Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-20T00:00:48.627Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of Direct and Inverse Cavity Scattering Problems

Published online by Cambridge University Press:  28 May 2015

Gang Bao ,
Jinglu Gao  and
Peijun Li
Gang Bao*
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, China Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Jinglu Gao*
Affiliation:
School of Mathematical Sciences, Jilin University, Changchun 130023, China
Peijun Li*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
*
Corresponding author.Email address:bao@math.msu.edu
Corresponding author.Email address:jinglugao@gmail.com
Corresponding author.Email address:lipeijun@math.purdue.edu
Get access

Abstract

Consider the scattering of a time-harmonic electromagnetic plane wave by an arbitrarily shaped and filled cavity embedded in a perfect electrically conducting infinite ground plane. A method of symmetric coupling of finite element and boundary integral equations is presented for the solutions of electromagnetic scattering in both transverse electric and magnetic polarization cases. Given the incident field, the direct problem is to determine the field distribution from the known shape of the cavity; while the inverse problem is to determine the shape of the cavity from the measurement of the field on an artificial boundary enclosing the cavity. In this paper, both the direct and inverse scattering problems are discussed based on a symmetric coupling method. Variational formulations for the direct scattering problem are presented, existence and uniqueness of weak solutions are studied, and the domain derivatives of the field with respect to the cavity shape are derived. Uniqueness and local stability results are established in terms of the inverse problem.

Keywords

78A4678M30Electromagnetic cavitydirect probleminverse problemfinite element methodsboundary integral equations
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 3 , August 2011 , pp. 335 - 358
Copyright
Copyright © Global Science Press Limited 2011

