ADJOINT VARIABLE METHODS FOR DESIGN SENSITIVITY ...
Transcript of ADJOINT VARIABLE METHODS FOR DESIGN SENSITIVITY ...
ADJOINT VARIABLE METHODS FOR DESIGN SENSITIVITY ANALYSIS WITH THE METHOD OF
MOMENTS
N.K. Georgieva, S. Glavic, M.H. Bakr and J.W. Bandler
McMaster University, CRL223, 1280 Main Street West, Hamilton, ON L8S 4K1, Canada
e-mail: [email protected]: (905) 525 9140 ext. 27141fax: (905) 523 4407
McMasterUniversity
Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
Outline
Introduction and Objectives
design sensitivity analysis
The Adjoint Variable Method
the direct differentiation method
the adjoint variable method
computational efficiency, feasibility, accuracy
Applications with Frequency-Domain Solvers
Conclusions
McMasterUniversity
Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
Introduction and Objectives
Design sensitivity analysis
sensitivity of the state variables
sensitivity of the response (or objective) function
Objectives
obtain the response and its gradient in the design variable space through a single full-wave analysis
applications with frequency-domain solvers
feasibility of the approach
McMasterUniversity
Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
The Adjoint Variable Method
the linear EM problem( )Z x I = V
1[ ]Tnx x=x - design parameters
1[ ]TmI I=I - state variables
define a scalar function (response function, objective function)
( ), ( )f x I xobjective
1 2 n
f f ffx x x
∂ ∂ ∂∇ = ∂ ∂ ∂
xsubject to ( )f∇x Z x I = V,
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Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
The Adjoint Variable Method
state variable sensitivity (direct differentiation method, DDM)E.J. Haug, K.K. Choi and V. Komkov, Design Sensitivity Analysis of Structural Systems, 1986J.W. Bandler, Optimization, vol. 1, Lecture Notes, 1994
McMasterUniversity
Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
1;
m
f ffI I
∂ ∂∇ = ∂ ∂
I
1 1
1
1
n
m m
n
I Ix x
I Ix x
∂ ∂ ∂ ∂
∇ = ∂ ∂ ∂ ∂
x I
1 ( )− ∇ = ∇ −∇ x x xI Z V ZI
ef f f∇ = ∇ +∇ ⋅∇x x I x I
1 , 1, ,i i i
i nx x x
− ∂ ∂ ∂= − = ∂ ∂ ∂
…I V ZZ I
The Adjoint Variable Method
response function sensitivity(adjoint variable method, AVM)
1 ( )ef f f − ∇ = ∇ +∇ ⋅ ∇ −∇ x x I x xZ V ZI
[ ]11ˆ T TTf f−− = ∇ ⋅ = ∇ I II Z Z
[ ]ˆ TT f= ∇ IZ I
ˆ ( )e Tf f ∇ = ∇ + ∇ −∇ x x x xI V ZI
ˆ , 1,2, ,Te
i i i i
ff i nx x x x
∂∂ ∂ ∂= + − = ∂ ∂ ∂ ∂
…V ZI I
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Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
The Adjoint Variable Method
computational efficiency (single excitation mode)
FDA 1n +
DDM
AVM
LU-decompositions Back-substitutions
1n +
1
1 1
n
McMasterUniversity
Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
The Adjoint Variable Method
feasibility and accuracy of the AVMfinite-difference approximations within the AVM
ˆ , 1, ,Te
i i i i
ff i nx x x x
∂∂ ∂ ∂= + − = ∂ ∂ ∂ ∂
…V ZI I
the matrix sensitivity
[ ]ˆ Tf= ∇IV
, 1, ,i i
i nx x∂
=∂
…Z Z
[ ]ˆ TT f= ∇ IZ I
1
ˆ , ,m
f fI I
…V
the adjoint excitation
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Applications
1. Input impedance of a dipole (Pocklington’s eqn., complex code)sensitivity with respect to the normalized length
/nL L λ=/in nR L∂ ∂ /in nX L∂ ∂ subject to =ZI V
(1) finite-difference approach (FDA):
( ) ( ) ( ) ( )
( )( ) ( ) ( )k k k k
in n in n n in nk
n n
Z L Z L L Z LL L
∂ + −∂
( ) ( )0.01k kn nL L=
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Applications
adjoint variable method (AVM):the matrix sensitivity
( ) ( ) ( )
( ) ( )
( ) ( )k k kij ij n n ij nk k
n n
Z Z L L Z LL L
+ − ( ) ( )0.01k kn nL L=
(2) complete re-meshing: full ∆Z matrix(3,4) boundary layer: sparse ∆Z matrix
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Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
( )kL
( ) ( )k kL L+
δ
δ δ+Fig. 1. The dipole and the boundary layer concept (S. Amari, 2001).
