rapid design optimization of antennas using space mapping and response surface approximation models

11
Rapid Design Optimization of Antennas Using Space Mapping and Response Surface Approximation Models Slawomir Koziel, Stanislav Ogurtsov Engineering Optimization and Modeling Center, School of Science and Engineering, Reykjavik University, Menntavegur 1, Reykjavik, IS-101 Iceland Received 14 March 2011; accepted 2 May 2011 ABSTRACT: A computationally efficient method for design optimization of antennas is dis- cussed. It combines space mapping, used as the optimization engine, and response surface approximation, used to create the fast surrogate model of the optimized antenna. The surro- gate is configured from the response of the coarse-mesh electromagnetic model of the antenna, and implemented through kriging interpolation. We provide a comprehensive nu- merical verification of this technique as well as demonstrate its capability to yield a satisfac- tory design after a few full-wave simulations of the original structure. V C 2011 Wiley Periodicals, Inc. Int J RF and Microwave CAE 21:611–621, 2011. Keywords: computer-aided design; antenna design; surrogate models; response surface approxi- mation; kriging interpolation; space mapping I. INTRODUCTION Computer-aided design (CAD) and electromagnetic (EM)- simulation-based design optimization become more and more important for the development of modern antennas since analytical models of the antennas can only be used—in many cases—to yield initial designs that need to be further tuned to meet given performance specifications and required model accuracy. In addition, no straightfor- ward design procedures exist for some emerging classes of antennas, for example, ultrawideband (UWB) antennas [1] and wideband dielectric resonator antennas [2]. However, a serious bottleneck of EM simulation-driven optimizations is their high computational cost. It makes approaches employing EM solvers directly in the optimi- zation loop very time-consuming or even impractical. This applies, in particular, to traditional gradient-based optimi- zation involving numerous evaluations of the EM-simula- tion-based objective function. Various modern techniques based on the multiagent search principle such as evolu- tionary algorithms [3–8], particle swarm optimizers [9– 12], or ant colony systems [13] are even more CPU-inten- sive although they permit to handle a number of issues, for example, lack of sensitivity information and presence of multiple local optima, which are problematic for the classical methods. Co-simulation [14, 15] is a partial solu- tion, because the circuits comprising EM components are still directly optimized and because this approach is diffi- cult to apply to antennas directly. Computationally efficient optimization of microwave structures, antennas in particular, can be carried out using surrogate-based (SBO) approaches [16]. According to the SBO paradigm, direct optimization of the computationally expensive high-fidelity (i.e., fine) EM model of the struc- ture is replaced by iterative optimization and correction of a computationally cheap, low-fidelity (i.e., coarse) model. The most successful method of this kind in microwave en- gineering so far is space mapping (SM) [17–24]. In con- trast to some other SBO approaches [25–27], SM exploits physics-based coarse models, typically circuit models of the structure, or, when available, models based on analyti- cal formulas [18]. This allows SM algorithms to yield sat- isfactory designs at the time costs corresponding to a few evaluations of the fine model. However, circuit models with reliable prediction capabilities may be difficult to de- velop for certain types of microwave devices, in particu- lar, antennas. Here, we describe a surrogate-based optimization framework for computationally efficient design optimiza- tion of antennas. We adopt a methodology [28] that exploits SM as the optimization engine. The coarse model Correspondence to: S. Koziel; e-mail: [email protected] V C 2011 Wiley Periodicals, Inc. DOI 10.1002/mmce.20555 Published online 26 September 2011 in Wiley Online Library (wileyonlinelibrary.com). 611

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Rapid Design Optimization of Antennas UsingSpace Mapping and Response SurfaceApproximation Models

Slawomir Koziel, Stanislav Ogurtsov

Engineering Optimization and Modeling Center, School of Science and Engineering,Reykjavik University, Menntavegur 1, Reykjavik, IS-101 Iceland

Received 14 March 2011; accepted 2 May 2011

ABSTRACT: A computationally efficient method for design optimization of antennas is dis-

cussed. It combines space mapping, used as the optimization engine, and response surface

approximation, used to create the fast surrogate model of the optimized antenna. The surro-

gate is configured from the response of the coarse-mesh electromagnetic model of the

antenna, and implemented through kriging interpolation. We provide a comprehensive nu-

merical verification of this technique as well as demonstrate its capability to yield a satisfac-

tory design after a few full-wave simulations of the original structure. VC 2011 Wiley

Periodicals, Inc. Int J RF and Microwave CAE 21:611–621, 2011.

