Skip to main content
SpringerLink
Log in

Application of Quaternion Neural Network to Time Reversal Based Nonlinear Elastic Wave Spectroscopy

  • Original Article
  • Published:
Transactions of the Indian National Academy of Engineering Aims and scope Submit manuscript

Abstract

Identification of crack positions or anomalous scattering centers in materials using the time reversal (TR) based nonlinear elastic wave spectroscopy (TR-NEWS) is an established method. We propose a system using transducers which emit forward propagating solitonic wave and time-reversed propagating solitonic wave produced by memristors placed on a side of a rectangle and scattered by cracks in the material and received by receivers which are placed on the opposite side of the rectangle. By measuring the convolution of the scattered forward propagating wave and the scattered TR wave, we get information of the position of the scattering center by using the neural network technique. Routes of the solitonic wave are expressed by quaternions projected on 2 dimensional planes, and the optimal route from signals are searched. We consider the wave is expressed by a soliton which is conformal, and discuss symmetry protected topological impurities and gravitational effects using the Atiyah-Patodi-Singer’s index theorem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

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

Similar content being viewed by others

Data availability

Machine Learning program can be provided when the subsequent monograph is completed in March 2023.

References

  • Abd-el-Malek MB, El-Mansi SMA (2000) Group theoretic methods applied to Burgers’ equation. J Comput Appl Math 115:1–12

    Article  MathSciNet  MATH  Google Scholar 

  • Aggarwal CC (2018) Neural network and deep learning, a textbook. Springer Nature, Switzerland

    Book  MATH  Google Scholar 

  • Altland A, Zirnbauer MR (1997) Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys Rev B 55:1142

    Article  Google Scholar 

  • Ell AT (2013) Quaternion Fourier Transform: Re-tooling Image and Signal Processing Analysis, In: “Quaternion and Clifford-Fourier Transforma and Wavelets”, Trends in Mathematics, 15-39, Springer, Basel

  • Atiyah MF, Singer IM (1963) The Index of Elliptic Operators on Compact Manifolds. Bull Amer Math Soc 69(3):422–433

    Article  MathSciNet  MATH  Google Scholar 

  • Atiyah MF, Bott R, Shapiro A (1964) Clifford Modules. Topology 3(I):3–38

    Article  MathSciNet  MATH  Google Scholar 

  • Atiyah MF, Bott R, Patodi VK (1973) On the Heat Equation and the Index Theorem. Inventiones Math 19:279–330

    Article  MathSciNet  MATH  Google Scholar 

  • Atiyah MF, Patodi VK, Singer IM (1975) Spectral Asymmetry and Riemannian Geometry, I. Math Proc Cambridge Philos Soc 77:43

    Article  MathSciNet  MATH  Google Scholar 

  • Atiyah MF, Bott R, Patodi VK (1975) Errata to thepaper : On the Heat Equation and the Index Theorem. Inventiones Math 28:277–280

    Article  MathSciNet  MATH  Google Scholar 

  • Bacot V, Labousse M, Eddi A, Fink M, Fort E (2016) Time reversal and holography with spacetime transformations. Nat Phys 12:972–977

    Article  Google Scholar 

  • Berthier M (2013) Spin Geometry and Image Processing, hal-00801224

  • Bilski A, Twardy M (2010) Hysteresis Modeling Using a Preisach Operator. Int J Electron Telecommun 56(4):473–478

    Article  Google Scholar 

  • Blanloeuil P, Rose LRF, Veidt M, Wang CH (2018) Time reversal invariance for a nonlinear scatterer exhibiting contact acoustic nonlinearity. J Sound Vib 417:413–431

    Article  Google Scholar 

  • Bodo B, Armand JS, Fouda E, Mvogo A, Tagne S (2018) Experimental hysteresis in memristor based Duffing oscillator. Chaos Solitons Fractals 115:190–195

    Article  Google Scholar 

  • Bou Matar O, Preobrazhenky V, Pernod P (2005) Two-dimensional axisymmetricbnumerical simulation of supercritical phase conjugation of ultrasound in active solid media. J Acoust Soc Am 118(5):2880–2890

