domain decomposition methods for finite periodic ...€¦ · master thesis domain decomposition...

Master Thesis

Domain Decomposition Methods forFinite Periodic Structures in

Electromagnetics

Gustav Casselbrant

Stockholm, Sweden 2012

XR-EE-ETK 2012:006

Domain Decomposition Methods for FinitePeriodic Structures in Electromagnetics

A Degree Project in Electromagnetic Theory

GUSTAV CASSELBRANT

Master’s Thesis at Department of Electromagnetic EngineeringSupervisors: Anders Ellgardt & Henrik Holter

Examiner: Martin Norgren

XR-EE-ETK 2012:006

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Abstract

Long computational times persist on being a limiting factor when designingcomplex electromagnetic structures. Even though computers get more andmore powerful, the most efficient improvement for reducing the necessary timewhen solving a numerical problem is to change or optimise the used numericalmethod. One numerical method introduced lately is the domain decompositionmethod (DDM).

This thesis explores the available algorithms for computational electro-magnetics which are based on DDM. The study is concentrated on algorithmsavailable for computational electromagnetics. The aim of the study is to findand further study one appropriate method for analysing finite periodic electro-magnetic structures like frequency selective surfaces and array antennas.

After an introducing study of DDM, the FACTOPOmethod, is selected andstudied in detail. The study includes an implementation of the method, whichis performed in three versions. The first deals with a 1-Dimensional waveguideproblem which follows the FACTOPO method for determining the scatteringparameters of a subdomain using admittance parameters of the correspondingdomain. By limiting the subdomain with perfect electric conductor (PEC)the computational problem is efficiently reduced. The second implementationcontinues to explore the 1-Dimensional problem but introduces the ability ofimporting scattering parameters of a specific domain from an electromagneticsimulation software. The third, and last, implementation is an extension of thesecond version which introduces a 2-D structure based on a 4-port unit cell.

The theoretical study and the results from the implementation show thatthat domain decomposition is a promising numerical technique which, whenused properly, can improve the numerical simulation software for electromag-netic structures. It is also concluded that domain decomposition is especiallypromising for periodic electromagnetic structures.

Contents

Contents iv

1 Introduction 11.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Theoretical Study . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.4 Reporting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Computational Electromagnetics 52.1 The Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . 62.3 The Finite-Difference Time-Domain Method . . . . . . . . . . . . . . 7

3 Domain Decomposition 93.1 Domain Decomposition and Electromagnetics . . . . . . . . . . . . . 10

3.1.1 Complex Source Expansion . . . . . . . . . . . . . . . . . . . 113.1.2 Factorisation Domain Decomposition Method . . . . . . . . . 123.1.3 Domain Decomposition - Finite Difference Time Domain . . . 133.1.4 Finite Element Tearing and Interconnecting . . . . . . . . . . 13

3.2 Selecting a Domain Decomposition Method . . . . . . . . . . . . . . 14

4 FACTOPO - The Selected Method 174.1 Introduction to the FACTOPO Method . . . . . . . . . . . . . . . . 174.2 Notations and Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Decomposition of Computational Volume . . . . . . . . . . . . . . . 184.4 Determining the Scattering Parameters of a Subdomain . . . . . . . 19

4.4.1 Specific Case Layout . . . . . . . . . . . . . . . . . . . . . . . 204.4.2 Determining the Induced Magnetic Field . . . . . . . . . . . . 214.4.3 Introducing the Test Function and Scalar Product . . . . . . 234.4.4 Introducing Matrix Notation . . . . . . . . . . . . . . . . . . 28

4.5 Assembling the Global Scattering Matrix . . . . . . . . . . . . . . . 30

iv

CONTENTS v

5 Implementation Example 335.1 Analytical Implementation . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1.1 Declare Variables . . . . . . . . . . . . . . . . . . . . . . . . . 345.1.2 Declare the EmStructure Instance . . . . . . . . . . . . . . . 345.1.3 Declare Waveguide Subdomain Instances . . . . . . . . . . . 355.1.4 Compute Wave Parameters . . . . . . . . . . . . . . . . . . . 355.1.5 Define the Admittance Matrices . . . . . . . . . . . . . . . . 355.1.6 Compute Scattering Parameters . . . . . . . . . . . . . . . . 355.1.7 Define the GSM (SΩ) . . . . . . . . . . . . . . . . . . . . . . 365.1.8 Define Sϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.1.9 Define and Excite Source Vector Bs . . . . . . . . . . . . . . 365.1.10 Compute Resulting Transmission and Reflection . . . . . . . 36

5.2 Analytical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2.1 Declare Variables . . . . . . . . . . . . . . . . . . . . . . . . . 375.2.2 Declare the EmStructure Instance . . . . . . . . . . . . . . . 375.2.3 Declare Waveguide Subdomain Instances . . . . . . . . . . . 375.2.4 Compute Wave Parameters . . . . . . . . . . . . . . . . . . . 385.2.5 Define the Admittance Matrices . . . . . . . . . . . . . . . . 385.2.6 Compute Scattering Parameters . . . . . . . . . . . . . . . . 385.2.7 Define the GSM (SΩ) . . . . . . . . . . . . . . . . . . . . . . 385.2.8 Define Sϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2.9 Define and Excite Source Vector Bs . . . . . . . . . . . . . . 415.2.10 Compute Resulting Transmission and Reflection . . . . . . . 415.2.11 Results From the Analytical Example . . . . . . . . . . . . . 41

5.3 1-Dimensional Implementation . . . . . . . . . . . . . . . . . . . . . 435.3.1 Importing CST Microwave Studio Data . . . . . . . . . . . . 44

5.4 1-Dimensional Example . . . . . . . . . . . . . . . . . . . . . . . . . 455.4.1 Results of the 1-Dimensional Implementation Example . . . . 45

5.5 2-Dimensional Implementation . . . . . . . . . . . . . . . . . . . . . 475.5.1 The Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.5.2 The 3x3 Unit Cell Array . . . . . . . . . . . . . . . . . . . . . 485.5.3 Defining the GSM (SΩ) . . . . . . . . . . . . . . . . . . . . . 495.5.4 Defining the (Sϕ) . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.6 2-Dimensional Implementation Example . . . . . . . . . . . . . . . . 505.6.1 Unit Cell in CST Microwave Studio . . . . . . . . . . . . . . 505.6.2 3x3 Unit Cell Array in CST Microwave Studio . . . . . . . . 515.6.3 Results of the 2-D Implementation . . . . . . . . . . . . . . . 51

6 Discussion 556.1 Domain Decomposition in General . . . . . . . . . . . . . . . . . . . 556.2 The FACTOPO Method . . . . . . . . . . . . . . . . . . . . . . . . . 566.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7 Conclusion 59

vi CONTENTS

8 Abbreviations 61

Bibliography 63

Chapter 1

Introduction

Even though we today possess powerful supercomputers, complex electromagneticproblems still continue to demand long computational time (days or even weeks).This is especially true when considering that during design processes for electromag-netic structures, similar problems are often solved repetitively, in order to optimisethe design. Over the years a large number of numerical algorithms for reducing thenecessary computational time have been introduced. Most of these algorithms havebeen applied to one out of four major electromagnetic computational methods usedtoday or to a combination of them.

During the last couple of decades numerical algorithms based on domain de-composition have become increasingly more popular. Originally a domain decom-position algorithm (DDM) was a method for solving partial differential equations(PDE) introduced by H. Schwartz in 1870 [36]. Even though the method was notintended as a numerical method, it has recently become a popular choice due to thefact that it is highly appropriate for computational parallelization.

1.1 Purpose

The purpose of this thesis is to summarise the current research and to evaluate thevalue of using methods based on domain decomposition in order to analyse anddesign finite periodic electromagnetic structures like frequency selective surfacesand array antennas. The purpose has been divided into the following objectives:

• Perform a theoretical study which summarises the current research status fordomain decomposition.

• Select one promising domain decomposition method and implement one sim-ple and one more advanced example of the method using electromagneticsimulation software and matlab.

• Present the project advances and its final result both orally and in writing.

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2 CHAPTER 1. INTRODUCTION

1.2 Methodology

The thesis project has been divided into four main phases. Section 1.2.1 - 1.2.4describe these phases and how the work has been performed within them.

1.2.1 Planning

An important part of the planning was the division of the project purpose intoproject objectives. The objectives were then further divided into requirements foreach phase. Based on the requirements, tasks could be identified and used forproducing a time plan. The project plan was the result of the planning phase.

1.2.2 Theoretical Study

The first part of the theoretical study was to study two preparative areas. The firstpreparative area covered three of the major numerical methods for conducting elec-tromagnetic simulations. These are essential for the thesis since any implementationof domain decomposition in a electromagnetic perspective will be based on one ora combination of these methods. The second preparative area was to study domaindecomposition in a general numerical analysis perspective. These two preparativeareas were studied mainly using literature.

The second part of the theoretical study concentrated on the different imple-mentations of domain decomposition methods for solving electromagnetic problems.This part was mainly performed using published articles which were found mainlyusing Google Scholar. Most articles were available through the IEEE and Sciencedirect databases. During the study it turned out that many methods were men-tioned in only a few scientific articles. Therefore the first task was to identifyand list all methods mentioned in the articles. The second task was to determinewhich methods could be categorised as established methods. In order to do so aselection criteria had to be chosen. The criteria taken was that if a method hadbeen mentioned in at least four articles it was to be considered established. Beforeneglecting any methods, a search for more articles of every method listed was per-formed. The last step in the study for established methods was to summarise theidentified methods.

An important milestone during the literature study was the selection of a domaindecomposition method to be used during the implementation phase of the project.The decision was mainly motivated by the intended application for the final resultas specified by the project supervisor Henrik Holter. The motivation of the selectionis listed in section 3.2.

The third part of the main study was to comprehensibly describe the selecteddomain decomposition method. An important part when describing the methodwas to develop a simple example which could illustrate the different steps in themethod.

1.2. METHODOLOGY 3

1.2.3 ImplementationThe implementation of the selected method was performed using Matlab. Thedevelopment process of the algorithm was more of evolving nature where smallchanges between the versions lead to an increasing complexity. Therefore a numberof different problems, with different complexity have been treated. The report willdeal with one analytical example, one 1-D example and one 2-D example. Theanalytical example exemplifies how the scattering parameters are determined bythe FACTOPO method. The two later examples makes use of data exported fromCST Microwave Studio.

1.2.4 ReportingEven though the official reporting phase did not start until the end of the projectthe reality is that the documentation as well as the reporting have been performedin parallel with the project. Besides the thesis report two shorter reports (includedin the thesis report as Chapter 4 & 5) have been produced together with the projectplan. The project plan as well as the theoretical study were also presented at theDepartment of Electromagnetic Engineering at KTH.

During the official report phase the last parts of the thesis report will be finished.Together with the presentation of the thesis they will mark the end of this thesisproject.

Chapter 2

Computational Electromagnetics

Traditionally electromagnetic structures were designed using analytical electrody-namic theory [16][26][4]. Analytical solutions methods are still important for un-derstanding the dynamics of simple structures. They are also commonly used forsolving problems with non complicated structures. However when the problemstructure is more complicated, or when it is necessary to determine the character-istics without simplifications, the analytical formulas get rather complex and timeconsuming. Therefore it is natural applying numerical methods in order to simulatethe design performance.

In the electromagnetic community the numerical methods are commonly referredto as Computational Electromagnetics. In similarity with most other numerical ex-pertise areas the four dominating methods are: the Method of Moments (MoM),the Finite Element Method (FEM), the Finite Difference Time Domain (FDTD)method and the Asymptotic method. These methods are today widely used bothwithin industrial and within academical contexts and constitute the absolute foun-dation of most available commercial simulation softwares. This chapter describesthe first three methods and how they are implemented in an electromagnetic per-spective. The fourth method, the Asymptotic Method, is a method used for solvingvery large problem. It is not commonly used together with domain decompositionand will therefore not be covered by this thesis.

