace f

with al1 possible Nu, i.e., FI.. LI, where LI is the total number of unloiowns. This

substitution generates a set of W equations in the form:

where (E) is a vector of E-field values along al1 the edges, and

The [a maûix is generated from the wave equation and al1 the boundary condition5 applicable to the problem of concem. However, the ( RI' } vector is constmcted fiom pofi boundary conditions only.

2.6 Scattering Matrix

The performance of any microwave device can be descnbed by a scattering rnaaix [q.

[SI relates the incident modes at al1 pons to the reflected ones:

{bl = [SI Cal

where (a) and {b) are column vectors of incident and reflected amplitudes,

respectively, of al1 the propagating modes at al1 the ports. The diagonal terms in [q are

the reflection coefficients, while the off-diagonal tems are the transmission coefficients.

Assuming again that the unit power mode m is incident at port i, while the other ports

were matched, then according to formula (2.26), the voltage of an outward wave of n ~ d e

n at port j is equal to the scattering parameter SC, i.e.

v Gl (E fi) ) = ~ 0 ) " n m nm

The voltage of the outward wave of mode m at pon i is equal to

v (E fi) ) = s(i)fi) m m mm + I

The general fonnula for the scattering parameten is, then :

This formula is used to obtain [S] fiom the computed solution E: .

2.7 Propagation Modes

The field distribution of a mode of an arbiîrarily shaped waveguide can be obtâined

using 2D finite element eigen-solver. Given the geometrical parameters and media

characteristics, the solver cornputes the field distribution and the propagation constants of

many propagating and evanescent modes. These can then be used by a 3D finite element

solver to impose port conditions and to calculate the scattering parameters.

However, for many uniform guided structures there are analytical fonnulae for both

the field distribution and the propagation constants [43][44]:

2.6.1 Rectangular Waveguide

Figure 2.6a shows a rectangular waveguide with its port placed on the x-y plane.

Assume that a TE or TM wave is propagating inside the waveguide in the positive z

direction. Then, the transverse components of the mnth TE mode are:

Cbl

Figure 2.6: (a) Rectangular waveguide (b) Circular waveguide

and the transverse components of the mnth TM mode are:

w here

2.6.2 Circular Waveguide

Figure 2.6b shows another example of a regu&irshaqepwaveg.&de, that is thecircular

waveguide. Again we assume that the port is placed on the r-@ plane, and the wave is

propagating in the positive z direction.

Then, the transverse components of the mnth TE mode are:

and the transverse components of the mnth TM mode are:

where a is the radius of the waveguide, p, and are the n-th roots of the m-th Bessel

f'unction and its derivat ive, respective1 y, and

CHAPTER 3

The Gradient of Scattering Parameten

3.1 Introduction

The scattering parameters of rnicrowave devices change as their geometrïes are

modified. To improve the performance of a device, one needs to modim some

geometrical parameters in a manner that leads to an optimum. To modify the geometry

properly, gradient information is very usefùl, Le. information about the rate of change of

the scattering parameten with respect to a set of geometrical parameters. The goal of this

chapter is to evaluate these gradients. using the finite element method.

3.2 The Adjoint Method

From formulae (2.29) and (2.1 7), the scattering parameter s!: can be expressed in a

general form as [2]:

where { E:) ) is the computed electric field within the device when port i is excited with

mode m. ( E:' ) is itself a hinction of g. Differentiating (3.1) with respect to g:

is the colwnn vector,

U is the total

Substituting (

number of unknowns. Di fferentiating both sides of formula (2.23):

13 -3) in (3.2) we obtain:

Fortnula (3.4) involves the calculation of the inverse of the symmetric [q matrix. To

avoid such an operation, we introduce the adjoint variable vector {L}, which is defined

as:

{A.) = [KI-' as:?i) ~ { E Z ' ) Substituting in ( 3 4 ,

Formula (3.5) can be rewritten as

asYi' (3.7) [KI {A} = a { ~ : ' )

Formula (3.6) indicates that the adjoint method requires two 3D solutions in order to

{ ( i ) ) calculate the denvative. First, formula (2.23) is solved to obtain E, , and then formula

(3.7) is solved to calculate the adjoint variable vector. However, once the adjoint solution

is found, derivatives with respect to any number of different g's can be found.

Assume that g is an intemal dimension which does not effect the location and the

shape of any port in the device of concern. Then the derivative of {R:' with respect to g

c m be dropped. Also since the scattering parameters are evaluated on the ports only, the

first partial denvative in formula (3.6) is equal to zero. Consequently, formula (3.6) can

be simplified to

Notice that formula (3.8) still requires two solutions to calculate the derivative.

3.3 The Nature of the Adjoint Variable

In Section (2.6) we explained how s:~'' could be extracted frorn the computed

electric field using [4] :

= Vf.(E')-c Sm (3 -9) Differentiating formula (3.9) with respect to one of the unknowns, say the field value

on the uth edge:

Substituting (2.17) into the right hand side of (3. IO),

The field inside any element can be descnbed using formula (2.22). Substituting (2.22)

into (3.1 1) yields,

But fiom (2.17) the nght hand side is simply v,

So now the adjoint problem in formula (3.7) is simply equal to,

Comparing this to (2.23) we see that

Le. it is in fact half the value of the field generated nom exciting port j with mode n. This

useful conclusion c m be used now in formula (3.8) to obtain,

Here we are assuming that g is an intemal parameter that does not affect the location

or the shape of any port.

