December  2019, 13(6): 1309-1325. doi: 10.3934/ipi.2019057

Unique determination of a transversely isotropic perturbation in a linearized inverse boundary value problem for elasticity

1. 

Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA

2. 

Institute for Advanced Study, The Hong Kong University of Science and Technology, Hong Kong, China

* Corresponding author

Received  January 2019 Revised  May 2019 Published  October 2019

Fund Project: The research of YY was partly supported by NSF Grant DMS-1715178, AMS-Simons travel grant, and start-up fund from Michigan State University.

We consider a linearized inverse boundary value problem for the elasticity system. From the linearized Dirichlet-to-Neumann map at zero frequency, we show that a transversely isotropic perturbation of a homogeneous isotropic elastic tensor can be uniquely determined. From the linearized Dirichlet-to-Neumann map at two distinct positive frequencies, we show that a transversely isotropic perturbation of a homogeneous isotropic density can be identified at the same time.

Citation: Yang Yang, Jian Zhai. Unique determination of a transversely isotropic perturbation in a linearized inverse boundary value problem for elasticity. Inverse Problems and Imaging, 2019, 13 (6) : 1309-1325. doi: 10.3934/ipi.2019057
References:
[1]

T. Alkhalifah, Velocity analysis using nonhyperbolic moveout in transversely isotropic media, SEG Technical Program Expanded Abstracts 1996, (1996), 1499–1502. doi: 10.1190/1.1826401.

[2]

T. AlkhalifahI. TsvankinK. Larner and J. Toldi, Velocity analysis and imaging in transversely isotropic media: Methodology and a case study, The Leading Edge, 15 (1996), 371-378.  doi: 10.1190/1.1437345.

[3]

J. A. BarcelóM. Folch-GabayetS. Pérez-EstevaA. Ruiz and M. C. Vilela, Uniqueness for inverse elastic medium problems, SIAM Journal on Mathematical Analysis, 50 (2018), 3939-3962.  doi: 10.1137/17M1138315.

[4]

O. A. Bauchau and J. I. Craig, Structural Analysis with Applications to Aerospace Structures, Springer-Verlag, New York, 2009.

[5]

E. Beretta, M. V. de Hoop, E. Francini, S. Vessella and J. Zhai, Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves, Inverse Problems, 33 (2017), 035013, 27 pp. doi: 10.1088/1361-6420/aa5bef.

[6]

E. Beretta, E. Francini, A. Morassi, E. Rosset and S. Vessella, Lipschitz continuous dependence of piecewise constant Lamé coefficients from boundary data: The case of non-flat interfaces, Inverse Problems, 30 (2014), 125005, 18 pp. doi: 10.1088/0266-5611/30/12/125005.

[7]

E. BerettaE. Francini and S. Vessella, Uniqueness and Lipschitz stability for the identification of Lamé parameters from boundary measurements, Inverse Probl. Imag., 8 (2014), 611-644.  doi: 10.3934/ipi.2014.8.611.

[8]

S. Bhattacharyya, Local uniqueness of the density from partial boundary data for isotropic elastodynamics, Inverse Problems, 34 (2018), 125001, 10 pp. doi: 10.1088/1361-6420/aade76.

[9]

A. P. Calderón, On an inverse boundary value problem, Computational & Applied Mathematics, 25 (2006), 133-138.  doi: 10.1590/S0101-82052006000200002.

[10]

C. I. CârsteaN. Honda and G. Nakamura, Uniqueness in the inverse boundary value problem for piecewise homogeneous anisotropic elasticity, SIAM J. Math. Anal., 50 (2018), 3291-3302.  doi: 10.1137/17M1125662.

[11]

M. V. de Hoop, G. Nakamura and J. Zhai, Unique Recovery of Piecewise Analytic Density and Stiffness Tensor from the Elastic-Wave Dirichlet-to-Neumann Map, 2018, arXiv: 1803.01091.

[12]

M. V. de Hoop, G. Uhlmann and A. Vasy, Recovery of Material Parameters in Transversely Isotropic Media, 2019, arXiv: 1902.09394.

[13]

G. Eskin and J. Ralston, On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems, 18 (2002), 907-921.  doi: 10.1088/0266-5611/18/3/324.

[14]

A. GreenleafM. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem, Mathematical Research Letters, 10 (2003), 685-693.  doi: 10.4310/MRL.2003.v10.n5.a11.

[15]

M. Ikehata, Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection, SIAM Journal on Applied Mathematics, 50 (1990), 1635-1644.  doi: 10.1137/0150097.

[16]

M. Ikehata, The linearization of the Dirichlet to Neumann map in anisotropic plate theory, Inverse Problems, 11 (1995), 165-181.  doi: 10.1088/0266-5611/11/1/009.

[17]

M. Ikehata, The linearization of the Dirichlet-to-Neumann map in the anisotropic Kirchhoff-Love plate theory, SIAM Journal on Applied Mathematics, 56 (1996), 1329-1352.  doi: 10.1137/S0036139994270437.

