analysis of waveguide discontinuities with the self ... · analysis of waveguide discontinuities...

Analysis of Waveguide Discontinuities with theSelf-Adaptive hp Finite Elements

Luis E. Garcıa-Castillo1∗, David Pardo2,4, Ignacio Gomez-Revuelto2,

Leszek F. Demkowicz3, Carlos Torres-Verdın4

1Departamento de Teorıa de la Senal y Comunicaciones.

Universidad Carlos III de Madrid.

Escuela Politecnica Superior (Edificio Torres Quevedo).

Avda. de la Universidad, 30. 28911 Leganes (Madrid) Spain.

Fax:+34-91-6248749, [email protected]

2Departamento de Ingenierıa Audiovisual y Comunicaciones.

Universidad Politecnica de Madrid, Madrid, Spain.

[email protected]

3ICES, University of Texas at Austin, Austin TX 78712, USA

[email protected], [email protected]

4Department of Petroleum and Geosystems Engineering,

The University of Texas at Austin, USA

[email protected], [email protected]

∗This work has been initiated during a stay of the first author at ICES supported by the Sec-retarıa de Estado de Educacion y Universidades of Ministerio de Educacion, Cultura y Deporte ofSpain. The authors want also to acknowledge the support of Ministerio de Educacion y Ciencia ofSpain under project TEC2004-06252/TCM.

AbstractThe accurate analysis and characterization of waveguide discontinuities is

an important issue in microwave engineering. In some cases, the discontinu-ities are an unavoidable result of mechanical or electric transitions which ef-fects have to be minimized; in other cases, the discontinuities are deliberatelyintroduced into the waveguide to perform a certain electric function.

This three parts report present the work on tha analysis of different types ofrectangular waveguide discontinuities by using a fully automatic hp-adaptivefinite element method.

In the first part, the general theory of rectangular waveguides is derivedand described as well as a very detailed mathematical analysis of rectangularwaveguide discontinuities for both, the E-plane and H-plane formulations. Wealso present a practical approach for computing scattering parameters usingFinite Elements. These scattering parameters are essential for characterizingwaveguide discontinuities, as we demonstrate throughout the paper.

In the second part, a fully automatic energy-norm based hp-adaptive FiniteElement (FE) strategy applied to a number of relevant waveguide structures, ispresented. The methodology produces exponential convergence rates in termsof the energy-norm error of the solution against the problem size (number ofdegrees of freedom). Extensive numerical results, including those related tomicrowave engineering devices of medium complexity, demonstrate the suit-ability of the hp-method for solving different rectangular waveguide disconti-nuities. These results illustrate the flexibility, reliability, and high-accuracy ofthe method.

In the third part, the work is extended by presenting a fully automaticgoal-oriented hp-adaptive FE in terms of the scattering or S-parameters. Themethodology produces exponential convergence rates in terms of an upperbound of a user-prescribed quantity of interest (in our case, the S-parameters)against the problem size (number of degrees of freedom). Specifically, thehp-methodology is applied to a fifth order microwave filter using a H-planestructure consisting on symmetric inductive irises. Numerical results illustratethe main differences between the energy-norm based and the goal-orientedbased hp-adaptivity, when applied to rectangular waveguide structures. Theyalso demonstrate the suitability of the goal-oriented hp-method for solvingproblems involving rectangular waveguide structures.

The self-adaptive hp-FEM, either the energy-norm based version or thegoal-oriented based approach, provides similar (sometimes more) accurate re-sults than those obtained with semi-analytical techniques such as the ModeMatching method, for problems where semi-analytical methods can be ap-plied. At the same time, the hp-FEM provides the flexibility of modeling morecomplex waveguide structures and including the effects of dielectrics, metal-lic screws, round corners, etc., which cannot be easily considered when usingsemi-analytical techniques.

CONTENTS i

Contents

I Waveguide Theory and Finite Element Formulation 1

I.1 Introduction 1

I.2 Overview of Electromagnetic Guided Wave Theory 3I.2.1 Waveguide propagation modes . . . . . . . . . . . . . . . . . . . . 6I.2.2 Rectangular Waveguides . . . . . . . . . . . . . . . . . . . . . . . 11

I.3 Analysis of Rectangular Waveguide Discontinuities 14I.3.1 Analysis of waveguide discontinuities in 2D . . . . . . . . . . . . . 15I.3.2 2D Variational Formulation . . . . . . . . . . . . . . . . . . . . . 16

I.3.2.1 H-plane variational formulation . . . . . . . . . . . . . . . 19I.3.2.2 E-plane variational formulation . . . . . . . . . . . . . . . 22

I.3.3 Computation of scattering parameters . . . . . . . . . . . . . . . 25I.3.3.1 H-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30I.3.3.2 E-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

I.4 Conclusions 31

II Energy-Norm Based Automatic hp-Adaptivity 32

II.1 Introduction 32

II.2 hp Finite Elements and Automatic Adaptivity 33II.2.1 hp-Finite Elements (FE) . . . . . . . . . . . . . . . . . . . . . . . 34II.2.2 Fully Automatic hp-Adaptivity . . . . . . . . . . . . . . . . . . . 34

II.2.2.1 The projection based interpolation operator . . . . . . . . 35II.2.2.2 The mesh optimization algorithm . . . . . . . . . . . . . 36

II.3 Numerical Results 39II.3.1 H-plane discontinuities . . . . . . . . . . . . . . . . . . . . . . . . 40

II.3.1.1 H-plane waveguide section . . . . . . . . . . . . . . . . . 40II.3.1.2 H-plane right angle bend . . . . . . . . . . . . . . . . . . 45II.3.1.3 H-plane symmetric inductive iris . . . . . . . . . . . . . . 50II.3.1.4 H-plane zero thickness septum . . . . . . . . . . . . . . . 55II.3.1.5 H-plane zero length septum . . . . . . . . . . . . . . . . . 59

CONTENTS ii

II.3.2 E-plane discontinuities . . . . . . . . . . . . . . . . . . . . . . . . 63II.3.2.1 E-plane right angle bend . . . . . . . . . . . . . . . . . . 63II.3.2.2 E-plane right angle bend with a round corner . . . . . . . 68II.3.2.3 E-plane capacitive symmetric iris . . . . . . . . . . . . . . 73II.3.2.4 E-plane double stub section . . . . . . . . . . . . . . . . . 78

II.4 Conclusions 87

II.5 Acknowledgment 87

III Goal-Oriented hp-Adaptivity 88

III.1 Introduction 88

III.2 A Fully Automatic Goal-Oriented hp-Adaptive FEM 89III.2.1 Variational Formulation and Goals . . . . . . . . . . . . . . . . . 89III.2.2 Goal-Oriented Adaptivity . . . . . . . . . . . . . . . . . . . . . . 91III.2.3 Projection based interpolation operator . . . . . . . . . . . . . . . 94III.2.4 Goal-Oriented hp-Mesh Optimization Algorithm . . . . . . . . . . 95III.2.5 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . 96

III.3 Numerical Results 96III.3.1 A Rectangular Waveguide Structure with Six Inductive Irises . . 97III.3.2 Results Obtained by Using the Energy-Norm hp-Adaptive Strategy 98III.3.3 Results Obtained by Using a Mode Matching Technique . . . . . 101III.3.4 Results Obtained by Using the Goal-Oriented hp-Adaptive Strategy101III.3.5 A Comparison Between the Energy-Norm and Goal-Oriented Self-

Adaptive hp-FE Strategies . . . . . . . . . . . . . . . . . . . . . . 105

III.4 Conclusions 109

LIST OF FIGURES iii

List of Figures

I.1 General waveguide discontinuity . . . . . . . . . . . . . . . . . . . . 3I.2 Examples of H-plane and E-plane discontinuities . . . . . . . . . . . 4I.3 General homogeneous waveguide with one conductor . . . . . . . . . 8I.4 Rectangular waveguide . . . . . . . . . . . . . . . . . . . . . . . . . 11I.5 Normalized propagation constant versus wavenumber for the modes

of a rectangular waveguide. The wavenumber is normalized to thecutoff frequency of the TE10 mode . . . . . . . . . . . . . . . . . . . 14

I.6 Example of 2-ports monomode rectangular waveguide discontinuity 16I.7 A rectangular H-plane discontinuity and its 2D FEM modeling . . . 17I.8 A rectangular E-plane discontinuity and its 2D FEM modeling . . . 18II.1 H-plane waveguide section . . . . . . . . . . . . . . . . . . . . . . . 42II.2 Initial mesh and some hp meshes for the H-plane waveguide section 43II.3 Magnitude of Hy, i.e., |Hy|, corresponding to the H-plane waveguide

section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44II.4 Convergence history for the H-plane waveguide section (energy norm

error for the magnetic field solution) . . . . . . . . . . . . . . . . . . 44II.5 H-plane 90 degrees bend . . . . . . . . . . . . . . . . . . . . . . . . 46II.6 Initial mesh for the H-plane 90 bend . . . . . . . . . . . . . . . . . 47II.7 Magnitude of Hy, i.e., |Hy|, corresponding to the H-plane 90 bend . 47II.8 Final hp mesh for the H-plane 90 bend . . . . . . . . . . . . . . . . 48II.9 Convergence history for the H-plane 90 bend . . . . . . . . . . . . 49II.10 H-plane symmetric inductive iris (l/a = 0.6, t/a = 0.2) . . . . . . . 51II.11 Initial mesh for the H-plane symmetric inductive iris . . . . . . . . . 52II.12 Convergence history for the H-plane inductive iris . . . . . . . . . . 53II.13 Magnitude of Hy for the H-plane inductive iris . . . . . . . . . . . . 53II.14 11th mesh for the H-plane symmetric inductive iris showing heavy

refinements around the left corners . . . . . . . . . . . . . . . . . . 54II.15 19th mesh for the H-plane symmetric inductive iris showing heavy

refinements around the left and also right corners . . . . . . . . . . 54II.16 H-plane zero thickness septum (l/a = 0.1) . . . . . . . . . . . . . . 56II.17 Initial mesh for the H-plane zero thickness septum . . . . . . . . . . 57II.18 Convergence history for the H-plane zero thickness septum . . . . . 57II.19 7th mesh for the zero thickness septum . . . . . . . . . . . . . . . . 58II.20 Magnitude of Hy corresponding to the H-plane zero thickness sep-

tum showing a stationary wave pattern at the input waveguide andsingular behavior of the field at the septum corners . . . . . . . . . 58

LIST OF FIGURES iv

II.21 H-plane zero length septum (t/a = 0.1) . . . . . . . . . . . . . . . . 60II.22 Initial mesh for the H-plane zero length septum . . . . . . . . . . . 61II.23 Convergence history for the H-plane zero length septum . . . . . . 61II.24 7th mesh for the zero length septum . . . . . . . . . . . . . . . . . . 62II.25 Magnitude of Hy corresponding to the H-plane zero length septum

showing a stationary wave pattern at the input waveguide and sin-gular behavior of the field at the septum corners . . . . . . . . . . . 62

II.26 E-plane 90 degrees bend . . . . . . . . . . . . . . . . . . . . . . . . 64II.27 hp mesh of 11th iteration of the E-plane 90 bend . . . . . . . . . . 65II.28 Magnitudes of Ey and Ex corresponding to the E-plane 90 bend

showing a singular behavior of the field at the corner . . . . . . . . 66II.29 Convergence history for the E-plane 90 bend . . . . . . . . . . . . 67II.30 E-plane 90 degrees bend with round corner (r/b = 0.2) . . . . . . . 69II.31 Initial mesh for the E-plane 90 bend with round corner . . . . . . . 70II.32 10th hp mesh for the E-plane 90 bend with round corner . . . . . . 70II.33 Magnitudes of Ey and Ex corresponding to the E-plane 90 . . . . . 71II.34 Convergence history for the E-plane 90 bend with round corner . . 72II.35 E-plane capacitive symmetric iris (d/b = 0.6, t/b = 0.2) . . . . . . . 74II.36 Initial mesh and mesh of the 4th iteration for the E-plane capacitive

symmetric iris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75II.37 Convergence history for the E-plane capacitive symmetric iris . . . . 76II.38 Magnitude of Ey and Ex corresponding to the E-plane capacitive

symmetric iris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77II.39 E-plane double stub structure (bs1/b = bs2/b = 5.0249, l/b = 1.2608) 81II.40 Frequency response of E-plane double stub section (|S11| in dB) . . 82II.41 Convergence history for the E-plane double stub section . . . . . . . 83II.42 Initial mesh and mesh corresponding to 1% energy error for the E-

plane double stub section . . . . . . . . . . . . . . . . . . . . . . . . 84II.43 Meshes corresponding to 1% energy error for the E-plane double stub

section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85II.44 Electric field

√|Ex|2 + |Ey|2 in the E-plane double stub section . . . 86

III.1 2D cross-section of the geometry of the waveguide problem with sixinductive irises. The initial grid is composed of 27 elements, asindicated by the black lines . . . . . . . . . . . . . . . . . . . . . . . 98

LIST OF FIGURES v

III.2 Convergence history using the fully automatic hp-adaptive strategyfor different initial grids. Different colors correspond to different ini-tial order of approximation. 27 is the minimum number of elementsneeded to reproduce the geometry, while 1620 is the minimum num-ber of elements needed to reproduce the geometry and to guaranteeconvergence of the iterative two grid solver described in [1]. . . . . . 99

III.3 Return loss of the waveguide structure . . . . . . . . . . . . . . . . 100III.4 | Hx | (upper figure), | Hy | (center figure), and

√| Hx |2 + | Hy |2

(lower figure) at 8.72 Ghz for the six irises waveguide problem. . . . 101III.5 | Hx | (upper figure), | Hy | (center figure), and

√| Hx |2 + | Hy |2


√| Hx |2 + | Hy |2


√| Hx |2 + | Hy |2

(lower figure) at 9.71 Ghz for the six irises waveguide problem. . . . 103III.8 Return loss of the waveguide structure. This graph has been com-

puted using a Mode Matching technique [2]. . . . . . . . . . . . . . 103III.9 Convergence history for the waveguide problem with six inductive

irises at different frequencies . . . . . . . . . . . . . . . . . . . . . . 106III.10 2D cross-section of the geometry of the waveguide problem with six

inductive irises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107III.11 Convergence history for the waveguide problem with six inductive

irises at 8.82 Ghz, with the three central cavities filled with a resistivematerial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

III.12 hp-grids obtained by using the fully automatic goal-oriented hp-adaptive finite element method . . . . . . . . . . . . . . . . . . . . . 109

LIST OF TABLES vi

List of Tables

I.1 Summary of results for the rectangular waveguide . . . . . . . . . . 13II.1 Scattering parameters for the H-plane waveguide section . . . . . . 43II.2 Scattering parameters for the H-plane 90 bend . . . . . . . . . . . 48II.3 Scattering parameters for the H-plane symmetric inductive iris . . . 52II.4 Scattering parameters for the H-plane zero thickness septum . . . . 55II.5 Scattering parameters for the H-plane zero length septum . . . . . . 59II.6 Scattering parameters for the E-plane 90 bend . . . . . . . . . . . 65II.7 Scattering parameters for the E-plane 90 bend with round corner . 68II.8 Scattering parameters for the E-plane capacitive symmetric iris . . . 73

1

Part I

Waveguide Theory and FiniteElement Formulation

I.1 Introduction

The accurate analysis and characterization of waveguide “discontinuities” is an im-portant issue in microwave engineering (see e.g., [3], [4]). A waveguide discontinu-ity is defined as any interruption in the translational symmetry of the waveguide.Other waveguide discontinuities are an unavoidable result of mechanical defects inthe waveguide, or mechanical and/or electric transitions between two or more waveg-uides. In these cases, the discontinuity effect is unwanted and its design is focusedon minimizing the effect of the discontinuity. This is the case of structures as bends,transitions, junctions, etc., where the goal is to achieve very low levels of energyreflected at the discontinuities. In other cases, discontinuities are deliberately in-troduced into the waveguide (producing significant levels of energy reflected at thediscontinuities) in order to perform a certain electric function (e.g., posts, iris, stubs,etc). Furthermore, these type of discontinuities are typically combined in order toperform a more sophisticated function as impedance matching and frequency filteringnetworks. Thus, the characterization of these microwave engineering devices impliesalso the analysis of “waveguide discontinuities”.

Among the different waveguide technologies encountered in the modern commu-nication systems, the rectangular waveguide technology is used in many of them; inparticular, in satellite systems and general communication systems working in theupper microwave and millimeter wave frequency bands. This is due to the simplicityof the geometry, low losses and capacity to handle high powers that characterize therectangular waveguides.

Within this context, the objective of this work is to analyze of rectangular waveg-uide discontinuities by means of a fully automatic hp-adaptive finite element method.The analysis of a general discontinuity problem means to deal with a problem domainas the one shown in Fig. I.1. It consists of a region, which may have an arbitrarygeometry and several materials inside, and a number of waveguides (i.e., structureswith translational symmetry) through which the energy transfer (inwards and out-wards) takes place. Thus, the “ports” of the structure are the apertures located atthe end of the waveguides. Actually, the structure shown in Fig. I.1 is so generalthat can describe any microwave passive device. In many cases, the geometry of the

I.1 INTRODUCTION 2

region is of the same type as the geometry of the waveguides used for the ports,and the problem is known in microwave engineering as a discontinuity problem. Arectangular waveguide discontinuity is a particular case in which the waveguides arerectangular. In practice, many rectangular waveguide discontinuities are invariantalong one direction and the analysis of those structures may be performed in two-dimensions (2D) by simply considering as the problem domain the intersection ofthe structure with a plane transverse to the invariant direction. This is the case ofthe so-called H-plane and E-plane rectangular waveguide discontinuities, dependingif the plane transverse to the invariant direction is the one that contains the mag-netic, or the electric field, respectively. More precisely, the objective of this work isthe 2D analysis of these H-plane and E-plane discontinuities. Several examples ofdiscontinuities that fit into this categories are shown in Fig. I.2. Results for some ofthem are included in the parts II and III.

Typically, a microwave engineer is interested in the characterization of the struc-ture in terms of its scattering parameters (S-parameters or S-matrix). That requiresthe solution of the Maxwell equations within the structure under certain excitationsand terminations at the ports, and the evaluation of the power waves ai and bi,whose physical meaning will be explained later. Also, the microwave engineer maybe interested in the field solution in certain regions in order to introduce modifica-tions in the structure. Therefore, the analysis of the discontinuities must include thecomputation of the S parameters.

Due to the extent and nature of the work in this area, the work is presentedin three parts. In the first one, the general theory of rectangular waveguides is de-rived and described as well as a very detailed mathematical analysis of rectangularwaveguide discontinuities for both, the E-plane and H-plane formulations, includ-ing stabilized variational formulations for both cases. We also describe a practicalapproach for computing scattering parameters using Finite Elements. In the sec-ond paper (part II), a numerical methodology based on a self-adaptive hp-FiniteElement Method [6, 7] is used, which has been extended for electromagnetic ap-plications [8, 9, 10, 11, 12, 13, 14, 7], is applied to the simulations of waveguidediscontinuities. Extensive numerical results illustrate the flexibility, reliability andhigh-accuracy simulations obtained with this methodology, providing more accurateresults than those obtained with semi-analytical (in particular, Mode-Matching [15],[16, Chapter 9]) techniques. Finally, in the third paper (part III), results obtained us-ing hp energy-norm adaptivity [14] are compared against those using a goal-orientedhp-adaptivity approach [18, 19, 20]. The results show that both methods are suitablefor simulation of waveguide discontinuities.