For an N-port model, to obtain the entire scattering matnx requires N field solutions,

given that one dominant mode is supported at each port. With the usual adjoint formula

(3.8), to get the gradient requires an additional N adjoint solutions. However, the new

formula (3.16) gives the gradients at no additional cost. Moreover, if just the retum loss

of the dominant mode of one port is to optimized, then only one solution is required to get

SI 1 and its gradient.

3.4 Evaluating the Derivative of [KI

In order to evaluate (3.16), we need the derivative of [a with respect to g. Using the

where the first summation is made over al1 vertices which move when parameter g is

changed, and the second summation is over the three coordinates x, y, and z of each

vertex. rl is the coordinate 1 of vertex k.

From formula (2.24), each entry of - a[K1 is equal to: a.:

a' 'vx Nr 'Nr * It cari Formula (3.19) involves three important derivative ternis; 7, ark ar: ' ar:

be s h o w that,

Further, using

we get

where N, and Nu are,

htroducing the following definitions:

Using (3.17) and (3.29), it is straightforward and inexpensive to obtain the denvatives

o f [KJ for any geometrical parameter g.

CHAPTER 4

Optimizatîon of Microwave Devices

4.1 Introduction

The main purpose of optirnizing microwave devices is to meet design objectives such

as return loss, insertion loss, and couphg. There are many techniques for optimizing

microwave devices, but, since we bave gradient information available at very little cost,

oniy optimization techniques that make use of gradient information are of interest.

In section 4.2 we present a generd formula for a coût functions used in microwave

design, and the correspondhg gradient formula. In section 4.3, we describe the conditions

for a point to be a local minimum, then in section 4.4 we discuss, in general, the theory of

the gradient methods. Different kinds of optimization techniques are discussed in section

4.5, while in section 4.6 we present the Quasi-Newton methods, which are the methods

used in this research. In section 4.7 the Line Search method i s presented in details.

4.2 Cost Function for Microwave Devices

A general cost fûnction of an N-port microwave device is [6]:

where ai and j3; are weight hctions, .!?,(j") are the desired scattering parameters,

and the frequency range has been discretized into N, fiequedes. The ai and

weights control the emphasis on the magnitude and phase repectively of the scattering

parameters. From this general formula, one can also deduce the formula of the gradient of

the cost h c t i o n with respect to any gwmetrical parameter, say g. The derivative of the

cost function is a fiinction of the scattering parameters and the derivative of the amplitude

W o r the derivative of the phase of the scattering parameters. The derivative of the

amplitude is equal to:

while the denvative of the phase is:

In terms of these, the general formula for the derivative of the cost fùnction is:

4.3 Conditions for a Local Minimum

In general, the cost function C can be written as a fiuiction of a vector {g) of

geometrical panuneters. Let (g}=(g*) be a local minimum, and C be continuously

differentiable at (g*). Then the following condition must be satisfied 145-481:

vqg* = O (4.5)

i.e. the dope of C is zero at (g*}. However, satis-g (4.5) does not necessarîly

mean that (g*) is a local minimum; it could be a local maximum. To obtain minima only,

an additional criterion is needed. The new criterion is derived fiom the second derivative

of the cost hc t ion C. -

Let C be twice continuously differentiable at (g*}. Then, one c m expand C in the

neighborhood of (g*) to obtain the formula:

where (6) is a very srnail perturbation vector, and the [H] ma& is the Hessian of C:

Higher order tenns are neglected in formula (4.6). For (g*) to be minimum, the left-hand

side of (4.6) should be greater than the cost function at (g*] . Consequently, the additional

requited criterion for minima is:

(4.8)

{ w [ ~ l @ b o ~ ( 6 ) # O Formula (4.8) can be satisfied if [HJ is a positive dennite Hessian matrix.

4.4 Multivariate Gradient Methods of MinSmkation

ln al1 gradient methods, a vector (g] of geomehical parameters is iteratively updated

until the objective cost, C, becomes lower than a prespecified threshold. The h l s t update

is of the following form:

where btk)} is ceriain search direction, and dk) is a step ske in the direction of @(k)).

Gradient methods use the dope (gradient) of C at (p') to construct a suitable search direction. Some gradient methods do not require denvative information, Wre Powell's

method. However, we are only interesteci in those techniques that make use of the

denvative information such as, steepest descent, Newton, quasi-Newton, and the

quickprop method.

The gradient methods M e r nom each other in the way the vector bR)} is updated.

The value vR) can be constant, or it can be calculated using single-variable opthkation

( h e search) technique.

In the steepest descent method, developed by Cauchy, the vector bfi)) is aven by:

i.e. the direction of search is dong the line of maximum rate of change of C. The steepest

descent method has a linear convergence rate for a quadratic function. This is due to the

fact that the second derivative is not used. Thus, the method is not usuaily recommended

for generai applications.

On the other hand, the Newton Raphson method uses the second denvative

information to update the search direction vector (pR)) . The formula used is:

where [d is the Hessian matrix of equation (4.7) which includes the second denvative

information.

The main disadvantage of the Newton-Raphson method is the cost of calculating [w, which nses dramatically as the number of variables increases. In addition, singular

Hessians, indefinite matrices, and overshooting could be possible sources of problems to

this technique.

