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. 2022 Jan 27;12:1485. doi: 10.1038/s41598-022-05547-7

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© The Author(s) 2022

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Planar piezoelectric ultrasonic transducer (PUT) applying the design method of the reference³⁰: (a) a problem definition of PUT to the 2D axisymmetric finite element simulation, (b) optimized structure and distance of piezoelectric ring (inner radius of n and outer radius m), (c) response of absolute pressure to the piezoelectric ring, (d) acoustic focusing response of the patterned piezoelectric ring array structure, and (e) topology design optimization procedure.