The Linear Sampling Method for Kirchhoff-Love infinite plates

Laurent Bourgeois; Arnaud Recoquillay

doi:10.3934/ipi.2020016

Article Contents

2020, Volume 14, Issue 2: 363-384. Doi: 10.3934/ipi.2020016

This issue Previous Article Non-local blind hyperspectral image super-resolution via 4d sparse tensor factorization and low-rank Next Article Integral formulation of the complete electrode model of electrical impedance tomography

The Linear Sampling Method for Kirchhoff-Love infinite plates

Laurent Bourgeois^1, , and
Arnaud Recoquillay^2,

1.
Laboratoire POEMS, ENSTA ParisTech, 828 Boulevard des Maréchaux, 91120 Palaiseau, France
2.
CEA, LIST, 91191 Gif-sur-Yvette, France

^* Corresponding author: Laurent Bourgeois
^* Corresponding author: Laurent Bourgeois

Received: June 30, 2019

Revised: September 30, 2019

Published: February 2020

Abstract Full Text(HTML) Figure(7) Related Papers Cited by

Abstract

This paper addresses the problem of identifying impenetrable obstacles in a Kirchhoff-Love infinite plate from multistatic near-field data. The Linear Sampling Method is introduced in this context. We firstly prove a uniqueness result for such an inverse problem. We secondly provide the classical theoretical foundation of the Linear Sampling Method. We lastly show the feasibility of the method with the help of numerical experiments.

Keywords:

Mathematics Subject Classification: Primary: 35R25, 35R30, 35R35, 74J15, 74K20.

Citation:

Full Text(HTML)

Figure 1. Validation of the artificial boundary condition. Left: scattering solution computed in $ \Omega_1 $. Right: scattering solution computed in $ \Omega_2 $

Download: Full-size image PowerPoint slide

Figure 2. Function $ \Psi $ given by (32) for a Dirichlet obstacle, exact data and various wave numbers $ k $. Top left: $ k = 10 $. Top right: $ k = 20 $. Bottom: $ k = 30 $

Download: Full-size image PowerPoint slide

Figure 3. Function $ \Psi $ for a Neumann obstacle, exact data and various wave numbers $ k $. Top left: $ k = 10 $. Top right: $ k = 20 $. Bottom: $ k = 30 $

Download: Full-size image PowerPoint slide

Figure 4. Left: Function $ \Psi $ for a Dirichlet obstacle formed by 3 circles, $ k = 30 $ and exact data. Right: Function $ \Psi $ for a Neumann kite-shaped obstacle, $ k = 20 $ and exact data

Download: Full-size image PowerPoint slide

Figure 5. Function $ \Psi $ for a Dirichlet obstacle, $ k = 20 $ and noisy data. Left: noise of amplitude $ 5\% $. Right: noise of amplitude $ 10\% $

Download: Full-size image PowerPoint slide

Figure 6. Function $ \Psi $ for a Dirichlet obstacle with less (exact) data, $ k = 30 $. Left: two circles. Right: three circles

Download: Full-size image PowerPoint slide

Figure 7. Function $ \Psi $ for a Dirichlet obstacle with less data in the presence of noise, $ k = 20 $. Left: noise of amplitude $ 5\% $. Right: noise of amplitude $ 10\% $

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	T. Arens, Linear sampling methods for 2D inverse elastic wave scattering, Inverse Problems, 17 (2001), 1445-1464. doi: 10.1088/0266-5611/17/5/314.
[2]	L. Bourgeois and C. Hazard, On Well-posedness of Scattering Problems in Kirchhoff-Love Infinite Plates, https://hal-ensta-paris.archives-ouvertes.fr//hal-02334004, submitted.
[3]	L. Bourgeois, F. Le Louër and E. Lunéville, On the use of Lamb modes in the linear sampling method for elastic waveguides, Inverse Problems, 27 (2011), 055001, 27pp. doi: 10.1088/0266-5611/27/5/055001.
[4]	V. Baronian, L. Bourgeois, B. Chapuis and A. Recoquillay, Linear sampling method applied to non destructive testing of an elastic waveguide: theory, numerics and experiments, Inverse Problems, 34 (2018), 075006, 34pp. doi: 10.1088/1361-6420/aac21e.
[5]	L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A formulation based on modes, Journal of Physics: Conference Series, 135 (2008), 012023.
[6]	C. Bernardi, Y. Maday and F. Rappetti, Discrétisations Variationnelles de Problèmes aux Limites Elliptiques, Springer-Verlag, Berlin, 2004.
[7]	D. Colton, M. Le Piana and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems, Inverse Problems, 13 (1997), 1477-1493. doi: 10.1088/0266-5611/13/6/005.
[8]	D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003.
[9]	F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, Berlin, 2006.
[10]	A. Charalambopoulos, D. Gintides and K. Kiriaki, Linear sampling methods for 2D inverse elastic wave scattering, Inverse Problems, 18 (2001), 547-558. doi: 10.1088/0266-5611/18/3/303.
[11]	F. Cakoni, G. Hsiao and W. Wendland, On the boundary integral equation method for a mixed boundary value problem of the biharmonic equation, Complex Var. Theory Appl., 50 (2005), 681-696. doi: 10.1080/02781070500087394.
[12]	N. Fata and B. Guzina, A linear sampling method for near-field inverse problems in elastodynamics, Inverse Problems, 20 (2001), 713-736.
[13]	M. Farhat, S. Guenneau and S. Enoch, Finite elements modelling of scattering problems for flexural waves in thin plates: application to elliptic invisibility cloaks, rotators and the mirage effect, Journal of Computational Physics, 230 (2011), 2237-2245. doi: 10.1016/j.jcp.2010.12.009.
[14]	G. Hsiao and W. Wendland, Boundary Integral Equations, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-68545-6.
[15]	J.-L. Lions and E. Magenes, "Problèmes Aux Limites non Homogènes et Applications, Vol. 1", Dunod, Paris, 1968.
[16]	P. Lascaux and E. Lesaint, Eléments finis non-conformes pour le problème de la flexion des plaques minces, Publications des Séminaires de Mathématiques et Informatique de Rennes, fascicule S4 "Journées éléments finis", (1974), 1-51.
[17]	A. Morassi and E. Rosset, Unique determination of unknown boundaries in an elastic plate by one measurement, Comptes Rendus Mécanique, 338 (2010), 450-460. doi: 10.1016/j.crme.2010.07.011.
[18]	A. Morassi, E. Rosset and S. Vessella, Optimal stability in the identification of a rigid inclusion in an isotropic Kirchhoff-Love plate, SIAM J. Math. Anal., 51 (2019), 731-747. doi: 10.1137/18M1203286.
[19]	L. S. D. Morley, The triangular equilibrium element in the solution of plate bending problems, Aero-Quart., 19 (1968), 149-169. doi: 10.1017/S0001925900004546.
[20]	M. J. A. Smith, M. H. Meylan and R. C. McPhedran, Scattering by cavities of arbitrary shape in an infinite plate and associated vibration problems, Journal of Sound and Vibration, 330 (2011), 4029-4046. doi: 10.1016/j.jsv.2011.03.019.
[21]	T. Tyni and V. Serov, Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Problems and Imaging, 12 (2018), 205-227. doi: 10.3934/ipi.2018008.