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.

[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.

[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.

[4]

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

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[23]

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

[24]

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

[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.

[26]

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

[27]

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[23]

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

[24]

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

[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.

[26]

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

[27]

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

[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.

[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.

[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.

[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.

[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.

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