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A Two-Dimensional Self-Adaptive hp Finite
Element Method for the Characterization of
Waveguide Discontinuities. Part III:
Goal-Oriented hp-Adaptivity.
David Pardo
Institute for Computational Engineering and Sciences (ICES).The University of Texas at Austin,
201 East 24th Street, ACES 4.102, Austin, TX 78712, USA
Luis E. Garcıa-Castillo 1,2
Departamento de Teorıa de la Senal y Comunicaciones. Universidad Carlos III deMadrid. Escuela Politecnica Superior (Edificio Torres Quevedo). Avda. de la
Universidad, 30. 28911 Leganes (Madrid) Spain. Fax:+34-91-6248749
Leszek F. Demkowicz
ICES, The University of Texas at Austin, USA
Carlos Torres-Verdın
Department of Petroleum and Geosystems Engineering,The University of Texas at Austin, USA
Abstract
This is the last of a series of three papers analyzing different types of rectangu-lar waveguide discontinuities by using a fully automatic hp-adaptive finite elementmethod.
In this paper, we describe a fully automatic goal-oriented hp-adaptive Finite El-ement (FE) strategy, which is applied to an H-plane structure consisting on aninductive waveguide with six irises. The methodology produces exponential conver-gence rates in terms of an upper bound of a user-prescribed quantity of interest (inour case, the S-parameters) against the problem size (number of degrees of freedom).
Numerical results illustrate the main differences between the energy-norm basedand the goal-oriented based hp-adaptivity, when applied to rectangular waveguidestructures. They also demonstrate the suitability of the goal-oriented hp-method forsolving problems involving rectangular waveguide structures.
Preprint submitted to Elsevier Science 1st March 2006
The goal-oriented self-adaptive hp-FEM provides more accurate results than thoseobtained with the Mode Matching method (a semi-analytical technique). At thesame time, the hp-FEM provides the flexibility of modeling more complex waveguidestructures possibly incorporating the effects of dielectrics, metallic screws, roundcorners, etc., which cannot be easily considered when using semi-analytical tech-niques.
Key words: Finite Element Method, hp-Adaptivity, Goal-Oriented Adaptivity,Rectangular Waveguides, Waveguide Discontinuities, S-parameters.
1 Introduction
As it was mentioned in the first two parts of this series [1,2], the accurate anal-ysis and characterization of “waveguide discontinuities” is an important issuein microwave engineering (see e.g., [3,4]). Specifically, H-plane and E-planerectangular waveguide discontinuities, which are the target of the presentedwork, play an important role in the communication systems working in theupper microwave and millimeter wave frequency bands.
A method based on the hp-Finite Elements, presented in [2], seems to be aperfect tool 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
in regions 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-methoddescribed in [2] is measured in terms of the energy-norm of the error functionagainst the problem size. Thus, the electromagnetic field is accurately known(with a user pre-specified level of accuracy) inside of the structure. This maybe useful to the microwave engineer in order to predict the location and sizeof tuning elements (e.g., screws, dielectric posts, etc.) and/or for the design ofmodifications to the original design. However, the microwave engineer is first
Email addresses: [email protected] (David Pardo), [email protected],[email protected] (Luis E. Garcıa-Castillo), [email protected] (LeszekF. Demkowicz), [email protected] (Carlos Torres-Verdın).1 This author want to acknowledge the support of Ministerio de Educacion y Cien-cia of Spain under project TEC2004-06252/TCM.2 On leave from Departamento de Teorıa de la Senal y Comunicaciones. Universidadde Alcala, Madrid, Spain
2
(and mainly) interested in accurately computing the S-parameters (see [1] fordetails) of the waveguide structure. Therefore, we should design a numericalmethod that is consistent with our goal, i.e., a high accuracy computation ofthe S-parameters, regardless of the quality of the solution in the rest of thedomain 3 . This type of methods are called weighted-based residual methodsor goal-oriented methods (see [5,6,7,8,9,10] 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 [2].
