Some applications of substructuring and domain decomposition techniques to radiation and scattering of time-harmonic electromagnetic waves

Nolwenn Balin; Abderrahmane Bendali; M'Barek Fares; Florence Millot; Nicolas Zerbib

doi:10.1016/j.crhy.2006.04.001

Some applications of substructuring and domain decomposition techniques to radiation and scattering of time-harmonic electromagnetic waves
[Quelques applications de techniques de sous-structuration et de décomposition de domaine au rayonnement et à la diffraction des ondes électromagnétiques en régime harmonique]

Nolwenn Balin ^1,² ; Abderrahmane Bendali ^1,³ ; M'Barek Fares ¹ ; Florence Millot ¹ ; Nicolas Zerbib ^1,⁴

¹ CERFACS, 42, avenue Gaspard-Coriolis, 31057 Toulouse cedex 1, France
² MBDA-France, 2, rue Béranger, 92323 Châtillon cedex, France
³ UMR 5640, laboratoire MIP, INSA–CNRS–UPS, 135, avenue de Rangueil, 31077 Toulouse cedex 4, France
⁴ CNES, Centre spatial de Toulouse, 18, avenue Édouard-Belin, 31401 Toulouse cedex 9, France

Comptes Rendus. Physique, Volume 7 (2006) no. 5, pp. 474-485.

Résumé
Abstract

Après une revue rapide des méthodes de décomposition de domaine, et, plus particulièment de leur application aux problèmes relatifs au rayonnement et à la diffraction des ondes électromagnétiques en régime harmonique, nous décrivons deux contributions des auteurs dans ce contexte. Les deux contributions sont liées à la diffraction d'une onde électromagnétique en régime harmonique par une structure parfaitement conductrice de grande taille comportant une cavité profonde. La première contribution est une technique de sous-structuration. Elle est utilisée pour accroître la vitesse de convergence de l'algorithme itératif dans le cadre d'une résolution par la méthode multipôle rapide multi-niveaux. Des expérimentations numériques illustrent l'efficacité de l'approche en ce que la méthode itérative de Krylov afférente converge en un nombre d'itérations qui reste quasiment constant lorsqu'on augmente une longueur caractéristique dans le problème. La seconde contribution propose une adaptation de la méthode de décomposition de domaine par recouvrement pour une formulation par équations intégrales de frontière. Elle est utilisée ici pour effectuer une hybridation d'une formulation exacte, utilisée à l'ouverture de la cavitée, avec une méthode asymptotique haute fréquence, employée pour le reste de la frontière extérieure de la structure. Des résultats numériques prouvent la fiabilité et l'efficacité de la méthode.

After a quick review of the domain decomposition methods, and, more particularly on their application to large size problems relative to radiation and scattering of time-harmonic waves, we describe two contributions of the authors in this context. The two contributions are related to the scattering of a time-harmonic electromagnetic wave by a large perfectly conducting structure including a deep cavity. The first contribution is a substructuring technique. It is used to increase the speed of the convergence of the iterative algorithm in a Multi-Level Fast Multipol Method (MLFMM) solution. Numerical experiments illustrate the effectiveness of the approach since the number of iterations of the underlying Krylov iterative method remains almost constant while increasing a characteristic length in the problem. The second contribution proposes an adaptation of the overlapping domain decomposition techniques for a boundary integral formulation. It is used here to perform a hybridization of an exact formulation, used at the opening of the cavity, with an asymptotic high-frequency method employed for the rest of the exterior boundary of the structure. Numerical results demonstrate the reliability and the efficiency of the method.

Publié le : 2006-06-01

DOI : 10.1016/j.crhy.2006.04.001

Keywords: Domain decomposition methods, Time-harmonic waves
Mot clés : Méthodes de décomposition de domaine, Régime harmonique

Affiliations des auteurs :

Nolwenn Balin ^{1, 2} ; Abderrahmane Bendali ^{1, 3} ; M'Barek Fares ¹ ; Florence Millot ¹ ; Nicolas Zerbib ^{1, 4}

¹ CERFACS, 42, avenue Gaspard-Coriolis, 31057 Toulouse cedex 1, France
² MBDA-France, 2, rue Béranger, 92323 Châtillon cedex, France
³ UMR 5640, laboratoire MIP, INSA–CNRS–UPS, 135, avenue de Rangueil, 31077 Toulouse cedex 4, France
⁴ CNES, Centre spatial de Toulouse, 18, avenue Édouard-Belin, 31401 Toulouse cedex 9, France