2.1 The Method of Moments

Perhaps the Boundary Element Method (BEM) is the traditionally most commonlyused numerical method within the electromagnetic community. However within thecommunity BEM is more known as the Method of Moments (MoM). The methodwas first introduced by R. F. Harington in 1967 [25]. In the article the authorpresents the method of moments as a concept for reducing the electromagnetic fieldequations into matrix equations. In order to demonstrate the method the article alsoincludes examples for computation of the electromagnetic fields caused by surfacesand wires.

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6 CHAPTER 2. COMPUTATIONAL ELECTROMAGNETICS

In contrast to most other numerical methods the MoM uses integral equationsinstead of differential equations in order to describe the dynamics of the problem.The method further utilises the fact that the electromagnetic fields are directlyrelated to the current distribution of an object. Integral equations describing thisrelationship is achieved by the use of Green’s functions which relates the electric fieldat an arbitrary observation point to the current distribution at an arbitrary sourcepoint [33]. By the use of orthogonal expansion the MoM expresses the integralequations as a system of linear equations. The method further uses test functionsin order to invoke the electric field boundary conditions. Using the matrix notationthe expansion coefficients associated with the basis functions are determined andthereby the current distribution can be found.

The most well known implementation of MoM is the Numerical ElectromagneticCode (NEC). The software was written in FORTRAN during the 1970:s. Thedevelopment was led by G. J. Burke and A. J. Poggio at the Lawrence LivermoreNational Laboratory and was sponsored by Naval Oceans Systems Center and theAir Force Weapons Laboratory. The theory behind as well as the code itself isdescribed in [11] and [12] while the usage of the code is described in [13].

2.2 The Finite Element MethodThe development of the Finite Element Method (FEM) is credited to R. L. Courantwho added an appendix to one of his papers [19] where he described how variationalmethods could be used in potential theory. In the description Courant introducedthe use of piecewise-linear approximants using sets of triangles. These triangleswere refereed to as elements by Courant and have, since then, been characteristicfor the method.

The use of finite element method in the computational electromagnetic com-munity began in the late sixties. One early paper, written by P. L. Bahrani [3] in1968, was written on the subject of treating Helmholtz’s equation using FEM. Aconcurrent paper was released the year after by P. Silvester [35] on the subject ofsolving homogeneous waveguide problems using FEM.

Today the use of FEM is well spread in the computational electromagnetic com-munity. A prominent writer in the subject is J. Jin who formulates the basic stepsof the algorithm as follows [27]:

1. Make a subdivision of the computational domain. For a 2D problem this isusually done using triangular or rectangular shaped elements.

2. Select the interpolation function. Usually a polynomial function of first, sec-ond or higher order is chosen.

3. Formulate the problem through system equations. The two most popularsolvers that are used for FEM is the Galerkin’s Method and the Ritz method.

4. Solve the problem by solving the system equations.

2.3. THE FINITE-DIFFERENCE TIME-DOMAIN METHOD 7

2.3 The Finite-Difference Time-Domain MethodThe Finite-Difference Time-Domain (FDTD) method was first introduced byK. S. Yee [56] who presented the algorithm now known as the Yee algorithm. Themethod was given its current name, Finite Differential Time Domain, as well as itsacronym (FD-TD at first but later FDTD) by Allen Taflove in his first article on theregarding the method [40]. According to [34] the Yee algorithm can be summarisedas follows:

1. Rewrite Ampere’s and Faraday’s laws to finite difference form.

2. Discretize space and time.

3. Express the future electric and magnetic field using the difference equationsbased on passed (known) values.

4. Evaluate the magnetic fields one time-step into the future.

5. Evaluate the electric fields one time-step into the future.

6. Repeat the last two steps until the fields are known for the desired time period.

Differential Time Domain (FDTD) it today one of the primary methods in Compu-tational Electromagnetics and is thoroughly described in [41]. Some of the reasonsfor its popularity are the flexibility together with its easiness to implement. Thefact that it operates in time domain means that it provides a solution for all fre-quencies only limited by the length of the sapling period as specified by the Nyquisttheorem. This makes it capable of handling impulsive and non linear behaviour ina natural way. On the drawback side is the fact that FDTD has a time step limitwhich for three dimensions can be expressed as ∆t < h/c

√3, where ∆t is the

time step, h is the cell size and c the speed of light. This limitation mainly appliesfor problems with time scales much longer than the wave propagation time for theregion of interest [10].

Chapter 3

Domain Decomposition

Domain decomposition methods (DDM) were originally developed for solving par-tial differential equations (PDE). The methods solve a problem by decomposing theinitial computational domain into smaller, less complicated, subdomains which aresolved independently. The solution of the full problem is then achieved by assem-bling the solutions of the subdomains. The reason for the recent popularity of themethod is that the subdomains are easily solved in parallel. This advantage makesthe method suitable of todays computers which in general utilise parallel processors.Three strong arguments, presented in [36], for using DDM are:

• simpler to parallelize with higher performance than the original problem

• provides simpler and less complicated problems and geometries

• good convergence properties

The earliest known method utilising domain decomposition, the Schwartz alternat-ing method, was introduced by the German mathematician H. Schwartz in 1870.Though the method was not intended as a numerical method it may be used forsolving elliptic boundary conditions. A numerical implementation of the methodcould be summarised as follows:

1. Start by determining an approximate solution of the problem for the full initialdomain.

2. Decompose the initial domain into subdomains.

3. Solve the the problem of each subdomain using the information in the over-lapping regions as boundary values.

4. Assemble the solutions of the subdomains.

5. Compare the improvement of the solution compared with the last solution.

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10 CHAPTER 3. DOMAIN DECOMPOSITION

6. If the problem has not reach the desired convergence condition, iterate fromstep 2.

A characterisation between different DDMs can be done by using two fundamentalproperties. The first property is used if the method is designed as an iterative ofdirect solvers. Iterative solvers has been given much attention lately since theydemand less computational memory and often are more efficient than the directversion. Theses iterative solvers are usually based on the Schwartz methods. Thedirect methods, also known as substructuring or tearing methods, on the otherhand do not demand the full problem to be solved for a change in one subdomain.An example of a substructuring method is the Schur complement method. It usesthe Schur complement, which is a linear algebraic method, in order to solve thesubdomain problems. A description of the major steps of the algorithm, based onAlgorithm 4.1.1 in [36], follows below:

1. Partition of the structure into nonoverlapping subdomains

2. Compute the Shur complement of each subdomain

3. Merge the Schur complements

4. Factor the Schur complement

The second property of a DDM is the decomposing strategy of the initial domainwhich can be performed with either non-overlapping or overlapping subdomains.Just as the naming of the two categories implies, each division is based on howthe decomposition of the original domain is performed. Figure 3.1 displays anoverlapping decomposition and a non overlapping decomposition of the respectiveinitial domain Ω. The popularity of methods based on the two decompositionstrategies has changed back and forth over time. Generally it can be said that theoverlapping methods have better convergence properties since neighbouring domainshare more information. On the other hand, since subdomains often are solvedin parallel and the communication in between the processors often is the limitingfactor, a faster convergence does not necessarily mean a faster solution.

3.1 Domain Decomposition and ElectromagneticsDuring the years a number of techniques involving Domain Decomposition for solv-ing numerical electromagnetic problems have been presented. An early attempt wasmade by A. Taflove and K. Umashankar in 1989 [45] where they introduced spatialdecomposition as an approach to reduce the dimensions of the matrices involvedwhen solving MoM systems. They also emphasised the improved condition of theresulting matrices which also contributed to the reduction of the computational bur-den. This early attempt was further followed by additional articles [44, 46] which

3.1. DOMAIN DECOMPOSITION AND ELECTROMAGNETICS 11

Ω1 Ω2

(a) A non overlapping decom-position, Ω = Ω1 ∪ Ω2, Ω1 ∩Ω2 = 0

Ω1 Ω2

(b) A overlapping decomposi-tion, Ω = Ω1∪Ω2, Ω1∩Ω2 6= 0

Figure 3.1: Examples of a overlapping respectively a non overlapping decompositionof an initial domain.

continued to explore the possibilities of spatial decomposition together with themethod of moments.

During the same time frame B. Despres also started to explore the possibilities ofusing Domain Decomposition in order to solve numerical electromagnetic problems.His first article on the subject [20] was released in 1991. It was later continued ina second article [21] released in 1993. In these articles he deals with solving theHelmholtz problem using a DDM. The articles also introduce the use of Robin typetransmission conditions, a method which later would be followed by many others[38, 18, 28, 32, 39, 31] Between the two articles B. Despres, together with a coupleof co-writers, released an article [22] in 1992 where they dealt with the harmonicMaxwell equations. The method introduced here is a direct development of the oneB. Despres earlier introduced in his article.

Today there are a number of domain decomposition methods used for solvingnumerical electromagnetic problem. Below follows a summary of the more commonones.

3.1.1 Complex Source ExpansionThe Complex Source Expansion (CSE) was introduced in an article in 2008 [15]by G. Carli et. al. The method is a development of the Complex Source PointExpansion (CSP) which in turn is based on the concept of CSP beams [24]. TheCSP beam concept is established as a way of constituting an exact solution of theMaxwells equations. The CSP method was first described by K. Tap et. al. in2007 [43] and then shortly after in a second article regarding the subject [42]. TheCSP method divides the object surface in subsectional basis function elements whichin turn are grouped together. The interactions between close neighbour groups aswell as elements within the groups are then determined using conventional e.g. FEMor MoM. The interaction between well separated groups are described using complexsource point beams [24]. This means that the fields of each basis functions enclosed


group are expanded. By then using the surface equivalent theorem an equivalentspherical surface can be used to analytically describe the group as a complex pointvalue.

The developments of the CSP method introduced by G. Carli et. al. in theCSE method provide a technique which is more adaptable to electromagnetic struc-tures. The writers have continued to develop the method further in [14, 30]. Themethod decomposes a complex domain into simpler interacting subdomains. Theinteractions are described by using a network formalism with CSP beams as prop-agators. The interactions inside each subdomain are described by using scatteringmatrices. This further provides that the interactions between all subdomains andthe source field can be described by correctly combining the scattering matrices andthe source vector. One advantage with the method is that the subdomains can becharacterised independently once and for all. This means that different interactionsbetween the subdomains can be described by correctly combining the scatteringmatrices. Another advantage is that the interactions within the subdomains can bedetermined by using different electromagnetic computational methods.

3.1.2 Factorisation Domain Decomposition Method

The factorisation domain decomposition method (FACTOPO) was first releasedby A. Barka et. al. in 2000[9]. The development of the method had primarilybeen made at the Electromagnetism and Radar Department of the Office Nationald’Etudes et de recherches Aerospatiales (ONERA), Toulouse. The basic was todevelop a method for collaborative studies involving independent manufacturers.In the article A. Barka et. al. describe the method when used for determining theradar cross section for a fighter jet engine intake. This usage of the method is lateralso covered in [37].

The method works by dividing the initial domain into multiple subdomains. Theelectromagnetic fields at each interface can then be expanded using basis functions inthe form of waveguide modes. By characterised each subdomain, using scatteringmatrices, the interactions in between subdomains and the exterior region can bedetermined by combining the scattering matrices into a Global Scattering Matrix(GSM).

In [5], released 2005, two of the original writers describe how the method can begeneralised and thus used as a solver for large electromagnetic problems. The articledescribes how the method can be used to optimise the positioning of a monopoleantenna on a Fokker 100. This is achieved by utilising the advantage of the fact thatmost of the subdomains remain unchanged and therefore do not need to be recalcu-lated when the antenna position is changed. The generalisations of the method areachieved by the use of deltagaps as basis functions for expanding the fields on theinterfaces. The method is further presented in a more comprehensive manner in [6].In [7] the method is used in order to evaluate the performance of a pod antennainstaller on a motor glider.