    Article  Google Scholar 

  • Chevalley C (1946) Theory of Lie Groups I. Princeton University Press, Princeton. Reprinted by Overseas Publications Ltd., Tokyo (1965)

    Book  MATH  Google Scholar 

  • Chua LO (1971) Memristor - The missing circuit element, IEEE Trans. Circuit Th. CT-18 507-519

  • Chua L (2011) Resistance switching memories are memristors. Appl Phys A 102:765–783

    Article  MATH  Google Scholar 

  • Chua L (2018) Five non-volatile memristor enigmas solved. Appl Phys A 124:563

    Article  Google Scholar 

  • Dirac PAM (1949) Forms of Relativistic Dynamics. Rev Mod Phys 21(3):392–399

    Article  MathSciNet  MATH  Google Scholar 

  • Dos Santos S (2004) Symmetry of nonlinear acoustic equations using group theoretic methods: a signal processing tool for extracting judicious physical variables. In: Proceedings of the joint congress CFA/DAGA ’04, Strasbourg, pp 549–550

  • Dos Santos S (2020) Advanced ground truth multiodal imaging using Time Reversal (TR) based Nonlinear Elastic Wave Spectroscopy (NEWS): medical imaging trends versus non-destructive testing applications, Chapter 4. In: Dos Santos S et al (eds) Recent Advances in Mathematics and Technology. Applied and Numerical Harmonic Analysis, Springer Nature, Switzerland AG

  • Dos Santos S, Plag C (2008) Excitation Symmetry Analysis Method (ESAM) for Calculation of Higher Order Nonlinearity. Int J Nonlinear Mech 43:114–118

    Google Scholar 

  • Dos Santos S, Choi BK, Sutin A, Sarvazyan A (2006) Nonlinear Imaging Based on Time Reversal Acoustic Focusing, CFA 2006 proceedings p.359-362

  • Dos Santos S, Furui S (2016) A memristor based ultrasonic transducer: the memoducer, In Ultrasonic Symposium (IUS) 2016 IEEE International (pp. 1-4)

  • Dos Santos S, Furui S, Nardoni G (2018) Self-calibration of multiscale hysteresis with memristors in nonlinear time reversal based processes, Proceedings of Biennial Baltic Electronic Conference (BEC)

  • Dos Santos S, Masood A (2018) Ultrasonic transducers self-calibration of nonlinear time reversal based experiments using memristor, Proceedings of 12th ECNDT, Goetheburg-Sweden

  • Dos Santos S, Prevorovsky Z (2020) The physical interpretation of the signal processing cross-correlation using TR-NEWS: an acoustic point of view, Forum Acousticum, p.2753

  • Dos Santos S, Vejvodova S, Prevorovsky Z (2009) Local nonlinear scatterers signatures using symbiosis of the time-reversal operator and symmetry analysis signal processing. J Acoust Soc Am 126(5)

  • Efimov SP (1979) Model of a nonreflecting anisotropic medium. Sov Phys Acoust 25(2):127–129

    Google Scholar 

  • Fierro GPM, Meo M (2022) Linear and nonlinear ultrasound time reversal using a condensing raster operation. Mech Syst Signal Process 168(1):108713

    Article  Google Scholar 

  • Fink M (1999) Time reversed acoustics, Scientific American, p.91 November and references therein

  • Furui S (2020) A Closer Look at Gluons, Chapter 6 of a book “Horizon in World Physics vol. 302”, Ed. by Albert Reimer, Nova Science Pub

  • Furui S (2020) Understanding Quaternions, The chapter 2 of “Understanding Quaternions” , Ed. by Peng Du et al., Nova Science Publishers

  • Furui S, Takano T (2014) On the amplitude of External Perturbation and Chaos via Devil’s Staircase in Muthuswamy-Chua System. Int J Bifurcation Chaos 23:1350136 arXiv:1406.4346 [nlin CD]

