December  2020, 14(6): 1025-1056. doi: 10.3934/ipi.2020054

Direct and inverse time-harmonic elastic scattering from point-like and extended obstacles

1. 

School of Mathematical Sciences, Nankai University, Tianjin 300071, China

2. 

Laboratoire de Mathématiques de Reims, UMR 9008 CNRS, France

3. 

RICAM, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040 Linz, Austria

4. 

Department of Computing & Mathematical Sciences, California Institute of Technology, 1200 East California Blvd., CA 91125, United States

* Corresponding author

Received  December 2019 Revised  July 2020 Published  August 2020

This paper is concerned with the time-harmonic direct and inverse elastic scattering by an extended rigid elastic body surrounded by a finite number of point-like obstacles. We first justify the point-interaction model for the Lamé operator within the singular perturbation approach. For a general family of pointwise-supported singular perturbations, including anisotropic and non-local interactions, we derive an explicit representation of the scattered field.

In the case of isotropic and local point-interactions, our result is consistent with the ones previously obtained by Foldy's formal method as well as by the renormalization technique. In the case of multiple scattering with pointwise and extended obstacles, we show that the scattered field consists of two parts: one is due to the diffusion by the extended scatterer and the other one is a linear combination of the interactions between the point-like obstacles and the interaction between the point-like obstacles with the extended one.

As to the inverse problem, the factorization method by Kirsch is adapted to recover simultaneously the shape of an extended elastic body and the location of point-like scatterers in the case of isotropic and local interactions. The inverse problems using only one type of elastic waves (i.e. pressure or shear waves) are also investigated and numerical examples are presented to confirm the inversion schemes.

Citation: Guanghui Hu, Andrea Mantile, Mourad Sini, Tao Yin. Direct and inverse time-harmonic elastic scattering from point-like and extended obstacles. Inverse Problems & Imaging, 2020, 14 (6) : 1025-1056. doi: 10.3934/ipi.2020054
References:
[1]

S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-88201-2.  Google Scholar

[2]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Sc. Norm. Super. Pisa Cl. Sci.(Ⅳ), 2 (1975), 151-218.   Google Scholar

[3]

C. J. Alves and R. Kress, On the far-field operator in elastic obstacle scattering, IMA J. Appl. Math., 67 (2002), 1-21.  doi: 10.1093/imamat/67.1.1.  Google Scholar

[4]

H. AmmariH. Kang and H. Lee, Asymptotic expansions for eigenvalues of the Lamé system in the presence of small inclusions, Communications in Partial Differential Equations, 32 (2007), 1715-1736.  doi: 10.1080/03605300600910266.  Google Scholar

[5]

T. Arens, Linear sampling method for 2D inverse elastic wave scattering, Inverse Problems, 17 (2001), 1445-1464.  doi: 10.1088/0266-5611/17/5/314.  Google Scholar

[6]

G. BaoG. HuJ. Sun and T. Yin, Direct and inverse elastic scattering from anisotropic media, J. Math. Pures Appl., 117 (2018), 263-301.  doi: 10.1016/j.matpur.2018.01.007.  Google Scholar

[7]

I. V. Blinova, A. A. Boitsev, I. Y. Popov, A. Froehly and H. Neidhardt, Point-like perturbation for Lame operator, to appear in: Complex Variables and Elliptic Equations, 2019. Available at: https: //doi.org/10.1080/17476933.2019.1579207 doi: 10.1080/17476933.2019.1579207.  Google Scholar

[8]

D. P. Challa and M. Sini, Inverse scattering by point-like scatterers in the Foldy regime, Inverse Problems, 28 (2012), 125006. doi: 10.1088/0266-5611/28/12/125006.  Google Scholar

[9]

A. CharalambopoulosA. KirschK. A. AnagnostopoulosD. Gintides and K. Kiriaki, The factorization method in inverse elastic scattering from penetrable bodies, Inverse Problems, 23 (2007), 27-51.  doi: 10.1088/0266-5611/23/1/002.  Google Scholar

[10]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, New York, 1998. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[11]

Y. Colin de Verdiére, Elastic wave equation, Actes du Séminaire de Théorie Spectrale et Géométrie, 25 (2008), 55-69.  doi: 10.5802/tsg.247.  Google Scholar

[12]

