Recovery of Cracks from Incomplete Boundary Data

We are interested in this article in recovering line segment (or planar) cracks from incomplete boundary measurements. Most recovery algorithms, and especially fast ones based on the reciprocity gap, however need a full set of boundary data, which compels us to extending the available data prior to achieving recovery. Both the extension and recovery problems being severely ill posed, their stacking might drive the whole process to a blowing up, unless the instabilities are properly handled. Numerical results actually prove the so-built composite algorithm to hold good accuracy and robustness features.

[1]  F. Delvare,et al.  Two Iterative Boundary Element Methods ForInverse Cauchy Problems , 2000 .

[2]  Mohamed Jaoua,et al.  Solution of the Cauchy problem using iterated Tikhonov regularization , 2001 .

[3]  J. Lions,et al.  Problèmes aux limites non homogènes et applications , 1968 .

[4]  Giovanni Alessandrini,et al.  Determining 2-dimensional cracks in 3-dimensional bodies: Uniqueness and stability , 1997 .

[5]  A. Tikhonov,et al.  Numerical Methods for the Solution of Ill-Posed Problems , 1995 .

[6]  An iterative boundary element method for Cauchy inverse problems , 2002 .

[7]  Identification of cracks in three-dimensional bodies by many boundary measurements , 1996 .

[8]  J. Dieudonné,et al.  Problèmes aux limites , 2003 .

[9]  Fadil Santosa,et al.  A computational algorithm to determine cracks from electrostatic boundary measurements , 1991 .

[10]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[11]  M. Dauge.  Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions , 1988 .

[12]  M. Dauge.  Elliptic boundary value problems on corner domains , 1988 .

[13]  Amel Ben Abda,et al.  Reciprocity principle and crack identification , 1999 .

[14]  J. Leblond,et al.  Parameter identification for Laplace equation and approximation in Hardy classes , 2003 .

[15]  On the inverse emergent plane crack problem , 1998 .

[16]  Elena Beretta,et al.  Determining linear cracks by boundary measurements: Lipschitz stability , 1996 .

[17]  Kurt Bryan,et al.  A uniqueness result concerning the identification of a collection of cracks from finitely many electrostatic boundary measurements , 1992 .

[18]  Hend Ben Ameur,et al.  Identification of 2D cracks by elastic boundary measurements , 1999 .

[19]  M. Vogelius,et al.  A computational algorithm to determine crack locations from electrostatic boundary measurements. The case of multiple cracks , 1994 .

[20]  Amel Ben Abda,et al.  A semi-explicit algorithm for the reconstruction of 3D planar cracks , 1997 .

[21]  Giovanni Alessandrini,et al.  Unique Determination of Multiple Cracks by Two Measurements , 1996 .

[22]  Avner Friedman,et al.  Determining Cracks by Boundary Measurements , 1989 .

[23]  A. B. Abda,et al.  Identification of planar cracks by complete overdetermined data: inversion formulae , 1996 .