In the presented work, we propose to use the fully automatic goal-orientedhp-FEM presented in [11,12,13] to compute the S-parameters of rectangularwaveguide structures. This method incorporates all desirable features men-tioned above (and described in [2]) with the additional advantage of obtainingexponential convergence in terms of a sharp upper bound of the quantity ofinterest (in this case, the S-parameters) against the problem size – numberof degrees of freedom. In this paper, we consider the application of this goal-oriented methodology to a rectangular (H-plane) waveguide structure with sixinductive irises. It is worth noting that this structure incorporates as manyas twenty-four singularities of different intensity, due to presence of re-entrantcorners in the geometry.
The organization of the paper is as follows. In Section 2 we introduce firstthe problem of interest: an H-plane structure with six inductive irises. InSection 3, we describe our numerical methodology based on the self-adaptivegoal-oriented hp-FEM. A brief discussion on implementation details is alsoprovided at this point. Section 4 is devoted towards numerical results. Inthe same section, we shall also analyze main advantages and disadvantagesof using the goal-oriented hp-adaptivity (as opposed to the energy-norm hp-adaptivity) for our problem of interest. Finally, we close with some conclusionsin Section 5.
2 A Rectangular Waveguide Structure with Six Inductive Irises
2.1 Geometry
We consider a six inductive irises filter 4 of dimensions ≈ 20 × 2 × 1 cm.,operating in the range of frequencies ≈ 8.8 − 9.6 Ghz. More precisely, our
3 For instance, it may be unnecessary to resolve all singularities in order to accu-rately compute the S-parameters.4 We thank Mr. Sergio Llorente for designing the waveguide filter structure.
3
computational domain is 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.,
• Ω4 = (x, y, z) : 8.9702 cm. ≤ x ≤ 9.1702 cm.,−1.143 cm. ≤ y ≤ −0.5078 cm., 0 cm. ≤z ≤ 1.016 cm.∪(x, y, z) : 8.9702 cm. ≤ x ≤ 9.1702 cm., 0.5078 cm. ≤ y ≤1.143 cm., 0 cm. ≤ z ≤ 1.016 cm.,
• Ω5 = (x, y, z) : 11.1211 cm. ≤ x ≤ 11.3211 cm.,−1.143 cm. ≤ y ≤−0.5078 cm., 0 cm. ≤ z ≤ 1.016 cm. ∪ (x, y, z) : 11.1211 cm. ≤ x ≤11.3211 cm., 0.5078 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,
• Ω7 = (x, y, z) : 15.0913 cm. ≤ x ≤ 15.2913 cm.,−1.143 cm. ≤ y ≤−0.7191 cm., 0 cm. ≤ z ≤ 1.016 cm. ∪ (x, y, z) : 15.0913 cm. ≤ x ≤15.2913 cm., 0.7191 cm. ≤ y ≤ 1.143 cm., 0 cm. ≤ z ≤ 1.016 cm.,
The geometry is shown in Fig. 1. The structure consists of five cavities actingas resonators coupled by themselves and with the waveguide sections by meansof the irises. Thus, a selective frequency response can be adjusted by a carefulchoice of the size of the cavities and the irises. Specifically, the structure hasbeen designed to work as a bandpass filter in the 8.8–9.6 GHz frequency range(see Fig. 3 in the section on numerical results).
Figure 1. 2D cross-section of the geometry of the waveguide problem with six in-ductive irises.
2.2 Variational Formulation
We excite the TE10-mode at the left hand side port of the waveguide structure(see Fig. 1), and we consider an H-plane discontinuity (see [1]). Thus, our
4
original 3D problem can be reduced to a 2D boundary value problem, whichwe can formulate in terms of the components of the magnetic field parallel tothe H-plane (denoted by HΩ) as follows.