[18]

M. Ikehata, A relationship between two Dirichlet to Neumann maps in anisotrpoic elastic plate theory, Journal of Inverse and Ill-Posed Problems, 4 (1996), 233-243.  doi: 10.1515/jiip.1996.4.3.233.

[19]

O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, On uniqueness of Lamé coefficients from partial Cauchy data in three dimensions, Inverse Problems, 28 (2012), 125002, 5 pp. doi: 10.1088/0266-5611/28/12/125002.

[20]

O. Y. Imanuvilov and M. Yamamoto, Global uniqueness in inverse boundary value problems for the Navier-Stokes equations and Lamé system in two dimensions, Inverse Problems, 31 (2015), 035004, 46 pp. doi: 10.1088/0266-5611/31/3/035004.

[21]

D. Kumar, M. K. Sen and R. J. Ferguson, Traveltime calculation and prestack depth migration in tilted transversely isotropic media, SEG Technical Program Expanded Abstracts 2003, (2003). doi: 10.1190/1.1818099.

[22]

Y.-H. Lin and G. Nakamura, Boundary determination of the Lamé moduli for the isotropic elasticity system, Inverse Problems, 33 (2017), 125004, 23 pp. doi: 10.1088/1361-6420/aa942d.

[23]

J.-P. Montagner and H.-C. Nataf, A simple method for inverting the azimuthal anisotropy of surface waves, Journal of Geophysical Research: Solid Earth, 91 (1986), 511-520.  doi: 10.1029/JB091iB01p00511.

[24]

A. I. Nachman, Reconstructions from boundary measurements, Annals of Mathematics, 128 (1988), 531-576.  doi: 10.2307/1971435.

[25]

G. Nakamura and G. Uhlmann, Inverse problems at the boundary for an elastic medium, SIAM J. Math. Anal., 26 (1995), 263-279.  doi: 10.1137/S0036141093247494.

[26]

G. Nakamura and G. Uhlmann, Layer stripping for a transversely isotropic elastic medium, SIAM J. Appl. Math., 59 (1999), 1879-1891.  doi: 10.1137/S0036139998337164.

[27]

G. Nakamura and G. Uhlmann, Erratum: Global uniqueness for an inverse boundary value problem arising in elasticity, Invent. Math., 152 (2003), 205-207.  doi: 10.1007/s00222-002-0276-1.

[28]

L. V. Rachele, An inverse problem in elastodynamics: Uniqueness of the wave speeds in the interior, Journal of Differential Equations, 162 (2000), 300-325.  doi: 10.1006/jdeq.1999.3657.

[29]

L. V. Rachele, Uniqueness of the density in an inverse problem for isotropic elastodynamics, Transactions of the American Mathematical Society, 355 (2003), 4781-4806.  doi: 10.1090/S0002-9947-03-03268-9.

[30]

J. N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, New York, second edition, 2002.

[31]

P. Stefanov, G. Uhlmann and A. Vasy, Local recovery of the compressional and shear speeds from the hyperbolic DN map, Inverse Problems, 34 (2018), 014003, 13 pp. doi: 10.1088/1361-6420/aa9833.

[32]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.

show all references

References:
[1]

T. Alkhalifah, Velocity analysis using nonhyperbolic moveout in transversely isotropic media, SEG Technical Program Expanded Abstracts 1996, (1996), 1499–1502. doi: 10.1190/1.1826401.

[2]

T. AlkhalifahI. TsvankinK. Larner and J. Toldi, Velocity analysis and imaging in transversely isotropic media: Methodology and a case study, The Leading Edge, 15 (1996), 371-378.  doi: 10.1190/1.1437345.

[3]

J. A. BarcelóM. Folch-GabayetS. Pérez-EstevaA. Ruiz and M. C. Vilela, Uniqueness for inverse elastic medium problems, SIAM Journal on Mathematical Analysis, 50 (2018), 3939-3962.  doi: 10.1137/17M1138315.

[4]

O. A. Bauchau and J. I. Craig, Structural Analysis with Applications to Aerospace Structures, Springer-Verlag, New York, 2009.

[5]

E. Beretta, M. V. de Hoop, E. Francini, S. Vessella and J. Zhai, Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves, Inverse Problems, 33 (2017), 035013, 27 pp. doi: 10.1088/1361-6420/aa5bef.

[6]

E. Beretta, E. Francini, A. Morassi, E. Rosset and S. Vessella, Lipschitz continuous dependence of piecewise constant Lamé coefficients from boundary data: The case of non-flat interfaces, Inverse Problems, 30 (2014), 125005, 18 pp. doi: 10.1088/0266-5611/30/12/125005.

[7]

E. BerettaE. Francini and S. Vessella, Uniqueness and Lipschitz stability for the identification of Lamé parameters from boundary measurements, Inverse Probl. Imag., 8 (2014), 611-644.  doi: 10.3934/ipi.2014.8.611.

[8]

S. Bhattacharyya, Local uniqueness of the density from partial boundary data for isotropic elastodynamics, Inverse Problems, 34 (2018), 125001, 10 pp. doi: 10.1088/1361-6420/aade76.

