Article Contents
Article Contents

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

• * Corresponding author
• 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.

Mathematics Subject Classification: 74B05, 78A45, 81Q10.

 Citation:

• 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] 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. Ammari, H. 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. Bao, G. Hu, J. 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. Charalambopoulos, A. Kirsch, K. A. Anagnostopoulos, D. 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. Gintides, M. 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. Hu, A. 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. Huang, K. 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. Mantile, A. 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 Vries, D. V. van Coevorden and A. Lagendijk, Point scatterers for classical waves, Rev. Modern Phys., 70 (1998), 447-466.  doi: 10.1103/RevModPhys.70.447.

Figures(5)