DESIGN OF DIELECTRIC RESONATOR ANTENNAS USING
SURROGATE-BASED OPTIMIZATION AND
ELECTROMAGNETIC MODELS
Slawomir Koziel, Stanislav Ogurtsov and Leifur Leifsson
Engineering Optimization & Modeling Center, School of Science and Engineering
Reykjavik University, 101 Reykjavik, Iceland
Keywords: Dielectric resonator antennas, Microwave design, Design optimization, Surrogate models.
Abstract: Design of dielectric resonator antennas (DRAs) is a challenging task because their analytical models are
only appropriate for estimation, i.e., to calculate the resonance frequency and radiation quality factor of an
isolated dielectric resonator or for obtaining an initial design. In practice, the geometry parameters that
ensure satisfaction of performance requirements are often obtained by repetitive electromagnetic (EM)
simulations guided by engineering experience. This is a tedious process, and it does not guarantee optimal
results. On the other hand, employing the EM solver directly in the optimization loop is typically
impractical because high-fidelity EM simulations are computationally expensive. Here, we describe several
techniques that allow designing DRAs in a computationally efficient way. All presented methods exploit
coarse-discretization EM models of the DRA. These models, after correction, serve as prediction tools that
guide the optimization process. As the low-fidelity models are computationally much cheaper than the
original, high-fidelity ones, the cost of the design process is greatly reduced. The approaches presented here
include adaptively adjusted design specifications, shape-preserving response prediction, and space mapping
with kriging-based coarse models. Antenna design examples are provided.
1 INTRODUCTION
Dielectric resonator antennas (DRAs) possess a
number of features which make them attractive for
engineers and designers (Kishk et al., 2007; Petosa,
2007): a wide frequency range of operation (1-to-44
GHz); compact size compared to their counterparts like
microstrip antennas; different radiation patterns for
various requirements; variety of feeding schemes (such
as probes, slots, microstrip, coplanar waveguide,
dielectric image guide); wider impendence bandwidth
compared with microstrip antennas and attributed to the
DRA radiation mechanism; wide temperature range of
operation; high power handling capability.
DRA design involves adjustment of geometry in
order to satisfy application specific requirements. In
most cases, available analytical models (Kishk et al.,
2007; Petosa, 2007) can only be used to estimate the
resonant frequency and radiation quality factor of the
isolated dielectric resonator. For accurate DRA
responses full-wave electromagnetic (EM) simulation
is necessary to account for the environment (e.g.,
installation platform, housing, feeding circuit).
Therefore, EM-simulation-based optimization seems to
be the only reliable option for DRA design. However,
the bottleneck is high computational cost: high-fidelity
simulation may take up to a few hours even for a single
set of design variables. As a result, approaches based
on the direct use of the EM solver in an optimization
loop are impractical. In practice, search for optimal
DRA dimensions is typically realized as a simulation-
based parametric study, e.g., (De Young et al., 2006;
Guo et al., 2005; Ong et al., 2002), or measurement of
multiple prototypes, e.g., (Petosa, 2007); unfortunately,
both approaches are tedious and do not guarantee
optimality of the final design.
Efficient simulation-driven design can be realized
using surrogate-based optimization (SBO) (Queipo
et al., 2005; Forrester et al., 2009). In SBO the
computational burden is shifted to a surrogate
model, a computationally cheap representation of the
optimized structure. SBO approaches, shown to be
successful in microwave area, include space
mapping (SM) (Bandler et al., 2004; Amari et al.,
2006; Koziel et al., 2006; Koziel et al., 2008), tuning
439
Koziel S., Ogurtsov S. and Leifsson L..
DESIGN OF DIELECTRIC RESONATOR ANTENNAS USING SURROGATE-BASED OPTIMIZATION AND ELECTROMAGNETIC MODELS.
DOI: 10.5220/0003571404390448
In Proceedings of 1st International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SDDOM-2011), pages
439-448
ISBN: 978-989-8425-78-2
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
(Rautio, 2007) and tuning SM (Koziel et al., 2009).
Unfortunately, applicability of these techniques for
antennas is limited. SM normally relies on a fast
coarse model, typically, circuit equivalent (Bandler
et al., 2004). Regrettably, reliable circuit equivalents
are not available for DRAs due to the underlying
EM phenomena. On the other hand, simulation-
based tuning is not directly applicable for radiating
structures.
