American Institute of Mathematical Sciences

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 & 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. Google Scholar [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. Google Scholar [3] A. Ben Abda, H. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] F. Cakoni and D. Colton, Qualitative Method in Inverse Scattering Theory, Berlin: Springer, 2006. Google Scholar [8] R. Chapko, R. 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. Google Scholar [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. Google Scholar [10] D. Colton and R. Kress. Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed, Springer Verlag: Berlin, 2013. Google Scholar [11] J. Guo, J. 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. Google Scholar [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. Google Scholar [13] G. Hu, A. Kirsch and M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Problems, 29 (2013), 015009, 21pp. Google Scholar [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. Google Scholar [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. Google Scholar [16] K. Kiriaki and V. Sevroglou, Integral equations methods in obstacle elastic scattering, Bull. Greek Math. Soc., 45 (2001), 57-69. Google Scholar [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. Google Scholar [18] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, 2008. Google Scholar [19] A. Kirsch, The factorization method for a class of inverse elliptic problems, Math. Nachr., 278 (2005), 258-277. doi: 10.1002/mana.200310239. Google Scholar [20] A. Kirsch, Properties of far field operators in acoustic scattering, Math. Meth. Appl. Sci., 11 (1989), 773-787. doi: 10.1002/mma.1670110604. Google Scholar [21] R. Kress, Inverse elastic scattering from a crack, Inverse Problems, 12 (1996), 667-684. doi: 10.1088/0266-5611/12/5/010. Google Scholar [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. Google Scholar [23] R. Kress, Linear Integral Equations, Springer: Berlin, 1989. Google Scholar [24] V. D. Kupradze, Potential Methods in the Theory of Elasticity, Israel Program for Scientific Translations, Jerusalem, 1965. Google Scholar [25] M. Mclean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. Google Scholar [26] P. M. Morse and H. Feshbach, Methods of Theoretical Physics vols I, II, New York: McGraw-Hill, 1953. Google Scholar [27] G. Nakamura, G. 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. Google Scholar [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. Google Scholar [29] K. Tanuma, Stroh formalism and Rayleigh waves, J. Elasticity, 89 (2007), ⅵ+159 pp. Google Scholar

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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. Google Scholar [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. Google Scholar [3] A. Ben Abda, H. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] F. Cakoni and D. Colton, Qualitative Method in Inverse Scattering Theory, Berlin: Springer, 2006. Google Scholar [8] R. Chapko, R. 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. Google Scholar [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. Google Scholar [10] D. Colton and R. Kress. Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed, Springer Verlag: Berlin, 2013. Google Scholar [11] J. Guo, J. 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. Google Scholar [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. Google Scholar [13] G. Hu, A. Kirsch and M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Problems, 29 (2013), 015009, 21pp. Google Scholar [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. Google Scholar [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. Google Scholar [16] K. Kiriaki and V. Sevroglou, Integral equations methods in obstacle elastic scattering, Bull. Greek Math. Soc., 45 (2001), 57-69. Google Scholar [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. Google Scholar [18] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, 2008. Google Scholar [19] A. Kirsch, The factorization method for a class of inverse elliptic problems, Math. Nachr., 278 (2005), 258-277. doi: 10.1002/mana.200310239. Google Scholar [20] A. Kirsch, Properties of far field operators in acoustic scattering, Math. Meth. Appl. Sci., 11 (1989), 773-787. doi: 10.1002/mma.1670110604. Google Scholar [21] R. Kress, Inverse elastic scattering from a crack, Inverse Problems, 12 (1996), 667-684. doi: 10.1088/0266-5611/12/5/010. Google Scholar [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. Google Scholar [23] R. Kress, Linear Integral Equations, Springer: Berlin, 1989. Google Scholar [24] V. D. Kupradze, Potential Methods in the Theory of Elasticity, Israel Program for Scientific Translations, Jerusalem, 1965. Google Scholar [25] M. Mclean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. Google Scholar [26] P. M. Morse and H. Feshbach, Methods of Theoretical Physics vols I, II, New York: McGraw-Hill, 1953. Google Scholar [27] G. Nakamura, G. 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. Google Scholar [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. Google Scholar [29] K. Tanuma, Stroh formalism and Rayleigh waves, J. Elasticity, 89 (2007), ⅵ+159 pp. Google Scholar
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 $\omega$.
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.
Inversion example for DIP: reconstruction of the curve (54) for $\omega = 4$, $\mu = 1$, $\lambda = 2$, noise level = $1\%$ with different polarization directions $\mathbf{p}$.
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 $\omega$.
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.
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 $\mathbf{p}$.
The exact objects: the shape of (52) (left), the shape of (53) (middle) and the shape of (54) (right)
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