L. Foldy, The multiple scattering of waves I. General theory of isotropic scattering by randomly distributed scatterers, Phys. Rev., 67 (1945), 107-119.  doi: 10.1103/PhysRev.67.107.  Google Scholar

[13]

D. GintidesM. Sini and N. T. Thanh, Detection of point-like scatterers using one type of scattered elastic waves, J. Comput. Appl. Math., 236 (2012), 2137-2145.  doi: 10.1016/j.cam.2011.09.036.  Google Scholar

[14]

D. Gintides and M. Sini, Identification of obstacles using only the scattered P-waves or the scattered S-waves, Inverse Probl. Imaging, 6 (2012), 39-55.  doi: 10.3934/ipi.2012.6.39.  Google Scholar

[15]

P. Hähner, On Acoustic, Electromagnetic, and Elastic Scattering Problems in Inhomogeneous Media, Habilitationsshrift, Göttingen, 1998. Google Scholar

[16]

G. HuA. Mantile and M. Sini, Direct and inverse acoustic scattering by a collection of extended and point-like scatterers, Multiscale Model. Simul., 12 (2014), 996-1027.  doi: 10.1137/130932107.  Google Scholar

[17]

G. Hu and M. Sini, Elastic scattering by finitely many point-like obstacles, J. Math. Phys., 54 (2013), 042901. doi: 10.1063/1.4799145.  Google Scholar

[18]

G. Hu, A. Kirsch and M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Problems, 29 (2013), 015009. doi: 10.1088/0266-5611/29/1/015009.  Google Scholar

[19]

K. Huang and P. Li, A two-scale multiple scattering problem, Multiscale Model. Simul., 8 (2010), 1511-1534.  doi: 10.1137/090771090.  Google Scholar

[20]

K. HuangK. Solna and H. Zhao, Generalized Foldy-Lax formulation, J. Comput. Phys., 229 (2010), 4544-4553.  doi: 10.1016/j.jcp.2010.02.021.  Google Scholar

[21]

M. Kar and M. Sini, On the inverse elastic scattering by interfaces using one type of scattered waves, J. Elasticity, 118(1) (2015), 15-38.  doi: 10.1007/s10659-014-9474-5.  Google Scholar

[22]

T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1976.  Google Scholar

[23]

A. Kirsch, Characterization of the shape of the scattering obstacle by the spectral data of the far-field operator, Inverse Problems, 14 (1998), 1489-1512.  doi: 10.1088/0266-5611/14/6/009.  Google Scholar

[24] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems (Oxford Lecture Series in Mathematics and its Applications), 36, Oxford, Oxford University Press, 2008.   Google Scholar
[25]

A. Komech and E. Kopylova, Dispersion Decay and Scattering Theory, John Wiley & Sons, Hoboken, New Jersey, 2012. doi: 10.1002/9781118382868.  Google Scholar

[26]

V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili and T. V. Burchuladze, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland Series in Applied Mathematics and Mechanics, 25, North-Holland Publishing Co., Amsterdam, 1979.  Google Scholar

[27]

A. MantileA. Posilicano and M. Sini, Limiting absorption principle, generalized eigenfunction and scattering matrix for Laplace operators with boundary conditions on hypersurfaces, J. Spectr. Theory, 8 (2018), 1443-1486.  doi: 10.4171/JST/231.  Google Scholar

[28]

A. Mantile and A. Posilicano, Asymptotic Completeness and S-Matrix for Singular Perturbations, preprint, arXiv: 1711.07556. doi: 10.1016/j.matpur.2019.01.017.  Google Scholar

[29] P. A. Martin, Multiple Scattering, Encyclopedia Math. Appl. 107, Cambridge University Press, Cambridge, UK, 2006.  doi: 10.1017/CBO9780511735110.  Google Scholar
[30]

A. Posilicano, A Kreĭn-like formula for singular perturbations of self-adjoint operators and applications, J. Funct. Anal., 183 (2001), 109-147.  doi: 10.1006/jfan.2000.3730.  Google Scholar

[31] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol I. Fourier Analysis, Self-adjointness, Academy Press, New York, 1972.   Google Scholar
[32] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol. Ⅱ: Fourier Analysis, Self-adjointness, Academic Press, New York, 1975.   Google Scholar
[33] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol. Ⅳ: Analysis of Operators, Academic Press, New York, 1978.   Google Scholar
[34]

