Reconstruction of thickness variation of a dielectric coating through the generalized impedance boundary conditions
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 4, p. 1011-1027

We deal with an inverse scattering problem whose aim is to determine the thickness variation of a dielectric thin coating located on a conducting structure of unknown shape. The inverse scattering problem is solved through the application of the Generalized Impedance Boundary Conditions (GIBCs) which contain the thickness, curvature as well as material properties of the coating and they have been obtained in the previous work [B. Aslanyürek, H. Haddar and H.Şahintürk, Wave Motion 48 (2011) 681-700] up to the third order with respect to the thickness. After proving uniqueness results for the inverse problem, the required total field as well as its higher order derivatives appearing in the GIBCs are obtained by the analytical continuation of the measured data to the coating surface through the single layer potential representation. The resulting system of non-linear differential equations for the unknown coating thickness is solved iteratively via the Newton-Raphson method after expanding the thickness function in a series of exponentials. Through the simulations it has been shown that the approach is effective under the validity conditions of the GIBCs.

DOI : https://doi.org/10.1051/m2an/2013131
Classification:  78A25,  78A46,  78A45,  65N21
Keywords: generalized impedance boundary conditions, thin coatings, inverse scattering problems, single layer potential, Newton−Raphson method
@article{M2AN_2014__48_4_1011_0,
author = {Aslany\"urek, Birol and Sahint\"urk, H\"ulya},
title = {Reconstruction of thickness variation of a dielectric coating through the generalized impedance boundary conditions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {48},
number = {4},
year = {2014},
pages = {1011-1027},
doi = {10.1051/m2an/2013131},
zbl = {1297.78004},
mrnumber = {3264344},
language = {en},
url = {http://www.numdam.org/item/M2AN_2014__48_4_1011_0}
}

Aslanyürek, Birol; Sahintürk, Hülya. Reconstruction of thickness variation of a dielectric coating through the generalized impedance boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 4, pp. 1011-1027. doi : 10.1051/m2an/2013131. http://www.numdam.org/item/M2AN_2014__48_4_1011_0/

[1] H. Ammari and J.C. Nedelec, Generalized impedance boundary conditions for the maxwell equations as singular perturbations problems. Commun. Partial Differ. Equ. 24 (1999) 24−38. | MR 1680917 | Zbl 0939.35180

[2] B. Aslanyürek, H. Haddar and H. Şahintürk, Generalized impedance boundary conditions for thin dielectric coatings with variable thickness. Wave Motion 48 (2011) 681−700. | MR 2835082 | Zbl 1239.78004

[3] L. Bourgeois, N. Chaulet and H. Haddar, Stable reconstruction of generalized impedance boundary conditions. Inverse Probl. 27 (2011) 19−38. | MR 2824761 | Zbl 1222.35208

[4] L. Bourgeois, N. Chaulet and H. Haddar, On Simultaneous Identification of the Shape and Generalized Impedance Boundary Condition in Obstacle Scattering. SIAM J. Sci. Comput. 34 (2012) A1824-A1848. | MR 2970275 | Zbl 1247.35199

[5] L. Bourgeois and H. Haddar, Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Probl. Imaging 4 (2010) 26. | MR 2592781 | Zbl 1189.35370

[6] M. Cayoren, I. Akduman, A. Yapar and L. Crocco, A new algorithm for the shape reconstruction of perfectly conducting objects. Inverse Probl. 23 (2007) 1087−1100. | MR 2329934 | Zbl 1118.35061

[7] R. Cicchetti, A class of exact and higer-order surface boundary conditions for layered structures. IEEE Trans. Antennas Propag. 44 (1996) 249−259. | MR 1373967 | Zbl 0945.78511

[8] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edn. Springer, Berlin (1999). | Zbl 0760.35053

[9] M. Duruflé, H. Haddar and P. Joly, Higher Order Generalized Impedance Boundary Conditions in Electromagnetic Scattering Problems. C.R. Phys. 7 (2006) 533−542.

[10] H. Haddar, P. Joly and H.M. Nguyen, Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case. Math. Models Methods Appl. Sci. 15 (2005) 1273−1300. | MR 2143271 | Zbl 1084.35102

[11] H. Haddar, P. Joly and H.M. Nguyen, Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: the case of Maxwell's equations. Math. Models Methods Appl. Sci. 18 (2008) 1787−1827. | MR 2463780 | Zbl 1170.35094

[12] H. Haddar and A. Lechleiter, Asymptotic models for scattering from unbounded media with high conductivity. Math. Mod. Numer. Anal. 44 (2010) 1295−1317. | Numdam | MR 2769059 | Zbl 1206.35066

[13] L. He, S. Kindermann and M. Sini, Reconstruction of shapes and impedance functions using few far-field measurements. J. Comput. Phys. 228 (2009) 717−730. | MR 2477785 | Zbl 1158.65044

[14] J.L. Holloway, E.F. Kuester, Impedance-type boundary conditions for a periodic interface between a dielectric and a highly conducting medium. IEEE Trans. Antennas Propag. 48 (2000) 1660-1672.

