February  2016, 10(1): 103-129. doi: 10.3934/ipi.2016.10.103

Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves

1. 

Weierstrass Institute, Mohrenstr. 39, 10117 Berlin

2. 

Department of Mathematics, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe

3. 

College of Mathematics and Statistics, Chongqing University, China

Received  November 2014 Revised  June 2015 Published  February 2016

Consider a time-harmonic acoustic wave incident onto a doubly periodic (biperiodic) interface from a homogeneous compressible inviscid fluid. The region below the interface is supposed to be an isotropic linearly elastic solid. This paper is concerned with the inverse fluid-solid interaction (FSI) problem of recovering the unbounded periodic interface separating the fluid and solid. We provide a theoretical justification of the factorization method for precisely characterizing the region occupied by the elastic solid by utilizing the scattered acoustic waves measured in the fluid. A computational criterion and a uniqueness result are presented with infinitely many incident acoustic waves having common quasiperiodicity parameters. Numerical examples in 2D are demonstrated to show the validity and accuracy of the inversion algorithm.
Citation: Guanghui Hu, Andreas Kirsch, Tao Yin. Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves. Inverse Problems & Imaging, 2016, 10 (1) : 103-129. doi: 10.3934/ipi.2016.10.103
References:
[1]

T. Arens and N. Grinberg, A complete factorization method for scattering by periodic surface,, Computing, 75 (2005), 111.  doi: 10.1007/s00607-004-0092-0.  Google Scholar

[2]

T. Arens and A. Kirsch, The factorization method in inverse scattering from periodic structures,, Inverse Problems, 19 (2003), 1195.  doi: 10.1088/0266-5611/19/5/311.  Google Scholar

[3]

A. S. Bonnet-Bendhia and P. Starling, Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem,, Math. Meth. Appl. Sci., 17 (1994), 305.  doi: 10.1002/mma.1670170502.  Google Scholar

[4]

P. Carney and J. Schotland, Three-dimensional total internal reflection microscopy,, Optics Letters, 26 (2001), 1072.   Google Scholar

[5]

J. M. Claeys, O. Leroy, A. Jungman and L. Adler, Diffraction of ultrasonic waves from periodically rough liquid-solid surface,, J. Appl. Phys., 54 (1983).  doi: 10.1063/1.331829.  Google Scholar

[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Berlin, (1998).  doi: 10.1007/978-3-662-03537-5.  Google Scholar

[7]

D. Courjon and C. Bainier, Near field microscopy and near field optics,, Rep. Prog. Phys., 57 (1994), 989.   Google Scholar

[8]

N. F. Declercq, J. Degrieck, R. Briers and O. Leroy, Diffraction of homogeneous and inhomogeneous plane waves on a doubly corrugated liquid/solid interface,, Ultrasonics, 43 (2005), 605.  doi: 10.1016/j.ultras.2005.03.008.  Google Scholar

[9]

J. Elschner and G. Hu, Variational approach to scattering of plane elastic waves by diffraction gratings,, Math. Meth. Appl. Sci., 33 (2010), 1924.  doi: 10.1002/mma.1305.  Google Scholar

[10]

J. Elschner and G. Hu, Scattering of plane elastic waves by three-dimensional diffraction gratings,, Mathematical Models and Methods in Applied Sciences, 22 (2012).  doi: 10.1142/S0218202511500199.  Google Scholar

[11]

J. Elschner, G. C. Hsiao and A. Rathsfeld, An inverse problem for fluid-solid interaction,, Inverse Problems and Imaging, 2 (2008), 83.  doi: 10.3934/ipi.2008.2.83.  Google Scholar

[12]

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), 168.  doi: 10.1137/080736922.  Google Scholar

[13]

J. Elschner and G. Schmidt, Diffraction in periodic structures and optimal design of binary gratings I. Direct problems and gradient formulas,, Math. Meth. Appl. Sci., 21 (1998), 1297.  doi: 10.1002/(SICI)1099-1476(19980925)21:14<1297::AID-MMA997>3.0.CO;2-C.  Google Scholar

[14]

N. Favretto-Anrès and G. Rabau, Excitation of the Stoneley-Scholte wave at the boundary between an ideal fluid and a viscoelastic solid,, Journal of Sound and Vibration, 203 (1997), 193.   Google Scholar

[15]

C. Girard and A. Dereux, Near-field optics theories,, Rep. Prog. Phys., 59 (1996), 657.   Google Scholar

[16]

