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A Two-Dimensional Self-Adaptive hp Finite
Element Method for the Characterization of
Waveguide Discontinuities. Part II:
Energy-Norm Based Automatic hp-Adaptivity
Luis E. Garćıa-Castillo 1,2
Departamento de Teoŕıa de la Señal y Comunicaciones.
Universidad Carlos III deMadrid. Escuela Politécnica Superior
(Edificio Torres Quevedo). Avda. de la
Universidad, 30. 28911 Leganés (Madrid) Spain.
Fax:+34-91-6248749
David Pardo
ICES, University of Texas at Austin, Austin TX 78712, USA
Ignacio Gómez-Revuelto
Departamento de Ingenieŕıa Audiovisual y Comunicaciones.
UniversidadPolitécnica de Madrid, Madrid, Spain.
Leszek F. Demkowicz
ICES, University of Texas at Austin, Austin TX 78712, USA
Abstract
This is the second 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, a fully automatic energy-norm based hp-adaptive
Finite Element(FE) strategy applied to a number of relevant
waveguide structures, is presented.The methodology produces
exponential convergence rates in terms of the energy-norm error of
the solution against the problem size (number of degrees of
freedom).
Extensive numerical results demonstrate the suitability of the
hp-method for solv-ing different rectangular waveguide
discontinuities. These results illustrate the flex-ibility,
reliability, and high-accuracy of the method.
The self-adaptive hp-FEM provides similar (sometimes more)
accurate resultsthan those obtained with semi-analytical techniques
such as the Mode Matchingmethod, for problems where semi-analytical
methods can be applied. At the sametime, the hp-FEM provides the
flexibility of modeling more complex waveguide
Preprint submitted to Elsevier Science 1st March 2006
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structures and including the effects of dielectrics, metallic
screws, round corners,etc., which cannot be easily considered when
using semi-analytical techniques.
Key words: Finite Element Method, hp-adaptivity, Energy-norm,
RectangularWaveguides, Waveguide Discontinuities, S-Parameters
1 Introduction
As it was mentioned in the first paper on this series [1], the
accurate analy-sis and characterization of “waveguide
discontinuities” is an important issuein microwave engineering (see
e.g., [2], [3]). Waveguide discontinuities, i.e.,the interruption
in the translational symmetry of the waveguide, may be aunavoidable
result of mechanical defects or electrical transition in
waveguidesystems, or they may be deliberately introduced in the
waveguide to performa certain electrical function. Specifically,
discontinuities in rectangular waveg-uide technology are very
common in the communication systems working inthe upper microwave
and millimeter wave frequency bands. In many cases,the rectangular
waveguide discontinuities can be analyzed in two-dimensions(2D)
because of the invariant nature of the geometry along one
direction. Thisis the case of the so called H-plane and E-plane
rectangular waveguide dis-continuities, 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 [4,5], which has been extended for
electromagnetic applications[6,7,8,9,10,11,12,5] is applied to a
number of relevant waveguide structures.The adaptive methodology
has a number of advantages that makes it suitablefor the analysis
of complex structures (containing several waveguide
sections,discontinuities, complex geometries, dielectrics, etc.) in
contrast to other semi-analytic and numerical techniques.
Namely:
• It automatically resolves different types of singularities
i.e., different types
Email addresses: [email protected], [email protected] (Luis
E.Garćıa-Castillo), [email protected] (Ignacio
Gómez-Revuelto),[email protected] (Leszek F. Demkowicz).1
This work has been initiated during a stay of the first author at
ICES supported bythe Secretaŕıa de Estado de Educación y
Universidades of Ministerio de Educación,Cultura y Deporte of
Spain. The authors wants also to acknowledge the support
ofMinisterio de Educación y Ciencia of Spain under project
TEC2004-06252/TCM.2 On leave from Departamento de Teoŕıa de la
Señal y Comunicaciones. Universidadde Alcalá, Madrid, Spain
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of discontinuities.• It efficiently deals with high frequencies,
that is, it delivers a low dispersion
error [13,14].• It provides high-accuracy results, so the
S-parameters (see comments below)
[1] can be accurately computed.• It allows for modeling of
complex (non-uniform) geometries.
The waveguide theory and a detailed analysis of rectangular
waveguide discon-tinuities (including the finite element
variational formulations used) were pre-sented in the first part of
the series [1]. Extensive numerical results presentedin this part
illustrate the flexibility, reliability, and high-accuracy
simulationsobtained with this methodology, providing more accurate
results than withsemi-analytical (in particular, Mode-Matching —MM—
[15], [16, Chapter 9])techniques. The adaptive methodology is shown
to produce exponential con-vergence rates in terms of the
energy-norm error of the solution against theproblem size (number
of degrees of freedom). Thus, the electromagnetic fieldis
accurately known (with a user pre-specified degree of accuracy)
inside thestructure. The high accuracy is essential in the
microwave engineering designaiming 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 pa-rameters of the structure, which are widely used in
microwave engineeringfor the characterization of microwave devices.
The notion of the scatteringparameters (or S-parameters), and their
computation using a finite elementsolution, have been explained in
the first paper of the series [1]. As the quan-tities of interest
for the microwave engineer are mainly the S-parameters,
agoal-oriented approach (in terms of the S-parameters) may be also
desirable.In the third paper of the series [17], results obtained
using hp energy-normadaptivity are compared against those using a
goal-oriented hp-adaptivity ap-proach [18,19,20]. Results show that
both methods are suitable for simulationof waveguide
discontinuities.
The organization of the paper is as follows. The hp finite
element discretizationand automatic adaptivity strategy are briefly
described in Section 2.1 and2.2, respectively. The refinement
strategy is based on the minimization of theprojection based
interpolation error, which is defined in Section 2.2.1. The stepsof
the mesh optimization algorithm are described in Section 2.2.2.
