Crack identification for transient heat operator by using domain decomposition method

Abstract

This work deals with cracks identification from over-determined boundary data. The consideration physical phenomena corresponds to the transient heat equation. we give a theoretical result of identifiability for the inverse problem under consideration. Then, we consider a recovering process based on coupling domain decomposition method and minimizing an energy-type error functional. The efficiency of the proposed approach is illustrated by several numerical results.

Keywords

• Inverse problem,crack,identifiability,transient heat equation,ill-posed problem,Cauchy problem,domain decomposition,virtual control

References

1. G. Alessandrini and A. Diaz Valenzuela. Unique determination of multiple cracks by two measurements. SIAM J. Control Optim., 34(3): 913–921, 1994.
2. S. Andrieux and T. N. Baranger. Energy methods for Cauchy problems of evolutions equations. In Journal of Physics: Conference Series, volume 135, page 012007. IOP Publishing, 2008.
3. S. Andrieux, T. N. Baranger, and A. Ben Abda. Solving Cauchy problems by minimizing an energy-like functional. Inverse problems, 22(1):115–133, 2006.
4. S. Andrieux and A. Ben Abda. Identification of planar cracks by complete overdetermined data: inversion formulae. Inverse problems, 12(5):553–563, 1996.
5. S. Andrieux, A. Ben Abda, and T. N. Baranger. Data completion via an energy error functional. Comptes Rendus M´ecanique, 33(2):171–177, 2005.
6. M. Azaiıez, F. Ben Belgacem, and H. El Fekih. On Cauchy’s problem: II. Completion, regularization and approximation. Inverse problems, 22(4): 1307–1336, 2006.
7. L. Baratchart, J. Leblond, F. Mandrea, and E. B. Saff. How can the meromorphic approximation help to solve some 2D inverse problems for the laplacian. Inverse Problems, 15(3): 79–90, 1999.
8. J. V. Beck, B. Blackwell, and Charles R. St. Clair Jr. Inverse heat conduction: Ill-posed problems. James Beck, 1985.

9. A. Ben Abda and H. D. Bui. Planar crack identification for the transient heat equation. Inv. Ill-Posed Problems, 11(11): 27–31, 2003.
10. F. Ben Belgacem. Why is the Cauchy problem severely ill-posed? Inverse Problems, 23(2): 823, 2007.
11. F. Ben Belgacem and H. El Fekih. On Cauchy’s problem: I. A variational Steklov-Poincar´e theory. Inverse Problems, 21(6):1915–1936, 2005.
12. M. Bertero, T. A. Poggio, and V. Torre. Ill-posed problems in early vision. Proceedings of the IEEE, 1988.
13. A. Björck. Numerical Methods for Least Squares Problems. SIAM Journal on Mathematical Analysis, 1996.
14. M. Brühl, M. Hanke, and M. Pidcock. Crack detection using electrostatic measurements. ESAIM: M2AN, 35(3): 595–605, 2001.
15. K. Bryan and M. Vogelius. A uniqueness result concerning the identification of a collection of cracks from nitely many elastostatic boundary measurements. SIAM J. Math. Anal., 23(4): 950–958, 1992.
16. A. Friedman and M. Vogelius. Determining cracks by boundary measurements. Indiana Univ. Math., 38:527–556, 1989.
17. P. Grisvard. Singularities in boundary value problems, volume 22. Springer, 1992.
18. J. Hadamard and Philip M. Morse. Lectures on Cauchy’s problem in linear partial differential equations. Physics Today, 6: 18, 1953.
19. D. N. Hao and D Lesnic. The Cauchy problem for Laplaces equation via the conjugate gradient method. IMA Journal of Applied Mathematics, 65(2): 199–217, 2000.
20. F. Hecht, O. Pironneau, A. Le Hyaric, and K. Ohtsuka. Freefem++. Numerical Mathematics and Scientific Computation. Laboratoire J. L. Lions, Universit´e Pierre et Marie Curie, 3, 2007.
21. V. Isakov. Inverse Problems for Partial Differential Equations. Springer Science, 1998.
22. S. Kowalevski. Zur theorie der partiellen difierentialgleichungen. J. Reine Angew. Math., 80:1–32, 1875.
23. V. A. Kozlov, V. G. Maz’ya, and A. V. Fomin. An iterative method for solving the Cauchy problem for elliptic equations. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 31(1): 64–74, 1991.
24. P. Le Tallec, Y. H. De Roeck, and M. Vidrascu. Domain decomposition methods for large linearly elliptic three-dimensional problems. Journal of Computational and Applied Mathematics, 34(1): 93–117, 1991.
25. K. Miller. Least squares methods for ill-posed problems with a prescribed bound. SIAM Journal on Mathematical Analysis, 1(1):52–74, 1970.
26. M. Moussaoui and B. K. Sadallah. R´egularite des coefficients de propagation de singularit´es pour l´equation de la chaleur dans un ouvert plan polygonal. CR Acad. Sci. Paris Ser. I Math, 293(5):297–300, 1981.
27. A. Quarteroni and A. Valli. Domain decomposition methods for partial differential equations. Oxford University Press, 1999.
28. T. Wei, Y. C. Hon, and L. Ling. Method of fundamental solutions with regularization techniques for cauchy problems of elliptic operators. Engineering Analysis with Boundary Elements, 31(4): 373–385, April 2007.