Recently, there has been tendency to use meta-
heuristic approaches (Yang, 2010) for antenna design,
e.g., genetic algorithms (Haupt, 2008), particle swarm
optimizers (Lizzi et al., 2006), and ant colony
optimization (Rocca et al., 2008). While these
techniques alleviate some optimization problems, e.g.,
handling multiple local optima, they normally require
massive amounts of objective function calls.
Therefore, meta-heuristic approaches are not well
suited for the DRA design purposes.
Here, we discuss three simulation-driven
optimization methodologies that are suitable for DRA
design. All of them exploit coarse-discretization EM
simulations as the low-fidelity models. Such models
are not as accurate as the original, high-fidelity
simulations, but they are computationally much
cheaper. After suitable correction, these low-fidelity
models can be used in place of high-fidelity ones in
the optimization process. We focus on three design
optimization approaches that are straightforward to
implement, and yet computationally efficient.
The first technique, shape-preserving response
prediction (SPRP) (Koziel, 2010b), creates a reliable
surrogate of the DRA by aligning the simulation
results of its low-fidelity model with that of its high-
fidelity model. The surrogate serves as a predictor
estimating the optimal geometry. The second
technique does not apply any corrections to the
coarse-discretization models directly. Instead, the
discrepancy between the low- and high-fidelity
models is accounted for by modification of design
specifications (Koziel, 2010a). The last methodology
exploits SM as the optimization engine with the
underlying coarse model created by kriging
interpolation of the coarse-discretization simulation
data (Koziel, 2009b).
Examples are provided for the described
techniques. In all cases, the final design is obtained
at a low computational cost corresponding to a few
high-fidelity EM simulations of the antenna under
consideration.
2 DRA DESIGN USING
SURROGATE MODELS
2.1 Design Problem Formulation
For the sake of this work, the DRA design is
formulated as a nonlinear minimization problem of
the form
(
)
*
arg min ( )
f
U=
f
x
xRx
(1)
where R
f
(x) R
m
is the response vector of a high-
fidelity (fine) model of the antenna of interest; U is a
given objective function (e.g., typically minimax
(Bandler et al., 2004)), whereas x R
n
is a vector of
design variables, typically, the geometry parameters.
The most common objective in the antenna design is to
minimize so-called reflection coefficient |S
11
| over
certain frequency band of interest. Other objectives
may concern the antenna gain or the shape of the
radiation pattern. It is assumed that the computational
cost of evaluating the high-fidelity model is high so that
solving (1) directly is impractical.
2.2 Surrogate-based Optimization.
Low-fidelity Models
The SBO techniques exploiting physics-based low-
fidelity models can be particularly efficient (Bandler
et at., 2004). As mentioned before, space mapping
(Koziel et al., 2008) and simulation-based tuning
(Rautio, 2007) are both highly efficient approaches,
however, their applicability is limited to devices
where fast circuit equivalents are readily available,
e.g., in the case of microstrip filters (Bandler et al.,
2004; Amari et al., 2006).
Here, we exploit the models obtained through
coarse-discretization of the original structure
(referred to as R
c
). The major advantage is that such
models are available for any DRA. Moreover,
coarse-discretization models can be implemented
with the same EM solver, as that of the
corresponding high-fidelity models, by applying
relaxed mesh requirements. The use of the same
solver simplifies implementation of the optimization
algorithm. On the other hand, coarse-discretization
EM models are still relatively expensive when
compared to circuit equivalents. For that reason, we
look for optimization techniques that are capable to
reduce not only the number of evaluations of the
high-fidelity model, but also the number of coarse-
discretization simulations so that the computational
overhead related to low-fidelity model evaluations
does not affect the total design cost significantly. In
SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and
Applications
440
particular, coarse-discretization EM models are
normally too expensive to serve as immediate coarse
models for efficient SM implementation. This would
be particularly problematic for the parameter
extraction step of SM which requires a substantial
number of model evaluations (Koziel et al., 2006).
In the remaining part of this section we discuss a
few optimization approaches utilizing coarse-
discretization EM models that are suitable for DRA
design. These methods are simple to implement and
computationally efficient. Application examples are
provided in Section 3.
2.3 Shape-Preserving Response
Prediction
The shape-preserving response prediction (SPRP)
method (Koziel, 2010b) exploits the iterative process
()
(1) ()
arg min ( )
ii
s
U
+
=
x
xRx
(2)
where R
s
(i)
is the surrogate model at iteration i,
whereas x
(i+1)
is the approximate solution to (1)
obtained by optimizing R
s
(i)
. In SPRP, the surrogate
is constructed under the assumption that the change
of the high-fidelity model response due to the
adjustment of the design variables can be predicted
using the actual changes of the low-fidelity model
response. This property is ensured by the low-
fidelity model being the coarse-mesh simulation of
the same DRA structure that represents the high-
fidelity one.