Extensivenumerical results, both for the E-plane and H-plane
simulations, are shown inSection 3. Finally, some conclusions are
given in Section 4.
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2 hp Finite Elements and Automatic Adaptivity
In order to solve the presented electromagnetic problems, a
numerical tech-nique that provides low discretization errors and,
simultaneously, solves thediscretized problem without prohibitive
computational cost, is needed. In thiscontext, an adaptive
hp-Finite Element Method satisfies both properties.
2.1 hp-Finite Elements (FE)
Each finite element is characterized by its size h and order of
approximationp. In the h-adaptive version of FE method, element
size h may vary fromelement to element, while order of
approximation p is fixed (usually p=1,2).In the p-adaptive version
of the FE method, p may vary locally, while h re-mains constant
throughout the adaptive procedure. Finally, a true
hp-adaptiveversion of FE method allows for varying both h and p
locally.
The hp-FE method used in this paper utilizes edge (Nédélec)
elements ofvariable order of approximation. FE spaces associated to
those elements havebeen carefully constructed (see [21] for
details) so in combination with theprojection based interpolation
operators (defined below), the commutativity ofthe de Rham diagram
is guaranteed. This commutativity property is essentialfor showing
convergence and stability of the FE method for
Electromagnetics[21].
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 ellip-tic problems with a piecewise
analytic solution, whereas h- or p-FEM convergeat best
algebraically” (see [22,23,24,25,26,27,28]).
Next, the fully automatic hp-adaptive strategy is presented.
Given a problemand a discretization tolerance error, the objective
is to generate automatically(without any user interaction) an
hp-grid that does not exceed the discretiza-tion error tolerance
and, at the same time, it employs a minimum number ofdegrees of
freedom (d.o.f.), by orchestrating an optimal distribution of
elementsize h and polynomial order of approximation p. By doing so,
it is possible toachieve exponential convergence rates in terms of
the error vs. the number ofd.o.f.
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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 element is broken into four element
sons (eight in 3D), and the orderof approximation is raised
uniformly by one. Then, the problem of interestis solved on the
fine mesh. The difference between the fine and coarse gridsolutions
is used to guide optimal refinements over the coarse grid. More
pre-cisely, the next optimal coarse mesh is then determined by
minimizing theprojection based interpolation error of the fine mesh
solution with respect tothe optimally refined coarse mesh (see
[4,29] for details).
The adaptive strategy is very general, and it applies to H1-,
H(curl)-, andH(div)-conforming discretizations. Moreover, since the
mesh optimization pro-cess is based on minimizing the interpolation
error rather than the residual,the algorithm is problem independent
and it can also be applied to nonlinearand eigenvalue problems.
The hp self-adaptive strategy incorporates also a two-grid
iterative solver,which allows to solve the fine grid problems
efficiently. Indeed, it has beenshown in [30,31] that it is
sufficient a partially converged fine grid solutionto guide optimal
hp-refinements. Thus, only few two-grid solver iterations areneeded
(below ten per grid).
In the remainder of this section, the projection based
interpolation operator[32,33], which is the main ingredient of the
mesh optimization algorithm, ispresented first. Then, the mesh
optimization algorithm is briefly described.
2.2.1 The projection based interpolation operator
The idea of projection based interpolation operator is based on
three proper-ties.
• Locality: Determination of element interpolant of a function
should involvethe values (and derivatives) of the interpolated
function in the element only.
• Conformity: The union of element interpolants should be
globally conform-ing.
• Optimality: The interpolation error should behave
asymptotically, both inh and p, in the same way as the actual
approximation error.
The H1-conforming projection based interpolation operator is
presented first.Let u ∈ H1+�(K) with � > 0. Locality and
conformity imply that the inter-
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polant w = Πu should match the interpolated function u at
vertexes:
w|vert = u|vert (1)
With the vertex values fixed, we project over each edge,
i.e.:
w := arg minv:(v−u)|vert=0
‖ v − u ‖edge (2)
This definition preserves locality and conformity. It also
preserves optimalityprovided that the optimal edge norm is
selected, which is dictated by the prob-lem being solved and the
Trace Theorem (see [21] for details). For example,in 1D, the
optimal edge norm is the H10 -norm. In 2D, the H
1/2-seminorm, andin 3D, the L2-norm, should be used.
Using the same argument, once vertex and edge values are fixed,
projectionover the interior of the element (faces in 3D) is
performed. Thus, the projectionbased interpolation 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
(3)
For a definition of projection based interpolation operator for
3D H1-problems,see [34].
Similarly, a projection based interpolation operator can be
defined for elementsin H(curl), which is the space of interest for
the electromagnetic field. GivenE 3 in H(curl), the projection
based interpolator Πcurl specialized to the 2Dcase (for the 3D see
[32,35]), is denoted by Ep = ΠcurlE, where Ep is given by:
‖ Ept − Et ‖−12
,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(4)
3 E is used here to abstractly denote an element in H(curl). In
this paper, discretiza-tions in H(curl) are used for the magnetic
field on the H-plane and the electric fieldon the E-plane of the
structures.
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Here, the bubble functions come from an appropriate polynomial
space mappedby the gradient operator onto the subspace of fields E
with zero curl and tan-gential trace on the element boundary.
A similar operator can be defined for H(div) problems (see
[21]).
Finally, it is important to mention that the de Rham diagram
equipped withthese projection based interpolation operators
commutes (see [21,12,32] fordetails), which is critical for proving
stability and convergence properties ofthe FEM for Maxwell
equations.
2.2.2 The mesh optimization algorithm
The mesh optimization algorithm in 2D follows the next
steps.
