Cavity detection by the electric method: The 3-dimensional case

ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

Volumn 84, Issue 2, 2004, Pages 75-85

  Ang, Dang Dinh ^a   Mennicken, Reinhard ^b   Thanh, Dinh Ngoc ^a   Trong, Dang Duc ^a  

^a VIETNAM NATIONAL UNIVERSITY   (Viet Nam)

^b UNIVERSITY OF REGENSBURG   (Germany)

Author keywords

Cauchy data; Cavity identification; Elliptic equation

Indexed keywords

EID: 1142288400     PISSN: 00442267     EISSN: None     Source Type: Journal    
DOI: 10.1002/zamm.200310119     Document Type: Article

References (29)

1. G. Alessandrini, Stable determination of a crack from boundary measurements, Proc. R. Soc. Edinb. A 123, 497-516 (1993).
2. G. Alessandrini, Examples of instability in inverse boundary-value problems, Inverse Probl. 13, 887-897 (1997).
3. G. Alessandrini, E. Beretta, E. Rosset, and S. Vessella, Optimal stability for inverse elliptic boundary value problems with unknown boundaries, Ann. Scuola Norm. Pisa, Cl. Sci. XXIX(4), 755-806 (2000).
4. G. Alessandrini and E. DiBenedetto, Determining 2-dimensional cracks in 3-dimensional body: Uniqueness & stability, Indiana Univ. Math. J. 46, 1-83 (1987).
5. G. Alessandrini and L. Rondi, Optimal stability for the inverse problem of multiple cavities, J. Differ. Equ. 176, 356-386 (2001).
6. G. Alessandrini and A. D. Valenzuela, Unique determination of multiple cracks by two measurements, SIAM J. Control Optim. 34(3), 913-921 (1996).
7. S. Andrieux, A. BenAbda, and H. D. Bui, Sur l'identification de fissures planes via le concept d'écart à la réciprocité en elasticité, C.R. Acad. Sci. I, Math. 324, 1431-1438 (1997).
8. D. D. Ang and D. D. Trong, Crack detection by the electric method: Uniqueness and approximation. Int. J. Fract. 93, 63-86 (1998).
9. D. D. Ang, D. D. Trong, and M. Yamamoto, Unique continuation and identification of boundary of an elastic body, J. Inverse ILL-Posed Probl. 3(6), 417-428 (1996).
10. D. D. Ang, D. D. Trong, and M. Yamamoto, identification of cavities inside two-dimensional heterogeneous Isotropic elastic body, J. Elast. 56, 199-212 (1999).
11. D. D. Ang and L. K. Vy, Domain identification for harmonic functions, Acta Applicandae Math. 38, 1995, 217-238.
12. H. D. Bui, Inverse Problems in the Mechanics of Materials: an Introduction (CRC Press Inc., Boca Raton, 1994).
13. A. L. Bukhgeim, J. Cheng, and M. Yamamoto, Uniqueness and stability for an inverse problem of determining parts of boundary, in: Inverse Problems in Engineering Mechanics, edited by M. Tanaka and G. S. Dulikravich (Elsevier, Amsterdam, 1998), pp. 327-336.
14. A. L. Bukhgeim, J. Cheng, and M. Yamamoto, On a sharp estimate in a nondestructive testing determination of unknown boundaries, preprint (1998).
15. A. L. Bukhgeim, J. Cheng, and M. Yamamoto, Stability for an inverse boundary problem of determining a part of boundary, Inverse Probl. 14, 1021-1032 (1999).
16. A. L. Bukhgeim, J. Cheng, and M. Yamamoto, Conditional stability in an inverse problem of determining non-smooth boundary, J. Math. Anal. Appl. 242, 57-74 (2000).
17. E. Beretta and S. Vessella, Stable determination of boundaries from Cauchy data, SIAM J. Math. Anal. 30, 220-232 (1998).
18. A. Friedman and M. Vogelius, Determining cracks by boundary measurements, Indiana Univ. Math. J. 38, 527-556 (1989).
19. M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Probl. 15, 1234-1241 (1999).
20. M. McIver, An inverse problem in electro-magnetic crack detection, IMA J. Appl. Math., 47, 127-141 (1991).
21. H. Kim and J. K. Seo, Uniqueness of determination of a collate of finite cracks from two boundary measurements. SIAM J. Math. Anal. 27, 1336-1340 (1996).
22. S. Kubo, Requirements for uniqueness of crack identification from electric potential distribution, in: Inverse Problems in Engineering Sciences, edited by M. Yamaguti et al. (Springer-Verlag, 1990), pp. 52-58.
23. R. N. Pederson, On the unique continuation theorem for certain second and fourth order elliptic equations, Commun. Pure Appl. Math. 11, 67-80 (1958).
24. L. Rondi, Uniqueness and Stability for the Determination of Boundary Defects by Electrostatic Measurements, Ph. D. thesis, S.I.S.S.A-I.S.A.S., Trieste (1999).
25. F. Santosa and M. Vogelius, A complete algorithm to determine cracks from electrical body measurements, Int. J. Eng. Sci. 29, 913-937 (1991).
26. I. N. Sneddon and M. Lowengrub, Crack Problems in the Classical Theory of Elasticity (John Wiley & Sons Inc., New York, 1969).
27. D. D. Trong, Domain identification for a nonlinear elliptic equation, Z. Analysis Anw. 17(4), 1021-1024 (1998).
28. D. D. Trong and D. D. Ang, Domain identification for semilinear elliptic equations in the plane: the zero flux case, Z. Analysis Anw. 19 (2000)1, 109-120.
29. H. Whitney, Geometric Integration Theory (Princeton Univ. Press, 1957).