0.005a λ=
Applications
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Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
the adjoint excitation (analytical)
2(1/ ) 1ˆ ˆ, 0 forin b
b jb b b
Z IV V j bI I I
∂ ∂= = = − = ≠∂ ∂
Fig. 2. Derivative of the input resistance of the dipole with respect to .nL
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-3000
-2000
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0
1000
2000
3000
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2/nL L λ=
/(
)in
nR
L∂
∂Ω
AVM, BL ( 0.5 )n nL δ=AVM, BL ( 0.1 )n nL δ=AVM, no BL ( 0.01 )n nL L=FDA ( 0.01 )n nL L=
Applications
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Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
Fig. 3. Derivative of the input reactance of the dipole with respect to .nL
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-3000
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-1000
0
1000
2000
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
/(
)in
nX
L∂
∂Ω
/nL L λ=
AVM, BL ( 0.5 )n nL δ=AVM, BL ( 0.1 )n nL δ=AVM, no BL ( 0.01 )n nL L=FDA ( 0.01 )n nL L=
Applications
2. Input impedance of a Yagi-Uda array
sensitivity with respect to the normalized separation distance driver-reflector and the normalized reflector length
1 1[ ]Tn nl s=x
1s2s
3s4s
5s
1l2l
3l4l
5l6l
2a
z y
McMasterUniversity
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0.0030.340.26070.4060.450.5243
2 /l λ /dl λ 1 /s λ /ds λ1 /l λ /a λ
3 4 5 6 2 3 4 5;d dl l l l l s s s s s= = = = = = = =
Fig. 4. The geometry of the Yagi-Uda array (initial design).
Applications
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Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
1/
()
inn
Rs
∂∂
Ω( )inR Ω
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-100
-50
0
50
100
150
200
250
300
350
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.540
45
50
55
60
65
70
75
80
85
1 1 /ns s λ=
inR
1/in nR s∂ ∂
FDA central FD 1%1AVM analytical / nZ s∂ ∂
1AVM / 1%nZ s1AVM / 5%nZ s
Fig. 5. Input resistance sensitivity with respect to s1n .
Applications
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-150
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-50
0
50
100
150
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-33
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-8
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2
7inX
1/in nX s∂ ∂
FDA central FD 1%1AVM analytical / nZ s∂ ∂
1AVM / 1%nZ s1AVM / 5%nZ s
1 1 /ns s λ=
1/
()
inn
Xs
∂∂
Ω( )inX Ω
Fig. 6. Input reactance sensitivity with respect to s1n .
Applications
3. Gain of a Yagi-Uda array (Pocklington’s eqn., real code)
sensitivity with respect to the normalized separation distances
/ , 1, ,5ks s kλ= = …
/ kG s∂ ∂ subject to =ZI V,Re Im =
II
I
ˆ , 1, ,5Te
nn ni ii
G is s
Gs
∂ ∂= = ∂ ∂
∂∂
…Z- I I
the gain sensitivity depends on explicitlynis
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Applications
Vthe adjoint excitation is a full vector
analytical
ˆRe( )
ˆIm( )
kk
k mk
GVIGV
I+
∂=∂
∂=∂
1, ,k m= …
ˆRe( )
ˆIm( )
kk
k mk
GVIGV
I+
finite differences
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Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
Applications
Fig. 7. Gain and gain sensitivity of the Yagi-Uda array with respect to .4ns
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-8
-6
-4
-2
0
2
4
6
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65
12.5
12.75
13
13.25
13.5
3/
nG
s∂
∂ Gain
3 3 /ns s λ=
ˆ(analytical )VAVM 1%
GainFDA 1%
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Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
Applications
Fig. 8. Gain sensitivity of the Yagi-Uda array with respect to ; finite-difference approximation of .