For conjugate gradient methods, the search direction bm), satisfies the conjugacy

condition [48]:

One can show that @"} vector can be generated iteratively without constmcting the

Hessian matrix [Hl, using formula:

Conjugate gradient methods are highly efficient when the number of variables is large,

and no information is available on the second denvatives, or it is very difficult to obtain

them. An improved convergence rate can be accomplished through exact line searches to

obtain the optimum step size v'~) in the direction of bfk)).

In the quickprop method [29], the cost function is approximated by a parabola which

has arms open upwards. The assumption made is that the slope of the cost with respect to

a variable is not affected by the change in the values of al1 other variables in the mode1 of

concern. Then, one can easily show that:

Rekanos showed that the QP technique is 3-4 times faster than the steepest descent

method. No cornparisons with Newton or quasi-Newton techniques are available yet.

4.5 Quasi-Newton Methods

To avoid the calculations of the exact Hessian matrix in the Newton-Raphson rnethod,

the quasi-Newton methods use an iterative scheme to approxirnate it. The gradient and the

cost function are used in the calculations, which take place at each iteration.

The search direction @&} is found by:

and the [Ml ma& is updated using the following formulae [45][47-51][52]:

Formula (4.16) is known as the Broyden-Fletcher-Goldfiab-Shamo (BFGS) method.

There is another quasi-Newton method called the Davidon-Fletcher-Powell @FP)

method. Powell [53] showed that BFGS perfonns better than DFP in rninimizing

quadratic huictions of two variables, which suggests that BFGS should be used also for

general non-linear hctions.

To improve the overall performance of the algorithm, a good initial value for [M] is

necessary [54]. A convenient choice is to set the initial value to ~(~'[l~], where [Io] is the

identity matrix. do' is the initial step size.

When usïng a line search, the value of the step size dk' converges to 1.

The algorithm of the quasi-Newton module has the following steps for minimization:

Begin with [MJ=[I,].

Using the initial geometry, calculate the cost and its derivative.

Call the line search module, and perform an approxïmate iine search along

{p'o'}= -~OC'O)} . The line search module returns: the ne* step do), the Cost and its derivative at do)

Set [M'O)]= dO)[l, 1.

Evaluate formulae (4.16).

Evaluate formula (4.15).

Update the geometrical parameten using formula (4.9).

Call Line Search which rehirns under three conditions:

8.1 If the cost or its derivative are better than the target values, r e m .

8.2 If the geometrical parameters were not changed, retum.

8.3 Otherwise proceed to step 9.

Redo 5-8.

4.6 Line Search

To improve the convergence rate of any gradient method, a line search is used to

update the value of the step size Jk) at each iteration. The goal of the line search is to find

the v'Y which makes C( {g(k)) + dk) (p'k)}) locally minimal, i.e.

Using the chah nile, this becomes

Formula (4.18) represents the condition for the termination of an exact search.

The line search module is composed of two consecutive steps: establishing bounds for the

search, and aliocating a minimum along the bn

The algorithm of the interval step includes the following steps:

1. Start with a=O.

2. If the slope of the cost function with respect to v is negative then,

2.1. Evaluate b using formula (4.19).

2.2. Evaluate formula (4.9) using v=b.

2.3. Solve the problem and calculate the cost hction and its denvative.

2.4. Ifthe cost function is below the target cost fhction Cr, retum.

2.5. If the derivative of the slope is negative, then,

Set a=b, C,=Cb, and CL = Ci, then go to 2.1

2.7 Else retum.

3. If the dope of the cost is positive then,

3.1. Set k, Cb=Ca, CL = C:

3.2. Evaluate a using formula (4.19), where a is replaced with b and b with a.

3.3. Evaluate formula (4.9) using v=a.

3.4. Solve the problem and calculate the cost fiuiction and its derivative.

3.5. Lfthe cost is below the target cost function Ci, r e m .

3.6. If the denvative of the slope is positive then,

Set b=a, G=Ca, and CL = Ci, then go to 3.1

3.7. Else retm.

4.6.2 Locating a Minimum

Once an interval bounding a minimum has been determineci, polywmial interpolation

methods can be used to approximate the cost fùnction by a polynomial, fiom which a

possible minimum is calculated. In this thesis, cubic interpolation is used The method

exhibits quaritatic convergence.

It cm be shown that if the cost function and its derivatives are known at the bounds of

the interval ([a,bJ), the minimum of a cubic polynomial fitting these values is given by:

Formulae (4.20) are used in an iterative process to obtain the real minimum of the cost

fiinction within the given interval. For each new "cubic" minimum, the cost bc t ioa and

its derivative are evaluated. If the derivative of the cost function is not smdl enough, then

the calculated point i s not a real minimum, However the new information is not wasted,

but used to n m w the interval from one of the two sides (lower or upper) depending on

the slope of the cost function at v. This process is repeated for the new interval, until the

slope of the cost is below certain threshold, or the interval is less than a given interval-

threshold.

The algorithm for the allocation of a minimum is:

Calculate v using formulae (4.20).

Evaluate formula (4.9).

If the differences between the new and old geometrical parameters are below the

geometrical tolerance, retum

Adjust the geometry and solve the new model.

Calculate the new cost function and its denvative.

6. If the new cost function or its derivative are better than the target cost Ct return.

7. If ihe derivative is negative then set a=v, C,=C, and Cg = Ci, and go to 1.

8. Else set 6=v, Cb=C, and CL = CL, and go to 1.

Computer Program lmplementation

5.1 Introduction

Using the theory discussed in the previous chapters, a Fortran-99 program was

developed.

