Microwave Design Optimization Exploiting Adjoint Sensitivity

Slawomir Koziel, Leifur Leifsson and Stanislav Ogurtsov

Engineering Optimization & Modeling Center, School of Science and Engineering, Reykjavik University, Reykjavik, Iceland

Keywords: Computer-aided Design (CAD), Simulation-driven Design, Microwave Design Optimization,

Electromagnetic Simulation, Adjoint Sensitivity.

Abstract: Adjustment of geometry and material parameters is an important step in the design of microwave devices

and circuits. Nowadays, it is typically performed using high-fidelity electromagnetic (EM) simulations,

which might be a challenging and time consuming process because accurate EM simulations are

computationally expensive. In particular, design automation by employing an EM solver in an numerical

optimization algorithm may be prohibitive. Recently, adjoint sensitivity techniques become available in

commercial EM simulation software packages. This makes it possible to speed up the EM-driven design

optimization process either by utilizing the sensitivity information in conventional, gradient-based

algorithms or by combining it with surrogate-based approaches. In this paper, we review several recent

methods and algorithms for microwave design optimization using adjoint sensitivity. We discuss advantages

and disadvantages of these techniques and illustrate them through numerical examples.

1 INTRODUCTION

Contemporary microwave engineering heavily relies on

electromagnetic (EM) simulation. EM simulation is not

only used for design verification but in the design

process itself, i.e., to adjust geometry and/or material

parameters of the structure under consideration.

Unfortunately, accurate EM simulation is CPU

intensive. A way to speed it up is parallelization

(OpenMP, MPI, GPU) or distributed computing.

However, the bottleneck in EM-simulation-based

optimization remains to be the large number of EM

simulations required by conventional optimization

algorithms. Another problem is related to the numerical

noise present in EM-based objective functions, due to

which local search methods often fail to find the

optimal design. While many commercial EM

simulation packages have implemented basic design

automation methods (mostly conventional gradient-

based and derivative-free approaches such as Quasi-

Newton or Nelder-Mead algorithms, or population-

based algorithms such as genetic algorithms), a

common practice is still to obtain satisfactory design

using tedious and time-consuming parameter sweeps

involving numerous full-wave EM simulations

combined with engineering experience.

Efficient simulation-driven design can be

performed using surrogate based optimization

(SBO). The most successful SBO techniques in

microwave engineering include space mapping (SM)

(Bandler et al., 2004); (Koziel et al., 2008a); (Amari

et al., 2006), simulation-based tuning (Swanson and

Macchiarella, 2007); (Rautio, 2008), manifold

mapping (MM) (Echeverria and Hemker, 2005), as

well as shape-preserving response prediction

(Koziel, 2010). While the SBO techniques can be

extremely efficient, they are not straightforward to

automate to make them reliable “push-button”-like

approaches that could work for a variety of

microwave problems. Their use typically requires

some experience (Koziel et al., 2008a) and most of

them are not globally convergent so that whether a

satisfactory design is obtained or not may depend on

a proper implementation, some parameter tuning, as

well as certain knowledge particularly while

constructing the surrogate model.

Another approach to improve efficiency of

simulation-driven design is by using adjoint sensitivity

that allows obtaining derivative information of the

system of interest with little or no extra computational

cost (Nair and Webb, 2003); (El Sabbagh et al., 2006);

(Kiziltas et al., 2003); (Uchida et al., 2009); (Bakr et

al., 2011). However, until recently, adjoint sensitivities

were not commercially available, which means that

they were not available for most engineers and

designers. Situation changed a few years ago when

499

Koziel S., Leifsson L. and Ogurtsov S..

Microwave Design Optimization Exploiting Adjoint Sensitivity.

DOI: 10.5220/0004164104990506

In Proceedings of the 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SDDOM-2012), pages

499-506

ISBN: 978-989-8565-20-4

Copyright

c

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

adjoint sensitivities were implemented for instance in

CST Microwave Studio (CST, 2011).

