ADJOINT VARIABLE METHODS FOR DESIGN SENSITIVITY ANALYSIS WITH THE METHOD OF MOMENTS

Transcription

1 ADJOINT VARIABLE METHODS FOR DESIGN SENSITIVITY ANALYSIS WITH THE METHOD OF MOMENTS N.K. Georgieva, S. Glavic, M.H. Bakr ad J.W. Badler, CRL223, 1280 Mai Street West, Hamilto, ON L8S 4K1, Caada tel: (905) ext fax: (905)

2 Outlie Itroductio ad Objectives desig sesitivity aalysis The Adjoit Variable Method the direct differetiatio method the adjoit variable method computatioal efficiecy, feasibility, accuracy Applicatios with Frequecy-Domai Solvers Coclusios

3 Itroductio ad Objectives Desig sesitivity aalysis sesitivity of the state variables sesitivity of the respose (or objective) fuctio Objectives obtai the respose ad its gradiet i the desig variable space through a sigle full-wave aalysis applicatios with frequecy-domai solvers feasibility of the approach

4 The Adjoit Variable Method the liear EM problem x = [ ] T x1 x Z( x) I =V - desig parameters I = [ ] T I1 I m - state variables defie a scalar fuctio (respose fuctio, objective fuctio) f xi, ( x) objective x f ( ) subject to Z( x) I =V, f f f x f = x1 x2 x

5 The Adjoit Variable Method state variable sesitivity (direct differetiatio method, DDM) E.J. Haug, K.K. Choi ad V. Komkov, Desig Sesitivity Aalysis of Structural Systems, 1986 J.W. Badler, Optimizatio, vol. 1, Lecture Notes, xi Z xv x( ZI) = I 1 V Z = Z I, i = 1,, xi xi xi e x f x f I f x I = + f f I f = ; I1 Im I1 I1 x1 x x I = Im I m x1 x

6 The Adjoit Variable Method respose fuctio sesitivity (adjoit variable method, AVM) e 1 f f f x x I Z xv x( ZI) = + 1 T T 1 f I Iˆ = Z = Z [ f ] T Z Iˆ = I e ˆ T x f = x f + I xv x( ZI) f e f ˆ T V Z = + I I, i= 1,2,, xi xi xi xi T [ f ] I T

7 The Adjoit Variable Method computatioal efficiecy (sigle excitatio mode) FDA + 1 LU-decompositios Back-substitutios +1 DDM 1 AVM 1 1

8 The Adjoit Variable Method feasibility ad accuracy of the AVM fiite-differece approximatios withi the AVM the matrix sesitivity f e f ˆ T V Z = + I I, i= 1,, xi xi xi xi Z Z, i = 1,, xi xi the adjoit excitatio T ˆ T ˆ T Z I = [ I f ] V = [ I f ] ˆ f f V,, I1 I m

9 Applicatios 1. Iput impedace of a dipole (Pockligto s eq., complex code) sesitivity with respect to the ormalized legth L = L/ λ R / L X / L subject to ZI = V i i (1) fiite-differece approach (FDA): ( k) ( k) ( k) ( k) i i i ( k) L L Z ( L ) Z ( L + L ) Z ( L ) L ( k) ( k) = 0.01L

10 Applicatios adjoit variable method (AVM): the matrix sesitivity ( k) ( k) ( k) ij ij + ij ( k) ( k) L Z Z ( L L ) Z ( L ) L L ( k) ( k) = 0.01L (2) complete re-meshig: full Z matrix (3,4) boudary layer: sparse Z matrix ( k ) L δ a = 0.005λ L ( k) ( k) + L δ + δ Fig. 1. The dipole ad the boudary layer cocept (S. Amari, 2001).

11 Applicatios the adjoit excitatio (aalytical) ˆ Zi (1/ Ib) 1 V, ˆ b = = = V 0 for 2 j = j b I I I b b b / ( ) i R L Ω FDA ( L = 0.01 L) AVM, o BL ( L = 0.01 L) AVM, BL ( L = 0.1 δ ) AVM, BL ( L = 0.5 δ ) L = L/ λ Fig. 2. Derivative of the iput resistace of the dipole with respect to. L

12 Applicatios / ( ) i X L Ω FDA ( L = 0.01 L) AVM, o BL ( L = 0.01 L) AVM, BL ( L = 0.1 δ ) AVM, BL ( L = 0.5 δ ) L = L/ λ Fig. 3. Derivative of the iput reactace of the dipole with respect to. L

13 Applicatios 2. Iput impedace of a Yagi-Uda array sesitivity with respect to the ormalized separatio distace driver-reflector ad the ormalized reflector legth x = [ l s ] T 1 1 z 2a y l 1 l 2 l 3 l 6 l 5 l 4 s 4 s 3 s 2 s 5 l λ l λ l / λ s 1 / λ s / λ a / λ 1 / 2 / d l = l = l = l = l ; s = s = s = s = s d d d s 1 Fig. 4. The geometry of the Yagi-Uda array (iitial desig).