The program consists of two main parts: the optimizer and the 3D nnite element

package. The optimizer is made of two modules: the Quasi-Newton and the Line Search

modules. The 3D finite element solver is made of four modules: the mesh generator, the

node-to-edge converter, the 3D finite element solver, and the remodeiier. The program

interface is via input and output files. The program needs four input files which are :

Geo.dat, 2D.dat, Input.dat, Ports.dat, Mats.&, and Param.dat. It generates three

temperory files: Tets-dat, Edges-dat, and Results.dat, and the output of the program is one

file: Tracing.log.

In Section 5.2 we describe the six modules of the program, then in section 5.3 we

present the different data files used during the run.

5.2 Program Structures

The flow chart shown in Figure 5.1 demonstrates the intercomection between the six

main modules of the program.

5.2.1 Quasi-Newton Optimize

The method implemented here uses the BFGS technique descnbed in the previous

chapter. The efficiency of this optimizer depends on how fast and accuratIy the cost

minimum, and in the second one, the cubic polynomial interpolation is implemented to

mùumize the cost function dong a search direction p.

The Line Search may not perform the two steps al1 the rime, as discussed in the previous

chapter. The user may also choose to discard the use of the Line Search option.

5.2.3 Mesb Generator

The mesh generator was written by Marc P. Choufani (Master Project) [54]. The user

supplies two data files: a module file and a design parameter file. The program generates

a data file which contains the (x,y,z) coordinates of al1 the points, elements material labels

and nodes, and the boundary conditions of each of the four faces of all the elements. The

program uses the Delauny triangulation method to construct the 2D mesh, then uses the

extrusion method to make the final 3D-nodal mesh.

5.2.4 Node-to-Edge Converter

The 3D mesher generates a 3D mesh in which each node is assigned a global number.

However, the 3D finite element solver is a 3D-edge solver. The Node-to-Edge converter

module assigns a global number to each edge in the 3D mesh. The output file, Edgesdat,

contains information on al1 edges and their associated nodes (hvo nodes per edge), and al1

elements and their associated edges (6 edges per element).

Some elements have the number zero assigned to some of their edges, this indicates

that theses edges are located on perfectly conducted surfaces, i.e., homogeneous Dirichlet

Boundaries.

5.2.5 3D Finite Element Solver

This module performs four different operations: constructing the [KI and [RI matrices,

solving the linear set of equations, calculating the scattering parameters, and finally

calculating the denvatives. Formula (3. la ) is used to constmct the w] and matrices.

The no-look-ahead QMR algorithm is used to solve the set of equations developed in

chapter 2. The preconditioning used in the QMR solver is the diagonal of the [KI matrix.

Formula (3.12) is used to calculate the derivatives of the scattering parameters. These

derivatives are then used to calculate the gradient of the cost function. Appendk .I

contains a copy of the QMR algorithm.

5.2.6 Remodeller

The remodeller modifies the geometry in accordance with the latest values of the

geometric parameters. It modifies two data mes: the model file, Geo-dat, which contains

the geometrical Iliformation and the derivative file, Param.dat, which contains

information on. the perturbed surfaces. The mesher uses the model file, while the 3D-

finite element solver uses the perturbeci file.

5.3 Data Files

The program communicates through a set of data files. Each procedure of the program

reads data fiom specific data files, and outputs the results to another set of data files.

5.3.1 Geo.dat

Geadat is the model file and it contains information about the geometrical dimensions,

the materials, and boundary conditions of the device under investigation. The Rernodeller

modifies Geo.dat prior to calling the FEM solver.

5.3.2 2D.dat

2D.dat is the mesh control file. 2D.dat contains some controlling parameters to be used

by the 3D Mesher. One important parameter in 2D.dat is the largest edge length allowed

in the system.

5.33 Inpatdat

The main program reads Input.& once at the beginning of the optimization process.

1nput.dat contains information about: the number of variables and their initiais, maximum

and minimum dimensions, fiequency points, QMR solver tolerances, Quasi and Line

Search tolerances, and the number of geomeûical precessions tolerated.

5.3.4 Ports.dat

Portsdat contains info~mation about the location, the shape, and the exited modes at

al1 the ports in the system. This file camot be changed by the Remodeller, since the

theory assumes that the dimensions, locations, and excited modes of al1 ports are ked.

5.3.5 Mats.dat

Matsdat contains a library of many materials with their pennitivity and penneability

parameters. Each matenal is given a label, and the same label should be used in the mode1

file.

5.3.6 Param.dat

G2mesh.dat is the derivative file. To calculate the denvatives of the scattering matrix

with respect to any geometrical parameter, the program needs to know which surfaces are

related to these geometrical parameters. G2mesh.dat contains information about al1

variables, the surfaces related to hem, and the direction of perturbation of each of these

surfaces. The program supports only rectangular surfaces at present.

5.3.7 Tracingdog :

Tracing-log contains information about dl intermediate results obtained during the

optimization. For each triai, the program saves the new values of the variables, the cost

fiuiction and the derivative of the cost.

5.3.8 Tekdat

Tetsdat is the output of the mesher. Tetsdat contains information on the cartesian

coordinates of al1 the points, the nodes of each element identified by theu global number,

the matdai label of each element, and the boundary conditions on each surface of each

element.

5.3.9 Edges.dat

Edges.dat is the output of the node-to-edge converter. Edges.dat contains Monnation

on the nodes of each edge and the edges of each element identified by their global

numbers.

