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A Review of Selected Works on Crack Identification

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Geometric Methods in Inverse Problems and PDE Control

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 137))

Abstract

We give a short survey of some of the results obtained within the last 10 years or so concerning crack identification using impedance imaging techniques. We touch upon uniqueness results, continuous dependence results, and computational algorithms.

This work was partially supported by the National Science Foundation under grant DMS-00-72556.

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References

  1. Adamjan V.M., Arov D.Z., and Krein M.G., Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem. Math. USSR Sbornik, 15, 1971, pp. 31–73.

    Article  Google Scholar 

  2. Alessandrini G., Examples of instability in inverse boundary-value problems, In verse Probl, 13(4), 1997, pp. 887–897.

    Article  MathSciNet  MATH  Google Scholar 

  3. Alessandrini G., Stable Determination of a crack from boundary measurement, Proc. Roy. Soc. Edinburgh Sect. A, 123, 1993, pp. 497–516.

    Article  MathSciNet  MATH  Google Scholar 

  4. Alessandrini A., Beretta E., and Vessella S., Determining linear cracks by boundary measurements: Lipschitz stability, Siam J. Math. Anal., 27(2), 1996, pp. 361–375.

    Article  MathSciNet  MATH  Google Scholar 

  5. Alessandrini G. and Dibenedetto E., Determining 2-dimensional crack in 3- dimensional bodies: Uniqueness and stability, Indiana U Math. J., 46(1), 1997, pp. 1–82.

    MathSciNet  MATH  Google Scholar 

  6. Alessandrini G. and Magnanini R., The index of isolated critical points and solutions of elliptic equations in the plane, Ann. Scuola. Norm. Sup. Pi. CI. Sci., 19(4), 1992, pp. 567–589.

    MathSciNet  MATH  Google Scholar 

  7. Alessandrini G. and Magnanini R., Elliptic equations in divergence form, geo metric critical points of solutions, and Stekloff eigenfunctions, Siam J. Math. Anal, 25, 1994, pp. 1259–1268.

    Article  MathSciNet  MATH  Google Scholar 

  8. Alessandrini G. and Rondi L., Stable determination of a crack in a planar inhomogeneous conductor, Siam J. Math. Anal., 30(2), 1998, pp. 326–340.

    Article  MathSciNet  Google Scholar 

  9. Alessandrini G. and Diaz Valenzuela A., Unique determination of multiple cracks by two measurements, Siam J. Cont. Opt., 34(3), 1996, pp. 913–921.

    Article  MATH  Google Scholar 

  10. Andersen K., Brooks S., and Hansen M., Bayesian approach to crack detec tion in electrically conducting media, Inverse ProbL, 17, 2001, pp. 121–136.

    Article  MathSciNet  MATH  Google Scholar 

  11. Andrieux S. and Ben Abda A., Identification de fissures planes par une donneé de bord unique; un procédé direct de localisation et d’identification, C.R. Acad. Sci., Paris I, 1992, 315, pp. 1323–1328.

    Google Scholar 

  12. Andrieux S., Ben Abda A., and Jaoua M., On the inverse emergent plane crack problem, Math. Method Appl. Sci., 21(10), 1998, pp. 895–906.

    Article  MathSciNet  MATH  Google Scholar 

  13. Andrieux S. and Ben Abda A., Identification of planar cracks by complete overdetermined data: inversion formulae. Inverse Probl, 12, 1996, pp. 553- 563.

    Article  MATH  Google Scholar 

  14. Andrieux S., Ben Abda A., and Bui H.D., Reciprocity principle and crack iden tification, Inverse Probl., 15(1), 1999, pp. 59–65.

    Article  MATH  Google Scholar 

  15. Aparicio N.D. and Pidcock M.K., On a class of free boundary problems for the Laplace equation in two dimensions, Inverse Probl., 14(1), 1998, pp. 9–18.

