Abstract

Most of the surface integral equation (SIE) formulations for composite conductor and/or penetrable objects suffer from balancing problems mainly because of the very different scales of the equivalent electric and magnetic currents. Consequently, the impedance matrix usually has high- or ill-condition number due to the imbalance between the different blocks. Using an efficient left and right preconditioner the elements of the impedance matrix are balanced. The proposed approach improves the matrix balance without modifying the underlying SIE formulation, which can be selected solely in terms of accuracy. The numerical complexity of this preconditioner is O(N) with N the number of unknowns, and it can be easily included on any existing SIE code implementation.

© 2012 OSA

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  23. Ö. Ergül and L. Gürel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antenn. Propag. 57(1), 176–187 (2009).
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  29. R. J. Adams, “Physical and analytical properties of a stabilized electric field integral equation,” IEEE Trans. Antenn. Propag. 52(2), 362–372 (2004).
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  30. F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative calderon preconditioner for the electric field integral equation,” IEEE Trans. Antenn. Propag. 56(8), 2398–2412 (2008).
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    [CrossRef]
  33. X.-Q. Sheng, J.-M. Jin, J. Song, W. C. Chew, and C.-C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antenn. Propag. 46(11), 1718–1726 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  38. S. Chen, J.-S. Zhao, and W. C. Chew, “Analyzing low-frequency electromagnetic scattering from a composite object,” IEEE Trans. Geosci. Rem. Sens. 40(2), 426–433 (2002).
    [CrossRef]
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2012 (2)

2011 (3)

J. M. Taboada, J. Rivero, F. Obelleiro, M. G. Araújo, and L. Landesa, “Method-of-moments formulation for the analysis of plasmonic nano-optical antennas,” J. Opt. Soc. Am. A 28(7), 1341–1348 (2011).
[CrossRef] [PubMed]

M. G. Araújo, J. M. Taboada, J. Rivero, and F. Obelleiro, “Comparison of surface integral equations for left-handed materials,” Prog. Electromagn. Res. 118, 425–440 (2011).
[CrossRef]

J. M. Bértolo, M. G. Araújo, J. M. Taboada, L. Landesa, F. Obelleiro, and J. L. Rodríguez, “Extended near field preconditioner for the analysis of large problems using the Nested-FMM-FFT algorithm,” Microw. Opt. Technol. Lett. 53(2), 430–433 (2011).
[CrossRef]

2010 (3)

2009 (1)

Ö. Ergül and L. Gürel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antenn. Propag. 57(1), 176–187 (2009).
[CrossRef]

2008 (1)

F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative calderon preconditioner for the electric field integral equation,” IEEE Trans. Antenn. Propag. 56(8), 2398–2412 (2008).
[CrossRef]

2007 (2)

P. Ylä-Oijala and M. Taskinen, “Improving conditioning of electromagnetic surface integral equations using normalized field quantities,” IEEE Trans. Antenn. Propag. 55(1), 178–185 (2007).
[CrossRef]

Y. A. Liu and W. C. Chew, “Stability of surface integral equation for left-handed materials,” IET Microwaves Antenn. Propag. 1(1), 84–89 (2007).
[CrossRef]

2005 (7)

P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40(6), RS6002 (2005).
[CrossRef]

P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antenn. Propag. 53(3), 1168–1173 (2005).
[CrossRef]

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general integral equation method for general composite metallic and dielectric structures with junctions,” Prog. Electromagn. Res. 52, 81–108 (2005).
[CrossRef]

D. L. Smith, L. N. Medgyesi-Mitschang, and D. W. Forester, “Surface integral equation formulations for left-handed materials,” Prog. Electromagn. Res. 51, 27–48 (2005).
[CrossRef]

T. W. Lloyd, J. M. Song, and M. Yang, “Numerical study of surface integral formulations for low-contrast objects,” IEEE Antennas Wirel. Propag. Lett. 4(1), 482–485 (2005).
[CrossRef]

A. Zhu, S. D. Gedney, and J. L. Visher, “A study of combined field formulations for material scattering for a locally corrected Nyström discretization,” IEEE Trans. Antenn. Propag. 53(12), 4111–4120 (2005).
[CrossRef]

P. Ylä-Oijala and M. Taskinen, “Well-conditioned Müller formulation for electromagnetic scattering by dielectric objects,” IEEE Trans. Antenn. Propag. 53(10), 3316–3323 (2005).
[CrossRef]

2004 (1)

R. J. Adams, “Physical and analytical properties of a stabilized electric field integral equation,” IEEE Trans. Antenn. Propag. 52(2), 362–372 (2004).
[CrossRef]

2003 (2)

J. Lee, J. Zhang, and C.-C. Lu, “Incomplete LU preconditioner for large scale dense complex linear systems from electromagnetic wave scattering problems,” J. Comput. Phys. 185(1), 158–175 (2003).
[CrossRef]

