An Efficient Numerical Technique An Efficient Numerical Technique for Gradient Computation with for Gradient Computation with

Full-Wave EM SolversFull-Wave EM Solvers

* e-mail: alis5@mcmaster.ca

tel: (905) 525 9140 ext. 27762

fax: (905) 523 4407

Shirook M. AliShirook M. Ali* * and Natalia K. Nikolovaand Natalia K. Nikolova

McMasterUniversity

Department of Electrical and Computer Engineering

Computational Electromagnetics Laboratory

Objectives and Outline

Applications with the frequency-domain TLM

Conclusions

Adjoint variable method in full-wave analysis

computational efficiency, feasibility, and accuracy

Optimization using gradient-based methods

adjoint-sensitivity analysis: objectives

obtain the response and its gradient with two obtain the response and its gradient with two full-full- wave analyses for all the design parameters, wave analyses for all the design parameters, re-re- meshing is not necessarymeshing is not necessary

* ( , ( ))arg min fx

x x I x

Optimization via gradient-based methods

The design problem

1[ ]TKx xx - design parameters

1[ ]TmI II - state variables

( , ( ))f x I x - scalar objective function

subject to ( ) ,fx Z x I V=

objective

1 K

f ff

x x

x

Fig. 1. Shape optimization process.

InputInitial shape, objective function, design

variables.

Perturbed system analysisMesh generation, numerical analysis

Design sensitivityanalysis

Stop

Yes

No

Optimizer

Original system analysisMesh generation, numerical analysis

fx

( )f x

timesK

Optimum designachieved ?

( )pkf x

The optimization process

K+1 analyses

2 analyses

InputInitial shape, objective function, design

variables.

Design sensitivityanalysis

Stop

Yes

No

Optimizer

Original system analysisMesh generation, numerical analysis

The AVM

fx

( )f x

Optimum designachieved ?

px

Adjoint Sensitivities of Linear Systems

response function sensitivity: the adjoint variable method (AVM) [E.J. Haug et al., Design Sensitivity Analysis of Structural Systems, 1986], [J.W.

Banler, Optimization, vol. 1, 1994]

( , ( ))f x I x

( ) Z x I V 1 ( ) x x xI Z V ZI

ef f f x x I xI

1 1

1

1

1

... ; K

mm m

K

I I

x xf f

fI I

I I

x x

I xI

1 ( )ef f f x xx x I Z V ZI

ˆ TT f IZ I

ˆ , 1,...,e

k k k k

Tf fk K

x x x x

V ZI Ior

11ˆ T TTf f I II Z Z

( )ˆe Tf f x x x xV ZII

Adjoint Sensitivities of Linear Systems

Adjoint Sensitivities of Linear Systems

feasibility and accuracy of the AVM with solvers on structured grids

ˆe

kTk

k k k k k

f f f

x x x x x

V ZI I

( ) ( )k k k I I x I x

L y

y

Lx

y

Fig. 3. Discrete perturbations.

(a) (b)

y x

1

32

A

2

1

3y

yx

A

L L

Fig. 2. Deformation and unwanted perturbations.

(a) (b)

Applications with the FD-TLM

Cavity

2* L

inL arg min Z

5

19

8

3

6

11

10

7

124

2

Fig. 5 (a). The SCN.

( ) is A x V V

x

yz

L

Fig. 4 (a). The initial cavity structure.

x

yz

5

1

3

6

11

10

7

12

9

8

4

2

Fig. 5 (b). The perturbed SCN.

( ) is A x x V V

L L

Fig. 4 (b). The perturbed cavity.

Applications with the FD-TLM

Cavity

0.04 0.045 0.05 0.055 0.06-2

-1

0

1

2x 10

7

L (m)

FFD with one cell perturbationSensitivities with the AVM approach

0.04 0.045 0.05 0.055 0.06

|Zin

|2 /L

(

)

1 2 3 4 5 6 0

1

2

3

4

5

6

7

8

iteration

Co

st f

un

ctio

n

f = | Zin |2

Fig. 6. Sensitivities of the cavity with respect to its length.

Fig. 7. The cost function during the optimization process of the cavity.

Applications with the FD-TLM

Single resonator filter (SRF)

2* *21 21 ( , ) ( , )i k

Li k

i

L arg min S x S x

Fig. 8. The SRF structure.

(a) initial filter

L

d

L

90 x 60z

1 y

Plan

e of

sym

met

ry

(b) perturbed filter

d

90 x 60z

1 y

Plan

e of

sym

met

ry

L

L

z

z

1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

iteration

Co

st f

un

ctio

n

f = | S21-S21t |2

Applications with the FD-TLM

Single resonator filter (SRF)

Fig. 9. Sensitivities of the SRF with respect to the length of the septa.

Fig. 10. The cost function during the optimization process of the SRF.

3 3.2 3.4 3.6 3.8 4 4.2x 10

9

-120

-100

-80

-60

-40

-20

0

frequency (Hz)

/ L

(m

-1)

FFD with one cell perturbationSensitivities with the AVM approach

f

Conclusions

The AVM is implemented into a feasible technique for frequency domain DSA of HF structures

Reduction in the CPU time requirement by a factor of K

Feasibility: does not require re-meshing during the optimization process

Improved accuracy and convergence

Factors affecting the accuracy

Perturbation step size Finite differences for the computation of the

gradients and ef f I x