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An Unpreconditioned Boundary-Integral for Iterative Solution of Scattering Problems with Non-Constant Leontovitch Impedance Boundary Conditions

Published online by Cambridge University Press:  03 June 2015

D. Levadoux ,
F. Millot  and
S. Pernet
D. Levadoux*
Affiliation:
ONERA, French Aerospace Lab, Chemin de la humière 91761 Palaiseau, France
F. Millot*
Affiliation:
CERFACS 42 avenue G. Coriolis31057 Toulouse, France
S. Pernet*
Affiliation:
ONERA, French Aerospace Lab, 2 Avenue Édouard Belin, 31000 Toulouse, France
*
Email:David.Levadoux@onera.fr
Email:millot@cerfacs.fr
Corresponding author.Email:Sebastien.Pernet@onera.fr
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Abstract

This paper concerns the electromagnetic scattering by arbitrary shaped three dimensional imperfectly conducting objects modeled with non-constant Leontovitch impedance boundary condition. It has two objectives. Firstly, the intrinsically well-conditioned integral equation (noted GCSIE) proposed in [30] is described focusing on its discretization. Secondly, we highlight the potential of this method by comparison with two other methods, the first being a two currents formulation in which the impedance condition is implicitly imposed and whose the convergence is quasi-optimal for Lipschitz polyhedron, the second being a CFIE-like formulation [14]. In particular, we prove that the new approach is less costly in term of CPU time and gives a more accurate solution than that obtained from the CFIE formulation. Finally, as expected, It is demonstrated that no preconditioner is needed for this formulation.

