Ultrasonic Beam-forming with the Genetic Algorithm Andrew Fiss, Vassar College Nathan Baxter, Ohio Northern University Jerry Magnan, Florida State University

We used Genetic Algorithm optimization of sparse array parameters to produce an ultrasonic beam with a higher, narrower main-lobe and lower side-lobes. The solutions obtained are neither unique nor trivial. Abstract

Ultrasonic Transducers: Piezoelectric elements generate sound waves which are used in a variety of applications—non-destructive testing and evaluation of material structure, sonar, diagnostic and therapeutic medicine, etc… Arrays of transducer elements provide advantages over single elements in forming and controlling the ultrasound beam. We explore the beam-shaping properties of sparse ultrasonic transducers by performing an optimization over the array’s structural parameters using a Genetic Algorithm. Background

Genetic Algorithm:  The Genetic Algorithm mimics natural evolution.  Real number strings called “chromosomes” contain parameters that can vary to produce desired results. Our chromosomes contained 63 parameters—30 positions for the 30 “active” elements, 30 apodizations, x-width of the elements, y-width of the elements, and center frequency.  Random initial “chromosomes” are used to populate the system. These “chromosomes” are assigned a “fitness” value based on their proximity to an optimal solution and produce subsequent child “chromosomes” through mutation or recombination operators, which respectively change or rearrange the number strings. Our fitness function is represented as follows: Background (cont.)

main-lobe height side-lobe height width at –3dB  The Field II program was used to generate the pressure fields. (see Method section)  The children replace the majority of the previous generation, leaving only the best parents.

We wanted to shape the beam to have lower side lobes and a higher, narrower main lobe. This combination produces greater clarity in imaging via improved directivity of the beam. Objective

Field II, developed by Jørgen Jensen, was used in generating the pressure data for the beam-patterns, which were observed along the y-axis at x=0 and a fixed z. These beam-patterns were assigned a “fitness” that rewarded higher main-lobes, narrower main-lobes, and lower side-lobes, and the Genetic Algorithm was used as an optimization technique with regular mutation, creep mutation, uniform list crossover, and average crossover operators. The solutions consisted of 30 active elements in a 30-by-30 array, with width and height constrained to lie between and, a center frequency between 1 MHz and 15MHz, and apodization varying between 0 and 1. Method

After 30 generations, the Genetic Algorithm improved the beam-pattern from a main-lobe height of 3.944× barr to × barr, a main- lobe width at -3 dB of 1.95mm to 1.35mm, and a side-lobe height of × barr to × barr (see figures). However, the algorithm converges quickly for this optimization problem (see figure), so similar results can be obtained after just five generations (see figures). The best configuration of a batch of 5,000 random chromosomes is presented, thus showing that similar beam-patterns could not be obtained through random generation. This is not surprising, since there are ~10 55 possible arrangements of transducer element positions. Also, one might expect a trivial solution to be a dense transducer, such as a large, single element, but that is not the case (see figure). Results

*Equations

*Equations (cont.)

Ultrasonic beams can be effectively shaped through use of the Genetic Algorithm, and the solution is neither trivial nor unique. Conclusion

In future work, we would like to try asymmetric beam forming and apodization reconfiguration with fixed positions in a sparse array. We would also try different optimization techniques and more precise programs for pressure calculations, and we would like to explore further the characteristics of the optimization problems through different Genetic Algorithm structure and operators. Future work

For more information, see Samuel Temkin’s Elements of Acoustics, Jon Mathews and R.L. Walker’s Mathematical Methods of Physics, or J  rgen Arendt Jensen’s “Calculation of Pressure Fields from Arbitrarily Shaped, Apodized, and Excited Ultrasound Transducers” published in the March 1992 edition of the IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control. References