5.3.10 Results.dat

The 3D finite element solver solves for the field value dong each edge in the model.

The 1 s t solution is saved by the system in Resultsdat.

CHAPTER 6

Results

6.1 Introduction

The twofold purpose of this chapter is (1) to verXy the theory of the gradient

developed in Chapter 3, and (2) to demonstrate the optimization teclmique desdbed in

Chapter 4.

Three examples are presented. The first example is a short-cùcuited rectangular

waveguide, the second is an E-plane miter bend, and the third is a single stage

waveguide transformer.

In the fhst example, the gradient is venfied by comparing numerical results with

analytical ones. In the second and third examples, the gradients are verified by numencal

perturbation.

In the ks t two examples, the devices have one variable and are optimized at only one

fiequency, while in the transformer case, three variables are optimized simultaneously

and at eight different fiequencies.

6.2 Short Circuited Rectangular Waveguide

Consider the unifonn rectangular waveguide in Figure 6.1 with inner dimensions: a

(in the x-direction) and b (in the y-direction). The waveguide is short-circuited at a

distance L h m the port. The waveguide is operating in the fûndarnental mode TElo. The

dimensions are as follows:

a/Â. = 2/3

b/A = o. 1

n p = 1.515

L is the design puameter. It is the wavelength of the operaîing fkequency.

Figure 6.1 : Rectangular waveguide short-circuited at a distance L tiom the port

The design cntenon is to change L until the desired phase of Si 1 is obtained. The cost

fùnction is defined as:

where the cost is evaluated at a single fiequency point f.. The gradient of the cost

hc t ion is:

The gradient of the cost with respect to L is a funetion of Si and its gradient with

respect to L

6.2.1 Verirying the Gradient of LS,,

Anaiyticd f o r d a e can be obtained for the gradient of Sri of a rectanguIar

waveguide short-circuited at a distance L fiom the port:

SI, = -cos(2k,L) - j sin(Zk,L) (6.3)

dS, 1 - = 2 k, (sin(2 k, L) - j cos(2kZ l)) dL

where k, is the phase constant of the TEio mode. Substituting (6.3-4) in (6.2) to obtain

the exact formula for the gradient of C. The values extracted fiom FEM solution using

formula (3.1 6) were compared with the exact values. The results are displayed in Figure

6.2 for L/ A 4.0667, . . ., 0.667. Good agreement between the finite elernent solutions

and the analytical ones can be concluded. The error in the phase of SIl is also displayed

in Figure 6.2 to dernonstrate the accuracy of the 3D finite element solver. Maximum

percentage error in the phase occwed at the location where the phase is almost zero, as

expected.

i i -Emin the? I Phase% 1

Figure 6.2: Error in % of the phase of SI[ and its gradient

6.2.2 O ptimization

The initial value for the opthkation is chosen to be:

&nitial /A = 113

The target phase is -0.7856 radians. The initial phase of SI 1 differs fiom the target by

146%. The results of the optimization are displayed in Figure 6.3, which shows the error

in the obtained phase after every 3D solution. Figure 6.3 shows how the program

converged h m the h t iteration to a solution that is very close to the final one. This

fa t convergence is because the cost function is a quadratic fùnction of L / A as shown in

Figure 6.4. Figure 6.4 presents the cost fiuiction in the vicinity of the optimum solution,

starting fkom the initial value. This result demonstrates both the accuracy of the gradient

calculations and the powafiilness of the optimization technique used (quasi-Newton +

line search). The final value of hirial A i s 0.47 1.

-- --

Figure 6.3: Error in the phase of SfI versus solution nurnber

Figure 6.4: Cost fùnction in the vicinity of the optimum solution

6.3 E-plane Miter Bend Waveguide

The second example is the miter-bend shown in Figure 6.5. The bend is in the E-

plane, Le. the 6-dimension. The device has two ports; port one is located on a y-z plane,

while port two is located on an x-z plane. The operating mode is again the hdamentai

TE10 mode.

The dimensions of the rectangular waveguide are:

dil = 2/3

6/A = 1/3

The guided wavelength is:

A is the free space wavelength at the operathg fkquency, and A, is the corresponding

guided wavelength. The design parameter g is shown in Figure 6.5, and the movable

bomdary related to it is the inched wall of the miter-bend. Increasing or decreasing the

design parameter g causes the inclinecl wall to move diagonally (45 degrees) outward or

înward respectively.

Figure 6.5: E-plane Miter Bend.

where the cost is evaluated at a single fiequency point f,. The gradient of the C Q S ~

fhction is

The gracüent of the cost fùnction with respect to g is a fûnction of the return loss SII

and its gradient with respect to g. In our case the desireci amplitude of Sll was chosen to

be O.

6.3.1 Verifying the Gradient

To calculate the gradient, the program needs to h o w in which direction each mesh

node on the inclinecl wall is moving when the geomeûical parameter g changes. For the

inclined waii of the miter bend there are many ways of moving the mesh nodes. The

method chosen is to divide the surface of the inclined wall into two parts: the mesh

nodes located in the lower part are moved in the y-direction, while the ones on the upper

part are moved in the x-direction, as demonstrateci in Figure 6.6.

