Nouvelles formulations intégrales pour les problèmes de diffraction d'ondes

David P. Levadoux; Bastiaan L. Michielsen

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • 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 38.1 (2010): 157-175. <http://eudml.org/doc/194204>.

@article{Levadoux2010,
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},
keywords = {Équations intégrales; opérateurs pseudo-différentiels; équation de Helmholtz.},
language = {eng},
month = {3},
number = {1},
pages = {157-175},
publisher = {EDP Sciences},
title = {Nouvelles formulations intégrales pour les problèmes de diffraction d'ondes},
url = {http://eudml.org/doc/194204},
volume = {38},
year = {2010},
}

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
DA - 2010/3//
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/194204
ER -

References

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  7. L. Hörmander, Fourier Integral Operators. Springer-Verlag (1994).  
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  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. 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).  
  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).  
  12. L. Schwartz, Théorie des Distributions. Hermann (1966).  

Citations in EuDML Documents

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  1. Xavier Antoine, Hélène Barucq, Approximation by generalized impedance boundary conditions of a transmission problem in acoustic scattering
  2. Xavier Antoine, Hélène Barucq, Approximation by generalized impedance boundary conditions of a transmission problem in acoustic scattering
  3. Xavier Antoine, Marion Darbas, Generalized combined field integral equations for the iterative solution of the three-dimensional Helmholtz equation
  4. David P. Levadoux, Proposition de préconditionneurs pseudo-différentiels pour l’équation CFIE de l’électromagnétisme
  5. David P. Levadoux, Proposition de préconditionneurs pseudo-différentiels pour l'équation CFIE de l'électromagnétisme

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