In this paper, we review several recent

techniques that exploit adjoint sensivity in order to

speed up the EM-simulation-driven microwave

design process. These techniques include gradient-

based search methods embedded in trust region

framework, as well as surrogate-based methods,

specifically space mapping (Koziel et al., 2008a) and

manifold mapping (Echeverria and Hemker, 2005),

enhanced by adjoint sensitivity in order to improve

their convergence properties and reduce the

computational cost of surrogate model optimization

step. The efficiency of the presented approaches is

demonstrated using several microwave design cases.

A performance comparison with other optimization

techniques, including Matlab’s fminimax (Matlab,

2008) and a Quasi-Newton type of algorithm

(Nocedal and Wright, 2000) is also provided.

2 DESIGN OPTIMIZATION WITH

TRUST-REGION AND ADJOINT

SENSITIVITIES

2.1 Design Problem Formulation

The microwave design task can be formulated as a

nonlinear minimization problem

 

*

argmin ( )

ff

U

x

x R x

(1)

where R

f

 R

m

denotes the response vector of a high-

fidelity (or fine) model of the microwave structure

of interest evaluated through expensive high-fidelity

EM simulation; x  R

n

is a vector of designable

variables. Typically, these are geometry and/or

material parameters. The response R

f

(x) might be,

e.g., the modulus of the transmission coefficient |S

21

|

evaluated at m different frequencies. In some cases,

R

f

may consists of several vectors representing, e.g.,

filter reflection and transmission coefficients, or an

antenna reflection, gain, etc. U is a given scalar

merit function, e.g., a norm, or a minimax function

with upper and lower specifications. U is formulated

so that a better design corresponds to a smaller value

of U. x

f

*

is the optimal design to be determined.

Direct solution of (1) using conventional

algorithm may be prohibitive because it usually

requires a large number of fine model evaluations,

each being computationally expensive by itself. For

many structures, the evaluation time may be as long

as a few hours.

2.2 Trust-Region-Based Optimization

with Adjoint Sensitivity

The algorithm proposed in (Koziel et al., 2012a)

uses the 1st-order model (or the surrogate) S(x) of

the high-fidelity model Rf. S(i)(x) is nothing else but

a linear function being a first-order Taylor expansion

of Rf at x(i) of the form:

( ) ( ) ( ) ( )

( ) ( ) ( )

( , , ( ), ( ))

( ) ( ) ( )

f

i i i i

f

i i i

f

J



   

R

S x x R x x

R x x x x

(2)

J

Rf

(x) is an estimated Jacobian of R

f

at x,

J

Rf

(x) = [∂R

fi

/∂x

j

]

i=1,…,m; j=1,…,n

, obtained using adjoint

sensitivity (if available) or finite differentiation

∂R

fi

/∂x

j

 [R

fi

([x

1

… x

j

+d

j

… x

n

]

T

) – R

fi

(x)]/d

j

for all

the other parameters.

The optimization algorithm framework is the

following (r

0

is the initial trust region radius)

1. i = 0; r = r

0

;

2. Optimize a linear model: x

tmp

= argmin{||x – x

(i)

||

≤ r : S

(i)

(x,x

(i)

,R

f

(x

(i)

),J

F

(x

(i)

))};

3. Calculate gain ratio:



= [U(R

f

(x

(i)

) –

U(R

f

(x

tmp

))]/[U(R

f

(x

(i)

) – U(S(x

tmp

))];

4. If U(R

f

(x

tmp

)) < U(R

f

(x

(i)

)) then x

(i+1)

= x

tmp

;

i = i + 1;

5. Update r:



< r

decr

then r = r/m

decr

; else if



> r

incr

then r = r·m

incr

;

6. If termination condition is not satisfied, go to 2;

else, END.

Here, r

decr

and r

incr

denote threshold values for

decreasing or increasing the trust region radius by

the corresponding factors m

decr

and m

incr

. The

algorithm is terminated if either of the following

conditions is satisfied: r <



r

, n

F

< n

Fmax

, ||x

(i)

– x

(i–1)

||

<



x

, or ||U(R

f

(x

(i)

)) – U(R

f

(x

(i–1)

))|| <



F

, where



r

,



x

,



F

, n

Fmax

, are user defined parameters, whereas n

F

is

the number of high-fidelity model evaluations.

