Carpio Rodríguez, Ana María and Rapún , M.L.
(2008)
*Domain reconstruction using photothermal techniques.*
Journal of Computational Physics, 227
(17).
pp. 8083-8106.
ISSN 0021-9991

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Official URL: http://www.sciencedirect.com/science/journal/00219991

## Abstract

A numerical method to detect objects buried in a medium by surface thermal measurements is presented. We propose a new approach combining the use of topological derivatives and Laplace transforms. The original optimization problem with time-dependent constraints is replaced by an equivalent problem with stationary constraints by means of Laplace transforms. The first step in the reconstruction scheme consists in discretizing the inversion formula to produce an approximate optimization problem with a finite set of constraints. Then, an explicit expression for the topological derivative of the approximate shape functional is given. This formula is evaluated at low cost using explicit expressions of the forward and adjoint fields involved. We apply this technique to a simple shape reconstruction problem set in a half space. Good approximations of the number, location and size of the obstacles are obtained. The description of their shapes can be improved by more expensive hybrid methods combining time averaging with topological derivative based iterative schemes.

Item Type: | Article |
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Uncontrolled Keywords: | Shape reconstruction; Nondestructive testing; Topological derivatives; Laplace transform; Boundary integral methods; Scattering; Heat equation; Thermal waves |

Subjects: | Sciences > Physics > Mathematical physics Sciences > Mathematics > Numerical analysis |

ID Code: | 14930 |

References: | [1] D.P. Almond, P.M. Patel, Photothermal Science and Techniques, Chapman and Hall, London, 1996. [2] H.T. Banks, F. Kojima, Boundary shape identification problems in two-dimensional domains related to thermal testing of materials, Quart. Appl. Math. 47 (1989) 273–293. [3] H.T. Banks, F. Kojima, W.P. Winfree, Boundary estimation problems arising in thermal tomography, Inverse Probl. 6 (1990) 897–921 [4] G. Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of causal generalized Radon transform, J. Math. Phys. 26 (1985) 99–108. [5] M. Burger, B. Hackl, W. Ring, Incorporating topological derivatives into level set methods, J. Comput. Phys. 194(2004) 344–362. [6] F. Cakoni, D. Colton, P. Monk, The determination of the surface conductivity of a partially coated dielectric, SIAM J. Appl. Math. 65 (2005) 767–789. [7] A. Carpio, M.-L. Rapu´ n, Solving inhomogeneous inverse problems by topological derivative methods, Inverse Probl., On line at stacks.iop.org/IP/24/045014, doi:10.1088/0266-5611/24/4/045014. [8] A. Carpio, M.-L. Rapu´ n, Topological derivatives for shape reconstruction, in: Inverse Problems and Imaging, Lecture Notes in Mathematics, Springer, 2008, pp. 85–134. [9] M. Cheney, D. Isaacson, J.C. Newell, Electrical impedance tomography, SIAM Rev. 41 (1999) 85–101. [10] M. Costabel, E. Stephan, A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl. 106 (1985)367–413. [11] O. Dorn, D. Lesselier, Level set methods for inverse scattering, Inverse Probl. 22 (2006) R67–R131. [12] L. Elden, F. Berntsson, T. Reginska, Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput. 21 (2000) 2187–2205. [13] G.R. Feijoo, A new method in inverse scattering based on the topological derivative, Inverse Probl. 20 (2004)1819–1840. [14] F. Garrido, A. Salazar, Thermal wave scattering from spheres, J. Appl. Phys. 95 (2004) 140–149. [15] B.B. Guzina, M. Bonnet, Topological derivative for the inverse scattering of elastic waves, Quart. J. Mech. Appl. Math. 57 (2004)161–179. [16] B.B. Guzina, M. Bonnet, Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics, Inverse Probl. 22 (2006)1761–1785. [17] D.M. Heath, C.S. Welch, W.P. Winfree, Quantitative thermal diffusivity measurements of composites, Review of Progress in Quantitative Non-Destructive Evaluation, vol. 5B, Plenum, New York, 1986, pp. 1125–1132. [18] T. Hohage, M.-L. Rapu´n, F.-J. Sayas, Detecting corrosion using thermal measurements, Inverse Probl. 23(2007) 53–72. [19] T. Hohage, F.-J. Sayas, Numerical approximation of a heat diffusion problem by boundary element methods using the Laplace transform, Numer. Math. 102 (2005) 67–92. [20] V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York, 1998. [21] A. Laliena, F.-J. Sayas, LDBEM in diffusion problems, in: Proceedings of XIX CEDYA/IX CMA, 2005 (electronic version). [22] M. Lo´pez-Ferna´ndez, C. Palencia, On the numerical inversion of the Laplace transform of certain holomorphic mappings, Appl. Numer. Math. 51 (2004) 289–303. [23] A. Mandelis, Diffusion-wave fields. Mathematical methods and Green functions, Springer, New York, 2001. [24] A. Mandelis, Diffusion waves and their uses, Phys. Today 53 (2000) 29–34. [25] L. Nicolaides, A. Mandelis, Image-enhanced thermal-wave slice diffraction tomography with numerically simulated reconstructions, Inverse Probl. 13 (1997) 1393–1412 [26] A. Oca´riz, A. Sa´nchez-Lavega, A. Salazar, Photothermal study of subsurface cylindrical structures: II. Experimental results, J. Appl. Phys. 81 (1997) 7561–7566. [27] M.A. O’Leary, D.A. Boas, B. Chance, A.G. Yodh, Refraction of diffusive photon density waves, Phys. Rev. Lett. 69 (1992) 2658–2662. [28] M.-L. Rapu´n, F.-J. Sayas, Boundary integral approximation of a heat diffusion problem in time-harmonic regime, Numer. Algor. 41(2006) 127–160. [29] M.-L. Rapu´n, F.-J. Sayas, Boundary element simulation of thermal waves, Arch. Comput. Methods Eng. 14 (2007) 3–46. [30] A. Salazar, A. Sa´nchez-Lavega, R. Celorrio, Scattering of cylindrical thermal waves in fiber composites: in-plane thermal diffusivity, J.Appl. Phys. 93(2003) 4536–4542. [31] A. Talbot, The accurate numerical inversion of Laplace transforms, J. Inst. Math. Appl. 23 (1979) 97–120. [32] J.M. Terro´n, A. Salazar, A. Sa´nchez-Lavega, General solution for the thermal wave scattering in fiber composites, J. Appl. Phys. 91 (2002) 1087–1098. |

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Last Modified: | 06 Feb 2014 10:12 |

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