Nouvelles formulations intégrales pour les problèmes de diffraction d’ondes

David P. Levadoux; Bastiaan L. Michielsen

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2004)

  • Volume: 38, Issue: 1, page 157-175
  • ISSN: 0764-583X

Abstract

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We present an integral equation method for solving boundary value problems of the Helmholtz equation in unbounded domains. The method relies on the factorisation of one of the Calderón projectors by an operator approximating the exterior admittance (Dirichlet to Neumann) operator of the scattering obstacle. We show how the pseudo-differential calculus allows us to construct such approximations and that this yields integral equations without internal resonances and being well-conditioned at all frequencies. An implementation technique is elaborated, where again reasonings from pseudo-differential calculus play an important rôle. Some numerical examples are presented which appear to confirm that the new integral equation leads to linear systems which are much better conditioned than the classical (“direct”) integral equations and hence have much better behaviour when solved with iterative techniques and matrix sparsification.

How to cite

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Levadoux, David P., and Michielsen, Bastiaan L.. "Nouvelles formulations intégrales pour les problèmes de diffraction d’ondes." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.1 (2004): 157-175. <http://eudml.org/doc/244809>.

@article{Levadoux2004,
abstract = {We present an integral equation method for solving boundary value problems of the Helmholtz equation in unbounded domains. The method relies on the factorisation of one of the Calderón projectors by an operator approximating the exterior admittance (Dirichlet to Neumann) operator of the scattering obstacle. We show how the pseudo-differential calculus allows us to construct such approximations and that this yields integral equations without internal resonances and being well-conditioned at all frequencies. An implementation technique is elaborated, where again reasonings from pseudo-differential calculus play an important rôle. Some numerical examples are presented which appear to confirm that the new integral equation leads to linear systems which are much better conditioned than the classical (“direct”) integral equations and hence have much better behaviour when solved with iterative techniques and matrix sparsification.},
author = {Levadoux, David P., Michielsen, Bastiaan L.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Équations intégrales; opérateurs pseudo-différentiels; équation de Helmholtz},
language = {eng},
number = {1},
pages = {157-175},
publisher = {EDP-Sciences},
title = {Nouvelles formulations intégrales pour les problèmes de diffraction d’ondes},
url = {http://eudml.org/doc/244809},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Levadoux, David P.
AU - Michielsen, Bastiaan L.
TI - Nouvelles formulations intégrales pour les problèmes de diffraction d’ondes
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 1
SP - 157
EP - 175
AB - We present an integral equation method for solving boundary value problems of the Helmholtz equation in unbounded domains. The method relies on the factorisation of one of the Calderón projectors by an operator approximating the exterior admittance (Dirichlet to Neumann) operator of the scattering obstacle. We show how the pseudo-differential calculus allows us to construct such approximations and that this yields integral equations without internal resonances and being well-conditioned at all frequencies. An implementation technique is elaborated, where again reasonings from pseudo-differential calculus play an important rôle. Some numerical examples are presented which appear to confirm that the new integral equation leads to linear systems which are much better conditioned than the classical (“direct”) integral equations and hence have much better behaviour when solved with iterative techniques and matrix sparsification.
LA - eng
KW - Équations intégrales; opérateurs pseudo-différentiels; équation de Helmholtz
UR - http://eudml.org/doc/244809
ER -

References

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  1. [1] N. Bartoli and F. Collino, Integral equations via saddle point problem for 2D electromagnetic problems. ESAIM: M2AN 34 (2000) 1023–1049. Zbl0964.78005
  2. [2] L. Boutet de Monvel, Boundary problems for pseudo-differential operators. Acta Math. 126 (1971) 11–51. Zbl0206.39401
  3. [3] F. Canning, Improved impedance matrix localisation method. IEEE Trans. Ant. Prop. 41 (1993) 659–667. 
  4. [4] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Springer-Verlag (1992). Zbl0760.35053MR1183732
  5. [5] A. de La Boudonnaye, High frequency approximation of integral equations modelling scattering phenomena. RAIRO Modél. Math. Anal. Numér. 28 (1994) 223–241. Zbl0822.65124
  6. [6] B. Després, Fonctionnelle quadratique et équations intégrales pour les équations de Maxwell harmoniques en domaine extérieur. C.R. Acad. Sciences, Série I 323 (1996) 547–552. Zbl0858.65126
  7. [7] L. Hörmander, Fourier Integral Operators. Springer-Verlag (1994). Zbl0212.46601MR1481433
  8. [8] F. Hu, A spectral boundary integral equation method for the 2 D Helmholtz equation. J. Comput. Phys. 120 (1995) 340–347. Zbl0840.65115
  9. [9] D. Levadoux, Étude d’une équation intégrale adaptée à la résolution hautes fréquences de l’équation de Helmholtz. Thèse de doctorat, Université Paris VI, France (2001). 
  10. [10] D. Levadoux and B. Michielsen, Analysis of a boundary integral equation for high frequency Helmholtz problems. Fourth International Conf. Mathematical and Numerical Aspects of Wave Propagation, Colorado, 1–5 June (1998). Zbl0940.65138
  11. [11] V. Rokhlin, Diagonal form of translation operators for the Helmholtz equation in three dimensions. Rapport technique YALEU/DCS/RR-894, Yale University, Department of Computer Science (1992). Zbl0795.35021MR1256528
  12. [12] L. Schwartz, Théorie des Distributions. Hermann (1966). Zbl0149.09501MR209834

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