1. Automatic design using FD TD simulator in an optimization loop
Presented by: Wojciech K.Gwarek
with contributions from : Malgorzata Celuch-Marcysiak, Przemyslaw Miazga, Maciej Sypniewski and Andrzej Wieckowski
the authors are with:
Institute of Radioelectronics
Warsaw University of Technology
Warszawa, Nowowiejska 15/19 Poland

2. Abstract :
We discuss the elements of a general optimization scheme consisting of : 1. A parametric shape editor
2. An FD-TD simulator
3. An optimizer
4. A goal function calculator
The aim is to discuss the possibilities provided to engineers by general-purpose tools available commercially. In particular we examine the possibilities provided by QWED’s QuickWave-3D FD-TD simulator and widely used Matlab ® toolbox. We discuss particular features of the FD-TD method applied in electromagnetic simulations for automatic design purposes, the sources of errors of analysis and ways to make it more efficient.
We show the results of automatic design on 5 examples:
a coaxial connector, a microstrip to waveguide transition, a waveguide diode mount, a septum polarizer and a waveguide filter. In each case we discuss the sources of errors, the ways to speed up the optimization process and to enhance the chances of convergence to a satisfactory solution. We also provide hints on practical applications.

3. General scheme for automatic design using FD TD simulator in an optimization loop

4. General requirements:
1. Simulator which:
- is fast enough to produce the single simulation results in seconds or minutes, not in hours
- allows smooth change of the results as a function of the change in dimensions
2. Parametric shape editor which is:
- universal, allowing a large variety of shapes and media
- easy to use
3. Comprehensive and programmable goal function calculator
4. Optimizer which
- is relatively insensitive to rough goal function
and preferably
- is able to skip local minimum to look for the global one.

5. Why should we use FD-TD method
When applied to linear circuits with pulse excitation it permits to extract wide-band circuit parameters after just one simulation
It is very fast especially when applied to a wide scope of deterministic problems concerning S-matrix or radiation pattern extraction. It produces acceptable computing times even for relatively large problems.
It has clear physical interpretations giving good insight into the operation of the circuit
It is easy to apply with a variety of media forming the analyzed structure
There are practically no problems with parasitic solutions and the algorithms are not sensitive to computer round-off errors
Disadvantages to be fought:
Fast, explicit integration schemes used in time domain enforce additional restrictions on meshing flexibility
High-Q circuits (like narrow band filters) need special consideration

6. Hints how to use FD-TD electromagnetic simulation in automatic design and/or optimization processes
whenever possible reduce the number of considered dimensions or use symmetry conditions
if possible perform segmentation and assign specific goals to optimization of the specific segments
check carefully the number of needed FD-TD iterations; avoid exciting off-band resonances
check the accuracy of the FD-TD approximation and continuity of the goal function with respect to change of variables
try to avoid the situation when the change of a dimension drastically modifies meshing in sensitive areas
try to economize the computing time by taking into account systematic errors
prior to final setting of the set of variables and scaling factors run sensitivity analysis and/or grid search to see the type of dependence on major variables

7. Whenever possible reduce the number of considered dimensions
Whenever possible reduce the number of considered dimensions. Problems presented below are all 2-D ( or more precisely vector 2D)

8. Example 1: Optimisation of N to LCM20 coaxial connector for 0-8GHz band Original commercial design:
Optimised design:

9. Comparison of the simulated S11 for the original and the optimized design:
Reference:P.Miazga, W.Gwarek, IEEE Trans MTT, May 1997, pp

10. Example 2 : Object: Microstrip to waveguide transition Design goal: Lowest possible S11 in the band GHz
Microstrip-to-waveguide transition considered in Example2. Picture shows half of the structure assuming magnetic symmetry plane in the middle. Upper and lower waveguide walls are not shown for clarity of the picture.

11. Example 2 (cont): The optimized variables in 2 options: A: One variable optimization with respect to lst=lst1=lst2=lst3 B: Six variable optimization with respect to lst1,lst2,lst3,hs2,hs3,sis
Side and upper view of the microstrip to waveguide transition with parameters as they are used in optimization