• Step 0: Compute an estimate of the approximation error on the
coarse grid.The approximation error on the coarse grid is estimated
by simply com-
puting the norm of the difference between the coarse and the
fine grid so-lutions. If the difference (relative to the fine grid
solution norm) is smallerthan a requested error tolerance, then the
fine mesh solution is delivered asthe final solution, and the
process stops.
• Step 1: For each edge in the coarse grid, compute the error
decrease ratefor the p refinement, and all possible
h-refinements.
Let p1, p2 be the order of the edge sons in the case of
h-refinement, andlet E = Eh/2,p+1 denote the fine grid solution.
Then, the error decrease rateis computed as:
Error decrease (ĥp) =‖Eh/2,p+1 −Πcurlhp E‖ − ‖Eh/2,p+1
−Πcurlĥp E‖
(p1 + p2 − p),
where ĥp = (ĥ, p̂) is such that ĥ ∈ {h, h/2}. If ĥ = h, then
p̂ = p + 1. Ifĥ = h/2, then p̂ = (p1, p2), where p1 + p2 − p >
0, max{p1, 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
corre-sponding to the p-refinement with all competitive
h-refinements. Compet-itive h-refinements are those that result in
the same increase in the num-ber of degrees-of-freedom (d.o.f.) as
the p-refinement, i.e., ĥ = h/2 andp1 + p2 − p = 1.
Next, the guaranteed rate with which the interpolation error
must de-crease over the edge is determined. That is, for each edge
the maximumof the error decrease rates for the p-refined edge and
all possible h-refinededges is computed.
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• Step 3: Select edges to be refined.Given the guaranteed rate
for each edge in the mesh, the maximum rate
for all elements is calculated
guaranteed ratemax = maxe (edge e guaranteed rate) .
All edges that produce a rate within 1/3 of the maximum
guaranteedrate, are selected 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 oneedge of the element is to be broken, the element is
refined accordingly. Asin [4,29], element isotropy flags are
computed. Isotropic h-refinement areenforced if the error function
within the element changes comparably inboth element
directions.
After this step, the topology of the new coarse mesh has been
determined,and it remains only to establish the optimal
distribution of orders of ap-proximation for the involuntarily
h-refined edges, and for the interior nodesof the element i.e.,
those nodes that are not located on the boundary ofthe element. For
the interior nodes, the starting point for the
minimizationprocedure will be based on the order of approximation p
for the adjacentedges 4 .
• Step 5: Determine the optimal orders of approximation p for
the refinededges and elements.
This step consists basically of p-adaptivity over a given grid
with thefine grid as a reference solution. Unfortunately, due to
the possible presenceof involuntary edge h-refinements and too low
p for h-refined elements, theinterpolation error of the coarse grid
after step 4 may actually be larger thanthe interpolation error for
the original coarse mesh. Thus, extra technicaldetails are
considered in order to guarantee interpolation error decrease.These
details are quite involving, and are described in [12].
Some remarks on the mesh optimization algorithm follow:
• A similar but yet more involved mesh optimization algorithm
has beenimplemented for 3D problems, although the 3D
electromagnetic version isstill under development.
• The main difference between the fully automatic hp-adaptive
strategy for
4 For triangles, the initial order of approximation will be
equal to the maximumof the three edges of the element. For
quadrilaterals, we have a horizontal and avertical order of
approximation p = (ph, pv). In this case, the starting point for
theminimization procedure will be the maximum of the two horizontal
edges for ph andthe maximum of the two vertical edges for pv.
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elliptic and electromagnetic problems resides in the definition
of the projec-tion based interpolation operator.
• A similar algorithm can be implemented for H(div)
problems.
3 Numerical Results
In the following, a number of rectangular H-plane and E-plane
waveguidediscontinuities, as well as more complex structures
obtained by combiningseveral discontinuities, are analyzed. The
analysis of all these structures isperformed by using the fully
automatic hp-adaptive FE strategy presentedabove.
TE10 mode excitation has been used in all the structures. Also,
the ratio ofthe broad dimension a to the narrow dimension b of the
rectangular waveg-uide sections is considered to be a/b = 2. The
results correspond to a givenfrequency which is chosen to be in the
middle of the monomode region, i.e.,k = 1.5Kc10. Exceptionally, the
structure analyzed in Section 3.2.4 is solvedfor a large number of
frequencies within a given frequency region, in order
tocharacterize its frequency response. The lengths of the waveguide
sections thatconnect the discontinuity to the ports of the
structure is typically around onewavelength for the H-plane
structures, and around half a wavelength for theE-plane structures.
This is enough for the first absorbing boundary conditionused at
the ports to perform correctly.
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 inwhich the initial mesh has to be as
coarse as possible in order to simplify themesh generation process.
The convergence history is always shown using a logscale 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 abscissas axis. Thus, according to [27] and
referencestherein, an straight line should appear in the plot
showing the theoreticalexponential convergence that can be achieved
with an optimal hp adaptivitystrategy. Note that the abscissas
scale corresponds to N
1/3dof while abscissas axis
tics should be read as Ndof in the plots.
The scattering parameters obtained using the hp-FEM are compared
withvalues 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. Itconsists of the decomposition of the domain
of the problem into several simpledomains, typically with
translational symmetry, in which, an analytical modalexpansion can
be performed. Imposing the tangential continuity of the fieldand
orthogonality of the modes yields a system of equations in which
the
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unknowns are the coefficients of the modal expansions.
The FEM scattering parameter results delivered from the hp
adaptivity aremore accurate than MM results. In this context, it is
important to pointout that MM results are typically considered as a
reference for the engineer-ing analysis of discontinuities in
rectangular waveguide technology, as for thestructures shown below.
In addition, the hp-FE technology allows for modelingof more
complex structures which cannot be solved using the MM.