Keywords

65R2015A1265N3865F1065Z05Integral equationboundary element methodimpedance boundary equationpreconditionerGMRESfast multipole method
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 5 , May 2014 , pp. 1431 - 1460
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Gay, J., Bendali, A. and Fares, M’B., A boundary-element solution of the leontovitch problem, IEEE Trans. Antennas Propagation, Octobre 1999.Google Scholar
[2]Alouges, F., Borel, S. and Levadoux, D., A stable well conditioned integral equation for elec-tromagnetism scattering, JCAM, 204(2) (2007).Google Scholar
[3]Andriulli, F. P., Well-Posed Boundary Element Formulations in Electromagnetics, PhD, University of Michigan, 2008.Google Scholar
[4]Bagci, K., Olyslager, H., Buffa, F., Christiansen, A., Michielssen, S., Andriulli, E. and Cools, F. P., A multiplicative calderon preconditioner for the electric field integral equation, IEEE Trans. Antennas Propag., 56(8) (2008).Google Scholar
[5]Bendali, A., Boundary element solution of scattering problems relative to a generalized impedance boundary condition, in Jäger, W., Nevcas, J., John, O., Najzar, K. and Stará, J., editors, Partial Differential Equations, Theory and Numerical Solution, volume 406, pages 10–24. Chapman & Hall/CRC, 1999.Google Scholar
[6]Bruno, O. P., Computational electromagnetics and acoustics: high-order solvers, highfre-quency configurations, high-order surface representations, Oberwolfach Reports, 5 (2007).Google Scholar
[7]Bruno, Oscar, Elling, Tim, Randy Paffenroth and Catalin Turc, Electromagnetic integral equations requiring small numbers of krylov-subspace iterations, J. Comput. Phys., 228(17) (2009).Google Scholar
[8]Buffa, A. and Hipmair, R., Galerkin boundary element methods for electromagnetic scattering, dans topics in computational wave propagation and inverse problems, Ainsworth, M.et al., eds., Springer-Verlag, Vol. 31, 2003.Google Scholar
[9]Buffa, Annalisa and Christiansen, Snorre H., A dual finite element complex on the barycentric refinement, Math. Comput., 76 (2007).Google Scholar
[10]Carpentieri, B., Sparse Preconditioners for Dense Complex Linear Systems in Electromagnetic Applications, Ph.D. dissertation, INPT, April 2002, TH/PA/02/48.Google Scholar
[11]Carpentieri, B., Duff, I. S. and Giraud, L., Sparse pattern selection strategies for robust frobenius-norm minimization preconditioners in electromagnetism, Numer. Linear Algebra Appl., 7(7-8) (2000), 667–685.Google Scholar
[12]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, Series Studies in Mathematics and its Applications, North-Holland, Amsterdam, 1978.Google Scholar
[13]Collino, F. and Despres, B., Integral equations via saddle point problems for time-harmonic maxwell’s, J. Comput. Appl. Math., 150 (2003), 157–192.Google Scholar
[14]Collino, F., Millot, F. and Pernet, S., Boundary-integral methods for iterative solution of scattering problems with variable impedance surface condition, Progress In Electromagnetics Research, PIER 80, 2008.CrossRefGoogle Scholar
[15]Colton, D. and Kreiss, P., Integral Equation Methods in Scattering, Wiley & Sons, New York, 1983.Google Scholar
[16]Millot, F., Levadoux, D. and Pernet, S., New trends in the preconditioning of integral equations of electromagnetism, in Mathematics in Industry-Scientific Computing in Electical Engineering, Springer, 2010.Google Scholar
[17]Darbas, M., Péconditionneurs Analytiques de Type Calderon pour les Formulations Intégrales des Problemes de Diffraction D’ondes, PhD Thesis INSA Toulouse, 2004.Google Scholar
[18]Darbas, M., Generalized cfie for the iterative solution of 3-d maxwell equations, Appl. Math. Lett., 19(8) (2006).Google Scholar
[19]Colton, D., Cakoni, F. and Monk, P., The electromagnetic inverse-scattering problem for partially coated lipschitz domains, Proc. Royal. Soc. Edinburgh, 134A (2004), 661–682.Google Scholar
[20]Frayssé, V., Giraud, L. and Gratton, S., A set of GMRES routines for real and complex arithmetics, Technical report, Cerfacs TR/PA/97/49, Toulouse, France, 1997.Google Scholar
[21]Lange, V., Equations Intégrales Espace-Temps pour les équations de Maxwell, Calcul du Champ Diffracté par un Obstacle Dissipatif, PhD Thesis Université de Bordeaux I, 1995.Google Scholar
[22]Leontovich, M. A., Approximate boundary condition for electromagnetic field on the surface of good conductor, Investigations on Radiowave Propagation part II Moscow, Academy of Sciences, Octobre 1978.Google Scholar
[23]Levadoux, D., Etude d’une équation Intégrale Adaptée a la Résolution Haute Fréquence de L’équation de Helmholtz, PhD Thesis Paris VI, 2001.Google Scholar
[24]Levadoux, D. and Michielsen, B. L., Analysis of a boundary integral equation for high frequency helmholtz problems, in Proceedings of 4th International Conference on Mathematical and Numerical Aspects of Wave Propagation, Golden, Colorado, pages 765–767, 1998.Google Scholar
[25]Levadoux, D. and Michielsen, B. L., Nouvelles formuations intégrales pour les problemes de diffraction d’ondes, Math. Model. Num. Anal., 38(1) (2004).Google Scholar
[26]Medgyesi-Mitschang, L. N. and Putnam, J. M., Integral equation formulations for imperfectly conducting scatterers, IEEE Trans. Antennas Propag., 33(2) (1985).CrossRefGoogle Scholar
[27]Milinazzo, F. A., Zala, C. A., and Brooke, G. H., Rational square-root approximations for parabolic equation algorithms, J. Acoust. Soc. Am., 101(2) (1997).Google Scholar
[28]Monk, P., Finite Element Methods for Maxwell’s Equations, Clarendon Press, Oxford, 2003.Google Scholar
[29]Kiminki, S. P., Yla-Oijala, P. and Jarvenpaa, S., Solving ibc-cfie with dual basis functions, IEEE Trans. Antennas Propag., 58(12) (2010).Google Scholar
[30]Pernet, S., A well-conditioned integral equation for iterative solution of scattering problems with a variable leontovitch boundary condition, Math. Model. Numer. Anal., 44(4), July 2010.Google Scholar
[31]Levadoux, D., Borel, S. and Alouges, F., A new well-conditioned integral formulation for maxwell equations in three-dimensions, IEEE Trans. Antennas Propag., 53(9) (2005).Google Scholar
[32]Kelley, C. T., Campbell, S. L., Ipsen, I. C. F. and Meyer, C. D., Gmres and the minimal polynomial, BIT Numerical Mathematics, Springer Netherlands, 36 (4), December 1996.Google Scholar
[33]Kelley, C. T., Meyer, C. D., Campbell, S. L., Ipsen, I. C. F. and Xue, Z. Q., Convergence estimates for solution of integral equations with gmres, Tech. Report CRSC-TR95-13, North Carolina State University, Center for Research in Scientific Computation, March 1995.Google Scholar