Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T01:32:40.448Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic models for scattering from unbounded media with high conductivity

Published online by Cambridge University Press:  15 April 2010

Houssem Haddar  and
Armin Lechleiter
Houssem Haddar
Affiliation:
INRIA Saclay Île-de-France/École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France. haddar@cmap.polytechnique.fr; alechle@cmap.polytechnique.fr
Armin Lechleiter
Affiliation:
INRIA Saclay Île-de-France/École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France. haddar@cmap.polytechnique.fr; alechle@cmap.polytechnique.fr
Get access

Abstract

We analyze the accuracy and well-posedness of generalized impedanceboundary value problems in the framework of scattering problemsfrom unbounded highly absorbing media. We restrict ourselves in this first workto the scalar problem (E-mode for electromagnetic scattering problems). Compared to earlier works, the unboundedness of the rough absorbing layer introduces severe difficultiesin the analysis for the generalized impedance boundary conditions, sinceclassical compactness arguments are no longer possible. Our new analysisis based on the use of Rellich-type estimates and boundedness of L2solution operators. We also discuss some numerical experimentsconcerning these boundary conditions.

Keywords

Scattering problemsunbounded domainsasymptotic modelsgeneralized impedance boundary conditionshigh conductivity
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 6 , November 2010 , pp. 1295 - 1317
Copyright
© EDP Sciences, SMAI, 2010

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

Artola, M. and Cessenat, M., Diffraction d'une onde électromagnétique par un obstacle borné à permittivité et perméabilité élevées. C. R. Acad. Sci. Paris Sér. I Math. 314 (1992) 349–354.
Chandler-Wilde, S.N. and Monk, P., Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces. SIAM. J. Math. Anal. 37 (2005) 598–618. CrossRef
Chandler-Wilde, S.N. and Monk, P., The pml for rough surface scattering. Appl. Numer. Math. 59 (2009) 2131–2154. CrossRef
Chandler-Wilde, S.N. and Ross, C.R., Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane. Math. Meth. Appl. Sci. 19 (1996) 959–976. 3.0.CO;2-R>CrossRef
Chandler-Wilde, S.N., Heinemeyer, E. and Potthast, R., Acoustic scattering by mildly rough surfaces in three dimensions. SIAM J. Appl. Math. 66 (2006) 1002–1026. CrossRef
Chandler-Wilde, S.N., Monk, P. and Thomas, M., The mathematics of scattering by unbounded, rough, inhomogeneous layers. J. Comput. Appl. Math. 204 (2007) 549–559. CrossRef
Cummings, P. and Feng, X., Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations. Math. Mod. Meth. Appl. Sci. 16 (2006) 139–160. CrossRef
Duruflé, M., Haddar, H. and Joly, P., Higher order generalized impedance boundary conditions in electromagnetic scattering problems. C. R. Phys. 7 (2006) 533–542. CrossRef
Haddar, H., Joly, P. and Nguyen, H.-M., Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case. Math. Mod. Meth. Appl. Sci. 15 (2005) 1273–1300. CrossRef
F. Ihlenburg, Finite element analysis of acoustic scattering. Springer (1998).
A. Lechleiter and S. Ritterbusch, A variational method for wave scattering from penetrable rough layers. IMA J. Appl. Math. (2009) doi:10.1093/imamat/hxp040.
W. McLean, Strongly Elliptic Systems and Boundary Integral Operators. Cambridge University Press, Cambridge (2000).
J.C. Nédélec, Acoustic and electromagnetic equations, Applied Mathematical Sciences 144. Springer-Verlag, Berlin (2001).
Rytov, S.M., Calcul du skin-effet par la méthode des perturbations. J. Phys. USSR 2 (1940) 233–242.
T.B.A. Senior and J.L. Volakis, Approximate boundary conditions in electromagnetics, IEE Electromagnetic waves series 41. The institution of Electrical Engineers, London (1995).
Zhang, B. and Chandler-Wilde, S.N., Integral equation methods for scattering by infinite rough surfaces. Math. Meth. Appl. Sci. 26 (2003) 463–488. CrossRef