    Google Scholar 

  • Georgiev S, Morais J, Kou KI, Sproessig W (2013) Bochner-Minlos Theorem and Quaternion Fourier Transform, In “Quaternion and Clifford-Fourier Transforma and Wavelets”, Trends in Mathematics, 15-39, Springer Basel

  • Goursolle T, Callé S, Dos Santos S, Bou Matar O (2007) A two-dimensional pseudospectral model for time reversal and nonlinear elastic wave spectroscopy. J Accoust Soc Am 122(6):3220–3229

    Article  Google Scholar 

  • Goursolle T, Dos Santos S, Bou Matar O, Calle S (2008) Non-linear based time reversal acoustic applied to crack detection: simulations and experiments. Int J Non-linear Mech 43:170–177

    Article  Google Scholar 

  • Green MB, Schwarz JH, Witten E (1985) Superstring theory, vol II. Cambridge University Press, Cambridge

    Google Scholar 

  • Gruzberg IA, Read N, Vishweshwara S (2005) Localization in disordered superconducting wires with broken spin-rotation symmetry. Phys Rev B 71:245124

    Article  Google Scholar 

  • Haldane FDM (2004) Berry Curvature on the Fermi Surface: Anoalous Hall Effect as a Topological Fermi-Liquid Property. Phys Rev Lett 93:206602

    Article  Google Scholar 

  • Hestenes D (1975) Observables, operators, and complex numbers in the Dirac theory. J Math Phys 16:556–572

    Article  MathSciNet  Google Scholar 

  • Hille E, Phillips R (1958) Functional analysis and semi-groups, Providence

  • Hitzer E, Sangwine SJ (2013) The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations, In “Quaternion and Clifford-Fourier Transforma and Wavelets”, Trends in Mathematics, 15-39, Springer Basel

  • Hitzer E (2022) Quaternion and Clifford Fourier Transforms. CRC Press

    MATH  Google Scholar 

  • Hörmander L (1983) The Analysis of Linear Partial Differential Operators I, Distribution Theory and Fourier Analysis. Springer-Verlag, Berlin Heidelberg NewYork Tokyo

    MATH  Google Scholar 

  • Hörmander L (1983) The Analysis of Linear Partial Differential Operators II, Differential Operators with Constant Coefficients. Springer-Verlag, Berlin Heidelberg NewYork Tokyo

    MATH  Google Scholar 

  • Hörmander L (1985) The Analysis of Linear Partial Differential Operators III, Pseiudo-Differential Operators. Springer-Verlag, Berlin Heidelberg NewYork Tokyo

    MATH  Google Scholar 

  • Hörmander L (1997) Lectures on nonlinear hyperbolic differential equations. Springer-Verlag, Berlin Heidelberg NewYork Tokyo

    MATH  Google Scholar 

  • Horváth J (1966) Topological Vector Spaces and Distributions, vol I, Addison-Wesley Publishing Company, Reading,

  • Kane CI, Mele EJ (2005) \(Z_2\) Topological Order and the Quantum Spin Hall Effect. Phys Rev Lett 95:146802

    Article  Google Scholar 

  • Kane CI, Mele EJ (2005) Quantum Spin Hall Effect in Graphene. Phys Rev Lett 95:226801

    Article  Google Scholar 

  • Khelil SB, Merlen A, Preobrazhensky V, Pernod Ph (2001) Numerical simulation of acoustic wave phase conjugation in active media. J Acoust Soc Am 109(1):75–83

    Article  MATH  Google Scholar 

  • Kittel C (1968) Introduction to Solid State Physics, 3rd Ed. John Wiley and Sons Inc., New York (1966), Translated to Japanese by Uno et al., Maruzen, Tokyo

  • Kryazhev FI, Kudyashov VM (1978) Spatial and temporal correlation functions of the sound field in a waveguide with rough boundaries. Sov Phys Acoust 24(2):118–121