An interesting development of the method is presented in [8]. In this article

3.1. DOMAIN DECOMPOSITION AND ELECTROMAGNETICS 13

A. Barka and F. X. Roux describe how the FETI-DP algorithm can be used todetermine the scattering parameters of the subdomains.

As can be noted the FACTOPO method is very similar to CSE. The mostsignificant difference is the choice of basis functions describing the relationship inbetween the domains. In the CSE case the choice of CSP beams makes the methodappropriate for problems with well separated subdomains. FACTOPO on the otherhand can make use of many different basis functions, which makes it appropriatefor problems where the subdomains are more tightly assembled.

3.1.3 Domain Decomposition - Finite Difference Time Domain

The method of Domain Decomposition - Finite Difference Time Domain (DD-FDTD) was presented by F. Xu and W. Hong in 2004 [52]. In the first articlethe writers introduce the method for analysing complicated horn antennas. Themethod works by decomposing the structure into overlapping subdomains. Thecomputation of each subdomain is carried out independently by the use of a localmesh and time step. The exchange of data in between the regions is performed inthe overlapping regions. Since each region has its local mesh and coordinate system,the meshes do not coincide the data exchange that is performed by interpolatingit from one region to another. An advantage with the method is that it providesthe possibility of having non uniform meshes. In this way it is possible to catch theelectromagnetic fields of fine structural details without the necessity of using thesame fine mesh over the full structure.

In a second article [51], also released in 2004, F. Xu and W. Hong develop theirmethod further by presenting an additional application. Here they make use of aconcept for directly computing the transient far field response of a two dimensionalstructure, earlier described in [53], in order to create an effective method of analysingscattering problems for two dimensional structures. This method is later continuedin[54] where a version for three dimensional structures is presented.

The next application of the method is presented in 2006 [55]. Here F. Xu andW. Hong together with K. Wu present a hybrid method, based on DD-FDTD, foranalysing substrate integrated circuits.

3.1.4 Finite Element Tearing and Interconnecting

The Finite Element Tearing and Interconnecting method (FETI) was first intro-duced in 1991 [23] by C. Farhat and F. X. Roux. The two writers were both involvedin the development of a FEM DDM within the mechanical engineering community.The general idea was to create a parallel method where each sub domain could beassigned a separate processor. This meant that each domain could be solved quickly.However they saw that the bottleneck with these methods was the slow informa-tion exchange in between the processors during the computation. In the articlethey suggested a FEM DDM [1] called FETI, that uses hybrid variational principle


based on Lagrange Multipliers. By this they achieved finite element domains thatare disconnected and also avoided the problem with ”floating” domains.

The explorations of the possibilities using FETI algorithm were continued bythe mechanical community and were later adopted by the electrophysics community.An early attempt was made by C. T. Wolfe et.al in 2000 [50]. They introduced themethod as a way of solving the vector wave equation and used perfectly matchedlayer (PML) in order to limit the problem domain.

The algorithm proposed by Wolfe can be summarised in the following steps:

1. Divide the computational domain into nonoverlapping subdomains

2. Discretisize the electric field testing functions in each subdomain

3. Solve each subdomain using Lagrange multipliers by enforcing the continuityof the tangential field across each boundary

4. Compute the electric field of each subdomain

5. Gather the subsolutions to determine the resulting fields

The FETI algorithm proved to be effective but could cause floating subdo-mains [1]. To deal with these early problems the FETI algorithm has been de-veloped further. One of the more common version today in the electromagneticcommunity is the DP-FETI (which also may be referred to as FETI-DP and FETI-DPEM) [49, 47, 59, 29, 58]. The algorithm was presented in 2004 [49] and intro-duces the Robin condition across the interfaces as well as the mortar method fornon-confirming triangulations.

Besides the DP-FETI algorithm, a very promising version of the FETI algorithmis the method presented in [48]. In similarity to DP-FETI this method also makesuse of ROBIN transmission conditions at the interfaces. However it also introducesa ”set of ”cement” unknowns on the inter-domain interfaces” [48] These unknownsallow the grid to be non-conforming/nonmatching. The method is further developedin [57] which also makes use of the results presented in [28].

3.2 Selecting a Domain Decomposition MethodThe method chosen for implementation is the FACTOPO algorithm. The motiva-tion why can be summarised as follows.

The FACTOPO method

• provides a solution of the system dynamics which is independent of the excita-tion since the excitation is not introduced the resulting resulting fields patternfor a specific excitation is desired.

• does not demand the solving of the full initial domain if a subdomain ischanged.

3.2. SELECTING A DOMAIN DECOMPOSITION METHOD 15

• can make use of different numerical solvers for different subdomains.

• is highly suitable for small near field problems.

The first two arguments are important since they make it possible to makechanges in a design, in order to find an optimum, without having to solve the fullproblem. The second is a practical one because it makes the method independentof the solver. This is an important feature since different solvers are suitable fordifferent parts in a design. The last motivation differs FACTOPO from the CSEmethod due to the fact that CSE is more suitable for far field dependencies. Thisis a direct consequence of CSE using source point expansion which makes use oftraditional methods for determining the close range dependencies.

Chapter 4

FACTOPO - The Selected Method

4.1 Introduction to the FACTOPO MethodAs earlier described in Section 3.1.2 the fundamental part of the FACTOPO methodis the possibility to characterise each subdomain using scattering matrices. Thisfurther provides the possibility of describing the subvolume’s interactions with eachother as well as with the surroundings by the use of a global scattering matrix(GSM). The main steps in the algorithm are as follows:

1. Decompose the initial computational volume into sub volumes

2. Determine the scattering matrices for each of the sub volumes

3. Assemble the scattering matrices in to a GSM

4. Solve the global problem by inverting the GSM

An important feature of the method is how the scattering matrices of the subdo-mains are determined. This is done by first determining the admittance matrix ofthe corresponding domain while limiting the domain using PEC. This limits thecomputational volume for each subdomain effectively. The admittance parametersare then converted to scattering parameters. The remainder of this chapter dealswith the described main steps of the algorithm in a more comprehensive manner.

4.2 Notations and Indexes

Notation Explanation

Ω Initial computational domain

P Number of computational subdomains

Continued on the next page

17

18 CHAPTER 4. FACTOPO - THE SELECTED METHOD

Notation Explanation

Vi Subdomain i

M Number of interfaces

Γ The total interface surrounding the current domain

Γj A part of the total Interface Γ corresponding to the interfacetowards a specific neighboring subdomain

Γ0 The perfectly conducting part of the total interface Γ

N Number of interfaces for a certain domain~Ej The tangential electric field of interface Γj~H j The tangential magnetic field of interface ΓjNj The number of basis functions expanding the fields of the

interface Γjajn The outgoing wave at interface Γjbjn The ingoing wave at interface Γj~gjn The magnetic basis function n at interface Γj~f jn The electric basis function n at interface Γj~H j exn The magnetic field induced in the domain by the electric

function ~f jn at one ot interfaces in the domain.

~nj Normal vector to interface Γj , is always directed inwardswith aspect to the current sub domain.

pn Port n.

4.3 Decomposition of Computational Volume

The decomposition of the initial volume into sub volumes may be directly correlatedto the physical division with the initial system in physical parts. However if the re-sulting subvolumes are complicated, the method also allows a further decompositionof the subvolumes in order to achieve subvolumes of equal complexity. Figure 4.1displays an example of how the decomposition could be performed.

4.4. DETERMINING THE SCATTERING PARAMETERS OF A SUBDOMAIN 19

V4

V3V2V1

V0

p1

p2p3

p4p5

p6

p7 p8

Γ1 Γ2 Γ3

Γ4

Figure 4.1: A decomposition example for a horn antenna with a monopole feedplaced in a cavity and a waveguide between the cavity and the horn.

The structure shows a horn antenna with a dipole feed, placed in a cavity, and awaveguide between the feed point and the horn. The initial volume Ω is decomposedinto the sub volumes V0-V4. The horn, waveguide and cavity correspond to subvolumes V1-V3 while V0 corresponds to the exterior region and V4 to a fictive regionrepresenting the feeding of the monopole.

The volumes are connected by ports (p1 − p8) at each of the interfaces Γ1-Γ4.The ports on the virtual interface Γ4 correspond to a coaxial feed connected to themonopole.

The tangential electric (~E) and magnetic field (~H ) of interface Γj are describedaccording to eq. 1 in[6] as:

~Ej =Nj∑n=1

(ajn + bjn

)~f jn (4.1)

~H j =Nj∑n=1

(ajn − bjn

)~gjn (4.2)

Where Nj is the number of basis functions at interface Γj , ajn and bjn representthe amplitude of the inward and outward travelling waves respectively and ~f jn ~gjnthe electric respectively the magnetic basis functions.

4.4 Determining the Scattering Parameters of aSubdomain

The scattering parameters for a specific subdomain can be determined from theadmittance matrix of the same subdomain. The process of determining the admit-tance matrix for a subdomain is shown in Section 4.4.2 - 4.4.3. Thereafter followsthe procedure of determining the scattering matrix based on the admittance matrixin Section 4.4.4. The method described follows the steps described by sections II.b- II.c in [6].


In order to visualise how the scattering parameters are determined an exampleinvolving the waveguide volume V2, shown in Figure 4.1, will be used. This examplewill be referred to as the specific case. Each step of the procedure will first bedescribed in a general case, followed by the same step for the specific case. Thelayout of the specific case is described in Section 4.4.1. The method directly followsthe steps described by sections II.b - II.c in [6].

4.4.1 Specific Case Layout

The specific case consists of a waveguide which is a part of the system displayedin Figure 4.1. The waveguide is displayed in detail in Figure 4.2. The walls of thewaveguide are perfect conductors. At the ends the waveguide has two interfaces,Γ2 & Γ3. The tangential electromagnetic fields at interface Γ2 are expanded bythree basis functions while interface Γ3 is expanded by two basis functions. For thisimplementation waveguide modes have been chosen as basis functions accordingto [9].

x

y

z

w

hV2

Γ2 Γ3

Figure 4.2: A waveguide with two interfaces.

The example layout is defined in such a way that only transverse electric (TE)modes exist through the waveguide. Later in the implementation of the example,objects will be placed within the waveguide. These objects will however be placedin such a way that the TE modes are maintained and no transverse magnetic (TM)modes exist. The electric and magnetic modes inside the waveguide for the TE caseare described by:

Em0,y(x) = sin(mπ

wx

)Hm0,x(x) = 1

Zm0 nj × Em0,y

m = 1, 2, 3, ...

The impedance Zm0 according to table 10.1 in [16] is given by:

Zm0 = η√1−

(fc

f

)2(4.3)


Where η is the intrinsic impedance of the guide medium while fc is the cut offrequency of the waveguide.

In the specific case the basis functions are chosen as:

~f jm = f jmy = Em0,y (4.4)~gjm = gjmx = Hm0,x (4.5)

4.4.2 Determining the Induced Magnetic Field

The process of determining the admittance matrix of a volume uses an expressionwhich determines the magnetic field at interface Γi induced by the electric field inthe volume Vj . This expression makes use of two different methods of determiningthe magnetic field which in turn are set to equal each other.

Determining the Magnetic Field, Method 1

General Case

The tangential magnetic field at interface Γj is given by Equation (4.2). This furtherprovides that the surface current can be expressed as:

nj × ~H j =Nj∑n=1

(ajn − bjn

)nj × ~gjn (4.6)

Where n is the normal vector to interface Γi inward to volume Vj .