We begin with a review of the theory of electromagnetic guided waves in Sec-

I.2 OVERVIEW OF ELECTROMAGNETIC GUIDED WAVE THEORY 3

Port N

Port 2

Port 1

Port 3

ε1, µ1

Port i

a1

b1

aN

b2

a2a3

ai

b3

bi

bN

ε3, µ3

ε2, µ2

Figure I.1: General waveguide discontinuity

tion I.2. The concept of “mode” is presented and developed for the particular caseof a rectangular waveguide. The discontinuity problem in rectangular waveguidetechnology is introduced in Section I.3, where the mathematical formulations for theH-plane (Section I.3.2.1) and E-plane (Section I.3.2.2) cases are derived. Stabilizedvariational formulations for both cases are presented. Procedures to compute thescattering parameters of the structure are described in Section I.3.3. Finally, someconclusions are given in Section I.4.

I.2 Overview of Electromagnetic Guided Wave The-

ory

In the following, a brief overview of the theory corresponding to electromagneticguided waves is presented. This theory is described in many microwave and electro-magnetic theory books (e.g., [21, 22, 23, 24, 25]).


E−plane structuresH−plane structures

Figure I.2: Examples of H-plane and E-plane discontinuities


Maxwell’s equations for a source free region are1:

∇× E = −∂B∂t

(I.1)

∇×H =∂D∂t

(I.2)

∇ · D = 0 (I.3)

∇ · B = 0 (I.4)

where E , H, D and B refer to the electric field intensity, magnetic field intensity,electric induction and magnetic induction, respectively.

Considering time harmonic fields, i.e., fields with time variation of the form ejωt,Maxwell’s equations can be written as

∇× E = −jωB (I.5)

∇×H = jωD (I.6)

∇ ·D = 0 (I.7)

∇ ·B = 0 (I.8)

where the magnitudes E, H, D, B refer now to complex-valued phasors. Thus, thetime variation is suppressed from the equations. The fields in the time domain aresimply obtained by multiplying the corresponding phasor with the time factor ejωt

and choosing a given part of the resulting complex number, e.g., E(r, t) = <(Eejωt)2. It is worth noting that, for ω 6= 0, the divergence equations (I.7) and (I.8) areimplicit in the curl equations (I.5) and (I.6).

The materials are introduced in the model by the constitutive relations that, forisotropic materials, are written as

D = εE (I.9)

B = µH (I.10)

1The original formulation of Maxwell’s used 20 equations and variables. The form of Maxwell’sequations, as they are known today, is due to expansions and modifications on the original equationsby Hertz, FitzGerald, Lodge and Heaviside. The interested reader is referred to [26].

2Thus, the phasor is a complex number in which its magnitude represents the maximum (rootmean square or peak) value of the amplitude of the field within a time cycle T = 2π/ω. The angleof the phasor (within 0 − 2π) indicates the relative time shift of the wave within the time periodT assuming a given time origin. It is worth noting that the choice of the real part of the phasormultiplied by the exponential time factor only implies the choice of the time origin.


where ε and µ stand for the electric permittivity and magnetic permeability, respec-tively. In case of lossy materials, ε and/or µ are complex. Otherwise, they are bothreal.

From the curl equations (I.5), (I.6), it is derived that the tangential componentsof the electric and magnetic field are continuous across material interfaces, i.e.,

n× (E2 − E1) = 0 (I.11)

n× (H2 −H1) = 0 (I.12)

where the subscripts 2 and 1 refer to the field at two different media, and n standsfor the unit vector normal to the material interface.

On the surface of perfect electric (and perfect magnetic3) conductors, it standsthat:

n× E = 0, on the surface of a perfect electric conductor (I.13)

n×H = 0, on the surface of a perfect magnetic conductor (I.14)

From the divergence equations (I.7), (I.8), it is obtained that the normal compo-nent of the electric and magnetic induction are continuous across material interfaces,i.e.,

n · (ε2E2 − ε1E1) = 0 (I.15)

n · (µ2H2 − µH1) = 0 (I.16)

On the surface of perfect electric (and perfect magnetic) conductors, it standsthat:

n · E = 0, on the surface of a perfect magnetic conductor (I.17)

n ·H = 0, on the surface of a perfect electric conductor (I.18)

I.2.1 Waveguide propagation modes

When applying Maxwell’s equations to a uniform waveguide, i.e., a structure withtranslational symmetry, solutions of the type

E(t1, t2, z) = E(t1, t2)e∓γz; H(t1, t2, z) = H(t1, t2)e

∓γz (I.19)

3Perfect magnetic conductors do not exist in nature. However, they can be used in the mathe-matical model, e.g., as symmetry walls.


are typically seeked, where t1 and t2 refer to the coordinates used on the transversesection, and z is the coordinate along the longitudinal direction (i.e., the direction inwhich translational symmetry applies), and γ is the complex propagation constant.The double sign of the exponential refers to field solutions propagating in the +zand −z direction, respectively. An example of waveguide illustrating the (t1, t2, z)coordinate system is shown in Fig. I.3.

By decomposing the electric and magnetic fields in terms of their transversecomponents Et, Ht, and longitudinal components Ezaz, hzaz, and performing the

same operation with the differential operator ∇ (i.e., ∇ = ∇t +∂

∂zaz), Maxwell’s

curl equations (I.5) and (I.6) are written as:

(∇tEz × az)− γ(az × Et) = −jωµHt (I.20)

(∇t × Et) = −jωµHzaz (I.21)

(∇tHz × az)− γ(az ×Ht) = jωεEt (I.22)

(∇t ×Ht) = jωεEzaz (I.23)

where constitutive relations (I.9) and (I.10) have been used, and a minus sign in(I.19) (i.e., waves propagating in the z direction) has been chosen.

From the expressions above, it is observed that the longitudinal components ofthe fields are obtained from the transverse components (equations (I.21) and (I.23)).Also, the transverse components may be obtained from the longitudinal componentsas:

Et =1

k2 + γ2(−jωµ∇tHz × az − γ∇tEz) (I.24)

Ht =1

k2 + γ2(jωε∇tEz × az − γ∇tHz) (I.25)

where k = ω√

εµ is referred to as the wavenumber.A classification of the field solutions for the waveguide, i.e., the modes, is typically

made depending on the existence of the longitudinal components:

• TEMz (Transverse Electro-Magnetic) modes: Ez = Hz = 0

• TMz (Transverse Magnetic) modes: Hz=0 and Ez 6= 0

• TEz (Transverse Electric) modes: Ez=0 and Hz 6= 0

For simplicity, the modes will be simply referred to as TEM, TM and TE, as it iscommon in the specialized literature.


z

cross section St

conductor

(t1, t2) planeε, µ

Figure I.3: General homogeneous waveguide with one conductor

The TEM modes can only be supported by waveguides with more than one con-ductor. Actually, there are as many TEM modes in a given waveguide as number ofconductors in the waveguide minus one. This fact occurs because the electric field isderived from a scalar potential (as the electric field is irrotational, see (I.21)). TheTEM, TM and TE modes can only be supported if the waveguide is homogeneous.For inhomogeneous waveguides these modes can not satisfy the boundary conditionsat the material interfaces. However, the classification in terms of TEM, TM, andTE modes is also useful as the propagating modes of a general waveguide are linearcombinations of TEM, TM and TE modes.

A general homogeneous waveguide is shown in Fig. I.3. Actually, the waveguideshown in the figure is a particular case of homogeneous waveguide because it in-volves only one conductor. The rectangular waveguide, which is the technology usedin the waveguide discontinuities object of the analysis, is a particular case of homo-geneous waveguide with one conductor. For those waveguides TM and TE modesare supported (but not TEM).

For homogeneous waveguides, (I.7) translates into ∇t ·Et = γEz, and (I.8) trans-


lates into ∇t ·Ht = γHz. Thus, by combining (I.21) and (I.22), and also (I.20) and(I.23), wave equations are obtained in terms of the longitudinal components:

∇2t Ez + (k2 + γ2)Ez = 0 (I.26)

∇2t Hz + (k2 + γ2)Hz = 0 (I.27)

the former valid for TM modes and the latter for TE modes. The case of TEMmodes will not be described because the rectangular waveguides do not supportTEM modes.

Thus, the TM and TE modes are obtained by solving the eigenvalue problems(I.26) and (I.27), respectively. Each mode n is associated with a positive eigenvaluethat we will refer to as k2

c (i.e., k2c = k2 + γ2). Appropriate boundary conditions

must also be enforced. Typically, (I.13) is used on conductor boundaries. Thus,Ez = 0 and ∂Hz/∂n = 0 (consequence of (I.24) —∂/∂n refers to the derivative inthe direction normal to the conductor—) must be enforced for TM and TE modes,respectively. Also, boundary conditions of perfect magnetic conductors (I.14) maybe considered: Hz = 0 and ∂Ez/∂n = 0 for TE and TM modes, respectively. Theseboundary conditions, together with real material parameters ε and µ imply a losslesscase4.

The complex propagation constant γ = α + jβ 5 is given by:

γ = jβ = ±j√

k2 − k2c for k > kc (I.28)

γ = α = ±√

k2c − k2 for k < kc (I.29)

where the cutoff effect is shown, i.e., propagation occurs only above a certain non-zero frequency fc = kc/(2π

√εµ). Below the cutoff frequency fc, the field amplitude

of the mode decays exponentially (even for the lossless case); it is an evanescentmode. Each mode field is associated with a (in general, different) cutoff wavenumberkc

6, i.e., a different cutoff frequency fc. Typically, the modes are ordered in termsof cutoff frequency fc in the ascending order (mode n, being n = 1, 2, . . .). The firstmode (n = 1) is referred to as the fundamental mode. The reason for that is becausemicrowave engineering is basically a monomode technology in which the working

4The lossless case is considered here for simplicity. For a low-loss case (the high-loss case wouldnot have sense for a practical waveguide structures), the solution can be calculated from the losslesssolution a posteriori by using a perturbation approach.

5In wave propagation α is know as the attenuation constant and β simply as the propagationconstant.

6A number of modes may have the same kc depending on the geometry (they are referred to asdegenerate modes).


frequency is chosen in such a way that only the first mode is propagating. Thus, thefrequency of the system must be higher than the cutoff frequency of the fundamentalmode f > fc1 but lower than the cutoff frequencies of the higher modes f < fcn,n = 2, 3 . . ..

Once the values of k2c n = k2 +γ2

n and the longitudinal components of the field (Ez

for TM, and Hz for TE) are known, the transverse components of the electric andmagnetic fields for each mode can be obtained from (I.24), and (I.25), consideringHz = 0 for the TM case, and Ez = 0 for the TE modes. Thus, it is easy to infer thatthe transverse components of the electric and magnetic field are related as follows:

TM case: Hnt =

az × Ent

ZnTM

; ZnTM =

γn

jωε(I.30)

TE case: Hnt =

az × Ent

ZnTE

; ZnTE =

jωµ

γn

(I.31)

where ZnTM or Zn

TE refers to the so called impedance of the mode.The power transported by the n-th mode is obtained by integrating the real part

of the Poynting vector, i.e., 7:

P nT = Re

[∫St

(Ent × Hn

t ) · dSt

](I.32)

where St refers to the cross-section of the waveguide and Hnt to the complex conjugate

of Hnt . The complex conjugate for is not really necessary because for a lossless

waveguide, the transverse electric and magnetic field may be chosen, without anyloss in generality, real8.

Taking into account (I.30) and (I.31), the power of the n-th mode may be writtenas

P nT = Re

[1

ZnTM/TE

∫St

|Ent |2 dSt

]= Re

[Zn

TM/TE

∫St

|Hnt |2 dSt

](I.33)

where ZnTM/TE refers to Zn

TM or ZnTE for the TM and TE cases, respectively.

7Root mean squares values for the phasor amplitudes of the field are assumed. If peak valuesare used, a (1/2) has to be added in front of the real part of (I.32).

8This statement is valid for any waveguide provided that the electric permittivity and magneticpermeability of the materials filling it are isotropic, as considered in (I.9) and (I.10). If the materialsare anisotropic, the statement is also true when the permittivity and/or permeability may becharacterized as tensors with null coupling between the transverse and longitudinal components ofthe fields and fluxes.


cross section St

conductor

b

a

η

ξ

ζ ≡ z

ε, µ

Figure I.4: Rectangular waveguide

I.2.2 Rectangular Waveguides

In the following, the particularization of the above results to a rectangular shapedwaveguide (as the one of Fig. I.4) is presented. The TM and TE modes of therectangular waveguide are obtained by solving (I.26), (I.27), i.e,

∂2Vz

∂ξ2+

∂2Vz

∂η2+ k2

cVz = 0 (I.34)

in a 2D rectangular region 0 < ξ < a, 0 < η < b. Vz is used here to denote Ez or Hz

depending of the type of mode (TM or TE).Due to the rectangular geometry, separation of variables can be used and solutions

of the typeVz(ξ, η) = X(ξ)Y (η) (I.35)

are substituted into (I.34) to obtain

1

X

d2X

dξ2+

1

Y

d2Y

dη2+ k2

c = 0 (I.36)

Then, by the usual separation of variables argument, each of the terms in (I.36)


must be a constant. Defining constants kξ and kη such that

d2X

dξ2+ k2

ξX = 0 (I.37)

d2Y

dη2+ k2

ηY = 0 (I.38)

and k2ξ + k2

η = k2c , the solution for Vz can be written as

Vz(ξ, η) = (A cos kξξ + B sin kξξ) (C cos kηη + D sin kηη) (I.39)

The constants are obtained by imposing the boundary conditions of the problem.Assuming perfect electric conductors on the boundary, the boundary conditions tobe applied are:

Vz = 0, on ξ = 0, ξ = a, η = 0, η = b (I.40)

for the TM case (Vz ≡ Ez) and

∂Vz/∂ξ = 0, on ξ = 0, ξ = a

∂Vz/∂η = 0, on η = 0, η = b(I.41)

for the TE case (Vz ≡ Hz).Thus, the solution for Vz is finally given as

TM → Vz(ξ, η) ≡ Ez(ξ, η) = V0 sin kξξ sin kηη

TE → Vz(ξ, η) ≡ Hz(ξ, η) = V0 cos kξξ cos kηη(I.42)

withkξ =

mπ

a, kη =

nπ

b(I.43)

where m, n are integer numbers. The index m refers to variations in the ξ directionand the index n in the η direction (variations in the sense of (I.43)). The TE and TMmodes with m variations in the ξ direction and n variations in the η direction will bereferred to as TMmn and TEmn modes, respectively. Note that now n refers to one ofthe two indexes used to denote the modes, specifically, to denote the variations in theη direction. Previously (page 9), n was used as the only index to denote the modesin an ordered fashion. However, it should be clear from the context the meaning ofn.

The transverse components are easily derived from (I.24) and (I.25). These andother results are summarized in Table I.1. In particular, the cutoff frequencies fc forthe TMmn and TEmn modes are equal to

fcmn =1

2π√

εµ

√(mπ

a

)2

+(nπ

b

)2

(I.44)


Quantity TEmn TMmn

k ω√

εµ ω√

εµ

kc

√(mπ

a

)2

+(nπ

b

)2√(mπ

a

)2

+(nπ

b

)2

β√

k2 − k2c

√k2 − k2

c

Ez ≡ Eζ 0 Bmn sinmπξ

asin

nπη

be−jβz

Hz ≡ Hζ Amn cosmπξ

acos

nπη

be−jβz 0

Eξjωµnπ

k2cb

Amn cosmπξ

asin

nπη

be−jβz −jβmπ

k2ca

Bmn cosmπξ

asin

nπη

be−jβz

Eη−jωµmπ

k2ca

Amn sinmπξ

acos

nπη

be−jβz −jβnπ

k2cb

Bmn sinmπξ

acos

nπη

be−jβz

Hξjβmπ

k2ca

Amn sinmπξ

acos

nπη

be−jβz jωεnπ

k2cb

Bmn sinmπξ

acos

nπη

be−jβz

Hηjβnπ

k2cb

Amn cosmπξ

asin

nπη

be−jβz −jωεmπ

k2ca

Bmn cosmπξ

asin

nπη

be−jβz

Z ZTE =ωµ

βZTM =

β

ωε

Table I.1: Summary of results for the rectangular waveguide

where 1/√

εµ is the speed of light in the medium.It is worth noting that for the TE case one of the indexes may be zero but not

for the TM case. If one of the indexes (or both) are zero for the TM case, a nullEz is obtained (see (I.42)) and, thus, a null electromagnetic field would be obtained.Also, the solution for the TE00 in (I.42) yields a constant field Hz and it must bediscarded.

For a > b, the first mode (i.e., the fundamental mode) is the TE10 mode. Itspropagation constant β10 (lossless case) is obtained from (I.28) by considering (I.43)with m = 1, n = 0:

β10 =

√k2 −

(π

a

)2

for k >π

a(I.45)

In the limiting case when a = b, the two first modes (TE10 and TE01) have thesame cutoff frequency. Actually, the cutoff frequencies of modes with indexes m, nand n,m are identical. The second mode depends on the aspect ratio a/b. Fig. I.5shows the dispersion chart of a rectangular waveguide for an aspect ratio a/b = 2. Asit was mentioned previously, microwave waveguides and related devices are typicallyused in a monomode way, i.e., only one mode is propagating. This can be achievedby a proper excitation, but it is more convenient to work in a frequency region whereonly the first mode (TE10) is propagating (marked as monomode region in the figure).

I.3 ANALYSIS OF RECTANGULAR WAVEGUIDE DISCONTINUITIES 14

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.5 1 1.5 2 2.5 3 3.5 4

β/k

k/kc10

mon

omod

e re

gion

TE10 TE01,TE20TE11,TM11TE21,TM21

Figure I.5: Normalized propagation constant versus wavenumber for the modes of a rect-angular waveguide. The wavenumber is normalized to the cutoff frequency of the TE10

mode

In the presented work, we limit ourselves to only frequencies in the monomode region.

I.3 Analysis of Rectangular Waveguide Disconti-

nuities

As mentioned in the Introduction, the objective of the paper is to analyze the scat-tering properties of rectangular waveguide discontinuities. The problem domain ofa general discontinuity problem may be thought as a region, which may have ar-bitrary geometry and several materials inside, and a number of waveguides (i.e.,structures with translational symmetry) through which the energy transfer (inwardsand outwards the region) takes place. Thus, the waveguides, and more preciselythe apertures at the end of the waveguides, are the ports of the structure (Fig. I.1).


A rectangular waveguide discontinuity is a particular case in which the ports arerectangular waveguides. An example with 2 ports is shown in Fig. I.6.

Let us consider a structure that is excited through the fundamental mode (TE10)at a given port (see Fig. I.6). The fundamental mode satisfies the Maxwell equationsand the boundary conditions for the waveguide. However, at the discontinuity thefield solution is no longer equal to the TE10 mode as this mode does not satisfy, ingeneral, the boundary conditions in this region. The field solution in the discontinuityregion will generate, in general, all the modes in the waveguides of the structure goingoutwards the region in order for the boundary conditions to be satisfied. Thus, partof the energy is reflected back into the excited port and the rest of the energy istransmitted to the other ports. If the frequency of the time-harmonic excitation iswithin the monomode region (below the cutoff frequency of the second mode) thefield amplitude of all the modes but the TE10 will be negligible in the waveguidesat a certain distance from the discontinuity region (for practical purposes, one TE10

wavelength is usually enough). Thus, at the ports of the structure only the TE10

mode is present. The discontinuity is simply characterized by the ratio of powercarried by the reflected TE10 wave (for the excited port) or by the transmittedTE10 wave (for the other ports) to power carried by the incident TE10 wave (at theincident port). These ratios are directly related to the so called scattering matrix orS-parameters. They will be rigorously defined later.