@article{CRPHYS_2006__7_5_474_0,
     author = {Nolwenn Balin and Abderrahmane Bendali and M'Barek Fares and Florence Millot and Nicolas Zerbib},
     title = {Some applications of substructuring and domain decomposition techniques to radiation and scattering of time-harmonic electromagnetic waves},
     journal = {Comptes Rendus. Physique},
     pages = {474--485},
     publisher = {Elsevier},
     volume = {7},
     number = {5},
     year = {2006},
     doi = {10.1016/j.crhy.2006.04.001},
     language = {en},
}

TY  - JOUR
AU  - Nolwenn Balin
AU  - Abderrahmane Bendali
AU  - M'Barek Fares
AU  - Florence Millot
AU  - Nicolas Zerbib
TI  - Some applications of substructuring and domain decomposition techniques to radiation and scattering of time-harmonic electromagnetic waves
JO  - Comptes Rendus. Physique
PY  - 2006
SP  - 474
EP  - 485
VL  - 7
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crhy.2006.04.001
LA  - en
ID  - CRPHYS_2006__7_5_474_0
ER  -

%0 Journal Article
%A Nolwenn Balin
%A Abderrahmane Bendali
%A M'Barek Fares
%A Florence Millot
%A Nicolas Zerbib
%T Some applications of substructuring and domain decomposition techniques to radiation and scattering of time-harmonic electromagnetic waves
%J Comptes Rendus. Physique
%D 2006
%P 474-485
%V 7
%N 5
%I Elsevier
%R 10.1016/j.crhy.2006.04.001
%G en
%F CRPHYS_2006__7_5_474_0

Nolwenn Balin; Abderrahmane Bendali; M'Barek Fares; Florence Millot; Nicolas Zerbib. Some applications of substructuring and domain decomposition techniques to radiation and scattering of time-harmonic electromagnetic waves. Comptes Rendus. Physique, Volume 7 (2006) no. 5, pp. 474-485. doi : 10.1016/j.crhy.2006.04.001. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2006.04.001/

Bibliographie
Cité par

[1] B. Smith; P. Bjorstad; W. Gropp Domain Decomposition, Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge Univ. Press, New York, 1996

[2] J.S. Przemieniecki Theory of Matrix Structural Analysis, Dover, New York, 1985

[3] C.R. Dohrmann A preconditioner for substructuring based on constrained energy minimisation, SIAM Journal of Scientific Computing, Volume 25 (2003) no. 1, pp. 246-258

[4] P.-L. Lions On the Schwarz alternating method. I (R. Glowinski; G.H. Golub; G.A. Meurant; J. Périaux, eds.), First International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988, pp. 1-42

[5] P.-L. Lions On the Schwarz alternating methods. II (T. Chan; R. Glowinski; J. Périaux; O. Widlund, eds.), Domain Decomposition Methods, SIAM, Philadelphia, 1989, pp. 47-70

[6] F. Farhat; M. Lesoinne; P. Le Tallec; K. Pierson; D. Rixen FETI-DP: A dual-primal unified FETI method—part i: A faster alternative to the two-level FETI method, International Journal for Numerical Methods in Engineering, Volume 50 (2001), pp. 1523-1544

[7] S. Silver Microwave Antenna Theory and Design, IEEE, London, 1997

[8] A. Barka; P. Soudais; D. Volpert Scattering from 3d cavities with a plug & play numerical scheme combining IE, PDE and modal techniques, IEEE Transactions on Antennas and Propagation, Volume 48 (2000) no. 5, pp. 704-712

[9] C.-C. Lu; W.C. Chew A near-resonance decoupling approach (NRDA) for scattering solution of near-resonant structures, IEEE Transactions on Antennas and Propagation, Volume 45 (1997) no. 12, pp. 1857-1862

[10] B. Després, Méthodes de décomposition de domaine pour les problèmes de propagation d'ondes en régime harmonique, PhD thesis, Université Paris IX Dauphine, 1991

[11] B. Després Domain decomposition method and the Helmholtz problem, Strasbourg, 1991 (G. Cohen; L. Halpern; P. Joly, eds.), SIAM, Philadelphia, PA (1991), pp. 44-52

[12] P.-L. Lions On the Schwarz alternating method. III, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, July–August 1989, vol. 22, no. 4, SIAM, Philadelphia, PA, 1990, pp. 202-223

[13] F. Collino; S. Ghanemi; P. Joly Domain decomposition method for harmonic wave propagation: a general presentation, Computer Methods in Applied Mechanics and Engineering, Volume 184 (2000), pp. 171-211

[14] M.J. Gander; F. Magoules; F. Nataf Optimized Schwarz methods without overlap for the Helmholtz equation, SIAM Journal of Scientific Computing, Volume 24 (2002), pp. 38-60

[15] F. Magoules; F.-X. Roux; S. Salmon Optimal discrete transmission conditions for a non-overlapping domain decomposition method for the Helmholtz equation, SIAM Journal on Scientific Computing, Volume 25 (2004) no. 5, pp. 1497-1515

