×

A high-order wideband direct solver for electromagnetic scattering from bodies of revolution. (English) Zbl 1452.78011

Summary: The generalized Debye source representation of time-harmonic electromagnetic fields yields well-conditioned second-kind integral equations for a variety of boundary value problems, including the problems of scattering from perfect electric conductors and dielectric bodies. Furthermore, these representations, and resulting integral equations, are fully stable in the static limit as \(\omega \rightarrow 0\) in multiply connected geometries. In this paper, we present the first high-order accurate solver based on this representation for bodies of revolution. The resulting solver uses a Nyström discretization of a one-dimensional generating curve and high-order integral equation methods for applying and inverting surface differentials. The accuracy and speed of the solvers are demonstrated in several numerical examples.

MSC:

78A45 Diffraction, scattering
65R20 Numerical methods for integral equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ahrens, J.; Geveci, B.; Law, C., Paraview: an end-user tool for large data visualization, (Hansen, C. D.; Johnson, C. R., The Visualization Handbook (2005), Elsevier)
[2] Alpert, B., Hybrid Gauss-trapezoidal quadrature rules, SIAM J. Sci. Comput., 20, 5, 1551-1584 (1999) · Zbl 0933.41019
[3] Atkinson, K. E., The Numerical Solution of Integral Equations of the Second Kind (1997), Cambridge University Press: Cambridge University Press New York, NY · Zbl 0899.65077
[4] Barnett, A. H.; Betcke, T., Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains, J. Comput. Phys., 227, 14, 7003-7026 (2008) · Zbl 1170.65082
[5] Bremer, J.; Gimbutas, Z., On the numerical evaluation of singular integrals of scattering theory, J. Comput. Phys., 251, 327-343 (2013) · Zbl 1349.65093
[6] Briggs, W. L.; Henson, V. E., The DFT: An Owner’s Manual for the Discrete Fourier Transform (1995), SIAM: SIAM Philadelphia, PA · Zbl 0827.65147
[7] Chew, W. C.; Michielssen, E.; Song, J. M.; Jin, J. M., Fast and Efficient Algorithms in Computational Electromagnetics (2001), Artech House, Inc.: Artech House, Inc. Norwood, MA
[8] Cohl, H. S.; Tohline, J. E., A compact cylindrical Green’s function expansion for the solution of potential problems, Astrophys. J., 527, 1, 86-101 (1999)
[9] Colton, D.; Kress, R., Integral Equation Methods in Scattering Theory (1983), John Wiley & Sons, Inc · Zbl 0522.35001
[10] Contopanagos, H.; Dembart, B.; Epton, M.; Ottusch, J. J.; Rokhlin, V.; Visher, J. L.; Wandzura, S. M., Well-conditioned boundary integral equations for three-dimensional electromagnetic scattering, IEEE Trans. Antennas Propag., 50, 12, 1824-1830 (2002)
[11] Conway, J. T.; Cohl, H. S., Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function, Z. Angew. Math. Phys., 61, 425-442 (2010) · Zbl 1200.35057
[12] Cools, K.; Andriulli, F. P.; Olyslager, F.; Michielssen, E., Nullspaces of MFIE and Calderon preconditioned EFIE operators applied to toroidal surfaces, IEEE Trans. Antennas Propag., 57, 10, 3205-3215 (2009) · Zbl 1369.78717
[13] Epstein, C. L.; Greengard, L., Debye sources and the numerical solution of the time harmonic Maxwell equations, Commun. Pure Appl. Math., 63, 4, 413-463 (2010) · Zbl 1190.35215
[14] Epstein, C. L.; Gimbutas, Z.; Greengard, L.; Klöckner, A.; O’Neil, M., A consistency condition for the vector potential in multiply-connected domains, IEEE Trans. Magn., 49, 3, 1072-1076 (2013)
[15] Epstein, C. L.; Greengard, L.; O’Neil, M., Debye sources and the numerical solution of the time harmonic Maxwell equations II, Commun. Pure Appl. Math., 66, 5, 753-789 (2013) · Zbl 1291.35383
[16] Epstein, C. L.; Greengard, L.; O’Neil, M., Debye sources, Beltrami fields, and a complex structure on Maxwell fields, Commun. Pure Appl. Math., 68, 2237-2280 (2015) · Zbl 1342.35368
[17] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9, 1-2, 69 (1998) · Zbl 0922.65074
[18] Frankel, T., The Geometry of Physics (2011), Cambridge University Press: Cambridge University Press New York, NY · Zbl 1250.58001
[19] Frittelli, M.; Sgura, I., Virtual element method for the Laplace-Beltrami equation on surfaces (2016)
[20] Gedney, S. D.; Mittra, R., The use of the FFT for the efficient solution of the problem of electromagnetic scattering by a body of revolution, IEEE Trans. Antennas Propag., 38, 3, 313-322 (1990)
[21] Gil, A.; Segura, J.; Temme, N. M., Numerical Methods for Special Functions (2007), SIAM: SIAM Philadelphia, PA · Zbl 1144.65016
[22] Gimbutas, Z.; Greengard, L., Fast multi-particle scattering: a hybrid solver for the Maxwell equations in microstructured materials, J. Comput. Phys., 232, 1, 22-32 (2013) · Zbl 1291.78058
[23] Gustafsson, M., Accurate and efficient evaluation of modal Green’s functions, J. Electromagn. Waves Appl., 24, 10, 1291-1301 (2010)