∇ ×(
1
ǫ∇ × HΩ
)
− ω2µHΩ = 0 in Ω
n× 1
ǫ∇ × HΩ =
jω2µ
β10
n × n × (2Hin −HΩ) on Γ1
n× 1
ǫ∇ × HΩ = −jω2µ
β10
n× n× HΩ on Γ2
n× 1
ǫ∇ × HΩ = 0 on Γ3 ,
(1)
where Γ1, Γ2, and Γ3 are the parts of the boundary corresponding to the excita-tion port (left port), non-excitation port (right port), and the perfect electricconductor, respectively; β10 refers to the propagation constant of the TE10
mode; Hin is the incident magnetic field at the excitation port; ω is the an-gular frequency; µ and ǫ are the magnetic permeability and the dielectricpermittivity of the media, respectively; n is the unit normal (outward) vector; and j =
√−1 is the imaginary unit. See [1] for details and derivation of this
formulation.
The corresponding variational formulation is given by:
Find HΩ ∈ HD(curl; Ω) such that
∫
Ω
1
ǫ(∇ ×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) .
(2)
In the above, HD(curl; Ω) is the Hilbert space of admissible solutions,
HD(curl; Ω) := HΩ ∈ L2(Ω) : curlHΩ ∈ L2(Ω) .
For the discussion on the stabilized variational formulation see [1].
2.3 Goal of Computations
The objective of the computations is to accurately determine the scatteringparameters Sij (1 ≤ i, j ≤ 2) for the range of frequencies (8.7 Ghz - 9.7 Ghz).
5
Thus, the quantity of interest (scattering parameters) for a two-ports waveg-uide consists of four numbers: Sij (1 ≤ i, j ≤ 2). Because of the reciprocityprinciple, we know that
S12 = S21 . (3)
We also know that
S11 = S22 (4)
because of the symmetry of the waveguide structure considered in this paper.Finally, we are assuming a loss-less media, so the S-matrix is unitary. Thus,we have:
S211 = S2
21 −S21
S21
, (5)
where S21 is the complex conjugate of S21. Therefore, one of the S-parametersis enough to determine the full scattering matrix (see [1] for details on S-parameters).
This problem may be solved by using semi-analytical techniques (for example,Mode Matching techniques [14], [15, Chapter 9]). Nevertheless, it would bedesirable to solve it by using purely numerical techniques, since a numericalmethod allows for simulation of more complex geometries and/or the effectsof dielectrics, metallic screws, round corners, etc., possibly needed for theconstruction of an actual waveguide.
3 A Fully Automatic Goal-Oriented hp-Adaptive Finite Element
Method
In this Section, we describe the self-adaptive goal-oriented hp-algorithm. Thealgorithm is an extension of the energy-norm based hp-adaptive algorithmdescribed in [2].
Given a problem, a quantity of interest (in this case, the S-parameters), and adiscretization tolerance error, the objective of the goal-oriented adaptivity isto construct, without any user interaction, an hp-grid containing a minimumnumber of unknowns, and such that the relative error in the quantity of interestis smaller than the (given) error tolerance.
Following [11,12], we first present the main ideas that make the practical useof goal-oriented adaptivity possible. Then, we introduce the projection basedinterpolation operator for 2D edge elements, which is a key component of thehp goal-oriented mesh optimization algorithm that is presented thereafter.
In the remainder of this paper, we shall work with 2D vectors (as opposed to3D vectors), and we shall use the symbol H when referring to HΩ (components
6
of the magnetic field parallel to the H-plane). Similarly, we will use symbol F
when referring to FΩ.
3.1 Goal-Oriented Adaptivity
Variational problem (2) can be stated here in terms of sesquilinear form b, andantilinear form f :
Find H ∈ V
b(H,F) = f(F) ∀F ∈ V ,
(6)
where
• V = HD(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
ǫ(∇×H) · (∇×F) dV
+jω2µ
β10
∫
Γ1∪Γ2
(n ×H) · (n× F )dS ,
c(H,F) = −∫
Ω
ω2µH · F dV .
