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Towards the Automated Design of Phased Array Ultrasonic Transducers – Using Particle Swarms to Find “Smart” Start Points Stephen Chen1 , Sarah Razzaqi2 , and Vincent Lupien3 1

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School of Information Technology, York University 4700 Keele Street, Toronto, Ontario M3J 1P3 sychen@yorku.ca Centre for Hypersonics, Division of Mechanical Engineering University of Queensland, Brisbane, Australia 4072 s.razzaqi@uq.edu.au 3 Acoustic Ideas Inc. 27 Eaton Street, Wakefield, MA 01880 vincent.lupien@acousticideas.com

Abstract. Continuum Probe DesignerTM by Acoustic Ideas Inc. is a tool that can help design the “best” phased array ultrasonic transducer for a given inspection task. Given a specific surface geometry for the ultrasonic transducer, one component of Continuum Probe DesignerTM can determine the number of elements, and the required size and shape of each element to meet a list of ultrasonic inspection goals. Using the number of elements as a cost function, an optimization problem to find the best surface geometry for the transducer is created. Previous work has demonstrated that a (1+λ)-evolution strategy (ES) can be a very effective search technique for this problem. The performance of this ES was improved by starting it from “smart” (i.e. better than random) start points. Particle swarm optimization (PSO) can be used to improve the “smart” start points, and the overall PSO-ES hybrid is capable of finding feasible transducer designs from all of the start points in a benchmark test suite. This level of performance is an important step towards the use of Continuum Probe DesignerTM as a fully automated tool for the design of phased array ultrasonic transducers.

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Introduction

An evolution strategy (ES) [1] has been developed for use with Continuum Probe DesignerTM – a tool developed by Acoustic Ideas Inc. that uses exclusive patent pending technology to design the best possible ultrasonic transducer (or probe) for a specified task [6,10]. In the developed ES, a high correlation was observed between the quality of the initial start point and the quality of the final solution. The performance of this ES was improved by using “smart” start points – start points that are better than random start points. Since the use of random search to find the “smart” start points was so successful [4], it is expected that the use H.G. Okuno and M. Ali (Eds.): IEA/AIE 2007, LNAI 4570, pp. 313–323, 2007. c Springer-Verlag Berlin Heidelberg 2007

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S. Chen, S. Razzaqi, and V. Lupien

of particle swarm optimization (PSO) [5] to find the “smart” start points will lead to an even better performance in the evolution strategy. The selection of PSO was inspired by WoSP (Waves of Swarm Particles) – a particle swam optimization technique that has been modified to find and explore multiple optima [7]. In a traditional PSO implementation, all of the particles will eventually converge and fully explore the region around a single optimum. In WoSP, as particles begin to converge, a short-range force can lead to the “ejection” of particles away from the neighbourhood of the current optimum into new waves of particles that will explore for additional optima. Compared to a traditional PSO, an implementation of WoSP will explore several optima coarsely rather than a single optimum exhaustively. Subsequently, WoSP benefits from being paired with a local optimizer that can find the exact minimum or maximum after the general neighbourhood of the optimum has been found. A local optimizer can also receive unique benefits from WoSP – WoSP can provide a small number of highly diverse and highly promising “smart” start points. The current implementation does not need WoSP to find multiple start points since random search is used to find the four initial start points. Therefore, four independent PSOs are used instead to improve upon these start points before applying a greedier local search technique (i.e. the evolution strategy). The use of WoSP with a local optimizer [7] appears to be quite different from other coarse search and local optimizer combinations. For example, simulated annealing [9] can be viewed as having a coarse search performed at higher temperatures for the start point that will eventually be optimized at lower temperatures [3]. However, simulated annealing (like a traditional PSO) eventually explores only a single optimum exhaustively. At the other end of the spectrum, memetic algorithms [11] perform a special kind of “search over the subspace of local optima” [12] – every search point is locally optimized. In the current application, a single local optimum can take over an hour to find with an evolution strategy. Therefore, it is only practical to explore a small number of optima neighbourhoods exhaustively. This small number (four in the current experiments) makes it impractical to maintain a population of local optima for a memetic algorithm to operate on. Conversely, significant benefits are still possible from exploring multiple optima. These two features of the current problem domain make it an ideal candidate for a coarse search-greedy search hybrid approach. The PSO-ES hybrid approach has been tested on a benchmark probe design problem. The experiments involve 30 initial probe geometries – one selected by an expert and 29 selected randomly. In the previous results, the ES was able to find feasible solutions for at most 25 of the 29 random start points. To have confidence in the ability of Continuum Probe DesignerTM to find acceptable solutions for any given real-world design problem when it is used by a non-expert designer, it is important to have an optimization technique that is robust enough to find feasible solutions for all 29 random start points on the benchmark design problem. The developed PSO-ES hybrid has been able to achieve this level of performance.