I.3.1 Analysis of waveguide discontinuities in 2D

In the general case, a 3D analysis is needed to obtain the electromagnetic field of thewaveguide discontinuity under analysis. However, most commonly, the geometry ofthe discontinuity is invariant along a transverse direction. Thus, the spatial variationof the fields in the discontinuity along that direction is known (equal to the variationof the field of the incident mode at the discontinuity along that invariant direction)and the scattering analysis may be done in 2D. This is the case of the so called H-plane and E-plane rectangular waveguide discontinuities. A large number of bends,junctions, phase shifters, filters, etc. in rectangular technology (see the examples ofFig. I.2) fit into this category.

A rectangular H-plane discontinuity (see Fig. I.7) has translational symmetryalong the narrow dimension of the waveguide (the η axis in Fig. I.4). In this case, theFEM analysis may be made by considering as the problem domain the intersectionof a plane orthogonal to η with the structure under analysis. That plane is the onecontaining the magnetic field of the TE10, and the analysis is made in the H-planeof the structure. Analogously, in a rectangular E-plane discontinuity (see Fig. I.8)


TEin

10

Excited port

TEout

10

TEout

10

ε, µ

ε1, µ1

ε2, µ2

ε3, µ3

TE10 + higher modes

TE10 + higher modes

Figure I.6: Example of 2-ports monomode rectangular waveguide discontinuity

there is translational symmetry along the broad dimension of the waveguide (the ξaxis in Fig. I.4). In this case, the FEM analysis may be made by considering as theproblem domain the intersection of a plane orthogonal to ξ with the structure underanalysis. That plane contains the electric field of the TE10, and the analysis is madein the E-plane of the structure.

I.3.2 2D Variational Formulation

The 2D variational formulation for the H-plane and E-plane discontinuity problemsare developed next. In both cases, the formulations are based on decomposing theelectric and magnetic fields in terms of their components parallel to the H-planeHΩ (or E-plane EΩ), and orthogonal to it H⊥ (or E⊥). Doing the same with thedifferential operator ∇ (i.e., ∇ = ∇Ω + ∇⊥), Maxwell’s curl equations (I.5), (I.6)are written as:

(∇Ω × E⊥) + (∇⊥ × EΩ) = −jωµHΩ (I.46)

(∇Ω × EΩ) = −jωµH⊥ (I.47)

(∇Ω ×H⊥) + (∇⊥ ×HΩ) = jωεEΩ (I.48)

(∇Ω ×HΩ) = jωεE⊥ (I.49)


Port 2

Port 3

Port 1

yx

ε, µ

H-p

lane

H-plane

Ω

ε, µΓ1

p

Γ2p

Γ3p

ΓN

ΓN

ΓN

y

x

Figure I.7: A rectangular H-plane discontinuity and its 2D FEM modeling


ε, µ

Port 1

E-plan

e

y

x

Port 2

Port 3

E-plane

Ω

ε, µ

y

x

Γ1p

ΓD

Γ2p

Γ3p

ΓD

ΓD

Figure I.8: A rectangular E-plane discontinuity and its 2D FEM modeling


Particularization to each case (H-plane or E-plane) is shown next.

I.3.2.1 H-plane variational formulation

In the H-plane discontinuities, the variation along the orthogonal direction to theplane, i.e., along the narrow dimension of the waveguide, is null for TE10 modeexcitation. Thus, ∇⊥ ≡ 0 and, by combining (I.46) and (I.49), a double curl vectorwave equation is obtained in term of HΩ:

∇× 1

ε∇×HΩ − ω2µHΩ = 0 (I.50)

or, equivalently,

∇× 1

εr

∇×HΩ − k20µrHΩ = 0 (I.51)

where εr = ε/εo, µr = µ/µo. For simplicity, the operator ∇Ω has been simplyreferred to as ∇. This convention will be kept in the rest of the paper except whenexplicitly noted.

The boundary conditions are (see Fig. I.7):

n×HΩ = 0, at perfect magnetic conductors (ΓD) (I.52)

n× 1

εr

∇×HΩ = 0, at perfect electric conductors (ΓN) (I.53)

n× 1

εr

∇×HΩ + jk2

εrβ10

n× n×HΩ = Uin, at the port boundaries (Γp) (I.54)

with Uin defined as

Uin = 2jk2

εrβ10

n× n×Hin (I.55)

and Hin being the magnetic field incident at the port corresponding to the TE10

mode. n stands for the outward unit vector normal to Γ.Perfect magnetic conductors are not present physically in the structure under

analysis (neither are shown in Fig. I.7) but they may be used in certain cases assymmetry boundary conditions for the FEM analysis, in order to reduce the numberof unknowns.

The boundary conditions at the ports are absorbing boundary conditions for theTE10 mode. They are obtained under the assumption that only the TE10 modeis present (with a non-negligible amplitude) at the ports (monomode propagation).This requires to truncate the FEM domain at a certain distance from the disconti-nuity.


Notice that the FEM domain may be truncated, if desired, closer to the discon-tinuity, by using a boundary condition at the ports different from the one of (I.54).This boundary condition must take into account the presence of a number of modesat the ports, i.e., it is a multimode boundary condition. The higher the number ofmodes considered the closer the port can be placed to the discontinuity. However,the discretization of the boundary condition yields then non-sparse matrices. Theinterested reader is referred to [27, Section 11.1].

Under the above assumption (only the TE10 mode is present at the ports), themagnetic field at a given port will be, in the general case, the sum of contributionsof two TE10 waves: one going inward and the other going outward the FEM domainΩ:

HΩ(ξ, η, ζ) = Hin(ξ, η, ζ) + Hout(ξ, η, ζ)

= H in0

(sin

πξ

aaξ +

π/a

jβ10

cosπξ

aaζ

)e−jβ10ζ

+ Hout0

(sin

πξ

aaξ +

π/a

−jβ10

cosπξ

aaζ

)ejβ10ζ , at the ports (I.56)

In the previous expression, the field configuration has been obtained from Tab. I.1(for the TE10 mode) by rewriting some terms. Also, a local (to the port considered)coordinate system is used. The coordinate system is the one used in Section I.2 todescribe the waveguide theory (see Fig. I.4).

Now, taking the curl of (I.56), it is obtained

∇×HΩ = H in0

(k2

jβ10

sinπξ

aaη

)e−jβ10ζ

+ Hout0

(k2

−jβ10

sinπξ

aaη

)ejβ10ζ , at the ports (I.57)

By expressing the outward wave in terms of HΩ, i.e., HoutΩ = HΩ − Hin, then

(I.54) is finally obtained. Note that the inward wave is due to the excitation of thestructure, i.e., it is part of the load data. Note also that n ≡ −ζ.

In electrical engineering terminology, the inward wave is referred to as the incidentwave. Typically, only one port is excited at each analysis. Thus, the outward wave isreferred to as the reflected wave for the excited port and as the transmitted wave forthe other ports. The complete characterization of the discontinuity requires severalanalysis exciting a different port of the structure each time.

The variational formulation is obtained by multiplying (I.51) with a test functionFΩ. Integration by parts (first vector Green’s theorem) is applied to the double curl


term and a boundary term appears in the variational formulation:∫Ω

FΩ · (∇× 1

εr

∇×HΩ) dΩ

=

∫Ω

(∇× FΩ) · ( 1

εr

∇×HΩ) dΩ−∫

Γ

n · (FΩ ×1

εr

∇×HΩ) dΓ (I.58)

Let us define the following spaces

W := A ∈ H(curl, Ω), n×A = 0 on ΓDV := p ∈ H1(Ω), p = 0 on ΓD

(I.59)

The boundary term for ΓD is zero by selecting FΩ ∈ W. Also, for ΓN theboundary term is set to zero in order to enforce, in a natural sense, (I.53). Thus,the boundary term is only non-zero at the boundary ports. Introducing (I.54) in theboundary terms for the ports, the following variational formulation is obtained:

Find HΩ ∈ W such that

c(FΩ,HΩ) = l(FΩ), ∀FΩ ∈ W (I.60)

where c(FΩ,HΩ) and l(FΩ) are defined in (I.62).This variational formulation is not uniformly stable with respect to (angular)

frequency ω. As ω → 0, i.e., k0 → 0, the term k2o

∫ΩFΩ ·µrHΩ dΩ becomes negligible

compared with the curl-product term∫

Ω(∇ × FΩ) · (

1

εr

∇ × HΩ) dΩ. Thus, for

small frequencies the problem becomes ill-posed, since numercially the term withthe product of the curls does not “see” the gradients, and the gradients remainundetermined. This situation is not an issue from the practical point of view for theH-plane formulation because, in order, for the TE10 mode to propagate, k0 has to bechosen in such a way that k0εrµr > π/a. However, the stability problem may alsoarise with very small size finite elements as the product of the curls term becomesdominant in the formulation.

As a remedy to this problem, a Lagrange multiplier p is introduced to enforce theweak form of the continuity equation (obtained by employing a gradient as a testfunction in (I.60)). Thus, the stabilized variational formulation with the Lagrangemultiplier is:

Find HΩ ∈ W, p ∈ V such that

c(FΩ,HΩ) + b(FΩ, ∇p) = l(FΩ) ∀FΩ ∈ W

b(∇q,HΩ) = g(∇q) ∀q ∈ V(I.61)


The sesquilinear and antilinear forms used in the variational formulations aboveare the following:

c(FΩ,HΩ) =

∫Ω

(∇× FΩ) · ( 1

εr

∇×HΩ) dΩ− k2o

∫Ω

FΩ · µrHΩ dΩ

+ jk2

εrβ10

∫∑

Γip

(n× FΩ) · (n×HΩ) dΓ

b(FΩ, ∇p) =− k2o

∫Ω

FΩ · µr∇p dΩ + jk2

εrβ10

∫∑

iΓip

(n× FΩ) · (n×∇p) dΓ

l(FΩ) = 2jk2

εrβ10

∫Γin

p

(n× FΩ) · (n×Hin) dΓ

g(∇q) = 2jk2

εrβ10

∫Γin

p

(n×∇q) · (n×Hin) dΓ

(I.62)

where F and q stands for the complex conjugate of F and q, respectively.By substituting FΩ = ∇q, q ∈ V , it is deduced that the Lagrange multiplier p

satisfies the weak form of a Laplace-like equation. Thus, if (homogeneous) Dirichletboundary conditions are present (ΓD 6= ∅), i.e., when symmetry walls are used in theanalysis, the Lagrange multiplier p identically vanishes. Notice that the multiplier pis an undefined constant when ΓD = 0. In that case, one degree of freedom of p isset to a given value (typically zero). Thus, the multiplier p identically vanishes forΓD = 0. Also, notice that b(FΩ, ∇p) and g(∇q) should both be divided by ko.

The stabilized formulation works because the gradients of scalar-valued potentialsfrom V form precisely the null space of the curl on vector functions in H. Thiscondition will have to be preserved at the discrete level by a careful construction ofthe finite element basis [28].

I.3.2.2 E-plane variational formulation

The E-plane discontinuities analysis is similar to the H-plane case in the sense thatthe problem domain is restricted to a plane. However, the formulation presents somerelevant differences with respect to the one used for the H-plane discontinuities. Thereason is that the orthogonal direction to the plane (E-plane) is the broad dimensionof the rectangular waveguide and, in contrast to the H-plane case, ∇⊥ 6= 0 for TE10

mode excitation.Thus, by combining (I.48) and (I.47), it is obtained

∇× 1

µ∇× EΩ + jω(∇⊥ ×HΩ)− ω2εEΩ = 0 (I.63)


With the local axis convention of Fig. I.4, ∇⊥ corresponds to the derivatives inthe ξ-direction (along the broad dimension of the waveguide), which are non-zero forTE10 mode excitation as mentioned above. More precisely, due to the translationalsymmetry in the orthogonal direction to the E-plane (ξ direction), the variation alongthe orthogonal direction to the E-plane (∇⊥) will be of the same type as the TE10

mode, i.e., as sin(πξ/a) or cos(πξ/a) depending on the vector component of the field.Furthermore, if the discontinuity is homogeneous9, it can be inferred that E⊥ = 0everywhere under TE10 mode excitation. Thus, (I.46) is written as

(∇⊥ × EΩ) = −jωµHΩ (I.64)

Thus, it is easy to obtain that the term jω(∇⊥ ×HΩ) of (I.63) is1

µ

(π

a

)2

EΩ.

By substituting this result in (I.63), a double-curl vector wave equation is obtained:

∇×∇× EΩ +((π/a)2 − ω2εµ

)EΩ = 0 (I.65)

where the term ((π/a)2 − ω2εµ) is identified as −β210 (see (I.45)). Thus, the double-

curl vector wave equation may be written as:

∇×∇× EΩ − β210EΩ = 0 (I.66)

It is worth noting that the above wave equation is not valid for the analysisof E-plane inhomogeneous structures, which requires, in the general case, the threecomponents of the electric field [29], [30]. The wave equation of (I.66) in terms of twocomponents (vector EΩ) has been used for comparison purposes with the formulationbased on (I.51) used for the H-plane case.

The boundary conditions of the problem are (see Fig. I.8):

n× EΩ = 0, at perfect electric conductors (ΓD) (I.67)

n×∇× EΩ = 0, at perfect magnetic conductors (ΓN) (I.68)

n×∇× EΩ + jβ10n× n× EΩ = Uin, at the port boundaries (Γp) (I.69)

with Uin defined asUin = 2jβ10n× n× Ein (I.70)

9A homogeneous discontinuity is defined as a discontinuity with only one material filling thediscontinuity. Note that the presence of perfect (electric or magnetic) conductors does not alter thehomogeneous character of the discontinuity, since the electromagnetic field is null inside perfect con-ductors. Thus, the definition of only one material stands for the regions where the electromagneticfield is non-null.


and Ein being the electric field incident at the port corresponding to the TE10 mode.n stands for the outward unit vector normal to Γ.

The boundary condition at the ports is obtained (as in the H-plane case) byassuming that only the TE10 mode is present at the ports (monomode propagation).The same comments made for the H-plane case (20) with respect to the use ofmultimode boundary conditions remain valid for the E-plane case.

Analogously to the H-plane, the field at a given port is given by the sum of twoTE10 waves: one going inward and another going outward the FEM domain Ω:

EΩ(ξ, η, ζ) = Ein(ξ, η, ζ) + Eout(ξ, η, ζ)

= Ein0 sin

πξ

aaη e−jβ10ζ + Eout

0 sinπξ

aaη ejβ10ζ , at the ports (I.71)

where the electric field configuration for the TE10 mode has been obtained fromTab. I.1. By taking the curl of (I.71) and expressing the outward wave as Eout =EΩ − Ein, (I.69) is finally obtained.

To obtain the variational formulation we begin by multiplying (I.66) with a testfunction FΩ. Integration by parts (first vector Green’s theorem) is applied to thedouble curl term and a boundary term appears in the variational formulation (anal-ogously as in (I.58)).∫

Ω

FΩ · (∇×∇× EΩ) dΩ

=

∫Ω

(∇× FΩ) · (∇× EΩ) dΩ−∫

Γ

n · (FΩ ×∇× EΩ) dΓ (I.72)

By selecting FΩ ∈ W (see (I.59)), we reduce the boundary term to zero except forthe boundary ports Γp, since (I.68) is enforced in a weak sense. Introducing (I.69) inthe boundary terms for the ports, the following variational formulation is obtained:

Find EΩ ∈ W such that

c(FΩ,EΩ) = l(FΩ), ∀FΩ ∈ W (I.73)

where c(FΩ,EΩ) and l(FΩ) are defined in (I.75).Analogously to the H-plane formulation of (I.60), formulation (I.73) is not uni-

formly stable with respect to β10. There is a significant difference with respect tothe H-plane formulation. In (I.73), the constant in front of the integral term of theproduct of the test and trial (unknown) functions (i.e., β2

10) may be chosen arbitrar-ily close to zero in the region where the TE10 mode still propagates. The physical


meaning of selecting β10 → 0 is that the frequency is chosen very close to cut-off. Inother words, the wavelength is very close to infinity and the problems of scales due tothe ratio of the finite element size to the wavelength in the double curl formulationarises, even with “reasonable” size finite elements.

As a remedy to this problem, a Lagrange multiplier p is introduced to enforcethe weak form of the continuity equation (obtained by employing a gradient as testfunction in (I.73)). Thus, the stabilized variational formulation with the Lagrangemultiplier, in the E-plane case, is obtained:

Find EΩ ∈ W, p ∈ V such that

c(FΩ,EΩ) + b(FΩ, ∇p) = l(FΩ) ∀FΩ ∈ W

b(∇q,EΩ) = g(∇q) ∀q ∈ V(I.74)

The sesquilinear and antilinear forms used in the variational formulations aboveare as follows:

c(FΩ,EΩ) =

∫Ω

(∇× FΩ) · (∇× EΩ) dΩ− β210

∫Ω

FΩ · EΩ dΩ

+ jβ10

∫∑

iΓip

(n× FΩ) · (n× EΩ) dΓ

b(FΩ, ∇p) =− β210

∫Ω

FΩ ·∇p dΩ + jβ10

∫∑

iΓip

(n× FΩ) · (n×∇p) dΓ

l(FΩ) = 2jβ10

∫Γin

p

(n× FΩ) · (n× Ein) dΓ

g(∇q) = 2jβ10

∫Γin

p

(n×∇q) · (n× Ein) dΓ

(I.75)

By substituting FΩ = ∇q, q ∈ V , it is deduced that the Lagrange multiplier psatisfies the weak form of a Laplace-like equation. In the E-plane case, in contrast tothe H-plane case, (homogeneous) Dirichlet boundary conditions are always present(ΓD 6= ∅) because of the presence of the conductors. Thus, the multiplier p identicallyvanishes.

Note that b(F, ∇p) and g(∇q) should both be divided by β10.

I.3.3 Computation of scattering parameters

As it was commented on page 15, a discontinuity (actually, any N -port microwavenetwork) may be characterized by the so called scattering matrix S. Thus, the


electrical behavior of a complex microwave system may be analyzed by a cascadeconnection of blocks, each block being characterized by its S matrix.

Provided that only one mode is propagating in the waveguide ports, the field ata given port, in the general case, will consist of the contributions of two waves: onegoing inward and the other going outward the port. Each of these waves is defined bya complex valued magnitude called power wave. For its definition in terms of circuitmagnitudes (voltages and currents), the reader is referred to [31] or a microwavetheory textbook (e.g., [24]). For understanding what follows, it is enough to knowthat the power wave is a complex number such that its square magnitude is thepower carried by the wave, and its argument is the phase of the wave. At each portΓi

p the power wave going inward is denoted as ai and the one going outward bi. Thus,ai will be proportional to H in

0 or Ein0 , and bi will be proportional to Hout

0 or Eout0 .

Therefore, it is easy to infer that the relation between the power waves is linear (forlinear media). These linear relation may be casted into matrix form as follows

b1

b2

.

.

bN

=

S11 S12 . . . S1N

S21 S22 . . . S2N

. . . . . .

. . . . . .

SN1 SN2 . . . SNN

a1

a2

.

.

aN

(I.76)

where Sij are the so-called scattering parameters, or simply S parameters. Note thatSii are reflection coefficients and Sji, j 6= i, are transmission coefficients.

Expression (I.76) completely determines the electrical behavior of the networkwhen only one mode is present (with non-negligible level) at the ports. In some cases(not for the structures analyzed in this paper) the discontinuities are multimode innature (multiple modes are propagating at the waveguide ports). Also, sometimesmonomode discontinuities must be characterized in terms of several modes; this is thecase when a network or circuit is the cascade of two or more discontinuities connectedby very short length waveguides, so that the evanescent modes of each discontinuityare also coupled. For these cases, for a given port, pairs of power waves as many asmodes at the port would have to be considered. Thus, a multimode S matrix wouldbe obtained when writing the linear relations between the power waves of the portsfor the different modes. However, for this paper, the common monomode S matrixof (I.76) is used.