(7)
We define an “energy” inner product on V as:
(H,F) := a(H,F) + c(H,F) ,
a(H,F) =∫
Ω
1
ǫ(∇×H) · (∇×F) dV
+|jω2µ
β10
|∫
Γ1∪Γ2
(n× H) · (n× F )dS ,
c(H,F) =∫
Ω
ω2µH · F dV ,
(8)
with the corresponding (energy) norm denoted by ‖H‖. Notice the inclusionof the material properties in the definition of the norm.
7
Using high-order Nedelec (edge) elements [16], we consider an hp-FE subspaceVhp ⊂ V. Then, we discretize (6) as follows:
Find Hhp ∈ Vhp
b(Hhp,Fhp) = f(Fhp) ∀Fhp ∈ Vhp .
(9)
We assume that our quantity of interest can be expressed as a continuous 5
and linear 6 functional L. By recalling the linearity of L, we have:
Error of interest = L(H) − L(Hhp) = L(H −Hhp) = L(e) , (10)
where e = H − Hhp denotes the error function. By defining the residualrhp belonging to 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 .
(11)
Problem (11) has a unique solution in V. The solution W, is usually referredto as the influence function.
By discretizing (11) via, for example, Vhp ⊂ V, we obtain:
Find Whp ∈ Vhp
b(Fhp,Whp) = L(Fhp) ∀Fhp ∈ Vhp .
(12)
Definition of the dual problem plus the Galerkin orthogonality for the originalproblem imply the final representation formula for the error in the quantityof interest, namely,
L(e) = b(e,W) = b(e,W − Fhp︸ ︷︷ ︸
ǫ
) = b(e, ǫ) .
5 If the quantity of interest is not continuous, we may consider a continuous ap-proximation L to the original quantity of interest.6 If the quantity of interest is represented by a non-linear functional, we linearize itaround a specific solution, and replace the original functional L with its linearizedversion.
8
At this point, Fhp ∈ Vhp is arbitrary, and b(e, ǫ) = b(e, ǫ) denotes the bilinearform corresponding to the original sesquilinear form.
Notice that, in practice, the dual problem is solved not for W but for itscomplex conjugate W utilizing the bilinear form and not the sesquilinear form.The linear system of equations is factorized only once, and the extra cost ofsolving (12) reduces to only one backward and one forward substitution (if adirect solver is used).
Once the error in the quantity of interest has been determined in terms ofbilinear form b, we wish to obtain a sharp upper bound for |L(e)| that dependsupon the mesh parameters (element size h and order of approximation p) only
locally. Then, a self-adaptive algorithm intended to minimize this bound willbe defined.
First, using a procedure similar to the one described in [17], we approximate H
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 theFE subspace Vh
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 = Hh
2, 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 elementcontributions. Let bK denote a contribution from element K to sesquilinearform b. It then follows that
|L(e)| = |b(e, ǫ)| ≤∑
K
|bK(e, ǫ)| , (13)
where summation over K indicates summation over elements.
3.2 Projection based interpolation operator
Once we have a representation formula for the error in the quantity of interestin terms of the sum of element contributions given by (13), we wish to expressthis upper bound in terms of local quantities, i.e. in terms of quantities thatdo not vary globally when we modify the grid locally. For this purpose, weintroduce the idea of the projection-based interpolation operator. More pre-cisely, we shall utilize the projection-based interpolation operator for hp-edgeelements Πcurl
hp : V −→ Vhp defined in [2]. This operator is local, it maintainsconformity and it is optimal in the sense that the error behaves asymptoti-cally, both in h and p, in the same way as the actual interpolation error (see[18] for details).
9
We shall also consider the Galerkin projection operator Pcurlhp : V −→ Vhp,
and we will denote Hhp = Pcurlhp H. Then, equation (13) becomes
|L(e)| ≤ ∑
K |bK(e, ǫ)| =
=∑
K
|bK(H− Πcurlhp H, ǫ) + bK(Πcurl
hp H − Pcurlhp H, ǫ)| .
