Abstract

We outline a generalized form of the method-of-moments technique. Integral equation formulations are developed for a diverse class of arbitrarily shaped three-dimensional scatterers. The scatterers may be totally or partially penetrable. Specific cases examined are scatterers with surfaces that are perfectly conducting, dielectric, resistive, or magnetically conducting or that satisfy the Leontovich (impedance) boundary condition. All the integral equation formulations are transformed into matrix equations expressed in terms of five general Galerkin (matrix) operators. This allows a unified numerical solution procedure to be implemented for the foregoing hierarchy of scatterers. The operators are general and apply to any arbitrarily shaped three-dimensional body. The operator calculus of the generalized approach is independent of geometry and basis or testing functions used in the method-of-moments approach. Representative numerical results for a number of scattering geometries modeled by triangularly faceted surfaces are given to illustrate the efficacy and the versatility of the present approach.

© 1994 Optical Society of America

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).
  2. N. A. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
    [CrossRef]
  3. P. W. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
    [CrossRef] [PubMed]
  4. D. S. Wang, P. W. Barber, “Scattering by inhomogeneous nonspherical objects,” Appl. Opt. 18, 1190–1197 (1979).
    [CrossRef] [PubMed]
  5. P. W. Barber, H. Massoudi, “Recent advances in light scattering calculations for nonspherical particles,” Aerosol Sci. Technol. 1, 303–315 (1982).
    [CrossRef]
  6. M. A. Morgan, ed., Finite Element and Finite Difference Methods in Electromagnetic Scattering (Elsevier, Amsterdam, 1990).
  7. A. Taflove, K. R. Umashankar, “Review of FDTD numerical modeling of electromagnetic wave scattering and radar cross-section,” Proc. IEEE 77, 682–699 (1989).
    [CrossRef]
  8. P. C. Waterman, “Scattering by dielectric obstacles,” Alta Freq. 38, 348–352 (1969).
  9. M. F. Iskander, S. C. Olson, R. E. Benner, D. Yoshida, “Optical scattering by metallic and carbon aerosols of high aspect ratio,” Appl. Opt. 25, 2514–2520 (1986).
    [CrossRef] [PubMed]
  10. A. Lakhtakia, M. F. Iskander, C. H. Durney, “An iterative extended boundary condition method for solving the absorption characteristics of lossy dielectric objects of large aspect ratio,” IEEE Trans. Microwave Theory Tech. MTT-31, 640–647 (1983).
    [CrossRef]
  11. A. C. Ludwig, “The generalized multipole method,” Comput. Phys. Commun. 68, 306–314 (1991).
    [CrossRef]
  12. R. C. Hansen, ed., Geometric Theory of Diffraction (Institute of Electrical and Electronics Engineers, New York, 1981).
  13. P. Y. Ufimtsev, “Approximate computation of the diffraction of plane electromagnetic waves at certain metal bodies,” Sov. Phys. Tech. Phys. 27, 1708–1718 (1957).
  14. P. Y. Ufimtsev, “Method of edge waves in physical theory of diffraction,” Foreign Tech. Div. Doc. ID FTD-HC-23-259-71, (U.S. Air Force System Command, Wright-Patterson Air Force Base, Ohio, 1971).
  15. N. A. Logan, “General research in diffraction theory,” Reps. LMSD-288087 and LMSD-288088, (Lockheed Aircraft Corporation, Burbank, Calif., 1959).
  16. R. F. Harrington, Field Computation by Moment Methods, 2nd ed. (Krieger, Malabar, Fla., 1982).
  17. E. K. Miller, L. N. Medgyesi-Mitschang, E. H. Newman, eds., Computational Electromagnetics Frequency Domain Method of Moments (Institute of Electrical and Electronics Engineers, New York, 1991).
  18. L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces (Pergamon, Oxford, 1964), pp. 586–587.
  19. L. V. Kantorovich, V. I. Krylov, Approximate Methods of Higher Analysis (Wiley, New York, 1964).
  20. B. G. Galerkin, Vestn. Inzh. Tekh. 19, 897 (1915).
  21. Ref. 16, p. 18.
  22. V. H. Rumsey, “The reaction concept in electromagnetic theory,” Phys. Rev. B 94, 1483–1491 (1954).
    [CrossRef]
  23. A. J. Poggio, E. K. Miller, “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed., 2nd ed. (Hemisphere, New York, 1987), pp. 159–264.
  24. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 8, pp. 424–481.
  25. J. R. Mautz, R. F. Harrington, “H-field, E-field, and combined field solutions for conducting bodies of revolutions,” Arch. Elek. Übertrag. 32, 157–164 (1978).
  26. P. L. Huddleston, L. N. Medgyesi-Mitschang, J. M. Putnam, “Combined field integral equation formulation for scattering from dielectrically coated conducting bodies,” IEEE Trans. Antennas Propag. AP-34,510–520 (1986).
    [CrossRef]
  27. J. M. Putnam, “General approach for treating boundary conditions on multi-region scatterer using the method of moments,” presented at the Seventh Annual Review of Progress in Applied Computational Electromagnetics, Monterey, Calif., March 1991.
  28. T. B. A. Senior, “Approximate boundary conditions,” IEEE Trans. Antennas Propag. AP-29, 826–829 (1981).
    [CrossRef]
  29. R. J. Garbacz, “Bistatic scattering from a class of lossy dielectric spheres with surface impedance boundary conditions,” Phys. Rev. A 133, 14–16 (1964).
  30. K. M. Mitzner, “An integral equation approach to scattering from a body of finite conductivity,” Radio Sci. 2, 1459–1470 (1967).
  31. D. S. Wang, “Limits and validity of the impedance boundary condition for penetrable surfaces,” IEEE Trans. Antennas Propag. AP-35, 453–457 (1987).
    [CrossRef]
  32. S. M. Rao, D. R. Wilton, A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. AP-30, 409–418 (1982).
    [CrossRef]
  33. J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. AP-20, 442–446 (1972).
    [CrossRef]
  34. L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from axially inhomogeneous bodies of revolution,” IEEE Trans. Antennas Propag. AP-32, 797–806 (1984).
    [CrossRef]
  35. J. M. Putnam, L. N. Medgyesi-Mitschang, M. B. Gedera, “carlos-3d™: Three-dimensional method of moments code,” McDonnell Douglas Aerospace Rep. (McDonnell Douglas Aerospace, Saint Louis, Mo., 1992), Vols. 1 and 2.
  36. J. M. Putnam, M. B. Gedera, “carlos-3d™: A general-purpose three-dimensional method-of-moments scattering code,” IEEE Antennas Propag. Mag. 35, 69–71 (1993).
    [CrossRef]
  37. D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. AP-32,276–281 (1984).
    [CrossRef]
  38. K. McInturff, P. S. Simon, “The Fourier transform of linearly varying functions with polygonal support,” IEEE Trans. Antennas Propag. 39, 1441–1443 (1991).
    [CrossRef]