3.1 H-plane discontinuities
The analysis of several H-plane discontinuities is considered
next. The bound-ary condition of the metallic conductors represents
a Neumann boundary con-dition for the H-plane formulation.
3.1.1 H-plane waveguide section
The first structure shown in Fig. 1 is a simple rectangular
waveguide section.This structure is chosen as a first verification
of the code, specifically for theboundary conditions at the ports
and the control of the dispersion error. Sincethere is no
discontinuity in the translational symmetry for this structure,
itmay be analyzed by means of either the H-plane or E-plane
formulations. Theresults shown in this section correspond to the
H-plane analysis. Also, becausethere is no discontinuity, the
scattering parameters of the structure are knownto be S11 = S22 = 0
and S21 = S12 = exp(−jβ10l), where l denotes thewaveguide section
length. In this case, l is equal to 2 wavelengths and, thus,S21 =
S12 = exp(−j4π) = 1. The field solution is also known: it
correspondsto the field TE10 mode inside of the waveguide
section.
The solution is smooth: a half-sine type variation in the y
direction (the ±ξlocal axis of the waveguide) and constant in
amplitude and phase variation asexp(−jβ10x) along the x direction
(the ±ζ local axis of the waveguide). Thus,the hp-adaptive strategy
is expected to deliver an increase in the polynomialorder of
approximation p. The initial mesh used for the analysis is shown
inFig. 2 together with some intermediate meshes. The colors
indicate, accordingto the scale on the right, the order p of the
elements. It is important to notethat the order corresponds to the
H1 Lagrange multiplier and that the field ofH(curl) is of order p−
1. As an example, the green color of the initial mesh ofFig. 2(a)
indicate that all elements are of order 3 for the Lagrange
multiplierand order 2 for the magnetic field. It is observed that
order p is increaseduntil the maximum p (p = 9) is reached in the
fine grid; from this moment, hrefinement is selected until the
specified error criterion is satisfied.
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The convergence history for the exact error and the estimated
error is plot-ted in Fig. 4, showing the quality of the error
estimation and the exponentialbehavior of the error. The
exponential convergence is deduced from the ob-servance of a
straight line in the plot for the log type ordinates axis and
theN
1/3dof scale in the abscissas axis set in the figure. It is
worth noting that the
slope change in the convergence history corresponds to the
moment when themaximum p is reached, so h refinements are forced.
In other words, this slow-down in convergence would not have
occurred if higher order elements wereallowed. A plot of the field
in the structure, specifically, |Hy|, is shown inFig. 3 where it is
clearly observed the sine and the zero variations along they and x
axis, respectively. Note that the the zero variation along the x
axisis because we are referring to |Hy| and not Hy. A constant
magnitude in thedirection of propagation means that there is only
one wave in the waveguideand, thus, |Hy| = |H in0 | | sin(πξ/a)|,
so it does not depend on the ζ ≡ ±xdirection. If a discontinuity
had generated a reflected wave, a stationary wavepattern would have
been observed at the input waveguide (as it is seen in theexamples
below).
The scattering parameters have been computed at each iteration
step of thehp strategy. Due to the reciprocity and symmetry of the
structure, the scatter-ing behavior of the discontinuity is
characterized by performing one analysis(exciting any of the two
ports).The results for the first iterations are shownin Tab. 1. A
fast convergence is observed.
Table 1Scattering 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◦
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y
x
x
H-plane
l
Port 2
y
ζPort 1
ξ
η
ξζ
η
ε0, µ0b
ε0, µ0Ω
Γ2pΓ1p
ΓN
H-planeζ
ξζ
ξ
a
l
ΓN
a
Figure 1. H-plane waveguide section
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x
y
z
(a) Initial mesh
11 12 13
x
y
z
(b) Mesh 4th iteration
14 15
16
17
28 29
30
31
42 43
44
45
x
y
z
(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
x
y
z
(d) Mesh 7th iteration
Figure 2. Initial mesh and some hp meshes for the H-plane
waveguide section
Figure 3. Magnitude of Hy, i.e., |Hy|, corresponding to the
H-plane waveguide sec-tion
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-5
-4
-3
-2
-1
0
1
2
228 448 778 1239 1854 2645 3634 4843 6294
log1
0 (r
elat
ive
erro
r %
)
Ndof
H-plane waveguide section
exact-energy-errorestimate-energy-error
Figure 4. Convergence history for the H-plane waveguide section
(energy norm errorfor the magnetic field solution)
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3.1.2 H-plane right angle bend
An H-plane 90◦ bend is analyzed next. The structure is shown in
Fig. 5. Thestructure is analyzed by exciting port 1 (on the left).
The bend is obviously acommon part of microwave circuits. The
initial mesh used for the analysis isshown in Fig. 6. Despite the
coarseness of this mesh, the hp-strategy achievesan energy error
lower than 1% error after 5 iterations. The convergence his-tory
(up to an error as low as 0.01%) is shown in Fig. 9. The final mesh
isshown in Fig. 8. It is observed how the mesh is refined around
the corner. Thehp-strategy in this case tends also to increase the
p. Actually, all elements ofthe final mesh (except those near the
corner) have reached the maximum porder. This is the right strategy
as the solution of the problem is smooth (theboundary condition at
the conductors for the H-plane formulation is of ho-mogeneous
Neumann type 5 ). The h-refinement around the corner is
preciselydue to the fact that the maximum p has been reached and,
in order, to reducethe error in this region, the elements must be
made smaller.
A plot of the field in the structure, specifically, |Hy|, is
shown in Fig. 7. They-component corresponds to the local ξ
component at the excitation portand the local ζ component at the
transmitted port. Notice the figure thestationary wave pattern in
the input waveguide (between the excitation portand the bend)
because of the combination of the two waves propagating inopposite
directions (the excited wave and the reflected wave at the bend).
