Development of numerical algorithms for solving the direct problem of propagation of ultrasonic waves in thin plates


  • Alexander S. Belyaev
  • Alexander V. Goncharsky
  • Sergey Y. Romanov


Mathematical modeling
ultrasound tomography
forward and inverse problems
vector wave model
non-destructive testing
Lamb waves


This article is devoted to the development of efficient numerical methods for solving direct problems of wave propagation in solids in vector mathematical models. Iterative methods for solving inverse problems of wave tomography use, at each iteration, the solution of the direct problem of wave propagation both in forward and backward time to calculate the gradient of the residual functional. Therefore, the solution of the direct problem of wave propagation in elastic media is an integral part of the solution of inverse problems of wave tomography. The purpose of the article is also to determine, using the methods of mathematical modeling characteristics of Lamb waves for ultrasonic diagnostics of defects in thin plates, determination of the ranges of values of the characteristic parameters of the experiment on tomographic diagnostics in thin plates on Lamb waves. The tools for mathematical modeling are the developed numerical methods and programs for solving direct problems. The ultimate goal of the research is to develop methods for solving inverse problems of tomographic non-destructive ultrasonic testing both on Lamb waves and on bulk waves.





Methods and algorithms of computational mathematics and their applications

Author Biographies

Alexander S. Belyaev

Alexander V. Goncharsky

Sergey Y. Romanov


  1. P. Huthwaite and F. Simonetti, “High-Resolution Guided Wave Tomography,” Wave Motion 50 (5), 979-993 (2013).
    doi 10.1016/j.wavemoti.2013.04.004
  2. E. Bazulin, A. Goncharsky, S. Romanov, and S. Seryozhnikov, “Ultrasound Transmission and Reflection Tomography for Nondestructive Testing Using Experimental Data,” Ultrasonics 124, Article Number 106765 (2022).
    doi 10.1016/j.ultras.2022.106765
  3. J. Virieux and S. Operto, “An Overview of Full-Waveform Inversion in Exploration Geophysics,” Geophysics 74 (6), WCC1-WCC26 (2009).
    doi 10.1190/1.3238367
  4. A. V. Goncharsky, S. Y. Romanov, and S. Y. Seryozhnikov, “Low-Frequency Ultrasonic Tomography: Mathematical Methods and Experimental Results,” Vestn. Mosk. Univ., Ser. 3: Fiz. Astron., No. 1, 40-47 (2019) [Moscow Univ. Phys. Bull. 74 (1), 43-51 (2019)].
    doi 10.3103/S0027134919010090
  5. F. Natterer, Numerical Solution of Bilinear Inverse Problems , Technical Report 19/96 N (Department of Mathematics, University of Münster, 1996).
  6. L. Beilina, M. V. Klibanov, and M. Yu. Kokurin, “Adaptivity with Relaxation for Ill-Posed Problems and Global Convergence for a Coefficient Inverse Problem,” J. Math. Sci. 167, 279-325 (2010).
    doi 10.1007/s10958-010-9921-1
  7. A. V. Goncharsky and S. Y. Romanov, “Iterative Methods for Solving Coefficient Inverse Problems of Wave Tomography in Models with Attenuation,” Inverse Probl. 33 (2), Article ID 025003 (2017).
    doi 10.1088/1361-6420/33/2/025003
  8. M. V. Klibanov and A. E. Kolesov, “Convexification of a 3-D Coefficient Inverse Scattering Problem,” Comput. Math. Appl. 77 (6), 1681-1702 (2019).
    doi 10.1016/j.camwa.2018.03.016
  9. A. V. Goncharsky and S. Y. Romanov, “A Method of Solving the Coefficient Inverse Problems of Wave Tomography,” Comput. Math. Appl. 77 (4), 967-980 (2019).
    doi 10.1016/j.camwa.2018.10.033
  10. R. Jirik, I. Peterlik, N. Ruiter, et al., “Sound-Speed Image Reconstruction in Sparse-Aperture 3-D Ultrasound Transmission Tomography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59 (2), 254-264 (2012).
    doi 10.1109/TUFFC.2012.2185
  11. M. Sak, N. Duric, P. Littrup, et al., “Using Speed of Sound Imaging to Characterize Breast Density,” Ultrasound Med. Biol. 43 (1), 91-103 (2017).
    doi 10.1016/j.ultrasmedbio.2016.08.021
  12. L. Liberti and N. Maculan (Eds.), Global Optimization: From Theory to Implementation (Springer, Berlin, 2006).
  13. I. M. Gel’fand and M. L. Tsetlin, “Some Methods of Control for Complex Systems,” Usp. Mat. Nauk 17 (1), 3-25 (1962) [Russ. Math. Surv. 17 (1), 95-117 (1962)].