F. Hettlich and A. Kirsch, Schiffer's theorem in inverse scattering for periodic structures,, Inverse Problems, 13 (1997), 351.  doi: 10.1088/0266-5611/13/2/010.  Google Scholar

[17]

G. C. Hsiao, R. E. Kleinman and G. F. Roach, Weak solution of fluid-solid interaction problem,, Math. Nachr., 218 (2000), 139.  doi: 10.1002/1522-2616(200010)218:1<139::AID-MANA139>3.0.CO;2-S.  Google Scholar

[18]

G. Hu, Y. L. Lu and B. Zhang, The factorization method for inverse elastic scattering from periodic structures,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/11/115005.  Google Scholar

[19]

G. Hu, J. Yang, B. Zhang and H. Zhang, Near-field imaging of scattering obstacles with the factorization method,, Inverse Problems, 30 (2014).  doi: 10.1088/0266-5611/30/9/095005.  Google Scholar

[20]

G. Hu, A. Rathsfeld and T. Yin, Finite element method for fluid-solid interaction problem with unbounded perioidc interfaces,, Numerical Methods for Partial Differential Equations, 32 (2016), 5.  doi: 10.1002/num.21980.  Google Scholar

[21]

S. W. Herbison, Ultrasonic Diffraction Effects on Periodic Surfaces,, Georgia Institute of Technology, (2011).   Google Scholar

[22]

A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator,, Inverse Problems, 14 (1998), 1489.  doi: 10.1088/0266-5611/14/6/009.  Google Scholar

[23]

A. Kirsch, Diffraction by periodic structures,, in Proc. Lapland Conf. Inverse Problems, 422 (1993), 87.  doi: 10.1007/3-540-57195-7_11.  Google Scholar

[24]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems,, New York, (2008).   Google Scholar

[25]

A. Kirsch and A. Ruiz, The factorization method for an inverse fluid-solid interaction scattering problem,, Inverse Problems and Imaging, 6 (2012), 681.  doi: 10.3934/ipi.2012.6.681.  Google Scholar

[26]

V. D. Kupradze, et al., Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity,, Amsterdam, (1979).   Google Scholar

[27]

A. Lechleiter, Factorization Methods for Photonics and Rough Surfaces,, PhD thesis, (2008).   Google Scholar

[28]

A. Lechleiter and D. L. Nguyen, Factorization method for electromagnetic inverse scattering from biperiodic structures,, SIAM Journal on Imaging Sciences, 6 (2013), 1111.  doi: 10.1137/120903968.  Google Scholar

[29]

C. J. Luke and P. A. Martin, Fluid-solid interaction: Acoustic scattering by a smooth elastic obstacle,, SIAM J. Appl. Math., 55 (1995), 904.  doi: 10.1137/S0036139993259027.  Google Scholar

[30]

K. Mampaert and O. Leroy, Reflection and transmission of normally incident ultasonic waves on periodic solid-liquid interfaces,, J. Acoust. Soc. Amer., 83 (1988), 1390.   Google Scholar

[31]

P. Monk and V. Selgas, An inverse fluid-solid interaction problem,, Inverse Probl. Imaging, 3 (2009), 173.  doi: 10.3934/ipi.2009.3.173.  Google Scholar

[32]

P. Monk and V. Selgas, Near field sampling type methods for the inverse fluid-solid interaction problem,, Inverse Probl. Imaging, 5 (2011), 465.  doi: 10.3934/ipi.2011.5.465.  Google Scholar

show all references

References:
[1]

T. Arens and N. Grinberg, A complete factorization method for scattering by periodic surface,, Computing, 75 (2005), 111.  doi: 10.1007/s00607-004-0092-0.  Google Scholar

[2]

T. Arens and A. Kirsch, The factorization method in inverse scattering from periodic structures,, Inverse Problems, 19 (2003), 1195.  doi: 10.1088/0266-5611/19/5/311.  Google Scholar

[3]

A. S. Bonnet-Bendhia and P. Starling, Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem,, Math. Meth. Appl. Sci., 17 (1994), 305.  doi: 10.1002/mma.1670170502.  Google Scholar

[4]

P. Carney and J. Schotland, Three-dimensional total internal reflection microscopy,, Optics Letters, 26 (2001), 1072.   Google Scholar

[5]

J. M. Claeys, O. Leroy, A. Jungman and L. Adler, Diffraction of ultrasonic waves from periodically rough liquid-solid surface,, J. Appl. Phys., 54 (1983).  doi: 10.1063/1.331829.  Google Scholar