K. C. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antenn. Propag. 51(10), 2814–2821 (2003).
[CrossRef]

2002 (1)

S. Chen, J.-S. Zhao, and W. C. Chew, “Analyzing low-frequency electromagnetic scattering from a composite object,” IEEE Trans. Geosci. Rem. Sens. 40(2), 426–433 (2002).
[CrossRef]

2000 (1)

K. Sertel and J. L. Volakis, “Incomplete LU preconditioner for FMM implementation,” Microw. Opt. Technol. Lett. 26(4), 265–267 (2000).
[CrossRef]

1999 (1)

M. S. Yeung, “Single integral equation for electromagnetic scattering by three-dimensional homogeneous dielectric objects,” IEEE Trans. Antenn. Propag. 47(10), 1615–1622 (1999).
[CrossRef]

1998 (3)

J. M. Song, C. C. Lu, W. C. Chew, and S. Lee, “Fast Illinois solver code (FISC),” IEEE Antenn. Propag. Mag. 40(3), 27–34 (1998).
[CrossRef]

X.-Q. Sheng, J.-M. Jin, J. Song, W. C. Chew, and C.-C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antenn. Propag. 46(11), 1718–1726 (1998).
[CrossRef]

X.-Q. Sheng, J.-M. Jin, J. Song, C.-C. Lu, and W. C. Chew, “On the formulation of hybrid finite-element and boundary-integral methods for 3-D scattering,” IEEE Trans. Antenn. Propag. 46(3), 303–311 (1998).
[CrossRef]

1997 (1)

J. M. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antenn. Propag. 45(10), 1488–1493 (1997).
[CrossRef]

1995 (1)

J. M. Song and W. C. Chew, “Multilevel fast multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10(1), 14–19 (1995).
[CrossRef]

1994 (1)

1993 (1)

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: a pedestrian prescription,” IEEE Antenn. Propag. Mag. 35(3), 7–12 (1993).
[CrossRef]

1990 (1)

S. M. Rao and D. R. Wilton, “E-field, H-field, and combined field solution for arbitrarily shaped three-dimensional dielectric bodies,” Electromagetics 10(4), 407–421 (1990).
[CrossRef]

1986 (1)

Y. Saad and M. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986).
[CrossRef]

1982 (1)

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).
[CrossRef]

1979 (1)

J. R. Mautz and R. F. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektron. Ubertragungstechn. (Electron. Commun.) 33, 71–80 (1979).

1977 (2)

Y. Chang and R. F. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antenn. Propag. AP-25(6), 789–795 (1977).
[CrossRef]

T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12(5), 709–718 (1977).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Adams, R. J.

R. J. Adams, “Physical and analytical properties of a stabilized electric field integral equation,” IEEE Trans. Antenn. Propag. 52(2), 362–372 (2004).
[CrossRef]

Andriulli, F. P.

F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative calderon preconditioner for the electric field integral equation,” IEEE Trans. Antenn. Propag. 56(8), 2398–2412 (2008).
[CrossRef]

Araújo, M. G.

Bagci, H.

F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative calderon preconditioner for the electric field integral equation,” IEEE Trans. Antenn. Propag. 56(8), 2398–2412 (2008).
[CrossRef]

Bértolo, J. M.

J. M. Bértolo, M. G. Araújo, J. M. Taboada, L. Landesa, F. Obelleiro, and J. L. Rodríguez, “Extended near field preconditioner for the analysis of large problems using the Nested-FMM-FFT algorithm,” Microw. Opt. Technol. Lett. 53(2), 430–433 (2011).
[CrossRef]

Buffa, A.

F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative calderon preconditioner for the electric field integral equation,” IEEE Trans. Antenn. Propag. 56(8), 2398–2412 (2008).
[CrossRef]

Chang, Y.

Y. Chang and R. F. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antenn. Propag. AP-25(6), 789–795 (1977).
[CrossRef]

Chen, S.

S. Chen, J.-S. Zhao, and W. C. Chew, “Analyzing low-frequency electromagnetic scattering from a composite object,” IEEE Trans. Geosci. Rem. Sens. 40(2), 426–433 (2002).
[CrossRef]

Chew, W. C.

