Identification of cracks with non linear impedances

Mohamed Jaoua; Serge Nicaise[1]; Luc Paquet

  • [1] Université de Valenciennes et du Hainaut Cambrésis, MACS, Le Mont Houy, 59313 Valenciennes Cedex 9, France. http://www.univ-valenciennes.fr/macs/Serge.Nicaise

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2003)

  • Volume: 37, Issue: 2, page 241-257
  • ISSN: 0764-583X

Abstract

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We consider the inverse problem of determining a crack submitted to a non linear impedance law. Identifiability and local Lipschitz stability results are proved for both the crack and the impedance.

How to cite

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Jaoua, Mohamed, Nicaise, Serge, and Paquet, Luc. "Identification of cracks with non linear impedances." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.2 (2003): 241-257. <http://eudml.org/doc/246102>.

@article{Jaoua2003,
abstract = {We consider the inverse problem of determining a crack submitted to a non linear impedance law. Identifiability and local Lipschitz stability results are proved for both the crack and the impedance.},
affiliation = {Université de Valenciennes et du Hainaut Cambrésis, MACS, Le Mont Houy, 59313 Valenciennes Cedex 9, France. http://www.univ-valenciennes.fr/macs/Serge.Nicaise},
author = {Jaoua, Mohamed, Nicaise, Serge, Paquet, Luc},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {inverse problems; cracks; nonlinear impedance law; local Lipschitz stability; identifiability},
language = {eng},
number = {2},
pages = {241-257},
publisher = {EDP-Sciences},
title = {Identification of cracks with non linear impedances},
url = {http://eudml.org/doc/246102},
volume = {37},
year = {2003},
}

TY - JOUR
AU - Jaoua, Mohamed
AU - Nicaise, Serge
AU - Paquet, Luc
TI - Identification of cracks with non linear impedances
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 2
SP - 241
EP - 257
AB - We consider the inverse problem of determining a crack submitted to a non linear impedance law. Identifiability and local Lipschitz stability results are proved for both the crack and the impedance.
LA - eng
KW - inverse problems; cracks; nonlinear impedance law; local Lipschitz stability; identifiability
UR - http://eudml.org/doc/246102
ER -

References

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  1. [1] G. Alessandrini, Stability for the crack determination problem, in Inverse problems in Mathematical Physics, L. Päıvaärinta and E. Somersalo Eds., Springer-Verlag, Berlin (1993) 1–8. Zbl0802.35152
  2. [2] G. Alessandrini, E. Beretta and S. Vessella, Determining linear cracks by boundary measurements: Lipschitz stability. SIAM J. Math. Anal. 27 (1996) 361–375. Zbl0855.35124
  3. [3] G. Alessandrini and A. Diaz Valenzuela, Unique determination of multiple cracks by two measurements. SIAM J. Control Optim. 34 (1996) 913–921. Zbl0864.35115
  4. [4] G. Alessandrini and A. DiBenedetto, Determining 2-dimensional cracks in 3-dimensional bodies: uniqueness and stability. Indiana Univ. Math. J. 46 (1997) 1–82. Zbl0879.35040
  5. [5] S. Andrieux and A. Ben Abda, Identification of planar cracks by overdetermined boundary data: inversion formulae. Inverse Problems 12 (1996) 553–563. Zbl0858.35131
  6. [6] S. Andrieux, A. Ben Abda and M. Jaoua, On the inverse emerging plane crack problem. Math. Methods Appl. Sci. 21 (1998) 895–907. Zbl0916.35129
  7. [7] A. Ben Abda, H. Ben Ameur and M. Jaoua, A semi-explicit algorithm for the reconstruction of 3D planar cracks. Inverse Problems 15 (1999) 67–78. Zbl0882.35128
  8. [8] R. Bellout and A. Friedman, Identification problems in potential theory. Arch. Rational Mech. Anal. 101 (1988) 143–160. Zbl0659.35102
  9. [9] M. Bonnet, Boundary Integral Equation Methods for Solids and Fluids. Wiley, New York (1995). Zbl0920.73001
  10. [10] K. Bryan and M. Vogelius, A uniqueness result concerning the identification of a collection of cracks from finitely many electrostatic boundary measurements. SIAM J. Math. Anal. 23 (1992) 950–958. Zbl0754.73077
  11. [11] M. Dauge, Elliptic boundary value problems in corner domains. Smoothness and asymptotics of solutions. Springer Verlag, Berlin, Lecture Notes in Math. 1341 (1988). Zbl0668.35001MR961439
  12. [12] C. Dellacherie and P.-A. Meyer, Probabilité et potentiel. Hermann (1975). Zbl0323.60039MR488194
  13. [13] P. Destuynder and M. Jaoua, Sur une interprétation mathématique de l’intégrale de Rice en théorie de la rupture fragile. Math. Methods Appl. Sci. 3 (1981) 70–87. Zbl0493.73087
  14. [14] R. Felfel, Étude de l’identifiabilité et de la stabilité d’une fissure présentant une résistivité de contact. DEA de Mathématiques Appliquées, ENIT, Tunis (1997). 
  15. [15] A. Friedman and M. Vogelius, Determining cracks by boundary measurements. Indiana Univ. Math. J. 38 (1989) 527–556. Zbl0697.35165
  16. [16] P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, Boston (1985). Zbl0695.35060MR775683
  17. [17] V.G. Maz’ya and B.A. Plamenevsky, On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points. Amer. Math. Soc. Transl. Ser. 2 123 (1984) 57–88. Zbl0554.35036
  18. [18] F. Murat and J. Simon, Quelques résultats sur le contrôle par un domaine géométrique. Preprint, Université de Paris VI (1974). MR437935
  19. [19] E.P. Stephan, Boundary integral equations for mixed boundary value problems, screen and transmission problems in 3 . Habilitationsschrift, TH Darmstadt, Germany (1984). 
  20. [20] E.P. Stephan, Boundary integral equations for screen problems in 3 . Integral Equations Operator Theory 10 (1987) 236–257. Zbl0653.35016
  21. [21] V.S. Vladimirov, Equations of Mathematical Physics. Marcel Dekker, New York (1971). Zbl0207.09101MR268497

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