[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Berlin, (1998).  doi: 10.1007/978-3-662-03537-5.  Google Scholar

[7]

D. Courjon and C. Bainier, Near field microscopy and near field optics,, Rep. Prog. Phys., 57 (1994), 989.   Google Scholar

[8]

N. F. Declercq, J. Degrieck, R. Briers and O. Leroy, Diffraction of homogeneous and inhomogeneous plane waves on a doubly corrugated liquid/solid interface,, Ultrasonics, 43 (2005), 605.  doi: 10.1016/j.ultras.2005.03.008.  Google Scholar

[9]

J. Elschner and G. Hu, Variational approach to scattering of plane elastic waves by diffraction gratings,, Math. Meth. Appl. Sci., 33 (2010), 1924.  doi: 10.1002/mma.1305.  Google Scholar

[10]

J. Elschner and G. Hu, Scattering of plane elastic waves by three-dimensional diffraction gratings,, Mathematical Models and Methods in Applied Sciences, 22 (2012).  doi: 10.1142/S0218202511500199.  Google Scholar

[11]

J. Elschner, G. C. Hsiao and A. Rathsfeld, An inverse problem for fluid-solid interaction,, Inverse Problems and Imaging, 2 (2008), 83.  doi: 10.3934/ipi.2008.2.83.  Google Scholar

[12]

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), 168.  doi: 10.1137/080736922.  Google Scholar

[13]

J. Elschner and G. Schmidt, Diffraction in periodic structures and optimal design of binary gratings I. Direct problems and gradient formulas,, Math. Meth. Appl. Sci., 21 (1998), 1297.  doi: 10.1002/(SICI)1099-1476(19980925)21:14<1297::AID-MMA997>3.0.CO;2-C.  Google Scholar

[14]

N. Favretto-Anrès and G. Rabau, Excitation of the Stoneley-Scholte wave at the boundary between an ideal fluid and a viscoelastic solid,, Journal of Sound and Vibration, 203 (1997), 193.   Google Scholar

[15]

C. Girard and A. Dereux, Near-field optics theories,, Rep. Prog. Phys., 59 (1996), 657.   Google Scholar

[16]

F. Hettlich and A. Kirsch, Schiffer's theorem in inverse scattering for periodic structures,, Inverse Problems, 13 (1997), 351.  doi: 10.1088/0266-5611/13/2/010.  Google Scholar

[17]

G. C. Hsiao, R. E. Kleinman and G. F. Roach, Weak solution of fluid-solid interaction problem,, Math. Nachr., 218 (2000), 139.  doi: 10.1002/1522-2616(200010)218:1<139::AID-MANA139>3.0.CO;2-S.  Google Scholar

[18]

G. Hu, Y. L. Lu and B. Zhang, The factorization method for inverse elastic scattering from periodic structures,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/11/115005.  Google Scholar

[19]

G. Hu, J. Yang, B. Zhang and H. Zhang, Near-field imaging of scattering obstacles with the factorization method,, Inverse Problems, 30 (2014).  doi: 10.1088/0266-5611/30/9/095005.  Google Scholar

[20]

G. Hu, A. Rathsfeld and T. Yin, Finite element method for fluid-solid interaction problem with unbounded perioidc interfaces,, Numerical Methods for Partial Differential Equations, 32 (2016), 5.  doi: 10.1002/num.21980.  Google Scholar

[21]

S. W. Herbison, Ultrasonic Diffraction Effects on Periodic Surfaces,, Georgia Institute of Technology, (2011).   Google Scholar

[22]

A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator,, Inverse Problems, 14 (1998), 1489.  doi: 10.1088/0266-5611/14/6/009.  Google Scholar

[23]

A. Kirsch, Diffraction by periodic structures,, in Proc. Lapland Conf. Inverse Problems, 422 (1993), 87.  doi: 10.1007/3-540-57195-7_11.  Google Scholar

[24]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems,, New York, (2008).   Google Scholar

[25]

A. Kirsch and A. Ruiz, The factorization method for an inverse fluid-solid interaction scattering problem,, Inverse Problems and Imaging, 6 (2012), 681.  doi: 10.3934/ipi.2012.6.681.  Google Scholar

[26]

V. D. Kupradze, et al., Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity,, Amsterdam, (1979).   Google Scholar

[27]

