Towards the Automated Design of Phased Array Ultrasonic Transducers – Using Particle Swarms to find “Smart” Start Points

Stephen Chen, York UniversitySarah Razzaqi, University of

QueenslandVincent Lupien, Acoustic Ideas

Inc.

June 26, 2007 IEA/AIE 2007

Phased Array Ultrasonic Transducers

A non-mechanical way to direct an energy beam Useful for Non-Destructive Evaluation

June 26, 2007 IEA/AIE 2007

Continuum Probe Designer™

Product of Acoustic Ideas Inc. Automated design tool that creates

an optimized probe for a given inspection task

Removes “art” of design

June 26, 2007 IEA/AIE 2007

Continuum Probe Designer™ Components

Cost function generator uses exclusive patent-pending technology to design an optimized probe

Continuum Probe Designer™

GUI

Optimization solver

Cost function generator

June 26, 2007 IEA/AIE 2007

Optimization Solver

The optimized probe is developed for a given probe geometry

Finding the best probe geometry is another optimization task

In this paper, the probe designer is treated as a “cost function generator”

June 26, 2007 IEA/AIE 2007

Optimization Objective

Probe costs are directly related to the number of elements used in a design

Existing instrumentation can only control 32 independent channels at a time

June 26, 2007 IEA/AIE 2007

An Evolution Strategy for the Optimization Solver (CEC2006)

Standard (1+λ)-ES with λ = 3 Performs significantly better than

gradient descent (i.e. fmincon) Note: fmincon takes about an hour

and uses about 300 evaluations

June 26, 2007 IEA/AIE 2007

Evolution Strategy vs. fmincon

Tested on one expert selected and 29 random start points ES results are much better and more consistent ES results are still not good enough

fmincon (1+λ)-ES

76.2 32.5

70.8 3.9

5 18

June 26, 2007 IEA/AIE 2007

Independent Parallel Runs

High standard deviation suggests that using multiple runs will lead to easy improvements Results are better, but still not good enough

(1+λ)-ESFour

parallel

32.5 31.3

3.9 2.9

18 22

June 26, 2007 IEA/AIE 2007

“Smart” Start Points

High correlation between ES solution and quality of random start point Use random search to find “smart” points Better results again

Four parallel

“Smart” start pts

31.3 30.1

2.9 3.2

22 25

June 26, 2007 IEA/AIE 2007

Analyzing “Smart” Start Points

Is perceived correlation significant? From 120 random start points, apply the (1+λ)-ES to the 30 worst and best

30 Worst 30 Best 30 Worst 30 Best

770.5 121.5 34.1 31.7

87.1 66.3 5.7 3.1

June 26, 2007 IEA/AIE 2007

“Smart” Start Points on the TSP

Is correlation an obvious/trivial observation? Correlation does not exist on TSP

30 Worst 30 Best 30 Worst 30 Best

1230% 1128% 11% 11%

18% 16% 1.3% 2.4%

June 26, 2007 IEA/AIE 2007

Coarse Search does not Help on TSP

Coarse search for better starting points does not improve the performance of two-opt on the TSP

Four parallel

“Smart” start pts

9.2% 8.8%

1.1% 1.4%

June 26, 2007 IEA/AIE 2007

Improve Coarse Search

Generate 50 random points Use best 4 to seed 4 PSOs Design PSOs to favour exploration

over convergence

June 26, 2007 IEA/AIE 2007

PSO vs. Random Searchto find “Smart” Start Points

PSO finds even better start points Improved “smart” start points lead to an even better performance

Random search

PSO

30.1 29.2

3.2 1.9

25 27

June 26, 2007 IEA/AIE 2007

Exploiting Global Convexity

Search space is globally convex Seek centre of search space by coordinating individual ESs with crossover

PSOWith

Crossover

29.2 28.5

1.9 1.3

27 30

June 26, 2007 IEA/AIE 2007

Current Work

Exploring Coarse Search – Greedy Search Inspired by WoSP (CEC2005) Different from memetic algorithms

(which apply greedy search to every search point)

Useful for expensive evaluations Useful for non-globally convex search

spaces

June 26, 2007 IEA/AIE 2007

Rastrigin function

Globally convex Average value of each “well” is directly related to the quality of the local optima

June 26, 2007 IEA/AIE 2007

Schwefel function

NOT globally convex Average value of each “well” should still be directly related to the quality of the local optima

June 26, 2007 IEA/AIE 2007

Summary

Achieved important level of performance on benchmark test suite for a difficult real-world problem

Demonstrated potential of coarse search-greedy search combinations