The change of the low-fidelity model response can
be described by the translation vectors corresponding
to so-called characteristic points of the model’s
response. These translation vectors are subsequently
used to predict the change of the high-fidelity model
response with the actual response of R
f
at the current
iteration point, R
f
(x
(i)
), treated as a reference.
Figure 1(a) shows the example low-fidelity
model response, |S
11
| versus frequency, at the design
x
(i)
, as well as the coarse model response at some
other design x. The responses come from the
dielectric resonator antenna considered in
Section 3.1. Circles denote characteristic points of
R
c
(x
(i)
), selected here to represent |S
11
| = –10 dB, |S
11
|
= –15 dB, and the local |S
11
| minimum. Squares
denote corresponding characteristic points for R
c
(x),
while line segments represent the translation vectors
(“shift”) of the characteristic points of R
c
when
changing the design variables from x
(i)
to x.
The high-fidelity model response at x can be
predicted using the same translation vectors applied
to the corresponding characteristic points of the
high-fidelity model response at x
(i)
, R
f
(x
(i)
). This is
illustrated in Fig. 1(b). Figure 1(c) shows the
predicted versus actual high-fidelity model response
at x. Rigorous formulation of SPRP can be found in
(Koziel, 2010b). It is omitted here for the sake of
brevity.
(a)
(b)
(c)
Figure 1: SPRP concept: (a) Low-fidelity model response
at the design x
(i)
, R
c
(x
(i)
) (—), the low-fidelity model
response at x, R
c
(x) (⋅⋅⋅⋅), characteristic points of R
c
(x
(i)
)
(o) and R
c
(x) (), and the translation vectors (); (b)
High-fidelity model response at x
(i)
, R
f
(x
(i)
) (—) and the
predicted high-fidelity model response at x (⋅⋅⋅⋅) obtained
using SPRP based on characteristic points of (a);
characteristic points of R
f
(x
(i)
) (o) and the translation
vectors () were used to find the characteristic points ()
of the predicted high-fidelity model response; (c) low-
fidelity model responses R
c
(x
(i)
) and R
c
(x) are plotted
using thin solid and dotted line, respectively.
2.4 Adaptively Adjusted Design
Specifications
It is not necessary to remove the discrepancies
5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9
-20
-15
-10
-5
Frequency [GHz]
|
S
11
| [dB]
5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9
-20
-15
-10
-5
Frequency [GHz]
|
S
11
| [dB]
5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9
-20
-15
-10
-5
Frequency [GHz]
|
S
11
| [dB]
DESIGN OF DIELECTRIC RESONATOR ANTENNAS USING SURROGATE-BASED OPTIMIZATION AND
ELECTROMAGNETIC MODELS
441
between the low- and high-fidelity models by
correcting the low-fidelity model. Another way is to
“absorb” the model misalignment by proper
adjustment of the design specifications. In
microwave engineering, most of the design tasks can
be formulated as minimax problems with upper and
lower specifications and it is easy to implement
modifications by, for example, shifting the
specification levels, corresponding frequency bands
(Koziel, 2010a).
The optimization procedure exploiting this idea
consists of the following two simple steps:
1. Modify the original design specifications to account
for the discrepancy between the low- and high-
fidelity models.
2. Obtain a new design by optimizing the low-
fidelity model with respect to the modified
specifications.
In Step 1, the design specifications are modified so
that the level of satisfying/violating the modified
specifications by the low-fidelity model response
corresponds to the satisfaction/violation levels of the
original specifications by the high-fidelity model
(Koziel, 2010a). The low-fidelity model is then
optimized in Step 2 with respect to the modified
specifications and the new design obtained this way
is treated as an approximated solution to the original
design problem, i.e., optimization of the high-
fidelity model with respect to the original
specifications. Because the low-fidelity model is
physics-based, the adjustment of the design variables
has similar effect on the response for both the low-
and high-fidelity models. As a result, the low-
fidelity model design obtained in Step 2 (i.e.,
optimal with respect to the modified specifications)
will be (almost) optimal for the high-fidelity model
with respect to the original specifications.
Steps 1 and 2 can be repeated if necessary.
Typically, a substantial improvement is observed after
the first iteration. Additional iterations may bring
further enhancement as the discrepancy between the
high- and low-fidelity models may change from one
design to another.