Y. A. Liu and W. C. Chew, “Stability of surface integral equation for left-handed materials,” IET Microwaves Antenn. Propag. 1(1), 84–89 (2007).
[CrossRef]

K. C. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antenn. Propag. 51(10), 2814–2821 (2003).
[CrossRef]

S. Chen, J.-S. Zhao, and W. C. Chew, “Analyzing low-frequency electromagnetic scattering from a composite object,” IEEE Trans. Geosci. Rem. Sens. 40(2), 426–433 (2002).
[CrossRef]

J. M. Song, C. C. Lu, W. C. Chew, and S. Lee, “Fast Illinois solver code (FISC),” IEEE Antenn. Propag. Mag. 40(3), 27–34 (1998).
[CrossRef]

X.-Q. Sheng, J.-M. Jin, J. Song, W. C. Chew, and C.-C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antenn. Propag. 46(11), 1718–1726 (1998).
[CrossRef]

X.-Q. Sheng, J.-M. Jin, J. Song, C.-C. Lu, and W. C. Chew, “On the formulation of hybrid finite-element and boundary-integral methods for 3-D scattering,” IEEE Trans. Antenn. Propag. 46(3), 303–311 (1998).
[CrossRef]

J. M. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antenn. Propag. 45(10), 1488–1493 (1997).
[CrossRef]

J. M. Song and W. C. Chew, “Multilevel fast multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10(1), 14–19 (1995).
[CrossRef]

Christiansen, S.

F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative calderon preconditioner for the electric field integral equation,” IEEE Trans. Antenn. Propag. 56(8), 2398–2412 (2008).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Coifman, R.

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: a pedestrian prescription,” IEEE Antenn. Propag. Mag. 35(3), 7–12 (1993).
[CrossRef]

Cools, K.

F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative calderon preconditioner for the electric field integral equation,” IEEE Trans. Antenn. Propag. 56(8), 2398–2412 (2008).
[CrossRef]

Donepudi, K. C.

K. C. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antenn. Propag. 51(10), 2814–2821 (2003).
[CrossRef]

Ergül, Ö.

Ö. Ergül and L. Gürel, “Efficient solutions of metamaterial problems using a low-frequency multilevel fast multipole algorithm,” Prog. Electromagn. Res. 108, 81–99 (2010).
[CrossRef]

Ö. Ergül and L. Gürel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antenn. Propag. 57(1), 176–187 (2009).
[CrossRef]

Forester, D. W.

D. L. Smith, L. N. Medgyesi-Mitschang, and D. W. Forester, “Surface integral equation formulations for left-handed materials,” Prog. Electromagn. Res. 51, 27–48 (2005).
[CrossRef]

Gallinet, B.

García-Tuñón, I.

Gedera, M. B.

Gedney, S. D.

A. Zhu, S. D. Gedney, and J. L. Visher, “A study of combined field formulations for material scattering for a locally corrected Nyström discretization,” IEEE Trans. Antenn. Propag. 53(12), 4111–4120 (2005).
[CrossRef]

Glisson, A. W.

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).
[CrossRef]

Gürel, L.

Ö. Ergül and L. Gürel, “Efficient solutions of metamaterial problems using a low-frequency multilevel fast multipole algorithm,” Prog. Electromagn. Res. 108, 81–99 (2010).
[CrossRef]

Ö. Ergül and L. Gürel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antenn. Propag. 57(1), 176–187 (2009).
[CrossRef]

Harrington, R. F.

J. R. Mautz and R. F. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektron. Ubertragungstechn. (Electron. Commun.) 33, 71–80 (1979).

Y. Chang and R. F. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antenn. Propag. AP-25(6), 789–795 (1977).
[CrossRef]

Järvenpää, S.

P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40(6), RS6002 (2005).
[CrossRef]

Jin, J.-M.

K. C. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antenn. Propag. 51(10), 2814–2821 (2003).
[CrossRef]

X.-Q. Sheng, J.-M. Jin, J. Song, W. C. Chew, and C.-C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antenn. Propag. 46(11), 1718–1726 (1998).
[CrossRef]

X.-Q. Sheng, J.-M. Jin, J. Song, C.-C. Lu, and W. C. Chew, “On the formulation of hybrid finite-element and boundary-integral methods for 3-D scattering,” IEEE Trans. Antenn. Propag. 46(3), 303–311 (1998).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Kern, A. M.

Landesa, L.

Lee, J.

J. Lee, J. Zhang, and C.-C. Lu, “Incomplete LU preconditioner for large scale dense complex linear systems from electromagnetic wave scattering problems,” J. Comput. Phys. 185(1), 158–175 (2003).
[CrossRef]

Lee, S.

J. M. Song, C. C. Lu, W. C. Chew, and S. Lee, “Fast Illinois solver code (FISC),” IEEE Antenn. Propag. Mag. 40(3), 27–34 (1998).
[CrossRef]

Liu, Y. A.

Y. A. Liu and W. C. Chew, “Stability of surface integral equation for left-handed materials,” IET Microwaves Antenn. Propag. 1(1), 84–89 (2007).
[CrossRef]

Lloyd, T. W.

T. W. Lloyd, J. M. Song, and M. Yang, “Numerical study of surface integral formulations for low-contrast objects,” IEEE Antennas Wirel. Propag. Lett. 4(1), 482–485 (2005).
[CrossRef]

Lu, C. C.