(14)
Given an element K, we conjecture that |bK(Πcurlhp H − PhpH, ǫ)| will be neg-
ligible compared to |bK(H−Πcurlhp H, ǫ)|. Under this assumption, we conclude
that:
|L(e)| ∑
K
|bK(H− Πcurlhp H, ǫ)| . (15)
In particular, for ǫ = W −Πcurlhp W, we have:
|L(e)| ∑
K
|bK(H −Πcurlhp H,W − Πcurl
hp W)| . (16)
By applying Cauchy-Schwartz inequality, we obtain the next upper bound for|L(e)|:
|L(e)| ∑
K
‖e‖K‖ǫ‖K , (17)
where e = H − Πcurlhp H, ǫ = W − Πcurl
hp W, and ‖ · ‖K denotes energy-norm‖ · ‖ restricted to element K.
3.3 Goal-Oriented hp-Mesh Optimization Algorithm
We describe a self-adaptive goal-oriented hp algorithm that utilizes the mainideas of the fully automatic (energy-norm based) hp-adaptive algorithm de-scribed in [17,19,2]. The goal-oriented mesh optimization algorithm iteratesalong the following steps.
• Step 0: Compute the upper-bound estimate of the coarse grid
approximation error in the quantity of interest. This estimate iscomputed using eq. (17). If the difference is smaller than a user-prescribederror tolerance, then we deliver the fine mesh solution as our final answer,and we stop the execution of our algorithm.
• Step 1: For each edge in the coarse grid, compute the error de-
crease rate for the p refinement, and all possible h-refinements.
Let p1, p2 be the order of the edge sons in the case of h-refinement. Then,
10
we compute:
Error decrease (hp) =∑
K
[‖H −Πcurlhp H‖K · ‖W − Πcurl
hp W‖K
(p1 + p2 − p)
−‖H − Πcurl
hpH‖K · ‖W −Πcurl
hpW‖K
(p1 + p2 − p)
,
(18)
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 [2] forenergy-norm hp-adaptivity. The remaining steps (2 through 5) are exactly thesame as those described in [2].
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-orientedalgorithm described above is identical with the energy-norm algorithm pre-sented in [2].
Similarities between the energy norm based hp-adaptivity and the goal-orientedhp-adaptivity are reflected well in the corresponding implementation. The lat-ter one is simply an extension of the first one.
3.4 Implementation details
In what follows, we discuss the main implementation details needed to extendthe fully automatic (energy-norm based) hp-adaptive algorithm [17,19] to afully automatic goal-oriented hp-adaptive algorithm.
(1) First, the solution W of the dual problem on the fine grid is necessary.This goal can be attained either by using a direct (frontal) solver or aniterative (two-grid) solver (see [20]).
(2) Subsequently, we should treat both solutions as satisfying two differentpartial differential equations (PDE’s). We select functions H and W asthe solutions of the system of two PDE’s.
(3) We proceed to redefine the evaluation of the error. The energy-normerror evaluation 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-adaptivealgorithm may now be utilized as a self-adaptive goal-oriented hp algo-rithm.
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4 Numerical Results
In this section, we first solve the waveguide problem of Section 2 by usingthree different methods.
• The fully automatic energy-norm based hp-adaptive strategy described in[2],
• a mode matching technique [14], and• the fully automatic goal-oriented hp-adaptive strategy described in Sec-
tion 2.
Then, results are compared against each other, and the main advantages anddisadvantages of each methodology are discussed.
4.1 Results Obtained by Using the Energy-Norm hp-Adaptive Strategy
First, we solve our waveguide problem using the fully automatic hp-adaptivestrategy. In order to construct an adequate initial grid for our adaptive algo-rithm, we notice the following limitation:
• We cannot guarantee the optimality of the fully automatic hp-adaptive strategy if the dispersion error is large, which may occur
in the pre-asymptotic regime. Since solution of the problem on thefine grid is used to guide optimal hp-refinements, we need to control thedispersion error on the fine grid. Thus, h needs to be sufficiently small orp sufficiently large. Otherwise, convergence results in the pre-asymptoticregime may not look as expected.