The response Jacobian is recalculated after each

successful iteration (i.e., when U(R

f

(x

tmp

)) <

U(R

f

(x

(i)

))) for those variables where adjoint

sensitivity is available. The finite-difference

sensitivity is not recalculated as long as the new

iteration is successful in order to reduce the number

of high fidelity function evaluations.

The above algorithm is a local-search method.

Assuming that the exact sensitivity of Rf at x(i) is

used to define the first-order model S(i), S(i)

satisfies both zero- and first-order consistency

conditions with the high-fidelity model Rf, i.e.,

S(i)(x(i)) = Rf(x(i)) and JS(x(i)) = JRf(x(i)). This is

sufficient for the global convergence of the

SIMULTECH 2012 - 2nd International Conference on Simulation and Modeling Methodologies, Technologies and

Applications

500

algorithm at least to a local optimum of the high-

fidelity model, provided that Rf is sufficiently

smooth (Alexandrov et al., 1998). In practice, the

high-fidelity model is noisy, nevertheless, the

performance of the algorithm is quite remarkable

(Koziel et al., 2012a).

Responses obtained using EM solvers are

inherently noisy (except, perhaps, when the mesh

topology is fixed). The major reason is that the mesh

topology is a discontinuous function of the design

variables (or, more general, of geometry of the

structure under considerations). The minor reason is

that the evaluation process itself is noisy (e.g., due to

finite tolerances used to terminate the EM

simulation). This poses some problems for the

optimization process. In particular, finite

differentiation with conventional small increments

(e.g., 10

–8

) will not work: the value of the derivative

obtained this way will be completely unreliable,

regardless of the model discretization density. The

reason is that the change of the response due to the

small perturbation of any given design variable will

be, most likely, much smaller than the amplitude of

the numerical noise. For noisy functions, better and

more consistent gradient estimation can be obtained

using larger finite differentiation step sizes. Based

on the above considerations, our algorithm uses

relatively large steps for finite differentiation,

typically, 10

–3

or larger (depending on the absolute

values of the design variables).

2.3 Example: Design of a Waveguide

Bandpass Filter

Consider the waveguide filter shown in Fig. 1.

Design variables are [h

1

h

2

h

3

s

1

s

2

s

3

s

4

w

1

w

2

w

3

w

4

]

T

. Design specifications are |S

11

|  –20 dB for

667.5 MHz 



 675 MHz. The initial design is x

(0)

= [163.5 172 165.3 160.5 160.5 160.5 130.5 –60.5 –

29.5 –28.5 –27.5]

T

(minimax specification error

+19.2 dB). Optimization results are shown in Table

1. Figure 2 shows the filter responses at the initial

design and at the optimized design found by the

algorithm of Section 2.2. In this case, the proposed

algorithm performs substantially better than the

methods used for comparison both with respect to

the computational cost of the design process and the

quality of the final design. It should be noted that the

computational complexity of our algorithm using

finite-differences derivatives is comparable to that of

Matlab’s fminimax, even though the latter exploits

adjoint sensitivity.

3 SURROGATE-BASED

OPTIMIZATION WITH

ADJOINT SENSITIVITY

3.1 Surrogate-based Optimization

A generic surrogate-based optimization (SBO)

algorithm (Koziel and Yang, 2011); (Forrester and

Keane, 2009) generates a sequence of approximate

solutions to (1), x

(i)

, as follows

 

( 1) ( )

argmin ( )

ii

s

U





x

x R x

(3)

Figure 1: Geometry of a waveguide bandpass filter.

(a)

(b)

Figure 2: Waveguide bandpass filter: (a) responses at the

initial design x

(0)

; (b) responses at the optimized design

found by the proposed algorithm using mixed adjoint and

finite-difference sensitivities.