Given this information the program can calculate the gradient of Si 1 with respect to

the design parameter g. Since there are no anaiyticai formulae for this problem, we use

the finite difference technique to calculate the gradient. To do that, the design parameter

To be able to compare properly both the finite difference and the f i t e element results,

the mesh nodes o f the inclined wall must be displaced in the direction that was adopted

when calculating the gradient. This means that the nodes of the lower part of the inclined

wail are displaced in the y-direction, while the ones on the upper part are displaced in

the x-direction.

Tm Port

Figure 6.6: Displacement of the me& nodes located on the surface of the inclined waii

Ushg f o d a (4.2), we caculated the gradient of the amplitude of Si1 for both the

finite difference and the finite element rnetbods. The resuits are presented in Figure 6.7.

Very good agreement between both solutions can be stated. gh was varied fiom -0.8 to

0.8. The locations of the bend for g b 0 . 8 and g M . 8 are shown in Figure 6.8.

- . - - pp --

Figure 6.7: % Error in the gradient of lSIIl of the miter-bend example

6.3.2 Optimization

The initiai value of g is:

The results of the optimization are displayed in Figure 6.9. The optimizer converged

after 6 solutions to within 0.01 of the final solution, i.e. 1 % of the 6 dimension. The

optimizer performed additional solutions before it hits the geometrical toierance (g/b

geometrical tolerance = 0.0001). Better performance cm be obtained by chooshg a

better initial vaiue, g m . 0 . The optimizer needed only 5 solutions before reaching the

geometricai tolerance.

Figure 6.8: Miter-bend at two different positions: (a) g b 0 . 8 , and (b) g M . 8

Figure 6.9: Miter-bend position as a fùnction of the solution number

Figure 6.10 presents the cost function versus solution number. The results indicate

that to within +/-0.01 of the final solution the cost fhction is around -40 dB, Le. the

retum loss is 40 dB. To get better performance a more accurate gh is required. In this

problem, up to 4" digit of precision was needed to reach beyond -60 dB. When the

initial value was -0.8, the optimizer needed extra six solutions to reach 4 0 dB, but for

the better initial value, i.e. 0.0, it used only three additional solutions.

l 1 1 2 3 4 5 6 7 8. 9 10 11 12

Solution #

Figure 6.10: Cost (dB) versus solution number

The single step E-plane miter-bend is a microwave component with a single resonmt

fiequency. The r e m loss over a range of frequencies is as shown in Figure 6.1 1. In

Figure 6.1 1, the optimum design was used to test the retwn loss for il /a fiom 1 .O72 to

1.875. For such a device one does not need to optimize at more than one fiequency. It is

sutficient to optimize the bend at one fiequency, usudly at the desired center of the

frequency-band, to get the best optimum dimensions. The miter-bend was optimized at

A /a=1.5, and hence the curve is centered at 1.5 as expected. From the graph, for 4ûdB

performance, the designeci miter-bend can be used with 24%-bandwidth systems.

8 Figure 6.1 1 : Retuni-loss of the optimum miter-bend

6.4 Single Stage Waveguide Impedance Transformer

The final example is the uniform rectangular waveguide transformer shown

6.12. The transformation is in both a- and b- dimensions. The device

the operating mode is the fundamental TElo mode.

The dimensions o f the input and output ports are

aiw= 2.0 m

b,-,, = 0.4 m

clo,,, = 2.4 m

bow, = 0.8 m

in Figure

has two ports, and

The optirnization is performed over 8 Ekequencies equally spaced fkom 95 MHZ to

105 MHz: 95.000,92.143,94.286,96.429, 98.572, 100.715, 102.858, 105.000 MHz.

Figure 6.12: Single stage rectangular transformer

There are three design parameters in the model: width, height, and Iength o f the middle

section. For the width and height, Le. the a-dimension and &-dimension, there are two

movable boundaries, while for the length there is only one movable surface.

The cost fimction is defined as:

The gradient of the cost function is sùnilar to ( 6 4 , with g replaced by a, b, or i. The

target value at al1 fiequencies was chosen to be equal to O.

6.4.1 Verifying the Gradient

Like the miter-bend example, there are no analytical f o d a e for the gradient of Si1

with respect to a, 6, or 1 dimensions of the middle stage. Consequently, the fiaite

dinerence technique is used again to calculate the gradient. To do that, the design

parameters were perturbai by + 0.5 x 1 o5 :

In Figure 6.13 both the finite difference and the h i t e element results are compared

for one geometrical position at eight different fiequencies. Notice that the difference

between FEM results and (6.9a) and (6.9~) were magnifieci 100 times. Good agreement

between both solutions can be observed.

90 95 100 105

Freq uency (Y Hz) t l

Figure 6.13: % Error in the Gradient of ISIII

6.4.3 Optimization

The initial values of a, b, and I of the middle section are:

ahil,-,,[ = 2.2 m

biniIiaf = 0.6 m

ZjniIiai = 1 .O m

liniIial was chosen to be roughly a quarter wavelength as it should be for a quarter-

wave transformer 1431. The results of the optimization are displayed in Figure 6.14,

which shows the cost as a hct ion of solution aumber. The results demonstrate how

choosing the proper initial values leads to ody few solutions before reaching the

optimum solution. In our case here, 3 solutions were enough to reach a very good

solution.

The ha1 dimensions are:

Figure 6.14: The cost (dB) as a fiinction of the solution number

Figure 6.15 presents the frequency response of the transfomer before and after the

optimization. A small improvement was observed at both ends of the band, while a

major improvement was accomplished for the center fkequencies. The final frequency

response is roughly symmetric about the center fiequency. Mode matching results were

also presented to validate the results obtained h m the f i t e element solver.

