April  2018, 12(2): 349-371. doi: 10.3934/ipi.2018016

The factorization method for cracks in elastic scattering

1. 

South-Central University For Nationalities, Wuhan 430074, China

2. 

School of Mathematics and Statistics, Hubei Engineering University, Xiaogan 43200, China

3. 

Central China Normal University, Wuhan 430079, China

* Corresponding author: Guozheng Yan

Received  February 2017 Revised  November 2017 Published  February 2018

Fund Project: This research is supported by National Natural Science Foundation of People's Republic of China, No.11571132 and No.11601138.

This paper is concerned with the scattering problems of a crack with Dirichlet or mixed impedance boundary conditions in two dimensional isotropic and linearized elasticity. The well posedness of the direct scattering problems for both situations are studied by the boundary integral equation method. The inverse scattering problems we are dealing with are the shape reconstruction of the crack from the knowledge of far field patterns due to the incident plane compressional and shear waves. We aim at extending the well known factorization method to crack determination in inverse elastic scattering, although it has been proved valid in acoustic and electromagnetic scattering, electrical impedance tomography and so on. The numerical examples are presented to illustrate the feasibility of this method.

Citation: Jun Guo, Qinghua Wu, Guozheng Yan. The factorization method for cracks in elastic scattering. Inverse Problems and Imaging, 2018, 12 (2) : 349-371. doi: 10.3934/ipi.2018016
References:
[1]

C. J. S. 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.

[2]

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.

[3]

A. Ben AbdaH. Ben Ameur and M. Jaoua, Identification of 2D cracks by elastic boundary measurements, Inverse problems, 15 (1999), 67-77.  doi: 10.1088/0266-5611/15/1/011.

[4]

Y. Boukari and H. Haddar, The factorization method applied to cracks with impedance boundary conditions, Inverse Probl. Imaging, 7 (2013), 1123-1138.  doi: 10.3934/ipi.2013.7.1123.

[5]

L. Bourgeois and E. Lunéville, On the use of the linear sampling method to identify cracks in elastic waveguides, Inverse Problems, 29 (2013), 0295017, 19pp.

[6]

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.

[7]

F. Cakoni and D. Colton, Qualitative Method in Inverse Scattering Theory, Berlin: Springer, 2006.

[8]

R. ChapkoR. Kress and L. Mönch, On the numerical solution of a hypersingular integral equation for elastic scattering from a planar crack, IMA J. Numer. Anal., 20 (2000), 601-619.  doi: 10.1093/imanum/20.4.601.

[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, 3rd ed, Springer Verlag: Berlin, 2013.

[11]

J. GuoJ. Hu and G. Yan, Application of the factorization method to retrieve a crack from near field data, J. Inverse Ill-posed Probl., 24 (2016), 527-541.  doi: 10.1515/jiip-2014-0073.

[12]

T. Ha-Duong, On the boundary integral equations for the crack opening displacement of flat cracks, Integral Eq. Oper. Theory, 15 (1992), 427-453.  doi: 10.1007/BF01200328.

[13]

G. Hu, A. Kirsch and M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Problems, 29 (2013), 015009, 21pp.

[14]

M. Ikehata and H. Itou, Reconstruction of a linear crack in an isotropic elastic body from a single set of measured data, Inverse Problems, 23 (2007), 589-607.  doi: 10.1088/0266-5611/23/2/008.

[15]

D. S. Jones, A uniqueness theorem in elastodynamics, Q. J. Mech. Appl. Math., 37 (1984), 121-142.  doi: 10.1093/qjmam/37.1.121.

[16]

K. Kiriaki and V. Sevroglou, Integral equations methods in obstacle elastic scattering, Bull. Greek Math. Soc., 45 (2001), 57-69. 

[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]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, 2008.

[19]

A. Kirsch, The factorization method for a class of inverse elliptic problems, Math. Nachr., 278 (2005), 258-277.  doi: 10.1002/mana.200310239.

[20]

A. Kirsch, Properties of far field operators in acoustic scattering, Math. Meth. Appl. Sci., 11 (1989), 773-787.  doi: 10.1002/mma.1670110604.

[21]

R. Kress, Inverse elastic scattering from a crack, Inverse Problems, 12 (1996), 667-684.  doi: 10.1088/0266-5611/12/5/010.

[22]

R. Kress, On the numerical solution of a hypersingular integral equation in scattering thoery, J. Comp. Appl. Math., 61 (1995), 345-360.  doi: 10.1016/0377-0427(94)00073-7.

[23]

R. Kress, Linear Integral Equations, Springer: Berlin, 1989.

[24]

V. D. Kupradze, Potential Methods in the Theory of Elasticity, Israel Program for Scientific Translations, Jerusalem, 1965.

[25]

M. Mclean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.

[26]

P. M. Morse and H. Feshbach, Methods of Theoretical Physics vols I, II, New York: McGraw-Hill, 1953.

[27]

G. NakamuraG. Uhlmann and J. N. Wang, Reconstruction of cracks in an inhomogeneous anisotropic elastic medium, J. Math. Pures Appl., 82 (2003), 1251-1276.  doi: 10.1016/S0021-7824(03)00072-2.

[28]

E. P. Stephan, Boundary Integral Equations for crack Problems in $\mathbb{R}^3$, Integral Eq. Oper. Theory, 10 (1987), 236-257.  doi: 10.1007/BF01199079.

[29]

K. Tanuma, Stroh formalism and Rayleigh waves, J. Elasticity, 89 (2007), ⅵ+159 pp.

show all references

References:
[1]

C. J. S. 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.

[2]

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.