Keywords: computer-aided design; antenna design; surrogate models; response surface approxi-

mation; kriging interpolation; space mapping

I. INTRODUCTION

Computer-aided design (CAD) and electromagnetic (EM)-

simulation-based design optimization become more and

more important for the development of modern antennas

since analytical models of the antennas can only be

used—in many cases—to yield initial designs that need to

be further tuned to meet given performance specifications

and required model accuracy. In addition, no straightfor-

ward design procedures exist for some emerging classes

of antennas, for example, ultrawideband (UWB) antennas

[1] and wideband dielectric resonator antennas [2].

However, a serious bottleneck of EM simulation-driven

optimizations is their high computational cost. It makes

approaches employing EM solvers directly in the optimi-

zation loop very time-consuming or even impractical. This

applies, in particular, to traditional gradient-based optimi-

zation involving numerous evaluations of the EM-simula-

tion-based objective function. Various modern techniques

based on the multiagent search principle such as evolu-

tionary algorithms [3–8], particle swarm optimizers [9–

12], or ant colony systems [13] are even more CPU-inten-

sive although they permit to handle a number of issues,

for example, lack of sensitivity information and presence

of multiple local optima, which are problematic for the

classical methods. Co-simulation [14, 15] is a partial solu-

tion, because the circuits comprising EM components are

still directly optimized and because this approach is diffi-

cult to apply to antennas directly.

Computationally efficient optimization of microwave

structures, antennas in particular, can be carried out using

surrogate-based (SBO) approaches [16]. According to the

SBO paradigm, direct optimization of the computationally

expensive high-fidelity (i.e., fine) EM model of the struc-

ture is replaced by iterative optimization and correction of

a computationally cheap, low-fidelity (i.e., coarse) model.

The most successful method of this kind in microwave en-

gineering so far is space mapping (SM) [17–24]. In con-

trast to some other SBO approaches [25–27], SM exploits

physics-based coarse models, typically circuit models of

the structure, or, when available, models based on analyti-

cal formulas [18]. This allows SM algorithms to yield sat-

isfactory designs at the time costs corresponding to a few

evaluations of the fine model. However, circuit models

with reliable prediction capabilities may be difficult to de-

velop for certain types of microwave devices, in particu-

lar, antennas.

Here, we describe a surrogate-based optimization

framework for computationally efficient design optimiza-

tion of antennas. We adopt a methodology [28] that

exploits SM as the optimization engine. The coarse model

Correspondence to: S. Koziel; e-mail: [email protected]

VC 2011 Wiley Periodicals, Inc.

DOI 10.1002/mmce.20555Published online 26 September 2011 in Wiley Online Library

(wileyonlinelibrary.com).

611

for the SM algorithm is constructed as response surface

approximation of the coarse-discretization antenna model

data, here implemented using kriging interpolation [16].

The coarse model built in this way is very fast and reli-

able. It is also continuous and differentiable, and, conse-

quently, it is easy to optimize. In addition, the kriging-

based SM algorithm does not resort to a circuit-based

coarse model. The robustness of our approach is demon-

strated through the design of several antenna structures.

II. SURROGATE-BASED OPTIMIZATION AND SM

A. Formulation of the Antenna DesignOptimization ProblemLet Rf denote the response vector of the fine antenna

model, for example, |S11|, gain, etc, usually evaluated at a

set of frequencies. Our goal is to solve

x�f ¼ argminx2X

U RfðxÞð Þ (1)

where x is a vector of the design variables, X is the search

space (defined by the design constraints), whereas U is a

given objective function, for example, minimax. In this arti-

cle, we consider a single-objective optimization. Multiobjec-

tive cases are treated either through a proper scalarization of

the objective function or by considering some of the figures

of interest as constraints. The fine model Rf is assumed to

be computationally expensive, which makes a direct (e.g.,

gradient-based) optimization of U(Rf(x)) prohibitive.

B. Surrogate-Based OptimizationTo solve (1), we consider an optimization algorithm that gen-

erates a sequence of designs x(i), i ¼ 0, 1, 2, … as follows:

xðiþ1Þ ¼ argminx

U RðiÞs ðxÞ

� �(2)

where {RðiÞs } is a family of surrogate models. Here, x(0) is

a starting point. RðiÞs is a representation of Rf, it is created

using available fine model data and updated after each

iteration.

C. Space MappingSM is a methodology that utilizes the algorithm (2) and

constructs the surrogate models based on the coarse model

Rc, a less accurate but computationally cheap representa-

tion of the fine model. Let �Rs be a generic SM surrogate

model, that is, one configured from Rc with suitable (usu-

ally linear) transformations. At iteration i, the surrogate

model RðiÞs is defined as

RðiÞs ðxÞ ¼ �Rsðx; pðiÞÞ (3)

where

pðiÞ ¼ argminp

Xi

k¼0wi:kjjRfðxðkÞÞ � �RsðxðkÞ; pÞjj (4)

is a vector of model parameters and wi.k are weights (one

of typical setups is wi.k ¼ 1 for all i and k).