Specific Case

The surface current of the two interfaces, limiting the volume V2, in the specificcase can be determined by expanding the sum in Equation (4.6). For the interfacesΓ2 and Γ3 the resulting components are as follows:

Γ2 : n2 × ~H 2 =(a2

1 − b21)n2 × ~g2

1 +(a2

2 − b22)n2 × ~g2

2 +(a2

3 − b23)n2 × ~g2

3

Γ3 : n3 × ~H 3 =(a3

1 − b31)n3 × ~g3

1 +(a3

2 − b32)n3 × ~g3

2

Determining the Magnetic Field, Method 2

General Case

It would be desirable to find a method which expresses the magnetic field of interfaceΓi as a function of the electric field in the volume Vj . This can be achieved by anumerical calculation of the resulting H-field (~H j ex

n ) at Γi induced by the respective


electrical basis function (~f jn), from Equation (4.1). Hence the magnetic field atinterface Γi can be written as:

nj × ~H j =∑i

Nj(i)∑n=1

(ain + bin

)nj × ~H i ex

n (4.7)

where i represents an iteration over all interfaces in the current volume. The equa-tion corresponds to the double sum formulation found at eq. 2 in [6].

Specific Case

The electric field of the two interfaces, based on Equation (4.1) are expressed as:

Γ2 : ~E2 =N2=3∑n=1

(a2n + b2n

)~f 2n

Γ3 : ~E3 =N3=2∑n=1

(a3n + b3n

)~f 3n

By expanding equation (4.7) the two interfaces Γ2 & Γ3 can be expressed by theirrespective components:

Γ2 : n2 × ~H 2 =(a2

1 + b21

)n2 × ~H 2 ex

1 +(a2

2 + b22

)n2 × ~H 2 ex

2 +(a2

3 + b23

)n2 × ~H 2 ex

3

+(a3

1 + b31

)n2 × ~H 3 ex

1 +(a3

2 + b32

)n2 × ~H 3 ex

2

Γ3 : n3 × ~H 3 =(a2

1 + b21

)n3 × ~H 2 ex

1 +(a2

2 + b22

)n3 × ~H 2 ex

2 +(a2

3 + b23

)n3 × ~H 2 ex

3

+(a3

1 + b31

)n3 × ~H 3 ex

1 +(a3

2 + b32

)n3 × ~H 3 ex

2

Equal the two Methods of Determining the Magnetic Field

General Case

Once the fields have been expressed in components it is possible to summarise thecontributions from each component in order to construct the total field applied onthe interfaces. This procedure corresponds to the complete Equation 2 found in [6]:

nj × ~H j =Nj∑n=1

(ajn − bjn

)~nj × ~gjn =

∑i

Nj(i)∑n=1

(ain + bin

)~nj × ~H i ex

n (4.8)

By the Equation above an expression has been found which describes the magneticfield at interface Γi induced by the electric field in the volume Vj .


Specific Case

For the specific case the resulting expressions for the interfaces Γ2 & Γ3 look asfollows:

Γ2 : n2 × ~H 2 =(a2

1 − b21)n2 × ~g2

1 +(a2

2 − b22)n2 × ~g2

2 +(a2

3 − b23)n2 × ~g2

3

=(a2

1 + b21

)n2 × ~H 2 ex

1 +(a2

2 + b22

)n2 × ~H 2 ex

2 +(a2

3 + b23

)n2 × ~H 2 ex

3

+(a3

1 + b31

)n2 × ~H 3 ex

1 +(a3

2 + b32

)n2 × ~H 3 ex

2

(4.9)

Γ3 : n3 × ~H 3 =(a3

1 − b31)n3 × ~g3

1 +(a3

2 − b32)n3 × ~g3

2

=(a2

1 + b21

)n3 × ~H 2 ex

1 +(a2

2 + b22

)n3 × ~H 2 ex

2 +(a2

3 + b23

)n3 × ~H 2 ex

3

+(a3

1 + b31

)n3 × ~H 3 ex

1 +(a3

2 + b32

)n3 × ~H 3 ex

2

(4.10)

4.4.3 Introducing the Test Function and Scalar ProductWhen the magnetic field has been described the next step, according to [6], is toevaluate the resulting field by applying a family of test function ~tjm = ~f jm on eachof the interfaces Γj . These are introduced together with the scalar product:

〈~a, ~b 〉 =∫

Γj

~a ·~b∗ dS

As described by the Gallerkin procedure, the resulting field will be given by theweakly formulated scalar product when applying the test functions ~tjm = ~f jm =sin

(mπw x

): to Equation (4.8).

Nj∑n=1

(ajn − bjn

)〈~nj × ~gjn, ~f jm 〉 =

∑i

Nj(i)∑n=1

(ain + bin

)〈~nj × ~H i ex

n , ~f jm 〉 (4.11)

The formulation described in Equation (4.11) will be treated in two steps corre-sponding to the left and right side of the expression.

Treating the Left Hand Side of Equation (4.11)

General Case

The general solution for each of the basis functions is given by:(ajn − bjn

)〈~nj × ~gjn, ~f jm 〉 =

(ajn − bjn

) ∫∫Γj

(~nj × ~gjn

)~f jm dS (4.12)


Specific Case

For the specific case each basis function can be treated in the following way(ajn − bjn

) ∫∫Γj

(~nj × ~gjn

)· ~f jm dS =

(ajn − bjn

) ∫∫Γj

(nj × gjn x

)·(f jm y

)dS (4.13)

For m 6= n the result of the integral in Equation (4.13) equals 0. For the case whenm = n and j = 2 → nj × x = z × x = y the result is:

(ajm − bjm

) ∫∫Γj

gjm · f jm dS = 1Zm

(ajm − bjm

) ∫ h

0

∫ w

0sin2

(mπ

wx

)dxdy

=

s = mπw x

dx = wmπds

= 1Zm

(ajm − bjm

) whmπ

∫ mπ

0sin2 (s) ds

= 1Zm

(ajm − bjm

) wh

2mπ [s− sin(s) cos(s)]mπ0

= 1Zm

(ajm − bjm

) wh2

(4.14)

Please observe that the solution makes use of the fact that the field direction inthe example is constant since the layout is defined for TE wave propagation. Thismeans that the resulting five cases in the specific case are given by :

(a2

1 − b21)〈~n2 × ~g2

1 ,~f 21 〉 =

(a2

1 − b21) wh

2(a2

2 − b22)〈~n2 × ~g2

2 ,~f 22 〉 =

(a2

2 − b22) wh

2(a2

3 − b23)〈~n2 × ~g2

3 ,~f 23 〉 =

(a2

3 − b23) wh

2(a3

1 − b31)〈~n3 × ~g3

1 ,~f 31 〉 =

(a3

1 − b31) wh

2(a3

2 − b32)〈~n3 × ~g3

2 ,~f 32 〉 =

(a3

2 − b32) wh

2

These five terms correspond to the five diagonal terms when expressing the lefthand side of Equation (4.11) using matrix notation.

Treating the Right Hand Side of Equation (4.11)

General Case

Now, treating the right side of the expression given in Equation (4.11), a generalsolution for the different terms can be written as:


∑i

Nj(i)∑n=1

(ain + bin

)〈~nj × ~H i ex

n , ~f jm 〉 =∑i

Nj(i)∑n=1

(ain + bin

)(∫∫Γj

~nj × ~H i exn · ~f jm dS

)(4.15)

This means that the magnetic field ~H i exn induced by the basis function ~f in sensed by

the test function ~f jm at interface Γj is expressed as:

〈~nj × ~H i exn , ~f jm 〉 =

∫∫Γj

~nj × ~H i exn · ~f jm dS (4.16)

Specific Case

Since the dynamics of an empty waveguide is well known, Equation (4.16) can besolved analytically. The first step in achieving this for the specific test function~f 21 (n = 1, i = 2) and the respective electric basis function is:

(a2

1 + b21

)〈~n2 × ~H 2 ex

1 , ~f 21 〉 =

(a2

1 + b21

) ∫∫Γ2~n2 × ~H 2 ex

1 · ~f 21 dS

=(a2

1 + b21

) ∫∫Γ2

z ×H 2 ex1 x · f 2

1 y dS

=(a2

1 + b21

) ∫∫Γ2

H 2 ex1 y · f 2

1 y dS

=(a2

1 + b21

) ∫∫Γ2

H 2 ex1 f 2

1 dS

(4.17)

The next step is to exchange the test function f 21 based on Equation (4.4).

f 21 = sin

(π

wx

)Concerning the well known wave guide, the magnetic field H 2 ex

1 induced by therespective electric basis function may be solved analytically. To do so the first stepis to provide an ansatz for expressing the electric and magnetic fields:

Ey =(Ae−jkZ +Bejkz

)sin(π

wx

)Hx = 1

Zm

(Ae−jkz −Bejkz

)sin(π

wx

) (4.18)

The next step is to provide the boundary conditions for the two interfaces. Theseare given by the following matrix relation:

[I] = [Y ][V ]

For a two dimensional case the admittance parameters are given by:


Y11 = I1V1

∣∣∣∣V2=0

Y12 = I1V2

∣∣∣∣V1=0

Since [I] and [V ] correspond to [H] and [E] respectively this can be achieved by theuse of a technique where the electric field at the interfaces is zero. This is achieved bythe concept of perfect electric conductors (PEC). Regarding the analytical case thismeans that the electric field at one interface (z = 0) is specified while the electricfield at the other interface (z = L) is set to zero. This provides the followingboundary conditions:

BC1 : Ey(z = 0) = E0

BC2 : Ey(z = L) = 0

Now A and B from the ansatz in equations (4.18) can be solved. The result is:

A = E0e−jkL

j2 sin (kL)

B = E0

(1− e−jkL

j2 sin (kL)

)

Finally this means that an expression for H 2 ex1 can be stated as:

H 2 ex1 = 1

Z1

(Ae−jkz −Bejkz

)sin(π

wx

)Where k = kzm depends on:

k20 = k2

xm + k2ym + k2

zm

kzm =√k2

0 − k2xm − k2

ym

Provided that:

k2xm + k2

ym =(mπ

w

)2+ 0

Then kz is:

kzm =

√k2

0 −(mπ

w

)2=k2

0 = 2πfc0

=

√(2πfc0

)2−(mπ

w

)2(4.19)

In this case both the test function and the excited electric basis function are at thesame interface so z = 0 → e−jkz = 1.


(a2

1 + b21

) ∫∫Γ2

H 2 ex1 f 2

1 dS =(a2

1 + b21

) ∫ h

0

∫ w

0

1Z1

(A−B) sin2(π

wx

)dxdy

=(a2

1 + b21

) h

Z1(A−B)

[x

2 −sin(πwx)

cos(πwx)

2(πwx) ]w

0

=(a2

1 + b21

) 1Z1

(A−B) wh2(4.20)

Please observe that regarding the simple waveguige case, the test function will onlysense the magnetic field induced by its respective electric basis function. In otherwords the result provided by Equation (4.20) is only valid when n = m. Whenm 6= n the result is 0.