In the following, the main properties of the S matrix are analyzed. The S matrixis symmetric if and only if the circuit is reciprocal10. This is the case of the structures

10In the simplest sense, a reciprocal system is such that the response of the system to a source is


analyzed in this paper. Non-reciprocal behavior would require a non-symmetrictensor for the electric permittivity and/or magnetic permeability. For a losslessstructure the S matrix is unitary.

Typically, the S parameters are obtained by making N “measurements”. Ateach measurement i, the port Γi

p is excited (ai 6= 0, Γip ≡ Γin

p ) and the others are“adapted”, i.e., no reflection back into the circuit occurs (aj = 0, j 6= i). Thus, thecoefficients on the i column (i = 1 . . . N) of the S parameters matrix are obtained as

Sji = bj/ai, j = 1 . . . N (I.77)

Obviously, the knowledge of some properties of the structure under analysis (recip-rocal or not, lossless or lossy, the presence of some symmetries, etc.) allows to infersome of the S parameters in terms of others, so the S matrix may be obtained witha lower number of experiments.

When the analysis is made in the computer, each of the “measurements” corre-spond to the assumption that there is a given incident field (inward to the structure)at a given port (i.e., set Hin in (I.62) or Ein in (I.75) for H-plane or E-plane dis-continuities, respectively) and zero incident field in the other ports. In order forthe analysis to be meaningful, the ports must absorb the waves, so no reflection isproduced by truncating the waveguides for the simulation. This condition is knownwithin the microwave literature as an “adaptation” condition and is achieved byabsorbing boundary conditions at the ports (as those of (I.54) and (I.69)11).

Once the magnetic and/or electric field is known in the structure, and, in partic-ular, its tangential component to each port, the S parameters may be computed asratios of power waves (equation (I.77)). If the comparison affects to the power wavesof the same port, which is the case for the reflection coefficients Sii, the comparisonmay be made directly in terms of the phasors H in

0 , Hout0 (see (I.56)), or Ein

0 , Eout0 (see

(I.71)). However, if the comparison affects to power waves at different ports, which isthe case for the transmission coefficients Sji (j 6= i), the comparison can not be madedirectly, in the general case, in terms of the phasors H in

0 , Hout0 , or Ein

0 , Eout0 . This is

due to the fact that the normalization of the power waves (the relation between theamplitude phasor and its power wave associated) may be different at each port (e.g.,because of different width and/or height, material, etc., at each port).

In the following, the process of obtaining the expressions to compute the S pa-rameters in the general case is described. In order to cover the H-plane and E-plane

interchanged when source and measurer are interchanged. It can be demonstrated that any systemin which the media are linear with permittivity and permeability characterized (in the general case)by symmetric tensors is reciprocal.

11Multimode boundary conditions may also be used for this purpose.


cases, Vt will denote the transverse component to the port of the magnetic field forthe H-plane case and the electric field for the E-plane case. Also, a superscript pi

will be used to specify the field at the i-th port.The transverse component of the field at the excited port Γi

p is given by the sumof contributions of two TE10 waves (one inward —incident— and the other outward—reflected—):

Vpit (ξ, η) = ai ν

pi10(ξ, η) + bi ν

pi10(ξ, η), (I.78)

where νpi10(ξ, η) is a real-valued vector function that represents the “shape” of the

field at the port.By comparing (I.78) with the transverse component of the magnetic field in (I.56)

(i.e., the ξ component) or the electric field in (I.71) (i.e., the η component)12, it isclear that νpi

10(ξ, η)13 is proportional to sin(πξ/a), being a the broad dimension ofthe waveguide at port i. Also, it is obvious that, as it was mentioned before, ai isproportional to the phasor of the inward wave and bi is proportional to the phasorof the outward wave. On the following, V in

0 will be used indistinctly to refer to themagnetic or electric field amplitude (phasor) of the inward wave. Analogously, V out

0

will be used to refer to the outward wave.Normalization of νpi

10(ξ, η) is obtained by enforcing the power wave definition, i.e.,the power of the wave is given by |ai|2, or |bi|2, accordingly. That requires νpi

10(ξ, η)to be normalized in such a way that gives unit power, i.e. (see (I.33)),

Re

[1

Ipi10

∫S

pit

|νpi10|2 dSpi

t

]= 1 (I.79)

where Ipi10 defines an inmittance such as

Ipi10 =

Zpi

TE10with Vt ≡ Et

1

Zpi

TE10

with Vt ≡ Ht(I.80)

Thus, νpi10 turns out to be

νpi10(ξ, η) =

√2Ipi

10

api bpisin

πξ

apiaξ (I.81)

and

ai = V in0

√api bpi

2Ipi10

, bi = V out0

√api bpi

2Ipi10

(I.82)

12The port may be considered at ζ = 0 without any loss of generality13Actually, for the TE10 mode, ν10 is not a function of η, i.e., ν10(ξ, η) ≡ ν10(ξ)


Note that, in the general case, each port can have different dimensions (denotedby api and bpi), and/or the TE10 mode at the port have different inmittance (denotedby Ipi

10).Now, multiplying (I.78) by (1/Ipi

10)νpi10(ξ, η) and integrating over the cross-section

of the rectangular waveguide corresponding to the i-th port, Spit , we obtain

1

Ipi10

∫S

pit

Vpit (ξ, η) · νpi

10(ξ, η) dSpit =

ai1

Ipi10

∫S

pit

|νpi10(ξ, η)|2 dSpi

t + bi1

Ipi10

∫S

pit

|νpi10(ξ, η)|2 dSpi

t (I.83)

Dividing (I.83) by ai, taking into account (I.79) and (I.82), and noting thatSii = bi/ai (see (I.77)), we get

Sii =

1

Ipi10

∫S

pit

Vpit (ξ, η) · νpi

10(ξ, η) dSpit

V in0

√api bpi

2Ipi10

− 1 (I.84)

The procedure to obtain the expression for Sji (j 6= i) is analogous. Considering(I.78) for the j-th port with only an outward wave (i.e., aj = 0), and multiplying by(1/I

pj

10)νpj

10(ξ, η) and integrating over Spj

t , the power wave bj is obtained as follows:

bj =1

Ipj

10

∫S

pjt

Vpj

t (ξ, η) · νpj

10(ξ, η) dSpj

t (I.85)

Thus, Sji is finally obtained as bj/ai

Sji =

1

Ipj

10

∫S

pjt

Vpj

t (ξ, η) · νpj

10(ξ, η) dSpj

t

V in0

√api bpi

2Ipi10

(I.86)

The expressions for Sii and Sji can be simplified as follows. The integrals involvedin (I.84), (I.86) are defined over the whole cross-section of the waveguide St. However,for the discontinuities analyzed in this paper, the structure is invariant in one of thedirections. Actually, this assumption has made the 2D analysis possible, as thevariation of the field in that direction is analytically known. Thus, the integrals ofthe field at the ports on that direction can be computed analytically and only theintegrals of Vt restricted to the boundary ports of the FEM domain Ω (i.e., Γi

p) needto be computed.


I.3.3.1 H-plane

Specifically, for the H-plane case, the transverse component of the magnetic fieldalong the η axis of the waveguide port is constant. Also, all the waveguide ports areof the same height b (bp1 = bp2 . . . bpN = b). Thus, taking into account (I.81), (I.82),the expression (I.84) for the reflection coefficient Sii is simplified as follows

Sii =

∫Γpi

HΩ(ξ) · sin πξ

apidΓpi

H in0

api

2

− 1 (I.87)

and, analogously, the reflection coefficient Sji (see (I.86)):

Sji =

1√I

pj

10

∫Γpj

HΩ(ξ) · sin πξ

apjdΓpj

H in0

api

2

1√Ipi10

(I.88)

The Sii of (I.87) corresponds to a definition with respect to admittance. Thereason is that the power waves used in its computation (consider (I.82) with V refer-ring to H) are proportional to values of magnetic field. Typically, the S parametersare defined in term of impedances, i.e., the power waves being proportional to theelectric field. It is easy to check that the relation between the reflection coefficient Sii

in terms of impedances or admittances is a 180 degrees turn in the complex plane.Thus, a minus sign should be added to (I.87) for a Sii definition with respect toimpedance. The transmission coefficients Sji (equation (I.88)) are not affected bythis issue.

I.3.3.2 E-plane

For the E-plane case, the variation of the transverse component of the electric fieldalong the ξ axis of the waveguide ports is known to be of the type sin(πξ/a) where a isthe broad dimension of the ports. Note that all the ports must have the same widtha (ap1 = ap2 . . . apN = a) for an E-plane discontinuity. Thus, taking into account(I.81), and (I.82), the expression (I.84) for the reflection coefficient Sii is simplifiedas follows.

Sii =

∫Γpi

EΩ(η) · dΓpi

Ein0 bpi

− 1 (I.89)

I.4 CONCLUSIONS 31

and, analogously, the reflection coefficient Sji (see (I.86)):

Sji =

1√I

pj

10

∫Γpj

EΩ(η) · dΓpj

Ein0 bpi

1√Ipi10

(I.90)

The above expression for the Sji is valid for an arbitrary E-plane discontinuity inrectangular waveguide. However, the specific E-plane formulation used in this paperis only valid for homogeneous structures. In this particular case, as the material andpropagation constant on all ports must be the same, the inmittances are identical inevery port. Thus, (I.90) may be further simplified as follows

Sji =

∫Γpj

EΩ(η) · dΓpj

Ein0 bpi

(homogeneous structure) (I.91)

It may be observed that, for both cases (H-plane and E-plane), the inmittancedoes not appear in the computation of Sii. This is due to the fact that the powerwaves compared correspond to the same port, and the inmittance associated to thetwo power waves are, obviously, identical. That does not occur for the transmissioncoefficients Sji as far as the power waves compared correspond to different ports and,in the general case, will have different inmittances.

I.4 Conclusions

The paper reviews a general theory of rectangular waveguides including a detaileddiscussion on rectangular waveguide discontinuities for both the E-plane and H-planeformulations. A large number of structures and devices (e.g., bends, junctions, phaseshifters, impedance matching circuits, filters, etc.) that are part of the communi-cation systems in the microwave and millimeter frequency bands are suitable to beanalyzed by the H-plane or E-plane formulations. Stabilized variational formulationsfor both H-plane and E-plane cases have been discussed. A practical approach forcomputing the scattering parameters using Finite Elements has also been presented.The scattering parameters completely characterize the behaviour of the structuresand devices from the microwave engineering point of view. The present theory laysdown a theoretical formulation for the simulations of waveguides using fully auto-matic hp-Finite Elements presented in the second and third parts of this work.

32

Part II

Energy-Norm Based Automatichp-Adaptivity

II.1 Introduction

As it was mentioned in the first paper (part I) on this series, the accurate analysis andcharacterization of “waveguide discontinuities” is an important issue in microwaveengineering (see e.g., [3], [4]). Waveguide discontinuities, i.e., the interruption in thetranslational symmetry of the waveguide, may be a unavoidable result of mechanicaldefects or electrical transition in waveguide systems, or they may be deliberatelyintroduced in the waveguide to perform a certain electrical function. Specifically,discontinuities in rectangular waveguide technology are very common in the com-munication systems working in the upper microwave and millimeter wave frequencybands. In many cases, the rectangular waveguide discontinuities can be analyzed intwo-dimensions (2D) because of the invariant nature of the geometry along one di-rection. This is the case of the so called H-plane and E-plane rectangular waveguidediscontinuities, which are the target of this work. It is worth noting that a largenumber of structures and devices fit into this category.

In this paper, a fully automatic energy-norm based hp-adaptive Finite Element(FE) strategy [6, 7], which has been extended for electromagnetic applications [8,9, 10, 11, 12, 13, 14, 7] is applied to a number of relevant waveguide structures.The adaptive methodology has a number of advantages that makes it suitable forthe analysis of complex structures (containing several waveguide sections, disconti-nuities, complex geometries, dielectrics, etc.) in contrast to other semi-analytic andnumerical techniques. Namely:

• It automatically resolves different types of singularities i.e., different types ofdiscontinuities.

• It efficiently deals with high frequencies, that is, it delivers a low dispersionerror [33, 34].

• It provides high-accuracy results, so the S-parameters (see comments below)can be accurately computed.

• It enables modeling of complex (non-uniform) geometries.

II.2 HP FINITE ELEMENTS AND AUTOMATIC ADAPTIVITY 33

The waveguide theory and a detailed analysis of rectangular waveguide disconti-nuities (including the finite element variational formulations used) were presented inthe first part of the series. Extensive numerical results presented in this part illustratethe flexibility, reliability, and high-accuracy simulations obtained with this methodol-ogy, providing more accurate results than with semi-analytical (in particular, Mode-Matching —MM— [15], [16, Chapter 9]) techniques. The adaptive methodology isshown to produce exponential convergence rates in terms of the energy-norm error ofthe solution against the problem size (number of degrees of freedom). Thus, the elec-tromagnetic field is accurately known (with a user pre-specified degree of accuracy)inside the structure. The high accuracy is essential in the microwave engineeringdesign aiming at finding optimum location and size of tuning elements (e.g., screws,dielectric posts, etc.) as well as for an a posteriori analysis of such structures.

The presented numerical results include computation of the scattering parametersof the structure, which are widely used in microwave engineering for the characteriza-tion of microwave devices. The notion of the scattering parameters (or S-parameters),and their computation using a finite element solution, have been explained in the firstpaper (part I) of the series. As the quantities of interest for the microwave engineerare mainly the S-parameters, a goal-oriented approach (in terms of the S-parameters)may be also desirable. In the third paper (part III) of the series, results obtainedusing hp energy-norm adaptivity are compared against those using a goal-orientedhp-adaptivity approach [18, 19, 20]. Results show that both methods are suitablefor simulation of waveguide discontinuities.

The organization of the paper is as follows. The hp finite element discretizationand automatic adaptivity strategy are briefly described in Section II.2.1 and II.2.2,respectively. The refinement strategy is based on the minimization of the projectionbased interpolation error, which is defined in Section II.2.2.1. The steps of the meshoptimization algorithm are described in Section II.2.2.2. Extensive numerical results,both for the E-plane and H-plane simulations, are shown in Section II.3. Finally,some conclusions are given in Section II.4.

II.2 hp Finite Elements and Automatic Adaptivity

In order to solve the presented electromagnetic problems, a numerical technique thatprovides low discretization errors and, simultaneously, solves the discretized problemwithout prohibitive computational cost, is needed. In this context, an adaptive hp-Finite Element Method satisfies both properties.


II.2.1 hp-Finite Elements (FE)

Each finite element is characterized by its size h and order of approximation p. In theh-adaptive version of FE method, element size h may vary from element to element,while order of approximation p is fixed (usually p=1,2). In the p-adaptive versionof the FE method, p may vary locally, while h remains constant throughout theadaptive procedure. Finally, a true hp-adaptive version of FE method allows forvarying both h and p locally.

The hp-FE method used in this paper utilizes edge (Nedelec) elements of variableorder of approximation. FE spaces associated to those elements have been carefullyconstructed (see [28] for details) so in combination with the projection based inter-polation operators (defined below), the commutativity of the de Rham diagram isguaranteed. This commutativity property is essential for showing convergence andstability of the FE method for electromagnetics [28].

The main motivation for the use of hp-FEM is given by the following result:“an optimal sequence of hp-grids can achieve exponential convergence for ellipticproblems with a piecewise analytic solution, whereas h- or p-FEM converge at bestalgebraically” (see [35, 36, 37, 38, 39, 40, 41]).

Next, the fully automatic hp-adaptive strategy is presented. Given a problem anda discretization tolerance error, the objective is to generate automatically (withoutany user interaction) an hp-grid that does not exceed the discretization error toler-ance and, at the same time, it employs a minimum number of degrees of freedom(d.o.f.), by orchestrating an optimal distribution of element size h and polynomial or-der of approximation p. By doing so, it is possible to achieve exponential convergencerates in terms of the error vs. the number of d.o.f.

II.2.2 Fully Automatic hp-Adaptivity

The self-adaptive strategy iterates along the following steps. First, a given (coarse)hp-mesh is globally refined both in h and p to yield a fine mesh, i.e., each elementis broken into four element sons (eight in 3D), and the order of approximation israised uniformly by one. Then, the problem of interest is solved on the fine mesh.The difference between the fine and coarse grid solutions is used to guide optimalrefinements over the coarse grid. More precisely, the next optimal coarse mesh is thendetermined by minimizing the projection based interpolation error of the fine meshsolution with respect to the optimally refined coarse mesh (see [6, 42] for details).

The adaptive strategy is very general, and it applies to H1-, H(curl)-, and H(div)-conforming discretizations. Moreover, since the mesh optimization process is based


on minimizing the interpolation error rather than the residual, the algorithm is prob-lem independent and it can also be applied to nonlinear and eigenvalue problems.

The hp self-adaptive strategy incorporates also a two-grid iterative solver, whichallows to solve the fine grid problems efficiently. Indeed, it has been shown in [43, 44]that it is sufficient a partially converged fine grid solution to guide optimal hp-refinements. Thus, only few two-grid solver iterations are needed (below ten pergrid).

In the remainder of this section, the projection based interpolation operator [45,46], which is the main ingredient of the mesh optimization algorithm, is presentedfirst. Then, the mesh optimization algorithm is briefly described.

II.2.2.1 The projection based interpolation operator

The idea of projection based interpolation operator is based on three properties.

• Locality: Determination of element interpolant of a function should involve thevalues (and derivatives) of the interpolated function in the element only.

• Conformity: The union of element interpolants should be globally conforming.

• Optimality: The interpolation error should behave asymptotically, both in hand p, in the same way as the actual approximation error.

The H1-conforming projection based interpolation operator is presented first. Letu ∈ H1+ε(K) with ε > 0. Locality and conformity imply that the interpolant w = Πushould match the interpolated function u at vertexes:

w|vert = u|vert (II.92)

With the vertex values fixed, we project over each edge, i.e.:

w := arg minv:(v−u)|vert=0

‖ v − u ‖edge (II.93)

This definition preserves locality and conformity. It also preserves optimalityprovided that the optimal edge norm is selected, which is dictated by the problembeing solved and the Trace Theorem (see [28] for details). For example, in 1D,the optimal edge norm is the H1

0 -norm. In 2D, the H1/2-seminorm, and in 3D, theL2-norm, should be used.


Using the same argument, once vertex and edge values are fixed, projection overthe interior of the element (faces in 3D) is performed. Thus, the projection basedinterpolation operator for 2D H1-problems is formally defined as:

w(v) := u(v) for each vertex v

|w − u| 12,e → min for each edge e

|w − u|1,K → min in the interior of element K

(II.94)

For a definition of projection based interpolation operator for 3D H1-problems,see [47].

Similarly, a projection based interpolation operator can be defined for elementsin H(curl), which is the space of interest for the electromagnetic field. Given E 14 inH(curl), the projection based interpolator Πcurl specialized to the 2D case (for the3D see [45, 48]), is denoted by Ep = ΠcurlE, where Ep is given by:

‖ Ept − Et ‖− 1

2,e→ min for each edge e

|∇× Ep −∇× E|0,K → min

(Ep − E, ∇φ)0,K = 0, for every “bubble” function, in the interior of element K

(II.95)

Here, the bubble functions come from an appropriate polynomial space mapped bythe gradient operator onto the subspace of fields E with zero curl and tangentialtrace on the element boundary.

A similar operator can be defined for H(div) problems (see [28]).Finally, it is important to mention that the de Rham diagram equipped with these

projection based interpolation operators commutes (see [28, 14, 45] for details), whichis critical for proving stability and convergence properties of the FEM for Maxwellequations.

II.2.2.2 The mesh optimization algorithm

The mesh optimization algorithm in 2D follows the next steps.