1993 (1)

J. M. Putnam, M. B. Gedera, “carlos-3d™: A general-purpose three-dimensional method-of-moments scattering code,” IEEE Antennas Propag. Mag. 35, 69–71 (1993).
[CrossRef]

1991 (2)

K. McInturff, P. S. Simon, “The Fourier transform of linearly varying functions with polygonal support,” IEEE Trans. Antennas Propag. 39, 1441–1443 (1991).
[CrossRef]

A. C. Ludwig, “The generalized multipole method,” Comput. Phys. Commun. 68, 306–314 (1991).
[CrossRef]

1989 (1)

A. Taflove, K. R. Umashankar, “Review of FDTD numerical modeling of electromagnetic wave scattering and radar cross-section,” Proc. IEEE 77, 682–699 (1989).
[CrossRef]

1987 (1)

D. S. Wang, “Limits and validity of the impedance boundary condition for penetrable surfaces,” IEEE Trans. Antennas Propag. AP-35, 453–457 (1987).
[CrossRef]

1986 (2)

P. L. Huddleston, L. N. Medgyesi-Mitschang, J. M. Putnam, “Combined field integral equation formulation for scattering from dielectrically coated conducting bodies,” IEEE Trans. Antennas Propag. AP-34,510–520 (1986).
[CrossRef]

M. F. Iskander, S. C. Olson, R. E. Benner, D. Yoshida, “Optical scattering by metallic and carbon aerosols of high aspect ratio,” Appl. Opt. 25, 2514–2520 (1986).
[CrossRef] [PubMed]

1984 (2)

L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from axially inhomogeneous bodies of revolution,” IEEE Trans. Antennas Propag. AP-32, 797–806 (1984).
[CrossRef]

D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. AP-32,276–281 (1984).
[CrossRef]

1983 (1)

A. Lakhtakia, M. F. Iskander, C. H. Durney, “An iterative extended boundary condition method for solving the absorption characteristics of lossy dielectric objects of large aspect ratio,” IEEE Trans. Microwave Theory Tech. MTT-31, 640–647 (1983).
[CrossRef]

1982 (2)

S. M. Rao, D. R. Wilton, A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. AP-30, 409–418 (1982).
[CrossRef]

P. W. Barber, H. Massoudi, “Recent advances in light scattering calculations for nonspherical particles,” Aerosol Sci. Technol. 1, 303–315 (1982).
[CrossRef]

1981 (1)

T. B. A. Senior, “Approximate boundary conditions,” IEEE Trans. Antennas Propag. AP-29, 826–829 (1981).
[CrossRef]

1979 (1)

1978 (1)

J. R. Mautz, R. F. Harrington, “H-field, E-field, and combined field solutions for conducting bodies of revolutions,” Arch. Elek. Übertrag. 32, 157–164 (1978).

1975 (1)

1972 (1)

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. AP-20, 442–446 (1972).
[CrossRef]

1969 (1)

P. C. Waterman, “Scattering by dielectric obstacles,” Alta Freq. 38, 348–352 (1969).

1967 (1)

K. M. Mitzner, “An integral equation approach to scattering from a body of finite conductivity,” Radio Sci. 2, 1459–1470 (1967).

1965 (1)

N. A. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
[CrossRef]

1964 (1)

R. J. Garbacz, “Bistatic scattering from a class of lossy dielectric spheres with surface impedance boundary conditions,” Phys. Rev. A 133, 14–16 (1964).

1957 (1)

P. Y. Ufimtsev, “Approximate computation of the diffraction of plane electromagnetic waves at certain metal bodies,” Sov. Phys. Tech. Phys. 27, 1708–1718 (1957).

1954 (1)

V. H. Rumsey, “The reaction concept in electromagnetic theory,” Phys. Rev. B 94, 1483–1491 (1954).
[CrossRef]

1915 (1)

B. G. Galerkin, Vestn. Inzh. Tekh. 19, 897 (1915).

Akilov, G. P.

