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Reconstruction in the inverse crack problem by variational methods

Published online by Cambridge University Press:  01 December 2008

LUCA RONDI*
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, via Valerio 12/1, I-34127 Trieste, Italy email: [email protected]

Abstract

We deal with a variational approach to the inverse crack problem, that is the detection and reconstruction of cracks, and other defects, inside a conducting body by performing boundary measurements of current and voltage type. We formulate such an inverse problem in a free-discontinuity problems framework and propose a novel method for the numerical reconstruction of the cracks by the available boundary data. The proposed method is amenable to numerical computations and it is justified by a convergence analysis, as the error on the measurements goes to zero. We further notice that we use the Γ-convergence approximation of the Mumford–Shah functional due to Ambrosio and Tortorelli as the required regularization term.

Type
Papers
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Alessandrini, G. & DiBenedetto, E. (1997) Determining 2-dimensional cracks in 3-dimensional bodies: Uniqueness and stability. Indiana Univ. Math. J. 46, 182.Google Scholar
[2]Ambrosio, L., Fusco, N. & Pallara, D. (2000) Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford.CrossRefGoogle Scholar
[3]Ambrosio, L. & Tortorelli, V.M. (1990) Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math. 43, 9991036.CrossRefGoogle Scholar
[4]Ambrosio, L. & Tortorelli, V. M. (1992) On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B 6, 105123.Google Scholar
[5]Bellettini, G. & Coscia, A. (1994) Discrete approximation of a free discontinuity problem. Numer. Funct. Anal. Optim. 15, 201224.CrossRefGoogle Scholar
[6]Bourdin, B. (1999) Image segmentation with a finite element method. M2AN Math. Model. Numer. Anal. 33, 229244.CrossRefGoogle Scholar
[7]Bourdin, B. (2007) Numerical implementation of the variational formulation for quasi-static brittle fracture. Inter. Free Bound. 9, 411430.CrossRefGoogle Scholar
[8]Braides, A. (1998) Approximation of Free-Discontinuity Problems, Springer-Verlag, Berlin, Heidelberg, New York.CrossRefGoogle Scholar
[9]Bryan, K. & Vogelius, M. S. (2004) A review of selected works on crack identification. In: Croke, C. B., Lasiecka, I., Uhlmann, G. & Vogelius, M. S. (editors), Geometric Methods in Inverse Problems and PDE Control, Springer-Verlag, Berlin, Heidelberg, New York, pp. 2546.CrossRefGoogle Scholar
[10]Dal Maso, G. (1993) An Introduction to Γ-convergence, Birkhäuser, Boston, Basel, Berlin.CrossRefGoogle Scholar
[11]Deny, J. & Lions, J. L. (1953–54) Les espaces du type de Beppo Levi. Ann. Inst. Fourier (Grenoble) 5, 305370.CrossRefGoogle Scholar
[12]Eller, M. (1996) Identification of cracks in three-dimensional bodies by many boundary measurements. Inv. Prob. 12, 395408.CrossRefGoogle Scholar
[13]Engl, H. W., Hanke, M. & Neubauer, A. (1996) Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, Boston, London.CrossRefGoogle Scholar
[14]Evans, L. C. & Gariepy, R. F. (1992) Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, Ann Arbor, London.Google Scholar
[15]March, R. (1992) Visual reconstruction with discontinuities using variational methods. Image Vis. Comput. 10, 3038.CrossRefGoogle Scholar
[16]Maz'ja, V. G. (1985) Sobolev Spaces, Springer-Verlag, Berlin, Heidelberg, New York.CrossRefGoogle Scholar
[17]Mumford, D. & Shah, J. (1989) Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42, 577685.CrossRefGoogle Scholar
[18]Rondi, L. (2006) Unique continuation from Cauchy data in unknown non-smooth domains. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5, 189218.Google Scholar
[19]Rondi, L. (2007) A variational approach to the reconstruction of cracks by boundary measurements. J. Math. Pure Appl. 87, 324342.CrossRefGoogle Scholar