Another consequence of this is, when using matrix notation for the right handside of Equation (4.11), the resulting matrix will not be fully populated. The firstcolumn in the matrix, corresponding to the magnetic field ~H 2 ex

1 induced by theelectric basis function ~f 2

1 , is given by:

(a2

1 + b21

)〈~n2 × ~H 2 ex

1 , ~f 21 〉 =

(a2

1 + b21

) ∫∫Γ2

H 2 ex1 f 2

1 dS

=(a2

1 + b21

) 1Z1

(A−B) wh2(a2

1 + b21

)〈~n2 × ~H 2 ex

1 , ~f 22 〉 =

(a2

1 + b21

) ∫∫Γ2

H 2 ex1 f 2

2 dS = 0(a2

1 + b21

)〈~n2 × ~H 2 ex

1 , ~f 23 〉 =

(a2

1 + b21

) ∫∫Γ2

H 2 ex1 f 2

3 dS = 0(a2

1 + b21

)〈~n3 × ~H 2 ex

1 , ~f 31 〉 = −

(a2

1 + b21

) ∫∫Γ2

H 2 ex1 f 3

1 dS

= −(a2

1 + b21

) 1Z1

(Ae−jkzL −BejkzL

) wh2(

a21 + b21

)〈~n3 × ~H 2 ex

1 , ~f 32 〉 = −

(a2

1 + b21

) ∫∫Γ2

H 2 ex1 f 3

2 dS = 0

The second column in the matrix, corresponding to the magnetic field ~H 2 ex2 induced

by the electric basis function ~f 22 , for this example would be:


(a2

2 + b22

)〈~n2 × ~H 2 ex

2 , ~f 21 〉 =

(a2

2 + b22

) ∫∫Γ2

H 2 ex2 f 2

1 dS = 0(a2

2 + b22

)〈~n2 × ~H 2 ex

2 , ~f 22 〉 =

(a2

2 + b22

) ∫∫Γ2

H 2 ex2 f 2

2 dS

=(a2

2 + b22

) 1Z2

(A−B) wh2(a2

2 + b22

)〈~n2 × ~H 2 ex

2 , ~f 23 〉 =

(a2

2 + b22

) ∫∫Γ2

H 2 ex2 f 2

3 dS = 0(a2

2 + b22

)〈~n3 × ~H 2 ex

2 , ~f 31 〉 = −

(a2

2 + b22

) ∫∫Γ2

H 2 ex2 f 3

1 dS = 0(a2

2 + b22

)〈~n3 × ~H 2 ex

2 , ~f 32 〉 = −

(a2

2 + b22

) ∫∫Γ2

H 2 ex2 f 3

2 dS

= −(a2

2 + b22

) 1Z2

(Ae−jkzL −BejkzL

) wh2

The third column in the matrix, corresponding to test function f 23 , for this example

would be:

(a2

3 + b23

)〈~n2 × ~H 2 ex

3 , ~f 21 〉 =

(a2

3 + b23

) ∫∫Γ2

H 2 ex3 f 2

1 dS = 0(a2

3 + b23

)〈~n2 × ~H 2 ex

3 , ~f 22 〉 =

(a2

3 + b23

) ∫∫Γ2

H 2 ex3 f 2

2 dS = 0(a2

3 + b23

)〈~n2 × ~H 2 ex

3 , ~f 23 〉 =

(a2

3 + b23

) ∫∫Γ2

H 2 ex3 f 2

3 dS

=(a2

3 + b23

) 1Z3

(A−B) wh2(a2

3 + b23

)〈~n3 × ~H 2 ex

3 , ~f 31 〉 = −

(a2

3 + 322

) ∫∫Γ2

H 2 ex3 f 3

1 dS = 0(a2

3 + b23

)〈~n3 × ~H 2 ex

3 , ~f 32 〉 = −

(a2

3 + b23

) ∫∫Γ2

H 2 ex3 f 3

2 dS = 0

4.4.4 Introducing Matrix Notation

General Case

In order to introduce a matrix notation it is convenient to once again have a look atthe field formulation described in Equation (4.11). According to [6] the two innerproduct expressions can be expressed in the form of admittance matrices whichfurther can be reformulated to a scattering matrix in the following manner:


Y jinm = −〈~nj × ~H i ex

n , ~f jm 〉 (4.21)Y 0 jjnm = −〈~nj × ~gjn, ~f jm 〉 (4.22)

[S ] =[

Y 0]

+ [Y ]−1 [

Y 0]− [Y ]

(4.23)

By then introducing the vectors ~a &~b corresponding to the ain & bin constants Equa-tion (4.11) can be rewritten in matrix notation:

[Y 0] (~a −~b

)= [Y ]

(~a +~b

)[Y 0]~a −

[Y 0]~b = [Y ]~a + [Y ]~b[

Y 0]

+ [Y ]~b =

[Y 0]− [Y ]

~a

~b =[

Y 0]

+ [Y ]−1 [

Y 0]− [Y ]

~a

~b = [S ]~a

(4.24)

Specific Case

In the simple waveguide example the admittance matrices, using the results deter-mined by Equation 4.20, will look as follows:

[Y ] = wh

2

−A−BZ1

0 0 Ae−jkzL−BejkzL

Z10

0 −A−BZ2


Z2

0 0 −A−BZ3

0 0Ae−jkzL−BejkzL

Z10 0 −A−B

Z10

0 Ae−jkzL−BejkzL

Z20 0 −A−B

Z2

(4.25)

[Y 0]

= −wh2

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

(4.26)


4.5 Assembling the Global Scattering Matrix

The scattering matrix of each subdomain links the ingoing wave amplitudes ~a =[a1a2...am]t of the subdomain to the outgoing wave amplitudes ~b = [b1b2...bm]t ofthe subdomain with the local source vector ~bl

s. The source vector is populated withthe amplitudes of each source.

[b] = [S ] [a] + [~bls]

When the scattering matrices for each of the included sub volumes have been de-termined, the global scattering matrix (GSM) can be assembled. The procedureis based on section 2.F in [6] however some changes are introduced by this thesis.The changes correspond to the removal of the fictive domains. Instead the relationbetween the sources and the input ports are treated by an additional matrix (Sϕ).The explanation of the suggested procedure will make use of the example layoutpresented in Figure 4.1.

Since the outgoing wave at one interface of a subdomain equalizes the ingoingwave of the subdomain at the other side of the interface it is convenient to introducethe port definition. The port vector contains all of the ports corresponding to theinternal ports and external output ports of the full computational volume ~P =[p1, p3, p4, p5, p6, p8]t. The external input ports, which are placed at the externalinterface of the full computational volume, excite the structure and are thereforeplaced in the source vector ~Bs = [p2, 0, 0, 0, 0, p7]t.

Now the GSM (SΩ) of the system can be defined together with the Sϕ matrix.In order to solve the system dynamics it is necessary to solve the following equation:

[SΩ ]× [~P] = [Sϕ]× [~Bs] (4.27)

The process of assembling the system displayed in Figure 4.1 exemplifies thisprocedure. First the scattering matrices of each sub domain are listed:

V1 :

p2

p3

=

S11 1 S1

1 2

S12 1 S1

2 2

p1

p4

V2 :

p4

p5

=

S21 1 S2

1 2

S22 1 S2

2 2

p3

p6

V3 :

p6

p7

=

S31 1 S3

1 2

S32 1 S3

2 2

p5

p8

Secondly the global system is assembled according to Equation 4.27:

4.5. ASSEMBLING THE GLOBAL SCATTERING MATRIX 31

1 0 −S11 2 0 0 0

0 1 −S12 2 0 0 0

0 −S21 1 1 0 −S2

1 2 0

0 −S22 1 0 1 −S2

2 2 0

0 0 0 −S31 1 1 0

0 0 0 −S32 1 0 1

×

p2

p3

p4

p5

p6

p7

=

−S11 1 0 0 0 0 0

−S12 1 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 −S31 2

0 0 0 0 0 −S32 2

×

p1

0

0

0

0

p8

Chapter 5

Implementation Example

This chapter deals with the implementation performed during the project in order toexemplify the use of the FACTOPO method. The implementation has mainly beenperformed using Matlab and has been performed as a stepwise procedure. Thisstepwise procedure has resulted in three versions of the implementation. The firstversion exemplifies the procedure using an analytical solution of a 1-D waveguidestructure. This version follows directly on the method described in chapter 4. Thesecond version of the implementation is used to solve the dynamics of a 1-D structurebased on numerical data exported from CST Microwave Studio. The last versionintroduces a 2-D structure using a 4 port unit cell.

5.1 Analytical Implementation

The first implementation was performed with the aim of accomplishing a workingalgorithm which, by determining the scattering parameters of each subdomain an-alytically, could determine the dynamics for a 1-D structure of empty waveguidesubdomains. Further on the algorithm supports a combination of an arbitrary num-ber of waveguide modes at the interfaces as well as an arbitrary number of waveguidesubdomains in the structure. The implementation follows the method described inchapter 4 directly.

The developed algorithm follows the process displayed in Figure 5.1. Theimplementation was performed using an object oriented approach. Two classes areused in the algorithm. The first is the Subdomain class, it is called both by thesecond class as well as by the main program. Each subdomain instance stores itsown properties, like length, width etcetera, of its respective waveguide subdomain.It also contains a method for computing the scattering matrix. The second class isthe EmStructure class which is only called directly from the main program. TheEmStructure creates the waveguide subdomains and then places them inside anobject array. This means that the object array fully corresponds to the the 1-dimensional waveguide structure. The EmStructure class also contains functionsfor determining the GSM. In Figure 5.1 the first column corresponds to the main

33

34 CHAPTER 5. IMPLEMENTATION EXAMPLE

Declare Variables

Declare the Em-Structure Instance

Declare WaveguideSubdomain Instances

Define GSM (SΩ)

Define Sϕ

Compute WaveParameters

Define Addmit-tance Mattrix

Compute Scatter-ing Parameters

Define and ExciteSource Vector Bs

Compute Result-ing Transmissionand Reflection

Figure 5.1: The algorithm of the Analytical Implementation.

algorithm, the second to the EmStructure class and the third to the SubDomainclass. Below follows an explanation to each step in the algorithm.

5.1.1 Declare Variables

The first step in the algorithm is to declare the variables used by the implementation.The first variables are the start and end frequencies of the frequency span togetherwith the number of frequencies within the span. The next variable, n_z, controls thenumber of length points in each subdomain. Thereafter follows m which describesthe number of waveguide modes at each interface. The last variables are relatedto the geometry of the waveguide subdomains. The analytical solution demandsthe same width and height on all waveguides therefore variables are common. Thelengths of the different waveguides are given in a single vector.

5.1.2 Declare the EmStructure Instance

During the declaration of an EmStructure instance all the variables of the instanceneed to be provided. The width and height are common for all subdomains whilethe lengths are given in a vector where the positions correspond to the placement

5.1. ANALYTICAL IMPLEMENTATION 35

in the structure. The constructor method of the EmStrucure instance will call themethods for creating a instances for each subdomain, the SΩ and the Sϕ matrices.

5.1.3 Declare Waveguide Subdomain InstancesThe subdomain class allows an object containing instances of the class to be declaredwithout having to provide the properties of each Subdomain. This is an importantfeature since it allows the EmStructure class to define the structure, i.e. an ob-ject vector representing a waveguide consisting of waveguide elements, containinginstances of the subdomain class. When the individual instance are declared theparameters have to be provided. They are used in the constructor method whichstores the parameters of the instance properties. It also calls the method for com-puting the scattering parameters of the current subdomain. This means that, aftera subdomain instance has been declared, all its properties are ready to be used.It is also possible to plot the scattering parameters, field strength and admittanceparameters of a specific domain by calling the corresponding plotting functions ofthe domain.

5.1.4 Compute Wave ParametersThis part of the algorithm computes the variables needed for declaring the admit-tance matrix (A,B, kz). It is based on the results from subsection 4.4.3. The waveparameters are dependent of the mode. Hence each parameter is stored in a sepa-rate matrix which contains the computed parameter as a function of frequency andmode number. As an addition the field strengths of the magnetic and electric fieldsare determined respectively as a function of length inside the subdomain.

5.1.5 Define the Admittance MatricesUsing the wave parameters computed in the section above the admittance matri-ces can be defined. This step follows the setup in the specific case described insubsection 4.4.4. However before they can be used for determining the scatteringparameter they need to be normalised. The normalisation is performed accordingto Equation (5.1) which is based on Equation 4.38b in[17].

Ynm = Ynm√YnYm

(5.1)

here Ynm is the admittance at position nm of the admittance matrix. Ynm is thenormalised position at position nm of the resulting normalized admittance matrixwhile Yn & Ym are the normalized admittance for each mode number.

5.1.6 Compute Scattering ParametersThe computation of scattering parameters are performed using Equation (4.23).The normalisation is performed once for every frequency in the frequency span.


5.1.7 Define the GSM (SΩ)The definition of the GSM in the EmStructure is dynamic in order to support a1-D structure with an arbitrary number of modes at the interfaces as well as anarbitrary number of subdomains. The construction follows the procedure describedin section section 4.5.