14E is used here to abstractly denote an element in H(curl). In this paper, discretizations inH(curl) are used for the magnetic field on the H-plane and the electric field on the E-plane of thestructures.


• Step 0: Compute an estimate of the approximation error on the coarse grid.

The approximation error on the coarse grid is estimated by simply computingthe norm of the difference between the coarse and the fine grid solutions. If thedifference (relative to the fine grid solution norm) is smaller than a requestederror tolerance, then the fine mesh solution is delivered as the final solution,and the optimization algorithm finishes.

• Step 1: For each edge in the coarse grid, compute the error decrease rate forthe p refinement, and all possible h-refinements.

Let p1, p2 be the order of the edge sons in the case of h-refinement, and letE = Eh/2,p+1 denote the fine grid solution. Then, the error decrease rate iscomputed as:

Error decrease (hp) =‖Eh/2,p+1 −Πcurl

hp E‖ − ‖Eh/2,p+1 −Πcurlhp

E‖(p1 + p2 − p)

,

where hp = (h, p) is such that h ∈ h, h/2. If h = h, then p = p + 1. Ifh = h/2, then p = (p1, p2), where p1 + p2 − p > 0, maxp1, p2 ≤ p + 1.

• Step 2: For each edge in the coarse mesh, choose between p and h refinement,and determine the guaranteed edge error decrease rate.

The optimal refinement is found by comparing the error decrease correspond-ing to the p-refinement with all competitive h-refinements. Competitive h-refinements are those that result in the same increase in the number of degrees-of-freedom (d.o.f.) as the p-refinement, i.e., h = h/2 and p1 + p2 − p = 1.

Next, the guaranteed rate with which the interpolation error must decreaseover the edge is determined. That is, for each edge, the maximum of theerror decrease rates for the p-refined edge and all possible h-refined edges iscomputed.

• Step 3: Select edges to be refined.

Given the guaranteed rate for each edge in the mesh, the maximum rate for allelements is calculated

guaranteed ratemax = maxe

(edge e guaranteed rate) .

All edges that produce a rate within 1/3 of the maximum guaranteed rate, areselected for a refinement. The factor 1/3 is somehow arbitrary.


• Step 4: Perform the requested h-refinements enforcing the 1-irregularity ruleof the mesh.

A loop through elements of the coarse grid is performed. If at least one edgeof the element is to be broken, the element is refined accordingly. As in [6, 42],element isotropy flags are computed. Isotropic h-refinement are enforced ifthe error function within the element changes comparably in both elementdirections.

After this step, the topology of the new coarse mesh has been determined, andit remains only to establish the optimal distribution of orders of approximationfor the involuntarily h-refined edges, and for the interior nodes of the elementi.e., those nodes that are not located on the boundary of the element. For theinterior nodes, the starting point for the minimization procedure will be basedon the order of approximation p for the adjacent edges15.

• Step 5: Determine the optimal orders of approximation p for the refined edgesand elements.

This step consists basically of p-adaptivity over a given grid with the fine grid asa reference solution. Unfortunately, due to the possible presence of involuntaryedge h-refinements and too low p for h-refined elements, the interpolation errorof the coarse grid after step 4 may actually be larger than the interpolationerror for the original coarse mesh. Thus, extra technical details are consideredin order to guarantee interpolation error decrease. These details are quiteinvolving, and are described in [14].

Some remarks on the mesh optimization algorithm follow:

• A similar but yet more involved mesh optimization algorithm has been imple-mented for 3D problems, although the 3D electromagnetic version is still underdevelopment.

• The main difference between the fully automatic hp-adaptive strategy for el-liptic and electromagnetic problems resides in the definition of the projectionbased interpolation operator.

• A similar algorithm can be implemented for H(div) problems.

15For triangles, the initial order of approximation will be equal to the maximum of the threeedges of the element. For quadrilaterals, we have a horizontal and a vertical order of approximationp = (ph, pv). In this case, the starting point for the minimization procedure will be the maximumof the two horizontal edges for ph and the maximum of the two vertical edges for pv.

II.3 NUMERICAL RESULTS 39

II.3 Numerical Results

In the following, a number of rectangular H-plane and E-plane waveguide disconti-nuities, as well as more complex structures obtained by combining several disconti-nuities, are analyzed. The analysis of all these structures is performed by using thefully automatic hp-adaptive FE strategy presented above.

TE10 mode excitation has been used in all the structures. Also, the ratio of thebroad dimension a to the narrow dimension b of the rectangular waveguide sectionsis considered to be a/b = 2. The results correspond to a given frequency which ischosen to be in the middle of the monomode region, i.e., k = 1.5kc10. Exceptionally,the structure analyzed in Section II.3.2.4 is solved for a large number of frequencieswithin a given frequency region, in order to characterize its frequency response.The lengths of the waveguide sections that connect the discontinuity to the ports ofthe structure are typically around one wavelength for the H-plane structures, andaround half a wavelength for the E-plane structures. This is enough for the firstabsorbing boundary condition used at the ports to perform correctly. The reasonwhy the length of the waveguide sections is lower for the E-plane structures is thefollowing. The same 2D domain (obviously with different b.c.) is used for the samekind of discontinuity, independently of analyzing it with the H-plane or E-planeformulation. Thus, a given length of a port in the 2D problem domain correspondsto the broad dimension a or narrow dimension b of the rectangular waveguide at thatport, depending on the formulation used. As a consequence, the cut-off frequency isaffected and, hence, the propagation constant β. Thus, the same physical length ofa given waveguide port correspond to different electrical lengths for the E-plane andH-plane cases; being half the electrical length of the E-plane case when comparedwith the H-plane case due to the relation a/b = 2 is satisfied.

Typically, quite coarse meshes are used as initial grids in order to assess therobustness of the hp strategy in the context of real engineering analysis in which theinitial mesh has to be as coarse as possible to simplify the mesh generation process.The convergence history is always shown using a log scale for the energy error (in

percent of the energy norm) in the ordinate axis and a scale corresponding to N1/3dof

(being Ndof the number of degrees of freedom in the mesh) in the abscissa axis.Thus, according to [40] and references therein, an straight line should appear in theplot showing the theoretical exponential convergence that can be achieved with anoptimal hp adaptivity strategy. Note that the abscissa scale corresponds to N

1/3dof

while abscissa axis tics should be read as Ndof in the plots.The scattering parameters obtained using the hp-FEM are compared with val-

ues computed with the Mode Matching (MM) method (see e.g., [15], [16, Chapter


9]). The MM method can be considered as a semi-analytic method. It consists ofthe decomposition of the domain of the problem into several simple domains, typi-cally with translational symmetry, in which, an analytical modal expansion can beperformed. Imposing the tangential continuity of the field and orthogonality of themodes yields a system of equations in which the unknowns are the coefficients of themodal expansions.

The FEM scattering parameter results delivered from the hp adaptivity are moreaccurate than MM results. In this context, it is important to point out that MMresults are typically considered as a reference for the engineering analysis of discon-tinuities in rectangular waveguide technology, as for the structures shown below. Inaddition, the hp-FE technology enables modeling of more complex structures whichcannot be solved using the MM.

II.3.1 H-plane discontinuities

The analysis of several H-plane discontinuities is considered next. The boundarycondition of the metallic conductors represents a Neumann boundary condition forthe H-plane formulation.

II.3.1.1 H-plane waveguide section

The first structure shown in Fig. II.1 is a simple rectangular waveguide section. Thisstructure is selected as a first verification of the code, specifically for the boundaryconditions at the ports and the control of the dispersion error. Since there is nodiscontinuity in the translational symmetry for this structure, it may be analyzedby means of either the H-plane or E-plane formulations. Results shown in thissection correspond to the H-plane analysis. Also, because there is no discontinuity,the scattering parameters of the structure are known to be S11 = S22 = 0 andS21 = S12 = exp(−jβ10l), where l denotes the waveguide section length. In this case,l is equal to 2 wavelengths and, thus, S21 = S12 = exp(−j4π) = 1. The field solutionis also known: it corresponds to the field TE10 mode inside of the waveguide section.

The solution is smooth: a half-sine type variation in the y direction (the ±ξ localaxis of the waveguide) and constant in amplitude and phase variation as exp(−jβ10x)along the x direction (the ±ζ local axis of the waveguide). Thus, the hp-adaptivestrategy is expected to deliver an increase in the polynomial order of approximationp. The initial mesh used for the analysis is shown in Fig. II.2 together with someintermediate meshes. The colors indicate, according to the scale on the right, theorder p of the elements. It is important to note that the order corresponds to the H1

Lagrange multiplier and that the field of H(curl) is of order p − 1. As an example,


the green color of the initial mesh of Fig. 2(a) indicate that all elements are of order3 for the Lagrange multiplier and order 2 for the magnetic field. It is observed thatorder p is increased until the maximum p (p = 9) is reached in the fine grid; fromthis moment, h refinement is selected until the specified error criterion is satisfied.

The convergence history for the exact error and the estimated error is plotted inFig. II.4, showing the quality of the error estimation and the exponential behavior ofthe error. The exponential convergence is deduced from the observance of a straightline in the plot for the log type ordinates axis and the N

1/3dof scale in the abscissa axis

set in the figure. It is worth noting that the slope change in the convergence historycorresponds to the moment when the maximum p is reached, so h refinements areforced. In other words, this slowdown in convergence would not have occurred ifhigher order elements were allowed. A plot of the field in the structure, specifically,|Hy|, is shown in Fig. II.3. In this field plot, as in the other field plots shown inthe paper, the colors indicate the intensity of the magnitude of the correspondingfield component (Hy in this case) in linear scale, corresponding the blue color to zerovalue and the orange/red color to the maximum value of the field.

It is clearly observed in Fig. II.3 the sine and the zero variations along the y andx axis, respectively. Note that the the zero variation along the x axis is becausewe are referring to |Hy| and not Hy. A constant magnitude in the direction ofpropagation means that there is only one wave in the waveguide and, thus, |Hy| =|H in

0 | | sin(πξ/a)|, so it does not depend on the ζ ≡ ±x direction. If a discontinuityhad generated a reflected wave, a stationary wave pattern would have been observedat the input waveguide (as it is seen in the examples below).

The scattering parameters have been computed at each iteration step of thehp strategy. Due to the reciprocity and symmetry of the structure, the scatteringbehavior of the discontinuity is characterized by performing one analysis (excitingany of the two ports).The results for the first iterations are shown in Tab. II.1. Afast convergence is observed.


y

x

x

H-plane

l

Port 2

y

ζPort 1

ξ

η

ξζ

η

ε0, µ0b

ε0, µ0

Ω

Γ2p

Γ1p

ΓN

H-planeζ

ξζ

ξ

a

l

ΓN

a

Figure II.1: H-plane waveguide section


xyz

(a) Initial mesh

11 12 13

xyz

(b) Mesh 4th iteration

14 15

16 17

28 29

30 31

42 43

44 45

xyz

(c) Mesh 5th iteration

58 59

60 61

70 71

72 73

82 83

84 85

94 95

96 97

110 111

112 113

122 123

124 125

134 135

136 137

146 147

148 149

162 163

164 165

174 175

176 177

188 189

190 191

202 203

204 205

xyz

(d) Mesh 7th iteration

Figure II.2: Initial mesh and some hp meshes for the H-plane waveguide section

Table II.1: Scattering parameters for the H-plane waveguide section

|S11| |S21| arg(S21)

Iter. 1 1.0302e-02 0.9991183 10.1204

Iter. 2 5.9652e-04 0.9994050 0.9797

Iter. 3 4.6872e-07 0.9999995 0.0402

Iter. 4 2.7117e-07 0.9999997 0.0013

Analytic 0.0 1.0 0.0


Figure II.3: Magnitude of Hy, i.e., |Hy|, corresponding to the H-plane waveguide section

-5

-4

-3

-2

-1

0

1

2

228 448 778 1239 1854 2645 3634 4843 6294

log

(rel

ativ

e er

ror %

)

Ndof

H-plane waveguide section

exact-energy-errorestimate-energy-error

Figure II.4: Convergence history for the H-plane waveguide section (energy norm error forthe magnetic field solution)


II.3.1.2 H-plane right angle bend

An H-plane 90 bend is analyzed next. The structure is shown in Fig. II.5. Thestructure is analyzed by exciting port 1 (on the left). The bend is a common partof microwave circuits. The initial mesh used for the analysis is shown in Fig. II.6.Despite the coarseness of this mesh, the hp-strategy achieves an energy error lowerthan 1% error after 5 iterations. The convergence history (up to an error as lowas 0.01%) is shown in Fig. II.9. The final mesh is shown in Fig. II.8. Heavy h-refinements of the mesh around the corner are observed. The hp-strategy in thiscase tends also to increase the p. Actually, all elements of the final mesh (exceptthose near the corner) have reached the maximum p order. This is the right strategysince the solution of the problem is smooth (the boundary condition at the conductorsfor the H-plane formulation is of homogeneous Neumann type16). The h-refinementaround the corner is precisely due to the fact that the maximum p has been reachedand, in order to reduce the error in this region, the elements must be made smaller.

A plot of the field in the structure, specifically, |Hy|, is shown in Fig. II.7. They-component corresponds to the local ξ component at the excitation port and thelocal ζ component at the transmitted port. Notice in the figure the stationary wavepattern in the input waveguide (between the excitation port and the bend) becauseof the combination of the two waves propagating in opposite directions (the excitedwave and the reflected wave at the bend). No stationary wave is observed in theoutput waveguide as there is only one wave propagating outward the transmittedport. As in the previous case, S21 = S12 and S22 = S11. The results for S11 andS21 are shown (for some of the hp meshes) in Tab. II.2. The scattering parameterscomputed with the hp-FEM method are compared with those obtained with a MMtechnique. Only four significant digits are shown in the table as the MM resultsare presumed to have no more than 4 digits of accuracy17. Observe the very goodagreement of the hp-FEM results with those provided by MM; better than 1% afterthe second iteration. After the fourth/fifth iteration, the FEM results seem to bemore accurate than those provided by the MM, as implied by the convergence patternshown in the table.

16The same domain but with Dirichlet boundary conditions corresponds to the E-plane bendwhich is analyzed in Section II.3.2.1.

17This is concluded after making higher the number of modes in the modal expansions andobserving fluctuations at the fifth digit level of the values of the S-parameters.


xy

y

x

H-plane

Port 1ξ

η

η

ξ

Port 2

b

l1

b

l2

aa

ζ

ζ

ε0, µ0

ε0, µ0

Ω

ΓN

ΓNH-plane

ζ

ξ

ξ

Γ2p

l1

l2

a

ζ

Γ1p

a

Figure II.5: H-plane 90 degrees bend


11 12

13

xyz

Figure II.6: Initial mesh for the H-plane 90 bend

Figure II.7: Magnitude of Hy, i.e., |Hy|, corresponding to the H-plane 90 bend


14 68

100 132 164 196 197 198 199 166 167 134 135 102 103

70 71

16 17

206 207 208 209 175 176 177 143 144 145 111

112 113 79

80 81

43

44 45

28 29

30

58 59

60 90 91

92 122 123

124 154 155 156 186 187 188 218 219 220 221

xyz

Figure II.8: Final hp mesh for the H-plane 90 bend

Table II.2: Scattering parameters for the H-plane 90 bend

|S11| |S21| arg(S11) arg(S21)

Iter. 1 0.5465 0.8372 10.049 112.615

Iter. 2 0.4459 0.8951 16.948 101.604

Iter. 3 0.4148 0.9099 5.639 95.665

Iter. 4 0.4156 0.9096 5.619 95.617

MM 0.4161 0.9093 5.345 95.345


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

208 387 647 1003 1471 2066 2802 3695 4760

log

(rel

ativ

e er

ror

%)

Ndof

H-plane 90 degrees Bend

energy error

Figure II.9: Convergence history for the H-plane 90 bend


II.3.1.3 H-plane symmetric inductive iris

The next structure is shown in Fig. II.10. The discontinuity consists of the narrow-ing of the broad dimension of the waveguide along a certain length. This regionis referred to as an iris18. Because the iris is centered with respect the waveguidebroad dimension, it is called a symmetric iris. Finally, the term inductive is alsoutilized because the discontinuity scattering behavior with respect to the planes ofthe discontinuity is equivalent to an inductance. The character of the discontinuity,inductive or capacitive, can be deduced by observing which field lines (electric ormagnetic) of the waveguide mode (the TE10 in this case) are “cut” by the discon-tinuity [3]. It is clear from Fig. II.10 that only the magnetic field lines are cut inthis case (the electric field of the TE10 is perpendicular to the H-plane of the waveg-uide). Thus, the symmetric iris of Fig. II.10 is of the inductive type. An analogousstructure, but of capacitive type, is considered in Section II.3.2.3.

The initial mesh used for the analysis is shown in Fig. II.11. The analysis is madeby exciting port 1 (at the left). The convergence history (up to an error as low as0.02%) is shown in Fig. II.12. Notice the exponential convergence of the method.

A plot of the field in the structure, specifically, |Hy|, is shown in Fig. II.13. They-component corresponds to the ±ξ component of the field modes in the waveguide.Observe the stationary wave pattern at the input port due to the wave reflectedfrom the discontinuity and a singular behavior of the field at the re-entrant corners.The magnitude of the fields is higher at the left corners. This is “caught” by the hpstrategy that refines around the left corners during the first few iterations, and oncethe error around the left corners is controlled (comparable to other regions of thestructure), it starts to “see” the error corresponding to the region around the rightcorners (see figures II.14 and II.15).

As in the previous case, S21 = S12 and S22 = S11. The results for S11 and S21

are shown (for some of the hp meshes) in Tab. II.3. Only the results of the firstiterations are shown as the hp FEM results for the consecutive meshes are, again,presumed to be more accurate than those of MM. Observations analogous to thosementioned in the previous case may be made here.

18The term iris is used in this context to refer to an aperture that connects two waveguidesections.


y

x

a

x

H-plane

Port 2

y

ζPort 1

ξ

η

ξζ

η

Ω

Γ2p

Γ1p

H-planeζ

ξζ

ξ

ΓN

a

b

l1l

l2

t

ε0, µ0

ε0, µ0

l

ΓN

l1 t l2

Figure II.10: H-plane symmetric inductive iris (l/a = 0.6, t/a = 0.2)


xyz

Figure II.11: Initial mesh for the H-plane symmetric inductive iris

Table II.3: Scattering parameters for the H-plane symmetric inductive iris

|S11| |S21| arg(S11) arg(S21)

Iter. 2 0.7333 0.6799 -157.92 -27.243

Iter. 4 0.7401 0.6725 -156.59 -26.325

Iter. 6 0.7386 0.6741 -156.48 -26.588

Iter. 8 0.7414 0.6711 -156.46 -26.287

MM 0.7417 0.6708 -156.51 -26.259


-2

-1.5

-1

-0.5

0

0.5

1

1.5

1035 1864 3046 4645 6725 9347 12575 16473 21102

log

(rel

ativ

e er

ror

%)

Ndof

H-plane iris inductive symmetric

energy-error

Figure II.12: Convergence history for the H-plane inductive iris

Figure II.13: Magnitude of Hy for the H-plane inductive iris


xyz

Figure II.14: 11th mesh for the H-plane symmetric inductive iris showing heavy refinementsaround the left corners

xyz

Figure II.15: 19th mesh for the H-plane symmetric inductive iris showing heavy refinementsaround the left and also right corners


II.3.1.4 H-plane zero thickness septum

The structure, shown in Fig. II.16, consists of an obstacle (the septum) placed at thecenter of the waveguide. The obstacle exhibits a translational symmetry along thenarrow dimension of the waveguide (η local axis of the waveguide ports). Thus, itcan be analyzed using the H-plane formulation. Also, as for the previous structure,the scattering behavior is basically inductive since the only field lines that are cutby the septum are those of the magnetic field.