    Article  MathSciNet  MATH  Google Scholar 

  16. Aparicio N.D. and Pidcock M.K., The boundary inverse problem for the Laplace equation in two dimensions, Inverse Probl., 12(5), 1996, pp. 565–577.

    Article  MathSciNet  MATH  Google Scholar 

  17. Bannour T., Ben Abda A., and Jaoua M., A semi-explicit algorithm for the reconstruction of 3D planar cracks, Inverse Probl., 13(4), 1997, pp. 899–917.

    Article  MATH  Google Scholar 

  18. Baratchart L., Leblond J., Mandrea F., and Saff E.B., How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian?, Inverse Probl., 15(1), 1999, pp. 79–90.

    Article  MathSciNet  MATH  Google Scholar 

  19. Ben Abda A., Ben Ameur H., and Jaoua M., Identification of 2D cracks by elastic boundary measurements, Inverse Probl., 15(1), 1999, pp. 67–77.

    Article  MATH  Google Scholar 

  20. Ben Abda A., Chaabane S., Dabaghi F. EL, and Jaoua M., On a non-linear ge ometrical inverse problem of Signorini type: Identifiablity and stability, Math. Method Appl. Sci., 21(15), 1998, pp. 1379–1398.

    Article  MATH  Google Scholar 

  21. Brühl M. Explicit characterization of inclusions in electrical impedance tomogra phy, preprint.

    Google Scholar 

  22. Brühl M. and Hanke M., Numerical implementation of two non-iterative meth ods for locating inclusions by impedance tomography, Inverse Probl., to appear.

    Google Scholar 

  23. Brühl M., Hanke M., and Pidcock M., Crack detection using electrostatic mea surements, M2AN Math. Model. Numer. Anal., 35, 2001, pp. 595–605.

    Article  MathSciNet  MATH  Google Scholar 

  24. Brühl M., Hanke M., and Vogelius M.S., A direct impedance tomography algo rithm for locating small inhomogeneities, Numerische Mathematik, to appear.

    Google Scholar 

  25. Bryan K. and Vogelius M., A uniqueness result concerning the identification of a collection of cracks from finitely many electrostatic boundary measurements, Siam J. Math. Anal., 23(4), 1992, pp. 950–958.

    Article  MathSciNet  MATH  Google Scholar 

  26. Bryan K. and Vogelius M., Computational algorithm to determine crack loca tions from electrostatic boundary measurements-the case of multiple cracks. Int. J. Engr. Sci., 32(4), 1994, pp. 579–603.

    Article  MathSciNet  MATH  Google Scholar 

  27. Bryan K. and Vogelius M., Effective behavior of clusters of microscopic cracks inside a homogeneous conductor. Asymptotic Anal., 16(2), 1998, pp. 141–179.

    MathSciNet  MATH  Google Scholar 

  28. Bryan K., Liepa V., and Vogelius M., Reconstruction of multiple cracks from experimental, electrostatic boundary measurements, proceedings of the conference on Inverse Problems and Optimal Design in Industry, July 8–10, 1993, Philadelphia PA.

    Google Scholar 

  29. Bui H.D., Constantinescu A., and Maigre H., Inverse scattering of a planar crack in 3D acoustics: closed form solution for a bounded body, Cr Acad. Sci. II B, 327(10), 1999, pp. 971–976.

    MATH  Google Scholar 

  30. Capdeboscq Y., and Vogelius M.S., A general representation formula for bound ary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, M2An Math. Model. Numer. Anal., to appear.

    Google Scholar 

  31. Diaz Valenzuela A., Unicita’ e stabilita per il problema inverse del crack perfet- tamenta isolante, thesis, Universita’ di Trieste, Trieste, Italy, 1993.

    Google Scholar 

  32. Elcrat A.R. and Hu C.L., Determination of surface and interior cracks from electrostatic boundary measurements using Schwarz-Christoffel transformations, Int J. Engr. Sci., 34(10), 1996, pp. 1165–1181.