Towards the Automated Design of Phased Array Ultrasonic Transducers

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Before presenting these recent developments towards the automated design of phased array ultrasonic transducers, an overview of Continuum Probe DesignerTM is presented in section 2 and previous work on its optimization solver is reviewed in section 3. A new analysis of “smart” start points in section 4 provides a foundation and justification for the current research with PSO that is introduced in section 5. Section 6 has the results for the new PSO-ES hybrid, and these results lead to a discussion of coarse search-greedy search hybrids in section 7. Conclusions and acknowledgements follow.

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Background

Ultrasound can be used to perform nondestructive evaluation (NDE) of structures for many industrial applications. For example, the benchmark design problem used in this paper represents an inspection challenge posed by the Federal Aviation Administration (FAA) for the inspection of titanium disks in aircraft engines. The beam requirements in this inspection challenge can be difficult to achieve with the existing instrumentation which can only control 32 independent channels at a time. The purpose of Continuum Probe DesignerTM is to help scientists and engineers develop feasible and even optimal (e.g. least cost) probes for any inspection task. Continuum Probe DesignerTM consists of two key components: a cost function generator and an optimization solver (see Fig. 1). For a given input (which represents an ultrasonic transducer’s surface geometry), the cost function generator can determine the fewest number of elements that a probe having the input shape will require to perform the specified inspection task. The role of the optimization solver is to find the optimal probe geometry – i.e. the surface geometry for the ultrasonic transducer that minimizes the cost function. The heuristic search techniques discussed in this paper are part of the optimization solver. For the purposes of this paper, the cost function generator of Continuum Probe DesignerTM is treated like a black box. The input into the cost function is a vector of rational numbers that represents the probe’s shape, and the output

TM

Continuum Probe Designer

GUI Optimization solver

Cost function generator

Fig. 1. The components of Continuum Probe DesignerTM

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that needs to be optimized is a rational value which varies with the number of elements. The number of elements required can be calculated by ceil(raw output / (π/3)) – the floating point output is provided mainly to help optimization methods search for the solution.

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Evolution Strategy

Evolution strategies [1] are a heuristic search technique within the broad category of evolutionary computation. In evolutionary computation, heuristic search is performed by simulating the effects of Darwinian evolution on a population of solutions. Each of these solutions has a fitness value – e.g. the objective value of a function that is being minimized or maximized. Using this fitness value, natural/artificial selection allows the fitter solutions to survive and reproduce, whereas the less fit solutions will be removed from the population. In an evolution strategy that has a population of one (e.g. a (1+λ)-ES), all of the offspring solutions are created from the best found solution. A (1+λ)-ES has been developed for the optimization solver in Continuum Probe DesignerTM [4]. This ES uses λ = 3, runs for 100 generations, and has a R mutation strength of σ = 0.01. Specifically, the randn function in MATLAB (which creates a random number with a Gaussian distribution) is called and multiplied by 0.01. The performance of the ES was evaluated from 30 start points – the origin (which is deemed by expert knowledge to be a reasonably good start point) and 29 additional start points which were generated by making each term a uniform R The random number from [-0.2, 0.2] (using the rand function in MATLAB). Table 1. Number of elements in the phased array ultrasonic transducers designed by Continuum Probe DesignerTM for the FAA inspection challenge. The origin and 29 random points were used as the initial probe geometries. Designs with 32 or fewer elements are considered “feasible”. A “better” design was recorded for one of the methods if it found a solution with fewer elements for a given start point. A paired, one-tailed t-test was used to test for significance since the experiments used the same 30 start points. A value of less than 5% shows that the result is significant to at least the 1 in 20 level. (1+λ)-ES average std. dev. maximum minimum feasible better t-test