[15] J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems 1st edn. Springer, New York (2004). | MR 2102218 | Zbl 1068.65022

[16] I. Knowles, A variational algorithm for electrical impedance tomography. Inverse Probl. 14 (1998) 1513−1525. | MR 1662464 | Zbl 0921.35191

[17] I. Knowles, Parameter identification for elliptic problems. J. Comput. Appl. Math. 131 (2001) 175-194. | MR 1835711 | Zbl 0983.65121

[18] M.J. Kong and B. Beker, Computation of resonant frequencies of cylindrical ferrite resonators using GIBCs. IEEE Trans. Microwave Theory Tech. 46 (1998) 1503-1507.

[19] J.J. Liu, G. Nakamura and M. Sini, Reconstruction of the shape and surface impedance from acoustic scattering data for an arbitrary cylinder. SIAM J. Appl. Math. 67 (2007) 1124−1146. | MR 2314199 | Zbl 1127.35080

[20] M. Ljalinov, Generalized impedance boundary conditions and model of a point conductor. Radio Sci. 30 (1995) 1777−1786.

[21] O. Marceaux and B. Stupfel, Higher order impedance boundary conditions for multilayer coated 3-D objects. IEEE Trans. Antennas Propag. 46 (2000) 429−436.

[22] O. Ozdemir, I. Akduman, A. Yapar and L. Crocco, Higher order inhomogeneous impedance boundary conditions for perfectly conducting objects. IEEE Trans. Geosci. Remote Sens. 45 (2007) 1291-1297.

[23] O. Ozdemir, H. Haddar and A. Yaka, Reconstruction of the electromagnetic field in layered media using the concept of approximate transmission conditions. IEEE Trans. Antennas Propag. 59 (2011) 2964−2972. | MR 2856164

[24] H.Y. Pao, Z. Zhu and S.L. Dvarok, Exact, closed-form representation for the time-domain surface impedances of a homogeneous, lossy half-space. IEEE Trans. Antennas Propag. 52 (2004) 2659−2665. | MR 2107317

[25] J.R. Poirier, A. Bendali and P. Borderies, Impedance boundary conditions for the scattering of time-harmonic waves by rapidly varying surfaces. IEEE Trans. Antennas Propag. 54 (2006) 995−1005.

[26] Z.G. Qian, W.C. Chew and R. Suaya, Generalized impedance boundary condition for conductor modeling in surface integral equation. IEEE Trans. Microwave Theory Tech. 55 (2007) 2354−2364.

[27] Z.G. Qian, M.S. Tong and W.C. Chew, Conductive medium modeling with an augmented GIBC formulation. PIER 99 (2009) 261−272.

[28] C.R. Rao, Linear Statistical Inference and Its Applications 2nd edn. Wiley, New York (1973). | MR 346957 | Zbl 0137.36203

[29] R.G. Rojas, Generalized impedance boundary conditions for EM scattering problems. Electron Lett. 24 (1998) 1093−1094.

[30] R.G. Rojas, Z. Al-hekail, Generalized impedance/resistive boundary conditions for EM scattering problems. Radio Sci. 24 (1989) 1−12.

[31] K. Schmidt and S. Tordeux, High order transmission conditions for thin conductive sheets in magneto-quasistatics. Math. Mod. Numer. Anal. 45 (2011) 1115−1140. | Numdam | MR 2833175 | Zbl 1273.78029

[32] T.B.A. Senior, J.L. Volakis and S.R. Legault, Higher order impedance and absorbing boundary conditions. IEEE Trans. Antennas Propag. 45 (1997) 107−114.

[33] M. Yousefi, S.K. Chaudhuri and S. Safavi-Naeini, GIBC formulation for the resonant frequencies and field distribution of a substrate-mounted dielectric resonator. IEEE Trans. Antennas Propag. 42 (1994) 38−46.

[34] S. Yuferev, L. Proekt and N. Ida, Surface impedance boundary conditions near corners and edges: Rigorous consideration. IEEE Trans. Magn. 37 (2001) 3466−3468.