[9]

A. P. Calderón, On an inverse boundary value problem, Computational & Applied Mathematics, 25 (2006), 133-138.  doi: 10.1590/S0101-82052006000200002.

[10]

C. I. CârsteaN. Honda and G. Nakamura, Uniqueness in the inverse boundary value problem for piecewise homogeneous anisotropic elasticity, SIAM J. Math. Anal., 50 (2018), 3291-3302.  doi: 10.1137/17M1125662.

[11]

M. V. de Hoop, G. Nakamura and J. Zhai, Unique Recovery of Piecewise Analytic Density and Stiffness Tensor from the Elastic-Wave Dirichlet-to-Neumann Map, 2018, arXiv: 1803.01091.

[12]

M. V. de Hoop, G. Uhlmann and A. Vasy, Recovery of Material Parameters in Transversely Isotropic Media, 2019, arXiv: 1902.09394.

[13]

G. Eskin and J. Ralston, On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems, 18 (2002), 907-921.  doi: 10.1088/0266-5611/18/3/324.

[14]

A. GreenleafM. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem, Mathematical Research Letters, 10 (2003), 685-693.  doi: 10.4310/MRL.2003.v10.n5.a11.

[15]

M. Ikehata, Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection, SIAM Journal on Applied Mathematics, 50 (1990), 1635-1644.  doi: 10.1137/0150097.

[16]

M. Ikehata, The linearization of the Dirichlet to Neumann map in anisotropic plate theory, Inverse Problems, 11 (1995), 165-181.  doi: 10.1088/0266-5611/11/1/009.

[17]

M. Ikehata, The linearization of the Dirichlet-to-Neumann map in the anisotropic Kirchhoff-Love plate theory, SIAM Journal on Applied Mathematics, 56 (1996), 1329-1352.  doi: 10.1137/S0036139994270437.

[18]

M. Ikehata, A relationship between two Dirichlet to Neumann maps in anisotrpoic elastic plate theory, Journal of Inverse and Ill-Posed Problems, 4 (1996), 233-243.  doi: 10.1515/jiip.1996.4.3.233.

[19]

O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, On uniqueness of Lamé coefficients from partial Cauchy data in three dimensions, Inverse Problems, 28 (2012), 125002, 5 pp. doi: 10.1088/0266-5611/28/12/125002.

[20]

O. Y. Imanuvilov and M. Yamamoto, Global uniqueness in inverse boundary value problems for the Navier-Stokes equations and Lamé system in two dimensions, Inverse Problems, 31 (2015), 035004, 46 pp. doi: 10.1088/0266-5611/31/3/035004.

[21]

D. Kumar, M. K. Sen and R. J. Ferguson, Traveltime calculation and prestack depth migration in tilted transversely isotropic media, SEG Technical Program Expanded Abstracts 2003, (2003). doi: 10.1190/1.1818099.

[22]

Y.-H. Lin and G. Nakamura, Boundary determination of the Lamé moduli for the isotropic elasticity system, Inverse Problems, 33 (2017), 125004, 23 pp. doi: 10.1088/1361-6420/aa942d.

[23]

J.-P. Montagner and H.-C. Nataf, A simple method for inverting the azimuthal anisotropy of surface waves, Journal of Geophysical Research: Solid Earth, 91 (1986), 511-520.  doi: 10.1029/JB091iB01p00511.

[24]

A. I. Nachman, Reconstructions from boundary measurements, Annals of Mathematics, 128 (1988), 531-576.  doi: 10.2307/1971435.

[25]

G. Nakamura and G. Uhlmann, Inverse problems at the boundary for an elastic medium, SIAM J. Math. Anal., 26 (1995), 263-279.  doi: 10.1137/S0036141093247494.

[26]

G. Nakamura and G. Uhlmann, Layer stripping for a transversely isotropic elastic medium, SIAM J. Appl. Math., 59 (1999), 1879-1891.  doi: 10.1137/S0036139998337164.

[27]

G. Nakamura and G. Uhlmann, Erratum: Global uniqueness for an inverse boundary value problem arising in elasticity, Invent. Math., 152 (2003), 205-207.  doi: 10.1007/s00222-002-0276-1.

[28]

L. V. Rachele, An inverse problem in elastodynamics: Uniqueness of the wave speeds in the interior, Journal of Differential Equations, 162 (2000), 300-325.  doi: 10.1006/jdeq.1999.3657.

[29]

L. V. Rachele, Uniqueness of the density in an inverse problem for isotropic elastodynamics, Transactions of the American Mathematical Society, 355 (2003), 4781-4806.  doi: 10.1090/S0002-9947-03-03268-9.

[30]

J. N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, New York, second edition, 2002.

[31]

P. Stefanov, G. Uhlmann and A. Vasy, Local recovery of the compressional and shear speeds from the hyperbolic DN map, Inverse Problems, 34 (2018), 014003, 13 pp. doi: 10.1088/1361-6420/aa9833.

[32]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.

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