A variety of SM surrogate models is available [17,

18], for example, the input SM, where the surrogate

model takes a form of

�Rsðx; pÞ ¼ �Rsðx;B; cÞ ¼ RcðB � xþ cÞ (5)

Here, parameters B and c have to be obtained by solv-

ing a (nonlinear) parameter extraction problem (4).

Another popular approach is the so-called output SM

�Rsðx; pÞ ¼ �Rsðx; dÞ ¼ RcðxÞ þ d (6)

where d is a correction term accounting for discrepancy

between the fine and coarse model responses at iteration i,so that d(i) ¼ Rf(x

(i)) – Rc(x(i)). The model (6) ensures

perfect alignment between the surrogate and the fine

model at x(i). It is often used on top of other SM mapping

transformations.

D. Coarse ModelsThe coarse model is a critical component of successful

SM optimization. The model should be physics-based, that

is, describing the same phenomena as the fine model. It

ensures that the surrogate model constructed using Rc has

good prediction capability [29]. In addition, Rc should be

computationally much cheaper than Rf so that the total

costs of surrogate model optimization (2) and parameter

extraction (4) problems is negligible.

The preferred choice for the coarse model is a circuit

equivalent, for example, implemented in Agilent ADS

[30]. Unfortunately, for many types of antennas, it is diffi-

cult to build a reliable circuit equivalent or the circuit

equivalent can be of insufficient accuracy resulting in

poor performance of the SM process. In addition, it might

be difficult to find a suitable combination of SM transfor-

mations to construct a reliable surrogate model [29].

In general, Rc can be implemented using the same EM

simulator as the one of the fine model by applying relaxed

mesh requirements. However, the coarse-mesh model may

have poor analytical properties (e.g., numerical noise, non-

differentiability and discontinuity of the response over the

design variables), which make optimization of the surro-

gate difficult [31]. In addition, it is not straightforward to

find an appropriate trade-off between the model accuracy

and evaluation time. The rule of thumb is that the evalua-

tion time of Rc should be at least two orders of magnitude

smaller than that of Rf to make the overhead of solving

(2) and (4) reasonably small.

It is worth mentioning that relaxed mesh is not the

only different feature of the coarse model compared to the

fine model. Other options include: (i) the reduced number

of cells in the perfectly matched layer absorbing boundary

conditions as well as reduced distance from the simulated

structure to the absorbing boundary conditions, in the case

of finite-volume EM simulators [32–34]; (ii) the lower

order of the basis functions, in the case of finite-element

and integral equation solvers [33, 34]; (iii) simplified exci-

tation [34, 35], for example, the discrete source of the

coarse model versus the waveguide port of the fine model

612 Koziel and Ogurtsov

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 6, November 2011

as it is implemented in the dual-band double-ring antenna

example presented below, Section V (B); (iv) zero thick-

ness of metallization; (v) the use of perfect electric con-

ductor in place of finite-conductivity metals.

III. RESPONSE-SURFACE-APPROXIMATION-BASEDCOARSE MODELS

A. General ConsiderationsIn this article, a coarse model is built using functional

approximation of data obtained with the same EM simula-

tor as the fine model but with a much coarser mesh. This

coarse-discretization EM-based model will be denoted as

Rcd. Note that the functional approximation model would

be normally set up using a sampled fine model data. Here,

to reduce the computational overhead, the surrogate is

constructed using its simplified representation, Rcd.

A variety of function approximation methods are avail-

able including polynomial approximation [16], neural net-

works [36–40], kriging [16, 41, 42], multidimensional

Cauchy approximation [43], or support vector regression

[44]. Here, the coarse model is constructed using kriging

interpolation. This choice follows not only from the fact

that kriging is a reliable and popular technique [16] but

also because the available Matlab kriging toolbox, DACE

[45], allows to configure kriging-based models in an effi-

cient way.

B. Coarse Model Construction Using KrigingInterpolationLet XB ¼ {x1, x2, …, xN} denote a base set, such that the

responses Rcd(xj) are known for j ¼ 1, 2, …, N. Let

Rcd(x) ¼ [Rcd.1(x) … Rcd.m(x)]T (components of the model

response vector may correspond to certain parameters, for

example, |S11| evaluated at m frequency points).

Here, we use ordinary kriging [28] that estimates deter-

ministic function f as fp(x) ¼ l þ e(x), where l is the

mean of the response at base points, and e is the error

with zero expected value, and with a correlation structure

being a function of a generalized distance between the

base points. We use a Gaussian correlation function of the

form

Rðxi; xjÞ ¼ expXN

k¼1hkjxik � xjkj2

h i(7)

Figure 1 Flowchart of the proposed design optimization proce-

dure exploiting a response-surface-approximation-based coarse

model and space mapping as the main optimization engine.