Nostationary wave is observed in the output waveguide as there is
only one wavepropagating outward the transmitted port. As in the
previous case, S21 = S12and S22 = S11. The results for S11 and S21
are shown (for some of the hpmeshes) in Tab. 2. The scattering
parameters computed with the hp-FEMmethod are compared with those
obtained with a MM technique. Only foursignificant digits are shown
in the table as the MM results are presumed tohave no more tan 4
digits of accuracy. Observe the very good agreement ofthe hp-FEM
results with those provided by MM; better than 1% after thesecond
iteration. After the fourth/fifth iteration, the FEM results seem
to bemore accurate than those provided by the MM, as implied by the
convergencepattern shown in the table.
5 The same domain but with Dirichlet boundary conditions
corresponds to theE-plane bend which is analyzed in Section
3.2.1.
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xy
y
x
H-plane
Port 1ξ
η
η
ξ
Port 2
b
l1
b
l2
aa
ζ
ζ
ε0, µ0
ε0, µ0Ω
ΓN
ΓNH-plane
ζ
ξ
ξ
Γ2p
l1
l2
a
ζ
Γ1p
a
Figure 5. H-plane 90 degrees bend
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11 12
13
x
y
z
Figure 6. Initial mesh for the H-plane 90◦ bend
Figure 7. Magnitude of Hy, i.e., |Hy|, corresponding to the
H-plane 90◦ bend
17
-
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
x
y
z
Figure 8. Final hp mesh for the H-plane 90◦ bend
-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 9. Convergence history for the H-plane 90◦ bend
18
-
Table 2Scattering 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◦
19
-
3.1.3 H-plane symmetric inductive iris
The structure is shown in Fig. 10. The discontinuity consists of
the narrowingof the broad dimension of the waveguide along a
certain length. This region isreferred to as an iris 6 . Because
the iris is centered with respect the waveguidebroad dimension, it
is called a symmetric iris. Finally, the term inductive isused
because the discontinuity scattering behavior with respect to the
planesof the discontinuity is equivalent to an inductance. The
character of the dis-continuity, inductive or capacitive, can be
deduced by observing which fieldlines (electric or magnetic) of the
waveguide mode (the TE10 in this case) are“cut” by the
discontinuity [2]. It is clear from Fig. 10 that only the
magneticfield lines are cut in this case (the electric field of the
TE10 is perpendicularto the H-plane of the waveguide). Thus, the
symmetric iris of Fig. 10 is of theinductive type. An analogous
structure, but of capacitive type, is consideredin Section
3.2.3.
The initial mesh used for the analysis is shown in Fig. 11. The
analysis ismade by exciting port 1 (at the left). The convergence
history (up to an erroras low as 0.02%) is shown in Fig. 12. Notice
the exponential convergence ofthe method.
A plot of the field in the structure, specifically, |Hy|, is
shown in Fig. 13.The y-component corresponds to the ±ξ component of
the field modes in thewaveguide. Observe the stationary wave
pattern at the input port due to thewave reflected from the
discontinuity and a singular behavior of the field atthe re-entrant
corners. The magnitude of the fields is higher at the left
corners.This is “caught” by the hp strategy that refines around the
left corners duringthe first few iterations, and once the error
around the left corners is controlled(comparable to other regions
of the structure), it starts to “see” the errorcorresponding to the
region around the right corners (see figures 14 and 15).
As in the previous case, S21 = S12 and S22 = S11. The results
for S11 and S21are shown (for some of the hp meshes) in Tab. 3.
Only the results of the firstiterations are shown as the hp FEM
results for the consecutive meshes arepresumed to be more accurate
than those of MM. Observations analogous tothose mentioned in the
previous case may be made here.
6 The term iris is used in this context to refer to an aperture
that connects twowaveguide sections.
20
-
y
x
a
x
H-plane
Port 2
y
ζPort 1
ξ
η
ξζ
η
Ω
Γ2pΓ1p
H-planeζ
ξζ
ξ
ΓN
a
b
l1l
l2
t
ε0, µ0
ε0, µ0
l
ΓN
l1 t l2
Figure 10. H-plane symmetric inductive iris (l/a = 0.6, t/a =
0.2)
21
-
x
y
z
Figure 11. Initial mesh for the H-plane symmetric inductive
iris
-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 12. Convergence history for the H-plane inductive
iris
22
-
Figure 13. Magnitude of Hy for the H-plane inductive iris
x
y
z
Figure 14. 11th mesh for the H-plane symmetric inductive iris
showing the refine-ments around the left corners
23
-
x
y
z
Figure 15. 19th mesh for the H-plane symmetric inductive iris
showing the refine-ments around the left and also right corners
Table 3Scattering 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◦
24
-
3.1.4 H-plane zero thickness septum
The structure, shown in Fig. 16, consists of an obstacle (the
septum) placed atthe center of the waveguide. The obstacle exhibits
a translational symmetryalong the narrow dimension of the waveguide
(η local axis of the waveguideports). Thus, it can be analyzed
using the H-plane formulation. Also, as forthe previous structure,
the scattering behavior is basically inductive since theonly field
lines that are cut by the septum are those of the magnetic
field.
The initial mesh used for the analysis is shown in Fig. 17. The
analysis is madeby exciting port 1 (on the left). The convergence
history is shown in Fig. 18.It is observed that the error
convergence (except for the first couple of meshes,due to a
deliberate coarseness of the initial mesh) behaves as predicted by
thetheory and a straight line is obtained in the plot (which means
exponentialconvergence).
An example of one of the hp meshes provided by the adaptivity,
specifically,the mesh of the 7th iteration, is shown in Fig. 19.
Again, as predicted by thetheory, it is observed the h-refinements
towards the corners where there is asingular behavior of the field
and the p-refinements in the regions where thefield variation is
smooth.