660 665 670 675 680 685

-30

-20

-10

0

Frequency [MHz]

|S

11

|

660 665 670 675 680 685

-30

-20

-10

0

Frequency [MHz]

|S

11

|

Microwave Design Optimization Exploiting Adjoint Sensitivity

501

Table 1: Optimization results for the waveguide bandpass

filter.

Optimization Algorithm

Final

Specification

Error

Number of

Function

Evaluations

Quasi-Newton optimizer

+5.3 dB

1454

Matlab’s fminimax

+1.2 dB

88

This work

(Algorithm of

Section 2)

Adjoint sensitivity

–2.2 dB

16

Mixed adjoint /

finite-difference

sensitivity

*

–1.3 dB

46

Finite-difference

sensitivity

–0.4 dB

107

* Adjoint sensitivity for the first seven variables, finite-

differences for the remaining four variables.

where R

s

(i)

is the surrogate model at iteration i. Here,

x

(0)

is the initial design. R

s

(i)

is assumed to be a

computationally cheap and sufficiently reliable

representation of R

f

, particularly in the neighborhood

of x

(i)

. Under these assumptions, the algorithm (3) is

likely to produce a sequence of designs that quickly

approach x

f

*

. Usually, R

f

is only evaluated once per

iteration (at every new design x

(i+1)

) for verification

purposes and to obtain the data necessary to update

the surrogate model. Because of the low

computational cost of the surrogate model, its

optimization cost can usually be neglected and the

total optimization cost is determined by the

evaluation of R

f

. The key point here is that the

number of evaluations of R

f

for a well performing

surrogate-based algorithm is substantially smaller

than for most conventional optimization methods.

3.2 Robustness of SBO Algorithms

Robustness of the surrogate-based optimization

process (3) depends on the quality of the surrogate

model R

s

(i)

. In general, in order to ensure

convergence of the algorithm (3) to at least local

optimum of the high-fidelity model, the first-order

consistency conditions have to be met (Alexandrov

and Lewis, 2001), i.e., one has to have R

s

(i)

(x

(i)

) =

R

f

(x

(i)

) and J

R

s

(i)

(x

(i)

) = J

R

f

(x

(i)

), where J stands for the

Jacobian of the respective model. Also, the process (3)

has to be embedded in the trust-region (TR)

framework (Conn et al., 2000), i.e., we have

( ) ( )

( 1) ( )

:|| ||

arg min ( ( ))

ii

s

U









x x x

x R x

(4)

where the TR radius



(i)

is updated using classical

rules (Conn et al., 2000). In general, the SBO

algorithm (4) can be successfully utilized without

satisfying the aforementioned conditions, see, e.g.

(Bandler et al., 2004); (Koziel et al., 2008a).

However, in these cases, the quality of the

underlying low-fidelity model may be critical for

performance (including the algorithm convergence)

(Koziel et al., 2008b) and accurate location of the

optimum design may not be possible.

Availability of cheap adjoint sensitivity (Nair

and Webb, 2003); (CST, 2011) makes it possible to

satisfy consistency conditions in a easy way (without

excessive computational cost by using, e.g., finite

differentiation). A few options exploiting this

possibility are discussed in the next section.

3.3 SBO with First-order Taylor Model

and Trust Regions

The simplest way of exploring adjoint sensitivity for

antenna optimization is to use the following

surrogate model for the SBO scheme (4):

( ) ( ) ( ) ( )

f

i i i i

sf

   

R

R x R x J x x x

(5)

where J

Rf

is the Jacobian of R

f

obtained using

adjoint sensitivity technique. The key point of the

algorithm is finding the new design x

(i)

and the

updating process for the search radius



(i)

. Here,

instead of the standard rules, we use the following

strategy (x

(i–1)

and



(i–1)

are the previous design and

the search radius, respectively):

1. For



k

= k



(i–1)

, k = 0, 1, 2, solve:

()

:|| ||

arg min ( ( ))

i

k

ki

s

U







x x x

x R x

. Note that x

0

= x

(i–1)

. The

values of



k

and U

k

= U(R

s

(i)

(x

k

)) are interpolated

using 2

nd

-order polynomial to find



*

that gives the

smallest (estimated) value of the specification error

(



*

is limited to 3



(i–1)

). Set



(i)

=



*

.

