Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-15T13:26:02.221Z Has data issue: false hasContentIssue false
  • English
  • Français

Reconstruction of thickness variation of a dielectric coatingthrough the generalized impedance boundary conditions∗

Published online by Cambridge University Press:  30 June 2014

Birol Aslanyürek  and
Hülya Sahintürk
Birol Aslanyürek
Affiliation:
Mathematical Engineering, Yildiz Technical University, Davutpasa, 34220 Istanbul, Turkey. [email protected]; [email protected]
Hülya Sahintürk
Affiliation:
Mathematical Engineering, Yildiz Technical University, Davutpasa, 34220 Istanbul, Turkey. [email protected]; [email protected]
Get access

Abstract

We deal with an inverse scattering problem whose aim is to determine the thicknessvariation of a dielectric thin coating located on a conducting structure of unknown shape.The inverse scattering problem is solved through the application of the GeneralizedImpedance Boundary Conditions (GIBCs) which contain the thickness, curvature as well asmaterial 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 uniquenessresults for the inverse problem, the required total field as well as its higher orderderivatives appearing in the GIBCs are obtained by the analytical continuation of themeasured data to the coating surface through the single layer potential representation.The resulting system of non-linear differential equations for the unknown coatingthickness is solved iteratively via the Newton−Raphson method after expanding thethickness function in a series of exponentials. Through the simulations it has been shownthat the approach is effective under the validity conditions of the GIBCs.