L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces (Pergamon, Oxford, 1964), pp. 586–587.

Al-Bundak, O. M.

D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. AP-32,276–281 (1984).
[CrossRef]

Barber, P. W.

Benner, R. E.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Butler, C. M.

D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. AP-32,276–281 (1984).
[CrossRef]

Durney, C. H.

A. Lakhtakia, M. F. Iskander, C. H. Durney, “An iterative extended boundary condition method for solving the absorption characteristics of lossy dielectric objects of large aspect ratio,” IEEE Trans. Microwave Theory Tech. MTT-31, 640–647 (1983).
[CrossRef]

Galerkin, B. G.

B. G. Galerkin, Vestn. Inzh. Tekh. 19, 897 (1915).

Garbacz, R. J.

R. J. Garbacz, “Bistatic scattering from a class of lossy dielectric spheres with surface impedance boundary conditions,” Phys. Rev. A 133, 14–16 (1964).

Gedera, M. B.

J. M. Putnam, M. B. Gedera, “carlos-3d™: A general-purpose three-dimensional method-of-moments scattering code,” IEEE Antennas Propag. Mag. 35, 69–71 (1993).
[CrossRef]

J. M. Putnam, L. N. Medgyesi-Mitschang, M. B. Gedera, “carlos-3d™: Three-dimensional method of moments code,” McDonnell Douglas Aerospace Rep. (McDonnell Douglas Aerospace, Saint Louis, Mo., 1992), Vols. 1 and 2.

Glisson, A. W.

D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. AP-32,276–281 (1984).
[CrossRef]

S. M. Rao, D. R. Wilton, A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. AP-30, 409–418 (1982).
[CrossRef]

Harrington, R. F.

J. R. Mautz, R. F. Harrington, “H-field, E-field, and combined field solutions for conducting bodies of revolutions,” Arch. Elek. Übertrag. 32, 157–164 (1978).

R. F. Harrington, Field Computation by Moment Methods, 2nd ed. (Krieger, Malabar, Fla., 1982).

Huddleston, P. L.

P. L. Huddleston, L. N. Medgyesi-Mitschang, J. M. Putnam, “Combined field integral equation formulation for scattering from dielectrically coated conducting bodies,” IEEE Trans. Antennas Propag. AP-34,510–520 (1986).
[CrossRef]

Iskander, M. F.

M. F. Iskander, S. C. Olson, R. E. Benner, D. Yoshida, “Optical scattering by metallic and carbon aerosols of high aspect ratio,” Appl. Opt. 25, 2514–2520 (1986).
[CrossRef] [PubMed]

A. Lakhtakia, M. F. Iskander, C. H. Durney, “An iterative extended boundary condition method for solving the absorption characteristics of lossy dielectric objects of large aspect ratio,” IEEE Trans. Microwave Theory Tech. MTT-31, 640–647 (1983).
[CrossRef]

Kantorovich, L. V.

L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces (Pergamon, Oxford, 1964), pp. 586–587.

L. V. Kantorovich, V. I. Krylov, Approximate Methods of Higher Analysis (Wiley, New York, 1964).

Krylov, V. I.

L. V. Kantorovich, V. I. Krylov, Approximate Methods of Higher Analysis (Wiley, New York, 1964).

Lakhtakia, A.

A. Lakhtakia, M. F. Iskander, C. H. Durney, “An iterative extended boundary condition method for solving the absorption characteristics of lossy dielectric objects of large aspect ratio,” IEEE Trans. Microwave Theory Tech. MTT-31, 640–647 (1983).
[CrossRef]

Logan, N. A.

N. A. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
[CrossRef]

N. A. Logan, “General research in diffraction theory,” Reps. LMSD-288087 and LMSD-288088, (Lockheed Aircraft Corporation, Burbank, Calif., 1959).

Ludwig, A. C.

A. C. Ludwig, “The generalized multipole method,” Comput. Phys. Commun. 68, 306–314 (1991).
[CrossRef]

Massoudi, H.

P. W. Barber, H. Massoudi, “Recent advances in light scattering calculations for nonspherical particles,” Aerosol Sci. Technol. 1, 303–315 (1982).
[CrossRef]

Mautz, J. R.

J. R. Mautz, R. F. Harrington, “H-field, E-field, and combined field solutions for conducting bodies of revolutions,” Arch. Elek. Übertrag. 32, 157–164 (1978).

McInturff, K.

K. McInturff, P. S. Simon, “The Fourier transform of linearly varying functions with polygonal support,” IEEE Trans. Antennas Propag. 39, 1441–1443 (1991).
[CrossRef]

Medgyesi-Mitschang, L. N.

P. L. Huddleston, L. N. Medgyesi-Mitschang, J. M. Putnam, “Combined field integral equation formulation for scattering from dielectrically coated conducting bodies,” IEEE Trans. Antennas Propag. AP-34,510–520 (1986).
[CrossRef]

L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from axially inhomogeneous bodies of revolution,” IEEE Trans. Antennas Propag. AP-32, 797–806 (1984).
[CrossRef]

J. M. Putnam, L. N. Medgyesi-Mitschang, M. B. Gedera, “carlos-3d™: Three-dimensional method of moments code,” McDonnell Douglas Aerospace Rep. (McDonnell Douglas Aerospace, Saint Louis, Mo., 1992), Vols. 1 and 2.