In Fig. 2, we compare the convergence history obtained by using thefully automatic hp-adaptive strategy starting with different initial grids. Forthird order elements, the dispersion error is under control (see estimates of[21,22,23]), and the fully automatic hp-adaptive strategy converges expo-nentially from the beginning, as indicated by a straight line in the algebraicvs logarithmic scales. We also observe that, in the asymptotic regime, allcurves present similar rates of convergence. These results indicate that thechoice of initial grid is not very important and, even in the case of verycoarse initial grids, the algorithm will eventually notice the asymptotic be-havior of the solution, and the overall convergence will be exponential.
We solved the six irises waveguide problem delivering a 0.3% relative error inthe energy-norm. Fig. 3 displays the magnitude of the S11 scattering parameter(on the decibel scale) with respect to the frequency. This quantity is usuallyreferred to as the return loss of the waveguide structure, and it is given by
12
125 1000 3375 8000 15625 27000 42875 64000 9112510
−1
100
101
102
103
Number of unknowns (algebraic scale)
Estim
ate
of th
e re
lativ
e er
ror (
%)
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 2. Convergence history using the fully automatic hp-adaptive strategy fordifferent initial grids. Different colors correspond to different initial order of approx-imation. 27 is the minimum number of elements needed to reproduce the geometry,while 1620 is the minimum number of elements needed to reproduce the geometryand to guarantee convergence of the iterative two grid solver described in [20].
equation (19), see [1].
S11 =
∫
Γ1
H(ξ) · sin πξ
adS
Hina
2
− 1 , (19)
where a is the size of the excitation port. For the frequency interval 8.8 − 9.6Ghz, the return loss is below −20dB, which indicates that almost all energypasses through the structure, and thus, the waveguide acts as a bandpass filter.The other scattering parameter of interest is S21 (S12 and S22 are obtainedusing (3) and (4), respectively) which is given in equation (20), see also [1].
S21 =
∫
Γ2
H(ξ) · sin πξ
adS
Hina
2
(20)
13
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 3. Return loss of the waveguide structure
14
Figure 4. | Hx | (upper figure), | Hy | (center figure), and√
| Hx |2 + | Hy |2 (lowerfigure) at 8.72 Ghz for the six irises waveguide problem.
Figure 5. | Hx | (upper figure), | Hy | (center figure), and√
| Hx |2 + | Hy |2 (lowerfigure) at 8.82 Ghz for the six irises waveguide problem.
Figures 4, 5, 6, and 7 display solution at different frequencies. For frequencies8.72 Ghz, and 9.71 Ghz, the return loss of the waveguide structure is large,and for frequencies 8.82 Ghz, and 9.58 Ghz the return loss is below −20dB.
15
Figure 6. | Hx | (upper figure), | Hy | (center figure), and√
| Hx |2 + | Hy |2 (lowerfigure) at 9.58 Ghz for the six irises waveguide problem.
Figure 7. | Hx | (upper figure), | Hy | (center figure), and√
| Hx |2 + | Hy |2 (lowerfigure) at 9.71 Ghz for the six irises waveguide problem.
16
Figure 8. Return loss of the waveguide structure. This graph has been computedusing a Mode Matching technique [14].
4.2 Results Obtained by Using a Mode Matching Technique
In this subsection, we solve our problem of interest with a Mode Matchingtechnique. More precisely, we compute the S-parameters by using the soft-ware available in [14]. In Fig. 8, we display the S-parameters as a function offrequency. Notice the similarities of these results with those presented in Fig.3.
4.3 Results Obtained by Using the Goal-Oriented hp-Adaptive Strategy
The final hp-grid delivered by the fully automatic, energy-norm based adap-tive algorithm is expected to produce a “reasonably good” approximation ofthe exact solution almost everywhere in the computational domain. Thus, wemay decide what quantity to compute (for example, the S11 parameter) aftersolving the problem.
However, when using the goal-oriented hp-adaptivity, we need to decide firstwhat quantity is of interest for us (for example, S11), and then construct agrid intended to obtain the best possible approximation of that quantity ofinterest with respect to the problem size. Notice that quality of the solutioneverywhere else in the domain may be poor.