    Google Scholar 

  • Landau LD, Lifshitz EM (1959) Theory of Elasticity, Translated from Russian by J.B. Sykes and W.H. Reid, Pergamon (1964); E. Lifshitz and V. Bérestetski, 4th edition, (1987) Translated to Japanese by J. Ishibashi, Tokyo Tosho Pub, Sato and Z, p 1989

  • Lapidus YR, Rudenko OV (1984) New approximations and results of the theory of nonlinear acoustic beams. Sov Phys Acoust 30(6):473–476

    MathSciNet  Google Scholar 

  • Lapidus YR, Rudenko OV (1992) An exact solution of the Khokhlov-Zaboltskaya equation. Sov Phys Acoust 38(2):197

    Google Scholar 

  • Lounesto P (1993) Clifford algebras and Hestenes Spinors. Found Phys 23(9):1203–1237

    Article  MathSciNet  Google Scholar 

  • Lounesto P (2001) Clifford Algebras and Spinors. Cambridge University Press

    Book  MATH  Google Scholar 

  • Lüscher M (2003) Lattice QCD and the Schwarz alternating procedure, arXiv:hep-lat/0304007 v1

  • Mackey GW (1968) Induced Representations of Groups and Quantum Mechanics, W.A. Benjamin, INC. New York, Editori Boringhieri, Torino,

  • Mathematica W 12™, Wolfram Research (2019)

  • Mayergoyz ID (1986) Mathematical Models of Hysteresis (Invited). IEEE Trans Magn 22:603

    Article  Google Scholar 

  • McCall KR, Guyer RA (1994) Equation of state and wave propagation in hysteretic nonlinear elastic materials. J Geophys Res 99:23887–23897

    Article  Google Scholar 

  • Morais JP, Georgiev S, Sproessig W (2014) Real Quaternionic Calculus Handbook. Birkhaeuser, Basel

    Book  MATH  Google Scholar 

  • Muthuswamy B, Chua LO (2010) Simplest Chaotic Circuit. Int J Bifurcation Chaos 20:1567–1590

    Article  Google Scholar 

  • Ostrovsky LA, Rudenko OV (2008) What Problems of Nonlinear Acoustics seems to be important and Interesting today? , AIP Conference Proceedings 1022 (1) p.9-16, AIP

  • Parcollet T, Ravanetti M, Morchid M, Linearès G, Trabelsi C, De Mori R, Bengio Y. Quaternion Recurrent Neural Networks, ICLR 2019 Proceedings; arXiv:1806.04418 v3 [stat.ML]

  • Patil GU, Matlack KH (2022) Review of exploiting nonlinearity in phononic materials to enable nonlinear wave responses. Acta Mech 233:1–46

    Article  MathSciNet  MATH  Google Scholar 

  • Pham H, Warin X, Germain M (2021) Neural network-based backward scheme for fully nonlinear PDEs. SN Partial Differ Equ Appl 2:16

    Article  MathSciNet  MATH  Google Scholar 

  • Raschka S, Liu Y, Mirjalili V (2022) Machine Learning with Pytorch and Scikit-Learn, Packt

  • Ryu S, Moore JE, Ludwig AWW (2012) Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors. Phys Rev B 85:045104

    Article  Google Scholar 

  • Samsonov AM, Semenova IV, Belashov AV (2017) Direct determination of bulk strain soliton parameters in solid polymeric waveguides. Wave Motion 71:120–126

    Article  MathSciNet  MATH  Google Scholar 

  • Sánchez-Morcillo VJ, Jiménez N, Chaline J, Bouaka A, Dos Santos S (2014) Spatio-temporal dynamics in a ring of coupled pendula: analogy with bubbles Localized Excitations in Nonlinear Complex Systems. Springer, New York, pp 251–262

    Book  MATH  Google Scholar 

  • Schnyder AP, Ryu S, Furusaki A, Ludwig AWW (2008) Classification of topological insulators and superconductors in three spatial dimensions. Phys Rev B 78:195125