A plot of |Hy| is shown in Fig. 20. The y-component corresponds,
as in theprevious case, 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 the singular
behavior of the field at the septumcorners.
The results for S11 and S21 corresponding to some of the
iterations of thehp adaptivity are shown in Tab. 4. As in the
previous cases, S21 = S12 andS22 = S11 due to reciprocity and
symmetry. Only the results up to the 10thiteration are shown, since
the error in the scattering parameters obtained fromthe hp meshes
(for iterations higher than the 10th) is expected to be lowerthan
the one of the MM results.
25
-
y
x
a
x
H-plane
Port 2
y
ζPort 1
ξ
η
ξζ
η
Ω
Γ2pΓ1p
H-planeζ
ξζ
ξ
ΓN
a
b
l1
l2
ε0, µ0
ΓN
l1 l2
lε0, µ0
l
ΓN
Figure 16. H-plane zero thickness septum (l/a = 0.1)
26
-
x
y
z
Figure 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 18. Convergence history for the H-plane zero thickness
septum
27
-
x
y
z
Figure 19. 7th mesh for the zero thickness septum
Figure 20. Magnitude of Hy corresponding to the H-plane zero
thickness septumshowing a stationary wave pattern at the input
waveguide and singular behavior ofthe field at the septum
corners
28
-
Table 4Scattering 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◦
29
-
3.1.5 H-plane zero length septum
This discontinuity is also a septum but it is placed transverse
to the wavepropagation. The dimension of the septum along the
propagation direction isconsidered zero. A representation of the
structure is shown in Fig. 21.
The analysis is made by exciting port 1 (on the left) and
considering thecoarse initial mesh of Fig. 22. As in the other
septum discontinuity, the errorconvergence (see Fig. 23) behaves as
predicted by the theory (exponentialconvergence) and a straight
line is obtained in the plot. Fig. 24 shows, as anexample, the mesh
corresponding to the 7th iteration where the refinementaround the
septum corners can be seen. This is what is expected from the
fieldsolution. The y component of the solution (its magnitude) is
shown in Fig. 25.Again, a stationary wave pattern is 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. 5. Only
results up to the 11th iterationare shown, since the error in the
scattering parameters obtained from the hpmeshes is expected to be
lower (for iterations higher than the 11th) than theone of the MM
results.
Table 5Scattering 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◦
30
-
y
x
a
a
x
H-plane
Port 2
y
ζPort 1
ξ
η
ξζ
η
Ω
Γ2pΓ1p
H-planeζ
ξζ
ξ
ΓN
b
l1
l2
ΓN
l1 l2
ε0, µ0
ε0, µ0
ΓN
t
t
Figure 21. H-plane zero length septum (t/a = 0.1)
31
-
x
y
z
Figure 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 23. Convergence history for the H-plane zero length
septum
32
-
x
y
z
Figure 24. 7th mesh for the zero length septum
Figure 25. Magnitude of Hy corresponding to the H-plane zero
length septum show-ing a stationary wave pattern at the input
waveguide and singular behavior of thefield at the septum
corners
33
-
3.2 E-plane discontinuities
The analysis of several E-plane discontinuities is considered
next. In contrastto the H-plane formulation, the boundary condition
of the metallic conductorsis of the Dirichlet type for the E-plane
formulation.
3.2.1 E-plane right angle bend
This discontinuity (Fig. 26) is as the one of Section 3.1.2, a
90 degrees bend.However, the plane of the bend in this case is the
E-plane. The domain shapeis the same as for the H-plane bend
(actually, the initial mesh is also the same,see Fig. 6—) but, this
time, the homogeneous Dirichlet boundary conditionat the conductors
are employed. This, apparently, produces a field singular-ity that
occurs at the corner. The hp-strategy behaves as expected and,
incontrast to the H-plane bend case, an h-refinement toward the
singularity isobserved while increasing the p backward. One of the
meshes obtained by thehp-adaptivity procedure is shown in Fig.
27.
Plots of the field component magnitudes |Ey| and |Ex| are shown
in Fig. 28. Astationary wave pattern is observed in the input
waveguide (between the ex-citation port and the bend). The
y-component corresponds to the componentalong the local −η axis of
the excitation port and the component along theζ local axis of the
transmitted port. Since the TE10 does not have ζ compo-nent, the Ey
component is null (numerically null provided that the port is
farenough from the discontinuity) at the transmitted port (port 2).
Analogously,the Ex component is null at the incident port.
The convergence history is shown in Fig. 29. Except for the peak
aroundthe third hp iteration (due to the coarseness of the initial
mesh), the errorshows an exponential decay. With respect to the
convergence of the scatteringparameters, Tab. 6 shows their values
(in magnitude and phase) for some ofthe iterations. The MM results
are shown for comparison purposes. A goodagreement is observed. It
is worth noting again that only the results up to the10th iteration
are shown, since the error in the scattering parameters
obtainedfrom the hp meshes (for iterations higher than the 10th) is
lower than the onecoming from the MM results.
34
-
y
x
xy
ηζ
ξ
ε0, µ0Ω
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 26. E-plane 90 degrees bend
35
-
x
y
z
Figure 27. hp mesh of 11th iteration of the E-plane 90◦ bend
Table 6Scattering 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◦
36
-
(a) |Ey|
(b) |Ex|
Figure 28. Magnitudes of Ey and Ex corresponding to the E-plane
90◦ bend showinga singular behavior of the field at the corner
37
-
-8
-7
-6
-5
-4
-3
-2
-1
4 5 6 7 8 9 10 11 12 13 14 15
log
(rel
ativ
e er
ror
%)
Ndof1/3
E-plane 90 degrees Bend
energy-error
Figure 29. Convergence history for the E-plane 90◦ bend
38
-
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(see Fig. 30). In practice, depending on the mechanical
process used to buildthe bends, the bends may either have sharp
corners or round corners (as inthis case). Although there is no
field singularity because of the roundnessof the corner, there is a
high variation of the fields around the corner, andthe adaptivity
behaves analogously to the case of a sharp corner (in the
pre-asymptotic regime, as a high variation in 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. 31. Fig. 32 shows a
sample mesh corresponding to the 10thiteration. A refinement
pattern similar to the one of Fig. 27 can be observed.The
exponential convergence history is shown in Fig. 34.
