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McMaster University. An Efficient Numerical Technique for Gradient Computation with Full-Wave EM Solvers. Shirook M. Ali * and Natalia K. Nikolova. * e-mail: [email protected] tel: (905) 525 9140 ext. 27762 fax: (905) 523 4407. Department of Electrical and Computer Engineering - PowerPoint PPT Presentation
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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: [email protected]
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