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

[1]Ammari, H. and Bao, G., Coupling of finite element and boundary element methods for the scattering by periodic chiral structures, J. Comput. Math., 26 (2008), pp. 261–283.Google Scholar
[2]Ammari, H., Bao, G., and Wood, A., An integral equation method for the electromagnetic scattering from cavities, Math. Meth. Appl. Sci., 23 (2000), pp. 1057–1072.3.0.CO;2-6>CrossRefGoogle Scholar
[3]Ammari, H., Bao, G., and Wood, A., Analysis of the electromagnetic scattering from a cavity, Jpn. J. Indus. Appl. Math., 19 (2001), pp. 301–308.CrossRefGoogle Scholar
[4]Ammari, H. and NÉdÉlec, J.-C., Couplage élements finis-équation intégrale pour les équations de Maxwell hétérogène, Equations aux Dérivées Partielles et Applications. Articles Dédiées à J.-L. Lions, Elsevier, 1998, pp. 19–34.Google Scholar
[5]Ammari, H. and NÉdÉlec, J.-C., Coupling of finite and boundary element methods for the time-harmonic Maxwell equations. Part II: a symmetric formulation, Operator Theory: Advances and Applications 108. The Maz’ya Anniversary Collection, Rossmann et al. (Eds), Volume 101 (1999), pp. 23–32.Google Scholar
[6]Bao, G. and Friedman, A., Inverse problems for scattering by periodic structures, Arch. Rational Mech. Anal., 132 (1995), pp. 49–72.CrossRefGoogle Scholar
[7]Bao, G. and Sun, W., A fast algorithm for the electromagnetic scattering from a large cavity, SIAM J. Sci. Comput., 27 (2005), pp. 553–574.CrossRefGoogle Scholar
[8]Brezzi, F. and Johnson, C., On the coupling of boundary integral and finite element methods, Calcolo, 16 (1979), pp. 189–201.CrossRefGoogle Scholar
[9]Cakoni, F. and Colton, D., The linear sampling method for cracks, Inverse Problems, 19 (2003), pp. 279–295.CrossRefGoogle Scholar
[10]Colton, D. and Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Appl. Math. Sci. 93, Springer-Verlag, Berlin, 1998.Google Scholar
[11]Costabel, M. and Stephan, E. P., Coupling of finite and boundary element methods for an elasto- plastic interface problem, SIAM J. Numer. Anal., 27 (1990), pp. 1212–1226.CrossRefGoogle Scholar
[12]Feng, L. and Ma, F., Uniqueness and local stability for the inverse scattering problem of determining the cavity, Science in China Ser. A Mathematics, 48 (2005), pp. 1113–1123.CrossRefGoogle Scholar
[13]Gatica, G. N. and Hisao, G. C., On the coupled BEM and FEMfor a nonlinear exterior Dirichlet problem in R2, Numer. Math., 61 (1992), pp. 171–214.CrossRefGoogle Scholar
[14]Gatica, G. N. and Wendland, W. L., Coupling of mixed finite elements and bounary elements for linear and nonlinear elliptic problems, Appl. Anal., 63 (1996), pp. 39–75.CrossRefGoogle Scholar
[15]Haddar, H. and Kress, R., On the Fréchet derivative for obstacle scattering with an impedance boundary condition, SIAM J. Appl. Math., 65 (2004), pp. 194–208.CrossRefGoogle Scholar
[16]Hettlich, F., Fréchet derivative in inverse obstacle scattering, Inverse Problems, 11 (1995), pp. 371–382.CrossRefGoogle Scholar
[17]Hsiao, G. C., The coupling of boundary element and finite element methods, Math. Mech., 6 (1990), pp. 493–503.Google Scholar
[18]Jin, J. M., Electromagnetic scattering from large, deep, and arbitrarily-shaped open cavities, Electromagnetics, 18 (1998), pp. 3–34.CrossRefGoogle Scholar
[19]Jin, J. M. and Volakis, J. L., A hybrid finite element method for scattering and radiation by micro strip patch antennas and arrays residing in a cavity, IEEE Trans. Antennas Propag., 39 (1991), pp. 1598–1604.CrossRefGoogle Scholar
[20]Liu, J.-J., On uniqueness and linearization of an inverse electromagnetic scattering problem, Appl. Math. Comput., 171 (2005), pp. 406–419.Google Scholar
[21]Liu, J. and Jin, J. M., A special higher order finite-element method for scattering by deep cavities, IEEE Trans. Antennas Propag., 48 (2000), pp. 694–703.Google Scholar
[22]Johnson, C. and Nédélec, J.-C., On the coupling of boundary integral and finite element methods, Math. Comp., 35 (1980), 1063–1079.CrossRefGoogle Scholar
[23]Kirsch, A., The domain derivative and two applications in inverse scattering theory, Inverse problems, 9 (1993), pp. 301–310.CrossRefGoogle Scholar
[24]Li, P., Coupling of finite element and boundary integral method for electromagnetic scattering in a two-layered medium, J. Comput. Phys., 229 (2010), pp. 481–497.CrossRefGoogle Scholar
[25]Mclean, W., Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.Google Scholar
[26]Meddahi, S., Márquez, A., and Selgas, V., Computing acoustic waves in an inhomogeneous medium of the plane by a coupling of spectral andfinite elements, SIAM J. Numer. Anal., 41 (2003), pp. 1729–1750.CrossRefGoogle Scholar
[27]Nédélec, J.-C., Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Springer, New York, 2000.Google Scholar
[28]Van, T. and Wood, A., Finite element analysis for 2-D cavity problem, IEEE Trans. Antennas Propag., 51 (2003), pp. 1–8.CrossRefGoogle Scholar
[29]Wood, A., Analysis of electromagnetic scattering from an overfilled cavity in the ground plane, J. Comput. Phys., 215 (2006), pp. 630–641.CrossRefGoogle Scholar
[30]Wood, W. D. and Wood, A. W., Development and numerical solution of integral equations for electromagnetic scattering from a trough in a ground plane, IEEE Trans. Antennas Propag., 47 (1999), pp. 1318–1322.CrossRefGoogle Scholar