Thus, we revisit the question on which of the S-parameters should be com-puted. For instance, will it be better to directly compute S11, or to compute S21
17
and then apply identity (5) to obtain S11? These questions are especially im-portant when considering goal-oriented adaptive algorithms, since the configu-ration of the final grid will depend upon the quantity of interest (S-parameter)that we select before refinements.
In order to select between the four different S-parameters relevant to ourwaveguide structure, we first notice that at the discrete level (when consideringfinite element approximation), identity (3) is also valid due to the fact thatreciprocity is also satisfied at the discrete level. Thus, the use of the goal-oriented adaptivity with S12 or S21 as our quantity of interest will provideidentical grids and results. However, identity (4) will hold at the discretelevel only if the grid is symmetric. Since in this paper we are consideringsymmetrical initial grids, the goal-oriented adaptive algorithm with S11 or S22
as our quantity of interest will provide identical grids and results. Finally,notice that identity (5) will not hold in general at the discrete level, even ifthe grid is symmetric. In summary, we shall consider two different quantitiesof interest,
(1) 7 L1(H) := S11(H) + 1, and,(2) L2(H) := S21(H).
Then, we will numerically study the self-adaptive goal-oriented hp-adaptivestrategy by executing it twice: first, using L1 as our quantity of interest, andthen considering L2 as our quantity of interest. Finally, in order to comparethe results, we will use for the second case (L2) equation (5) in order to post-process S11(H) + 1 from S21(H).
Convergence results are displayed in Fig. 9 at different 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). These frequencies correspond to solutionsshown in figures 4, 5, 6, and 7, respectively. The red curve displays the relativeerror in percentage of S11 + 1 with respect to the number of unknowns, whenexecuting the goal-oriented hp-adaptive algorithm with L1 as our quantityof interest. If we consider L2 as our quantity of interest, the correspondingrelative error of S11 + 1 -computed using (5)- is displayed by the black curve.Finally, the blue curve corresponds to upper bound (17) used for minimization,when considering |L1(H)| as our quantity of interest.
From results of Fig. 9, we conclude the following.
• The self-adaptive goal-oriented hp-FEM delivers exponential converges ratesin terms of upper bound (17) of quantity of interest |L1(H)| against the
7 Notice that S11(H) is not a linear functional. However, S11(H)+1 is a linear andcontinuous functional, and therefore, we may use it as our quantity of interest forthe goal-oriented optimization algorithm.
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Figure 9. Convergence history for the waveguide problem with six inductive irises atdifferent 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 curvesdisplay the relative error in percentage of S11 + 1 with respect to the number ofunknowns, when executing the goal-oriented hp-adaptive algorithm with L1 and L2
as our quantity of interest, respectively. The blue curve corresponds to upper bound(17) used for minimization, when considering |L1(H)| as our quantity of interest.
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problem 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 boundestimate of the red curve, the latter curve should also display an overallexponential convergence behavior (or at least it is bounded by a curve thatexponentially converges to zero), regardless of the fact that the error in thequantity of interest may 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 execut-ing the goal-oriented adaptive algorithm is not essential for this problem.Numerical results are similar in both cases.
4.4 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 func-tional f representing the right hand side of the original problem. Thus, solutionof the dual problem W is also equal (up to a multiplicative constant) to thesolution of the original problem H. For this particular case, the goal-orientedadaptivity coincides exactly with the energy driven adaptivity, and the cor-responding numerical results are identical. In other words, for this particularwaveguide problem, energy-norm adaptivity is optimal for approximating L1.
