Challenges of Sensitivity Analysis with Time Domain EM Solvers

8/10/2019 Challenges of Sensitivity Analysis with Time Domain EM Solvers

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Challenges of SensitivityAnalysis with Time DomainEM Solvers

Guoqiang Shen, Helen Tam and Natalia Georgieva

Computational Electromagnetics Laboratory

Department of Electrical and Computer EngineeringMcMaster UniversityCanada

Outline

n Background

n Reviewed existing results

n Our current resultsn Challenges

n Conclusions


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Outline

n Backgroundn Reviewed existing results

n Our current results

n Challenges

n Conclusions

Backgroundn Design Sensitivity Analysis (DSA)

DSA concerns the relationship between the objectivefunction and design variables that describe the shapeof geometry or the material properties to be

optimized.

( ) ( )0

, ,

T

z y x

F G dtdxdydz= p pe eObjective function:

T: fixed final time

G: arbitrary differentiable function

e: field variable vector, p: design variable vector


3/16

Purpose of DSA: to evaluate the derivative of thesystems response with respect to variations in thedesign parameters

0

1, 2, ... ,

T e

p

i i iz y x

F G Gdtdxdydz i n

p p p

= + =

e

e

Design sensitivity:

np: the total number of the design variable pi

/ :eG p the explicit dependence of G on pi

n Adjoint Variable Method (AVM)AVM is an efficient design approach to complex linearand nonlinear problems and it has been proposed inmany areas, such as structural design, circuit theoryand control theory, etc.

Comparing with direct difference method (DDM), AVMoffers significant reduction in CPU time by producingthe response and its gradient through a single full-wave analysis.n DDM: requires solving the system equation for each design

variable.

n AVM: only need to solve the adjoint variable equation in theintroduced adjoint system and it requires only a moderateamount of computation.


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Implementation of AVM: by introducing adjoint systemand solving the adjoint variable equation as well asthe original system.

Vector wave equation:2

02 2

1 r

r c t t

+ =

E E J

Original system:

( )

( )

2

2,

0 0,

00

t

t

+ =

=

=

M K Qe

e

e

e

2M /r c=

1K r

=

0Q / t= J

( )

( )

2

2

0

0

TG

t

T

Tt

+ = =

=

M Ke

( )

( )

2

2

0 0

0 0

TG

+ =

=

=

M Ke

T t=

Adjoint system:

Terminal condition for t Initial condition for

Backward time scheme


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Design sensitivity evaluated by AVM:

T

0

eT

i i iz y x

F Gdtdxdydz

p p p

= +

R

2

2t

=

R Q M K

%%e e

%e is independent of the design variable ip

.

Outline

n Background



n Conclusions


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Published achievements

n Results by FETD(Y.S. Chung et al. 2001)Design microwave passive devices with small return

loss by FETD technique

AVM is used to evaluate design sensitivity

Unstructured triangular elements used

n Results by FDTD(Y.S. Chung et al. 2000, 2001)

Using uniqueness theorem, adjoint variable equationsfrom FETD are derived as coupled Maxwells curlequations and can be solved by the FDTD method.

AVM is used to evaluate design sensitivity.

Unstructured quadrilateral grids and DSI techniqueare used to model the design space, and PML isemployed as ABC.


7/16

Outline

n Backgroundn Reviewed existing results

n Our current results

n Challenges

n Conclusions

Our current results

n Methodology

To evaluate the design sensitivity of the design variablesby employing AVM in the FDTD method.

execute the

original system

execute the

adjoint system

execute the

design

sensitivity

algorithmi

F


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Original system

Inside the original

system

Compute and

record the adjoint

current

Compute the

objective function

for convergence

check

Record the electric

field in the

perturbation

domain

Adjoint system

Inside the adjointsystem

Reverse the signof Maxwellsequations

Record the adjointfield in the

perturbationdomain


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Design sensitivity algorithm

Inside the design

sensitivity algorithm

Retrieve the data files of

the original field and the

adjoint field; save the

data into arrays

Update the adjoint

variable equation with

the corresponding fields

at each time step

n Numerical results and comparisonStep waveguide junction structure

The flow of the width variation in the step waveguide junction structure(symmetric about y-axis)

10

10y

300z

z

yPort 1

Port 2

Mid-section


10/16

Objective function:

2

0

2

0 1

2

2

T

x y

ymaxT

y

F zdxdydt

F z xdydt=

=

=

x

x

e

e

Design variable: the width of the mid section y.

Port 2Mid-section

Port 1

LengthWidthHeight

1 x

1 x

1 x

10 y

10 y

5 13y y

147 z

10 z

147 z

Comparison of the design sensitivity between the adjointvariable method and the central finite difference


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Waveguide H-plane filter

y

z

the middle septum

28 y

600 z

The layout of the waveguide H-plane filter(symmetric about y-axis)

Objective function:

2

0

2

0 1

2

2

T

x y

ymaxT

y

F zdxdydt

F z xdydt=

=

=

x

x

e

e

Design variable: the width of the mid section y.

4th

3rd and 5th

2nd and 6th

1st and 7thLengthWidthHeightSepta

1

1

1

1

7 y

10 y

6 15 y

9 y

1 z

1 z

1 z

1 z


12/16

The design sensitivity of the septum waveguide

Single-section transformer

The layout of the single-section transformer(symmetric about y-axis)

10z

12y

600z

20y

Mid-section

Port 2

Port 1

z

y


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The design sensitivity of the single-section transformer

Outline

n Background



n Conclusions


14/16

Challenges

n Advantages of AVMComputationally efficient as it only requires two

simulations regardless of the number of design

variables.

AVM with FDTD can provide broad-band sensitivityestimation.

n Limitations of current algorithmExcitation cannot have a DC component.

The design variables can not be freely chosen.

The two ports are expected to have same width inorder to reduce the reflection.

For those with few field points, the results will be less

accurate.Absorbing boundary conditions affect the accuracy.


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n Ways to improve the algorithm Increase the number of observation field points, i.e.,

increase the observation domain.

Use more efficient absorbing boundary conditions,such as PML (Perfectly Matched Layer) as ABC. Theaccuracy of the excitation of the adjoint systemdepends much on the efficiency of ABC.

Outline

n Background



n Conclusions


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Conclusions

n Overview of DSA and AVM techniques.n Implementation with FETD and FDTD methods.

n Advantages and limitations.

n Challenges and improvements.

References

n Y.S. Chung, J. Ryu, C. Cheon, I.H. Park and S.Y. Hahn, Optimaldesign method for microwave device using time domain method anddesign sensitivity analysis Part I: FETD case, pp. 3289-3293,IEEE Trans. Magnetics, vol. 37, No. 5, Sep. 2001.

n Y.S. Chung, C. Cheon, I.H. Park and S.Y. Hahn, Optimal designmethod for microwave device using time domain method and designsensitivity analysis Part I: FDTD case, pp. 3255-3259, IEEE Trans.Magnetics, vol. 37, No. 5, Sep. 2001.

n Y.S. Chung, C. Cheon, I.H. Park and S.Y. Hahn, Optimal shapedesign of microwave device using FDTD and design sensitivityanalysis, pp. 2289-2296, IEEE Trans. Microwave Theory andTechniques, vol. 48, No. 12, Dec. 2000.

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Challenges of Sensitivity Analysis with Time Domain EM Solvers

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