12. Example of a parser type parametrized input of a 3-D shape Editor for optimization purposes
Microstrip-to-waveguide transition described in the UDO language by QWED. Header appearing in the Editor (below) and parts of the UDO code (right)
aacomment="Microstrip to waveguide transition";
bitmap="nobitmap.bmp";
PAR("Name ",oname,"mstowg");
PAR("wg width (wga)",wga,23);
PAR("wg height (wgb)",wgb,11);
PAR("ms width (msw)",msw,3.75);
PAR("ms height (msh)",msh,1.27);
PAR("ms substr. width",mssw,30);
PAR("ms substrate",med,substr);
PAR("ref. dist. (rd)",rd,7);
PAR("length of step1 (lst1)",lst1,6);
PAR("length of step2 (lst2)",lst2,6);
PAR("length of step3 (lst3)",lst3,6);
PAR("height of step2 (hs2)",hs2,7);
PAR("height of step3 (hs3)",hs3,3);
PAR("substrate insert (sis)",sis,0);
ENDHEADER;
msl= rd*3;
wgl=lst1+lst2+lst3+rd*2;
wgah=wga*0.5;
mswh=msw*0.5;
msswh=mssw*0.5;
ltot=msl+wgl;
hs1=wgb-msh;
OPENOBJECT(oname);
sis1=0;
if sis<0 do sis1=-sis;
endif;
ELEMENT(z,msh,0,med,substr,IN);
NEWLINE(x,y,x+msl-sis1,y);
ADDY(msswh);
ADDX(-msl+sis1);
CLOSELINE;
ENDELEM;
if sis>0 do
NEWLINE(x+msl,y,x+msl+sis,y);
ADDY(wgah);
ADDX(-sis);
endif;
ELEMENT(z+msh,0,0,metal,strip,IN);
NEWLINE(x,y,x+msl+lst1,y);
ADDY(mswh);
ADDX(-msl-lst1);
....
PORT(z,wgb,INPTEMPLATE,UP,inpms,rinpms);
NEWLINE(x,y,x,y+msswh);
NEWLINE(x,y+0.5*mswh,x,y);
PORTEXC(msh,msh);
GETIOPAR("mstowgi.iop");
ENDPORT;

13. Choice of optimizer :
Option A: an optimizer specially prepared to work with a particular simulator (as exemplified belowby the one from QWED)
Advantage: simplicity in use with prepared dialogues
Disadvantage: limited flexibility in choice of the optimization method and the goal function

14. Choice of optimizer :
Option B: a general purpose optimizer (as exemplified below by the one from Matlab® Toolbox). The text on this and next slide is a complete code for running a minimax optimization on 6 variables of the microstrip to waveguide transition using an FD-TD Editor and Simulator activated from a DOS command line
% Microstrip to waveguide trans.
% minimax method
x0=[ ];
global spar;
global start_ch;
global best_ch;
global fbest;
global first;
global iterac;
fbest=1;
first=1;
iterac=1;
OPTIONS(1)=0; %display intermediate results
OPTIONS(2)=0.0001; %termination tolerance x
OPTIONS(3)=0.0001; %termination tolerance F
OPTIONS(14)=2000; %max function call
OPTIONS(16)=0.003; %min change for grad
OPTIONS(17)=0.1; %max change for grad
OPTIONS(18)=1; %step length
grid on
minimax('ms2wgb',x0,OPTIONS);

15. continuation of Matlab® script for 6 variables optimization of the microstrip to waveguide transition
function [w,g] = ms2wgb(x)
global spar;global start_ch;
global best_ch;global fbest;
global first;global iterac;
save 'c:\3dex\matopt\ms2wgb\paramsb' x -ASCII;
dos('"c:\Program Files\Qwed\Qw_3d\Qw_edi\bin\zed.exe" -p"c:\3dex\matopt\ms2wgb\ms2wgb.pro" -m -o1000 -e -q -i');
dos('"c:\Program Files\Qwed\Qw_3d\Qw_sim\bin\ker1.exe" -t "c:\3dex\matopt\ms2wgb\ms2wgb.ta3"');
spar=load('c:\3dex\matopt\ms2wgb\ms2wgb.spl');
w=spar(41:81,2);g=zeros(1,41);
if(first)
first=0;
start_ch=spar(:,2);
end
if(max(w)<fbest)
fbest=max(w); best_ch=spar(:,2); x
fprintf('iter=%d min=%6.2f dB\n',iterac,20*log10(fbest));
plot(spar(:,1)./10^9,20*log10(spar(:,2)),'k-',spar(:,1)./10^9,20* log10(start_ch),'b-',spar(:,1)./10^9,20*log10(best_ch),'g-');
grid on
iterac=iterac+1;

16. Example 2 (cont): The results of optimization of S11[dB] versus frequency : thin blue line - initial solution
medium green line - results of one variable optimization
thick magenta line - results of six variables optimization

17. Example 3 : Matching of a driving point impedande of a waveguide diode mount
Half of the structure is taken due to magnetic symmetry. We try to match the impedance to 100  (corresponding to 50 for the entire structure)
We consider two cases
A) a post made entirely of metal and short-circuited to the waveguide base
B) lower part of the post made of teflon

18. The case A (purely metal post) is resonant and difficult to match wide band. Figure below presents the results of calculation after 1000 iterations for two different distances from the guide short.
In the case of a resonant structure we should assure that
- the number of FD-TD iterations is big enough to limit the ripples caused by truncated Fourier transform
- the step for gradient calculation of the optimizer is big enough to get out of the local minima formed by the ripples
Otherwise we are likely to finish optimization at a local minimum as in the presented case in which we tried to tune the matching frequency from 11.6 to 10.44

19. Prony Postprocessing
One of the solutions to smoothen the ripples is by using a special postprocessing like for example the Prony method.
Picture below presents the result of simulation of S11 versus frequency with 1000 FD-TD iterations of the structure from the previous slides. The results of direct analysis (blue) are compared to the results obtained using QProny algorithm authored by M. Mrozowski . Application of such a procedure is very valuable in the analysis of resonators, filters and multiplexers.