Figure 2 illustrates an iteration of our technique used
for design of a CBCPW-to-SIW transition (Deslandes
and Wu, 2005). One can observe that the absolute
matching between the low- and high-fidelity models is
not as important as the shape similarity.
2.5 Optimization using Space Mapping
and Kriging-based Coarse Models
Similarly as SPRP, space mapping (SM) (Bandler et
al., 2004) solves the original design problem (1)
(a)
(b)
(c)
(d)
Figure 2: Adaptively adjusted design specification technique
applied to optimize CBCPW-to-SIW transitions. High- and
low-fidelity model response denoted as solid and dashed
lines, respectively. |S
22
| distinguished from |S
11
| using circles.
Design specifications denoted by thick horizontal lines. (a)
High- and low-fidelity model responses at the beginning of
the iteration as well as original design specifications; (b)
High- and low-fidelity model responses and modified design
specifications that reflect the differences between the
responses; (c) Low-fidelity model optimized to meet the
modified specifications; (d) high-fidelity model at the low-
fidelity model optimum shown versus original specifications.
Thick horizontal lines indicate the design specifications.
using an iterative procedure (2) (cf. Section 2.3). SM
surrogate is also constructed from the low-fidelity
model R
c
by applying suitable transformation of the
model parameter space and/or response. A variety of
SM surrogate models are available (Koziel et al.,
2006). A specific model used in this work is defined
as R
s
(i)
(x) = R
c
(x + c
(i)
) + d
(i)
. The vector c
(i)
is
obtained in the parameter extraction process c
(i)
=
argmin{c : ||R
f
(x
(i)
) – R
c
(x
(i)
+ c)||} which aims at
reducing misalignment between the high- and SM-
mapped low-fidelity model responses at x
(i)
. The
vector d
(i)
is calculated as d
(i)
= R
f
(x) – R
c
(x + c
(i)
).
The parameter shift x + c
(i)
is referred to as input
6 7 8 9 10 11 12
-30
-20
-10
0
Frequency [GHz]
|S
11
|, |S
22
|
6 7 8 9 10 11 12
-30
-20
-10
0
Frequency [GHz]
|S
11
|, |S
22
|
6 7 8 9 10 11 12
-30
-20
-10
0
Frequency [GHz]
|S
11
|, |S
22
|
6 7 8 9 10 11 12
-30
-20
-10
0
Frequency [GHz]
|S
11
|, |S
22
|
SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and
Applications
442
SM, while the response correction through the
vector d
(i)
is called output SM (Bandler et al., 2004).
Space mapping is a flexible and general surrogate-
based optimization methodology (Bandler et al., 2004),
however, its application to DRA design may not be
straightforward because SM requires a physically-
based and yet computationally cheap low-fidelity
model (preferably, an equivalent circuit). The use of
coarse-discretization EM models as the low-fidelity
model may be problematic because the SM algorithm
typically requires a large number of low-fidelity model
evaluations, particularly in the parameter extraction
step (Koziel et al., 2008). As EM simulations, even
low-fidelity ones, are relatively expensive (typically,
only 10 to 50 times faster than the high-fidelity
models), the efficiency of the SM algorithm could be
compromised because of numerous evaluations of the
coarse-discretization model.
The workaround is to build a (local) function
approximation model using coarse-discretization model
data, and treat it as a low-fidelity model for the space
mapping algorithm (Koziel, 2009b). This approach has
several advantages: (i) R
c
model is computationally
cheap, smooth, and therefore, easy to optimize, (ii)
there is no need for circuit-equivalent model, and,
consequently, no extra simulation software needs to be
involved; the space mapping algorithm implementation
is simpler and exploits a single EM solver, (iii) it is
possible to apply SM for DRA design problems where
finding reliable and fast low-fidelity models is difficult
or impossible. Here, we use kriging (Queipo et al.,
2005) as a function approximation technique.
The design procedure is the following:
1. Starting from x
init
, find an approximate optimal
design x
(0)
of the coarse-discretization model R
cd
.
In this work, we use a pattern search algorithm
(Kolda et al., 2003).
2. Sample R
cd
in the vicinity of x
(0)
and construct a
response surface approximation model R
c
.
3. Find a high-fidelity model optimum by applying
the algorithm (2) with the surrogate model created
using R
c
as an underlying low-fidelity model.
The surrogate constructed by means of coarse-
discretization model data and the SM alignment is a
good prediction tool. It allows us to locate the high-
fidelity model optimum in a few iterations (each
iteration amounts to just one evaluation of the high-
fidelity model) so that the entire design process is
computationally inexpensive.