Plots of the field component magnitudes |Ey| and |Ex| are shown
in Fig. 33.Comments analogous to those made on the E-plane bend
with a sharp arevalid for this case as well. Tab. 7 shows the
values (in magnitude and phase)of S11 and S21 for the first few
iterations. A fast convergence is observed andseven digits are
needed in order to be able to observe the convergence of
thescattering parameters. No MM results are shown for this case.
The analysisof this structure by MM requires the use of special
functions and somehowdiffers of what it is usually referred to as
the MM method.
Table 7Scattering 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◦
39
-
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 30. E-plane 90 degrees bend with round corner (r/b =
0.2)
40
-
x
y
z
Figure 31. Initial mesh for the E-plane 90◦ bend with round
corner
x
y
z
Figure 32. 10th hp mesh for the E-plane 90◦ bend with round
corner
41
-
(a) |Ey|
(b) |Ex|
Figure 33. Magnitudes of Ey and Ex corresponding to the E-plane
90◦
42
-
-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 34. Convergence history for the E-plane 90◦ bend with
round corner
43
-
3.2.3 E-plane capacitive symmetric iris
This discontinuity (Fig. 35) is due to a symmetric iris (as the
one of Sec-tion 3.1.3), but in the E-plane. Thus, the FEM domain is
identical to the oneused for the H-plane inductive symmetric iris,
but with different boundaryconditions on the conductors boundaries
(of Dirichlet type for this case). Theanalysis is made by exciting
port 1 (on the left).
The initial mesh is shown in Fig. 36(a). Fig. 36(b) shows a
sample meshcorresponding to the 4th iteration. A refinement pattern
around the corners ofthe iris is observed due to the presence of
field singularities at those locations.The convergence history (up
to an error as low as 0.1%) is shown in Fig. 37.Exponential
convergence is again observed.
Plots of the field component magnitudes |Ey| and |Ex| are shown
in Fig. 38.A stationary wave pattern is observed in the input
waveguide (between theexcitation port and the iris) as well as a
singular behavior of the field at thecorners of the iris. The
y-component of the field in the structure correspondsto the ∓η
component of the waveguide modes. Analogously, the
x-componentcorresponds to the ±ζ component of the waveguide. Thus,
the Ex componentof the field is generated at the discontinuity, and
it is only significant close toit.
The results for S11 and S21 corresponding to some of the
iterations of the hpadaptivity are shown in Tab. 8. The equalities
S21 = S12 and S22 = S11 holddue to reciprocity and symmetry of the
structure. Only the results of the firstiterations are shown as the
hp FEM results for the consecutive meshes arepresumed to be more
accurate than those of MM.
Table 8Scattering 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◦
44
-
y
x
Ω
Γ2pΓ1p
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 35. E-plane capacitive symmetric iris (d/b = 0.6, t/b =
0.2)
45
-
48 49
52 53
54 55
62 63
60
61
56 57
58 59
64 65
50 51
x
y
z
(a) Initial mesh
x
y
z
(b) Mesh of 4th iteration
Figure 36. Initial mesh and mesh of the 4th iteration for the
E-plane capacitivesymmetric iris
46
-
-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 37. Convergence history for the E-plane capacitive
symmetric iris
47
-
(a) |Ey|
(b) |Ex|
Figure 38. Magnitude of Ey and Ex corresponding to the E-plane
capacitive sym-metric iris
48
-
3.2.4 E-plane double stub section
The structure is shown in Fig. 39. It consists of a main
waveguide going fromport 1 to port 2 and two waveguides loading it
(referred to as stubs). Thestubs are closed at their ends causing a
total reflection of the energy at theirinputs. Thus, the load
impedance that they present to the main waveguideis purely
imaginary. The behavior of this structure may be roughly
explainedas follows. The load of each stub is like a discontinuity
in the waveguide,producing a reflected wave (and also a transmitted
wave) with a given phase.As there are two stubs, i.e., two
discontinuities, the contributions from thetwo stubs may (totally
or partially) add or cancel, depending on the relativephase between
the corresponding waves. The relative phase depends (for givenstubs
dimensions) on the electrical distance θd between them. As the
electricaldistance depends on the frequency for a given physical
distance d, i.e., θd =β10d, the frequency response can be adjusted
for several applications. Forexample, the double stub section can
be designed to work as a phase shifter,i.e., causing an extra shift
in the phase of the wave at the transmitted port for agiven
frequency band. This is done by designing the double stub in such a
waythat there is an adding interference at the transmitted port of
the two wavesgenerated by the stubs (with a given phase). Another
usual application is animpedance matching network, i.e., the double
stub is designed to compensatethe reflection present at a given
port, due to, e.g., a change in the height of thewaveguide, in a
given frequency band. This is done by adjusting the design sothere
is a cancellation of the two reflected 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 thecut-off frequency of the TE10
mode). Notice in Fig. 39 that the waveguidesections at the two
ports are identical. Thus, if the two stubs were not presentthere
would be no reflection (S11 ideally null) as the structure would
simplyconsists of a waveguide section. The reason we choose to
analyze the structurein Fig. 39 (which has a little practical
application) is because this is a goodtest case. The idea is that
for the null reflection frequency, the fields in themain waveguide
have to be basically the same as those in a single
waveguidesection. 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 the left).