J. M. Song, C. C. Lu, W. C. Chew, and S. Lee, “Fast Illinois solver code (FISC),” IEEE Antenn. Propag. Mag. 40(3), 27–34 (1998).
[CrossRef]

J. M. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antenn. Propag. 45(10), 1488–1493 (1997).
[CrossRef]

Lu, C.-C.

J. Lee, J. Zhang, and C.-C. Lu, “Incomplete LU preconditioner for large scale dense complex linear systems from electromagnetic wave scattering problems,” J. Comput. Phys. 185(1), 158–175 (2003).
[CrossRef]

X.-Q. Sheng, J.-M. Jin, J. Song, W. C. Chew, and C.-C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antenn. Propag. 46(11), 1718–1726 (1998).
[CrossRef]

X.-Q. Sheng, J.-M. Jin, J. Song, C.-C. Lu, and W. C. Chew, “On the formulation of hybrid finite-element and boundary-integral methods for 3-D scattering,” IEEE Trans. Antenn. Propag. 46(3), 303–311 (1998).
[CrossRef]

Martin, O. J. F.

Mautz, J. R.

J. R. Mautz and R. F. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektron. Ubertragungstechn. (Electron. Commun.) 33, 71–80 (1979).

Medgyesi-Mitschang, L. N.

D. L. Smith, L. N. Medgyesi-Mitschang, and D. W. Forester, “Surface integral equation formulations for left-handed materials,” Prog. Electromagn. Res. 51, 27–48 (2005).
[CrossRef]

L. N. Medgyesi-Mitschang, J. M. Putnam, and M. B. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Soc. Am. A 11(4), 1383–1398 (1994).
[CrossRef]

Michielssen, E.

F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative calderon preconditioner for the electric field integral equation,” IEEE Trans. Antenn. Propag. 56(8), 2398–2412 (2008).
[CrossRef]

Obelleiro, F.

Olyslager, F.

F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative calderon preconditioner for the electric field integral equation,” IEEE Trans. Antenn. Propag. 56(8), 2398–2412 (2008).
[CrossRef]

Putnam, J. M.

Rao, S. M.

S. M. Rao and D. R. Wilton, “E-field, H-field, and combined field solution for arbitrarily shaped three-dimensional dielectric bodies,” Electromagetics 10(4), 407–421 (1990).
[CrossRef]

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).
[CrossRef]

Rivero, J.

Rodríguez, J. L.

J. M. Bértolo, M. G. Araújo, J. M. Taboada, L. Landesa, F. Obelleiro, and J. L. Rodríguez, “Extended near field preconditioner for the analysis of large problems using the Nested-FMM-FFT algorithm,” Microw. Opt. Technol. Lett. 53(2), 430–433 (2011).
[CrossRef]

Rokhlin, V.

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: a pedestrian prescription,” IEEE Antenn. Propag. Mag. 35(3), 7–12 (1993).
[CrossRef]

Saad, Y.

Y. Saad and M. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986).
[CrossRef]

Sarvas, J.

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general integral equation method for general composite metallic and dielectric structures with junctions,” Prog. Electromagn. Res. 52, 81–108 (2005).
[CrossRef]

Schultz, M.

Y. Saad and M. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986).
[CrossRef]

Sertel, K.

K. Sertel and J. L. Volakis, “Incomplete LU preconditioner for FMM implementation,” Microw. Opt. Technol. Lett. 26(4), 265–267 (2000).
[CrossRef]

Sheng, X.-Q.

X.-Q. Sheng, J.-M. Jin, J. Song, W. C. Chew, and C.-C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antenn. Propag. 46(11), 1718–1726 (1998).
[CrossRef]

X.-Q. Sheng, J.-M. Jin, J. Song, C.-C. Lu, and W. C. Chew, “On the formulation of hybrid finite-element and boundary-integral methods for 3-D scattering,” IEEE Trans. Antenn. Propag. 46(3), 303–311 (1998).
[CrossRef]

Smith, D. L.

D. L. Smith, L. N. Medgyesi-Mitschang, and D. W. Forester, “Surface integral equation formulations for left-handed materials,” Prog. Electromagn. Res. 51, 27–48 (2005).
[CrossRef]

Solís, D. M.

Song, J.

X.-Q. Sheng, J.-M. Jin, J. Song, C.-C. Lu, and W. C. Chew, “On the formulation of hybrid finite-element and boundary-integral methods for 3-D scattering,” IEEE Trans. Antenn. Propag. 46(3), 303–311 (1998).
[CrossRef]

X.-Q. Sheng, J.-M. Jin, J. Song, W. C. Chew, and C.-C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antenn. Propag. 46(11), 1718–1726 (1998).
[CrossRef]

Song, J. M.