The initial mesh used for the analysis is shown in Fig. II.17. The analysis is madeby exciting port 1 (on the left). The convergence history is shown in Fig. II.18. It isobserved that the error convergence (except for the first couple of meshes, due to adeliberate coarseness of the initial mesh) behaves as predicted by the theory and astraight line is obtained in the plot (which means exponential convergence).

An example of one of the hp meshes provided by the adaptivity, specifically, themesh of the 7th iteration, is shown in Fig. II.19. Again, as predicted by the theory, itis observed the h-refinements towards the corners where there is a singular behaviorof the field and the p-refinements in the regions where the field variation is smooth.

A plot of |Hy| is shown in Fig. II.20. The y-component corresponds, as in theprevious case, to the ±ξ component of the field modes in the waveguide. Observethe stationary wave pattern at the input port due to the wave reflected from thediscontinuity and the singular behavior of the field at the septum corners.

The results for S11 and S21 corresponding to some of the iterations of the hpadaptivity are shown in Tab. II.4. As in the previous cases, S21 = S12 and S22 = S11

due to reciprocity and symmetry. Only the results up to the 10th iteration areshown, since the error in the scattering parameters obtained from the hp meshes (foriterations higher than the 10th) is expected to be lower than the one of the MMresults.

Table II.4: Scattering parameters for the H-plane zero thickness septum

|S11| |S21| arg(S11) arg(S21)

Iter. 1 0.7383 0.6743 206.84 -53.120

Iter. 4 0.7466 0.6653 205.58 -53.801

Iter. 7 0.7740 0.6332 208.64 -51.486

Iter. 10 0.7785 0.6277 209.03 -50.962

MM 0.7788 0.6273 209.03 -50.939


y

x

a

x

H-plane

Port 2

y

ζPort 1

ξ

η

ξζ

η

Ω

Γ2p

Γ1p

H-planeζ

ξζ

ξ

ΓN

a

b

l1

l2

ε0, µ0

ΓN

l1 l2

lε0, µ0

l

ΓN

Figure II.16: H-plane zero thickness septum (l/a = 0.1)


xyz

Figure II.17: Initial mesh for the H-plane zero thickness septum

-1.5

-1

-0.5

0

0.5

1

1.5

566 984 1569 2351 3355 4612 6149 7995 10177

log

(rel

ativ

e er

ror

%)

Ndof

H-plane zero thickness septum

energy-error

Figure II.18: Convergence history for the H-plane zero thickness septum


xyz

Figure II.19: 7th mesh for the zero thickness septum

Figure II.20: Magnitude of Hy corresponding to the H-plane zero thickness septum showinga stationary wave pattern at the input waveguide and singular behavior of the field at theseptum corners


II.3.1.5 H-plane zero length septum

This discontinuity is also a septum but it is placed transverse to the wave propa-gation. The dimension of the septum along the propagation direction is consideredzero. A representation of the structure is shown in Fig. II.21.

The analysis is made by exciting port 1 (on the left) and considering the coarseinitial mesh of Fig. II.22. As in the other septum discontinuity, the error convergence(see Fig. II.23) behaves as predicted by the theory (exponential convergence) and astraight line is obtained in the plot. Fig. II.24 shows, as an example, the meshcorresponding to the 7th iteration where the refinement around the septum cornerscan be seen. This is what is expected from the field solution. The y component ofthe solution (its magnitude) is shown in Fig. II.25. Again, a stationary wave patternis observed in the input waveguide section.

Finally, the results for S11 and S21 corresponding to some of the iterations ofthe hp adaptivity are shown in Tab. II.5. Only results up to the 11th iteration areshown, since the error in the scattering parameters obtained from the hp meshes isexpected to be lower (for iterations higher than the 11th) than the one of the MMresults.

Table II.5: Scattering parameters for the H-plane zero length septum

|S11| |S21| arg(S11) arg(S21)

Iter. 2 0.7200 0.6939 50.520 -37.987

Iter. 5 0.7817 0.6236 56.432 -33.550

Iter. 8 0.7883 0.6153 57.042 -32.962

Iter. 11 0.7896 0.6136 57.165 -32.840

MM 0.7897 0.6135 57.171 -32.829


y

x

a

a

x

H-plane

Port 2

y

ζPort 1

ξ

η

ξζ

η

Ω

Γ2p

Γ1p

H-planeζ

ξζ

ξ

ΓN

b

l1

l2

ΓN

l1 l2

ε0, µ0

ε0, µ0

ΓN

t

t

Figure II.21: H-plane zero length septum (t/a = 0.1)


xyz

Figure II.22: Initial mesh for the H-plane zero length septum

-2

-1.5

-1

-0.5

0

0.5

1

1.5

795 1555 2689 4272 6379 9086 12466 16596 21549

log

(rel

ativ

e er

ror

%)

Ndof

H-plane zero length septum

energy-error

Figure II.23: Convergence history for the H-plane zero length septum


xyz

Figure II.24: 7th mesh for the zero length septum

Figure II.25: Magnitude of Hy corresponding to the H-plane zero length septum showinga stationary wave pattern at the input waveguide and singular behavior of the field at theseptum corners


II.3.2 E-plane discontinuities

The analysis of several E-plane discontinuities is considered next. In contrast to theH-plane formulation, the boundary condition of the metallic conductors is of theDirichlet type for the E-plane formulation.

II.3.2.1 E-plane right angle bend

This discontinuity (Fig. II.26) is as the one of Section II.3.1.2, a 90 degrees bend.However, the plane of the bend in this case is the E-plane. The domain shape is thesame as for the H-plane bend (actually, the initial mesh is also the same, see Fig. II.6)but, this time, the homogeneous Dirichlet boundary condition at the conductors areemployed. This, apparently, produces a field singularity that occurs at the corner.The hp-strategy behaves as expected and, in contrast to the H-plane bend case, anh-refinement toward the singularity is observed while increasing the p backward. Oneof the meshes obtained by the hp-adaptivity procedure is shown in Fig. II.27.

Plots of the field component magnitudes |Ey| and |Ex| are shown in Fig. II.28. Astationary wave pattern is observed in the input waveguide (between the excitationport and the bend). The y-component corresponds to the component along thelocal −η axis of the excitation port and the component along the ζ local axis of thetransmitted port. Since the TE10 does not have ζ component, the Ey component isnull (numerically null provided that the port is far enough from the discontinuity) atthe transmitted port (port 2). Analogously, the Ex component is null at the incidentport.

The convergence history is shown in Fig. II.29. Except for the peak around thethird hp iteration (due to the coarseness of the initial mesh), the error shows anexponential decay. With respect to the convergence of the scattering parameters,Tab. II.6 shows their values (in magnitude and phase) for some of the iterations.The MM results are shown for comparison purposes. A good agreement is observed.It is worth noting again that only the results up to the 10th iteration are shown, sincethe error in the scattering parameters obtained from the hp meshes (for iterationshigher than the 10th) is lower than the one coming from the MM results.


y

x

xy

ηζ

ξ

ε0, µ0

Ω

Γ2p

l1

l2

Γ1p

η

ξζ

l2

Port 1

ε0, µ0

Port 2

E-plane

b

b

ΓD

ΓD

η

ζ

ζ

η

E-plane

bb

l1

a

a

Figure II.26: E-plane 90 degrees bend


xyz

Figure II.27: hp mesh of 11th iteration of the E-plane 90 bend

Table II.6: Scattering parameters for the E-plane 90 bend

|S11| |S21| arg(S11) arg(S21)

Iter. 1 0.5542 0.8323 -47.810 -137.81

Iter. 4 0.5387 0.8425 -46.794 -136.94

Iter. 7 0.5487 0.8360 -48.590 -138.45

Iter. 10 0.5499 0.8352 -48.558 -138.49

MM 0.5507 0.8347 -48.462 -138.46


(a) |Ey|

(b) |Ex|

Figure II.28: Magnitudes of Ey and Ex corresponding to the E-plane 90 bend showing asingular behavior of the field at the corner


-1.5

-1

-0.5

0

0.5

1

1.5

2806221917211303960682464298178

log

(rel

ativ

e er

ror %

)

Ndof

E-plane 90 degrees Bend

energy-error

Figure II.29: Convergence history for the E-plane 90 bend


II.3.2.2 E-plane right angle bend with a round corner

This structure is also a 90 bend in the E-plane but with a round corner (seeFig. II.30). In practice, depending on the mechanical process used to build thebends, the bends may either have sharp corners or round corners (as in this case).Although there is no field singularity because of the roundness of the corner, thereis a high variation of the fields around the corner, and the adaptivity behaves anal-ogously to the case of a sharp corner (in the pre-asymptotic regime, a high variationin the fields is “seen” as a singularity).

The analysis is made by exciting port 1 (on the left) and considering the coarseinitial mesh of Fig. II.31. Fig. II.32 shows a sample mesh corresponding to the 10thiteration. A refinement pattern similar to the one of Fig. II.27 can be observed. Theexponential convergence history is shown in Fig. II.34.

Plots of the field component magnitudes |Ey| and |Ex| are shown in Fig. II.33.Comments analogous to those made on the E-plane bend with a sharp are valid forthis case as well. Tab. II.7 shows the values (in magnitude and phase) of S11 and S21

for the first few iterations. A fast convergence is observed and seven digits are neededin order to be able to observe the convergence of the scattering parameters. No MMresults are shown for this case. The analysis of this structure by MM requires theuse of special functions and somehow differs of what it is usually referred to as theMM method.

Table II.7: Scattering parameters for the E-plane 90 bend with round corner

|S11| |S21| arg(S11) arg(S21)

Iter. 1 0.5835441 0.8120788 -86.29061 -176.29024

Iter. 2 0.5835960 0.8120416 -86.27239 -176.27328

Iter. 3 0.5836238 0.8120215 -86.26833 -176.26855

Iter. 4 0.5836237 0.8120217 -86.26894 -176.26880


y

x

xy

Ω

Γ2p

Γ1p

η

ξζ

l2

Port 1

ε0, µ0

Port 2

E-plane

ΓD

η

ζ

ζ

η

E-plane

l1

a

a

b

b

ε0, µ0

b b

l2

l1

ΓD r

r ηζ

ξ

Figure II.30: E-plane 90 degrees bend with round corner (r/b = 0.2)


xyz

Figure II.31: Initial mesh for the E-plane 90 bend with round corner

xyz

Figure II.32: 10th hp mesh for the E-plane 90 bend with round corner


(a) |Ey|

(b) |Ex|

Figure II.33: Magnitudes of Ey and Ex corresponding to the E-plane 90


-1

-0.5

0

0.5

1

1.5

2

900 1191 1540 1951 2430 2980 3608 4319 5118

log

(rel

ativ

e er

ror

%)

Ndof

E-plane 90 degrees bend with round corner

energy error

Figure II.34: Convergence history for the E-plane 90 bend with round corner


II.3.2.3 E-plane capacitive symmetric iris

This discontinuity (Fig. II.35) is due to a symmetric iris (as the one of Section II.3.1.3),but in the E-plane. Thus, the FEM domain is identical to the one used for the H-planeinductive symmetric iris, but with different boundary conditions on the conductorsboundaries (of Dirichlet type for this case). The analysis is made by exciting port 1(on the left).

The initial mesh is shown in Fig. 36(a). Fig. 36(b) shows a sample mesh corre-sponding to the 4th iteration. A refinement pattern around the corners of the irisis observed due to the presence of field singularities at those locations. The conver-gence history (up to an error as low as 0.1%) is shown in Fig. II.37. Exponentialconvergence is again observed.

Plots of the field component magnitudes |Ey| and |Ex| are shown in Fig. II.38. Astationary wave pattern is observed in the input waveguide (between the excitationport and the iris) as well as a singular behavior of the field at the corners of the iris.The y-component of the field in the structure corresponds to the ∓η component ofthe waveguide modes. Analogously, the x-component corresponds to the ±ζ com-ponent of the waveguide. Thus, the Ex component of the field is generated at thediscontinuity, and it is only significant close to it.

The results for S11 and S21 corresponding to some of the iterations of the hpadaptivity are shown in Tab. II.8. The equalities S21 = S12 and S22 = S11 hold dueto reciprocity and symmetry of the structure. Only the results of the first iterationsare shown as the hp FEM results for the consecutive meshes are presumed to bemore accurate than those of MM.

Table II.8: Scattering parameters for the E-plane capacitive symmetric iris

|S11| |S21| arg(S11) arg(S21)

Iter. 1 0.3180 0.9481 -163.31 -53.24

Iter. 2 0.3070 0.9517 -162.71 -52.60

Iter. 5 0.3013 0.9535 -162.37 -52.25

Iter. 6 0.3009 0.9537 -162.35 -52.22

MM 0.3008 0.9537 -162.36 -52.23


y

x

Ω

Γ2p

Γ1p

ε0, µ0

l1 t l2

E-plane

d

ΓD

ΓD

b

b

a

l

l2

l1

t ε0, µ0

ζ

ξ

ζ

ξη

η

ηζ

ζη

y

E-plane

x

Port 1

Port 2

Figure II.35: E-plane capacitive symmetric iris (d/b = 0.6, t/b = 0.2)


48 49

52 53

54 55

62 63

60

61

56 57

58 59

64 65

50 51

xyz

(a) Initial mesh

xyz

(b) Mesh of 4th iteration

Figure II.36: Initial mesh and mesh of the 4th iteration for the E-plane capacitive sym-metric iris


-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

88437147568344343383251218061246816

log

(rel

ativ

e er

ror %

)

Ndof

E-plane capacitive symmetric iris

energy error

Figure II.37: Convergence history for the E-plane capacitive symmetric iris


(a) |Ey|

(b) |Ex|

Figure II.38: Magnitude of Ey and Ex corresponding to the E-plane capacitive symmetriciris


II.3.2.4 E-plane double stub section

The structure is shown in Fig. II.39. It consists of a main waveguide going fromport 1 to port 2 and two waveguides loading it (referred to as stubs). The stubs areclosed at their ends causing a total reflection of the energy at their inputs. Thus,the load impedance that they present to the main waveguide is purely imaginary.The behavior of this structure may be roughly explained as follows. The load ofeach stub is like a discontinuity in the waveguide, producing a reflected wave (andalso a transmitted wave) with a given phase. As there are two stubs, i.e., twodiscontinuities, the contributions from the two stubs may (totally or partially) addor cancel, depending on the relative phase between the corresponding waves. Therelative phase depends (for given stubs dimensions) on the electrical distance θd

between them. As the electrical distance depends on the frequency for a givenphysical distance d, i.e., θd = β10d, the frequency response can be adjusted for severalapplications. For example, the double stub section can be designed to work as a phaseshifter, i.e., causing an extra shift in the phase of the wave at the transmitted port fora given frequency band. This is done by designing the double stub in such a way thatthere is an adding interference at the transmitted port of the two waves generated bythe stubs (with a given phase). Another usual application is an impedance matchingnetwork, i.e., the double stub is designed to compensate the reflection present at agiven port, due to, e.g., a change in the height of the waveguide, in a given frequencyband. This is done by adjusting the design so there is a cancellation of the tworeflected waves at the stubs junctions (180 out of phase with respect to each other).

In here, the double stub has been designed for the latter application, i.e., tohave a null reflection around a frequency given by k0 = 1.39kc (kc being the cut-offfrequency of the TE10 mode). Notice in Fig. II.39 that the waveguide sections atthe two ports are identical. Thus, if the two stubs were not present there would beno reflection (S11 ideally null) as the structure would simply consists of a waveguidesection. The reason we analyze the structure in Fig. II.39 (which has little practicalapplication) is because this is a good test case. The idea is that for the null reflectionfrequency, the fields in the main waveguide have to be basically the same as those ina single waveguide section. Since the stubs are identical, the structure is symmetrical(S11 = S22). As in the other cases, the analysis is made by exciting port 1 (on theleft).

The frequency response of the structure around the frequency corresponding tok = 1.39kc is shown in Fig. 40(a). The ordinate axis corresponds to S11 in dB, i.e.,10 log10 |S11|2 19. The results have been obtained by running the hp-adaptivity until

19The dB is a logarithmic unit for dimensionless magnitudes, but it is always with respect to the


an energy error of 1% is achieved and computing S11 using the final hp-mesh. Theexpected low values of the reflection coefficient S11 around k0/kc = 1.39 is observed.For a comparison, Fig. II.40 presents also results obtained using a MM technique. Avery good agreement is observed.

Figure 40(b) shows the frequency response over a broad frequency interval thatconsists of a centered band covering the 60% of the monomode frequency band.A very good agreement between the hp-FEM and MM results is observed. Theonly exception is around the frequency corresponding to k0/kc = 1.74. For thesefrequencies, a more refined mesh seems to be needed.

Convergence properties have been studied by executing the hp-adaptivity with anenergy-norm error tolerance of 0.01% (and a maximum number of iterations equalto 20). Results corresponding to five significant frequency points (symmetricallychosen around the value of k0/kc = 1.39): k0/kc=1.32, 1.36, 1.42, 1.46 are displayedin Fig. II.41. For k0/kc = 1.32, 1.46 there is a high reflection of the energy at theinput waveguide; for k0/kc = 1.39 there is a very low (almost null) reflection atthe input; and for k0/kc = 1.36, 1.42 an intermediate situation occurs. Except forthe first few iterations, the plots follow approximately a straight line, reflecting anexponential decrease of the energy-norm error. The erratic behavior of the errorduring the first few iterations is due to the coarseness of the initial mesh (shown inFig. 42(a)).

The final meshes for k0/kc = 1.32 (high reflection at the input), k0/kc = 1.39 (lowreflection at the input), and k0/kc = 1.36 (intermediate reflection at the input) aredisplayed in Figures II.42 and II.43. For the case of high reflection, the mesh (shownin Fig. 42(b)) displays a typical refinement pattern around the corners (junctions ofthe stubs with the main waveguide). The electric field20 for this case is plotted inFig. 44(a). A stationary wave pattern in the input waveguide is observed due to theinterference between the incident and the reflected waves. On the other hand, themesh for the low reflection case (shown in Fig. 43(b)) displays a situation very similarto the situation of the smooth field solution inside a waveguide section (i.e., withoutsingularities). As it was explained above, this occurs because the stubs do not loadthe main waveguide (it is like if they were not present in the structure). This is best

power (and not the field) magnitudes. Thus, when applied to the scattering parameters that relatefield quantities, the “10 log10” factor has to be applied to the square of the scattering coefficient.For example, S11 = −40dB means that the power reflected at the input waveguide is 40 dB belowthe power of the excitation at the input waveguide, i.e., 104 lower (or equivalently, the electric andmagnetic fields are 102 lower).

