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  December 2020 Early access  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 and 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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[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.

[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. 
[25]

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

[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.

[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.

[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.

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

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

[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.

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[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.

[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. 
[25]

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

[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.

[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.

[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.

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

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

[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.

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) $
[1]

Olha Ivanyshyn. Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Problems and Imaging, 2007, 1 (4) : 609-622. doi: 10.3934/ipi.2007.1.609

[2]

Mark S. Gockenbach, Akhtar A. Khan. Identification of Lamé parameters in linear elasticity: a fixed point approach. Journal of Industrial and Management Optimization, 2005, 1 (4) : 487-497. doi: 10.3934/jimo.2005.1.487

[3]

Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001

[4]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks and Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617

[5]

Nicolas Van Goethem. The Frank tensor as a boundary condition in intrinsic linearized elasticity. Journal of Geometric Mechanics, 2016, 8 (4) : 391-411. doi: 10.3934/jgm.2016013

[6]

Roland Griesmaier, Nuutti Hyvönen, Otto Seiskari. A note on analyticity properties of far field patterns. Inverse Problems and Imaging, 2013, 7 (2) : 491-498. doi: 10.3934/ipi.2013.7.491

[7]

Isaac Harris, Dinh-Liem Nguyen, Thi-Phong Nguyen. Direct sampling methods for isotropic and anisotropic scatterers with point source measurements. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022015

[8]

Mohameden Ahmedou, Mohamed Ben Ayed, Marcello Lucia. On a resonant mean field type equation: A "critical point at Infinity" approach. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1789-1818. doi: 10.3934/dcds.2017075

[9]

Daniele Boffi, Franco Brezzi, Michel Fortin. Reduced symmetry elements in linear elasticity. Communications on Pure and Applied Analysis, 2009, 8 (1) : 95-121. doi: 10.3934/cpaa.2009.8.95

[10]

Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 131-155. doi: 10.3934/dcds.2019006

[11]

Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015

[12]

Huai-An Diao, Peijun Li, Xiaokai Yuan. Inverse elastic surface scattering with far-field data. Inverse Problems and Imaging, 2019, 13 (4) : 721-744. doi: 10.3934/ipi.2019033

[13]

Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems and Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757

[14]

Huey-Er Lin, Jian-Guo Liu, Wen-Qing Xu. Effects of small viscosity and far field boundary conditions for hyperbolic systems. Communications on Pure and Applied Analysis, 2004, 3 (2) : 267-290. doi: 10.3934/cpaa.2004.3.267

[15]

Giovanni Alessandrini, Eva Sincich, Sergio Vessella. Stable determination of surface impedance on a rough obstacle by far field data. Inverse Problems and Imaging, 2013, 7 (2) : 341-351. doi: 10.3934/ipi.2013.7.341

[16]

Qi Wang, Yanren Hou. Determining an obstacle by far-field data measured at a few spots. Inverse Problems and Imaging, 2015, 9 (2) : 591-600. doi: 10.3934/ipi.2015.9.591

[17]

Tielei Zhu, Jiaqing Yang, Bo Zhang. Recovering a bounded elastic body by electromagnetic far-field measurements. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022012

[18]

Shenglong Hu. A note on the solvability of a tensor equation. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021146

[19]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks and Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007

[20]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791

2020 Impact Factor: 1.639

Metrics

  • PDF downloads (241)
  • HTML views (183)
  • Cited by (0)

Other articles
by authors

[Back to Top]