3 DESIGN EXAMPLES
In this section, we present several DRA design
examples exploiting the methodologies described in
Section 2.
3.1 DRA Design using SPRP
Consider a rectangular DRA at a metal ground
(Petosa, 2007) shown in Fig. 3. The DRA is fed with
a 50 ohm microstrip through a ground plane slot.
The design variables are x =[a
x
a
y
a
z
a
y0
u
s
w
s
y
s
]
T
,
where a
x
, a
y
, and a
z
are dimensions of the dielectric
resonator (DR), a
y0
stands for the shift of the DR
center in Y-direction relative to the slot center, u
s
is
the slot width, w
s
is the slot length, and y
s
is the
length of the microstrip stub. Relative permittivity
and loss tangent of the DR are 10 and 10
–4
,
respectively. Substrate thickness is 0.5 mm. The
width of the microstrip signal trace is 1.17 mm.
Metallization is with 1.5 oz copper.
The design objective for reflection coefficient is
|S
11
|
–10 dB for at least 8% fractional bandwidth
centered at 5.5 GHz (5.28 GHz to 5.72 GHz). The
initial design is x
(0)
= [8.0 14.0 8.0 0.0 1.7 8.4 8.3]
T
mm, and it is obtained for 5.5 GHz with available
design guidelines and data curves of (Petosa, 2007).
However, this initial design does not meet the
specifications (dot and dash lines in Fig. 4).
Requirements to the DRA radiation are the
following: realized gain not less than 3 dB for zero
zenith angle; and, realized gain in directions down
the substrate (back radiation) not greater than 15
dB, all over the frequencies where |S
11
|
meets the
specifications.
In the optimization process, the |S
11
| requirements
are handled directly (through the objective function).
The radiation requirements are treated as constraints
and included into the objective function through the
appropriate penalty terms.
The high-fidelity model R
f
is simulated in 10 min
47 s using the CST MWS transient solver (CST,
2010) (505,250 mesh cells at the initial design). The
low-fidelity model R
cd
is also evaluated using CST
MWS but with a coarser mesh (14,800 mesh cells at
x
(0)
, 24 seconds).
The final design x
(2)
= [8.2 14.2 8.3 0.0 1.8 9.4
7.6]
T
mm is obtained after two iterations of the
SPRP-based optimization with the total cost
corresponding to about seven evaluations of the
high-fidelity model (Table 1). Figure 5 shows the
reflection of R
f
at both the initial and the final
design, as well as the response of R
cd
at x
(0)
.
DESIGN OF DIELECTRIC RESONATOR ANTENNAS USING SURROGATE-BASED OPTIMIZATION AND
ELECTROMAGNETIC MODELS
443
X
YZ
X
h
s
w
0
w
s
u
s
y
s
a
y
a
x
a
z
GND
(a) (b)
Figure 3: Dielectric resonator antenna (Petosa, 2007): (a)
top and (b) side views.
Figure 4: Dielectric resonator antenna: high- (dashed line)
and low-fidelity fidelity model response (dotted line) at
the initial design x
(0)
, and high-fidelity (solid line) model
response at the final design.
Figure 5: Realized gain of the DRA at the final design: for
zenith angle of 0
0
(thick solid line); and back radiation,
zenith angles of 135
0
(positive Y-direction, thin solid line),
180
0
(dash line), and 135
0
(negative Y-direction dash-dot
line). Design constrains are shown with the upper
horizontal line of 3 dB level and the lower line of 15 dB
level.
Table 1: Rectangular DRA design: optimization cost.
Algorithm
Component
Number of
Model
Evaluations
Evaluation Time
Absolute
[min]
Relative
to
R
f
Evaluation
of
R
cd
*
105 × R
cd
42 3.9
Evaluation
of
R
f
#
3 × R
f
32 3.0
Total design
time
N/A 74 6.9
*
Includes optimization of SPRP surrogate (based on R
cd
).
#
Excludes evaluation of R
f
at the initial design.
3.2 Stacked Ring DRA for Two
Installation Scenarios using AADS
Consider an axi-symmetric DRA structure (Shum et
al., 1995) shown in Fig. 6. It comprises: two TM
01δ
ring dielectric resonators with relative permittivity,
ε
r1
, of 36, two supporting Teflon rings, Teflon filling,
finite ground (t
g
= 1mm). Teflon permittivity, ε
r2
,
is
2.08. The DRA is covered by a polycarbonate
(ε
r3
= 2.7) dome. Thickness of the dome shell, d, is 2
mm. Loss tangents are: 10
–4
for the DRs, 410
–4
for
Teflon, and 10
–2
for the dome. Dielectrics are
described using the 1
st
order Debye model;
permittivity and loss tangent values are listed for 6
GHz. The radii of the supporting rings are equal to the
radii of the DR above them. Metal parts are of copper.