T. W. Lloyd, J. M. Song, and M. Yang, “Numerical study of surface integral formulations for low-contrast objects,” IEEE Antennas Wirel. Propag. Lett. 4(1), 482–485 (2005).
[CrossRef]

J. M. Song, C. C. Lu, W. C. Chew, and S. Lee, “Fast Illinois solver code (FISC),” IEEE Antenn. Propag. Mag. 40(3), 27–34 (1998).
[CrossRef]

J. M. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antenn. Propag. 45(10), 1488–1493 (1997).
[CrossRef]

J. M. Song and W. C. Chew, “Multilevel fast multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10(1), 14–19 (1995).
[CrossRef]

Taboada, J. M.

Taskinen, M.

P. Ylä-Oijala and M. Taskinen, “Improving conditioning of electromagnetic surface integral equations using normalized field quantities,” IEEE Trans. Antenn. Propag. 55(1), 178–185 (2007).
[CrossRef]

P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40(6), RS6002 (2005).
[CrossRef]

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general integral equation method for general composite metallic and dielectric structures with junctions,” Prog. Electromagn. Res. 52, 81–108 (2005).
[CrossRef]

P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antenn. Propag. 53(3), 1168–1173 (2005).
[CrossRef]

P. Ylä-Oijala and M. Taskinen, “Well-conditioned Müller formulation for electromagnetic scattering by dielectric objects,” IEEE Trans. Antenn. Propag. 53(10), 3316–3323 (2005).
[CrossRef]

Tsai, L. L.

T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12(5), 709–718 (1977).
[CrossRef]

Visher, J. L.

A. Zhu, S. D. Gedney, and J. L. Visher, “A study of combined field formulations for material scattering for a locally corrected Nyström discretization,” IEEE Trans. Antenn. Propag. 53(12), 4111–4120 (2005).
[CrossRef]

Volakis, J. L.

K. Sertel and J. L. Volakis, “Incomplete LU preconditioner for FMM implementation,” Microw. Opt. Technol. Lett. 26(4), 265–267 (2000).
[CrossRef]

Wandzura, S.

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: a pedestrian prescription,” IEEE Antenn. Propag. Mag. 35(3), 7–12 (1993).
[CrossRef]

Wilton, D. R.

S. M. Rao and D. R. Wilton, “E-field, H-field, and combined field solution for arbitrarily shaped three-dimensional dielectric bodies,” Electromagetics 10(4), 407–421 (1990).
[CrossRef]

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).
[CrossRef]

Wu, T. K.

T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12(5), 709–718 (1977).
[CrossRef]

Yang, M.

T. W. Lloyd, J. M. Song, and M. Yang, “Numerical study of surface integral formulations for low-contrast objects,” IEEE Antennas Wirel. Propag. Lett. 4(1), 482–485 (2005).
[CrossRef]

Yeung, M. S.

M. S. Yeung, “Single integral equation for electromagnetic scattering by three-dimensional homogeneous dielectric objects,” IEEE Trans. Antenn. Propag. 47(10), 1615–1622 (1999).
[CrossRef]

Ylä-Oijala, P.

P. Ylä-Oijala and M. Taskinen, “Improving conditioning of electromagnetic surface integral equations using normalized field quantities,” IEEE Trans. Antenn. Propag. 55(1), 178–185 (2007).
[CrossRef]

P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40(6), RS6002 (2005).
[CrossRef]

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general integral equation method for general composite metallic and dielectric structures with junctions,” Prog. Electromagn. Res. 52, 81–108 (2005).
[CrossRef]

P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antenn. Propag. 53(3), 1168–1173 (2005).
[CrossRef]

P. Ylä-Oijala and M. Taskinen, “Well-conditioned Müller formulation for electromagnetic scattering by dielectric objects,” IEEE Trans. Antenn. Propag. 53(10), 3316–3323 (2005).
[CrossRef]

Zhang, J.

J. Lee, J. Zhang, and C.-C. Lu, “Incomplete LU preconditioner for large scale dense complex linear systems from electromagnetic wave scattering problems,” J. Comput. Phys. 185(1), 158–175 (2003).
[CrossRef]

Zhao, J.-S.

S. Chen, J.-S. Zhao, and W. C. Chew, “Analyzing low-frequency electromagnetic scattering from a composite object,” IEEE Trans. Geosci. Rem. Sens. 40(2), 426–433 (2002).
[CrossRef]

Zhu, A.

A. Zhu, S. D. Gedney, and J. L. Visher, “A study of combined field formulations for material scattering for a locally corrected Nyström discretization,” IEEE Trans. Antenn. Propag. 53(12), 4111–4120 (2005).
[CrossRef]

Arch. Elektron. Ubertragungstechn. (Electron. Commun.) (1)

J. R. Mautz and R. F. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektron. Ubertragungstechn. (Electron. Commun.) 33, 71–80 (1979).