Figure 2 DRA: (a) 3D view; (b) top view; and (c) front view; substrate shown transparent.

Figure 3 DRA, |S11| versus frequency: Fine model Rf at the ini-

tial design (- - -), optimized coarse-discretization model Rcd (�����),and Rf at the optimum of Rcd (—). [Color figure can be viewed

in the online issue, which is available at wileyonlinelibrary.com.]

Rapid Design Optimization of Antennas 613

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

where yk are unknown correlation parameters used to fit

the model, while xik and xjk are the kth components of the

base points xi and xj.The kriging-based coarse model Rc is defined as

RcðxÞ ¼ Rc:1ðxÞ ::: Rc:mðxÞ½ �T (8)

where

Rc:jðxÞ ¼ �lj þ rTðxÞR�1ðf j � 1�ljÞ (9)

Here 1 denotes an N-vector of ones,

f j ¼ ½Rcd:jðx1Þ ::: :::Rcd:jðxNÞ�T (10)

r is the correlation vector between the point x and base

points

rTðxÞ ¼ ½Rðx; x1Þ ::: :::Rðx; xNÞ�T (11)

whereas R is the correlation matrix between the base

points

R ¼Rðx1; x1Þ Rðx1; x2Þ � � � Rðx1; xNÞRðx2; x1Þ Rðx2; x2Þ � � � Rðx1; xNÞ

..

. ... . .

. ...

RðxN ; x1Þ RðxN ; x2Þ � � � RðxN ; xNÞ

26664

37775 (12)

The mean �lj is given by

�lj ¼ ð1TR�11Þ�11TR�1f j (13)

Correlation parameters yk are found by maximixing

[28]

� ½N lnð�r2Þ þ ln jRj�=2 (14)

in which the variance

�r2j ¼ ðf j � 1�ljÞTR�1ðf j � 1�ljÞ=N (15)

and |R| are both functions of yk. Here, the kriging model

is implemented using the DACE Toolbox [45].

C. Advantages of Response Surface ApproximationCoarse ModelsResponse surface approximation in general (and kriging

interpolation in particular) as a method of generating the

coarse model for SM algorithm has a number of

advantages:

Figure 4 DRA, |S11| versus frequency: Rf at the final design.

[Color figure can be viewed in the online issue, which is avail-

able at wileyonlinelibrary.com.]

Figure 5 DRA, realized gain versus frequency: (—) is for the

zero zenith angle (y ¼ 0�); (- - -) is back radiation for y ¼ 180�.Here, only y-polarization (u ¼ 90�) contributes to the gain for

the listed directions. [Color figure can be viewed in the online

issue, which is available at wileyonlinelibrary.com.]

TABLE I DRA: Optimization Cost

Algorithm

Component

Number of

Model

Evaluations

CPU Time

Absolute

Relative

to Rf

Optimization of Rcd 150 � Rcd 105 min 2.9

Setting up Rc 200 � Rcd 140 min 3.9

Evaluation of Rf 4 � Rf 145 min 4.0

Total cost N/A 390 min 10.8

Figure 6 Geometry of the double ring antenna.

614 Koziel and Ogurtsov

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 6, November 2011

1. The resulting model is computationally cheap, smooth,

and therefore, easy to optimize.

2. There is no need for circuit-equivalent model, and,

consequently, no extra simulation software needs to be

involved; the SM algorithm implementation is simpler

and exploits a single EM solver.

3. It is possible to apply SM for antenna design problems

where finding reliable and fast coarse models is diffi-

cult or impossible.

4. Although the response surface coarse model uses func-

tion approximation, it retains the features of the physi-

cally based model if the number of base points is

sufficiently large.

5. Overall accuracy of the response-surface-based coarse

model may be much better than the accuracy of other

possible coarse models because the model is built using

the data from the same simulator as the one used to

evaluate the fine model.

6. Initial design obtained through optimization of the

coarse-mesh EM model is usually better than the initial

design that could be possibly obtained by means of

other methods.

IV. DESIGN OPTIMIZATION PROCEDURE

A. Design Optimization AlgorithmThe design optimization procedure proposed here can be

summarized as follows (Fig. 1).

1. Take initial design xinit.2. Find the starting point x(0) for SM algorithm by opti-

mizing the coarse-discretization model Rcd.

3. Allocate N base designs, XB ¼ {x1, …, xN}.4. Evaluate Rcd at each design xj, j ¼ 1, 2, …, N.5. Build the coarse model Rc as a kriging interpolation

of data pairs {(xj, Rcd(xj))}j ¼ 1, …, N.