A. Lechleiter, Factorization Methods for Photonics and Rough Surfaces,, PhD thesis, (2008).   Google Scholar

[28]

A. Lechleiter and D. L. Nguyen, Factorization method for electromagnetic inverse scattering from biperiodic structures,, SIAM Journal on Imaging Sciences, 6 (2013), 1111.  doi: 10.1137/120903968.  Google Scholar

[29]

C. J. Luke and P. A. Martin, Fluid-solid interaction: Acoustic scattering by a smooth elastic obstacle,, SIAM J. Appl. Math., 55 (1995), 904.  doi: 10.1137/S0036139993259027.  Google Scholar

[30]

K. Mampaert and O. Leroy, Reflection and transmission of normally incident ultasonic waves on periodic solid-liquid interfaces,, J. Acoust. Soc. Amer., 83 (1988), 1390.   Google Scholar

[31]

P. Monk and V. Selgas, An inverse fluid-solid interaction problem,, Inverse Probl. Imaging, 3 (2009), 173.  doi: 10.3934/ipi.2009.3.173.  Google Scholar

[32]

P. Monk and V. Selgas, Near field sampling type methods for the inverse fluid-solid interaction problem,, Inverse Probl. Imaging, 5 (2011), 465.  doi: 10.3934/ipi.2011.5.465.  Google Scholar

[1]

Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems & Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681

[2]

Johannes Elschner, George C. Hsiao, Andreas Rathsfeld. An inverse problem for fluid-solid interaction. Inverse Problems & Imaging, 2008, 2 (1) : 83-120. doi: 10.3934/ipi.2008.2.83

[3]

Beatrice Bugert, Gunther Schmidt. Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 435-473. doi: 10.3934/dcdss.2015.8.435

[4]

Francesca Bucci, Irena Lasiecka. Regularity of boundary traces for a fluid-solid interaction model. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 505-521. doi: 10.3934/dcdss.2011.4.505

[5]

Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013

[6]

Peter Monk, Virginia Selgas. An inverse fluid--solid interaction problem. Inverse Problems & Imaging, 2009, 3 (2) : 173-198. doi: 10.3934/ipi.2009.3.173

[7]

Qinghua Wu, Guozheng Yan. The factorization method for a partially coated cavity in inverse scattering. Inverse Problems & Imaging, 2016, 10 (1) : 263-279. doi: 10.3934/ipi.2016.10.263

[8]

S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604

[9]

David Bourne, Howard Elman, John E. Osborn. A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part II: Analysis of Convergence. Communications on Pure & Applied Analysis, 2009, 8 (1) : 143-160. doi: 10.3934/cpaa.2009.8.143

[10]

Stuart S. Antman, David Bourne. A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part I: Formulation, Analysis, and Computations. Communications on Pure & Applied Analysis, 2009, 8 (1) : 123-142. doi: 10.3934/cpaa.2009.8.123

[11]

Jun Guo, Qinghua Wu, Guozheng Yan. The factorization method for cracks in elastic scattering. Inverse Problems & Imaging, 2018, 12 (2) : 349-371. doi: 10.3934/ipi.2018016

[12]

Peter Monk, Virginia Selgas. Near field sampling type methods for the inverse fluid--solid interaction problem. Inverse Problems & Imaging, 2011, 5 (2) : 465-483. doi: 10.3934/ipi.2011.5.465

[13]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[14]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations & Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83

[15]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations & Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59

[16]

Stéphane Brull, Pierre Charrier, Luc Mieussens. Gas-surface interaction and boundary conditions for the Boltzmann equation. Kinetic & Related Models, 2014, 7 (2) : 219-251. doi: 10.3934/krm.2014.7.219

[17]

John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181

[18]

Lorena Bociu, Jean-Paul Zolésio. Sensitivity analysis for a free boundary fluid-elasticity interaction. Evolution Equations & Control Theory, 2013, 2 (1) : 55-79. doi: 10.3934/eect.2013.2.55

[19]

Lorena Bociu, Jean-Paul Zolésio. Existence for the linearization of a steady state fluid/nonlinear elasticity interaction. Conference Publications, 2011, 2011 (Special) : 184-197. doi: 10.3934/proc.2011.2011.184

[20]

Lorena Bociu, Lucas Castle, Kristina Martin, Daniel Toundykov. Optimal control in a free boundary fluid-elasticity interaction. Conference Publications, 2015, 2015 (special) : 122-131. doi: 10.3934/proc.2015.0122

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]