Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-01T19:37:46.195Z Has data issue: false hasContentIssue false
  • English
  • Français

Nouvelles formulations intégrales pour les problèmesde diffraction d'ondes

Published online by Cambridge University Press:  15 February 2004

David P. Levadoux  and
Bastiaan L. Michielsen
David P. Levadoux
Affiliation:
ONERA, centre de Palaiseau, Chemin de la Hunière, 91761 Palaiseau, France. [email protected].
Bastiaan L. Michielsen
Affiliation:
ONERA, centre de Palaiseau, Chemin de la Hunière, 91761 Palaiseau, France. [email protected].
Get access

Abstract

We present an integral equation method for solving boundary valueproblems of the Helmholtz equation in unbounded domains. Themethod relies on the factorisation of one of the Calderón projectors by an operator approximating the exterioradmittance (Dirichlet to Neumann) operator of the scatteringobstacle. We show how the pseudo-differential calculus allows usto construct such approximations and that this yields integralequations without internal resonances and being well-conditionedat all frequencies. An implementation technique is elaborated,where again reasonings from pseudo-differential calculus play animportant rôle. Some numerical examples are presented which appearto confirm that the new integral equation leads to linear systemswhich are much better conditioned than the classical ("direct")integral equations and hence have much better behaviour whensolved with iterative techniques and matrix sparsification.

Keywords

Équations intégralesopérateurs pseudo-différentielséquation de Helmholtz.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 1 , January 2004 , pp. 157 - 175
Copyright
© EDP Sciences, SMAI, 2004

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

Bartoli, N. and Collino, F., Integral equations via saddle point problem for 2D electromagnetic problems. ESAIM: M2AN 34 (2000) 1023–1049. CrossRef
Boutet de, L. Monvel, Boundary problems for pseudo-differential operators. Acta Math. 126 (1971) 11–51.
Canning, F., Improved impedance matrix localisation method. IEEE Trans. Ant. Prop. 41 (1993) 659–667. CrossRef
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Springer-Verlag (1992).
de La Boudonnaye, A., High frequency approximation of integral equations modelling scattering phenomena. RAIRO Modél. Math. Anal. Numér. 28 (1994) 223–241. CrossRef
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.
L. Hörmander, Fourier Integral Operators. Springer-Verlag (1994).
Hu, F., A spectral boundary integral equation method for the 2D Helmholtz equation. J. Comput. Phys. 120 (1995) 340–347. CrossRef
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).
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).
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).
L. Schwartz, Théorie des Distributions. Hermann (1966).