20Note that the magnitude plotted is√|Ex|2 + |Ey|2 that, although does not correspond to |E|

or other physically meaningful magnitude, it is useful for visualizing in one plot the field in themain waveguide and in the stubs.


understood by seeing Fig. 44(b), which displays the electric field in the structure forthis case. The stationary wave pattern at the input waveguide can hardly be seen,which means that the level of the reflected wave is very low. Finally, the mesh for theintermediate case (k0/kc = 1.36) is shown in Fig. 43(a). It is observed how effectivelythe mesh corresponds to an intermediate case between the meshes of Figures 42(b)and 43(b).


xy

η

ξζ

Port 1

l1

ξ

ζ

η

a

b

b

b

l2

bs1

bs2

l

E-plane

Port 2

y

x

Ω

Γ1

p

b

ζ

η

b b

bε0, µ0

bs1 bs2

Γ2

p

ΓD

ΓD

E-plane

ζ

ηΓD

ΓDΓD

ΓD

ΓD ΓD

l

l1 l2

Figure II.39: E-plane double stub structure (bs1/b = bs2/b = 5.0249, l/b = 1.2608)


-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

1.3 1.35 1.4 1.45 1.5

dB

K0/Kc

E-plane double stub

|S11||S11| (MM)

(a) Frequency response in the band of interest

-50

-40

-30

-20

-10

0

1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8

dB

K0/Kc

E-plane double stub

|S11||S11| (MM)

(b) Frequency response over the 60% of the monomode region

Figure II.40: Frequency response of E-plane double stub section (|S11| in dB)


-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

571 1067 1790 2783 4087 5747 7804 10301 13281 16786 20860

log

(rel

ativ

e er

ror

%)

Ndof

E-plane double stub

energy error (K0/KC=1.32)energy error (K0/KC=1.36)energy error (K0/KC=1.39)energy error (K0/KC=1.42)energy error (K0/KC=1.46)

Figure II.41: Convergence history for the E-plane double stub section


59 60 61 62

63

64

65

66

67 68 69

70

71

72

73

74 75 76 77

xyz

(a) Initial mesh

59 60 61 78 79

102 103 176 177 334 335 412 413 508 509 510 511 415 337 179 105

81

592 593 594 595 519 520 521 345 346 347 231 232 233 209

210 211 274 486 570 571 572 573 488 489

276 277

154 155

64

65

66

90 91

186 187 288 289 498 499 582 583 584 585 501 291 189

166 167 168 220 221

222 354 355 356 422 423 424 530 531 532 533

112 113

242 243 432 433 604 605 606 607 435 245

196 197 198 254 255

256 442 443 444 540 541 542 543

126 127

312 313 402 403 550 551 552 553 405 315

322 323 324 452 453

454 614 615 616 617 624 625 626 627 463 464 465 367

368 369 378 474 636 637 638 639 476 477

380 381

300 301

71

72

73

138 139

140 264 265 266 392 393

394 558 559 560 561

75 76 77

xyz

(b) Mesh for k0/kc = 1.32

Figure II.42: Initial mesh and mesh corresponding to 1% energy error for the E-planedouble stub section


59 60 61 78 79

142 143 256 257

258 259 145 81

354 355 356 357 309

310 311 286 366 367

368 369 288 289

186 187

64

65

66

90 91

198 199 298 299

300 301 201 208 209

210 322 323 324 325

102 103

152 153 378 379

380 381 155 162 163

164 266 267 268 269

116 117

234 235 344 345

346 347 237 244 245

246 414 415 416 417

400 401 402 403

332 388 389 390 391

334 335

222 223

71

72

73

128 129

130 174 175 176 276 277

278 279

75 76 77

xyz

(a) Mesh for k0/kc = 1.36

59 60 61 78 79

80 81

63

64

65

66

90 91

92 93

102 103

104 105

116 117

118 119

70

71

72

73

128 129

130 131

75 76 77

xyz

(b) Mesh for k0/kc = 1.39

Figure II.43: Meshes corresponding to 1% energy error for the E-plane double stub section


(a) High reflection at the input waveguide (k0/kc = 1.32)

(b) Low reflection at the input waveguide (k0/kc = 1.39)

Figure II.44: Electric field√|Ex|2 + |Ey|2 in the E-plane double stub section

II.4 CONCLUSIONS 87

II.4 Conclusions

An hp-adaptive Finite Element Method for studying the characterization of mi-crowave rectangular waveguide discontinuities with a geometry invariant along onedirection (a common situation in rectangular waveguide technology), has been pre-sented. The assumption on the geometry of the discontinuity enables a 2D analysisin the so-called H-plane or E-plane of the structure.

A fully automatic hp-adaptive strategy based on maximizing the rate of decreaseof the (projection-based) interpolation error of the fine grid solution has been appliedto a number of important engineering examples. Computation of the scatteringmatrix that characterize the electromagnetic behavior of the discontinuities for themicrowave engineer has been implemented as a post-processing of the solution.

A wide variety of structures have been analyzed, including microwave engineeringdevices of medium complexity. The hp adaptivity has shown to deliver exponentialconvergence rates for the error for both regular and singular solutions. A consistentconvergence pattern indicates that the results are more accurate than those obtainedwith semi-analytical techniques. At the same time, this hp-methodology presents theimportant advantage of being a purely numerical method, which allows for model-ing complex waveguide structures that cannot be simulated using semi-analyticaltechniques.

II.5 Acknowledgment

The authors would like to thank Sergio Llorente-Romano at the Universidad Politecnicade Madrid for their helpful discussions on the MM techniques and for letting us usetheir MM codes that have been used to produce some of the MM results shown inthis paper.

88

Part III

Goal-Oriented hp-Adaptivity

III.1 Introduction

As it was mentioned in the first part, the accurate analysis and characterizationof “waveguide discontinuities” is an important issue in microwave engineering (seee.g., [3, 4]). Specifically, H-plane and E-plane rectangular waveguide discontinuities,which are the target of the presented work, play an important role in the commu-nication systems working in the upper microwave and millimeter wave frequencybands.

A method based on hp-Finite Elements, presented in Part I, seems to be a perfecttool for solving the waveguide discontinuity problem because:

• the method automatically refines the grid around the singularities,

• the method automatically increases the polynomial order of approximation inregions where the solution is smooth, and,

• the overall convergence of the method is fast (exponential convergence).

It is critical to notice that the exponential convergence of the hp-method describedin part II is measured in terms of the energy-norm of the error function againstthe problem size. Thus, the electromagnetic field is accurately known (with a userpre-specified level of accuracy) inside of the structure. This may be useful to themicrowave engineer in order to predict the location and size of tuning elements(e.g., screws, dielectric posts, etc.) and/or for the design of modifications to theoriginal design. However, the microwave engineer is first (and mainly) interestedin accurately computing the S-parameters (see part I) for details) of the waveguidestructure. Therefore, we should design a numerical method that is consistent withour goal, i.e., a high accuracy computation of the S-parameters, regardless of thequality of the solution in the rest of the domain21. This type of methods are calledweighted-based residual methods or goal-oriented methods (see [50, 51, 52, 53, 54, 55]for details).

At the same time, we would like to maintain all desirable properties (such asexponential convergence) of the self-adaptive hp-FEM described in part II.

21For instance, it may be unnecessary to resolve all singularities in order to accurately computethe S-parameters.

III.2 A FULLY AUTOMATIC GOAL-ORIENTED HP -ADAPTIVE FEM 89

In this work, we propose to use the fully automatic goal-oriented hp-FEM pre-sented in [18, 19, 20] to compute the S-parameters of rectangular waveguide struc-tures. This method incorporates all desirable features mentioned above with theadditional advantage of obtaining exponential convergence in terms of a sharp upperbound of the quantity of interest (in this case, the S-parameters) against the problemsize – number of degrees of freedom. In this paper, we consider the application ofthis goal-oriented methodology to a rectangular (H-plane) waveguide structure withsix inductive irises. It is worth noting that this structure incorporates as many astwenty-four singularities of different intensity, due to presence of re-entrant cornersin the geometry.

The organization of the paper is as follows. In Section III.2, we describe ournumerical methodology based on the self-adaptive goal-oriented hp-FEM. The vari-ational formulation together with relevant properties of the quantities of interest areshown first. Following [18, 19], we present the main ideas that make the practicaluse of goal-oriented adaptivity possible. Then, we introduce the projection basedinterpolation operator for 2D edge elements, which is a key component of the hpgoal-oriented mesh optimization algorithm that is presented thereafter. A brief dis-cussion on implementation details is also provided at this point. Section III.3 isdevoted towards numerical results. We introduce first the problem of interest in sub-section III.3.1. Then, main advantages and disadvantages of using the goal-orientedhp-adaptivity (as opposed to the energy-norm hp-adaptivity) for our problem of in-terest are carefully analyzed. Finally, we close with some conclusions in Section III.4.

III.2 A Fully Automatic Goal-Oriented hp-Adaptive

Finite Element Method

In this Section, we present the self-adaptive goal-oriented hp-FE method. The algo-rithm is an extension of the energy-norm based hp-adaptive FE method presentedin part II.

III.2.1 Variational Formulation and Goals

We consider the variational formulation presented in Part I. For convenience, it isbriefly reviewed here serving also as an introduction to the notation used in theremainder of the paper. Specifically, the H-plane formulation is shown since is theone used to obtain the numerical results presented later in Section III.3.


We excite the TE10-mode at one of the ports of the H-plane structure. As ex-plained in part I, the original 3D problem can be reduced to a 2D boundary valueproblem, which we can formulate in terms of the components of the magnetic fieldparallel to the H-plane (denoted by HΩ) as follows:

∇×(

1

ε∇×HΩ

)− ω2µHΩ = 0 in Ω

n× 1

ε∇×HΩ =

jω2µ

β10

n× n× (2Hin −HΩ) on Γ1

n× 1

ε∇×HΩ = −jω2µ

β10

n× n×HΩ on Γ2

n× 1

ε∇×HΩ = 0 on Γ3 ,

(III.96)

which is a particularization of expressions of part I; specifically, equations (I.51) and(I.52)–(I.54).

In (III.96), Γ1, Γ2, and Γ3 stand for the parts of the boundary corresponding to theexcitation port, non-excitation port, and the perfect electric conductor, respectively.That is, the formulation has been reduced to the case of a two ports structure;the Cauchy boundary condition at the ports has been expressed for the case ofexcitation port and non excitation port. Note that no perfect magnetic conductors,i.e., Dirichlet boundary conditions (used to implement symmetry planes) have beenconsidered. The symbol β10 refers to the propagation constant of the TE10 mode;Hin is the incident magnetic field at the excitation port; ω is the angular frequency;µ and ε are the magnetic permeability and the dielectric permittivity of the media,respectively; n is the unit normal (outward) vector ; and j =

√−1 is the imaginary

unit.The corresponding variational formulation is given by:

Find HΩ ∈ HD(curl; Ω) such that∫Ω

1

ε(∇×HΩ) · (∇× F Ω)dV −

∫Ω

ω2µHΩ · F ΩdV

+jω2µ

β10

∫Γ1∪Γ2

(n×HΩ) · (n× F Ω)dS =

2jω2µ

β10

∫Γ1

(n×Hin) · (n× F Ω)dS for all F Ω ∈ HD(curl; Ω) .

(III.97)

In the above, H(curl; Ω) is the Hilbert space of admissible solutions,

H(curl; Ω) := HΩ ∈ L2(Ω) : curlHΩ ∈ L2(Ω) .


In the remainder of this paper, we shall work with 2D vectors, and we shall usethe symbol H when referring to HΩ (components of the magnetic field parallel tothe H-plane). Similarly, we will use symbol F when referring to FΩ.

It is worth noting that the final objective of the computations is not to accuratelydetermine H but the scattering parameters Sij (1 ≤ i, j ≤ 2) (see part I for detailson S-parameters). Thus, the quantity of interest (scattering parameters) for a two-ports waveguide consists of four complex numbers: Sij (1 ≤ i, j ≤ 2). Because of thereciprocity principle, we know that

S12 = S21 . (III.98)

We also know thatS11 = S22 (III.99)

for symmetric structures.For a loss-less media, the S-matrix of the structure is unitary. Thus, we have:

S211 = S2

21 −S21

S21

, (III.100)

where S21 is the complex conjugate of S21. Therefore, one of the S-parameters isenough to determine the full scattering matrix. Also, for loss-less media, |S11| = |S22|in a two ports structure. The phases are equal only if there is symmetry (which means(III.98) holds).

This problem may be solved by using semi-analytical techniques (for example,Mode Matching techniques [2], [16, Chapter 9]). Nevertheless, it would be desirableto solve it by using purely numerical techniques, since a numerical method allowsfor simulation of more complex geometries and/or the effects of dielectrics, metal-lic screws, round corners, etc., possibly needed for the construction of an actualwaveguide.

III.2.2 Goal-Oriented Adaptivity

Given a problem, a quantity of interest (in this case, the S-parameters), and a dis-cretization tolerance error, the objective of the goal-oriented adaptivity is to con-struct, without any user interaction, an hp-grid containing a minimum number ofunknowns, and such that the relative error in the quantity of interest is smaller thanthe (given) error tolerance.


Variational problem (III.97) can be stated here in terms of sesquilinear form b,and antilinear form f : Find H ∈ V

b(H,F) = f(F) ∀F ∈ V ,(III.101)

where

• V = H(curl; Ω) is a Hilbert space.

• f(F) = 2jω2µ

β10

∫Γ1

(n ×Hin) · (n × F )dS ∈ V′ is an antilinear and continuous

functional on V.

• b is a sesquilinear form. We have:

b(H,F) = a(H,F) + c(H,F) ,

a(H,F) =

∫Ω

1

ε(∇×H) · (∇×F) dV

+jω2µ

β10

∫Γ1∪Γ2

(n×H) · (n× F )dS ,

c(H,F) = −∫

Ω

ω2µH · F dV .

(III.102)

We define an “energy” inner product on V as:

(H,F) := a(H,F) + c(H,F) ,

a(H,F) =

∫Ω

1

ε(∇×H) · (∇×F) dV

+|jω2µ

β10

|∫

Γ1∪Γ2

(n×H) · (n× F )dS ,

c(H,F) =

∫Ω

ω2µH · F dV ,

(III.103)with the corresponding (energy) norm denoted by ‖H‖. Notice the inclusionof the material properties in the definition of the norm.


Using high-order Nedelec (edge) elements [28], we consider an hp-FE subspaceVhp ⊂ V. Then, we discretize (III.101) as follows: Find Hhp ∈ Vhp

b(Hhp,Fhp) = f(Fhp) ∀Fhp ∈ Vhp .(III.104)

We assume that our quantity of interest can be expressed as a continuous22 andlinear23 functional L. By recalling the linearity of L, we have:

Error of interest = L(H)− L(Hhp) = L(H−Hhp) = L(e) , (III.105)

where e = H−Hhp denotes the error function. By defining the residual rhp belongingto the dual space of V (denoted by V′) as rhp(F) = f(F) − b(Hhp,F) = b(H −Hhp,F) = b(e,F), we look for the solution of the dual problem: Find W ∈ V

b(F,W) = L(F) ∀F ∈ V .(III.106)

Problem (III.106) has a unique solution in V. The solution W, is usually referredto as the influence function.

By discretizing (III.106) via, for example, Vhp ⊂ V, we obtain: Find Whp ∈ Vhp

b(Fhp,Whp) = L(Fhp) ∀Fhp ∈ Vhp .(III.107)

Definition of the dual problem plus the Galerkin orthogonality for the originalproblem imply the final representation formula for the error in the quantity of inter-est, namely,

L(e) = b(e,W) = b(e,W − Fhp︸︷︷︸ε

) = b(e, ε) .

At this point, Fhp ∈ Vhp is arbitrary, and b(e, ε) = b(e, ε) denotes the bilinearform corresponding to the original sesquilinear form.

22If the quantity of interest is not continuous, we may consider a continuous approximation L tothe original quantity of interest.

23If the quantity of interest is represented by a non-linear functional, we linearize it around aspecific solution, and replace the original functional L with its linearized version.


Notice that, in practice, the dual problem is solved not for W but for its complexconjugate W utilizing the bilinear form and not the sesquilinear form. The linearsystem of equations is factorized only once, and the extra cost of solving (III.107)reduces to only one backward and one forward substitution (if a direct solver is used).

Once the error in the quantity of interest has been determined in terms of bilinearform b, we wish to obtain a sharp upper bound for |L(e)| that depends upon themesh parameters (element size h and order of approximation p) only locally. Then,a self-adaptive algorithm intended to minimize this bound will be defined.

First, using a procedure similar to the one described in [56], we approximateH and W with fine grid functions Hh

2, p+1, Wh

2, p+1, which have been obtained by

solving the corresponding linear system of equations associated with the FE subspaceVh

2, p+1. In the remainder of this article, H and W will denote the fine grid solutions

of the direct and dual problems (H = Hh2, p+1, and W = Wh

2, p+1, respectively), and

we will restrict ourselves to discrete FE spaces only.Next, we bound the error in the quantity of interest by a sum of element con-

tributions. Let bK denote a contribution from element K to sesquilinear form b. Itthen follows that

|L(e)| = |b(e, ε)| ≤∑K

|bK(e, ε)| , (III.108)

where summation over K indicates summation over elements.

III.2.3 Projection based interpolation operator

Once we have a representation formula for the error in the quantity of interest interms of the sum of element contributions given by (III.108), we wish to expressthis upper bound in terms of local quantities, i.e. in terms of quantities that do notvary globally when we modify the grid locally. For this purpose, we introduce theidea of the projection-based interpolation operator. More precisely, we shall utilizethe projection-based interpolation operator for hp-edge elements Πcurl

hp : V −→ Vhp

defined in part II. This operator is local, it maintains conformity and it is optimalin the sense that the error behaves asymptotically, both in h and p, in the same wayas the actual interpolation error (see [47] for details).

We shall also consider the Galerkin projection operator Pcurlhp : V −→ Vhp, and

we will denote Hhp = Pcurlhp H. Then, equation (III.108) becomes

|L(e)| ≤∑

K |bK(e, ε)| =

=∑K

|bK(H−Πcurlhp H, ε) + bK(Πcurl

hp H−Pcurlhp H, ε)| .

(III.109)


Given an element K, we conjecture that |bK(Πcurlhp H − PhpH, ε)| will be negligible

compared to |bK(H−Πcurlhp H, ε)|. Under this assumption, we conclude that:

|L(e)| ∑K

|bK(H−Πcurlhp H, ε)| . (III.110)

In particular, for ε = W −Πcurlhp W, we have:

|L(e)| ∑K

|bK(H−Πcurlhp H,W −Πcurl

hp W)| . (III.111)

By applying Cauchy-Schwartz inequality, we obtain the next upper bound for|L(e)|:

|L(e)| ∑K

‖e‖K‖ε‖K , (III.112)

where e = H − Πcurlhp H, ε = W − Πcurl

hp W, and ‖ · ‖K denotes energy-norm ‖ · ‖restricted to element K.

III.2.4 Goal-Oriented hp-Mesh Optimization Algorithm

We describe a self-adaptive goal-oriented hp algorithm that utilizes the main ideasof the fully automatic (energy-norm based) hp-adaptive algorithm described in [56,42]and part II. The goal-oriented mesh optimization algorithm iterates along thefollowing steps.

• Step 0: Compute the upper-bound estimate of the coarse grid ap-proximation error in the quantity of interest. This estimate is computedusing eq. (III.112). If the difference is smaller than a user-prescribed error tol-erance, then we deliver the fine mesh solution as our final answer, and we stopthe execution of our algorithm.

• Step 1: For each edge in the coarse grid, compute the error decreaserate for the p refinement, and all possible h-refinements. Let p1, p2 bethe order of the edge sons in the case of h-refinement. Then, we compute:

Error decrease (hp) =∑K

[‖H−Πcurl

hp H‖K · ‖W −Πcurlhp W‖K

(p1 + p2 − p)

−‖H−Πcurl

hpH‖K · ‖W −Πcurl

hpW‖K

(p1 + p2 − p)

],

(III.113)

III.3 NUMERICAL RESULTS 96

where hp = (h, p) is such that h ∈ h, h/2. If h = h, then p = p + 1. Ifh = h/2, then p = (p1, p2), where p1 + p2 − p > 0, maxp1, p2 ≤ p + 1.

Steps 0 and 1 described above are analogous to those presented in part II forenergy-norm hp-adaptivity. The remaining steps (2 through 5) are exactly the sameas those described in part II.

REMARK: If functional L describing the quantity of interest is equal to theload f of the original (direct) problem, then H = W, and the goal-oriented algorithmdescribed above is identical with the energy-norm algorithm presented in part II.