    doi 10.1070/rm1962v017n01abeh001124
  14. A. V. Sulimov, D. A. Zheltkov, I. V. Oferkin, et al., “Tensor Train Global Optimization: Application to Docking in the Configuration Space with a Large Number of Dimensions,” in Communications in Computer and Information Science (Springer, Cham, 2017), Vol. 793, pp. 151-167.
    doi 10.1007/978-3-319-71255-0_12
  15. A. V. Goncharsky, S. Yu. Romanov, and S. Yu. Seryozhnikov, “Low-Frequency 3D Ultrasound Tomography: Dual-Frequency Method,” Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie) 19 (4), 479-495 (2018). doi10.26089/NumMet.v19r443.
  16. F. Natterer, “Possibilities and Limitations of Time Domain Wave Equation Imaging,” in: Contemporary Mathematics (American Mathematical Society, Providence, 2011), Vol. 559, pp. 151-162.
    doi 10.1090/conm/559
  17. A. Backushinsky, A. Goncharsky, S. Romanov, and S. Seatzu, “On the Identification of Velocity in Seismics and in Acoustic Sounding,” Pubblicazioni Dell’istituto di Analisa Globale e Applicazioni, Serie &quotProblemi non ben posti e inversi&quot, Florence (1994), Issue 71, pp. 1-14.
  18. A. B. Bakushinskii, A. I. Kozlov, and M. Y. Kokurin, “On Some Inverse Problem for a Three-Dimensional Wave Equation,” Zh. Vychisl. Mat. Mat. Fiz. 43 (8), 1201-1209 (2003) [Comput. Math. Math. Phys. 43 (8), 1149-1158 (2003)].
  19. E. G. Bazulin, A. V. Goncharsky, S. Yu. Romanov, and S. Yu. Seryozhnikov, “Inverse Problems of Ultrasonic Tomography in Nondestructive Testing: Mathematical Methods and Experiment,” Defektoskopiya, No. 6, 30-39 (2019) [Russ. J. Nondestruct. Test. 55 (6), 453-462 (2019)].
    doi 10.1134/S1061830919060020
  20. E. G. Bazulin, A. V. Goncharsky, S. Y. Romanov, and S. Y. Seryozhnikov, “Parallel CPU- and GPU-Algorithms for Inverse Problems in Nondestructive Testing,” Lobachevskii J. Math. 39 (4), 486-493 (2018).
    doi 10.1134/S1995080218040030
  21. S. Y. Romanov, “Supercomputer Simulations of Nondestructive Tomographic Imaging with Rotating Transducers,” Supercomput. Front. Innov. 5 (3), 98-102 (2018).
    doi 10.14529/jsfi180318
  22. L. Yu, V. Giurgiutiu, and P. Pollock, “A Multi-Mode Sensing System for Corrosion Detection Using Piezoelectric Wafer Active Sensors,” Proc. SPIE, Vol. 6932 (2008).
    doi 10.1117/12.776670
  23. J. Rao, M. Ratassepp, and Z. Fan, “Guided Wave Tomography Based on Full Waveform Inversion,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 63 (5), 737-745 (2016).
    doi 10.1109/TUFFC.2016.2536144
  24. X. Zhao and J. L. Rose, “Ultrasonic Guided Wave Tomography for Ice Detection,” Ultrasonics 67, 212-219 (2016).
    doi 10.1016/j.ultras.2015.12.005
  25. J. Tong, M. Lin, X. Wang, et al., “Deep Learning Inversion with Supervision: A Rapid and Cascaded Imaging Technique,” Ultrasonics 122, Article Number 106686 (2022).
    doi 10.1016/j.ultras.2022.106686
  26. S. Rodriguez, M. Deschamps, M. Castaings, and E. Ducasse, “Guided Wave Topological Imaging of Isotropic Plates,” Ultrasonics 54 (7), 1880-1890 (2014).
    doi 10.1016/j.ultras.2013.10.001
  27. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics , Vol. 7: Theory of Elasticity (Nauka, Moscow, 1987; Pergamon, Oxford, 1995).
  28. J. Virieux, “P-SV Wave Propagation in Heterogeneous Media: Velocity-Stress Finite-Difference method,” Geophysics 51 (4), 889-901 (1986).
    doi 10.1190/1.1442147
  29. V. V. Lisitsa, Numerical Methods and Algorithms for Calculating Wave Seismic Fields in Media with Local Complicating Factors Doctoral Thesis in Physics and Mathematics (Trofimuk Institute of Petroleum Geology and Geophysics SB RAS, Novosibirsk, 2017). . Cited July 10, 2023.
  30. I. A. Viktorov, “Lamb Ultrasonic Waves,” Akust. Zh. 11 (1), 1-18 (1965).
  31. A. V. Goncharsky, S. Y. Romanov, and S. Y. Seryozhnikov, “Supercomputer Technologies in Tomographic Imaging Applications,” Supercomput. Front. Innov. 3 (1), 41-66 (2016).
    doi 10.14529/jsfi160103
  32. Vad. V. Voevodin, S. L. Ovchinnikov, and S. Yu. Romanov, “Development of High-Performance Scalable Software for Ultrasound Tomography,” Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 13 (2), 307-315 (2012).