+ Initial Response +Final Response

! -t Mode Matching (30 modes)

Figure 6.15: Return Loss (dB) for the initial and the optimized geometry

6.5 Summary

In this chapter we presented three examples: a short-circuited rectangular waveguide,

an E-plane miter bend, and a single stage waveguide transformer. The design cntenon

for the first example was to change the length of the waveguide until the desired phase

of Si, is obtained, and for the second and third examples the design criterion was to

change some geometrical parameters to optimize the return loss response for one or

group of fiequencies.

For the three examples presented here, the gradients of the amplitude or phase of Sir

were verified versus malytical or finite difference results, and the agreement were

temarkable.

The optimization performed on the three casa twk between 4 and 12 3D finite

element solutions to converge. It was demonstrated that good initial values would lead to

fast convergence. However, even when the initial values were bad, as in the first case for

the miter-bend, the optimizer managed in 5 solutions to get close to the final solution.

Such results would not been obtained if the gradient information was not available, or

was not calculated accurately enough.

CHAPTER 7

CONCLUSlON

7.1 Summary

The adjoint method provides a way of calculating the derivatives of scattering

parameters with respect to geometric parameters. However, straightforward application of

the method requires additional adjoint solutions. In this thesis, a new fomula for the

derivatives was presented requiring no additional solutions.

A finite element program package was created for the design of microwave devices.

The package uses a quasi-Newton method, specifically the BFGS method, to do the

0 optimization and makes use of the efficiently calculated derivatives. The line search

method was used also to calculate the step size, to improve the convergence rate of the

quasi-Newton module.

The new derivative formula was verified using three different examples: a short-

circuited waveguide, a miter bend, and a 3D transformer. The first model has analytical

solutions for both its return loss and the gradient, while for the second and third exarnples

a finite difference approach was used to calculate the gradient. The values calculated

fiom the derivative fomula closely matched the values obtained fiom analytical or finite

difference calculations.

The three models were then optimized under different criteria. The critenon in the

short-circuited waveguide model was a specific phase of Si 1, while for the miter bend and

the transformer, the criterion was minimum amplitude of Si 1. For the three models, the

optimizer needed only 2-5 solutions to get very close to the final solution, and only 4-12

iterations to reach the h a 1 solution.

7.2 Original Contributions

The original contributions of this work are two fold:

(a) An improved version of the adjoint formula to calculate the gradient of the

scattenng parameters. The new formula requires half the number of solutions needed by

the standard adjoint method.

(b) A simple and inexpensive formula for the calculation of the derivatives of [Kj with

respect to any geometrical parameter, for the widely used tetrahedral edge element.

7.3 Suggestions for further work

(a) Allowing Ports to vary: In this dissertation, the ports are assumed to be

unchangeable: the shape, the location, and the excited modes do not vary with the

geometric parameters. One could look into removing some or al1 of these conditions.

(b) Antenna Problems: The approach may be extended e a d y to antema problems.

In such problems, one can assume that the ports and the absorbing boundary are

unchangeable. Such an assurnption does not effect in any way the design of the antenna,

since the feed port is usually known, and the absorbing boundaries are fictitious boundary

located at a distance fiom the antenna.

(c) Curvilinear Elements: Instead of using tetrahedral elements, one could study the

effect of using curvilinear elements. These elements have the advantages of

approximating more curved boundaries more accurately [55-571.

(d) Software Improvement: In the QMR matrix equation solver used, the

preconditioning was simply the diagonal terms of the [ K ] matrix. Better preconditioning

[ S a ] cm be implemented to improve the convergence rate. Also replacing both the nodal-

mesher and the nodal-to-edge modules, with a fast edge-mesher could improve the speed

of the whole package.

(e) Mesh Refinement + Optimization: One interesting subject to look into is how to

simultaneously, optimize the device and refine the mesh. Such a combination is very

promising, since the refinement of the mesh is a time-consuming step in optimization.

(f) Discontinuous Boundaries: It was observed by many authors [59], that there is

an inherent possibility of converging to an oscillating or saw-toothed boundary as an

optimal solution. This can be overcome by constraining the geometrïc parameters by

smooth functions, e.g. constraining the geometrical parameters to a Bezier curve [60].

Matrix Assembly for 30 Edge Elements

Hi& fiequency software generally uses edge elements rather than nodal elements,

because nodal solutions suffer h m the effects of non-physicai spurious solutions [l], the

inconvenience in imposing boundary conditions at material interfaces, and the difficulty

Ui treating conducting and dielectric edges.

For a tetrahedral element, with nodes labeled locally fkom 1 to 4 as shown in Figure

Al. 1, one can define six basic fbnctious Ni, i=l, . . ., 6. Using these six basic hctions,

the field inside a tetrahedral element can be described in the fouowing form:

i=l

The definition of Ni is,

Table A 1.1 presents the values of a and b for i =1 to 6.

Table A 1.1 Edge Defini tions for a Tetrahedral Element

1 Edge Number 1 a 1 b 1

(Al. 1)

Given these dennitions and formulae (3.23-24), the fïrst term on the right hand side of

formula (2.24) c m be rewritten in more compact form as [l]:

The reader cm refer to Jin's book [l] for the dennitions of b, c, and d terms.

In Table A1.2 we include the closed form formulae for the second term.