V. Sevroglou, The far-field operator for penetrable and absorbing obstacles in 2D inverse elastic scattering, Inverse Problems, 21 (2005), 717-738.  doi: 10.1088/0266-5611/21/2/017.  Google Scholar

[35]

P. de VriesD. V. van Coevorden and A. Lagendijk, Point scatterers for classical waves, Rev. Modern Phys., 70 (1998), 447-466.  doi: 10.1103/RevModPhys.70.447.  Google Scholar

show all references

References:
[1]

S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-88201-2.  Google Scholar

[2]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Sc. Norm. Super. Pisa Cl. Sci.(Ⅳ), 2 (1975), 151-218.   Google Scholar

[3]

C. J. Alves and R. Kress, On the far-field operator in elastic obstacle scattering, IMA J. Appl. Math., 67 (2002), 1-21.  doi: 10.1093/imamat/67.1.1.  Google Scholar

[4]

H. AmmariH. Kang and H. Lee, Asymptotic expansions for eigenvalues of the Lamé system in the presence of small inclusions, Communications in Partial Differential Equations, 32 (2007), 1715-1736.  doi: 10.1080/03605300600910266.  Google Scholar

[5]

T. Arens, Linear sampling method for 2D inverse elastic wave scattering, Inverse Problems, 17 (2001), 1445-1464.  doi: 10.1088/0266-5611/17/5/314.  Google Scholar

[6]

G. BaoG. HuJ. Sun and T. Yin, Direct and inverse elastic scattering from anisotropic media, J. Math. Pures Appl., 117 (2018), 263-301.  doi: 10.1016/j.matpur.2018.01.007.  Google Scholar

[7]

I. V. Blinova, A. A. Boitsev, I. Y. Popov, A. Froehly and H. Neidhardt, Point-like perturbation for Lame operator, to appear in: Complex Variables and Elliptic Equations, 2019. Available at: https: //doi.org/10.1080/17476933.2019.1579207 doi: 10.1080/17476933.2019.1579207.  Google Scholar

[8]

D. P. Challa and M. Sini, Inverse scattering by point-like scatterers in the Foldy regime, Inverse Problems, 28 (2012), 125006. doi: 10.1088/0266-5611/28/12/125006.  Google Scholar

[9]

A. CharalambopoulosA. KirschK. A. AnagnostopoulosD. Gintides and K. Kiriaki, The factorization method in inverse elastic scattering from penetrable bodies, Inverse Problems, 23 (2007), 27-51.  doi: 10.1088/0266-5611/23/1/002.  Google Scholar

[10]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, New York, 1998. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[11]

Y. Colin de Verdiére, Elastic wave equation, Actes du Séminaire de Théorie Spectrale et Géométrie, 25 (2008), 55-69.  doi: 10.5802/tsg.247.  Google Scholar

[12]

L. Foldy, The multiple scattering of waves I. General theory of isotropic scattering by randomly distributed scatterers, Phys. Rev., 67 (1945), 107-119.  doi: 10.1103/PhysRev.67.107.  Google Scholar

[13]

D. GintidesM. Sini and N. T. Thanh, Detection of point-like scatterers using one type of scattered elastic waves, J. Comput. Appl. Math., 236 (2012), 2137-2145.  doi: 10.1016/j.cam.2011.09.036.  Google Scholar

[14]

D. Gintides and M. Sini, Identification of obstacles using only the scattered P-waves or the scattered S-waves, Inverse Probl. Imaging, 6 (2012), 39-55.  doi: 10.3934/ipi.2012.6.39.  Google Scholar

[15]

P. Hähner, On Acoustic, Electromagnetic, and Elastic Scattering Problems in Inhomogeneous Media, Habilitationsshrift, Göttingen, 1998. Google Scholar

[16]

G. HuA. Mantile and M. Sini, Direct and inverse acoustic scattering by a collection of extended and point-like scatterers, Multiscale Model. Simul., 12 (2014), 996-1027.  doi: 10.1137/130932107.  Google Scholar

[17]

G. Hu and M. Sini, Elastic scattering by finitely many point-like obstacles, J. Math. Phys., 54 (2013), 042901. doi: 10.1063/1.4799145.  Google Scholar

[18]

G. Hu, A. Kirsch and M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Problems, 29 (2013), 015009. doi: 10.1088/0266-5611/29/1/015009.  Google Scholar

[19]