5.1.8 Define Sϕ

In difference to the process described in section 4.3 no virtual domains are used inthe implementation. Instead the source vector is directly related to the input andoutput ports using the Sϕ matrix which is defined using the scattering parametersfor the external input ports. The structure of the resulting SΩ and Sϕ matrices,as well as how they are related to the scattering parameters of each subdomain, isbest understood in the example provided in section 5.2.

5.1.9 Define and Excite Source Vector Bs

The source vector Bs keeps the field strength for each of the ports and is used toexcite the external input ports.

5.1.10 Compute Resulting Transmission and ReflectionThe reflection and transmission are computed using the following equation:

P = S−1Ω · Sϕ ·Bs (5.2)

Where P will contain the solution for all ports in the network.

5.2 Analytical ExampleBelow follows an example of how the analytical version of the algorithm will workfor a specific case. The procedure is described step by step following the explanationof the algorithm given in section 5.1.

V1

P1(1)

P1(2)

P1(3)

P2(1)

P2(2)

P2(3)

P3(1)

P3(2)

P3(3)

P4(1)

P4(2)

P4(3)

V2

P5(1)

P5(2)

P5(3)

P6(1)

P6(2)

P6(3)

V3

P7(1)

P7(2)

P7(3)

P8(1)

P8(2)

P8(3)

Figure 5.2: The geometry used for the analytical implementation.

The geometry of the analytical example is displayed in Figure 5.2. As can beseen the structure consists of three waveguide subdomains, each with its individual

5.2. ANALYTICAL EXAMPLE 37

length. The example makes use of three waveguide modes at each interface. Theport numbering uses the notation pm(n) where m is the port and n the modenumber.

5.2.1 Declare VariablesThe variables and constants used in the example are listed in the table below. Thenumber of computed frequency points are set to 301.

Variable Value Comment

c0 3 · 108 m/s Speed of light

η 377 Ω Impedance in vacuum

w 15 · 10−3 m Waveguide width (x)

h 7.5 · 10−3 m Waveguide height (y)

L1 5 · 10−3 m Waveguide length (z)



fstart 10 GHz Frequency spectrum start

fend 25 GHz Frequency spectrum end

E0 1 V/m Defined electric field strength.

m 1 Mode number, please observe that only TEm0modes are used.

5.2.2 Declare the EmStructure Instance

In order to achieve the structure displayed in Figure 5.2 the EmStrucure instancehas to be declared using a length vector containing the three different lengthsof the respective waveguides. In this example the length vector is declared as:lVec = [5e-3 23e-3 17e-3];.

5.2.3 Declare Waveguide Subdomain Instances

The EmStrucuture constructor function calls the function which creates the subdo-mains. The subdomains are declared one after the other, each with its individuallength.


5.2.4 Compute Wave Parameters

The computation of waveguide domains results in this case in matrices containingdata for the 3 modes and 301 frequency points used by the example.

5.2.5 Define the Admittance Matrices

The admittance matrix uses the data computed for the waveparameters. Belowfollows an example admittance matrix for one frequency point. As can be seenall parameters which depend on more than a single mode are set to 0. This is aconsequence of the fact that only TEm0 modes are used in the example.

[Y ] =wh

2

− A−BZ1


Z10 0

0 − A−BZ2


Z20

0 0 − A−BZ3


Z3Ae−jkzL−BejkzL

Z10 0 − A−B

Z10 0

0 Ae−jkzL−BejkzL

Z20 0 − A−B

Z20


Z30 0 − A−B

Z3

(5.3)

[Y 0]

= −wh2

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 1 0 1 0

0 0 0 0 0 1

(5.4)

5.2.6 Compute Scattering Parameters

The normalisation step within the computation of the scattering parameters is, inthis example, simplified since no mixed waveguide modes exist within the structure.This means that the equation for the normalization is reduced to:

Ym = Y

Ym(5.5)

5.2.7 Define the GSM (SΩ)

Below follows the deriving of the SΩ for the example case with the 3 subdomainsand 3 waveguide modes.


The first step is to write the port equations for each subdomain using the scat-tering matrices.

V1 :

P2(1)

P2(2)

P2(3)

P3(1)

P3(2)

P3(3)

=

S111 S1

12 S113 S1

14 S115 S1

16

S121 S1

22 S123 S1

24 S125 S1

26

S131 S1

32 S133 S1

34 S135 S1

36

S141 S1

42 S143 S1

44 S145 S1

46

S151 S1

52 S153 S1

54 S155 S1

56

S161 S1

62 S163 S1

64 S165 S1

66

·

P1(1)

P1(2)

P1(3)

P4(1)

P4(2)

P4(3)

V2 :

P4(1)

P4(2)

P4(3)

P5(1)

P5(2)

P5(3)

=

S211 S2

12 S213 S2

14 S215 S2

16

S221 S2

22 S223 S2

24 S225 S2

26

S231 S2

32 S233 S2

34 S235 S2

36

S241 S2

42 S243 S2

44 S245 S2

46

S251 S2

52 S253 S2

54 S255 S2

56

S261 S2

62 S263 S2

64 S265 S2

66

·

P3(1)

P3(2)

P3(3)

P6(1)

P6(2)

P6(3)

V3 :

P6(1)

P6(2)

P6(3)

P7(1)

P7(2)

P7(3)

=

S311 S3

12 S313 S3

14 S315 S3

16

S321 S3

22 S323 S3

24 S325 S3

26

S331 S3

32 S333 S3

34 S335 S3

36

S341 S3

42 S343 S3

44 S345 S3

46

S351 S3

52 S353 S3

54 S355 S3

56

S361 S3

62 S363 S3

64 S365 S3

66

·

P5(1)

P5(2)

P5(3)

P8(1)

P8(2)

P8(3)

In the next step the matrix equations are expanded to systems of linear equations:

V1 :

P2(1) = S111P1(1) + S1

12P1(2) + S113P1(3) + S1

14P4(1) + S115P4(2) + S1

16P4(3)

P2(2) = S121P1(1) + S1

22P1(2) + S123P1(3) + S1

24P4(1) + S125P4(2) + S1

26P4(3)

P2(3) = S131P1(1) + S1

32P1(2) + S133P1(3) + S1

34P4(1) + S135P4(2) + S1

36P4(3)

P3(1) = S141P1(1) + S1

42P1(2) + S143P1(3) + S1

44P4(1) + S145P4(2) + S1

46P4(3)

P3(2) = S151P1(1) + S1

52P1(2) + S153P1(3) + S1

54P4(1) + S155P4(2) + S1

56P4(3)

P3(3) = S161P1(1) + S1

62P1(2) + S163P1(3) + S1

64P4(1) + S165P4(2) + S1

66P4(3)


V2 :

P4(1) = S211P3(1) + S2

12P3(2) + S213P3(3) + S2

14P6(1) + S215P6(2) + S2

16P6(3)

P4(2) = S221P3(1) + S2

22P3(2) + S223P3(3) + S2

24P6(1) + S225P6(2) + S2

26P6(3)

P4(3) = S231P3(1) + S2

32P3(2) + S233P3(3) + S2

34P6(1) + S235P6(2) + S2

36P6(3)

P5(1) = S241P3(1) + S2

42P3(2) + S243P3(3) + S2

44P6(1) + S245P6(2) + S2

46P6(3)

P5(2) = S251P3(1) + S2

52P3(2) + S253P3(3) + S2

54P6(1) + S255P6(2) + S2

56P6(3)

P5(3) = S261P3(1) + S2

62P3(2) + S263P3(3) + S2

64P6(1) + S265P6(2) + S2

66P6(3)

V3 :

P6(1) = S311P5(1) + S3

12P5(2) + S313P5(3) + S3

14P8(1) + S315P8(2) + S3

16P8(3)

P6(2) = S321P5(1) + S3

22P5(2) + S323P5(3) + S3

24P8(1) + S325P8(2) + S3

26P8(3)

P6(3) = S331P5(1) + S3

32P5(2) + S333P5(3) + S3

34P8(1) + S335P8(2) + S3

36P8(3)

P7(1) = S341P5(1) + S3

42P5(2) + S343P5(3) + S3

44P8(1) + S345P8(2) + S3

46P8(3)

P7(2) = S351P5(1) + S3

52P5(2) + S353P5(3) + S3

54P8(1) + S355P8(2) + S3

56P8(3)

P7(3) = S361P5(1) + S3

62P5(2) + S363P5(3) + S3

64P8(1) + S365P8(2) + S3

66P8(3)

The columns and rows of the GSM will correspond to the internal ports andoutput ports of the structure. The GSM is then filled by the scattering parametersdescribing the relations between these ports. This gives the GSM (SΩ) matrixbelow:

SΩ =

1 0 0 0 0 0 −S114 −S1

15 −S116 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 −S124 −S1

25 −S126 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 −S134 −S1

35 −S136 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 −S144 −S1

45 −S146 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 −S154 −S1

55 −S156 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 −S164 −S1

65 −S166 0 0 0 0 0 0 0 0 0

0 0 0 −S211 −S2

12 −S213 1 0 0 0 0 0 −S2

14 −S215 −S2

16 0 0 00 0 0 −S2

21 −S222 −S2

23 0 1 0 0 0 0 −S224 −S2

25 −S226 0 0 0

0 0 0 −S231 −S2

32 −S233 0 0 1 0 0 0 −S2

34 −S235 −S2

36 0 0 00 0 0 −S2

41 −S242 −S2

43 0 0 0 1 0 0 −S244 −S2

45 −S246 0 0 0

0 0 0 −S251 −S2

52 −S253 0 0 0 0 1 0 −S2

54 −S255 −S2

56 0 0 00 0 0 −S2

61 −S262 −S2

63 0 0 0 0 0 1 −S264 −S2

65 −S266 0 0 0

0 0 0 0 0 0 0 0 0 −S311 −S3

12 −S313 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 −S321 −S3

22 −S323 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 −S331 −S3

32 −S333 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 −S341 −S3

42 −S343 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 −S351 −S3

52 −S353 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 −S361 −S3

62 −S363 0 0 0 0 0 1

(5.6)


5.2.8 Define Sϕ

When the GSM has been created the next step is to create the Svarphi matrix.Following the procedure from step 2 it is given by:

Sϕ =

S111 S1

12 S113 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

S121 S1

22 S123 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

S131 S1

32 S133 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

S141 S1

42 S143 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

S151 S1

52 S153 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

S161 S1

62 S163 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S3

14 S315 S3

16

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S324 S3

25 S326

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S334 S3

35 S336

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S344 S3

45 S346

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S354 S3

55 S356

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S364 S3

65 S366

(5.7)

5.2.9 Define and Excite Source Vector Bs

In the example case (P1 & P8) correspond to the external input ports of the sys-tem. The computation of the reflection and transmission of a specific mode fora specific port is performed by setting the corresponding position in the Bs =[P1(1)P1(2)P1(3)P8(1)P8(2)P8(3)] vector to one while the remaining positions re-main at zero.

5.2.10 Compute Resulting Transmission and Reflection

The example computation is performed for mode one and two. As a reference anextra computation is performed for a single domain with the same length as thesum of the three others. This domain will however not be a part of the EmStrucutreinstance. Instead its scattering parameters are plotted directly.

5.2.11 Results From the Analytical Example

The computed reflection and transmission of waveguide mode 1 are displayed inFigure 5.3. As can be seen the results achieved by the domain decompositionmethod are very close to those provided by a single domain. Figure 5.9 displaysthe results achieved for mode 2. Here the reference has changed in order to reflectthat it is computed for the second mode. At lower frequencies all energy of mode


2 is reflected. As soon as the cut off frequency (20 GHz) is passed the energy istransmitted to the other interface.