[16] B. Stupfel; B. Després A domain decomposition method for the solution of large electromagnetic scattering problems, Journal of Computational Waves and Applications, Volume 13 (1999), pp. 1553-1568

[17] D. Goudin; M. Mandallena; K. Mer-Nkonga; B. Stupfel A domain decomposition method for the solution of large electromagnetic problems using a massively parallel hybrid finite element—integral equation MLFMA, Antennas and Propagation Society Symposium, Volume 1 (2004), pp. 337-338

[18] C. Farhat; A. Macedo; M. Lesoinne; F.-X. Roux; F. Magoules; A. de la Bourdonnaye Two-level domain decomposition methods with Lagrange multipliers for the fast iterative solution of acoustic scattering problems, Computer Methods in Applied Mechanics and Engineering, Volume 184 (2000) no. 2, pp. 213-240

[19] A. Bendali, Y. Boubendir, M. Fares, A FETI-like domain decomposition method for coupling finite elements and boundary elements in large-size scattering problems of acoustic scattering, Computers & Structures, 2005, in press

[20] P. Soudais, A. Barka, Sub-domain methods for collaborative electromagnetic computations, this issue

[21] A. Bendali, Y. Boubendir, Non-overlapping domain decomposition method for a nodal finite element method, Numerisch Mathematik, published electronically | DOI

[22] A. Bendali; Y. Boubendir Méthode de décomposition de domaine et éléments finis nodaux pour la résolution de l'équation d'Helmholtz, C. R. Acad. Sci. Paris, Ser. I, Volume 339 (2004), pp. 229-234

[23] Youcef Saad Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston, 1996

[24] J. Jin The Finite Element Method in Electromagnetics, John Wiley & Sons, New York, 2002

[25] J.-M. Jin; J. Liu; Z. Lou; C.S.T. Liang A fully high-order finite-element simulation of scattering by deep cavities, IEEE Transactions on Antennas and Propagation, Volume 51 (2003) no. 9, pp. 2420-2429

[26] N. Balin, Etude de méthodes de couplage pour la résolution des équations de Maxwell—Application à la signature radar d'aéronefs par hybridation de méthodes exactes et asymptotiques, PhD thesis, INSA de Toulouse, 2005

[27] N. Balin, A. Bendali, A substructuring approach in the boundary integral solution of a scattering problem involving a deep cavity, in preparation, 2005

[28] N. Balin, F. Millot, M. Fares, N. Zerbib, Domain decomposition method for electromagnetic scattering by electrically deep cavities, under revision

[29] V.H. Rumsey Reaction concept in electromagnetic theory, Physical Review, Volume 94 (1954), pp. 1483-1491

[30] D. Colton; R. Kress Integral Equation Methods in Scattering Theory, John Wiley and Sons, New York, 1983

[31] B.H. Jung; T.K. Sarkar; Y.-S. Chung A survey of various frequency domain integral equations for the analysis of scattering from three-dimensional dielectric objects, Progress in Electromagnetic Research, PIER, Volume 36 (2002), pp. 193-246

[32] G. Alléon; M. Benzi; L. Giraud Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics, numerical algorithms, Numerical Algorithms, Volume 16 (1997) no. 1, pp. 1-15

[33] A.F. Peterson; S.L. Ray; R. Raj Mittra Computational Methods for Electromagnetics, IEEE Press, Piscataway, NJ, 1998

[34] M. Balabane; V. Tirel Décomposition de domaine pour un calcul hybride de l'équation d'Helmholtz, C. R. Acad. Sci. Paris, Ser. I, Volume 324 (1997), pp. 281-286

[35] K. Eriksson; D. Estep; P. Hansbo; C. Johnson Computational Differential Equations, Cambridge University Press, Lund, New York and Melbourne, 1996

[36] S. Alfonzetti; G. Borzi Accuracy of the Robin boundary condition iteration method for the finite element solution of scattering problems, International Journal for Numerical Modelling, Volume 13 (2000), pp. 217-231

[37] A. Bendali; M. Fares; J. Gay A boundary-element solution of the Leontovitch problem, IEEE Transactions on Antennas and Propagation, Volume 47 (1999) no. 10, pp. 1597-1605

[38] S.H. Christiansen; J.C. Nédélec A preconditioner for the electric field integral equation based on Calderón formulas, SIAM Journal of Numerical Analysis, Volume 40 (2002) no. 3, pp. 1100-1135

[39] A. Buffa; S.H. Christiansen A dual finite element complex on the barycentric refinement, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2005), pp. 461-464

Cité par Sources :

Commentaires - Politique