K. Huang and P. Li, A two-scale multiple scattering problem, Multiscale Model. Simul., 8 (2010), 1511-1534.  doi: 10.1137/090771090.  Google Scholar

[20]

K. HuangK. Solna and H. Zhao, Generalized Foldy-Lax formulation, J. Comput. Phys., 229 (2010), 4544-4553.  doi: 10.1016/j.jcp.2010.02.021.  Google Scholar

[21]

M. Kar and M. Sini, On the inverse elastic scattering by interfaces using one type of scattered waves, J. Elasticity, 118(1) (2015), 15-38.  doi: 10.1007/s10659-014-9474-5.  Google Scholar

[22]

T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1976.  Google Scholar

[23]

A. Kirsch, Characterization of the shape of the scattering obstacle by the spectral data of the far-field operator, Inverse Problems, 14 (1998), 1489-1512.  doi: 10.1088/0266-5611/14/6/009.  Google Scholar

[24] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems (Oxford Lecture Series in Mathematics and its Applications), 36, Oxford, Oxford University Press, 2008.   Google Scholar
[25]

A. Komech and E. Kopylova, Dispersion Decay and Scattering Theory, John Wiley & Sons, Hoboken, New Jersey, 2012. doi: 10.1002/9781118382868.  Google Scholar

[26]

V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili and T. V. Burchuladze, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland Series in Applied Mathematics and Mechanics, 25, North-Holland Publishing Co., Amsterdam, 1979.  Google Scholar

[27]

A. MantileA. Posilicano and M. Sini, Limiting absorption principle, generalized eigenfunction and scattering matrix for Laplace operators with boundary conditions on hypersurfaces, J. Spectr. Theory, 8 (2018), 1443-1486.  doi: 10.4171/JST/231.  Google Scholar

[28]

A. Mantile and A. Posilicano, Asymptotic Completeness and S-Matrix for Singular Perturbations, preprint, arXiv: 1711.07556. doi: 10.1016/j.matpur.2019.01.017.  Google Scholar

[29] P. A. Martin, Multiple Scattering, Encyclopedia Math. Appl. 107, Cambridge University Press, Cambridge, UK, 2006.  doi: 10.1017/CBO9780511735110.  Google Scholar
[30]

A. Posilicano, A Kreĭn-like formula for singular perturbations of self-adjoint operators and applications, J. Funct. Anal., 183 (2001), 109-147.  doi: 10.1006/jfan.2000.3730.  Google Scholar

[31] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol I. Fourier Analysis, Self-adjointness, Academy Press, New York, 1972.   Google Scholar
[32] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol. Ⅱ: Fourier Analysis, Self-adjointness, Academic Press, New York, 1975.   Google Scholar
[33] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol. Ⅳ: Analysis of Operators, Academic Press, New York, 1978.   Google Scholar
[34]

V. Sevroglou, The far-field operator for penetrable and absorbing obstacles in 2D inverse elastic scattering, Inverse Problems, 21 (2005), 717-738.  doi: 10.1088/0266-5611/21/2/017.  Google Scholar

[35]

P. de VriesD. V. van Coevorden and A. Lagendijk, Point scatterers for classical waves, Rev. Modern Phys., 70 (1998), 447-466.  doi: 10.1103/RevModPhys.70.447.  Google Scholar

Figure 1.  The kite-shaped extended obstacle
Figure 2.  Reconstruction of the kite-shaped obstacle and 6 point-like scatterers for Example 1 with different polarization vectors $ \mathbf{a} = (\cos\beta,\sin\beta) $. We set $ \beta = 0 $ in (a, c, e) and $ \beta = \pi/2 $ in (b, d, f)
Figure 3.  Reconstruction of the kite-shaped obstacle and 11 point-like scatterers for Example 1 with different polarization vectors $ \mathbf{a} = (\cos\beta,\sin\beta) $. $ \alpha = 0 $ in (a, c, e), $ \beta = \pi/2 $ in (b, d, f)
Figure 4.  Reconstruction of the kite-shaped obstacle and 6 point-like scatterers for Example 2 with different "impedance'' coefficients $ \alpha _{j}, j = 1,\cdots,M $.
Figure 5.  Reconstruction of the kite-shaped obstacle and 20 point-like scatterers for Example 3 with different polarization vectors $ \mathbf{a} = (\cos\beta,\sin\beta) $
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