Meixner, J.

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. AP-20, 442–446 (1972).
[CrossRef]

Miller, E. K.

A. J. Poggio, E. K. Miller, “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed., 2nd ed. (Hemisphere, New York, 1987), pp. 159–264.

Mitzner, K. M.

K. M. Mitzner, “An integral equation approach to scattering from a body of finite conductivity,” Radio Sci. 2, 1459–1470 (1967).

Olson, S. C.

Poggio, A. J.

A. J. Poggio, E. K. Miller, “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed., 2nd ed. (Hemisphere, New York, 1987), pp. 159–264.

Putnam, J. M.

J. M. Putnam, M. B. Gedera, “carlos-3d™: A general-purpose three-dimensional method-of-moments scattering code,” IEEE Antennas Propag. Mag. 35, 69–71 (1993).
[CrossRef]

P. L. Huddleston, L. N. Medgyesi-Mitschang, J. M. Putnam, “Combined field integral equation formulation for scattering from dielectrically coated conducting bodies,” IEEE Trans. Antennas Propag. AP-34,510–520 (1986).
[CrossRef]

L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from axially inhomogeneous bodies of revolution,” IEEE Trans. Antennas Propag. AP-32, 797–806 (1984).
[CrossRef]

J. M. Putnam, L. N. Medgyesi-Mitschang, M. B. Gedera, “carlos-3d™: Three-dimensional method of moments code,” McDonnell Douglas Aerospace Rep. (McDonnell Douglas Aerospace, Saint Louis, Mo., 1992), Vols. 1 and 2.

J. M. Putnam, “General approach for treating boundary conditions on multi-region scatterer using the method of moments,” presented at the Seventh Annual Review of Progress in Applied Computational Electromagnetics, Monterey, Calif., March 1991.

Rao, S. M.

D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. AP-32,276–281 (1984).
[CrossRef]

S. M. Rao, D. R. Wilton, A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. AP-30, 409–418 (1982).
[CrossRef]

Rumsey, V. H.

V. H. Rumsey, “The reaction concept in electromagnetic theory,” Phys. Rev. B 94, 1483–1491 (1954).
[CrossRef]

Schaubert, D. H.

D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. AP-32,276–281 (1984).
[CrossRef]

Senior, T. B. A.

T. B. A. Senior, “Approximate boundary conditions,” IEEE Trans. Antennas Propag. AP-29, 826–829 (1981).
[CrossRef]

Simon, P. S.

K. McInturff, P. S. Simon, “The Fourier transform of linearly varying functions with polygonal support,” IEEE Trans. Antennas Propag. 39, 1441–1443 (1991).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 8, pp. 424–481.

Taflove, A.

A. Taflove, K. R. Umashankar, “Review of FDTD numerical modeling of electromagnetic wave scattering and radar cross-section,” Proc. IEEE 77, 682–699 (1989).
[CrossRef]

Ufimtsev, P. Y.

P. Y. Ufimtsev, “Approximate computation of the diffraction of plane electromagnetic waves at certain metal bodies,” Sov. Phys. Tech. Phys. 27, 1708–1718 (1957).

P. Y. Ufimtsev, “Method of edge waves in physical theory of diffraction,” Foreign Tech. Div. Doc. ID FTD-HC-23-259-71, (U.S. Air Force System Command, Wright-Patterson Air Force Base, Ohio, 1971).

Umashankar, K. R.

A. Taflove, K. R. Umashankar, “Review of FDTD numerical modeling of electromagnetic wave scattering and radar cross-section,” Proc. IEEE 77, 682–699 (1989).
[CrossRef]

Wang, D. S.

D. S. Wang, “Limits and validity of the impedance boundary condition for penetrable surfaces,” IEEE Trans. Antennas Propag. AP-35, 453–457 (1987).
[CrossRef]

D. S. Wang, P. W. Barber, “Scattering by inhomogeneous nonspherical objects,” Appl. Opt. 18, 1190–1197 (1979).
[CrossRef] [PubMed]

Waterman, P. C.

P. C. Waterman, “Scattering by dielectric obstacles,” Alta Freq. 38, 348–352 (1969).

Wilton, D. R.

D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. AP-32,276–281 (1984).
[CrossRef]

S. M. Rao, D. R. Wilton, A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. AP-30, 409–418 (1982).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Yeh, C.

Yoshida, D.

Aerosol Sci. Technol. (1)

P. W. Barber, H. Massoudi, “Recent advances in light scattering calculations for nonspherical particles,” Aerosol Sci. Technol. 1, 303–315 (1982).
[CrossRef]

Alta Freq. (1)

P. C. Waterman, “Scattering by dielectric obstacles,” Alta Freq. 38, 348–352 (1969).

Appl. Opt. (3)

Arch. Elek. Übertrag. (1)

J. R. Mautz, R. F. Harrington, “H-field, E-field, and combined field solutions for conducting bodies of revolutions,” Arch. Elek. Übertrag. 32, 157–164 (1978).

