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April  2020, 14(2): 363-384. doi: 10.3934/ipi.2020016

The Linear Sampling Method for Kirchhoff-Love infinite plates

1. 

Laboratoire POEMS, ENSTA ParisTech, 828 Boulevard des Maréchaux, 91120 Palaiseau, France

2. 

CEA, LIST, 91191 Gif-sur-Yvette, France

* Corresponding author: Laurent Bourgeois

Received  July 2019 Revised  October 2019 Published  February 2020

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.

Citation: Laurent Bourgeois, Arnaud Recoquillay. The Linear Sampling Method for Kirchhoff-Love infinite plates. Inverse Problems and Imaging, 2020, 14 (2) : 363-384. doi: 10.3934/ipi.2020016
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. ColtonM. 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. CharalambopoulosD. 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. CakoniG. 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. FarhatS. 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. MorassiE. 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. SmithM. 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.

show all references

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. ColtonM. 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. CharalambopoulosD. 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. CakoniG. 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. FarhatS. 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. MorassiE. 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. SmithM. 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.

Figure 1.  Validation of the artificial boundary condition. Left: scattering solution computed in $ \Omega_1 $. Right: scattering solution computed in $ \Omega_2 $
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 $
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 $
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
Figure 5.  Function $ \Psi $ for a Dirichlet obstacle, $ k = 20 $ and noisy data. Left: noise of amplitude $ 5\% $. Right: noise of amplitude $ 10\% $
Figure 6.  Function $ \Psi $ for a Dirichlet obstacle with less (exact) data, $ k = 30 $. Left: two circles. Right: three circles
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\% $
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