32.5 3.9 45 27 18 8

Four independent ES runs 31.3 2.9 39 27 22 15 2.1%


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high variation in the results for the (1+λ)-ES (see Table 1) suggested that its performance could be easily improved by using several independent runs and returning the best overall solution. For all of the previous 30 start points, three additional start points were created by adding a uniform random mutation of [-1, 1]/6 to each term. In each set, the same ES as before was run from all four start points. The use of the three additional independent runs achieved a less than expected reduction in the average number of elements used. A quick analysis suggested that there was a high correlation between the quality of start and end points – the high variation in results might be directly related to the high variation in the quality of the initial start points. Using 49 random start points (rather than just three), a follow-up experiment had the ES runs started from the four best of the 50 candidate start points – the initial geometry (e.g. the origin) was not necessarily used. These “smart” start points led to an even larger reduction in the average number of elements used (see Table 2). Table 2. Results comparing the use of “smart” start points with four independent ES runs. See Table 1 for explanations of the data.

average std. dev. maximum minimum feasible better t-test

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Four independent ES runs 31.3 2.9 39 27 22 8

“Smart” Start Points 30.1 3.2 42 27 25 18

2.2%

An Analysis of “Smart” Start Points

To verify the observation that the quality of the start and end points is indeed correlated, 90 addition runs of the (1+λ)-ES have been performed starting from random start points with a [-0.2,0.2] range. (Note: data from the original parallel runs are not useful since the grouped start points are clustered around the 30 random start points.) Using the 90 new and 30 original ES runs, Table 3 shows that there is a significant improvement in the performance of the (1+λ)-ES when it is started from better (i.e. “smart”) start points. The verification that the current problem domain can benefit from “smart” start points is not a trivial addition – all problem domains and/or local optimization techniques will not necessarily have this high correlation between the quality of a random solution and the quality of its local optimum. To demonstrate, “smart” start points have been used on the LIN318 instance of the travelling salesman problem (TSP) using two-opt as the local optimizer. Generating

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Table 3. Results comparing the quality of the 30 best and 30 worst of 120 random start points, and the subsequent quality of their ES solutions on the benchmark probe design problem. A two-tailed, homoscedastic t-test shows that the difference in the quality of the ES solutions is significant at the 1 in 20 level.

average start std. dev. t-test average end std. dev. feasible t-test

30 worst 770.5 87.1

30 best 121.5 66.3 0.0%

34.1 5.7 22

31.7 3.1 25 4.2%

Table 4. Results comparing the quality of the 30 best and 30 worst of 120 random initial tours, and the subsequent quality of their two-opt solutions for the TSP. A twotailed, homoscedastic t-test shows no significant difference in the quality of the two-opt solutions.

average start std. dev. t-test average end std. dev. t-test

30 worst 559138 7443

30 best 516242 6683 0.0%

46651 560

46795 988 48.8%

Table 5. Results comparing the performance of using “smart” start points and using four random initial tours on the TSP. A two-tailed, homoscedastic t-test shows no significant difference in the quality of the best two-opt solution found in each set.

average std. dev. t-test

Four independent “Smart” Start two-opt runs Points 45876 45747 485 577 35.4%

120 random initial tours and two-opt minima, it can be seen that the (inverse) correlation between the quality of the final solutions for the 30 best and 30 worst initial tours is not significant (see Table 4). Without a correlation between the quality of the start and end solutions, there is no expected benefit of using “smart” start points for independent parallel runs. This expectation is confirmed by grouping the 120 previous TSP runs into 30 sets of four (e.g. set 1 is runs 1, 31, 61, and 91) and comparing them against


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runs using “smart” start points – 50 random TSP solutions are generated, and the four best are optimized with two-opt. The results in Table 5 show that no significant improvement was achieved by using “smart” start points on the TSP.

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Using PSO to Find “Smart” Start Points

The above analysis of “smart” start points suggests that the performance of the ES on the probe design problem can be improved by pairing it with a coarse search strategy. Specifically, the role of the coarse search strategy is to find better initial start points. Building upon the success of random search (the coarsest of coarse search strategies) and inspired by the coarse search-greedy search combination presented with WoSP [7], particle swarm optimization is hereby tried as the coarse search technique. Finding "Smart" Start Points with PSO Begin start with one initial point (e.g. the origin) create a pool of 50 points by generating 49 random points choose the 4 best points from this pool of points For each of the four best points choose 4 random points from the pool (no replacement) give all 5 points/particles a random initial velocity For i = 1 to n (number of iterations for each particle) For each particle calculate and announce fitness End For update best particle for each wave For each particle calculate new velocity calculate new location End For End For End For return the best solution from each of the four PSOs End

line 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Fig. 2. Pseudocode for the coarse search strategy to find “smart” start points

The pseudocode for the coarse search strategy is presented in Fig. 2. In line 3, the 49 additional start points are created by adding a uniform random mutation of [-1, 1]/12 to each term of the initial start point. After line 10, the best particle location B of each PSO is updated. The calculation of particle velocities on line 14 is performed with the following equation: ¯ t + GU ¯B ¯ t+1 = M V (1) V

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In (1), the new velocity of a particle is a weighted average of its previous ¯ B is the unit vector velocity and a velocity towards the best particle location. U from the current particle to the best particle location. The terms M = 0.5 and G = 0.9 were chosen such that the average influence of the momentum term (M ) is slightly larger than the influence of the gravity term (G). This weighting promotes exploration over convergence.