Comput. Phys. Commun. (1)

A. C. Ludwig, “The generalized multipole method,” Comput. Phys. Commun. 68, 306–314 (1991).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

J. M. Putnam, M. B. Gedera, “carlos-3d™: A general-purpose three-dimensional method-of-moments scattering code,” IEEE Antennas Propag. Mag. 35, 69–71 (1993).
[CrossRef]

IEEE Trans. Antennas Propag. (8)

D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. AP-32,276–281 (1984).
[CrossRef]

K. McInturff, P. S. Simon, “The Fourier transform of linearly varying functions with polygonal support,” IEEE Trans. Antennas Propag. 39, 1441–1443 (1991).
[CrossRef]

T. B. A. Senior, “Approximate boundary conditions,” IEEE Trans. Antennas Propag. AP-29, 826–829 (1981).
[CrossRef]

P. L. Huddleston, L. N. Medgyesi-Mitschang, J. M. Putnam, “Combined field integral equation formulation for scattering from dielectrically coated conducting bodies,” IEEE Trans. Antennas Propag. AP-34,510–520 (1986).
[CrossRef]

D. S. Wang, “Limits and validity of the impedance boundary condition for penetrable surfaces,” IEEE Trans. Antennas Propag. AP-35, 453–457 (1987).
[CrossRef]

S. M. Rao, D. R. Wilton, A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. AP-30, 409–418 (1982).
[CrossRef]

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. AP-20, 442–446 (1972).
[CrossRef]

L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from axially inhomogeneous bodies of revolution,” IEEE Trans. Antennas Propag. AP-32, 797–806 (1984).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. Lakhtakia, M. F. Iskander, C. H. Durney, “An iterative extended boundary condition method for solving the absorption characteristics of lossy dielectric objects of large aspect ratio,” IEEE Trans. Microwave Theory Tech. MTT-31, 640–647 (1983).
[CrossRef]

Phys. Rev. A (1)

R. J. Garbacz, “Bistatic scattering from a class of lossy dielectric spheres with surface impedance boundary conditions,” Phys. Rev. A 133, 14–16 (1964).

Phys. Rev. B (1)

V. H. Rumsey, “The reaction concept in electromagnetic theory,” Phys. Rev. B 94, 1483–1491 (1954).
[CrossRef]

Proc. IEEE (2)

A. Taflove, K. R. Umashankar, “Review of FDTD numerical modeling of electromagnetic wave scattering and radar cross-section,” Proc. IEEE 77, 682–699 (1989).
[CrossRef]

N. A. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
[CrossRef]

Radio Sci. (1)

K. M. Mitzner, “An integral equation approach to scattering from a body of finite conductivity,” Radio Sci. 2, 1459–1470 (1967).

Sov. Phys. Tech. Phys. (1)

P. Y. Ufimtsev, “Approximate computation of the diffraction of plane electromagnetic waves at certain metal bodies,” Sov. Phys. Tech. Phys. 27, 1708–1718 (1957).

Vestn. Inzh. Tekh. (1)

B. G. Galerkin, Vestn. Inzh. Tekh. 19, 897 (1915).

Other (14)

Ref. 16, p. 18.

R. C. Hansen, ed., Geometric Theory of Diffraction (Institute of Electrical and Electronics Engineers, New York, 1981).

P. Y. Ufimtsev, “Method of edge waves in physical theory of diffraction,” Foreign Tech. Div. Doc. ID FTD-HC-23-259-71, (U.S. Air Force System Command, Wright-Patterson Air Force Base, Ohio, 1971).

N. A. Logan, “General research in diffraction theory,” Reps. LMSD-288087 and LMSD-288088, (Lockheed Aircraft Corporation, Burbank, Calif., 1959).

R. F. Harrington, Field Computation by Moment Methods, 2nd ed. (Krieger, Malabar, Fla., 1982).

E. K. Miller, L. N. Medgyesi-Mitschang, E. H. Newman, eds., Computational Electromagnetics Frequency Domain Method of Moments (Institute of Electrical and Electronics Engineers, New York, 1991).

L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces (Pergamon, Oxford, 1964), pp. 586–587.

L. V. Kantorovich, V. I. Krylov, Approximate Methods of Higher Analysis (Wiley, New York, 1964).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

M. A. Morgan, ed., Finite Element and Finite Difference Methods in Electromagnetic Scattering (Elsevier, Amsterdam, 1990).

A. J. Poggio, E. K. Miller, “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed., 2nd ed. (Hemisphere, New York, 1987), pp. 159–264.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 8, pp. 424–481.

J. M. Putnam, “General approach for treating boundary conditions on multi-region scatterer using the method of moments,” presented at the Seventh Annual Review of Progress in Applied Computational Electromagnetics, Monterey, Calif., March 1991.

J. M. Putnam, L. N. Medgyesi-Mitschang, M. B. Gedera, “carlos-3d™: Three-dimensional method of moments code,” McDonnell Douglas Aerospace Rep. (McDonnell Douglas Aerospace, Saint Louis, Mo., 1992), Vols. 1 and 2.

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Figures (13)

Fig. 1
Fig. 1

Classes of penetrable or partially penetrable arbitrary 3-D scatterers analyzed with the present method. Objects are shown generically in two dimensions for clarity: (a) connected discrete regions, (b) multiple homogeneous coatings, (c) generic combination of cases (a) and (b).

Fig. 2
Fig. 2

Generic penetrable 3-D scatterer: (a) original problem, (b) equivalent problem exterior to S, (c) equivalent problem interior to S.

Fig. 3
Fig. 3

Faceted 3-D scatterer with details of patch coordinates used in rooftop expansion functions.

Fig. 4
Fig. 4

Some representative surface-junction conditions implemented in the 3-D patch formulation.