6. Set i ¼ 0.

7. Evaluate the fine model Rf at x(i).

8. Construct the surrogate model RðiÞs as in (3) and (4).

9. Find a new design x(i þ 1) as in (2).

10. Set i ¼ i þ 1.

11. If the termination condition is not satisfied go to 7.

12. END.

The first phase of the design process is to find an opti-

mized design of the coarse-discretization model. The opti-

mum of Rcd is usually the best design we can get at a rea-

sonably low computational cost. This cost can be further

reduced by relaxing tolerance requirements while search-

ing for x(0): due to a limited accuracy of Rcd it is suffi-

cient to find only a rough approximation of its optimum.

Steps 3–5 describe the construction of the kriging-based

coarse model as explained in Section III. Some details

concerning the allocation of the base points are given in

Section IV (B). Steps 6–12 describe the flow of the

SM algorithm with particular operations detailed in

Section II(C).

In this work, the algorithm is terminated when no

improvement of the fine model objective function is

obtained in a given iteration.

B. Some Practical ConsiderationsThe mesh density for the coarse-discretization model Rcd

should be adjusted so that its evaluation time is substan-

tially smaller than that of the fine model and, at the same

Figure 7 Double-ring antenna, |S11| versus frequency: Fine

model Rf at the initial design (- - -), optimized coarse-discretiza-

tion model Rcd (����), and Rf at the optimum of Rcd (—). [Color

figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com.]

Figure 8 Double-ring antenna, |S11| versus frequency: Rf at the

optimum of Rcd (—); kriging model Rc (- - -) and surrogate

model Rs (����). [Color figure can be viewed in the online issue,

which is available at wileyonlinelibrary.com.]

Figure 9 Double-ring antenna, |S11| versus frequency: Rf at the

final design. [Color figure can be viewed in the online issue,

which is available at wileyonlinelibrary.com.]

Rapid Design Optimization of Antennas 615

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

time, its accuracy is still decent. Typically, if Rcd is set up

so that it is 20–60 times faster than Rf its accuracy is ac-

ceptable for the purpose of the proposed design

procedure.

The coarse model is created in the neighbourhood XN

of x(0), the approximate optimum of the coarse-discretiza-

tion model Rcd. The relative size of this neighbourhood

depends on the sensitivity of the antenna response to

design variables as well as the discrepancy between the

fine and coarse-discretization models, and may vary from

a few to 20%. The number of base designs N depends on

the problem dimensionality; typical values are 50–200.

The base points should be allocated as uniformly as

possible in XN. Here, we use a modified Latin Hypercube

Sampling [46] algorithm that gives uniform distribution of

samples regardless of the number of design variables.

V. VERIFICATION EXAMPLES

In this section, we verify the robustness and computational

efficiency of the proposed optimization procedure using

three antenna designs: a dielectric resonator antenna, dual-

band double ring microstrip antenna, and UWB stripline

monopole.

A. Dielectric Resonator AntennaConsider a rectangular DRA [2], see Figure 2 for its ge-

ometry. The DRA comprises: a rectangular dielectric reso-

nator (DR) estimated to operate at the perturbed TEd11

mode [2], supporting RO4003C [47] slabs, and polycar-

bonate housing. The housing is fixed to the circuit board

with four through M1 bolts. The DRA is energized with a

50 X microstrip through a slot made in the metal ground.

Substrate is 0.5 mm thick RO4003C. Design specifications

are |S11| � –15 dB for 5.1–to–5.9 GHz; also, the DRA is

required to have antenna gain better than 5 dBi for the

zero zenith angle over bandwidth of interest.

There are nine design variables: x ¼ [ax, ay, az, ay0, us,ws, ys, g1, y1]

T, where ax, ay, and az are dimensions of the

DR; ay0 stands for the offset of the DR center relative to

the slot center (marked by black dot in Fig. 2b) in Y-direction; us and ws are the slot dimensions; ys is the

length of the microstrip stub; and g1 and y1 are slabs

dimensions. Relative permittivity and loss tangent of the

DR are 10 and 1e-4, respectively, at 6.5 GHz.

The width of the microstrip signal trace is 1.15 mm.

Metallization of the trace and ground is with 50 lm cop-

per. Relative permittivity and loss tangent of the polycar-

bonate housing are 2.8 and 0.01 at 6.5 GHz, respectively.

DRA models are defined with the CST MWS [48], and

the built in single-pole Debye model is used for all dielec-

trics to describe their dispersion properties. Other dimen-

sions are fixed as follows: hx ¼ hy ¼ hz ¼ 1, bx ¼ 7.5, sx¼ 2, and ty ¼ ay � ay0 � 1, all in mm.