Electromagetics (1)

S. M. Rao and D. R. Wilton, “E-field, H-field, and combined field solution for arbitrarily shaped three-dimensional dielectric bodies,” Electromagetics 10(4), 407–421 (1990).
[CrossRef]

IEEE Antenn. Propag. Mag. (2)

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: a pedestrian prescription,” IEEE Antenn. Propag. Mag. 35(3), 7–12 (1993).
[CrossRef]

J. M. Song, C. C. Lu, W. C. Chew, and S. Lee, “Fast Illinois solver code (FISC),” IEEE Antenn. Propag. Mag. 40(3), 27–34 (1998).
[CrossRef]

IEEE Antennas Wirel. Propag. Lett. (1)

T. W. Lloyd, J. M. Song, and M. Yang, “Numerical study of surface integral formulations for low-contrast objects,” IEEE Antennas Wirel. Propag. Lett. 4(1), 482–485 (2005).
[CrossRef]

IEEE Trans. Antenn. Propag. (14)

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).
[CrossRef]

P. Ylä-Oijala and M. Taskinen, “Well-conditioned Müller formulation for electromagnetic scattering by dielectric objects,” IEEE Trans. Antenn. Propag. 53(10), 3316–3323 (2005).
[CrossRef]

P. Ylä-Oijala and M. Taskinen, “Improving conditioning of electromagnetic surface integral equations using normalized field quantities,” IEEE Trans. Antenn. Propag. 55(1), 178–185 (2007).
[CrossRef]

X.-Q. Sheng, J.-M. Jin, J. Song, C.-C. Lu, and W. C. Chew, “On the formulation of hybrid finite-element and boundary-integral methods for 3-D scattering,” IEEE Trans. Antenn. Propag. 46(3), 303–311 (1998).
[CrossRef]

A. Zhu, S. D. Gedney, and J. L. Visher, “A study of combined field formulations for material scattering for a locally corrected Nyström discretization,” IEEE Trans. Antenn. Propag. 53(12), 4111–4120 (2005).
[CrossRef]

J. M. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antenn. Propag. 45(10), 1488–1493 (1997).
[CrossRef]

R. J. Adams, “Physical and analytical properties of a stabilized electric field integral equation,” IEEE Trans. Antenn. Propag. 52(2), 362–372 (2004).
[CrossRef]

F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative calderon preconditioner for the electric field integral equation,” IEEE Trans. Antenn. Propag. 56(8), 2398–2412 (2008).
[CrossRef]

K. C. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antenn. Propag. 51(10), 2814–2821 (2003).
[CrossRef]

Ö. Ergül and L. Gürel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antenn. Propag. 57(1), 176–187 (2009).
[CrossRef]

M. S. Yeung, “Single integral equation for electromagnetic scattering by three-dimensional homogeneous dielectric objects,” IEEE Trans. Antenn. Propag. 47(10), 1615–1622 (1999).
[CrossRef]

P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antenn. Propag. 53(3), 1168–1173 (2005).
[CrossRef]

Y. Chang and R. F. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antenn. Propag. AP-25(6), 789–795 (1977).
[CrossRef]

X.-Q. Sheng, J.-M. Jin, J. Song, W. C. Chew, and C.-C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antenn. Propag. 46(11), 1718–1726 (1998).
[CrossRef]

IEEE Trans. Geosci. Rem. Sens. (1)

S. Chen, J.-S. Zhao, and W. C. Chew, “Analyzing low-frequency electromagnetic scattering from a composite object,” IEEE Trans. Geosci. Rem. Sens. 40(2), 426–433 (2002).
[CrossRef]

IET Microwaves Antenn. Propag. (1)

Y. A. Liu and W. C. Chew, “Stability of surface integral equation for left-handed materials,” IET Microwaves Antenn. Propag. 1(1), 84–89 (2007).
[CrossRef]

J. Comput. Phys. (1)

J. Lee, J. Zhang, and C.-C. Lu, “Incomplete LU preconditioner for large scale dense complex linear systems from electromagnetic wave scattering problems,” J. Comput. Phys. 185(1), 158–175 (2003).
[CrossRef]

J. Opt. Soc. Am. A (3)

Microw. Opt. Technol. Lett. (3)

J. M. Bértolo, M. G. Araújo, J. M. Taboada, L. Landesa, F. Obelleiro, and J. L. Rodríguez, “Extended near field preconditioner for the analysis of large problems using the Nested-FMM-FFT algorithm,” Microw. Opt. Technol. Lett. 53(2), 430–433 (2011).
[CrossRef]

K. Sertel and J. L. Volakis, “Incomplete LU preconditioner for FMM implementation,” Microw. Opt. Technol. Lett. 26(4), 265–267 (2000).
[CrossRef]

J. M. Song and W. C. Chew, “Multilevel fast multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10(1), 14–19 (1995).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Prog. Electromagn. Res. (4)