The inner conductor of the 50 ohm coax is extended
as a probe (h
0
above the ground), and its diameter is
1.27 mm. Coax filling is Teflon.
Design variables are inner and outer radii of the
DRs, heights of the DRs and the supporting rings, the
probe length, dome height and radius, and radius of
the DRA ground, namely, x = [a
1
a
2
b
1
b
2
h
1
h
2
g
1
g
2
h
0
h
d
r
d
r
g
]
T
. The design objective is |S
11
| –15 dB in
the band of 4-to-6 GHz for the DRA that is to be
installed in two environments, see Fig. 6, one is with
an infinite metal ground plane and the other is only
with the DRA ground (the radius of r
g
).
It should be emphasized that the above design
problem is challenging for the following reasons: (i)
the large number of design variables, 12, (ii) high-
computational cost of simulation, (iii) design for two
installation scenarios at the same time.
(a)
(b)
Figure 6: DRA side views: (a) DRA installed at the infinite
metal ground; (b) the same DRA with its finite ground only.
The feeding cable is shown on (b). Teflon filling is not
shown. The dome and DRA rings are shown transparent.
5 5.2 5.4 5.6 5.8 6
-25
-20
-15
-10
-5
0
Frequency [GHz]
|S
11
|
5 5.2 5.4 5.6 5.8 6
-30
-20
-10
0
10
Frequency [GHz]
[dB]
2r
ε
1r
ε
1r
ε
2r
0
ε
0
ε
3r
ε
0
ε
0
ε
SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and
Applications
444
The last issue not only increases the
computational cost of the design process but also
requires finding a trade-off between optimal designs
for each environment taken separately. Solving this
problem either by using parameter sweeps or by
direct EM-based optimization involving high-
fidelity simulations seems to be hardly feasible.
The EM models of the DRA are defined using
CST MWS software and simulated using the
transient solver (CST, 2010). The low-fidelity
models are much faster than high-fidelity ones (in
our case, about 15 times), however, they are also
less accurate: the discrepancy in |S11| between low-
and high-fidelity models depends on frequency and
can be as large as 5 to 10 dB.
The AADS algorithm comprises the following
steps:
1. Starting from x
init
, find an approximate
optimal design x
(0)
of the coarse-discretization
model R
cd
. Here we use a pattern search.
2. Modify the original design specifications to
take into account the difference between the
responses of the low- and high-fidelity
models. Obtain a new design by optimizing
the low-fidelity model with respect to the
modified specifications.
Design starts from x
in
= [a
1
a
2
b
1
b
2
h
1
h
2
g
1
g
2
h
0
h
d
r
d
r
g
]
T
= [6.9 6.9 1.05 1.05 6.2 6.2 2.0 2.0 6.8 12.0
10 16.5]
T
which is far from meeting the design
requirements (see Fig. 3(a)). At the initial design, the
high-fidelity model with the finite ground has
4,369,634 mesh-cells and that with the infinite
ground has 4,006,017 mesh-cells; their run times are
10,088 s and 8,697 s, respectively. The coarse-
discretization model with the finite ground has
696,135 mesh-cells and that with the infinite ground
has 600,848 mesh-cells; their run times are 684 s
and 577 s, respectively.
The final design is x
*
= [5.9 1.05 7.825 5.9 1.8
7.95 4.75 0.90 7.75 13.50 10.0 18.40]
T
. Figure 7
shows the reflection responses of the DRA at the
optimized design. The far field responses of the final
design at selected frequencies are shown in Fig. 8.
The total design cost corresponds to about 20
high-fidelity model evaluations, which shows that
our optimization procedure is quite efficient taking
into account the number of design variables. It
follows from the responses shown in Fig. 7(b) that it
would be possible to obtain better designs for the
DRA for each installation case considered
separately. Our final design is a compromise
ensuring that the DRA satisfies the design
specifications for both considered scenarios.
(a)
(b)
Figure 7: |S
11
| of the initial, (a), and optimized, (b),
designs: at the finite () and infinite ( ) ground.
Specifications are shown with the thick solid line.