In order to interpret the result it is wise to keep in mind that an amplitude of≈ -310 dB means that the amplitude is close to −∞ meaning that no energy isreflected. An amplitude of ≈ 0 dB means that the electromagnetic energy inputat one of the external interfaces is transmitted through the waveguide. Since noelectromagnetic energy is reflected, the reflection phase will not be well defined.

10 15 20 25−200

−150

−100

−50

0

50

100

150

200

Frequency [GHz]

Arg

[°]

Reflection phase mode: 1

Ref

DD

10 15 20 25−380

−360

−340

−320

−300

−280

−260

−240

Frequency [GHz]

Am

p d

B

Reflection amplitude mode: 1

Ref

DD

10 15 20 25−200

−150

−100

−50

0

50

100

150

200

Frequency [GHz]

Arg

[°]

Transmission phase mode: 1

Ref

DD

10 15 20 25−120

−100

−80

−60

−40

−20

0

20

Frequency [GHz]

Am

p d

B

Transmission amplitude mode: 1

Ref

DD

Figure 5.3: The resulting reflection and transmission of the analytical examplecorresponding to waveguide mode 1.

5.3. 1-DIMENSIONAL IMPLEMENTATION 43

10 15 20 25−200

−150

−100

−50

0

50

100

150

200

Frequency [GHz]

Arg

[°]

Reflection phase mode: 2

Ref

DD

10 15 20 25−600

−550

−500

−450

−400

−350

−300

−250

Frequency [GHz]

Am

p d

B

Reflection amplitude mode: 2

Ref

DD

10 15 20 25−200

−150

−100

−50

0

50

100

150

200

Frequency [GHz]

Arg

[°]

Transmission phase mode: 2

Ref

DD

10 15 20 25−160

−140

−120

−100

−80

−60

−40

−20

0

20

Frequency [GHz]

Am

p d

B

Transmission amplitude mode: 2

Ref

DD

Figure 5.4: The resulting reflection and transmission of the analytical examplecorresponding to waveguide mode 2. The cut of frequency of mode 2 is 20 GHz.

5.3 1-Dimensional Implementation

The next step in the implementation was to achieve an algorithm capable of im-porting scattering parameters provided by simulations performed with CST Mi-crowave Studio. This means that the analytical determination of scattering pa-rameters will no longer be used. Instead the scattering parameters will be exportedfrom CST Microwave Studio into a text file which is then imported to Matlab.The changes are introduced in the Subdomain class. In exception to these changesthe algorithm remain the same as in the analytical implementation. The process ofthe 1-D algorithm is displayed in Figure 5.5. However the change from an analyticalsolution to a numerical one allows the the waveguides to be nonuniform in widthand height. It also allows objects to be placed within the subdomains. The onlydemand is that the interfaces between two subdomains are of the same size.


Declare Variables

Declare the Em-Structure Instance

Declare WaveguideSubdomain Instances

Define GSM (SΩ)

Define Sϕ

Immport Scatter-ing Parameters

Define and ExciteSource Vector Bs

Compute Result-ing Transmissionand Reflection

Figure 5.5: The algorithm of the 1-D implementation.

5.3.1 Importing CST Microwave Studio DataText output from CST Microwave Studio is formatted in blocks such a way thateach one corresponds to a scattering parameter. At the beginning of each blockthere is one header line of text which describes which scattering parameter the datacorresponds to. The data block contains phase and amplitude of the scatteringparameter for each of the frequencies in the frequency span. By interpreting theheader line the data is placed at the right position of the scattering matrix. Somesample data are visible below.

1 Frequency / GHz S1(1),1(1)/abs,linear S1(1),1(1)/arg,degrees2 ------------------------------------------------------------------3 11 0.00027493845 76.3170654 11.006999 0.00027257186 76.0885185 11.013999 0.00027024363 75.8642016 11.020999 0.00026795333 75.6441677 11.028 0.00026570056 75.4283888 11.035 0.00026348491 75.2168919 11.042 0.00026130593 75.009675

10 11.049 0.00025916333 74.806738

5.4. 1-DIMENSIONAL EXAMPLE 45

5.4 1-Dimensional ExampleIn order to validate the method the aim of the 1-D implementation is to use struc-tures where coupling between different modes occur. The structure consists of thethree waveguide section, displayed in Figure 5.6. Each of the structures contain ametallic object in order to achieve coupling between the modes. The result is vali-dated using a reference structure which is simulated directly in CST MicrowaveStudio. The reference waveguide corresponds to the three waveguide sections whenplaced after each other and is displayed in Figure 5.7. The height of the wavelengthwas 6 mm and the width 15 mm.

(a) V1 with L = 6mm (b) V2 with L = 12mm (c) V3 with L = 18mm

Figure 5.6: The three subdomains which are assembled using the FACTOPOmethod.

Figure 5.7: The reference waveguide used by the 1-D implementation.

5.4.1 Results of the 1-Dimensional Implementation Example

The reflection and transmission of waveguide mode 1 and 2 are displayed in Fig-ure 5.8 & Figure 5.9 respectively. As can be seen the results achieved with domaindecomposition using 8 modes are very close to those achieved when analyzing the


whole structure directly in CST Microwave Studio. If the number of modesis reduces to 1 the accuracy is decreased. The careful reader may note that theamplitude of mode 2, just below cut of frequency, is above 0 dB. This is a numericalerror originating from the CST Microwave Studio simulation and can be seenboth for the domain decomposition results as well as for the reference. The error iscaused by the fact that when determining the scattering parameters it is necessaryto divide the energy which is transmitted or reflected in the input energy. How-ever, since the electromagnetic energy is at resonance, no energy will be transportedwhich causes the error.

10 12 14 16 18 20 22−200

−150

−100

−50

0

50

100

150

200Reflection phase mode: 1

Frequency [GHz]

Arg

[°]

Ref

DD 8 modes

DD 1 mode

10 12 14 16 18 20 22−30

−25

−20

−15

−10

−5

0

5Reflection amplitude mode: 1

Frequency [GHz]

Am

p d

B

Ref

DD 8 modes

DD 1 mode

10 12 14 16 18 20 22−200

−150

−100

−50

0

50

100

150

200Transmission phase mode: 1

Frequency [GHz]

Arg

[°]

Ref

DD 8 modes

DD 1 mode

10 12 14 16 18 20 22−120

−100

−80

−60

−40

−20

0

20Transmission amplitude mode: 1

Frequency [GHz]

Am

p d

B

Ref

DD 8 modes

DD 1 mode

Figure 5.8: The resulting reflection and transmission corresponding to waveguidemode 1 of the 1-D example.


10 12 14 16 18 20 22−200

−150

−100

−50

0

50

100

150

200Reflection phase mode: 2

Frequency [GHz]

Arg

[°]

Ref

DD 8 modes

DD 2 modes

10 12 14 16 18 20 22−80

−70

−60

−50

−40

−30

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0

10

20Reflection amplitude mode: 2

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Am

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Ref

DD 8 modes

DD 2 modes

10 12 14 16 18 20 22−200

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0

50

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200Transmission phase mode: 2

Frequency [GHz]

Arg

[°]

Ref

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DD 2 modes

10 12 14 16 18 20 22−120

−100

−80

−60

−40

−20

0

20Transmission amplitude mode: 2

Frequency [GHz]

Am

p d

B

Ref

DD 8 modes

DD 2 modes

Figure 5.9: The resulting reflection and transmission corresponding to waveguidemode 2 of the 1-D example. The cut off frequency of mode 2 is 20 GHz.

5.5 2-Dimensional Implementation

The 2-D implementation aims on an algorithm capable of analysing 2-dimensionalfinite structures, e.g. an array antenna or frequency selective surface, which makesuse of periodicity in the structure. The algorithm will be built on the earlier de-velopment performed in the thesis but an important change is that the algorithmnow has to support the 2D structure. The implementation of 2D structures is basedon the introduction of 4-port unit cells which the 2D array consist of. These unitcells are, in similarity to the waveguides of the 1-D implementation, analysed usingCST Microwave STudio. The scattering parameters for the respective unit cellare imported by the Matlab algorithm. The most important change introduced inthe algorithm, in order to support the 2-D structure, is the steps for declaring theGSM (SΩ) and Sϕ matrices. These steps are described below. First however it isnecessary to describe the geometry of the 4-port unit cell and how it can be usedfor designing larger structures.


5.5.1 The Unit Cell

The 4-port unit cell used by the algorithm is displayed in Figure 5.10. In principle itconsists of two waveguides crossing each other. The ports are numbered from up todown and left to right. The scattering parameters matrix for the unit cell is writtenas follows when 2 waveguide modes (in the example of the 2-D implementation 8modes will be used) are used at each port :

P1(1)

P1(2)

P3(1)

P3(2)

P5(1)

P5(2)

P7(1)

P7(2)

=

S11 S12 S13 S14 S15 S16 S17 S18

S21 S22 S23 S24 S25 S26 S27 S28

S31 S32 S33 S34 S35 S36 S37 S38

S41 S42 S43 S44 S45 S46 S47 S48

S51 S52 S53 S54 S55 S56 S57 S58

S61 S62 S63 S64 S65 S66 S67 S68

S71 S72 S73 S74 S75 S76 S77 S78

S81 S82 S83 S84 S85 S86 S87 S88

·

P2(1)

P2(2)

P4(1)

P4(2)

P6(1)

P6(2)

P8(1)

P8(2)

(5.8)

P1(1)

P1(2)

P2(1)

P2(2)

P4(2)P4(1)P3(2)P3(1)

P5(1)

P5(2)

P6(1)

P6(2)

P8(2)P8(1)P7(2)P7(1)

Figure 5.10: The 4-port unit cell.

5.5.2 The 3x3 Unit Cell Array

A good example of the kind of structures which can be designed using the unit cellis the 3x3 unit cell array, displayed in Figure 5.11, with two waveguide modes ateach interface. The port numbering of the structure is essential since the algorithmof the EmStructure class uses it when defining the SΩ and Sϕ matrices. Howeverthe algorithm supports an arbitrary number of modes. It is also possible to definewhich modes should be used when importing the scattering parameters.


V11

P1(1)

P1(2)

P2(1)

P2(2)

P4(2)P4(1)P3(2)P3(1)

P5(1)

P5(2)

P6(1)

P6(2)

P8(2)P8(1)P7(2)P7(1)

V12

P10(2)P10(1)P9(2)P9(1)

P11(1)

P11(2)

P12(1)

P12(2)

P14(2)P14(1)P13(2)P13(1)

V13

P16(2)P16(1)P15(2)P15(1)

P17(1)

P17(2)

P18(1)

P18(2)

P20(2)P20(1)P19(2)P19(1)

V21

P21(1)

P21(2)

P22(1)

P22(2)

P23(1)

P23(2)

P24(1)

P24(2)

P26(2)P26(1)P25(2)P25(1)

V22

P27(1)

P27(2)

P28(1)

P28(2)

P30(2)P30(1)P29(2)P29(1)

V23

P31(1)

P31(2)

P32(1)

P32(2)

P34(2)P34(1)P33(2)P33(1)

V31

P35(1)

P35(2)

P36(1)

P36(2)

P37(1)

P37(2)

P38(1)

P38(2)

P40(2)P40(1)P39(2)P39(1)

V32

P41(1)

P41(2)

P42(1)

P42(2)

P44(2)P44(1)P43(2)P43(1)

V33

P45(1)

P45(2)

P46(1)

P46(2)

P48(2)P48(1)P47(2)P47(1)

Figure 5.11: An example of a 3x3 unit cells array.

5.5.3 Defining the GSM (SΩ)The step in the algorithm for defining the GSM can be divided into the followingsteps:

1. Define a matrix which covers all ports in the structure, including the externalinputs.

2. Identify the input and output port numbers of each unit cell.

3. Place the scattering parameters in the matrix based on the input ports andoutput ports of the current cell, output ports corresponds to the row and


input ports to the column

4. Identify the external inputs in the structure.

5. Redefine the matrix keeping only the internal ports and outputs ports, exter-nal ports are removed.

6. Add the matrix with the identity matrix of the same size.

As can be seen the port numbering of the cells is essential for the algorithm. Itis important to understand that the algorithm handles each mode as an individualport at the interface. This means that in reality the numbering of the ports for aunit cell may has changed in order to reflect the fact that the unit cell will contain8 ·m ports. Where m corresponds to the number of modes at the interfaces.