Fig. 5
Fig. 5

Bistatic scattering cross section of spherical penetrable (εr = 1.75 − j0.3) scatterers. Comparison of Mie and MM solutions: (a) dielectric sphere, a = 0.2λ; (b) hollow spherical shell, a = 0.2λ, b = 0.18λ; (c) coated PEC sphere, a = 0.2λ, b = 0.18λ.

Fig. 6
Fig. 6

Monostatic scattering cross section of a partially coated (εr = 2.6) right circular cylinder (l = 1.22λ, a = 0.455λ): (a) θθ polarization, (b) ϕϕ polarization. MM BOR solution versus 3-D patch MM formulation.

Fig. 7
Fig. 7

Bistatic scattering cross section of penetrable (εr = 1.75 − j0.3) oblate spheroidal scatterers (a = 0.2λ, b = 0.1λ): (a) dielectric spheroid illustrating sensitivity of MM solution with facetization, (b) penetrable spheroidal shell with spherical void (r = 0.05λ), (c) penetrable spheroid with spherical conducting core (r = 0.05λ).

Fig. 8
Fig. 8

Bistatic scattering cross section of penetrable (εr = 1.75 − j0.3) prolate spheroidal scatterers (a = 0.1λ, b = 0.2λ): (a) dielectric spheroid illustrating sensitivity of MM solution with facetization, (b) penetrable spheroidal shell with spherical void (r = 0.05λ), (c) penetrable spheroid with spherical conducting core (r = 0.05λ).

Fig. 9
Fig. 9

Bistatic scattering cross section of penetrable (εr = 1.75 − j0.3) oblate spheroid (a = 0.2λ, b = 0.1λ): (a) θθ polarization, (b) ϕϕ polarization. Spherical conducting core (r = 0.05λ) centered and offset by 2r = 0.1λ.

Fig. 10
Fig. 10

Bistatic scattering cross section of penetrable (εr = 1.75 − j0.3) prolate spheroid (a = 0.1λ, b = 0.2λ): (a) θθ polarization, (b) ϕϕ polarization. Spherical conducting core (r = 0.05λ) centered and offset by 2r = 0.1λ.

Fig. 11
Fig. 11

Bistatic scattering cross section of multiple agglomerated conducting spheres (r = 0.4λ): (a) θθ polarization, (b) ϕϕ polarization. MM solutions incorporating radiative coupling.

Fig. 12
Fig. 12

Bistatic scattering cross section of multiple agglomerated penetrable spheres (εr = 1.75 − j0.3, r = 0.4λ): (a) θθ polarization; (b) ϕϕ polarization. MM solutions incorporating radiative coupling.

Fig. 13
Fig. 13

Bistatic scattering cross section of multiple agglomerated penetrable spheres (ε1 = 1.75 − j0.3, ε2 = 2.25 − j0.5, r = 0.4λ): (a) θθ polarization; (b) ϕϕ polarization. MM solutions incorporating radiative coupling.

Tables (1)

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Table 1 Transformation Matrices for Generalized Galerkin Method

Equations (113)