The initial design is xin ¼ [8.000, 14.000, 9.000, 0,

1.750, 10.000, 3.000, 1.500, 6.000]T mm. The fine

(1,099,490 mesh cells at xin) and coarse-discretization

(26,796 mesh cells at xin) antenna models are evaluated

Figure 10 Gain of the double-ring antenna: (a) co-pol. in the E-plane (XOZ) at 1.7325 GHz (—) and at 2.1325 GHz (- - -). The left and

right sectors are for u ¼ 0� and 180�, respectivelly; (b) �-pol. in the H-plane (YOZ) at 1.7325 GHz (—) and at 2.1325 GHz (- - -).

TABLE II Double-Ring Antenna: Optimization Cost

Algorithm

Component

Number of

Model

Evaluations

CPU Time

Absolute

Relative

to Rf

Optimization of Rcd 190 � Rcd 5.8 hours 3.5

Setting up Rc 100 � Rcd 3.1 hours 1.8

Evaluation of Rf 4 � Rf 6.7 hours 4.0

Total cost N/A 15.6 hours 9.3

Figure 11 Geometry of the UWB stripline monopole antenna.

The dash-dot lines show the magnetic symmetry walls (XOZ and

YOZ).

616 Koziel and Ogurtsov

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 6, November 2011

with CST MWS transient solver in 2,175 and 42 seconds,

respectively.

A coarse-discretization model optimum is x(0) ¼[7.444, 13.556, 9.167, 0.250, 1.750, 10.500, 2.500, 1.500,

6.000]T mm. Figure 3 shows the fine model reflection

response at the initial design as well as that of the fine

and coarse-discretization model Rcd at x(0). The kriging

coarse model is set up using 200 samples of Rcd allocated

in the vicinity of x(0) of the size [0.5, 0.5, 0.5, 0.25, 0.5,

0.25, 0.25, 0.25, 0.5]T mm.

The final design, x(4)¼[7.556, 13.278, 9.630, 0.472,

1.287, 10.593, 2.667, 1.722, 6.482]T mm, is obtained after

four SM iterations, its reflection response is in Figure 4.

The far-field response of the final design is shown in Fig-

ure 5. For the bandwidth of interest, the peak gain is

above 5 dBi, and the back radiation level is bellow –14

dB (relative to the maximum). All responses shown

include the effect of 25 mm input microstrip. The surro-

gate model used by the optimization algorithm exploited

input and output SM of the form Rs(x) ¼ Rc(x þ c) þ d.Optimization costs are summarized in Table I. The total

design time corresponds to about 11 evaluations of the

fine model.

B. Dual-Band Double-Ring AntennaConsider a double ring antenna [49] shown in Figure 6. It

is made with three dielectric layers with permittivity of er1¼ 2.2, er2 ¼ 1.07, and er3 ¼ 2.2. The dielectric loss tan-

gent is set to 0.001 for all layers. The ground plane is

modeled as infinite. All metal parts have conductivity of

copper, 5.8e7 S/m. Thickness of the rings is 0.05 mm.

Design variables are inner and outer radii of the rings,

location of the feed’s pin, thicknesses of the first and sec-

ond dielectric layers, and lateral extends of these dielec-

trics, namely x ¼ [a1, a2, b1, b2, q1, d1, d2, l1, l2]T mm.

The radius of the pin and thickness of the topmost dielec-

tric are fixed to r0 ¼ 0.325 mm and d3 ¼ 0.254 mm,

respectively. The fine model is fed through 50 X coaxial

waveguide port, whereas the coarse-discretization model

is excited by 50 X discrete line source applied at the gap

(0.5 mm) between the ground and the pin. Design specifi-

cations for AWS dual-band operation [50] are: |S11| � –10

dB for 1.710–1.755 GHz and 2.110–2.155 GHz. Require-

ment on IEEE gain to be not less than 7.5 dB for zero ele-

vation angle over the dual frequency band is imposed as

an optimization constrain.

Design follows methodology described in Section IV,

and it starts from xin ¼ [10, 15, 30, 30, 20, 6, 8, 100,

100]T mm; its response is shown in Figure 7. The antenna

models are evaluated with the CST MWS transient solver:

the coarse-discretization model with 55,000 mesh cells at

xin (evaluation time 110 seconds), and the fine model with

1,240,896 mesh cells at xin (evaluation time 100 minutes).

Figure 12 Monopole antenna, |S11| versus frequency: Fine

model Rf at the initial design (- - -), optimized coarse model

Rcd (����), and Rf at the optimum of Rcd (—). [Color figure can

be viewed in the online issue, which is available at

wileyonlinelibrary.com.]

Figure 13 Monopole antenna, |S11| versus frequency: fine

model at the final design. [Color figure can be viewed in the

online issue, which is available at wileyonlinelibrary.com.]

