Identification of the interface between acoustic and elastic waves from internal measurements

  • 1 School of Mathematics and Informational Science, Yantai University, Shandong, 264005, Yantai, P. R. China
  • 2 School of Mathematics and Informational Science, Yantai University, Shandong, 264005, Yantai, P. R. China
Yanli Cui
  • School of Mathematics and Informational Science, Yantai University, Yantai, Shandong, 264005, P. R. China
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and Fenglong QuORCID iD: https://orcid.org/0000-0003-2935-1253

Abstract

Consider the fluid-solid interaction problem for a two-layered non-penetrable cavity. We provide a novel fundamental proof for a uniqueness theorem on the determination of the interface between acoustic and elastic waves from many internal measurements, disregarding the boundary conditions imposed on the exterior non-penetrable boundary. The proof depends on a uniform H1-norm boundedness for the elastic wave fields and the construction of the coupled interior transmission problem related to the acoustic and elastic wave fields.

  • [1]

    F. Cakoni, D. Colton and S. Meng, The inverse scattering problem for a penetrable cavity with internal measurements, Inverse Problems and Applications, Contemp. Math. 615, American Mathematical Society, Providence (2014), 71–88.

  • [2]

    D. Colton, R. Kress and P. Monk, Inverse scattering from an orthotropic medium, J. Comput. Appl. Math. 81 (1997), no. 2, 269–298.

    • Crossref
    • Export Citation
  • [3]

    J. Elschner, G. C. Hsiao and A. Rathsfeld, An inverse problem for fluid-solid interaction, Inverse Probl. Imaging 2 (2008), no. 1, 83–119.

    • Crossref
    • Export Citation
  • [4]

    J. Elschner, G. C. Hsiao and A. Rathsfeld, An optimization method in inverse acoustic scattering by an elastic obstacle, SIAM J. Appl. Math. 70 (2009), no. 1, 168–187.

    • Crossref
    • Export Citation
  • [5]

    G. C. Hsiao, R. E. Kleinman and G. F. Roach, Weak solutions of fluid-solid interaction problems, Math. Nachr. 218 (2000), 139–163.

    • Crossref
    • Export Citation
  • [6]

    Y. Hu, F. Cakoni and J. Liu, The inverse scattering problem for a partially coated cavity with interior measurements, Appl. Anal. 93 (2014), no. 5, 936–956.

    • Crossref
    • Export Citation
  • [7]

    P. Jakubik and R. Potthast, Testing the integrity of some cavity—the Cauchy problem and the range test, Appl. Numer. Math. 58 (2008), no. 6, 899–914.

    • Crossref
    • Export Citation
  • [8]

    A. Kimeswenger, O. Steinbach and G. Unger, Coupled finite and boundary element methods for fluid-solid interaction eigenvalue problems, SIAM J. Numer. Anal. 52 (2014), no. 5, 2400–2414.

    • Crossref
    • Export Citation
  • [9]

    A. Kirsch and A. Ruiz, The factorization method for an inverse fluid-solid interaction scattering problem, Inverse Probl. Imaging 6 (2012), no. 4, 681–695.

    • Crossref
    • Export Citation
  • [10]

    X. Liu, The factorization method for cavities, Inverse Problems 30 (2014), no. 1, Article ID 015006.

  • [11]

    C. J. Luke and P. A. Martin, Fluid-solid interaction: Acoustic scattering by a smooth elastic obstacle, SIAM J. Appl. Math. 55 (1995), no. 4, 904–922.

    • Crossref
    • Export Citation
  • [12]

    S. Meng, H. Haddar and F. Cakoni, The factorization method for a cavity in an inhomogeneous medium, Inverse Problems 30 (2014), no. 4, Article ID 045008.

  • [13]

    P. Monk and V. Selgas, An inverse fluid-solid interaction problem, Inverse Probl. Imaging 3 (2009), no. 2, 173–198.

    • Crossref
    • Export Citation
  • [14]

    P. Monk and V. Selgas, Near field sampling type methods for the inverse fluid-solid interaction problem, Inverse Probl. Imaging 5 (2011), no. 2, 465–483.