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× E = j ω μ H M ,
× H = j ω ɛ E + J ,
· E = ρ / ɛ ,
· H = m / μ ,
× × E k 2 E = j ω μ J × M
× × H k 2 H = j ω ɛ M + × J ,
θ 1 ( r ) E 1 ( r ) = E inc ( r ) S { j ω μ 1 ( n × H 1 ) Φ 1 ( n × E 1 ) × Φ 1 ( n · E 1 ) Φ 1 } d s ,
θ 1 ( r ) H 1 ( r ) = H inc ( r ) + S { j ω ɛ 1 ( n × E 1 ) Φ 1 + ( n × H 1 ) × Φ 1 + ( n · H 1 ) Φ 1 } d s ,
θ 1 ( r ) = { 1 for r R 1 1 / 2 for r R 1 , 0 otherwise
Φ 1 ( r r ) exp ( j k 1 | r r | ) 4 π | r r | ,
J i = n ̂ i × H i | R i ,
M i = E i × n ̂ i | R i
n ̂ · E i = 1 j ω ɛ · ( n ̂ × H i ) = 1 j ω ɛ · J i ,
n ̂ · H i = 1 j ω μ · ( n ̂ × E i ) = 1 j ω μ · M i .
θ 1 ( r ) E 1 ( r ) = E inc ( r ) L 1 J + ( r ) + K 1 M + ( r ) ,
θ 1 ( r ) H 1 ( r ) = H inc ( r ) K 1 J + ( r ) 1 η 1 2 L 1 M + ( r ) .
θ 2 ( r ) E 2 ( r ) = L 2 J ( r ) + K 2 M ( r ) ,
θ 2 ( r ) H 2 ( r ) = K 2 J ( r ) 1 η 2 2 L 2 M ( r ) .
L i X ( r ) = R i [ j ω μ i X ( r ) + j ω ɛ i · X ( r ) ] Φ i ( r r ) d s ,
K i X ( r ) = R i X ( r ) × Φ i ( r r ) d s ,
n ̂ × ( E 1 E 2 ) | S = 0 ,
n ̂ × ( H 1 H 2 ) | S = 0 ,
E inc ( r ) | tan = ( L 1 + L 2 ) J ( r ) | tan ( K 1 + K 2 ) M ( r ) | tan ,
H inc ( r ) | tan = ( K 1 + K 2 ) J ( r ) | tan + ( 1 η 1 2 L 1 + 1 η 2 2 L 2 ) M ( r ) | tan .
J ± ( r ) = n J n ± f n ( r ) , r S ,
M ± ( r ) = η 0 n M n ± f n ( r ) , r S ,
A , B S = S A * · B d s ,
[ 1 + 2 K 1 K 2 K 1 + K 2 ɛ 1 μ 1 1 + ɛ 2 μ 2 2 ] [ J + M + ] = [ ] ,
E ± ( r ) = η 0 n E n ± f n ( r ) , r S ,
H ± ( r ) = n H n ± f n ( r ) , r S ,
I + Ĩ = V ,
I = [ I 1 I 2 ] ,
( A + B ) I 1 = V .
( A + B ) I 1 = V .
1 J + = .
[ K x 1 + K ] J + = x ,
[ α 1 + ( 1 α ) ( K x 1 + K ) ] J + = α + ( 1 α ) x ,
E tan + = R s η 0 ( J + + J ) E + = R s ( J + + J ) ,
E tan = E tan + E = R s ( J + + J ) ,
J + = n ̂ × H + H + X J + ,
H tan = H tan + 1 R s η 0 M + H X J + 1 R s M + .
[ 1 + R s K K 1 R s K K 1 ɛ 2 μ 2 2 + ɛ 1 μ 1 1 + 1 R s K K 2 R s K K 2 2 + R s K ] [ J + M + J ] = [ 0 ] .
[ 1 + 2 R s K ¯ ] J = ,
n ̂ + ( H + H ) = 1 R s η 0 E tan = 1 2 R s η 0 ( E + + E ) tan ,
n ̂ + × ( E + E ) = η 0 G s H tan = η 0 2 G s ( H + + H ) tan .
( J + + J ) = 1 2 R s η 0 ( E + + E ) tan ,
( M + + M ) = η 0 2 G s ( H + + H ) tan .
( H + H ) tan = 1 2 R s η 0 ( M + M ) ,
( E + E ) tan = 1 2 G s ( J + J ) .
E + | tan = ( R s + 1 4 G s ) η 0 J + + ( R s 1 4 G s ) η 0 J ,
E | tan = ( R s 1 4 G s ) η 0 J + + ( R s + 1 4 G s ) η 0 J ,
H + | tan = ( G s + 1 4 R s ) 1 η 0 M + + ( G s 1 4 G s ) 1 η 0 M ,
H | tan = ( G s 1 4 R s ) 1 η 0 M + + ( G s + 1 4 R s ) 1 η 0 M .
[ 1 + ( R s + 1 4 G s ) K K 1 ( R s 1 4 G s ) K 0 K 1 ɛ 1 μ 1 1 + ( G s 1 4 R s ) K 0 ( G s 1 4 R s ) K ( R s 1 4 G s ) K 0 2 + ( R s + 1 4 G s ) K K 2 0 ( G s 1 4 R s ) K K 2 ɛ 2 μ 2 2 + ( G s + 1 4 R s ) K ] [ J + M + J M ] = [ 0 0 ] .
[ 1 + 2 R s K K 1 K 1 ɛ 1 μ 1 1 + 2 G s K ] [ J M ] = [ ] .
J ( r ) n = 1 N i + N j I n f n ( r ) ,
M ( r ) η 0 n = 1 N i + N j K n f n ( r ) ,
f n ( r ) = { l n 2 A n + ( r ν n + ) r T n + l n A n ( r ν n ) , r T n 0 otherwise .
( S 2 , S 1 ; R ) = t ( S 1 , S 2 ; R ) ,
K ( S 2 , S 1 ; R ) = K t ( S 1 , S 2 ; R ) ,
( S 2 , S 1 i ; R ) = t ( S 1 , S 2 i ; R ) ,
K ( S 2 , S 1 i ; R ) = K t ( S 1 , S 2 i ; R ) ,
l l = r r , l r = r l , l s = r s , s l = s r , K l l = K r r K l r = K r l , K l s , = K r s ,
K s l = K s r .
A
B
[ 1 0 0 0 ]
[ 0 X 0 0 ]
[ J + M + ]
[ 1 0 1 0 0 1 0 1 ]
[ 0 X 0 X X 0 X 0 ]
[ J + M + J ]
[ 1 0 0 0 1 0 0 0 1 0 1 0 ]
[ R s 0 R s X 0 0 R s 0 R s X 1 / R s 0 ]
[ J + M + J M ]
[ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ]
[ R s + 1 4 G s 0 R s 1 4 G s 0 0 G s + 1 4 R s 0 G s 1 4 R s R s 1 4 G s 0 R s + 1 4 G s 0 0 G s 1 4 R s 0 G s + 1 4 R s ]
[ J + M + ]
[ 1 0 0 1 0 0 0 0 ]
[ η s 0 0 1 / η s 0 0 0 0 ]
k l ( S 1 , S 2 ; R ) = f k , L ( f l ) S 1 .
k l ( ) = j S k d s S l d s { ω μ f k ( r ) · f l ( r ) Φ ( r r ) + 1 ω ɛ [ f k ( r ) · Φ ( r r ) ] · f l ( r ) }
k l ( ) = j S k d s S l d s { ω μ f k ( r ) · f l ( r ) 1 ω ɛ [ · f k ( r ) ] [ · f l ( r ) ] } Φ ( r r ) .
· f n ( r ) = { l n A n + r T n + l n A n , r T n 0 otherwise .
k l ( ) = j l k l l 4 p = ± q = ± p q A k p A l q T k p d s T l q d s × [ ω μ ( r ν k p ) · ( r ν l q ) 4 ω ɛ ] Φ ( r r ) .
K k l ( S 1 , S 2 ; R ) = η 0 f k , K ( f l ) S 1 .
K k l ( ) = η 0 S k d s S l d s f k ( r ) · f l ( r ) × Φ ( r r ) ,
Φ ( r r ) = ( 1 + j k | r r | ) 4 π | r r | 3 ( r r ) exp ( j k | r r | ) , = ( r r ) Ψ ( r r ) .
K k l ( ) = η 0 S k d s S k d s f k ( r ) · f l ( r ) × ( r r ) Ψ ( r r ) = η 0 l k l l 4 p = ± q = ± p q A k p A l q T k p d s T l q d s × ( r ν k p ) · ( r ν l q ) × ( r r ) Ψ ( r r ) .
( r r ) · ( ν k p × ν l q ) + ( r × r ) · ( ν k p ν l q ) .
x k l ( S 1 , S 2 ; R ) = f k , n ̂ × L ( f l ) S 1 = n ̂ × f k , L ( f l ) S 1 .
x k l ( ) = j S k d s S l d s { ω μ n ̂ × f k ( r ) · f l ( r ) Φ ( r r ) + 1 ω ɛ [ n ̂ × f k ( r ) · Φ ( r r ) ] · f l ( r ) } .
x k l ( ) = j l k l l 4 p = ± q = ± p q A k p A l q T k p d s T l q d s × { ω μ n ̂ × ( r ν k p ) · ( r ν l q ) Φ ( r r ) + 2 ω ɛ [ n ̂ × ( r ν k p ) · Φ ( r r ) ] } .
T d s n ̂ × ( r ν ) · Φ = T d t t ̂ · ( r ν ) Φ
x k l ( ) = j l k l l 4 p = ± q = ± p q A k p A l q [ ω μ T k p d s T l q d s n ̂ × ( r ν k p ) · ( r ν l q ) Φ ( r r ) 2 ω ɛ T k p d t T l q d s t ̂ · ( r ν k p ) Φ ( r r ) ] .
K x k l ( S 1 , S 2 ; R ) = η 0 f k , n ̂ × K ( f l ) S 1 = η 0 n ̂ × f k , K ( f l ) S 1 .
K x k l ( ) = η 0 S k d s S l d s [ n ̂ × f k ( r ) ] · f l ( r ) × Φ ( r r ) = η 0 S k d s S l d s [ n ̂ × f k ( r ) ] · f l ( r ) × ( r r ) Ψ ( r r ) = η 0 l k l l 4 p = ± q = ± p q A k p A l q T k p d s T l q d s × [ n ̂ × ( r ν k p ) ] · ( r ν l q ) × ( r r ) Ψ ( r r ) .
[ n ̂ × ( r ν k p ) ] · ( r ν l p ) × ( r r ) = n ̂ · r × ( r × r ) [ ( n ̂ × ν k p ) × ν l q ] · ( r r ) ( n ̂ × ν k p ) · ( r × r ) ν l q · [ ( n ̂ × r ) × ( r r ) ] .
K k l ( S ) = η 0 2 f k , f l S .
K k l ( ) = η 0 2 S k S l d s f k ( r ) · f l ( r ) = η 0 l k l l 8 p = ± q = ± p q A k p A l q T k p T l q d s ( r ν k p ) · ( r ν l q ) .
R S K k l ( ) = η 0 2 S k S l d s R s ( r ) f k ( r ) · f l ( r ) = η 0 l k l l 8 p = ± q = ± p q A k p A l q R k p T k p T l q d s × ( r ν k p ) · ( r ν l q ) ,
k ( S ; k , α ) = f k , E α inc S = f k ( r ) , α ̂ exp ( j k · r ) S ,
k ( S ; k , α ) = f k , η 0 H α inc S = f k ( r ) , β ̂ exp ( j k · r ) S ,
( S ; k , θ ) = ( S ; k , ϕ ) ,
( S ; k , ϕ ) = ( S ; k , θ ) .
k ( S ; k , θ ) = l k 2 p = ± p A k p × α ̂ · [ T k p d s ( r ν k p ) exp ( j k · r ) ] ,
θ ̂ = cos ( θ ) [ cos ( ϕ ) x ̂ + sin ( θ ) y ̂ ] sin ( θ ) z ̂ ,
ϕ ̂ = sin ( ϕ ) x ̂ + cos ( ϕ ) y ̂ ,
k = k [ sin ( θ ) cos ( ϕ ) x ̂ + sin ( θ ) sin ( ϕ ) y ̂ + cos ( θ ) z ̂ ] ,
exp ( j k · r ) = exp ( j k { sin ( θ ) [ x cos ( ϕ ) + y sin ( ϕ ) ] + z cos ( θ ) } ) .
x ( S ; k , α ) = f k , n ̂ × E α inc S ,
x ( S ; k , α ) = f k , η 0 n ̂ × H α inc S .
x ( S ; k , α ) = l k 2 p = ± p A k p ( n ̂ × α ̂ ) · [ T k p d s ( r ν k p ) exp ( j k · r ) ] .

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