Similarities between the energy norm based hp-adaptivity and the goal-orientedhp-adaptivity are well reflected in the corresponding implementation. The latter oneis simply an extension of the first one.

III.2.5 Implementation details

In what follows, we discuss the main implementation details needed to extend thefully automatic (energy-norm based) hp-adaptive algorithm [56, 42] to a fully auto-matic goal-oriented hp-adaptive algorithm.

1. First, solution W of the dual problem on the fine grid is necessary. This goalcan be attained either by using a direct (frontal) solver or an iterative (two-grid) solver (see [1]).

2. Subsequently, we should treat both solutions as satisfying two different partialdifferential equations (PDE’s). We select functions H and W as the solutionsof the system of two PDE’s.

3. We proceed to redefine the evaluation of the error. The energy-norm errorevaluation of a two dimensional function is replaced by the product ‖ H −Πcurl

hp H ‖ · ‖ W −Πcurlhp W ‖.

4. After these simple modifications, the energy-norm based self-adaptive algo-rithm may now be utilized as a self-adaptive goal-oriented hp algorithm.

III.3 Numerical Results

In this section, we use three different methods.

• The fully automatic energy-norm based hp-adaptive strategy described in partII,


• a mode matching technique [2], and

• the fully automatic goal-oriented hp-adaptive strategy described in Section III.3.1.

The methods are applied to a fifth order filter consisting on a H-plane structurewith six symmetric inductive irises (as the one analyzed in the first paper ). This isa challenging problem with 24 singularities.

Then, results are compared against each other, and the main advantages anddisadvantages of each methodology are discussed.

III.3.1 A Rectangular Waveguide Structure with Six Induc-tive Irises

We consider a six inductive irises filter24 of dimensions ≈ 20×2×1 cm., operating inthe range of frequencies ≈ 8.8− 9.6 Ghz. More precisely, our computational domainis given by Ω = Ω1 − (Ω2 ∪ Ω3 ∪ Ω4 ∪ Ω5 ∪ Ω6 ∪ Ω7), where:

• Ω1 = (x, y, z) : 0 cm. ≤ x ≤ 20.2913 cm.,−1.143 cm. ≤ y ≤ 1.143 cm., 0 cm. ≤z ≤ 1.016 cm.,

• Ω2 = (x, y, z) : 5 cm. ≤ x ≤ 5.2 cm.,−1.143 cm. ≤ y ≤ −0.7191 cm., 0 cm. ≤z ≤ 1.016 cm.∪(x, y, z) : 5 cm. ≤ x ≤ 5.2 cm., 0.7191 cm. ≤ y ≤ 1.143 cm., 0 cm. ≤z ≤ 1.016 cm.,

• Ω3 = (x, y, z) : 6.8641 cm. ≤ x ≤ 7.0641 cm.,−1.143 cm. ≤ y ≤ −0.54975 cm., 0 cm. ≤z ≤ 1.016 cm. ∪ (x, y, z) : 6.8641 cm. ≤ x ≤ 7.0641 cm., 0.54975 cm. ≤ y ≤1.143 cm., 0 cm. ≤ z ≤ 1.016 cm.,



• Ω6 = (x, y, z) : 13.2272 cm. ≤ x ≤ 13.4272 cm.,−1.143 cm. ≤ y ≤ −0.54975 cm., 0 cm. ≤z ≤ 1.016 cm.∪(x, y, z) : 13.2272 cm. ≤ x ≤ 13.4272 cm., 0.54975 cm. ≤ y ≤1.143 cm., 0 cm. ≤ z ≤ 1.016 cm., and,

24We thank Mr. Sergio Llorente for designing the waveguide filter structure.



The geometry is shown in Fig. III.1. The structure consists of five cavities actingas resonators coupled by themselves and with the waveguide sections by means ofthe irises. Thus, a selective frequency response can be adjusted by a careful choiceof the size of the cavities and the irises. Specifically, the structure has been designedto work as a bandpass filter in the 8.8–9.6 GHz frequency range (see Fig. III.3.2 inthe section on numerical results).

xyz

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hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic 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code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic 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code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code2Dhp90: A Fully automatic hp-adaptive Finite Element code

Figure III.1: 2D cross-section of the geometry of the waveguide problem with six inductiveirises. The initial grid is composed of 27 elements, as indicated by the black lines

III.3.2 Results Obtained by Using the Energy-Norm hp-AdaptiveStrategy

First, we solve our waveguide problem using the fully automatic hp-adaptive strategy.In order to construct an adequate initial grid for our adaptive algorithm, we noticethe following limitation:

• We cannot guarantee the optimality of the fully automatic hp-adaptivestrategy if the dispersion error is large, which may occur in the pre-asymptotic regime. Since solution of the problem on the fine grid is usedto guide optimal hp-refinements, we need to control the dispersion error onthe fine grid. Thus, h needs to be sufficiently small or p sufficiently large.Otherwise, convergence results in the pre-asymptotic regime may not look asexpected.

In Fig. III.3.2, we compare the convergence history obtained by using the fullyautomatic hp-adaptive strategy starting with different initial grids. For thirdorder elements, the dispersion error is under control (see estimates of [57, 58,59]), and the fully automatic hp-adaptive strategy converges exponentially from


125 1000 3375 8000 15625 27000 42875 64000 9112510

−1

100

101

102

103

Number of unknowns (algebraic scale)

Est

imat

e of

the

rela

tive

erro

r (%

)

p=1, 1620 Initial grid elementsp=2, 1620 Initial grid elementsp=3, 1620 Initial grid elementsp=1, 27 Initial grid elementsp=2, 27 Initial grid elementsp=3, 27 Initial grid elements

Figure III.2: Convergence history using the fully automatic hp-adaptive strategy for dif-ferent initial grids. Different colors correspond to different initial order of approximation.27 is the minimum number of elements needed to reproduce the geometry, while 1620 isthe minimum number of elements needed to reproduce the geometry and to guaranteeconvergence of the iterative two grid solver described in [1].

the beginning, as indicated by a straight line in the algebraic vs logarithmicscales. We also observe that, in the asymptotic regime, all curves presentsimilar rates of convergence. These results indicate that the choice of initialgrid is not very important and, even in the case of very coarse initial grids, thealgorithm will eventually notice the asymptotic behavior of the solution, andthe overall convergence will be exponential.

We solved the six irises waveguide problem delivering a 0.3% relative error in theenergy-norm. Fig. III.3.2 displays the magnitude of the S11 scattering parameter(on the decibel scale) with respect to the frequency. This quantity is usually referredto as the return loss of the waveguide structure, and it is given by equation (III.114),


8.8 8.9 9 9.1 9.2 9.3 9.4 9.5 9.6−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency (Ghz)

|S11

| (dB

)

Figure III.3: Return loss of the waveguide structure

see part I.

S11 =

∫Γ1

H(ξ) · sin πξ

adS

Hina

2

− 1 , (III.114)

where a is the size of the excitation port. For the frequency interval 8.8−9.6 Ghz, thereturn loss is below −20dB, which indicates that almost all energy passes through thestructure, and thus, the waveguide acts as a bandpass filter. The other scatteringparameter of interest is S21 (S12 and S22 are obtained using (III.98) and (III.99),respectively) which is given in equation (III.115), see also part I.

S21 =

∫Γ2

H(ξ) · sin πξ

adS

Hina

2

(III.115)

Figures III.4, III.5, III.6, and III.7 display solution at different frequencies. Forfrequencies 8.72 Ghz, and 9.71 Ghz, the return loss of the waveguide structure is


Figure III.4: | Hx | (upper figure), | Hy | (center figure), and√| Hx |2 + | Hy |2 (lower

figure) at 8.72 Ghz for the six irises waveguide problem.

large, and for frequencies 8.82 Ghz, and 9.58 Ghz the return loss is below −20dB.

III.3.3 Results Obtained by Using a Mode Matching Tech-nique

In this subsection, we solve our problem of interest with a Mode Matching technique.More precisely, we compute the S-parameters by using the software available in [2].In Fig. III.8, we display the S-parameters as a function of frequency. Notice thesimilarities of these results with those presented in Fig. III.3.2.

III.3.4 Results Obtained by Using the Goal-Oriented hp-Adaptive Strategy

The final hp-grid delivered by the fully automatic, energy-norm based adaptive al-gorithm is expected to produce a “reasonably good” approximation of the exactsolution almost everywhere in the computational domain. Thus, we may decide whatquantity to compute (for example, the S11 parameter) after solving the problem.

However, when using the goal-oriented hp-adaptivity, we need to decide first whatquantity is of interest for us (for example, S11), and then construct a grid intendedto obtain the best possible approximation of that quantity of interest with respect tothe problem size. Notice that quality of the solution everywhere else in the domain




Figure III.8: Return loss of the waveguide structure. This graph has been computed usinga Mode Matching technique [2].


may be poor.We note that it is possible to define a single quantity of interest relating different

S-parameters. Thus, an accurate solution in the quantity of interest will imply accu-racy of several S-parameters simultaneously (multiple goals). However, it is unclearfor these authors how to optimally select such a quantity of interest. For example,if we select the sum of different S-parameters, then the goal-oriented algorithm willfirst approximate the S-parameters with higher value. Perhaps a weighted sum ofthe S-parameters is an adequate selection for the quantity of interest, where theweights are inversely proportional to the values of the S-parameters. For details onmulti-goal oriented adaptivity, see [60].

Thus, we revisit the question on which of the S-parameters should be computed.For instance, will it be better to directly compute S11, or to compute S21 and thenapply identity (III.100) to obtain S11? These questions are especially importantwhen considering goal-oriented adaptive algorithms, since the configuration of thefinal grid will depend upon the quantity of interest (S-parameter) that we selectbefore refinements.

In order to select between the four different S-parameters relevant to our waveg-uide structure, we first notice that at the discrete level (when considering finite ele-ment approximation), identity (III.98) is also valid due to the fact that reciprocityis also satisfied at the discrete level. Thus, the use of the goal-oriented adaptivitywith S12 or S21 as our quantity of interest will provide identical grids and results.However, identity (III.99) will hold at the discrete level only if the grid is symmetric.Since in this paper we are considering symmetrical initial grids, the goal-orientedadaptive algorithm with S11 or S22 as our quantity of interest will provide identicalgrids and results. Finally, notice that identity (III.100) will not hold in general atthe discrete level, even if the grid is symmetric. In summary, we shall consider twodifferent quantities of interest,

1. L1(H) := S11(H) + 1 25, and,

2. L2(H) := S21(H).

Then, we will numerically study the self-adaptive goal-oriented hp-adaptive strat-egy by executing it twice: first, using L1 as our quantity of interest, and then consid-ering L2 as our quantity of interest. Finally, in order to compare the results, we willuse for the second case (L2) equation (III.100) in order to post-process S11(H) + 1from S21(H).

25Notice that S11(H) is not a linear functional. However, S11(H) + 1 is a linear and continuousfunctional, and therefore, we may use it as our quantity of interest for the goal-oriented optimizationalgorithm.


Convergence results are displayed in Fig. III.9 at different frequencies: 8.72 Ghz(top-left panel), 8.82 Ghz (top-right panel), 9.58 Ghz (bottom-left panel), and 9.71Ghz (bottom-right panel). These frequencies correspond to solutions shown in figuresIII.4, III.5, III.6, and III.7, respectively. The red curve displays the relative error inpercentage of S11 + 1 with respect to the number of unknowns, when executing thegoal-oriented hp-adaptive algorithm with L1 as our quantity of interest. If we considerL2 as our quantity of interest, the corresponding relative error of S11 + 1 -computedusing (III.100)- is displayed by the black curve. Finally, the blue curve correspondsto upper bound (III.112) used for minimization, when considering |L1(H)| as ourquantity of interest.

From results of Fig. III.9, we conclude the following.

• The self-adaptive goal-oriented hp-FEM delivers exponential converges ratesin terms of upper bound (III.112) of quantity of interest |L1(H)| against theproblem size (number of d.o.f.), as indicated by the straight line (blue curve)in the algebraic (number of d.o.f. to the power of 1/3) vs. logarithmic scale.

• Since the blue curve converges exponentially, and it is an upper bound estimateof the red curve, the latter curve should also display an overall exponentialconvergence behavior (or at least it is bounded by a curve that exponentiallyconverges to zero), regardless of the fact that the error in the quantity of interestmay temporary increase when executing the refinements.

• The final relative error in the quantity of interest remains below 0.1% in allcases.

• To utilize quantity of interest L2 as opposed to L1 (or viceversa) for executingthe goal-oriented adaptive algorithm is not essential for this problem. Numer-ical results are similar in both cases.

III.3.5 A Comparison Between the Energy-Norm and Goal-Oriented Self-Adaptive hp-FE Strategies

Quantity of interest L1 is equal (up to a multiplicative constant) to the functionalf representing the right hand side of the original problem. Thus, solution of thedual problem W is also equal (up to a multiplicative constant) to the solution of theoriginal problem H. For this particular case, the goal-oriented adaptivity coincidesexactly with the energy driven adaptivity, and the corresponding numerical resultsare identical. In other words, for this particular waveguide problem, energy-normadaptivity is optimal for approximating L1.


1000 3375 8000 15625 27000 42875 6400010−2

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Figure III.9: Convergence history for the waveguide problem with six inductive irises at dif-ferent frequencies: 8.72 Ghz (top-left panel), 8.82 Ghz (top-right panel), 9.58 Ghz (bottom-left panel), and 9.71 Ghz (bottom-right panel). The red and black curves display the rela-tive error in percentage of S11 +1 with respect to the number of unknowns, when executingthe goal-oriented hp-adaptive algorithm with L1 and L2 as our quantity of interest, respec-tively. The blue curve corresponds to upper bound (III.112) used for minimization, whenconsidering |L1(H)| as our quantity of interest.


p=1

p=2

p=3

p=4

p=5

p=6

p=7

p=8

-6.689440 209.6024 -66.89439

66.89439

Figure III.10: 2D cross-section of the geometry of the waveguide problem with six inductiveirises. The red color indicates that the three central cavities are filled with a resistivematerial.

On the other hand, if we are interested in computing S21, then results obtainedfrom the goal-oriented algorithm are different from those obtained with energy-normadaptivity. Nevertheless, Fig. III.9 illustrates that the results coming from both al-gorithms, although different, are quite similar to each other, since S21 is also stronglyrelated to the energy of the solution.

Differences between energy-norm adaptivity and goal-oriented adaptivity shallbecome larger as we consider resistive materials and/or more complex waveguidestructures with possibly several ports, in which the quantity of interest is not stronglyrelated to the conservation of energy. To illustrate this effect, we consider our originalproblem with the three central cavities filled by a lossy material (see Fig. III.10)with ε = ε0εr, where ε0 = 8.854 ∗ 10−12 F/m is the permittivity of the vacuum,and εr = (1 − 0.07805j/(ωε0)). Results at 8.82 Ghz, i.e. εr = (1 − j/(2π)) aredisplayed in Fig. III.11. When the goal of computations is to approximate S11, thegrid obtained with the goal-oriented adaptivity for S11 + 1 provides results that areone order of magnitude more accurate than those obtained with the goal-orientedadaptivity and S21 as our quantity of interest. If the goal of computations is toapproximate S21, the grid obtained with the goal-oriented adaptivity for S21 providesresults that are up to five hundred times more accurate than those obtained with thegoal-oriented adaptivity and S11 +1 as our quantity of interest. Since grids obtainedfrom using the goal-oriented adaptivity for S11 + 1 are equal to those obtained fromusing energy-norm adaptivity, we conclude that the goal-oriented adaptivity for S21

provides approximations to S21 that are far more accurate than those obtained withthe energy-norm adaptivity. Eventually, for a resistive enough material, the use ofgoal-oriented adaptivity will become essential (see [18, 19] for details) for accuratelyapproximating S21.

Two hp-grids for our original problem with the three central cavities filled by alossy material are shown in Fig. III.12. The two hp-grids displayed in Fig. III.12 areessentially different:


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Figure III.11: Convergence history for the waveguide problem with six inductive irises at8.82 Ghz, with the three central cavities filled with a resistive material. The left paneldescribes the convergence history for S11, and the right panel for S21. The blue curveshave been obtained by using goal-oriented adaptivity with S21 as our quantity of interest,and the red curves by using goal-oriented adaptivity with S11+1 as our quantity of interest.

1. The top panel displays a grid obtained by using L1 as our quantity of interest.Therefore, we observe severe refinements towards the excitation (left) port.

2. The bottom panel displays a grid obtained by using L2 as our quantity ofinterest. In this case, we obtain a symmetric grid with several refinementstowards both the excitation (left) and the output (right) ports.

Notice that the final mesh is symmetric with respect to the center of thewaveguide structure because the initial grid is symmetric and the upper bound∑

K |bK(e, ε)|, used for the minimization, is also symmetric since solutions ofthe original and dual problems are symmetric with respect to each other (upto a constant).

To summarize, the energy-norm adaptivity can be seen as a particular case ofthe goal-oriented adaptivity. For the waveguide problem that we are consideringin this paper, results for (the particular case of) energy-norm adaptivity are closeto optimal. Indeed, they are exactly optimal if we consider L1 as our quantity ofinterest. Thus, there is no need to use the goal-oriented adaptivity for this problem.Nevertheless, it is important to realize that for some other problems (for instance,waveguide problems with lossy media, or waveguide problems with several ports),the use of goal-oriented adaptivity may become essential. In any case, results coming

III.4 CONCLUSIONS 109

p=1

p=2

p=3

p=4

p=5

p=6

p=7

p=8

-6.689440 209.6024 -66.89439

66.89439 2Dhp90: A Fully automatic hp-adaptive Finite Element code

p=1

p=2

p=3

p=4

p=5

p=6

p=7

p=8

-6.689440 209.6024 -66.89439

66.89439 2Dhp90: A Fully automatic hp-adaptive Finite Element code

Figure III.12: hp-grids obtained by using the fully automatic goal-oriented hp-adaptivefinite element method with S11 + 1 (top panel) and S21 (bottom panel) as our quantitiesof interest, respectively. Different colors indicate different polynomials orders of approxi-mation, ranging from 1 (dark blue) up to 8 (pink). The hp-grids contain 12110 (top) and13054 (bottom) unknowns, respectively.

from the goal-oriented adaptivity should never be worse than those obtained withthe energy-norm adaptivity.

III.4 Conclusions

In this paper, a fully automatic goal-oriented hp-FEM has been presented for theanalysis of rectangular waveguide discontinuities. A challenging problem containingsix inductive irises (twenty-four singularities) in the H-plane has been solved.

Numerical results indicate that,

• the method converges exponentially (regardless of the initial grid used),

• accuracy of the method is comparable to that obtained with a Mode-Matching(semi-analytical) technique, and at the same time, the finite element methodallows for modeling of more complex waveguide structures,

• the goal-oriented adaptivity is a generalization of the energy-norm adaptivitywhich may (or may not) be essential in order to accurately solve a problem,and,

• from the physical point of view, the waveguide acts as a bandpass filter.

Summarizing the two papers we conclude the following. A relevant microwave en-gineering problem involving the analysis of H-plane and E-plane rectangular waveg-uide discontinuities (and the computation of their S-parameters), has been solved

III.4 CONCLUSIONS 110

by using a fully automatic hp-adaptive FEM. The hp adaptivity has been shownto deliver exponential convergence rates for the error, even in the presence of sin-gularities, for a wide variety of relevant structures including microwave engineeringdevices of medium complexity. While keeping the advantage of being a purely numer-ical method, the hp-adaptivity has shown to be more accurate than semi-analyticaltechniques. The suitability of a goal-oriented approach in terms of the S-parameters,in comparison with a “conventional” approach in terms of energy-norm, has beenclarified.

REFERENCES 111

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