    Article  Google Scholar 

  • Schwartz L (1966) Méthodes mathématiques pour les science physique, Hermann, Paris ; Translated to japanese by K. Yosida and J. Watanabe, Iwanami Pub. (1966)

  • Siong GW (2021) Time Reversal Acoustics, Springernature com

  • Sutin AM, TenCate JA, Johnson PA (2004) Single-channel time reversal in elastic solids. J Accoust Soc Am 116(5):2779–2784

    Article  Google Scholar 

  • Thomas J-L, Roux P, Fink M (1994) Inverse Scattering Analysis with an Acoustic Time-Reversal Mirror. Phys Rev Lett 72(5):637–640

    Article  Google Scholar 

  • Trotter HF (1959) On the product of semi-groups of operators, Proceedings of American Mathematical Society, p.545-551

  • Trotter HF (1958) Approximation of semi-groups of operators. Pacific J Math 8:887–919

    Article  MathSciNet  MATH  Google Scholar 

  • Vejvodova S, Prevorovsky Z, Dos Santos S (2009) Nonlinear Time Reversal Tomography of Structural Defects, Proceedings of Meetings on Acoustics, 3

  • Walsh JB (1965) The Effect of Cracks on the Uniaxial Elastic Compression of Rocks. J Geophys Res 70(2):399–411

    Article  MATH  Google Scholar 

  • Wikipedia (2019) Kniznik-Zamolodchikov equation, https:en.wikipedia.org

  • Wikipedia (2020) Bochner-Minlos theorem

  • Witten E (2016) Fermion path integrals and topological phases, Rev. Mod. Phys. 88, 035001, 1-40 arXiv:1508.04715 [cond-mat.mes-hall]

  • Witten E (2016) The “parity” anomaly on an unorientable manifold, Phys. Rev. B 94, 195150 : arXiv:1605.02391 v3 [hep-th]

  • Witten E (1985) Global Gravitational Anomalies. Commun Math Phys 100:197–229

    Article  MathSciNet  MATH  Google Scholar 

  • Yosida K (1948) On the differentiability and the representation of one-parameter semi-group of linear operators. J Math Soc Japan 1:15–21

    Article  MathSciNet  MATH  Google Scholar 

  • Yosida K (1958) On the differentiability of semi-groups of linear operators. Proc Jap Acad 34:337–340

    MATH  Google Scholar 

  • Yosida K, Kawada K, Iwamura T (1967) Basics of Topological Analysis, In Japanese, Iwanami Shoten Pub. 6th Ed

  • Yue Y, Yong-Shi W, Xie X (2017) Bulk-edge correspondence, spectral flow and Atiyah-Patodi-Singer theorem for the \({{\cal{Z} }}_2\) invariant in topological insulators. Nucl Phys B 916:550–566

    Article  MATH  Google Scholar 

  • Zabolotskaya EA, Khokhlov RV (1969) Quasi-plane waves in the nonlinear acoustics of confined beams. Sov Phys Acoust 15(1):35–40

    Google Scholar 

Download references

Acknowledgements

S.F. thanks Prof. Stan Brodsky and Prof. Guy de Téramond for valuable information on conformal field theory. Thanks are also due to the RCNP of Osaka University for allowing checking FFT programs using its super computer, and Tokyo Institute of Technology for consulting references and a guidance of supercomputer programmings.

Author information

Authors and Affiliations

  1. Faculty of Science and Engineering, Teikyo University, Utsunomiya, 320, Japan

    Sadataka Furui

  2. INSA Centre  Val de Loire, Inserm U1253, Université de Tours, Imagerie et Cerveau, Imaging and brain: iBrain, Blois, France

    Serge Dos Santos

Authors
  1. Sadataka Furui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Serge Dos Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sadataka Furui.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Furui, S., Dos Santos, S. Application of Quaternion Neural Network to Time Reversal Based Nonlinear Elastic Wave Spectroscopy. Trans Indian Natl. Acad. Eng. 8, 183–199 (2023). https://doi.org/10.1007/s41403-023-00388-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41403-023-00388-w

Keywords

Search

Navigation