2. Find a new design x

(i)

by solving (4) with the

current



(i)

.

3. Calculate the gain ratio



= [U(R

f

(x

(i)

)) – U

0

]/

[U(R

s

(i)

(x

(i)

)) – U

0

]; If



< 0.25 then



(i)

=



(i)

/3; else

if



> 0.75 then



(i)

= 2



(i)

;

4. If



< 0 go to 2;

5. Return x

(i)

and



(i)

;

The trial points x

k

are used to find the best value of

the search radius, which is further updated based on

the gain ratio



(actual versus expected objective

function improvement). If the new design is worse

than the previous one, the search radius is reduced to

find x

(i)

again, which eventually will bring the

improvement of U as R

s

(i)

and R

f

are first-order

SIMULTECH 2012 - 2nd International Conference on Simulation and Modeling Methodologies, Technologies and

Applications

502

consistent (Alexandrov and Lewis, 2001). This

precaution is necessary because the procedure in

Step 1 only gives an estimation of the search radius.

As an example, consider a wideband hybrid

antenna (Petosa, 2007) shown in Fig. 3, a quarter-

wavelength monopole loaded by dielectric ring

resonator. The design goal is to have |S

11

| ≤ −20 dB

for 8-to-13 GHz. The design variables are x = [h

1

h

2

r

1

r

2

g]

T

. The initial design is x

(0)

= [2.5 9.4 2.3 3.0

0.5]

T

mm. Other parameters are fixed. The final

design with the proposed algorithm is x

(0)

= [3.94

10.01 2.23 3.68 0.0]

T

mm. Table 2 and Fig. 4

compare the design cost and quality of the final

design found by the algorithm described above and

Matlab’s fminimax. It can be observed that our

algorithm yields better design at significantly

smaller computational cost (75 percent design time

reduction).

3.4 Space Mapping and Manifold

Mapping

Construction of the surrogate model can also be

based on the underlying low-fidelity (or coarse)

model R

c

, e.g., obtained from coarse-discretization

EM simulation data. The two methods considered

here that use this approach are space mapping (SM)

(Koziel et al., 2008a) and manifold mapping (MM)

(Echeverria and Hemker, 2005). Usually, the

knowledge about the system embedded in the low-

fidelity model allows us to reduce the number of

high-fidelity model evaluations necessary to find an

optimum design.

The SM surrogate considered here is constructed

using input and output SM (Bandler et al., 2004) of

the form:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

i i i i i

sc

    R x R x c d E x x

(6)

Here, only the input SM vector c

(i)

is obtained

through the nonlinear parameter extraction process

( ) ( ) ( )

argmin|| ( ) ( )||

i i i

fc

  

c

c R x R x c

(7)

Output SM parameters are calculated as

( ) ( ) ( ) ( )

( ) ( )

i i i i

fc

  d R x R x c

(8)

and

( ) ( ) ( ) ( )

( ) ( )

fc

i i i i

  

RR

E J x J x c

(9)

Formulation (6)-(9) ensures zero- and first-order

consistency (Alexandrov and Lewis, 2001) between the

surrogate and the fine model.

The manifold mapping (MM) surrogate model is

defined as (Echeverria and Hemker, 2005)

 

( ) ( ) ( ) ( )

i i i i

s f c c

  R x R x S R x R x

(10)

where S

(i)

is the mm correction matrix defined as

( ) ( ) ( ) †

( ) ( )

fc

i i i



RR

S J x J x

(11)

Figure 3: Wideband hybrid antenna: geometry.

Figure 4: Wideband hybrid antenna: reflection response at

the initial design (  ), at the final design by Matlab’s

fminimax (- - -), and by the proposed algorithm (—).

Table 2: Wideband hybrid antenna: design results.

