Comptes Rendus
Identification of internal cracks in a three-dimensional solid body via Steklov–Poincaré approaches
Comptes Rendus. Mécanique, Volume 339 (2011) no. 10, pp. 674-681.

This Note deals with the identification of internal planar cracks inside a three-dimensional elastic body via two approaches relying on domain decomposition using elastostatic measurements. These approaches consist in recasting the problem in terms of primal or dual Steklov–Poincaré equations. The primal approach is a straightforward continuation to the elastic Cauchy problem of the work presented in J. Ben Abdallah (2007) [1] which is devoted to the Cauchy problem for the scalar Laplace equation. The numerical performances of these formulations are compared.

Cette Note concerne la résolution numérique du problème dʼidentification de fissures planes dans un solide élastique tridimentionnel, via deux approches issues de la décomposition de domaine, à lʼaide de mesures élastostatiques. Ces approches consistent à reformuler le problème en termes dʼéquations de Steklov–Poincaré primale et duale. Lʼapproche primale a fait lʼobjet dʼune étude numérique dans le cadre de lʼéquation de Laplace J. Ben Abdallah (2007) [1]. Les performances numériques des deux approches sont comparées entre elles.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2011.06.004
Keywords: Computational solid mechanics, Crack identification, Data completion, Inverse problem, Cauchy problem, Steklov–Poincaré equations
Mot clés : Mécanique des solides numérique, Identification de fissures, Complétion de données, Problème inverse, Problème de Cauchy, Équations de Steklov–Poincaré

Mohamed Larbi Kadri 1; Jalel Ben Abdallah 1; Thouraya Nouri Baranger 2

1 Université El Manar, LAMSIN, École nationale dʼingénieurs de Tunis, BP no. 37, 1002 Tunis, Tunisia
2 Université de Lyon, CNRS, université Lyon 1, LaMCoS UMR 5259, INSA-Lyon, 18-20, rue des sciences, 69621 Villeurbanne cedex, France
@article{CRMECA_2011__339_10_674_0,
     author = {Mohamed Larbi Kadri and Jalel Ben Abdallah and Thouraya Nouri Baranger},
     title = {Identification of internal cracks in a three-dimensional solid body via {Steklov{\textendash}Poincar\'e} approaches},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {674--681},
     publisher = {Elsevier},
     volume = {339},
     number = {10},
     year = {2011},
     doi = {10.1016/j.crme.2011.06.004},
     language = {en},
}
TY  - JOUR
AU  - Mohamed Larbi Kadri
AU  - Jalel Ben Abdallah
AU  - Thouraya Nouri Baranger
TI  - Identification of internal cracks in a three-dimensional solid body via Steklov–Poincaré approaches
JO  - Comptes Rendus. Mécanique
PY  - 2011
SP  - 674
EP  - 681
VL  - 339
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crme.2011.06.004
LA  - en
ID  - CRMECA_2011__339_10_674_0
ER  - 
%0 Journal Article
%A Mohamed Larbi Kadri
%A Jalel Ben Abdallah
%A Thouraya Nouri Baranger
%T Identification of internal cracks in a three-dimensional solid body via Steklov–Poincaré approaches
%J Comptes Rendus. Mécanique
%D 2011
%P 674-681
%V 339
%N 10
%I Elsevier
%R 10.1016/j.crme.2011.06.004
%G en
%F CRMECA_2011__339_10_674_0
Mohamed Larbi Kadri; Jalel Ben Abdallah; Thouraya Nouri Baranger. Identification of internal cracks in a three-dimensional solid body via Steklov–Poincaré approaches. Comptes Rendus. Mécanique, Volume 339 (2011) no. 10, pp. 674-681. doi : 10.1016/j.crme.2011.06.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2011.06.004/

[1] J. Ben Abdallah A conjugate gradient type method for the Steklov–Poincaré formulation of the Cauchy–Poisson problem, International Journal of Applied Mathematics and Mechanics, Volume 3 (2007), pp. 27-40

[2] J. Hadamard Lectures on Cauchyʼs Problem in Linear Partial Differential Equation, Dover, New York, 1953

[3] A. Quarteroni; A. Valli Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, Oxford, 1999

[4] F. Ben Belgacem; H. El Fekih On Cauchyʼs problem: I. A variational Steklov–Poincaré theory, Inverse Problem, Volume 21 (2005), pp. 1915-1936

[5] M. Azaiez; F. Ben Belgacem; H. El Fekih On Cauchyʼs problem: II. Completion, regularization and approximation, Inverse Problems, Volume 22 (2006), pp. 1307-1336

[6] F. Ben Belgacem Why is the Cauchyʼs problem severely ill-posed?, Inverse Problems, Volume 23 (2008), pp. 823-836

[7] T.N. Baranger; S. Andrieux Constitutive law gap functionals to solve Cauchy problem for a linear elliptic PDE: a review, 2010 | HAL

[8] V.A. Kozlov; V.G. Mazʼya; A.V. Fomin An iterative method for solving the Cauchy problem for elliptic equations, Computational Mathematics and Mathematical Physics, Volume 31 (1991), pp. 64-74

[9] W.C. Yeih; T. Koya; T. Mura An inverse problem in elasticity with partially overspecified boundary conditions. I. Theoretical approach, Transactions of the ASME Journal of Applied Mechanics, Volume 60 (1993), pp. 595-600

[10] S. Andrieux; T.N. Baranger An energy error-based method for the resolution of the Cauchy problem in 3D linear elasticity, Computer Methods in Applied Mechanics and Engineering, Volume 197 (2008), pp. 902-920

[11] T.N. Baranger; S. Andrieux Data completion for linear symmetric operators as a Cauchy problem: an efficient method via energy like error minimization, Vietnam Journal of Mechanics, VAST, Volume 31 (2009), pp. 247-261

[12] T.N. Baranger; S. Andrieux An optimization approach to solve Cauchy problem in linear elasticity, Journal of Structural and Multidisciplinary Optimization, Volume 35 (2008), pp. 141-152

[13] W. Weikl; H. Andrä; E. Schnack An alternating iterative algorithm for the reconstruction of internal cracks in a three-dimensional solid body, Inverse Problems, Volume 17 (2001), pp. 1957-1975

[14] C. Bellis; M. Bonnet Crack identification by 3D time-domain elastic or acoustic topological sensitivity, C. R. Mecanique, Volume 337 (2009), pp. 124-130

[15] M. Hanke Conjugate Gradient Type Methods for Ill-Posed Problems, Pitman Research Notes in Mathematics, vol. 327, CRC Press, United States, 1995

[16] P.C. Hansen Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia, 1998

[17] D. Calvetti; B. Lewis; L. Reichel GMRES, L-curves, and discrete ill-posed problems, BIT, Volume 42 (2002), pp. 44-65

Cited by Sources:

Comments - Policy