(a) (b)
Figure 8: Gain [dBi] of the final design in the elevation
plane (a) DRA at the infinite ground, and (b) DRA with
the finite ground at 4.5 GHz (), 5.0 GHz ( ), and 5.5
GHz ( · ·).
3.3 Dual Rectangular DRA with a
Substrate Integrated Cavity using
Kriging and SM
Consider a DRA shown in Fig. 9 and 10. It has two
mutually coupled rectangular DRs (Deng et al.,
2004) which are installed at a printed circuit board
(PCB) layer. The layer has the upper and lower
metal grounds, and its dielectric substrate is 2.5 mm
thick RT6010. The relative permittivity and loss
tangent of the DRs are 36 and 10
–4
. The DRs are in
polycarbonate housing (relative permittivity of 2.7
and dielectric loss tangent of 0.01). The housing is
fixed to the board with four bolts. Feeding of the
3.5 4 4.5 5 5.5 6 6.5
-40
-30
-20
-10
0
Frequency [GHz]
|S
11
|, | S
22
| [dB]
3.5 4 4.5 5 5.5 6 6.5
-40
-30
-20
-10
0
Frequency [GHz]
|
S
11
|, |
S
22
| [dB]
-20
-10
0
10
60
120
30
150
0
180
30
150
60
120
90
90
-20
-10
0
10
60
120
30
150
0
180
30
150
60
120
90
90
DESIGN OF DIELECTRIC RESONATOR ANTENNAS USING SURROGATE-BASED OPTIMIZATION AND
ELECTROMAGNETIC MODELS
445
DRA is with a 50 ohm grounded coplanar
waveguide (GCPW). The GCPW is terminated by
two symmetrical slots (width s
1
and length x
1,
see
Fig. 10(a)). Figures 9 and 10(a) also show vias
forming a substrate integrated cavity. The TE
x
δ11
mode is excited in the DRA.
Dimensions of the DRA are to be adjusted for the
following design requirements: input reflection
coefficient, |S
11
|, should be better than –20 dB, and
gain is to be higher than 3dBi for
θ
= 0
0
(Z-
direction), both over the 2.4-to-2.5 GHz band.
There are eleven design variables: x = [x
0
y
0
x
d
y
d
z
d
s
1
x
1
x
v
y
v
s
x
s
y
]
T
, where x
0
and y
0
are location of
the center of one DR relative to the origin of the
coordinate system marked by O in Fig. 10; x
d
, y
d
,
and z
d
are dimensions of the DRs (ceramic body); s
1
and x
1
are dimensions of the slots energizing the
DRs; x
v
, y
v
, s
x
, and s
y
describe via locations and in
row spacing as shown in Fig. 10(a). The substrate
integrated cavity is defined with ten vias in the lower
(horizontal) row, eleven vias in the upper
(horizontal) row, and nine vias in the vertical rows,
see Fig. 10(a). Other dimensions are fixed as
follows. Dimensions of the GCPW are signal trace
width, w
0
, of 1.5 mm and spacing, s
0
, of 1mm.
Diameter of the vias, d
v
, is 1.5 mm.
Thicknesses of the polycarbonate housing, x
h
, y
h
,
and z
h
, are 2 mm. Location of the mounting bolts are
described by x
h
= s
x
and y
h
= 1 mm. The heads of the
bolts are 4 mm in diameters and 1 mm tick. Lateral
extension of the housing is l
h
= x
v
+5s
x
+3 [mm]. The
whole structure has a magnetic symmetry plane
which is shown with vertical dash-dot lines in Fig.
10. Ground plane and GCPW signal trace
metallization is with 1.5 oz (0.05 mm thick) copper.
Design starts from x
init
= [x
0
y
0
x
d
y
d
z
d
s
1
x
1
x
v
y
v
s
x
s
y
]
T
= [7.75 5 6 16.5 18 2 10.75 6 14 4 6]
T
mm.
The final design was found to be x* = [7.62 5.70 6.2
16.43 17.9 1.9 10.45 6.08 13.83 4.37 6.03]
T
mm.
The design response meets the specifications; its |S
11
|
is shown in Fig. 11, the gain versus frequency for
θ
= 0
0
is shown in Fig. 12, and the gain pattern cuts
at 2.45 GHz are shown in Fig. 13.
For the purpose of comparison, the DRA without
substrate integrated cavity was also considered. In
this case there were seven design variables
x*
,n.v.
