SPACE MAPPING OPTIMIZATION

J.W. Bandler

SOS-98-14-V

June 30, 1998

© J.W. Bandler 1998

No part of this document may be copied, translated, transcribed or entered in any form into any machine without written permission. Address enquiries in this regard to Dr. J.W. Bandler. Excerpts may be quoted for scholarly purposes with full acknowledgement of source. This document may not be lent or circulated without this title page and its original cover.

SPACE MAPPING OPTIMIZATION

John W. Bandler

Simulation Optimization Systems Research Laboratory

and Department of Electrical and Computer Engineering

McMaster University, Hamilton, Canada L8S 4L7

bandler@mcmaster.ca

soya.sos.mcmaster.ca

presented at

Workshop on Microwave Filters and Multiplexers: Theory, Practice and CAD

28th European Microwave Conference, Amsterdam, The Netherlands, October 5, 1998

Introduction

commercial EM simulators are becoming increasingly faster and more accurate

in order to realize their full potential, EM simulators have to be optimizer-driven to automatically adjust designable parameters

the new thrust is to integrate EM simulations directly into the

linear/nonlinear circuit design process transparently to the designer

numerical methods offer excellent accuracy at the expense of

heavy demand on computer resources

practical use of EM simulators is often limited to design

validation

it is commonly perceived that iterative optimization methods

would require too many EM simulations and consequently

consume excessive CPU time

the objective of Space Mapping is to avoid direct optimization

of computationally intensive models

Overview of Presentation

electromagnetic (EM) optimization

Space Mapping (SM) optimization

SM optimization - original approach

SM optimization - aggressive method

SM optimization - trust region method (TRASM)

Frequency Space Mapping (FSM)

combining decomposition and SM in a coherent strategy

optimization of an interdigital filter

TRASM optimization of an HTS filter

literature review

Space Mapping

The Advantages of Space Mapping

the aim is to avoid direct optimization in the CPU-intensive Xem space

the bulk of the computation involved in optimization is carried out in the Xos space

the optimal solution is mapped from the Xos space to the Xem space using the inverse mapping P-1

we expect to obtain a rapidly improved design after each fine model simulation

significantly more efficient than the "brute force" direct EM optimization

a fundamentally new concept in engineering-oriented

optimization practice

Background

assume that Xos (optimization space) and Xem (EM space) have the same dimensionality, i.e.,

xos IRn and xem IRn,

but may not represent the same parameters

the Xos-space model can be comprised of empirical models, or an efficient coarse-grid EM model

the Xem-space model is typically a fine-grid EM model but, ultimately, can represent actual hardware prototypes

we assume that the Xos-space model responses, Ros(xos), are much faster to calculate but less accurate than the Xem-space model responses, Rem(xem)

we initially perform optimization in Xos to obtain the optimal design , for instance in the minimax sense

subsequently, apply SM to find the mapped solution in Xem to reproduce the optimal performance predicted by the

empirical model

The Concept of Space Mapping

(Bandler, Biernacki, Chen, Grobelny and Hemmers, 1994)

our aim is to find an appropriate mapping, P, from the

Xem-space to the Xos-space, i.e.,

such that

we assume that such a mapping exists and is one-to-one within some local modeling region encompassing our SM solution

once the mapping is established, the SM solution is

Original Space Mapping Method

the mapping is established through an iterative process

to obtain the initial approximation to the mapping, , we perform EM analyses at a preselected set of base points in Xem around the starting point

as the first base point we may select the starting point, i.e.,

assuming and represent the same physical parameters, followed by additional base points chosen by perturbation as

this is followed by parameter extraction optimization in Xos to obtain the set of corresponding base points according to

for , where indicates a suitable norm

Original Space Mapping Method (continued)

at the jth iteration, both sets may be expanded to contain mj points which are used to establish the updated mapping

the current approximation is used to estimate as

the process continues until the termination condition

is satisfied, where is a small positive constant, then is our desired P

if not, the set of base points in Xem is augmented by and correspondingly, determined by parameter extraction augments the set of base points in Xos

upon termination, we set as the SM solution

Aggressive Approach to Space Mapping

(Bandler, Biernacki, Chen, Hemmers and Madsen, 1995)

at the SM solution, will closely match ,

where M is the number of iterations needed to converge to an SM solution

hence, after an additional parameter extraction optimization in Xos, the resulting point

approaches the point (optimal solution in Xos), or

as

where is a small positive constant

by setting to 0, we consider the set of n nonlinear equations

of the form

where is a given vector

Aggressive Space Mapping - Quasi-Newton Iteration

let be the jth approximation to the solution and written for

the next iterate is found by a quasi-Newton iteration

by solving the linear system

is an approximation to the Jacobian matrix

in our implementation, is set to the identity matrix

the approximation to the Jacobian matrix is updated by the classic Broyden formula (Broyden, 1965)

Aggressive Space Mapping - Implementation

begin with a point, , representing the optimal design in Xos where is some appropriate objective function

Step 0	initialize , , ,	; stop if

Step 1	solve for

Step 2	set

Step 3	evaluate

Step 4	compute ; if , stop

Step 5	update to

Step 6	set ; go to Step 1

Fully Automated Space Mapping Optimization

two-level Datapipe architecture

Frequency Space Mapping for Parameter Extraction

parameter extraction can be a serious challenge, especially at the starting point, if the model responses are misaligned

Re{S11} using OSA90/hope ( ) and em (---) at

straightforward optimization from such a starting point can lead to a local minimum