    • Crossref
    • Export Citation
  • [15]

    D. Natroshvili, S. Kharibegashvili and Z. Tediashvili, Direct and inverse fluid-structure interaction problems, Rend. Mat. Appl. (7) 20 (2000), no. 1–4, 57–92.

  • [16]

    R. Potthast, A point source method for inverse acoustic and electromagnetic obstacle scattering problems, IMA J. Appl. Math. 61 (1998), no. 2, 119–140.

    • Crossref
    • Export Citation
  • [17]

    R. Potthast, On the convergence of a new Newton-type method in inverse scattering, Inverse Problems 17 (2001), no. 5, 1419–1434.

    • Crossref
    • Export Citation
  • [18]

    H.-H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems 27 (2011), no. 3, Article ID 035005.

  • [19]

    H.-H. Qin and D. Colton, The inverse scattering problem for cavities, Appl. Numer. Math. 62 (2012), no. 6, 699–708.

    • Crossref
    • Export Citation
  • [20]

    H.-H. Qin and D. Colton, The inverse scattering problem for cavities with impedance boundary condition, Adv. Comput. Math. 36 (2012), no. 2, 157–174.

    • Crossref
    • Export Citation
  • [21]

    H.-H. Qin and X. Liu, The interior inverse scattering problem for cavities with an artificial obstacle, Appl. Numer. Math. 88 (2015), 18–30.

    • Crossref
    • Export Citation
  • [22]

    F. Qu and J. Yang, On recovery of an inhomogeneous cavity in inverse acoustic scattering, Inverse Probl. Imaging 12 (2018), no. 2, 281–291.

    • Crossref
    • Export Citation
  • [23]

    F. Qu, J. Yang and B. Zhang, Recovering an elastic obstacle containing embedded objects by the acoustic far-field measurements, Inverse Problems 34 (2018), no. 1, Article ID 015002.

  • [24]

    F. Qu, J. Yang and H. Zhang, Shape reconstruction in inverse scattering by an inhomogeneous cavity with internal measurements, SIAM J. Imaging Sci. 12 (2019), no. 2, 788–808.

    • Crossref
    • Export Citation
  • [25]

    J. Sanchez Hubert and E. Sánchez-Palencia, Vibration and Coupling of Continuous Systems. Asymptotic Methods, Springer, Berlin, 1989.

  • [26]

    Q. Wu and G. Yan, The factorization method for a partially coated cavity in inverse scattering, Inverse Probl. Imaging 10 (2016), no. 1, 263–279.

    • Crossref
    • Export Citation
  • [27]

    J. Yang, B. Zhang and H. Zhang, Uniqueness in inverse acoustic and electromagnetic scattering by penetrable obstacles with embedded objects, J. Differential Equations 265 (2018), no. 12, 6352–6383.

    • Crossref
    • Export Citation
  • [28]

    T. Yin, G. C. Hsiao and L. Xu, Boundary integral equation methods for the two-dimensional fluid-solid interaction problem, SIAM J. Numer. Anal. 55 (2017), no. 5, 2361–2393.

    • Crossref
    • Export Citation
  • [29]

    T. Yin, G. Hu, L. Xu and B. Zhang, Near-field imaging of obstacles with the factorization method: fluid-solid interaction, Inverse Problems 32 (2016), no. 1, Article ID 015003.

  • [30]

    F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity, Inverse Problems 27 (2011), no. 12, Article ID 125002.

  • [31]

    F. Zeng, X. Liu, J. Sun and L. Xu, The reciprocity gap method for a cavity in an inhomogeneous medium, Inverse Probl. Imaging 10 (2016), no. 3, 855–868.

    • Crossref
    • Export Citation
  • [32]

    F. Zeng, P. Suarez and J. Sun, A decomposition method for an interior inverse scattering problem, Inverse Probl. Imaging 7 (2013), no. 1, 291–303.

    • Crossref
    • Export Citation
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This journal presents original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published.

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