[3]

A. Ben AbdaH. Ben Ameur and M. Jaoua, Identification of 2D cracks by elastic boundary measurements, Inverse problems, 15 (1999), 67-77.  doi: 10.1088/0266-5611/15/1/011.

[4]

Y. Boukari and H. Haddar, The factorization method applied to cracks with impedance boundary conditions, Inverse Probl. Imaging, 7 (2013), 1123-1138.  doi: 10.3934/ipi.2013.7.1123.

[5]

L. Bourgeois and E. Lunéville, On the use of the linear sampling method to identify cracks in elastic waveguides, Inverse Problems, 29 (2013), 0295017, 19pp.

[6]

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.

[7]

F. Cakoni and D. Colton, Qualitative Method in Inverse Scattering Theory, Berlin: Springer, 2006.

[8]

R. ChapkoR. Kress and L. Mönch, On the numerical solution of a hypersingular integral equation for elastic scattering from a planar crack, IMA J. Numer. Anal., 20 (2000), 601-619.  doi: 10.1093/imanum/20.4.601.

[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, 3rd ed, Springer Verlag: Berlin, 2013.

[11]

J. GuoJ. Hu and G. Yan, Application of the factorization method to retrieve a crack from near field data, J. Inverse Ill-posed Probl., 24 (2016), 527-541.  doi: 10.1515/jiip-2014-0073.

[12]

T. Ha-Duong, On the boundary integral equations for the crack opening displacement of flat cracks, Integral Eq. Oper. Theory, 15 (1992), 427-453.  doi: 10.1007/BF01200328.

[13]

G. Hu, A. Kirsch and M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Problems, 29 (2013), 015009, 21pp.

[14]

M. Ikehata and H. Itou, Reconstruction of a linear crack in an isotropic elastic body from a single set of measured data, Inverse Problems, 23 (2007), 589-607.  doi: 10.1088/0266-5611/23/2/008.

[15]

D. S. Jones, A uniqueness theorem in elastodynamics, Q. J. Mech. Appl. Math., 37 (1984), 121-142.  doi: 10.1093/qjmam/37.1.121.

[16]

K. Kiriaki and V. Sevroglou, Integral equations methods in obstacle elastic scattering, Bull. Greek Math. Soc., 45 (2001), 57-69. 

[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]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, 2008.

[19]

A. Kirsch, The factorization method for a class of inverse elliptic problems, Math. Nachr., 278 (2005), 258-277.  doi: 10.1002/mana.200310239.

[20]

A. Kirsch, Properties of far field operators in acoustic scattering, Math. Meth. Appl. Sci., 11 (1989), 773-787.  doi: 10.1002/mma.1670110604.

[21]

R. Kress, Inverse elastic scattering from a crack, Inverse Problems, 12 (1996), 667-684.  doi: 10.1088/0266-5611/12/5/010.

[22]

R. Kress, On the numerical solution of a hypersingular integral equation in scattering thoery, J. Comp. Appl. Math., 61 (1995), 345-360.  doi: 10.1016/0377-0427(94)00073-7.

[23]

R. Kress, Linear Integral Equations, Springer: Berlin, 1989.

[24]

V. D. Kupradze, Potential Methods in the Theory of Elasticity, Israel Program for Scientific Translations, Jerusalem, 1965.

[25]

M. Mclean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.

[26]

P. M. Morse and H. Feshbach, Methods of Theoretical Physics vols I, II, New York: McGraw-Hill, 1953.

[27]

G. NakamuraG. Uhlmann and J. N. Wang, Reconstruction of cracks in an inhomogeneous anisotropic elastic medium, J. Math. Pures Appl., 82 (2003), 1251-1276.  doi: 10.1016/S0021-7824(03)00072-2.

[28]

E. P. Stephan, Boundary Integral Equations for crack Problems in $\mathbb{R}^3$, Integral Eq. Oper. Theory, 10 (1987), 236-257.  doi: 10.1007/BF01199079.

[29]

K. Tanuma, Stroh formalism and Rayleigh waves, J. Elasticity, 89 (2007), ⅵ+159 pp.

Figure 2.  Inversion example for DIP: reconstruction of the semi-circle (52) for $\mu = 2$, $\lambda = 1, \mathbf{p} = [1/2, \sqrt{3}/2]^\top$, noise level = $5\%$ with different $\omega$.
Figure 4.  Inversion example for DIP: reconstruction of the line (53) for $\omega = 4$, $\mu = 1$, $\lambda = 2$, $\mathbf{p} = [-1/2, \sqrt{3}/2]^\top$ with different noise levels.
Figure 6.  Inversion example for DIP: reconstruction of the curve (54) for $\omega = 4$, $\mu = 1$, $\lambda = 2$, noise level = $1\%$ with different polarization directions $\mathbf{p}$.
Figure 3.  Inversion example for MIP: reconstruction of the semi-circle (52) for $\mu=1$, $\lambda=2$, $\eta=2$, $\mathbf{p}=[0,1]^\top$, noise level=$1\%$ with different $\omega$.
Figure 5.  Inversion example for MIP: reconstruction of the line (53) for $\omega = 5$, $\mu = 1$, $\lambda = 3$, $\eta = 1$, $\mathbf{p} = [\sqrt{3}/2, 1/2]^\top$ with different noise levels.
Figure 7.  Inversion example for MIP: reconstruction of the curve (54) for $\omega = 5$, $\mu = 1$, $\lambda = 2$, $\eta = 1$, noise level = $1\%$ with different polarization directions $\mathbf{p}$.
Figure 1.  The exact objects: the shape of (52) (left), the shape of (53) (middle) and the shape of (54) (right)
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