4nsV
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-10
-8
-6
-4
-2
0
2
4
6
8
10
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.653 3 /ns s λ=
3/
nG
s∂
∂
AVM 1%
ˆanalytical Vˆfinite-difference 1%Vˆfinite-difference 5%V
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Applications
4. Optimization of the Yagi-Uda array for maximum gain and an input impedance of 73 Ω
design parameters
1 2 3 4 5[ ]Tn n n n ns s s s s=x
objective function
( ) ( )2 2 2( ) 0.5 Re 73 Im 0.5in inf Z Z G = − + − x
we start from a design already optimized for gain only
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Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
Applications
iteration
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0
50
100
150
200
250
1 2 3 4 5 6 7 8 9 10 11 12 13
obje
ctiv
e fu
nctio
n
Fig. 9. The progress of the objective function during the optimization of the input impedance and the gain of the Yagi-Uda array.
McMasterUniversity
Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
Applications TABLE I
DESIGN PARAMETERS, INPUT IMPEDANCE AND GAIN OF THE YAGI-UDA ARRAY DESIGN
1ns 2ns 3ns 4ns 5ns inR inX
13.750.3871.800.39660.40460.37710.41680.290613
13.770.9271.510.39370.40640.37490.41750.288412
13.59-1.2772.260.38250.40570.36850.41930.287411
13.45-1.0072.990.36270.42050.37770.40610.299910
13.250.9775.630.36450.36070.37940.43570.25319
13.415.4672.850.33620.48220.38440.39230.30628
12.91-5.7772.360.34320.46530.35350.39860.32147
12.87-5.9970.230.39090.42040.39530.37440.34506
13.9211.1865.850.40230.45190.42320.36130.30865
13.0813.2573.330.41580.45910.42290.37200.31584
11.19-16.2681.020.35440.41220.36390.42940.35443
11.08-23.5277.770.37650.38530.33010.40500.34552
15.08-4.1547.100.43530.44710.37350.34000.26071
G
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Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
Applications
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Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
5. Optimization of a patch antenna for an input impedance of 50 Ω
design parameters
objective function
W
L
Tw
2.321.59 mm5 mm2.5 mm15 mm90 mm
14 mm
rhwTLWS
ε =======
S[ ]TL W S=x
( ) ( )2 2( ) Re 50 Im in inf Z Z= − +x
Applications (0) [50 90 14] (mm)T=x (4) [51.51 96.39 15.004] (mm)T=x
1 2 3 4--10
0
10
20
30
40
50
60
0
500
1000
1500
2000
2500
iteration
Re inZIm inZobjective function
resi
stan
ce/re
acta
nce
()
Ω objective function
Fig. 10. The progress of the objective function during the optimization of the input impedance of the patch antenna.
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Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
Conclusions
The AVM is implemented into a feasible technique for the frequency-domain DSA of HF structures
reduction of the CPU time requirements for the DSA by a factor of n to (n+1)
improved accuracy and convergence
feasibility: does not require significant modification of existing codes
Factors affecting the accuracyfinite differences with the matrix: insignificantix∂ Zfinite differences with : significantkI f∂
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Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory
Conclusions
Applications of the DSA based on the AVMoptimizationmodelingstatistical and yield analysis
Limitations
linear frequency-domain analysisextension to nonlinear frequency-domain analysis is straightforward
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Department of Electrical and Computer EngineeringComputational Electromagnetics Laboratory