6 V m Table A1.2: Closed Formula for - !(li2e N,.N, )fa J ~ G

2 2 2 2 2

2 3 4 5 6

(fi3-fi3+fi 1)/(360V) (2f34-fi yf14+fi i)/(720V) (f33-f23-fi3+2fi2)/(720V) (f23-f34-fi 2+f14)/(720V) (f* 3-43-2fi4+fM)/(720V) -

where fij= bi bj + Ci + di dl.

The third tenn which contains V(N,) is evaiuated using Gaussian Quadrature [61]:

(AL .4)

A i=I

i.e. the integral is quai to the sllrnmatiou of the values of the hction at nq-point,

multiplied by weight numbers, wi. In our case nq=3, aiI wi were equal to 1/3, and the

simplex values were chosen according to Table A1.3.

Table A 1.3 : The Simplex Coordinates for the Gaussian Quadrature Rule, nq=3

Formula (2.25) is also evaiuated using the Gaussian Quadrature d e .

Quasi-Minimal Residuai Method

The Problem is to solve the set of linear equations defined as [A] (x) = (b ) , where [A]

is a large, sparse, symmetric, complex matrix.

The iterative solver used in the finite element package, is called a no-look-ahead Quasi

Minimai Etesdual Method (QMR).

The algorithm is presented below 1621:

[Ml1 = d W M )

COMP~TE (r(')) = (b) - [A] {x"'} for rom initiai mess {x'O)) .

(c(") = (r'"'} ; SOLVE [M, ] b) = F'") ; p, = l ly l12 . y, =1;q , = l

FOR i=1,2, ...

ifp,=Oor 5, = O thenmthodfaüs

(~ "7 ) = (Y('-)} / pi ; U>) = U>) 1 pi

S O L E LM, 1 63 = O.}

sj = ('1' {Y} IF ( i=l) THEN

{P"') = (Si) ELSE

(p"') = {y) - (p,3, l ci-,) bci-'))

END IF

APPENDIX 3

Derivations

A3.1 Deriving Formula (2.18)

Consider an N-port device, where port i is excited with mode m, while the other ports

are ternllnated b y perféctly matched loads, i.e. only a:' = 1 . The transverse part of the H-

field on port j can be then described as (eqn 2. l Sb):

where the summation is over the set of propagating and evanescent modes. But nom

Y,'" is computed nom the field distribution using formula (2.17), and hence

1:) = 2 - y,") (~(9 ) m

Substitute (A3.3) in (A.3.1) to get,

= x(2@ h, - V,(~)(E?) h,) n

But al1 a:' are zeros except a:' = 1, consequentiy,

H ( ~ ~ = ;?sl b, -cv,

A3.2 Deriving Formula (3.19)

Define ri as,

Le. the x, y, or z component of a point in 913 . For a point inside a tetrahedral element, its

x, y, or z component can be expressed in ternis of the sixnplex coordinates as,

where ri' 's are the coordinates of the 4 vertices of the tetrahedron. Using the following

relation,

the summation of formula (A3.7) can be reduce to three terms only,

one can denve the corresponding Jacobian maûix, whose entries are dehed as,

The inverse Jacobian ma& is then,

It follows that,

For k=l , formula (A3.13) becomes

SimiTa. results can be obtained for k=2, and 3. For k=4,

Using again the foliowing relation,

in fonnula (A3.15) to obtain,

Applying the results h m (A3.16) and (M. 17) to ail components, i.e.

a a g a K a K a K a %k 'Ca ' C , -(-,--,-;-> = --- --- & ûr' ar ôr &' al' &' ar2' drf i3r3

which can be rewrïtten in more compact form as

A3.3 DeWing Formula (3.20)

The volume of a tetrahedral element can be written as:

Then formuta (3.19) can be written as:

av a -=- at: ar; (6~5,0753 VSi II-'

But,

Substitute (3.19) in (A3.22) to get,

For k=l formula (A3.23) becomes,

The first and second terms on the right hand-side of formula (A3.24) are equal to zero.

Consequently,

A similar result is obtained for k=2 and k3. For k4,

However,

Substitute (A3.27) in (A3.26),

The first, third, fourth, fifth, eighth, and ninth terms are al1 equal to zero. So formula

A3.5 ûeriving Formula (3.21)

The basic funetion N, is equal to,

N, = L V Y b -CbVCa

Applying the curl operation,

V x N r = v x ( ~ a v ~ b ) - v x ( ~ b v ~ a )

Consider the following identity operation,

V X ( ~ A ) = V ~ X A + W V X A

U A is the gradient of a scaiar funetion, as in (A3.3 l), then the second terni in (A3.32)

can be eliminated uskg the identity operation,

vx(vv)= O

Consequently, (A3.3 1) becomes,

V x N, = Vca x VGb -VGb x V c a

V x N, = 2 ( ~ < , XVC,)

Differentiating with respect to r: .

which is formula (3.20).

A3.2 Deriving Formula (3.22)

The basic fùnction N, is equal to,

Di fferentiating with respect to r: ,

which is formula (3.2 1).

A3.6 Deriving Formula (4.1)

Starting nnt with the denvative of the amplitude terni, we h d that

However,

where S = Sr + jSi . Substiîute (A3.42) in (A3.41) to obtain the first term of (4.1).

Next is the phase term,

However,

where S = Sr + jSi . Substitute (A3.44) in (A3.43) to obtain fomiula (4.3).

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