5.5.4 Defining the (Sϕ)

The Sϕ matrix is defined similarly to the SΩ. In fact the first step in defining theSϕ is to copy the SΩ matrix after step 3 above. The second step is to remove theexternal output ports from the matrix by redefining the matrix keeping the rowsand columns containing the scattering parameters which correspond to the externaloutputs at their respective positions.

5.6 2-Dimensional Implementation Example

The 2-D example simulated by the algorithm corresponds to the structure describedin subsection 5.5.2. The unit cell was simulated in CST Microwave Studio andanalysed by the algorithm. The simulation is compared to a 3x3 strucutre simulateddirectly in CST Microwave Studio.

5.6.1 Unit Cell in CST Microwave Studio

The unit cell designed in CST Microwave Studio is displayed in Figure 5.12.The corner posts are 1x1 mm. The length and width of each cell, including thecorner posts, are 22 mm. The height is 15 mm. The CST Microwave Studiosimulation used 16 waveguide modes, however only 8 were excited.

5.6. 2-DIMENSIONAL IMPLEMENTATION EXAMPLE 51

Figure 5.12: The unit cell as it was designed in CST Microwave Studio.

5.6.2 3x3 Unit Cell Array in CST Microwave Studio

The reference structure is displayed in Figure 5.13. It corresponds to the 3x3 struc-ture. However, since it is simulated as a single structure, it will have 12 ports.

Figure 5.13: The unit cell structure, consisting of a 3x3 array, which was studied inthe example.

5.6.3 Results of the 2-D Implementation

The results of the 2-D simulation is displayed in Figure 5.14. As can be seen theresult is very noisy. This is a consequence of the fact that the cut off frequencyof mode 2 is in the middle of the analysed frequency span. This means that thelongitudinal wavenumber will be close to 0 which in turn means that the pulse usedto analyse the structure in time domain will decay very slowly due to resonance.


As an alternative a curve corresponding to a coputation performed without thescattering parameters corresponding to mode 2 has been added. It shows that thenoise disappears. However the accuracy is decreased since mode 2 is importantfor the computation. As a last attempt the simulation was performed once moreusing CST Microwave Studio. This time the frequency domain solver was usedinstead of the time domain solver used before. The results based on the frequencydomain determined scattering parameters are displayed in Figure 5.15. The figureshows a clear improvement regarding noise as well as accuracy. In this figure a thirdcurve has been added which displays the result for a computation performed withonly one mode.

8 10 12 14 16 18 20−200

−150

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0

50

100

150

200Reflection phase port: 1

Frequency [GHz]

Arg

[°]

DD 8 modes

DD 7 modes

Ref.

8 10 12 14 16 18 20−50

−40

−30

−20

−10

0

10Reflection amplitude port: 1

Frequency [GHz]

Am

p d

B

DD 8 modes

DD 7 modes

Ref.

8 10 12 14 16 18 20−200

−150

−100

−50

0

50

100

150

200Transmission phase port: 9

Frequency [GHz]

Arg

[°]

DD 8 modes

DD 7 modes

Ref.

8 10 12 14 16 18 20−25

−20

−15

−10

−5

0

5Transmission amplitude port: 9

Frequency [GHz]

Am

p d

B

DD 8 modes

DD 7 modes

Ref.

Figure 5.14: The resulting reflection and transmission corresponding to waveguidemode 1 of the 2-D example using the first 8 waveguide modes, the 7 mode curveexcludes mode 2 from the computation.

5.6. 2-DIMENSIONAL IMPLEMENTATION EXAMPLE 53

8 10 12 14 16 18 20−200

−150

−100

−50

0

50

100

150

200Reflection phase port: 1

Frequency [GHz]

Arg

[°]

DD 8 modes

DD 1 mode

Ref.

8 10 12 14 16 18 20−25

−20

−15

−10

−5

0Reflection amplitude port: 1

Frequency [GHz]

Am

p d

B

DD 8 modes

DD 1 mode

Ref.

8 10 12 14 16 18 20−200

−150

−100

−50

0

50

100

150

200Transmission phase port: 9

Frequency [GHz]

Arg

[°]

DD 8 modes

DD 1 mode

Ref.

8 10 12 14 16 18 20−16

−14

−12

−10

−8

−6

−4

−2Transmission amplitude port: 9

Frequency [GHz]

Am

p d

B

DD 8 modes

DD 1 mode

Ref.

Figure 5.15: The resulting reflection and transmission corresponding to waveguidemode 1 of the 2-D example based on scattering parameters determined with afrequency domain solver using 1 and 8 modes respectively.

Chapter 6

Discussion

This chapter will discuss domain decomposition in general and the FACTOPO do-main decomposition method in particular. It also includes a section of suggestionsfor future work in order to achieve a software, based on FACTOPO, for analysingantenna arrays and FSS.

6.1 Domain Decomposition in General

The theoretical study shows that it is obvious that domain decomposition is anemerging numerical technique within the computational electromagnetic commu-nity. This is especially true for large structures which demands parallelisation asdescribed in [28]: "Without parallelization, it would be prohibitive to solve large fi-nite antenna arrays using traditional methods... However, due to many repetitionsin the structure, it is natural to apply DDMs to model large finite antenna arrays".An important sign of domain decomposition as an emerging technique is the factthat the commercial producers of electromagnetic simulation software have startedto implement the method [2].

One complication when implementing the method is that, in order to take fulladvantage of the method, it is necessary to use a highly specialised algorithm totreat a specific problem. Luckily a number of different implementations of themethod exist. In order to determine which implementation is appropriate for aspecific problem it is necessary to not only consider the structure it self but alsowhat data is desired from the simulation.

These two arguments bring the discussion to another different question namelythe convenience of developing specialised algorithms. In general it can be said thatthere are two main reasons for developing a specialised DDM algorithm instead ofusing a more general simulation software. Either because it is necessary to analysea large number of similar problems or because the problem is very large or complexand demands an algorithm which allows for parallel processing. Now both of thesecriterion could be met by problems which would be candidates for a unique domaindecomposition algorithm. However it may be more efficient to use a commercial

55

56 CHAPTER 6. DISCUSSION

software, even though it is not optimal for the problem, since such software does notdemand a specialised algorithm to be developed in order to analyse the structure.

6.2 The FACTOPO MethodThe use of the FACTOPO method has proven to imply both advantages and disad-vantages. One advantage is how the method deals with a periodic structure sinceonly one example of every unique kind of subdomain needs to be analysed. Thedynamics of the full structure is then analysed using the GSM. This also means thatit is relatively easy to introduce small changes in the structure by changing one or afew of the included subdomains. This is a method of changing the dynamics whichalso can be introduced in order to change the external problem or internal problemindependently from each other. This in turn could mean that the dynamics of thestructure surrounding an antenna can be analysed once and for all.

One disadvantage with the FACTOPO method is caused by the time domainsolver when determining the admittance matrix. The problem is that the admit-tance matrix is calculated by treating each subdomain as a metallic cavity. If thelosses in the cavity are low, which usually is the case, the energy in the computationdomain will decrease very slowly. This will in turn result in a very long simulationtime. Since frequency domain based solvers usually does not suffer from this prob-lem, they may be the only usable alternative for calculating the admittance matrixusing the FACTOPO method.

Another advantage when using the method for designing finite periodic struc-tures, i.e. antenna arrays, is the fact that the computation is independent of theexcitation. This is a clear advantage if it is desired to find the dynamics as a functionof the excitation which is usually the case for array antennas.

6.3 Future WorkBelow follows a number of suggestions for future work to develop the implementa-tion of the FACTOPO method in order to solve problems based on finite periodicelectromagnetic structures.

The choice of basis for the expansion of the electromagnetic field on the bound-ary between the subdomains usually has a major impact on the computational timeand accuracy. The least number of basis functions required, and thereby shortestcomputational times, is obtained by solving an eigenvalue problem that results inan orthonormal basis. The waveguide modes used in the thesis represents suchbasis functions. However waveguide modes cannot be used for elements in an ar-ray antenna or in a frequency selective surface. It should be possible to find anorthonormal basis for the array antenna/frequency selective surface problem. Thiswould probably be the next step in implementing the FACTOPO method.

Another important aspect of the implementation related to the choice of anappropriate basis function is the determination of admittance parameters. As earlier

6.3. FUTURE WORK 57

described the determination is based on the induced magnetic field caused by anexcited electric field on a limited part of the subdomain interface while the electricfield is set to zero at the remaining interface using PEC. The main purpose of thisproject has been to investigate structures with well defined interfaces and ports.Before implementing this step it is important to investigate the need of supportingopen structures i.e. structures without walls or posts limiting the subdomains. Ageneral method has to support such structures and, in that case, therefore alsoneeds to support a non zero electric field at the remaining part of the subdomainboundary. One method in achieving this is to implement an alternative magneticfield boundary condition using the magnetic counterpart to PEC.

In order to achieve an algorithm which can treat antenna array problems it isalso necessary to introduce the coupling between the structure and the surroundingair. This could be achieved by introducing a fifth port to the unit cell, whichin turn would be connected to a surrounding common air box. It is however firstnecessary to investigate how the air port coupling should be described. This will alsocomplicate the domain decomposition since it will introduce a common boundaryto the unit cells.

Chapter 7

Conclusion

In this thesis the numerical method of domain decomposition has been studied. Thestudy has been concentrated on algorithms available for computational electromag-netics. The aim has been to find and further study one appropriate method foranalysing finite periodic electromagnetic structures like frequency selective surfacesand array antennas.

One DDM, the FACTOPO method, has been selected and studied in detail.The study has included an implantation of the method. The implementation hasbeen performed in three versions. The first deals with a 1-Dimensional waveguideproblem which follows the FACTOPO method for determining the scattering pa-rameters of a subdomain using admittance parameters of the corresponding domain.By limiting the subdomain with perfect electric conductor (PEC) the computationalproblem is efficiently reduced. The second implementation continues to explore the1-Dimensional problem but introduces the ability of importing scattering parame-ters of a specific domain from an electromagnetic simulation software. The third,and last, implementation is an extension of the second version which introduces a2-D structure based on a 4-port unit cell. At last the use of domain decompositionin general and FACTOPO in specific has been discussed.

Judging by the theoretical study as well as the performed implementation it isevident that domain decomposition is a promising numerical method for computa-tional electromagnetics. It is also clear that much implementation work remainsbefore an in house developed software can be used for designing electromagneticstructures. Further on both Ansys and CST, the companies behind the electro-magnetic simulation software HFSS and Microwave Studio respectively, haveeither already introduced domain decomposition or are working on an implemen-tation of their commercial simulation software [2]. Hence it can be concluded thatfurther studies aiming on developing an in house simulation software based on DDMonly should be undertaken if it is desired to analyse very specific problems involvingeither changing excitations or large complex periodic structures.

59

Chapter 8

Abbreviations

Abbreviation Explanation

BEM Boundary Element Method

CSE Complex Source Expansion

CSP Complex Source Point

DD Domain Decomposition

DDM Domain Decomposition Method

DP-FETI Dual Primal Finite Element Tearing and Interconnect

EM Electro Magnetics

FEM Finite Element Method

FETI Finite Element Tearing and Interconnect

FETI-DPEM Finite Element Tearing and Interconnect Dual Primal Elec-tro Magnetics

GSM Global Scattering Matrix

MoM Method of Moments

PEC Perfect Electric Conductor

Ref Reference

TE Transverse Electric field

TM Transverse Magnetic field

61

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