The factorization method applied to cracks with impedance boundary conditions

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November 2013, 7(4): 1123-1138. doi: 10.3934/ipi.2013.7.1123

The factorization method applied to cracks with impedance boundary conditions

Yosra Boukari ^1, and Houssem Haddar ^2,

1.	INRIA Saclay Ile de France/Ecole Polytechnique, CMAP, Route de Saclay, 91128 Palaiseau Cedex, France
2.	INRIA Saclay Ile de France / CMAP Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex

Received August 2012 Revised June 2013 Published November 2013

Abstract
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We use the Factorization method to retrieve the shape of cracks with impedance boundary conditions from farfields associated with incident plane waves at a fixed frequency. This work is an extension of the study initiated by Kirsch and Ritter [Inverse Problems, 16, pp. 89-105, 2000] where the case of sound soft cracks is considered. We address here the scalar problem and provide theoretical validation of the method when the impedance boundary conditions hold on both sides of the crack. We then deduce an inversion algorithm and present some validating numerical results in the case of simply and multiply connected cracks.

Keywords: the Factorization method., sampling methods for cracks, Inverse scattering problems, impedance boundary conditions.

Mathematics Subject Classification: 35R30, 35J05, 78A4.

Citation: Yosra Boukari, Houssem Haddar. The factorization method applied to cracks with impedance boundary conditions. Inverse Problems and Imaging, 2013, 7 (4) : 1123-1138. doi: 10.3934/ipi.2013.7.1123

References:

[1]	G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proceedings of the American Mathematical Society, 133 (2005), 1685-1691, (electronic). doi: 10.1090/S0002-9939-05-07810-X.
[2]	H. Ammari, J. Garnier, H. Kang, W. K. Park and K. Solna, Imaging schemes for perfectly conducting cracks, SIAM, J. Appl. Math., 71 (2011), 68-91. doi: 10.1137/100800130.
[3]	A. Ben Abda, F. Delbary and H. Haddar, On the use of the reciprocity-gap functional in inverse scattering from planar cracks, Math. Models Methods Appl. Sci.,15 (2005), 1553-1574. doi: 10.1142/S0218202505000819.
[4]	F. Ben Hassen, Y. Boukari and H. Haddar, Application of the linear sampling method to retrieve cracks with impedance boundary conditions, Inverse Problems in Science and Engineering, 2012. doi: 10.1080/17415977.2012.686997.
[5]	M. Bonnet, Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems, Eng. Anal. Bound. Elem. 35 (2011), 223-235. doi: 10.1016/j.enganabound.2010.08.007.
[6]	M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements, M2AN Math. Model. Numer. Anal. 35 (2001), 595-605. doi: 10.1051/m2an:2001128.
[7]	K. Bryan and M. S. Vogelius, A review of selected works on crack identification, in Geometric Methods in Inverse Problems and PDE Control (Springer, New York), MA Vol. 137, Math. Appl., 2004, 25-46. doi: 10.1007/978-1-4684-9375-7_3.
[8]	F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, 2006.
[9]	F. Cakoni and D. Colton, The linear sampling method for cracks, Inverse Problems, 19 (2003), 279-295. doi: 10.1088/0266-5611/19/2/303.
[10]	N. Zeev and F. Cakoni, The identification of thin dielectric objects from far field or near field scattering data, SIAM J. Appl. Math., 69 (2009), 1024-1042. doi: 10.1137/070711542.
[11]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, second edition, Heidelberg: Springer, 1998.
[12]	D. Koyama, Error estimates of the DtN finite element method for the exterior Helmholtz problem, J. Comput. Appl. Math., 200 (2007), 21-31. doi: 10.1016/j.cam.2005.12.004.
[13]	T. Johansson and B. D. Sleeman, Reconstruction of an acoustically sound-soft obstacle from one incident field and the far field pattern, IMA Journal of Applied Mathematics, 72 (2007), 96-112. doi: 10.1093/imamat/hxl026.
[14]	O. Ivanyshyn and R. Kress, Inverse scattering for planar cracks via nonlinear integral equations, Math. Methods Appl. Sci., 31 (2008), 1221-1232. doi: 10.1002/mma.970.
[15]	N. Grinberg and A. Kirsch, The linear sampling method in inverse obstacle scattering for impedance boundary conditions, J. Inverse Ill-Posed Probl., 10 (2002), 171-185. doi: 10.1515/jiip.2002.10.2.171.
[16]	A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture in Mathematics and Its Applications 36, (2008).
[17]	A. Kirsch and S. Ritter, A linear sampling method for inverse scattering from an open arc, Inverse Problems, 16 (2000), 89-105. doi: 10.1088/0266-5611/16/1/308.
[18]	R. Kress and P. Serranho, A hybrid method for two-dimensional crack reconstruction, Inverse Problems, 21 (2005), 773-784. doi: 10.1088/0266-5611/21/2/020.
[19]	J. J. Liu, P. A. Krutitskii and M. Sini, Numerical solution of the scattering problem for acoustic waves by a two-sided crack in 2-dimensional space, J. Comput. Math., 29 (2011), 141-166. doi: 10.4208/jams.012111.012811a.
[20]	A. Lechleiter, The factorization method is independent of transmission eigenvalues, Inverse Probl. Imaging, 3 (2009), 123-138. doi: 10.3934/ipi.2009.3.123.
[21]	J. Liu and M. Sini, Reconstruction of cracks of different types from far-field measurements, Math. Meth. Appl. Sci., 33 (2010), 950-973. doi: 10.1002/mma.1203.
[22]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
[23]	J. C. Nédélec, Acoustic and Electromagnetic Equations, Applied Matimatical Sciences. Springer-Verlag, Berlin, 2001.