[24] Hao, S.; Barnett, A. H.; Martinsson, P.-G.; Young, P., High-order accurate Nyström discretization of integral equations with weakly singular kernels on smooth curves in the plane, Adv. Comput. Math., 40, 245-272 (2014) · Zbl 1300.65093
[25] Helsing, J.; Karlsson, A., An explicit kernel-split panel-based Nyström scheme for integral equations on axially symmetric surfaces, J. Comput. Phys., 272, 686-703 (2014) · Zbl 1349.65709
[26] Helsing, J.; Karlsson, A., Determination of normalized magnetic eigenfields in microwave cavities, IEEE Trans. Microw. Theory Tech., 63, 5, 1457-1467 (2015)
[27] Imbert-Gerard, L.-M.; Greengard, L., Pseudo-spectral methods for the Laplace-Beltrami equation and the Hodge decomposition on surfaces of genus one, Numer. Methods Partial Differ. Equ., 33, 3, 941-955 (2017) · Zbl 1397.65303
[28] Jackson, J. D., Classical Electrodynamics (1999), Wiley: Wiley New York, NY · Zbl 0920.00012
[29] Jin, J.-M., Theory and Computation of Electromagnetic Fields (2010), IEEE Press: IEEE Press Piscataway, NJ
[30] Kapur, S.; Long, D. E., IES3: efficient electrostatic and electromagnetic simulation, Comput. Sci. Eng., 60-67 (1998)
[31] Kirsch, A.; Monk, P., A finite element/spectral method for approximating the time-harmonic Maxwell system in \(R^3\), SIAM J. Appl. Math., 55, 5, 1324-1344 (1995) · Zbl 0840.65128
[32] Kucharski, A. A., A method of moments solution for electromagnetic scattering by inhomogeneous dielectric bodies of revolution, IEEE Trans. Antennas Propag., 48, 8, 1202-1210 (2000) · Zbl 1368.78048
[33] Liu, Y.; Barnett, A. H., Efficient numerical solution of acoustic scattering from doubly-periodic arrays of axisymmetric objects, J. Comput. Phys., 324, 226-245 (2016) · Zbl 1360.65297
[34] Mautz, J. R.; Harrington, R. F., Electromagnetic scattering from a homogeneous material body of revolution, Archiv Elektronik und Uebertragungstechnik, 33, 71-80 (1979)
[35] Monk, P., A finite element method for approximating the time-harmonic Maxwell equations, Numer. Math., 63, 1, 243-261 (1992) · Zbl 0757.65126
[36] Müller, C., Foundations of the Mathematical Theory of Electromagnetic Waves (1969), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0181.57203
[37] Nedelec, J.-C., Acoustic and Electromagnetic Equations (2001), Springer: Springer New York, NY · Zbl 0981.35002
[38] Olver, F. W.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W., NIST Handbook of Mathematical Functions (2010), Cambridge University Press: Cambridge University Press New York, NY, USA, ISBN 0521140633, 9780521140638 · Zbl 1198.00002
[39] O’Neil, M., Second-kind integral equations for the Laplace-Beltrami problem on surfaces in three dimensions, Adv. Comput. Math., 44, 1385-1409 (2018) · Zbl 1404.35121
[40] O’Neil, M.; Cerfon, A. J., An integral equation-based numerical solver for Taylor states in toroidal geometries, J. Comput. Phys., 359, 263-282 (2018) · Zbl 1383.76550
[41] Papas, C. H., Theory of Electromagnetic Wave Propagation (1988), Dover: Dover New York, NY · Zbl 0113.20602
[42] Sidi, A.; Israeli, M., Quadrature methods for periodic singular and weakly singular Fredholm integral equations, J. Sci. Comput., 3, 2, 201-231 (1988) · Zbl 0662.65122
[43] Sifuentes, J.; Gimbutas, Z.; Greengard, L., Randomized methods for rank-deficient linear systems, Electron. Trans. Numer. Anal., 44, 177-188 (2015) · Zbl 1312.65057
[44] Taskinen, M.; Yla-Oijala, P., Current and charge integral equation formulation, IEEE Trans. Antennas Propag., 54, 1, 58-67 (2006) · Zbl 1369.78309
[45] Trefethen, L. N., Spectral Methods in MATLAB (2000), SIAM: SIAM Philadelphia, PA · Zbl 0953.68643
[46] Vico, F.; Gimbutas, Z.; Greengard, L.; Ferrando-Bataller, M., Overcoming low-frequency breakdown of the magnetic field integral equation, IEEE Trans. Antennas Propag., 61, 3, 1285-1290 (2013) · Zbl 1370.78158
[47] Vico, F.; Ferrando, M.; Greengard, L.; Gimbutas, Z., The decoupled potential integral equation for time-harmonic electromagnetic scattering, Commun. Pure Appl. Math., 69, 771-812 (2016) · Zbl 1342.78030
[48] Viola, M. S., A new electric field integral equation for heterogeneous dielectric bodies of revolution, IEEE Trans. Microw. Theory Tech., 43, 230-233 (1995)
[49] Wright, G. B.; Javed, M.; Montanelli, H.; Trefethen, L. N., Extension of Chebfun to periodic functions, SIAM J. Sci. Comput., 37, C554-C573 (2015) · Zbl 1348.42003
[50] Yee, K., Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propag., 14, 302-307 (1966) · Zbl 1155.78304
[51] Young, P.; Hao, S.; Martinsson, P.-G., A high-order Nyström discretization scheme for boundary integral equations defined on rotationally symmetric surfaces, J. Comput. Phys., 231, 11, 4142-4159 (2012) · Zbl 1250.65146
[52] Youssef, N. N., Radar cross section of complex targets, Proc. IEEE, 77, 722-734 (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.