Detecting the localization of elastic inclusions and Lamé coefficients

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August 2014, 8(3): 779-794. doi: 10.3934/ipi.2014.8.779

Detecting the localization of elastic inclusions and Lamé coefficients

Nuno F. M. Martins ^1,

1.	CEMAT-IST and Departamento de Matemática, Faculdade de Ciências e Tecnologia (NULisbon), Universidade Nova de Lisboa, Quinta da Torre, Caparica, Portugal

Received July 2013 Revised July 2014 Published August 2014

Abstract
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In this paper we develop and analyse a direct method, based on the so called reciprocity gap functional, for retrieving the location of an elastic inclusion. The method requires a pair of displacement (imposed) and traction (measured) data, on an accessible part of the boundary. We provide a criterion for the choice of displacement that, in one hand, provides more accurate results and on the other, does not require the Lamé parameters of background medium. We use this property to develop a two boundary measurements direct method for retrieving both Lamé parameters. Several numerical examples are presented in order to illustrate the accuracy and stability of the proposed methods.

Keywords: Lamé parameters, linear elasticity, Inverse obstacle problems, reciprocity gap functional., center of mass.

Mathematics Subject Classification: 65N21, 65N35, 35J2.

Citation: Nuno F. M. Martins. Detecting the localization of elastic inclusions and Lamé coefficients. Inverse Problems and Imaging, 2014, 8 (3) : 779-794. doi: 10.3934/ipi.2014.8.779

References:

[1]	C. J. S. Alves and H. Ammari, Boundary integral formulae for the reconstruction of imperfections of small diameter in an elastic medium, SIAM Journal on Applied Mathematics, 62 (2001), 94-106. doi: 10.1137/S0036139900369266.
[2]	C. J. S. Alves, M. J. Colaço, V. M. A. Leitão, N. F. M. Martins, H. R. B. Orlande and N. C. Roberty, Recovering the source term in a linear diffusion problem by the Method of Fundamental Solutions, Inverse Problems in Science and Engineering, 16 (2008), 1005-1021. doi: 10.1080/17415970802083243.
[3]	C. J. S. Alves and N. F. M. Martins, Reconstruction of inclusions or cavities in potential problems using the MFS, in The Method of Fundamental Solutions-A Meshless Method (eds. C. S. Chen, A. Karageorghis and Y. S. Smyrlis), Dynamic Publishers Inc., 2008, 51-71.
[4]	C. J. S. Alves and N. F. M. Martins, The direct method of fundamental solutions and the inverse Kirsch-Kress method for the reconstruction of elastic inclusions or cavities, Journal of Integral Equations and Applications, 21 (2009), 153-178. doi: 10.1216/JIE-2009-21-2-153.
[5]	C. J. S. Alves, N. F. M. Martins and N. C. Roberty, Full identification of acoustic sources with multiple frequencies and boundary measurements, Inverse Problems and Imaging, 3 (2009), 275-294. doi: 10.3934/ipi.2009.3.275.
[6]	C. J. S. Alves and N. C. Roberty, On the identification of star shape sources from boundary measurements using a reciprocity functional, Inverse Problems in Science and Engineering, 17 (2009), 187-202. doi: 10.1080/17415970802082799.
[7]	S. Andrieux and A. B. Abda, The reciprocity gap: A general concept for flaws identification, Mechanics Research Communication, 20 (1993), 415-420. doi: 10.1016/0093-6413(93)90032-J.
[8]	G. Chen and J. Zhou, Boundary Element Methods, Computational Mathematics and Applications, Academic Press, London, 1992.
[9]	M. J. Colaço and C. J. S. Alves, A fast non-intrusive method for estimating spatial thermal contact conductante by means of the reciprocity functional approach and the method of fundamental solutions, International Journal of Heat and Mass Transfer, 60 (2013), {653-663}.
[10]	S. Cox and M. Gockenbach, Recovering planar Lamé moduli from a single traction experiment, Mathematics and Mechanics of Solids, 2 (1997), 297-306. doi: 10.1177/108128659700200304.
[11]	A. El-Badia and T. Ha Duong, Some remarks on the problem of source identification from boundary measurements, Inverse Problems, 14 (1998), 883-891. doi: 10.1088/0266-5611/14/4/008.
[12]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 1998.
[13]	M. S. Gockenbach, B. Jadamba and A. A. Khan, Equation error approach for elliptic inverse problems with an application to the identification of Lamé parameters, Inverse Problems in Science and Engineering, 16 (2008), 349-367. doi: 10.1080/17415970701602580.
[14]	B. Jadamba, A. A. Khan and F. Raciti, On the inverse problem of identifying Lamé coefficients in linear elasticity, Computers and Mathematics with Applications, 56 (2008), 431-443. doi: 10.1016/j.camwa.2007.12.016.
[15]	Y. Liu, L. Z. Sun and G. Wang, Tomography based 3D anisotropic elastography using boundary measurements, IEEE Transactions on Medical Imaging, 24 (2005), 1323-1333.
[16]	L. Marin, L. L. Elliot, D. B. Ingham and D. Lesnic, Identification of material properties and cavities in two-dimensional linear elasticity, Computational Mechanics, 31 (2003), 293-300.
[17]	N. F. M. Martins and D. Soares, Localization of Immersed Obstacles from a Pair of Boundary Data, Proceedings of Jornadas do Mar, Escola Naval, Portugal (2012), (IN CDROM).
[18]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
[19]	G. Nakamura and G. Uhlmann, Identification of Lamé parameters by boundary measurements, American Journal of Mathematics, 115 (1993), 1161-1187. doi: 10.2307/2375069.
[20]	N. Tlatli-Hariga, R. Bouhlila and A. B. Abda, Recovering data in groundwater: Boundary conditions and well's positions and fluxes, Computational Geosciences, 15 (2011), 637-645. doi: 10.1007/s10596-011-9231-9.