Figure 14 Monopole antenna, realized gain versus frequency:

Fine model at the final design (optimized for reflection). [Color

figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com.]

TABLE III UWB Monopole: Optimization Cost (Case I:Optimization of Reflection)

Algorithm

Component

Number of

Model

Evaluations

CPU Time

Absolute

Relative

to Rf

Optimization of Rcd 130 � Rcd 3.3 hours 3.9

Setting up Rc 100 � Rcd 2.5 hours 3.0

Evaluation of Rf 3 � Rf 2.5 hours 3.0

Total cost N/A 8.3 hours 9.9

Rapid Design Optimization of Antennas 617

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

A coarse-discretization model optimum is found to be

x(0)¼[8.48, 13.32, 28.77, 29.42, 20.10, 7.62, 8.65, 101.06,

97.40]T mm; its reflection response is also shown in Fig-

ure 7. The kriging coarse model is set up using 100 sam-

ples of Rcd allocated in the vicinity of x(0) of the size [1,

1, 1, 1, 1, 1, 1, 2, 2]T mm. Responses of the kriging

coarse model before and after extraction of SM surrogate

model parameters at x(0) are shown in Figure 8. The final

design, x(3) ¼ [8.54, 12.86, 29.23, 29.96, 19.15, 8.47,

8.74, 100.62, 95.88]T mm, is obtained after three iterations

of the SM algorithm; its reflection response is in Figure 9

and its gain at the central frequencies of the dual-bands is

in Figure 10. The surrogate model used by the optimiza-

tion algorithm exploits input, output and frequency SM

[17]. Optimization costs are summarized in Table II.

C. UWB Monopole AntennaConsider a stripline monopole antenna shown in Figure

11. Design variables are x ¼ [h1, h2, h3, h4, h5, w1, w2,

w3]T. Other parameters are fixed: tm ¼ 0.05, td ¼ 0.45,

hg1 ¼ 22.5, hg2 ¼ 1, hg3 ¼ 5, he ¼ 10, wg1 ¼ 30, wg2 ¼25, ws ¼ 0.4 (all in mm). FR-408 dielectric with the max-

imal losses at 10 GHz is used for the substrate. We

require the antenna to be well matched and have omnidir-

ectional realized gain in the XOY-plane over the band-

width of interest. Two separate cases of design were con-

sidered: (i) the antenna is optimized for minimal reflection

(single-objective case) and (ii) the antenna is optimized

for minimal reflections and, at the same time, to minimize

variations of the realized gain simultaneously over fre-

quency and a number of radiation directions (two-objec-

tive case). Realized gain is recorded for the nine equidis-

tant azimuth angles in the 0� � 90� sector, that is, 0�,11.25�, …, and 90� and the elevation angle of 90�. Thedesign specifications for reflection are |S11| � –10 dB for

3.1 to 10.6 GHz.

For the first design case, the initial design is xin ¼[0.5, 5, 5, 5, 5, 10, 15, 10]T mm. The evaluation times of

its coarse-discretization (57,120 mesh cells of subgrids)

and fine (1,689,941mesh cells of subgrids) models are 1.5

min and 50 min using the CST MWS transient solver,

respectively. The response of the fine model at the initial

design as well as responses of the optimized coarse-dis-

cretization model (optimized for reflection), x(0) ¼[0.4580, 1.8020, 9.0280, 5.0700, 0.1000, 12.7340,

20.9470, 11.7160]T, and the |S11| response of the fine

model at this coarse-discretization model optimum are

shown in the Figure 12. The kriging coarse model is set

up using 100 samples of the coarse-discretization model

Rcd allocated in the vicinity of x(0) of the size [0.1, 0.1,

0.5, 0.5, 0.2, 0.5, 0.5, 0.5]T mm. The fine model at the

final design of the first case (i.e., optimized for reflection

only using the methodology of Section IV) is x(2) ¼[0.389, 1.829, 8.905, 5.114, 0.198, 12.444, 20.611,

12.160]T. Its reflection response with |S11| � –17.7 dB for

3.1–10.6 GHz is shown in Figure 13, and its far-field

response with the realized gain variations of 15.2 dB for

3.1–10.6 GHz is shown in Figure 14. Design costs are

summarized in Table III. Note that for this example, the

design specifications are already satisfied at the coarse-dis-

cretization model optimum (|S11| � –15.5 dB); however,

the SM stage is still capable to ‘‘push’’ |S11| down by over

2 dB.

The second design case started from x(0) of the first

design. The gain of the coarse model was optimized first

over the frequency and a number of radiation directions

Figure 15 Monopole stripline antenna, |S11| versus frequency:

Fine model at the final design optimized for both gain and reflec-

tion (—), and fine model at the design optimized for reflection

only (����). [Color figure can be viewed in the online issue, which

is available at wileyonlinelibrary.com.]

