In this paper we extend recent work on the detection of inclusions using electrostatic measurements to the problem of crack detection in a two-dimensional object. As in the inclusion case our method is based on a factorization of the difference between two Neumann-Dirichlet operators. The factorization possible in the case of cracks is much simpler than that for inclusions and the analysis is greatly simplified. However, the directional information carried by the crack makes the practical implementation of our algorithm more computationally demanding.

Keywords: inverse boundary value problem, nondestructive testing, crack

@article{M2AN_2001__35_3_595_0, author = {Br\"uhl, Martin and Hanke, Martin and Pidcock, Michael}, title = {Crack detection using electrostatic measurements}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {35}, number = {3}, year = {2001}, pages = {595-605}, zbl = {0985.35103}, mrnumber = {1837086}, language = {en}, url = {http://www.numdam.org/item/M2AN_2001__35_3_595_0} }

Brühl, Martin; Hanke, Martin; Pidcock, Michael. Crack detection using electrostatic measurements. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 3, pp. 595-605. http://www.numdam.org/item/M2AN_2001__35_3_595_0/

[1] Unique determination of multiple cracks by two measurements. SIAM J. Control Optim. 34 (1996) 913-921. | Zbl 0864.35115

and ,[2] Explicit characterization of inclusions in electrical impedance tomography. SIAM J. Math. Anal. 32 (2001) 1327-1341. | Zbl 0980.35170

,[3] Numerical implementation of two noniterative methods for locating inclusions by impedance tomography. Inverse Problems 16 (2000) 1029-1042. | Zbl 0955.35076

and ,[4] A computational algorithm to determine crack locations from electrostatic boundary measurements. The case of multiple cracks. Internat. J. Engrg. Sci. 32 (1994) 579-603. | Zbl 0924.73179

and ,[5] Regularization of inverse problems. Kluwer, Dordrecht (1996). | MR 1408680 | Zbl 0859.65054

, and ,[6] Determining cracks by boundary measurements. Indiana Univ. Math. J. 38 (1989) 527-556. | Zbl 0697.35165

and ,[7] Unique determination of a collection of a finite number of cracks from two boundary measurements. SIAM J. Math. Anal. 27 (1996) 1336-1340. | Zbl 0859.35138

and ,[8] Characterization of the shape of a scattering obstacle using the spectral data of the far field operator. Inverse Problems 14 (1998) 1489-1512. | Zbl 0919.35147

,[9] A linear sampling method for inverse scattering from an open arc. Inverse Problems 16 (2000) 89-105. | Zbl 0968.35129

and ,[10] Linear integral equations. 2nd edn., Springer, New York (1999).

,[11] Partial differential equations of elliptic type. 2nd edn., Springer, Berlin (1970). | MR 284700 | Zbl 0198.14101

,[12] On the numerical solution of the direct scattering problem for an open sound-hard arc. J. Comput. Appl. Math. 71 (1996) 343-356. | Zbl 0854.65106

,[13] A boundary integral equation method for an inverse problem related to crack detection. Internat. J. Numer. Methods Engrg. 32 (1991) 1371-1387. | Zbl 0760.73072

and ,[14] A computational algorithm to determine cracks from electrostatic boundary measurements. Internat. J. Engrg. Sci. 29 (1991) 917-937. | Zbl 0825.73761

and ,