Ö. Ergül and L. Gürel, “Efficient solutions of metamaterial problems using a low-frequency multilevel fast multipole algorithm,” Prog. Electromagn. Res. 108, 81–99 (2010).
[CrossRef]

M. G. Araújo, J. M. Taboada, J. Rivero, and F. Obelleiro, “Comparison of surface integral equations for left-handed materials,” Prog. Electromagn. Res. 118, 425–440 (2011).
[CrossRef]

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general integral equation method for general composite metallic and dielectric structures with junctions,” Prog. Electromagn. Res. 52, 81–108 (2005).
[CrossRef]

D. L. Smith, L. N. Medgyesi-Mitschang, and D. W. Forester, “Surface integral equation formulations for left-handed materials,” Prog. Electromagn. Res. 51, 27–48 (2005).
[CrossRef]

Radio Sci. (2)

P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40(6), RS6002 (2005).
[CrossRef]

T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12(5), 709–718 (1977).
[CrossRef]

SIAM J. Sci. Stat. Comput. (1)

Y. Saad and M. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986).
[CrossRef]

Other (6)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

B. M. Kolundzija and A. R. Djordjevic, Electromagnetic Modeling of Composite Metallic and Dielectric Structures (Artech House, 2002).

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer, 1969).

A. J. Poggio and E. K. Miller, Computer Techniques for Electromagnetics (Permagon, 1973), Chap. 4.

Y. Saad, Iterative Methods for Sparse Linear Systems (PWS Publishing, 1996).

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Figures (4)

Fig. 1
Fig. 1

Condition number without/with preconditioner vs. sphere size for dielectric spheres of εr = 2.1 analyzed with the five formulations.

Fig. 2
Fig. 2

Condition number without/with preconditioner vs. sphere size for Au spheres (εr = −5.84−j2.11) analyzed with the five formulations.

Fig. 3
Fig. 3

Condition number without/with preconditioner vs. sphere size for dielectric spheres of εr = 25 analyzed with the five formulations.

Fig. 4
Fig. 4

Iterative convergence of PMCHWT formulation and CTF solved with GMRES(30) with and without preconditioner.

Tables (2)

Tables Icon

Table 1 Combination parameters a i , b i , c i , d i and O norm ( Z ¯ ) for the five formulations considered

Tables Icon

Table 2 Combination ( a,b,c,d ) and preconditioning ( α 11 , α 22 , β 11 , β 22 ) coefficients for the five formulations considered

Equations (44)

Equations on this page are rendered with MathJax. Learn more.