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-norm adaptivity. Nevertheless, Fig. 9 illustrates that the results coming fromboth algorithms, although different, are quite similar to each other, since S21
is also strongly related 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 waveg-uide structures with possibly several ports, in which the quantity of interest isnot strongly related to the conservation of energy. To illustrate this effect, weconsider our original problem with the three central cavities filled by a lossymaterial (see Fig. 10) with ǫ = ǫ0ǫr, where ǫ0 = 8.854 ∗ 10−12 F/mis the per-mittivity of the vacuum, and ǫr = (1 − 0.07805j/(ωǫ0)). Results at 8.82 Ghz,i.e. ǫr = (1− j/(2π)) are displayed in Fig. 11. When the goal of computationsis to approximate S11, the grid obtained with the goal-oriented adaptivityfor S11 + 1 provides results that are one order of magnitude more accuratethan those obtained with the goal-oriented adaptivity and S21 as our quantityof interest. If the goal of computations is to approximate S21, the grid ob-tained with the goal-oriented adaptivity for S21 provides results that are up to
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Figure 10. 2D cross-section of the geometry of the waveguide problem with sixinductive irises. The red color indicates that the three central cavities are filledwith a resistive material.
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Figure 11. Convergence history for the waveguide problem with six inductive irisesat 8.82 Ghz, with the three central cavities filled with a resistive material. Theleft panel describes the convergence history for S11, and the right panel for S21.The blue curves have been obtained by using goal-oriented adaptivity with S21 asour quantity of interest, and the red curves by using goal-oriented adaptivity withS11 + 1 as our quantity of interest.
five hundred times more accurate than those obtained with the goal-orientedadaptivity and S11 + 1 as our quantity of interest. Since grids obtained fromusing the goal-oriented adaptivity for S11 +1 are equal to those obtained fromusing energy-norm adaptivity, we conclude that the goal-oriented adaptivityfor S21 provides approximations to S21 that are far more accurate than thoseobtained with the energy-norm adaptivity. Eventually, for a resistive enoughmaterial, the use of goal-oriented adaptivity will become essential (see [11,12]for details) for accurately approximating S21.
Two hp-grids for our original problem with the three central cavities filled bya lossy material are shown in Fig. 12. The two hp-grids displayed in Fig. 12are essentially different:
(1) The top panel displays a grid obtained by using L1 as our quantity of
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p=1
p=2
p=3
p=4
p=5
p=6
p=7
p=8
Figure 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 ourquantities of interest, respectively. Different colors indicate different polynomialsorders of approximation, ranging from 1 (dark blue) up to 8 (pink). The hp-gridscontain 12110 (top) and 13054 (bottom) unknowns, respectively.
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 upperbound
∑
K |bK(e, ǫ)|, used for the minimization, is also symmetric sincesolutions of the original and dual problems are symmetric with respectto each other (up to 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 consider-ing in this paper, results for (the particular case of) energy-norm adaptivityare close to optimal. Indeed, they are exactly optimal if we consider L1 as ourquantity of interest. Thus, there is no need to use the goal-oriented adaptivityfor this problem. Nevertheless, it is important to realize that for some otherproblems (for instance, waveguide problems with lossy media, or waveguideproblems with several ports), the use of goal-oriented adaptivity may becomeessential. In any case, results coming from the goal-oriented adaptivity shouldnever be worse than those obtained with the energy-norm adaptivity.
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5 Conclusions
In this paper, a fully automatic goal-oriented hp-FEM has been presented forthe analysis of rectangular waveguide discontinuities. A challenging waveguideproblem containing six inductive irises and 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 adap-tivity which may (or may not) be essential in order to accurately solve aproblem, and,
• from the physical point of view, the waveguide acts as a bandpass filter.
Summarizing the whole series of the three papers we conclude the following.A relevant microwave engineering problem involving the analysis of H-planeand E-plane rectangular waveguide discontinuities (and the computation oftheir S-parameters), has been solved by using a fully automatic hp-adaptiveFEM. The hp adaptivity has been shown to deliver exponential convergencerates for the error, even in the presence of singularities, for a wide variety ofrelevant structures including microwave engineering devices of medium com-plexity. While keeping the advantage of being a purely numerical method, thehp-adaptivity has shown to be more accurate than semi-analytical techniques.The suitability of a goal-oriented approach in terms of the S-parameters, incomparison with a “conventional” approach in terms of energy-norm, has beenclarified.
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