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

 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 ( )

xRx

(1)

where R

(x) ∈ R

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

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

| 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

). 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

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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 ( )

xRx

(2)

where R

(i)

is the surrogate model at iteration i,

whereas x

(i+1)

is the approximate solution to (1)

obtained by optimizing R

(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

at the current

iteration point, R

(i)

), treated as a reference.

Figure 1(a) shows the example low-fidelity

model response, |S

| versus frequency, at the design

(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

(i)

), selected here to represent |S

| = –10 dB, |S

= –15 dB, and the local |S

| minimum. Squares

denote corresponding characteristic points for R

(x),

while line segments represent the translation vectors

(“shift”) of the characteristic points of R

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

(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

(i)

) (—), the low-fidelity model

response at x, R

(x) (⋅⋅⋅⋅), characteristic points of R

(i)

)

(o) and R

(x) (□), and the translation vectors (▬); (b)

High-fidelity model response at x

(i)

, R

(i)

) (—) and the

predicted high-fidelity model response at x (⋅⋅⋅⋅) obtained

using SPRP based on characteristic points of (a);

characteristic points of R

(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

(i)

) and R

(x) are plotted

using thin solid and dotted line, respectively.

2.4 Adaptively Adjusted Design

Specifications

It is not necessary to remove the discrepancies

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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

| distinguished from |S

| 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

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

(i)

(x) = R

(x + c

(i)

) + d

(i)

. The vector c

(i)

obtained in the parameter extraction process c

(i)

argmin{c : ||R

(i)

) – R

(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

(x) – R

(x + c

(i)

The parameter shift x + c

(i)

is referred to as input

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|, |S

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|, |S

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-30

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|, |S

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|, |S

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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

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

In this work, we use a pattern search algorithm

(Kolda et al., 2003).

2. Sample R

in the vicinity of x

(0)

and construct a

response surface approximation model R

3. Find a high-fidelity model optimum by applying

the algorithm (2) with the surrogate model created

using R

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

]

where a

, a

, and a

are dimensions of the dielectric

resonator (DR), a

stands for the shift of the DR

center in Y-direction relative to the slot center, u

the slot width, w

is the slot length, and y

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

≤

–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]

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

meets the

specifications.

In the optimization process, the |S

| 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

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

is also evaluated using CST

MWS but with a coarser mesh (14,800 mesh cells at

(0)

, 24 seconds).

The final design x

(2)

= [8.2 14.2 8.3 0.0 1.8 9.4

7.6]

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

at both the initial and the final

design, as well as the response of R

at x

(0)

DESIGN OF DIELECTRIC RESONATOR ANTENNAS USING SURROGATE-BASED OPTIMIZATION AND

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443

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

(thick solid line); and back radiation,

zenith angles of 135

(positive Y-direction, thin solid line),

180

(dash line), and 135

(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

Evaluation

105 × R

42 3.9

Evaluation

3 × R

32 3.0

Total design

time

N/A 74 6.9

Includes optimization of SPRP surrogate (based on R

Excludes evaluation of R

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,

, of 36, two supporting Teflon rings, Teflon filling,

finite ground (t

= 1mm). Teflon permittivity, ε

2.08. The DRA is covered by a polycarbonate

(ε

= 2.7) dome. Thickness of the dome shell, d, is 2

mm. Loss tangents are: 10

–4

for the DRs, 4⋅10

–4

for

Teflon, and 10

–2

for the dome. Dielectrics are

described using the 1

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

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

]

. The design objective is |S

| ≤ –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

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.

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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

. 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

= [a

]

= [6.9 6.9 1.05 1.05 6.2 6.2 2.0 2.0 6.8 12.0

10 16.5]

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]

. 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

| 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

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|, | S

| [dB]

3.5 4 4.5 5 5.5 6 6.5

-40

-30

-20

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Frequency [GHz]

|, |

| [dB]

-20

-10

120

150

180

150

120

-20

-10

120

150

180

150

120

DESIGN OF DIELECTRIC RESONATOR ANTENNAS USING SURROGATE-BASED OPTIMIZATION AND

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445

DRA is with a 50 ohm grounded coplanar

waveguide (GCPW). The GCPW is terminated by

two symmetrical slots (width s

and length x

see

Fig. 10(a)). Figures 9 and 10(a) also show vias

forming a substrate integrated cavity. The TE

δ11

mode is excited in the DRA.

Dimensions of the DRA are to be adjusted for the

following design requirements: input reflection

coefficient, |S

|, should be better than –20 dB, and

gain is to be higher than 3dBi for

= 0

(Z-

direction), both over the 2.4-to-2.5 GHz band.

There are eleven design variables: x = [x

]

, where x

and y

are location of

the center of one DR relative to the origin of the

coordinate system marked by O in Fig. 10; x

, y

and z

are dimensions of the DRs (ceramic body); s

and x

are dimensions of the slots energizing the

DRs; x

, y

, s

, and s

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

, of 1.5 mm and spacing, s

, of 1mm.

Diameter of the vias, d

, is 1.5 mm.

Thicknesses of the polycarbonate housing, x

, y

and z

, are 2 mm. Location of the mounting bolts are

described by x

= s

and y

= 1 mm. The heads of the

bolts are 4 mm in diameters and 1 mm tick. Lateral

extension of the housing is l

= x

+5s

+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

]

= [7.75 5 6 16.5 18 2 10.75 6 14 4 6]

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]

mm.

The design response meets the specifications; its |S

is shown in Fig. 11, the gain versus frequency for

= 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

,n.v.

= [x

]

. 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]

mm (|S

< –11.5 dB, gain (

= 0

) > 2.5dBi) and x**

,n.v

[6.79 5.25 5.68 16.22 19.97 0.250 9.46]

mm (|S

| <

–13.5 dB, gain (

= 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

| response of the coarse-discretization DRA

model at the initial design (⋅ ⋅ ⋅), |S

| of the coarse-

discretization model at its optimized design (- - -), and |S

of the high-fidelity model at the coarse-discretization

model optimum (—). Specifications are shown with the

horizontal line.

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SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and

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446

Figure 12: Simulation-driven design procedure (second

stage): |S

| 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

| 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,

,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,

| 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

|, 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

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| [dB]

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

-5

Frequency [GHz]

[dBi]

-10-10 00 1010

[dBi]

-10-10 00 1010

[dBi]

DESIGN OF DIELECTRIC RESONATOR ANTENNAS USING SURROGATE-BASED OPTIMIZATION AND

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