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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

Guanghui Hu 1, , Andreas Kirsch 2, and  Tao Yin 3,

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.
Keywords: linear elasticity., biperiodic surface, Factorization method, Helmholtz equation, Navier equation, inverse scattering, fluid-solid interaction.
Mathematics Subject Classification: Primary: 74J25; Secondary: 74J20, 35R30, 35Q7.
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), 111-132. 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), 1195-1211. 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), 305-338. doi: 10.1002/mma.1670170502.  Google Scholar

[4]

P. Carney and J. Schotland, Three-dimensional total internal reflection microscopy, Optics Letters, 26 (2001), 1072-1074. 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), 5657. doi: 10.1063/1.331829.  Google Scholar

[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Berlin, Springer, 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), 989-1028. 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), 605-618. 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), 1924-1941. 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), 1150019, 34pp. 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), 83-119. 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), 168-187. 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), 1297-1342. 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), 193-208. Google Scholar

[15]

C. Girard and A. Dereux, Near-field optics theories, Rep. Prog. Phys., 59 (1996), 657-699. Google Scholar

[16]

F. Hettlich and A. Kirsch, Schiffer's theorem in inverse scattering for periodic structures, Inverse Problems, 13 (1997), 351-361. 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), 139-163. 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), 115005, 25pp. 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), 095005, 25pp. 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), 5-35. doi: 10.1002/num.21980.  Google Scholar

[21]

S. W. Herbison, Ultrasonic Diffraction Effects on Periodic Surfaces, Georgia Institute of Technology, PhD Thesis, 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), 1489-1512. doi: 10.1088/0266-5611/14/6/009.  Google Scholar

[23]

A. Kirsch, Diffraction by periodic structures, in Proc. Lapland Conf. Inverse Problems, (eds. L. Päivärinta et al), Berlin, Springer, 422 (1993), 87-102. doi: 10.1007/3-540-57195-7_11.  Google Scholar

[24]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, New York, Oxford Univ. Press, 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), 681-695. 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, North-Holland, 1979.  Google Scholar

[27]

A. Lechleiter, Factorization Methods for Photonics and Rough Surfaces, PhD thesis, University of Karlsruhe, 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), 1111-1139. 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), 904-922. 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), 1390-1398. Google Scholar

[31]

P. Monk and V. Selgas, An inverse fluid-solid interaction problem, Inverse Probl. Imaging, 3 (2009), 173-198. 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), 465-483. 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), 111-132. 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), 1195-1211. 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), 305-338. doi: 10.1002/mma.1670170502.  Google Scholar

[4]

P. Carney and J. Schotland, Three-dimensional total internal reflection microscopy, Optics Letters, 26 (2001), 1072-1074. 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), 5657. doi: 10.1063/1.331829.  Google Scholar

[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Berlin, Springer, 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), 989-1028. 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), 605-618. 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), 1924-1941. 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), 1150019, 34pp. 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), 83-119. 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), 168-187. 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), 1297-1342. 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), 193-208. Google Scholar

[15]

C. Girard and A. Dereux, Near-field optics theories, Rep. Prog. Phys., 59 (1996), 657-699. Google Scholar

[16]

F. Hettlich and A. Kirsch, Schiffer's theorem in inverse scattering for periodic structures, Inverse Problems, 13 (1997), 351-361. 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), 139-163. 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), 115005, 25pp. 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), 095005, 25pp. 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), 5-35. doi: 10.1002/num.21980.  Google Scholar

[21]

S. W. Herbison, Ultrasonic Diffraction Effects on Periodic Surfaces, Georgia Institute of Technology, PhD Thesis, 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), 1489-1512. doi: 10.1088/0266-5611/14/6/009.  Google Scholar

[23]

A. Kirsch, Diffraction by periodic structures, in Proc. Lapland Conf. Inverse Problems, (eds. L. Päivärinta et al), Berlin, Springer, 422 (1993), 87-102. doi: 10.1007/3-540-57195-7_11.  Google Scholar

[24]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, New York, Oxford Univ. Press, 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), 681-695. 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, North-Holland, 1979.  Google Scholar

[27]

A. Lechleiter, Factorization Methods for Photonics and Rough Surfaces, PhD thesis, University of Karlsruhe, 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), 1111-1139. 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), 904-922. 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), 1390-1398. Google Scholar

[31]

P. Monk and V. Selgas, An inverse fluid-solid interaction problem, Inverse Probl. Imaging, 3 (2009), 173-198. 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), 465-483. doi: 10.3934/ipi.2011.5.465.  Google Scholar

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