14 Applicatios Ri / s1 ( Ω) R i R i / s 1 Ri ( Ω) FDA cetral FD 1% AVM aalytical Z / s1 AVM Z/ s1 1% AVM Z/ s 5% s1 = s1/ λ Fig. 5. Iput resistace sesitivity with respect to s 1.

15 Applicatios Xi / s1 ( Ω) X i X i / s 1 FDA cetral FD 1% AVM aalytical Z / s1 AVM Z/ s1 1% AVM Z/ s 5% 1 Xi ( Ω) s1 = s1/ λ Fig. 6. Iput reactace sesitivity with respect to s 1.

16 Applicatios 3. Gai of a Yagi-Uda array (Pockligto s eq., real code) sesitivity with respect to the ormalized separatio distaces sk = s/ λ, k = 1,,5 Re{ I} G/ sk subject to ZI = V, I = Im{ } I the gai sesitivity depeds o explicitly G s i s i eg ˆ T Z = -I I, i = 1,,5 si si

17 Applicatios the adjoit excitatio aalytical Vˆ k Vˆ G = Re( I ) k+ m k G = Im( I ) k Vˆ is a full vector fiite differeces Vˆ k Vˆ G Re( I ) k+ m k G Im( I ) k k = 1,, m

18 Applicatios G/ s Gai FDA 1% AVM 1% (aalytical Vˆ ) Gai s3 = s3/ λ Fig. 7. Gai ad gai sesitivity of the Yagi-Uda array with respect to s 4.

19 Applicatios G/ s AVM 1% aalytical Vˆ fiite-differece Vˆ 1% fiite-differece Vˆ 5% s3 = s3/ λ Fig. 8. Gai sesitivity of the Yagi-Uda array with respect to s 4 ; fiitedifferece approximatio of Vˆ.

20 Applicatios 4. Optimizatio of the Yagi-Uda array for maximum gai ad a iput impedace of 73 Ω desig parameters x = [ s s s s s ] T objective fuctio ( ) ( ) i f( x) = 0.5 Re{ Zi} 73 + Im{ Z } 0.5G we start from a desig already optimized for gai oly

21 Applicatios objective fuctio iteratio Fig. 9. The progress of the objective fuctio durig the optimizatio of the iput impedace ad the gai of the Yagi-Uda array.

22 Applicatios TABLE I DESIGN PARAMETERS, INPUT IMPEDANCE AND GAIN OF THE YAGI-UDA ARRAY DESIGN s1 s2 s3 s4 s5 Ri X i G

23 Applicatios 5. Optimizatio of a patch atea for a iput impedace of 50 Ω desig parameters x = objective fuctio [ LWS] T ε r = 2.32 h = 1.59 mm w = 5mm T = 2.5 mm L = 15 mm W = 90 mm S = 14 mm ( ) ( ) 2 2 i f( x) = Re{ Z } 50 + Im{ Z } i w T S L W

24 Applicatios (0) [ ] T (4) T x = (mm) x = [ ] (mm) resistace/reactace ( Ω ) Re{ Z i } Im{ Z i } objective fuctio objective fuctio Fig iteratio The progress of the objective fuctio durig the optimizatio of the iput impedace of the patch atea. 0

25 Coclusios The AVM is implemeted ito a feasible techique for the frequecy-domai DSA of HF structures reductio of the CPU time requiremets for the DSA by a factor of to (+1) improved accuracy ad covergece feasibility: does ot require sigificat modificatio of existig codes Factors affectig the accuracy fiite differeces with the x i Z matrix: isigificat fiite differeces with I f : sigificat k

26 Coclusios Applicatios of the DSA based o the AVM optimizatio modelig statistical ad yield aalysis Limitatios liear frequecy-domai aalysis extesio to oliear frequecy-domai aalysis is straightforward