    Article  MATH  Google Scholar 

  33. Elcrat A.R., Isakov V., and Neculoiu O., On finding a surface crack from boundary measurements, Inverse Probl., 11(2), 1995, pp. 343–351.

    Article  MathSciNet  MATH  Google Scholar 

  34. Eller M., Identification of cracks in three-dimensional bodies by many boundary measurements. Inverse Probl., 12(4), 1996, pp. 395–408.

    Article  MathSciNet  MATH  Google Scholar 

  35. Friedman A. and Vogelius M., Determining cracks by boundary measurements, Ind. U Math. J., 38, 1989, pp. 527–556.

    Article  MathSciNet  MATH  Google Scholar 

  36. Kim H. and Seo J., Unique determination of a collection of a finite number of cracks from two boundary measurements, Siam J. Math. Anal. 27, 1996, pp. 1336–1340.

    Article  MathSciNet  MATH  Google Scholar 

  37. Kirsch A., Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Probl, 14, 1998, pp. 1489–1512.

    Article  MathSciNet  MATH  Google Scholar 

  38. Kolehmainen V., Arridge S.R., Et AL., Recovery of region boundaries of piece- wise constant coefficients of an elliptic Pde from boundary data, Inverse Probl, 15(5), 1999, pp. 1375–1391.

    Article  MATH  Google Scholar 

  39. Liepa V., Santosa F., and Vogelius M., Crack determination from bound ary measurements-reconstruction using experimental data, J. Nondestructive Evaluation, 12, 1993, pp. 163–174.

    Article  Google Scholar 

  40. Mellings S.C. and Aliabadi M.H., Three-dimensional flaw identification using inverse analysis, Int. J. Engr. Sci., 34(4), 1996, pp. 453–469.

    Article  MathSciNet  MATH  Google Scholar 

  41. Miller K., Stabilized numerical analytic prolongation with poles, Siam J. Appl. Math., 18, 1970, pp. 346–363.

    Article  MathSciNet  MATH  Google Scholar 

  42. Nishimura N. and Kobayashi S., Determination of cracks having arbitrary shapes with the boundary integral-equation method, Eng. Anal. Bound. Elem., 15(2), 1995, pp. 189–195.

    Article  Google Scholar 

  43. Rondi L., Optimal stability estimates for the determination of defects by electro static measurements, Inverse Probl. 15(5), 1999 pp. 1193–1212.

    Article  MathSciNet  MATH  Google Scholar 

  44. Santos A F. and Vogelius M., A computational algorithm to determine cracks from electrostatic boundary measurements, Int. J. Engr. Sci., 29, 1991, pp. 917–937.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Rose-Hulman Institute of Technology, 47803, Terre Haute, IN, USA

    Kurt Bryan

  2. Rutgers University, 08903, New Brunswick, NJ, USA

    Michael S. Vogelius

Authors
  1. Kurt Bryan
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  2. Michael S. Vogelius
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Pennsylvania, 19104-6395, Philadelphia, PA, USA

    Christopher B. Croke

  2. Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA

    Michael S. Vogelius

  3. Department of Mathematics, University of Washington, 98195-4350, Seattle, WA, USA

    Gunther Uhlmann

  4. Department of Mathematics, University of Virginia, 22903, Charlottesville, VA, USA

    Irena Lasiecka

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© 2004 Springer-Verlag New York, Inc.

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Bryan, K., Vogelius, M.S. (2004). A Review of Selected Works on Crack Identification. In: Croke, C.B., Vogelius, M.S., Uhlmann, G., Lasiecka, I. (eds) Geometric Methods in Inverse Problems and PDE Control. The IMA Volumes in Mathematics and its Applications, vol 137. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9375-7_3

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  • DOI: https://doi.org/10.1007/978-1-4684-9375-7_3

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-2341-7

  • Online ISBN: 978-1-4684-9375-7

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