T-EFIE i : ( η i i ( J i ) K i ( M i ) ) tan + 1 2 n ^ i × M i = ( E i inc ) tan
T-MFIE i : ( K i ( J i )+ η i 1 i ( M i ) ) tan 1 2 n ^ i × J i = ( H i inc ) tan
N-EFIE i : n ^ i ×( η i i ( J i ) K i ( M i ) ) 1 2 M i = n ^ i × E i inc
N-MFIE i : n ^ i ×( K i ( J i )+ η i 1 i ( M i ) )+ 1 2 J i = n ^ i × H i inc
i ( X i )=j k i [ S X i (r') G i (r,r')dS'+ 1 k i 2 S ' X i G i (r,r')dS' ]
K i ( X i )= S,PV X i (r')× G i (r,r')dS'
G i (r,r')= exp(j k i | rr' |) 4π| rr' |
JCFIE i = a i η i 1 T-EFIE i + b i N-MFIE i
MCFIE i = c i N-EFIE i + d i η i T-MFIE i
JCFIE: a 1 ( 1 (J) η 1 1 K 1 (M) ) tan + a 2 ( 2 (J) η 2 1 K 2 (M) ) tan + b 1 n ^ ×( K 1 (J)+ η 1 1 1 (M) ) b 2 n ^ ×( K 2 (J)+ η 2 1 2 (M) ) + 1 2 ( a 1 η 1 1 a 2 η 2 1 ) n ^ ×M+ 1 2 ( b 1 + b 2 )J = a 1 η 1 1 ( E 1 inc ) tan a 2 η 2 1 ( E 2 inc ) tan + b 1 n ^ × H 1 inc + b 2 n ^ × H 2 inc ,rS
MCFIE: c 1 n ^ ×( η 1 1 (J) K 1 (M) )+ c 2 n ^ ×( η 2 2 (J) K 2 (M) ) + d 1 ( η 1 K 1 (J)+ 1 (M) ) tan + d 2 ( η 2 K 2 (J)+ 2 (M) ) tan + 1 2 ( c 1 + c 2 )M 1 2 ( d 1 η 1 d 2 η 2 ) n ^ ×J = c 1 n ^ × E 1 inc c 2 n ^ × E 2 inc + d 1 η 1 ( H 1 inc ) tan d 2 η 2 ( H 2 inc ) tan ,rS
J= n J n f n ;M= n M n f n ;r S ij
a 1 nS ( A mn 1 J n η 1 1 B mn 1 M n ) + a 2 nS ( A mn 2 J n η 2 1 B mn 2 M n ) + b 1 nS ( B ' mn 1 J n + η 1 1 A ' mn 1 M n ) b 2 nS ( B ' mn 2 J n + η 2 1 A ' mn 2 M n ) + 1 2 nS [ ( a 1 η 1 1 a 2 η 2 1 )I ' mn M n ] + 1 2 nS [ ( b 1 + b 2 ) I mn J n ] = E m +H ' m ,mS
c 1 nS ( η 1 A ' mn 1 J n B ' mn 1 M n ) + c 2 nS ( η 2 A ' mn 2 J n B ' mn 2 M n ) + d 1 nS ( η 1 B mn 1 J n + A mn 1 M n ) + d 2 nS ( η 2 B mn 2 J n + A mn 2 M n ) + 1 2 nS [ ( c 1 + c 2 ) I mn M n ] 1 2 nS [ ( d 1 η 1 d 2 η 2 )I ' mn J n ] =E ' m + H m ,mS
A mn i = Δ m f m i ( f n )dS
B mn i = Δ m f m K i ( f n )dS
A ' mn i = Δ m f m n ^ m × i ( f n )dS
B ' mn i = Δ m f m n ^ m × K i ( f n )dS
I mn = Δ m f m f n dS
I ' mn = Δ m f m n ^ m × f n dS .
E m = Δ m f m [ a 1 ( η i 1 E i inc ) tan a 2 ( η 2 1 E j inc ) tan ]dS
H m = Δ m f m [ d 1 ( η 1 H 1 inc ) tan d 2 ( η 2 H 2 inc ) tan ]dS
E ' m = Δ m f m [ c 1 n ^ m × E 1 inc + c 2 n ^ m × E 2 inc ]dS
H ' m = Δ m f m [ b 1 n ^ m × H 1 inc + b 2 n ^ m × H 2 inc ]dS
Z ¯ I=V
Z ¯ =[ Z ¯ 1J Z ¯ 1M Z ¯ 2J Z ¯ 2M ]
Z ¯ mn 1J = a 1 A mn 1 + a 2 A mn 2 + b 1 B ' mn 1 b 2 B ' mn 2 + 1 2 ( b 1 + b 2 ) I mn
Z ¯ mn 1M = a 1 η 1 1 B mn 1 a 2 η 2 1 B mn 2 + b 1 η 1 1 A ' mn 1 b 2 η 2 1 A ' mn 2 + 1 2 ( a 1 η 1 1 a 2 η 2 1 )I ' mn
Z ¯ mn 2J = d 1 η 1 B mn 1 + d 2 η 2 B mn 2 c 1 η 1 A ' mn 1 + c 2 η 2 A ' mn 2 1 2 ( d 1 η 1 d 2 η 2 )I ' mn
Z ¯ mn 2M = d 1 A mn 1 + d 2 A mn 2 + c 1 B ' mn 1 c 2 B ' mn 2 + 1 2 ( c 1 + c 2 ) I mn
V m 1 = E m +H ' m
V m 2 =E ' m + H m .
O( Z ¯ )=O( [ Z ¯ 1J Z ¯ 1M Z ¯ 2J Z ¯ 2M ] )( Θf( a 1 , a 2 , b 1 , b 2 ) Θg( a 1 , a 2 , b 1 , b 2 , η 1 , η 2 ) Θh( c 1 , c 2 , d 1 , d 2 , η 1 , η 2 ) Θf( c 1 , c 2 , d 1 , d 2 ) )
O( [ Z ¯ 1J Z ¯ 1M Z ¯ 2J Z ¯ 2M ] )( Θ(a+b) Θ(a+b) η Θ(c+d)η Θ(c+d) )
O norm ( Z ¯ )= O( Z ¯ ) O( Z ¯ 1J ) ( 1 1 η (c+d) (a+b) η (c+d) (a+b) )
Z ˜ =[ Z ¯ 1J Z ¯ 1M Z ¯ 2J Z ¯ 2M ]( 1 η (a+b) (c+d) 1 η (a+b) (c+d) )
I ˜ = M R 1 I
V ˜ = M ¯ L V
M ¯ L =diag{ [ α 11 ... α 11 ] 1×N , [ α 22 ... α 22 ] 1×N }
M ¯ R =diag{ [ β 11 ... β 11 ] 1×N , [ β 22 ... β 22 ] 1×N }
Z ˜ I ˜ = V ˜
Z ˜ = M ¯ L Z ¯ M ¯ R
α 11 = β 11 =1 β 22 =η α 22 = a+b c+d 1 η
e rms = ( σ Mie σ ) 2 /N max( σ Mie )

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