Algorithm

max|S

11

| for 8 to 13 GHz

at Final Design

Design Cost

(Number of EM

Analyses)

Matlab’s

fminimax

–22.6 dB

98

This work

–24.6 dB

24

The pseudoinverse, denoted by

†

, is defined as

††

c

c c c

T



R R R

R J J J

JVΣU

(12)

where U

J

Rc

, 

J

Rc

, and V

J

Rc

are the factors in the

singular value decomposition of J

R

c

. The matrix



J

Rc

†

is the result of inverting the nonzero entries in



J

Rc

, leaving the zeroes invariant (Echeverria and

Hemker, 2005). Using the sensitivity data as in (12)

h

2

h

1

g

GND

50 Ω coax

r

1

r

2

ε

1

ε

2

ε

3

ε

1

ε

2

r

0

d

6 8 10 12 14 16

-35

-30

-25

-20

-15

-10

-5

0

Frequency [GHz]

|S

11

| [dB]

Microwave Design Optimization Exploiting Adjoint Sensitivity

503

ensures that the surrogate model (10) is first-order

consistent with the fine model. In our

implementation, the coarse model is preconditioned

using input space mapping of the form (7) in order

to improve its initial alignment with the fine model.

Both the parameter extraction (7) and surrogate

model optimization processes (4) are implemented

by exploiting adjoint sensitivity data of the low-

fidelity model, which allows for further cost savings.

The details of these implementations can be found in

(Koziel et al., 2012b).

In order to illustrate the operation and

performance of the SM and MM algorithms, let us

consider an UWB antenna shown in Fig. 5. The

antenna and its models include: a microstrip

monopole, housing, edge mount SMA connector,

section of the feeding coax. The design variables are

x = [l

1

l

2

l

3

w

1

]

T

. Simulation time of the low-fidelity

model R

c

(156,000 mesh cells) is 1 min, and that of

the high-fidelity model R

f

(1,992,060 mesh cells) is

40 min (both at the initial design). Both models are

simulated with the transient solver of CST

Microwave Studio (CST, 2011). The design

specifications for reflection are |S

11

|

≤

–12 dB for 3.1

GHz to 10.6 GHz. The initial design is x

init

= [20 2 0

25]

T

mm.

The antenna was optimized using the SBO

algorithm (4) with both the SM and MM surrogate

models. Fig. 6(a) shows the responses of R

f

and R

c

at

x

init

. Fig. 6(b) shows the response of the high-fidelity

model at the final design x

(2)

= [20.22 2.43 0.128

19.48]

T

(|S

11

|

≤

–12.5 dB for 3.1 to 10.6 GHz) obtained

after only two SBO iterations with MM surrogate, i.e.

only 4 evaluations of the high-fidelity model (Table

3). The number of function evaluations is larger than

the number of MM iterations because some designs

can be rejected by the TR mechanism. The algorithm

using SM surrogate required three iterations and the

final design is x

(3)

= [20.29 2.27 0.058 19.63]

T

(|S

11

|

≤

–12.8 dB for 3.1 to 10.6 GHz) obtained after three

SM iterations. The total optimization cost (Table 4) is

equivalent to around 6 evaluations of the fine model.

Figure 7 shows the evolution of the specification

algorithm for the manifold mapping algorithm.

As another example, consider the third-order

Chebyshev bandpass filter (Kuo et al., 2003) shown in

Fig. 8. The design variables are x = [L

1

L

2

S

1

S

2

]

T

mm.

Other parameters are: W

1

= W

2

= 0.4 mm. Both fine

(396,550 mesh cells, evaluation time 45 min) and

coarse (82,350 mesh cells, evaluation time 1 min)

models are evaluated by the CST MWS transient solver

(CST, 2011).

The design specifications are |S

21

|  –3 dB for

1.8 GHz 



 2.2 GHz, and |S

21

|  –20 dB

for

1.0

GHz







1.55GHz and 2.45 GHz 



 3.0 GHz. The

initial design is the coarse model optimal solution

x

init

= [16 16 1 1]

T

mm.