show all references

References:

[1]	G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proceedings of the American Mathematical Society, 133 (2005), 1685-1691, (electronic). doi: 10.1090/S0002-9939-05-07810-X.
[2]	H. Ammari, J. Garnier, H. Kang, W. K. Park and K. Solna, Imaging schemes for perfectly conducting cracks, SIAM, J. Appl. Math., 71 (2011), 68-91. doi: 10.1137/100800130.
[3]	A. Ben Abda, F. Delbary and H. Haddar, On the use of the reciprocity-gap functional in inverse scattering from planar cracks, Math. Models Methods Appl. Sci.,15 (2005), 1553-1574. doi: 10.1142/S0218202505000819.
[4]	F. Ben Hassen, Y. Boukari and H. Haddar, Application of the linear sampling method to retrieve cracks with impedance boundary conditions, Inverse Problems in Science and Engineering, 2012. doi: 10.1080/17415977.2012.686997.
[5]	M. Bonnet, Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems, Eng. Anal. Bound. Elem. 35 (2011), 223-235. doi: 10.1016/j.enganabound.2010.08.007.
[6]	M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements, M2AN Math. Model. Numer. Anal. 35 (2001), 595-605. doi: 10.1051/m2an:2001128.
[7]	K. Bryan and M. S. Vogelius, A review of selected works on crack identification, in Geometric Methods in Inverse Problems and PDE Control (Springer, New York), MA Vol. 137, Math. Appl., 2004, 25-46. doi: 10.1007/978-1-4684-9375-7_3.
[8]	F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, 2006.
[9]	F. Cakoni and D. Colton, The linear sampling method for cracks, Inverse Problems, 19 (2003), 279-295. doi: 10.1088/0266-5611/19/2/303.
[10]	N. Zeev and F. Cakoni, The identification of thin dielectric objects from far field or near field scattering data, SIAM J. Appl. Math., 69 (2009), 1024-1042. doi: 10.1137/070711542.
[11]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, second edition, Heidelberg: Springer, 1998.
[12]	D. Koyama, Error estimates of the DtN finite element method for the exterior Helmholtz problem, J. Comput. Appl. Math., 200 (2007), 21-31. doi: 10.1016/j.cam.2005.12.004.
[13]	T. Johansson and B. D. Sleeman, Reconstruction of an acoustically sound-soft obstacle from one incident field and the far field pattern, IMA Journal of Applied Mathematics, 72 (2007), 96-112. doi: 10.1093/imamat/hxl026.
[14]	O. Ivanyshyn and R. Kress, Inverse scattering for planar cracks via nonlinear integral equations, Math. Methods Appl. Sci., 31 (2008), 1221-1232. doi: 10.1002/mma.970.
[15]	N. Grinberg and A. Kirsch, The linear sampling method in inverse obstacle scattering for impedance boundary conditions, J. Inverse Ill-Posed Probl., 10 (2002), 171-185. doi: 10.1515/jiip.2002.10.2.171.
[16]	A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture in Mathematics and Its Applications 36, (2008).
[17]	A. Kirsch and S. Ritter, A linear sampling method for inverse scattering from an open arc, Inverse Problems, 16 (2000), 89-105. doi: 10.1088/0266-5611/16/1/308.
[18]	R. Kress and P. Serranho, A hybrid method for two-dimensional crack reconstruction, Inverse Problems, 21 (2005), 773-784. doi: 10.1088/0266-5611/21/2/020.
[19]	J. J. Liu, P. A. Krutitskii and M. Sini, Numerical solution of the scattering problem for acoustic waves by a two-sided crack in 2-dimensional space, J. Comput. Math., 29 (2011), 141-166. doi: 10.4208/jams.012111.012811a.
[20]	A. Lechleiter, The factorization method is independent of transmission eigenvalues, Inverse Probl. Imaging, 3 (2009), 123-138. doi: 10.3934/ipi.2009.3.123.
[21]	J. Liu and M. Sini, Reconstruction of cracks of different types from far-field measurements, Math. Meth. Appl. Sci., 33 (2010), 950-973. doi: 10.1002/mma.1203.
[22]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
[23]	J. C. Nédélec, Acoustic and Electromagnetic Equations, Applied Matimatical Sciences. Springer-Verlag, Berlin, 2001.

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