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PSO-ES Results

Similar to the previous experiments with four random start points and four “smart” start points found by random search, the four “smart” start points found by PSO are used as the initial start points for four independent, parallel (1+λ)-evolution strategies. Each ES has λ = 3, runs for 100 generations, and has a mutation strength of σ = 0.01. Thus, the results shown in Table 6 isolate the benefits of the improved start points found by using PSO. The key features of these results are a 3.1% reduction in the average number of elements required and an increase by two in the number of feasible solutions found. Table 6. Results comparing the use of “smart” start points found randomly and by PSO. See Table 1 for explanations of the data.


Random “Smart” PSO “Smart” Start Points Start Points 30.1 29.2 3.2 1.9 42 34 27 27 25 27 7 15 3.1%

The search space of probe geometries is known to be globally convex [4]. Thus, better solutions can often be found by searching between other (locally optimal) solutions. If two solutions are on opposite sides of the “big valley”, a point in between these two solutions may be closer to the centre of the “big valley” where better solutions (and the global optimum) are found. Crossover (which generates an offspring that is the arithmetic mean of two solutions) has been added to the parallel ES runs – two random parents are mated after each generation, and the worst of the four original solutions is replaced if the offspring produced by crossover is fitter. The results with the addition of crossover finally achieve the desired result (see Table 7) – feasible solutions are found for all 30 start points. This achievement is critical for the future use of Continuum Probe DesignerTM by non-expert users. In addition to the 2.6% reduction in the average number of elements used,


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Table 7. Results comparing independent parallel ES runs and ES runs coordinated by using crossover. See Table 1 for explanations of the data.


PSO “Smart” PSO-ES with Start Points crossover 29.2 28.5 1.9 1.3 34 32 27 27 27 30 6 13 1.1%

the large reduction in the standard deviation of the results also provides an important level of confidence in the performance of the optimization solver.

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Discussion

The idea from WoSP of combining coarse global search and greedy local search techniques is quite different from the ideas behind metaheuristics like memetic algorithms [11]. In memetic algorithms and other restart techniques, the local search technique is applied to every start point that is generated. These search strategies subsequently perform a special kind of “search over the subspace of local optima” [12]. If this subspace of local optima is significantly smaller than the overall search space (as it is in globally convex search spaces [2]), then these search strategies can be very efficient. Conversely, the use of coarse search makes the assumption that there is a correlation between the quality of a random and/or intermediate start point and the final quality of its local optimum. This correlation is independent of a search spaces global convexity – it exists for probe geometries, and it does not exist for the travelling salesman problem (even though both search spaces are globally convex [2,4]). In a non-globally convex search space, this correlation could be particularly useful since memetic algorithms and other restart techniques should receive little benefit from exploring in the neighbourhood of other local optima. Coarse search also has advantages over memetic algorithms in problem domains (like the one currently studied) where the local optimization technique is too expensive to run a large number of times. Since the most commonly used benchmark optimization problem (i.e. the TSP) does not show the required correlation between the quality of a random start point and the quality of its local optimum, it is understandable why coarse search and “smart” start points have not previously received extensive analysis. It is hypothesized that the correlation required to make “smart” start points effective is more likely to occur in continuous rather than discrete search spaces. This hypothesis is a promising topic for future research.

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Conclusions

Particle swarm optimization has been used to find “smart” start points for a greedier local optimization technique (e.g. an evolution strategy). These “smart” start points have led to an important objective in the development of Continuum Probe DesignerTM as an industry-leading commercial application. The ability to find feasible solutions from all of the random starting points in a benchmark test suite represents the minimum level of robustness required to have confidence that Continuum Probe DesignerTM is ready for use by non-expert designers. The results also demonstrate the potential benefits of coarse search-greedy search hybrids in search spaces that have a correlation between the quality of a random start point and the quality of its local optimum.

Acknowledgements This work has received funding support from the Natural Sciences and Engineering Research Council of Canada and the Atkinson Faculty of Liberal and Professional Studies, York University

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