(a)

(b)

Figure 5: UWB monopole: (a) 3D view; (b) top view. The

housing is shown transparent.

(a)

(b)

Figure 6: UWB monopole optimized using manifold

mapping algorithm: (a) responses of R

f

(—) and R

c

(- - -)

at the initial design x

init

; (b) response of R

f

(—) at the

final design.

The filter was optimized using the SM algorithm.

Optimization results are shown in Fig. 9 and Table

5. The final design x

(5)

= [14.58 14.57 0.93 0.56]

T

is

obtained after five SM iterations. As before,

optimization cost is very low. Also, thanks to

sensitivity information as well as trust region, the

algorithm improves the specification error at each

iteration, see Fig. 10. This is not the case for

GND

l

1

l

2

l

3

w

1

2 4 6 8 10 12

-20

-15

-10

-5

Frequency [GHz]

|S

11

| [dB]

2 4 6 8 10 12

-20

-15

-10

-5

Frequency [GHz]

|S

11

| [dB]

SIMULTECH 2012 - 2nd International Conference on Simulation and Modeling Methodologies, Technologies and

Applications

504

conventional space mapping (Bandler et al., 2004).

Figure 7: UWB monopole: Minimax specification error

versus manifold mapping algorithm iteration index.

Table 3: UWB monopole antenna: optimization results

using manifold mapping.

Algorithm

Component

Number of Model

Evaluations

*

CPU Time

Absolute

Relative to

R

f

Evaluation of R

c

31

31 min

0.8

Evaluation of R

f

4

120 min

4.0

Total cost

*

N/A

151 min

4.8

* Includes R

f

evaluation at the initial design.

Table 4: UWB monopole antenna: optimization results

using space mapping.

Algorithm

Component

Number of Model

Evaluations

*

CPU Time

Absolute

Relative to R

f

Evaluation of R

c

45

45 min

1.1

Evaluation of R

f

5

200 min

5.0

Total cost

*

N/A

205 min

6.1

* Includes R

f

evaluation at the initial design.

Figure 8: Third-order Chebyshev bandpass filter: geometry.

4 CONCLUSIONS

A review of recent microwave design optimization

techniques exploiting adjoint sensitivity has been

presented. We have demonstrated that by exploiting

cheap derivative information, the EM-simulation-

driven design process can be performed efficiently

and in a robust way. Adjoint sensitivity can also be

used to improve performance of the surrogate-based

optimization algorithm as illustrated on the example

of space mapping and manifold mapping techniques.

(a)

(b)

Figure 9: Third-order Chebyshev filter: (a) responses of R

f

(—) and R

c

(- - -) at the initial design x

init

; (b) response of

R

f

(—) at the final design.

Figure 10: Third-order Chebyshev filter: minimax

specification error versus SM iteration index.

Table 5: Third-order Chebyshev filter: optimization

results using space mapping.

Algorithm

Component

Number of

Model

Evaluations

*

CPU Time

Absolute

Relative to R

f

Evaluation of R

c

67

67 min

1.5

Evaluation of R

f

6

270 min

6.0

Total cost

*

N/A

337 min

7.5

* Includes R

f

evaluation at the initial design.

0 0.5 1 1.5 2

-1

0

1

2

3

4

5

Iteration index

Specification error [dB]

L

1

W

1

S

1

L

2

L

1

L

2

S

2

S

2

S

1

W

2

W

2

W

1

W

2

1.4 1.6 1.8 2 2.2 2.4 2.6

-30

-20

-10

0

Frequency [GHz]

|S

21

| [dB]

1.4 1.6 1.8 2 2.2 2.4 2.6

-30

-20

-10

0

Frequency [GHz]

|S

21

| [dB]

0 1 2 3 4 5

-1

0

1

2

3

4

5

Iteration index

Specification error [dB]

Microwave Design Optimization Exploiting Adjoint Sensitivity

505

ACKNOWLEDGEMENTS

The authors would like to thank CST AG for making

CST Microwave Studio available. This work was

supported in part by the Icelandic Centre for

Research (RANNIS) Grant 110034021.

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