The frequency response of the structure around the frequency
correspondingto k = 1.39Kc is shown in Fig. 40(a). The ordinate
axis corresponds to S11in dB, i.e., 10 log10 |S11|2 7 . The results
have been obtained by running the
7 The dB is a logarithmic unit for dimensionless magnitudes, but
it is always withrespect to the power (and not the field)
magnitudes. Thus, when applied to thescattering parameters that
relate field quantities, the “10 log10” factor has to beapplied to
the square of the scattering coefficient. For example, S11 = −40dB
means
49
-
hp-adaptivity until an energy error of 1% is achieved and
computing the S11using the final hp-mesh. The expected low values
of the reflection coefficientS11 around k0/kc = 1.39 is observed.
For a comparison, Fig. 40 presents alsoresults obtained using a MM
technique. A very 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. Forthese frequencies, a
more refined mesh seems to be needed.
The convergence has been studied by running the hp-adaptivity
with anenergy-norm error tolerance of 0.01% (and a maximum number
of iterationsequal to 20). Results corresponding to five
significant frequency points (sym-metrically chosen around the
value of k0/kc = 1.39): k0/kc=1.32, 1.36, 1.42,1.46 are displayed
in Fig. 41. For k0/kc = 1.32, 1.46 there is a high reflection ofthe
energy at the input waveguide; for k0/kc = 1.39 there is a very low
(almostnull) reflection at the input; and for k0/kc = 1.36, 1.42 an
intermediate situa-tion occurs. Except for the first few
iterations, the plots follow approximatelya straight line,
reflecting an exponential decrease of the energy-norm error.The
erratic behavior of the error during the first few iterations is
due to thecoarseness of the initial mesh (shown in Fig. 42(a)).
The final meshes for k0/kc = 1.32 (high reflection at the
input), k0/kc = 1.39(low reflection at the input), and k0/kc = 1.36
(intermediate reflection at theinput) are displayed in Figures 42
and 43. For the case of high reflection, themesh (shown in Fig.
42(b)) displays a typical refinement pattern around thecorners
(junctions of the stubs with the main waveguide). The electric
field 8
for this case is plotted in Fig. 44(a). A stationary wave
pattern in the inputwaveguide is observed due to the interference
between the incident and thereflected waves. On the other hand, the
mesh for the low reflection case (shownin Fig. 43(b)) displays a
situation very similar to the situation of the smoothfield solution
inside a waveguide section (i.e., without singularities). As it
wasexplained above, this occurs because the stubs do not load the
main waveguide(it is like if they were not present in the
structure). This is best understood byseeing Fig. 44(b), which
displays the electric field in the structure for this case.The
stationary wave pattern at the input waveguide can hardly be seen,
whichmeans that the level of the reflected wave is very low.
Finally, the mesh forthe intermediate case (k0/kc = 1.36) is shown
in Fig. 43(a). It is observed how
that the power reflected at the input waveguide is 40 dB below
the power of theexcitation at the input waveguide, i.e., 104 lower
(or equivalently, the electric andmagnetic fields are 102 lower).8
Note that the magnitude plotted is
√|Ex|2 + |Ey|2 that, although does not cor-
respond to |E| or other physically meaningful magnitude, it is
useful for visualizingin one plot the field in the main waveguide
and in the stubs.
50
-
effectively the mesh corresponds to an intermediate case between
the meshesof 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 39. E-plane double stub structure (bs1/b = bs2/b =
5.0249, l/b = 1.2608)
51
-
-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 40. Frequency response of E-plane double stub section
(|S11| in dB)
52
-
-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 41. Convergence history for the E-plane double stub
section
53
-
59 60 61 62
63
64
65
66
67 68 69
70
71
72
73
74 75 76 77
x
y
z
(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
x
y
z
(b) Mesh for K0/Kc = 1.32
Figure 42. Initial mesh and mesh corresponding to 1% energy
error for the E-planedouble stub section
54
-
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
x
y
z
(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
x
y
z
(b) Mesh for K0/Kc = 1.39
Figure 43. Meshes corresponding to 1% energy error for the
E-plane double stubsection
55
-
(a) High reflection at the input waveguide (K0/Kc = 1.32)
(b) Low reflection at the input waveguide (K0/Kc = 1.39)
Figure 44. Electric field√|Ex|2 + |Ey|2 in the E-plane double
stub section
56
-
4 Conclusions
An hp-adaptive Finite Element Method for studying the
characterization ofmicrowave rectangular waveguide discontinuities
with a geometry invariantalong one direction (a common situation in
rectangular waveguide technology),has been presented. The
assumption on the geometry of the discontinuityenables a 2D
analysis in so called H-plane or E-plane of the structure.
A fully automatic hp-adaptive strategy based on maximizing the
rate of de-crease of the (projection-based) interpolation error of
the fine grid solution,has been applied to a number of important
engineering examples. Computa-tion of the scattering matrix that
characterize the electromagnetic behaviorof the discontinuities for
the microwave engineer has been implemented as apost-processing of
the solution.
A wide variety of structures have been analyzed, including
microwave engi-neering devices of medium complexity. The hp
adaptivity has shown to deliverexponential convergence rates for
the error for both regular and singular solu-tions. A consistent
convergence pattern makes us believe that the results aremore
accurate than those obtained with semi-analytical techniques. At
thesame time, this hp-methodology presents the important advantage
of beinga purely numerical method, which allows for modeling of
complex waveguidestructures that cannot be solved by using
semi-analytical techniques.
5 Acknowledgment
The authors would like to thank Sergio Llorente-Romano at the
UniversidadPolitécnica de Madrid for their helpful discussions on
the MM techniques andfor letting us use their MM codes that have
been used to produce some of theMM results shown in this paper.
57
-
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