Keywords

Generalized impedance boundary conditionsthin coatingsinverse scattering problemssingle layer potentialNewton−Raphson method
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014 , pp. 1011 - 1027
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ammari, H. and Nedelec, J.C., Generalized impedance boundary conditions for the maxwell equations as singular perturbations problems. Commun. Partial Differ. Equ. 24 (1999) 24−38. Google Scholar
Aslanyürek, B., Haddar, H. and Şahintürk, H., Generalized impedance boundary conditions for thin dielectric coatings with variable thickness. Wave Motion 48 (2011) 681−700. Google Scholar
Bourgeois, L., Chaulet, N. and Haddar, H., Stable reconstruction of generalized impedance boundary conditions. Inverse Probl. 27 (2011) 19−38. Google Scholar
Bourgeois, L., Chaulet, N. and Haddar, H., On Simultaneous Identification of the Shape and Generalized Impedance Boundary Condition in Obstacle Scattering. SIAM J. Sci. Comput. 34 (2012) A1824–A1848. Google Scholar
Bourgeois, L. and Haddar, H., Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Probl. Imaging 4 (2010) 26. Google Scholar
Cayoren, M., Akduman, I., Yapar, A. and Crocco, L., A new algorithm for the shape reconstruction of perfectly conducting objects. Inverse Probl. 23 (2007) 1087−1100. Google Scholar
Cicchetti, R., A class of exact and higer-order surface boundary conditions for layered structures. IEEE Trans. Antennas Propag. 44 (1996) 249−259. Google Scholar
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edn. Springer, Berlin (1999).
Duruflé, M., Haddar, H. and Joly, P., Higher Order Generalized Impedance Boundary Conditions in Electromagnetic Scattering Problems. C.R. Phys. 7 (2006) 533−542. Google Scholar
Haddar, H., Joly, P. and Nguyen, H.M., Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case. Math. Models Methods Appl. Sci. 15 (2005) 1273−1300. Google Scholar
Haddar, H., Joly, P. and Nguyen, H.M., 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. Google Scholar
Haddar, H. and Lechleiter, A., Asymptotic models for scattering from unbounded media with high conductivity. Math. Mod. Numer. Anal. 44 (2010) 1295−1317. Google Scholar
He, L., Kindermann, S. and Sini, M., Reconstruction of shapes and impedance functions using few far-field measurements. J. Comput. Phys. 228 (2009) 717−730. Google Scholar
Holloway, J.L., Kuester, E.F., Impedance-type boundary conditions for a periodic interface between a dielectric and a highly conducting medium. IEEE Trans. Antennas Propag. 48 (2000) 1660–1672. Google Scholar
J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems 1st edn. Springer, New York (2004).
Knowles, I., A variational algorithm for electrical impedance tomography. Inverse Probl. 14 (1998) 1513−1525. Google Scholar
Knowles, I., Parameter identification for elliptic problems. J. Comput. Appl. Math. 131 (2001) 175–194. Google Scholar
Kong, M.J. and Beker, B., Computation of resonant frequencies of cylindrical ferrite resonators using GIBCs. IEEE Trans. Microwave Theory Tech. 46 (1998) 1503–1507. Google Scholar
Liu, J.J., Nakamura, G. and Sini, M., Reconstruction of the shape and surface impedance from acoustic scattering data for an arbitrary cylinder. SIAM J. Appl. Math. 67 (2007) 1124−1146. Google Scholar
Ljalinov, M., Generalized impedance boundary conditions and model of a point conductor. Radio Sci. 30 (1995) 1777−1786. Google Scholar
Marceaux, O. and Stupfel, B., Higher order impedance boundary conditions for multilayer coated 3-D objects. IEEE Trans. Antennas Propag. 46 (2000) 429−436. Google Scholar
Ozdemir, O., Akduman, I., Yapar, A. and Crocco, L., Higher order inhomogeneous impedance boundary conditions for perfectly conducting objects. IEEE Trans. Geosci. Remote Sens. 45 (2007) 1291–1297. Google Scholar
Ozdemir, O., Haddar, H. and Yaka, A., Reconstruction of the electromagnetic field in layered media using the concept of approximate transmission conditions. IEEE Trans. Antennas Propag. 59 (2011) 2964−2972. Google Scholar
Pao, H.Y., Zhu, Z. and Dvarok, S.L., Exact, closed-form representation for the time-domain surface impedances of a homogeneous, lossy half-space. IEEE Trans. Antennas Propag. 52 (2004) 2659−2665. Google Scholar
Poirier, J.R., Bendali, A. and Borderies, P., Impedance boundary conditions for the scattering of time-harmonic waves by rapidly varying surfaces. IEEE Trans. Antennas Propag. 54 (2006) 995−1005. Google Scholar
Qian, Z.G., Chew, W.C. and Suaya, R., Generalized impedance boundary condition for conductor modeling in surface integral equation. IEEE Trans. Microwave Theory Tech. 55 (2007) 2354−2364. Google Scholar
Qian, Z.G., Tong, M.S. and Chew, W.C., Conductive medium modeling with an augmented GIBC formulation. PIER 99 (2009) 261−272. Google Scholar
C.R. Rao, Linear Statistical Inference and Its Applications 2nd edn. Wiley, New York (1973).
Rojas, R.G., Generalized impedance boundary conditions for EM scattering problems. Electron Lett. 24 (1998) 1093−1094. Google Scholar
Rojas, R.G., Al-hekail, Z., Generalized impedance/resistive boundary conditions for EM scattering problems. Radio Sci. 24 (1989) 1−12. Google Scholar
Schmidt, K. and Tordeux, S., High order transmission conditions for thin conductive sheets in magneto-quasistatics. Math. Mod. Numer. Anal. 45 (2011) 1115−1140. Google Scholar
Senior, T.B.A., Volakis, J.L. and Legault, S.R., Higher order impedance and absorbing boundary conditions. IEEE Trans. Antennas Propag. 45 (1997) 107−114. Google Scholar
Yousefi, M., Chaudhuri, S.K. and Safavi-Naeini, S., GIBC formulation for the resonant frequencies and field distribution of a substrate-mounted dielectric resonator. IEEE Trans. Antennas Propag. 42 (1994) 38−46. Google Scholar
Yuferev, S., Proekt, L. and Ida, N., Surface impedance boundary conditions near corners and edges: Rigorous consideration. IEEE Trans. Magn. 37 (2001) 3466−3468. Google Scholar