20. Results of optimization of the diode mount for the band 10.5-11.5 GHz
Initial (left) and optimized (right) dimensions of the diode mount and comparison of their characteristics of S11 versus frequency . Optimization was performed on 2 variables : distance of the post from the waveguide short and height of the teflon part of the post

21. Optimization preprocessing - A grid search
When the character of the goal function is difficult to predict or when we suspect to have multiple minima, it may be is advisable to run preliminary calculations on a predefined grid of a pair (or several pairs) of sensitive variables and plot the results on a 3-D or 2-D graph as exemplified below for the Example 3. The waveguide short distance was assumed to be (3,5,7,9,11) and the height of the dielectric part of the post to be (2,4,6,8).
White spot on the right picture presents the point where the minimum has been found in the optimization process.

22. Example 4: A septum polarizer
This and the following slide show a particular design by Saab Ericsson Space.
Next slides show a process of automatic design of such a device using an electromagnetic simulator in an optimization loop.

23. Comparison of measurements by Saab Ericsson Space and simulations by QWED of the phase difference between the output signal of horizontal and vertical polarizations (below) and pictures illustrating propagation of two polarizations at center frequency (right)

24. Segmentation of the polarizer permits to extract the region of the septum which decides on the most important parameter - the axial ratio of circular polarization. Optimization is performed on seven variables as indicated below.
Imperfect matching can be corrected outside the septum region in the segment of rectangular waveguide.

25. spar=load('c:\3dex\matopt\septm3\septm3.spl');
angdif=abs(rem(spar(:,9)- spar(:,12)+360,360.0)-95)*1.745;
modrat=abs(spar(:,8)./spar(:,11)-1)*100;
aux=zeros(1,61);
aux1=zeros(1,61);
aux=atan(angdif(71:131,1)./modrat(71:131,1));
aux1=abs(sin(aux));
aux=abs(cos(aux));
w=angdif(71:131,1).*aux1(1:61,1)+modrat(71:131,1).*aux(1:61,1);
Matlab® script for axial ratio as a goal function
phase difference
between vertical and horizontal polarizations of the optimized polarizer
amplitude ratio
of vertical and horizontal polarisations of the optimized polarizer

26. S11 [dB] versus frequency of the septum part (right) obtained from its optimization for the best axial ratio.
If the matching of the septum part is not considered satisfactory it can be compensated in the guide bent (below left)

27. General scheme of segmentation
Attention: Segmentation may be in general multimodal with higher modes being evanescent. In such a case we face problems of
a) definition of S-matrix with complex reference impedance
b) nonorthogonality of the incident and reflected power waves
Those problems have been thoroughly treated in the recommended paper:
R.Marks, B.Williams - A General Waveguide Circuit Theory, Journal of Research of NIST, Vol97, No.5 Sept.-Oct. 1992, pp

28. Change of dimensions may result in modification of meshing which causes some discontinuity of the goal function even when conformal FD-TD methods are applied. We should take care that with changed dimensions the meshing of sensitive areas does not change its character. Such sensitive areas are metal corners like one of the corners of a septum polarizer presented below. It is better to use mesh snapping planes to reproduce the corner always in a similar way (a and b) than let the software make its arbitrary decisions about its approximation (c) . Change between a) and b) is not so drastic as the change between a and c) a) b) c)

29. Results of calculation of the phase difference between horizontal and vertical polarization produced by the septum polarizer.
1. thin blue curve - initial dimensions of the polarizer, FD-TD cell size 0.5
2. medium green curve - initial dimensions of the polarizer, FD-TD cell size 0.15
3. thick magenta curve - optimized polarizer, FD-TD cell size 0.5
4. v.thick red curve - optimized polarizer, FD-TD cell size 0.15
Optimization was conducted with cell size 0.5 and goal phase difference 950

30. Illustration of the influence of the spectrum of the exciting pulse on the convergence of the FD-TD algorithm.
S11 versus frequency of the septum polarizer excited by a wide-band ( type) pulse observed after 2000, 3000 and FD-TD iterations
S11 versus frequency of the septum polarizer excited by a narrow -band pulse
observed after 1100, 1500 and 3000 FD-TD iterations

31. Example 5 : A 3-resonator waveguide filter
The images present the process of automatic design of the filter with four variables by a minimax method.
S11 (up) and S21 (left), both in dB of the considered filter.
Blue lines show the characteristics for starting conditions and the red lines the characteristics for final design.

32. Conclusions:
The level of state-of-art hardware and software has reached the stage, at which automatic design of a variety of types microwave circuits using a general purpose FD-TD simulator and a general purpose optimizer is a practical alternative for engineers determined to get better design much faster than by the classical approaches. However, the success of such an operation is in many cases not a priori guaranteed and requires a very conscious use of both tools and careful choice of their parameters.This underlines importance of the exchange of experience of different users and software vendors like that provided by contributors to this workshop.