= [x
0
y
0
x
d
y
d
z
d
s
1
x
1
]
T
. Figures 14 and 15 give
responses of the two alternative designs, x*
,n.v
=
[7.65 5.51 5.39 16.20 19.45 0.263 10.05]
T
mm (|S
11
|
< –11.5 dB, gain (
θ
= 0
0
) > 2.5dBi) and x**
,n.v
=
[6.79 5.25 5.68 16.22 19.97 0.250 9.46]
T
mm (|S
11
| <
–13.5 dB, gain (
θ
= 0
0
) > 0.5dBi). The difference in
the gains (Figs. 14 and 15) is due to the parasitic
signal emission into substrate happening in the via-
less designs.
Figure 9: DRA, 3D view: two rectangular DRs in a
housing; feeding is with a GCPW.
(a)
(b)
Figure 10: DRA layout: (a) top view; (b) front view (vias
forming substrate integrated cavity not shown).
Figure 11: Simulation-driven design procedure (first
stage): |S
11
| response of the coarse-discretization DRA
model at the initial design ( ), |S
11
| of the coarse-
discretization model at its optimized design (- - -), and |S
11
|
of the high-fidelity model at the coarse-discretization
model optimum (—). Specifications are shown with the
horizontal line.
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
-30
-20
-10
0
Frequency [GHz]
|S
11
|
SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and
Applications
446
Figure 12: Simulation-driven design procedure (second
stage): |S
11
| response of the high-fidelity model at the
coarse-discretization model optimum (- - -), and at the
final design obtained using space-mapping optimization
with kriging coarse model (—).
Figure 13: DRA, |S
11
| response at the final design: with
substrate integrated cavity, x* (—); no vias, x*
,n.v
(- - -);
and no vias, x**
,n.v.
( ).
Figure 14: DRA, gain response in Z-direction at the final
design: with substrate integrated cavity, x* (—); no vias,
x*
,n.v
(- - -); and no vias, x**
,n.v.
( ). Design
specifications shown with the horizontal line.
(a)
(b)
Figure 15: DRA, gain at 2.45 GHz (a) co-pol. in the E-plane
(YOZ), the right sector is for the positive Y-direction; (b) x-
pol. in the H-plane (XOZ). Design with substrate integrated
cavity, x*, (—); designs without vias, x*
,n.v
(- - -) and
x**
,n.v.
( ).
3.4 Discussion
All the methods considered in this paper have been
demonstrated (Section 3) to yield an optimized DRA
design at the computational cost corresponding to a
few high-fidelity EM simulations of the antenna
structure of interest.
The first two methods, SPRP and AADS are
simple to implement, however, they both require the
responses of the low- and high-fidelity model (here,
|S
11
| versus frequency) to be similar in shape. SPRP is
a response correction technique and it requires that
the distinctive features of responses for both models
correspond to each other (Koziel, 2010b). AADS is
not that restrictive with respect to the relationship
between the model responses, however, it is most
suitable for simple design specifications (e.g., single
requirement for |S
11
|, see Section 3.2). With SPRP, on
the other hand, it is more straightforward to handle
multiple objectives and constraints.
The last method, space mapping with kriging-
based coarse models (Section 2.3) is more general
than SPRP and AADS in the sense that it can handle
the cases when the low- and high-fidelity model
responses are more misaligned. However, SM is more
difficult to implement and requires more experience
from the user in order to set it up properly (Koziel et
al., 2008).
4 CONCLUSIONS
Computationally efficient simulation-driven design
of dielectric resonator antennas is discussed. The
techniques described here exploit low-fidelity DRA
models obtained through coarse-discretization EM
simulations as well as various correction methods
that aim at constructing a reliable surrogate model of
the DRA structure under consideration. We
demonstrate that the optimized designs can be
obtained at a low computational cost corresponding
to a few high-fidelity full-wave electromagnetic
simulations of the DRA of interest.
ACKNOWLEDGEMENTS
The authors thank CST AG, Darmstadt, Germany,
for making CST Microwave Studio available. This
work was supported in part by the Icelandic Centre
for Research (RANNIS) Grant 110034021.
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
-30
-20
-10
0
Frequency [GHz]
|S
11
|
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
-30
-20
-10
0
Frequency [GHz]
|S
11
| [dB]
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
-5
0
5
10
Frequency [GHz]
[dBi]
-10-10 00 1010
90
60
30
0
30
60
90
[dBi]
-10-10 00 1010
90
60
30
0
30
60
90
[dBi]
DESIGN OF DIELECTRIC RESONATOR ANTENNAS USING SURROGATE-BASED OPTIMIZATION AND
ELECTROMAGNETIC MODELS
447
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Applications
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