Frequency Space Mapping - Mapping and Alignment

to better condition the parameter extraction subproblem

first, we align Ros and Rem along the frequency axis using

this frequency space mapping can be as simple as

at the starting point, we determine and by

where xos and xem are fixed and xos = xem

resulting alignment between OSA90/hope ( ) and em (---):

Frequency Space Mapping:

Sequential FSM (SFSM) Algorithm

we perform a sequence of optimizations to gradually achieve the identity Frequency Space Mapping

we optimize xos to match Ros and Rem:

the values and are updated according to

and

,

respectively, for

K determines the number of steps in the sequence

larger values of K increase the probability of success in the parameter extraction subproblem at the expense of longer optimization time

Frequency Space Mapping:

Exact Penalty Function (EPF) Algorithm

we perform only one optimization to achieve the identity Frequency Space Mapping and optimize xos to match Ros to Rem

the 1 norm version of the EPF formulation is given by

the minimax version is given by

where

and

in both EPF formulations, are kept fixed and must be sufficiently large to obtain the identity mapping and hence the solution to the parameter extraction problem

Frequency Space Mapping - Results

Re{S11} using OSA90/hope ( ) and em (---)

resulting match after applying the FSM algorithm

Decomposition

(Bandler, Biernacki, Chen and Huang, 1997)

partitions a complex structure into a few smaller substructures

each substructure is analyzed separately

the results are combined to obtain the response of the overall structure

2D analytical methods or even empirical formulas can be used for some non-critical regions

full-wave 3D models are adopted for the analysis of the key substructures

A Five-Pole C-Band Interdigital Filter

15 mil thick alumina substrate with

the width of each microstrip is chosen to be 10 mil

quarter wavelength resonators

Interdigital Filter Design

specifications

passband cutoff	f1 = 4.9 GHz, f2 = 5.3 GHz

passband ripple	r = 0.1 dB

isolation bandwidth	BWI = 0.95 GHz

isolation	DBI = 30 dB

the order of the filter is determined as 5

all other dimensions including the gaps and the positions of the tapped lines are obtained by synthesis (Matthaei et al., 1964)

design variables include two gaps between the resonators and four lengths of microstrip lines from an appropriate position of each resonator to its ends

the size of the vias is fixed

The Fine Model of the Interdigital Filter

full-wave EM simulations of the whole structure using Sonnet's em

for good accuracy the grid size has to be sufficiently small

selected grid size: 1 × 1 mil

about 1.5 CPU hours per frequency point on a Sun SPARCstation 10

much longer if losses are included

this translates into considerable EM simulation time for fine frequency sweeps

direct optimization would require many EM analyses and consequently excessive CPU time

Dimensions and Material Parameters of the Filter

FILTER MATERIAL PARAMETERS AND

GEOMETRICAL DIMENSIONS

———————————————————————————————

Parameter	Value

———————————————————————————————

substrate dielectric constant	9.8

substrate thickness (mil)	15

conducting metal thickness (mil)	0

substrate dielectric loss tangent	0/0.001*

conductivity of the metal	∞/5.8×107*

shielding cover height (mil)	75

width of input/output lines (mil)	10

width of each resonator (mil)	10

via diameter (mil)	13

via pad dimensions (mil × mil)	25 × 25

———————————————————————————————

*	loss tangent and conductivity for simulations

without and with losses, respectively

———————————————————————————————

Decomposition of the Interdigital Filter

the coarse model is constructed using decomposition

the substructures are analyzed separately using either EM models with a coarse grid or empirical models

the partial results are then combined through circuit theory to obtain the response of the overall filter

The Coarse Model of the Interdigital Filter

the center shaded 12-port network is analyzed by em with a very coarse grid: 5 × 10 mil

the vias have fixed dimensions - one via is analyzed by em with a grid of 1 × 1 mil only once; in subsequent simulations all vias are represented by their reflection coefficient

all other parts including the microstrip line sections and the open ends are analyzed using the empirical models of OSA90/hope

less than 1 CPU minute per frequency point on a Sun SPARCstation 10

off-grid responses, when needed during optimization, are obtained by interpolation

the coarse model retains most of the adjacent and non-adjacent couplings, thus it provides reasonably accurate results at dramatically faster speed

Design Procedure

first, we optimize the filter using the coarse model

minimax solution is obtained

we check this coarse model solution using the fine model at a few selected frequencies

solid curves	optimized S11 and S21 responses of the

coarse model at the optimal point

circles	fine model responses at

Results of EM Validation

the fine model responses deviate significantly from the optimized coarse model responses

the passband return loss is only about 11 dB and the bandwidth is wider than specified

discrepancies may be due to the coarse grid and some couplings not taken into account by the coarse model

WHAT'S NEXT?

typically, engineers manually tune the design and try to meet design specifications

we offer an automated approach using Space Mapping

Space Mapping Optimization of the Interdigital Filter

SM optimization starts with

after the first iteration, a new point in the Xem space is obtained

the fine model responses of this new point are compared with the coarse model optimal responses

return loss is improved and the bandwidth is reduced at the lower frequency end

Second Iteration of Space Mapping

another iteration of SM produces

the fine model responses at at 13 frequency points are compared with the coarse model optimal responses

only three EM simulations of the fine model were needed

Final EM Validation

a dense frequency sweep is desired

here, simulation includes the conductor and dielectric losses

the fine model responses at

the passband return loss is better than 18.5 dB