show all references

References:

[1]	C. J. S. Alves and H. Ammari, Boundary integral formulae for the reconstruction of imperfections of small diameter in an elastic medium, SIAM Journal on Applied Mathematics, 62 (2001), 94-106. doi: 10.1137/S0036139900369266.
[2]	C. J. S. Alves, M. J. Colaço, V. M. A. Leitão, N. F. M. Martins, H. R. B. Orlande and N. C. Roberty, Recovering the source term in a linear diffusion problem by the Method of Fundamental Solutions, Inverse Problems in Science and Engineering, 16 (2008), 1005-1021. doi: 10.1080/17415970802083243.
[3]	C. J. S. Alves and N. F. M. Martins, Reconstruction of inclusions or cavities in potential problems using the MFS, in The Method of Fundamental Solutions-A Meshless Method (eds. C. S. Chen, A. Karageorghis and Y. S. Smyrlis), Dynamic Publishers Inc., 2008, 51-71.
[4]	C. J. S. Alves and N. F. M. Martins, The direct method of fundamental solutions and the inverse Kirsch-Kress method for the reconstruction of elastic inclusions or cavities, Journal of Integral Equations and Applications, 21 (2009), 153-178. doi: 10.1216/JIE-2009-21-2-153.
[5]	C. J. S. Alves, N. F. M. Martins and N. C. Roberty, Full identification of acoustic sources with multiple frequencies and boundary measurements, Inverse Problems and Imaging, 3 (2009), 275-294. doi: 10.3934/ipi.2009.3.275.
[6]	C. J. S. Alves and N. C. Roberty, On the identification of star shape sources from boundary measurements using a reciprocity functional, Inverse Problems in Science and Engineering, 17 (2009), 187-202. doi: 10.1080/17415970802082799.
[7]	S. Andrieux and A. B. Abda, The reciprocity gap: A general concept for flaws identification, Mechanics Research Communication, 20 (1993), 415-420. doi: 10.1016/0093-6413(93)90032-J.
[8]	G. Chen and J. Zhou, Boundary Element Methods, Computational Mathematics and Applications, Academic Press, London, 1992.
[9]	M. J. Colaço and C. J. S. Alves, A fast non-intrusive method for estimating spatial thermal contact conductante by means of the reciprocity functional approach and the method of fundamental solutions, International Journal of Heat and Mass Transfer, 60 (2013), {653-663}.
[10]	S. Cox and M. Gockenbach, Recovering planar Lamé moduli from a single traction experiment, Mathematics and Mechanics of Solids, 2 (1997), 297-306. doi: 10.1177/108128659700200304.
[11]	A. El-Badia and T. Ha Duong, Some remarks on the problem of source identification from boundary measurements, Inverse Problems, 14 (1998), 883-891. doi: 10.1088/0266-5611/14/4/008.
[12]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 1998.
[13]	M. S. Gockenbach, B. Jadamba and A. A. Khan, Equation error approach for elliptic inverse problems with an application to the identification of Lamé parameters, Inverse Problems in Science and Engineering, 16 (2008), 349-367. doi: 10.1080/17415970701602580.
[14]	B. Jadamba, A. A. Khan and F. Raciti, On the inverse problem of identifying Lamé coefficients in linear elasticity, Computers and Mathematics with Applications, 56 (2008), 431-443. doi: 10.1016/j.camwa.2007.12.016.
[15]	Y. Liu, L. Z. Sun and G. Wang, Tomography based 3D anisotropic elastography using boundary measurements, IEEE Transactions on Medical Imaging, 24 (2005), 1323-1333.
[16]	L. Marin, L. L. Elliot, D. B. Ingham and D. Lesnic, Identification of material properties and cavities in two-dimensional linear elasticity, Computational Mechanics, 31 (2003), 293-300.
[17]	N. F. M. Martins and D. Soares, Localization of Immersed Obstacles from a Pair of Boundary Data, Proceedings of Jornadas do Mar, Escola Naval, Portugal (2012), (IN CDROM).
[18]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
[19]	G. Nakamura and G. Uhlmann, Identification of Lamé parameters by boundary measurements, American Journal of Mathematics, 115 (1993), 1161-1187. doi: 10.2307/2375069.
[20]	N. Tlatli-Hariga, R. Bouhlila and A. B. Abda, Recovering data in groundwater: Boundary conditions and well's positions and fluxes, Computational Geosciences, 15 (2011), 637-645. doi: 10.1007/s10596-011-9231-9.

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