Figure 16 Monopole stripline antenna, realized gain versus

frequency: Fine model at the final design (optimized for

both gain and reflection). Fine model gain at the design optimized

for reflection only shown using thin dotted line. Bandwidth of

interest denoted using vertical dotted lines. [Color figure can

be viewed in the online issue, which is available at

wileyonlinelibrary.com.]

TABLE IV UWB Monopole: Optimization Cost (Case II:Optimization of Reflection and Gain)

Algorithm

Component

Number of

Model

Evaluations

CPU Time

Absolute

Relative

to Rf

Optimization of Rcd 109 � Rcd 2.7 hours 3.3

Setting up Rc 100 � Rcd 2.5 hours 3.0

Evaluation of Rf 3 � Rf 2.5 hours 3.0

Total cost N/A 7.7 hours 9.3

618 Koziel and Ogurtsov

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 6, November 2011

(keeping |S11| � � 10 dB as a constraint). Then the fine

model reflection has been re-optimizing using the method-

ology of Section IV with the maximum variation on the

gain of 7.5 dB imposed as a constraint. The final design

of this case (i.e., optimized for realized gain and reflec-

tion) is x(*) ¼ [0.373, 1.511, 10.222, 1.597, 0.432, 12.021,

20.803, 12.760]T. Its reflection response with |S11| � –13.5

dB for 3.1 to 10.6 GHz is shown in Figure 15, and its far-

field response with the realized gain variations of 7.5 dB

for 3.1–10.6 GHz is shown in Figure 16. The optimization

cost is summarized in Table IV. Realized gain pattern at

selected frequencies is shown in Figure 17 for both

designs.

D. DiscussionThe numerical results provided in Sections V (A–C) give

a consistent picture regarding the performance of the

design optimization method described in Sections III and

IV. In all cases, the optimized design is obtained at the

computational cost corresponding to a few evaluations of

the high-fidelity model (typically, similar to the number of

design variables). This indicates the robustness of the pre-

sented approach and its suitability for antenna design.

Although direct high-fidelity model optimization was not

performed, the data concerning the optimization of the

coarse-discretization model Rcd (the first step of the

design procedure) can give a good estimate of the compu-

tational cost of such a process. Having in mind that the

optimization of Rcd was not performed until convergence

but just to get an approximate optimum of the model [see

Section IV (A)], the direct high-fidelity model optimiza-

tion would require—depending on the example—from 200

to 300 model evaluations. Thus, the approach exploited

here reduces the design cost by at least 95%, which is

significant.

VI. CONCLUSION

An efficient procedure for design optimization of antennas

has been presented. Our technique exploits a kriging-

based interpolation of the coarse-discretization EM simu-

lation data to create a coarse model of the antenna struc-

ture under consideration. This allows us to take advantage

of the computational efficiency of SM which is used here

as the optimization engine. The robustness of our

approach is demonstrated through the design optimization

of three planar antennas. In each case, the optimized

design is obtained at the low computational cost corre-

sponding to a few full-wave simulations of the considered

structures.

ACKNOWLEDGMENT

The authors thank CST AG, Darmstadt, Germany, for mak-

ing CST Microwave Studio available. This work was sup-

ported in part by the Icelandic Centre for Research

(RANNIS) Grant 110034021.

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International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 6, November 2011

620 Koziel and Ogurtsov

BIOGRAPHIES

Slawomir Koziel

Slawomir Koziel received the M.Sc.

and Ph.D. degrees in electronic engi-

neering from Gdansk University of

Technology, Poland, in 1995 and 2000,

respectively. He also received the

M.Sc. degrees in theoretical physics

and in mathematics, in 2000 and 2002,

respectively, as well as the PhD in mathematics in 2003,

from the University of Gdansk, Poland. He is currently an

Associate Professor with the School of Science and Engi-

neering, Reykjavik University, Iceland. His research inter-

ests include CAD and modeling of microwave circuits,

simulation-driven design, surrogate-based optimization,

space mapping, circuit theory, analog signal processing,

evolutionary computation and numerical analysis.

Stanislav Ogurtsov

Stanislav Ogurtsov received the

degree of physicist from Novosibirsk

State University, Novosibirsk, Russia,

in 1993, and the Ph.D. degree in

electrical engineering from Arizona

State University, Tempe, in 2007.

He is currently a postdoctoral

researcher at the Electromagnetic Optimization and

Modeling Center, Reykjavik University, Iceland. His

research interests include simulation-driven computer

aided design of RF, microwave, and millimeter-wave

circuits, ultrawideband antennas, computational electro-

magnetics, modeling of high-speed digital circuits, and

material characterization.

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

Rapid Design Optimization of Antennas 621