## Improving condition number and convergence of the surface integral-equation method of moments for penetrable bodies |

Optics Express, Vol. 20, Issue 15, pp. 17237-17249 (2012)

http://dx.doi.org/10.1364/OE.20.017237

Acrobat PDF (1142 KB)

### 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

## 1. Introduction

10. 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]

11. 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]

18. M. G. Araújo, J. M. Taboada, D. M. Solís, J. Rivero, L. Landesa, and F. Obelleiro, “Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers,” Opt. Express **20**(8), 9161–9171 (2012). [CrossRef] [PubMed]

19. 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]

20. 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]

21. 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]

22. 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]

24. M. G. Araújo, J. M. Taboada, J. Rivero, D. M. Solís, and F. Obelleiro, “Solution of large-scale plasmonic problems with the multilevel fast multipole algorithm,” Opt. Lett. **37**(3), 416–418 (2012). [CrossRef] [PubMed]

26. 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]

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

28. 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]

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

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). [CrossRef]

31. 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]

32. 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]

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]

36. 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]

37. 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]

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]

8. 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]

39. 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]

18. M. G. Araújo, J. M. Taboada, D. M. Solís, J. Rivero, L. Landesa, and F. Obelleiro, “Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers,” Opt. Express **20**(8), 9161–9171 (2012). [CrossRef] [PubMed]

40. 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]

9. 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]

## 2. Surface integral-equation formulations for composite penetrable bodies

*i*= 1,2.

*jωt*) is assumed and suppressed from the formulation. Let us denote with

*S*the interface surface between regions

*S*and pointing toward

8. 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]

**r**denotes the observation points approaching to

_{r'∈S}denotes the source points.

*PV*denotes the principal value of the integral in Eq. (6),

*S*, as given by the equivalence principle for each region

*S*), the surface current densities in both sides of

*S*should satisfy

*S*, namely

7. 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]

**J**and

**M**:

## 3. Discretization of the integral equations

*T*indicating vector transposition, and where

## 4. Numerical balance of the impedance matrix

*f, g*and

*h*are functions which note linear combinations of the combination parameters (

_{ai, bi,ci,di}) and the intrinsic impedances (

_{ηi}) of the media on both sides of surface

_{S}(

*i*= 1,2) according to the expressions of Eqs. (27)–(30).

*η*with the diagonal blocks. In the case of CTF, the diagonal blocks are the same order (which is the main objective of this formulation, thus greatly improving the convergence with respect to PMCHWT), but consequently the off-diagonal blocks differ by a factor of

*η*with respect to the diagonal blocks. Otherwise, CNF behaves similar to CTF, with equal diagonal blocks but with off-diagonal blocks differing by a factor of

*η*with regard to the diagonal blocks. Finally, Müller formulation behaves in the same way that the PMCHWT formulation. It can be concluded that in all formulations the strong imbalance between the impedance matrix blocks leads to high condition numbers, which causes bad convergence and/or lose of accuracy.

*a*,

*b*,

*c*,

*d*and

*η*. In this work, we have selected for these parameters the outer medium values:

*a*=

*a*

_{1},

*b*=

*b*

_{1},

*c*=

*c*

_{1},

*d*=

*d*

_{1}, and

*η*=

*η*

_{1}, although other possibilities also work properly, such as choosing the arithmetic or geometric means or the vacuum equivalent values, among others.

## 5. Left and right preconditioner

*N*× 2

*N*) as follows: With these changes Eq. (25) can be written aswhere we have defined

*μ*=

*μ*

_{1}and

*ε*=

*ε*

_{1}according to the previous choice made for

*a*,

*b*,

*c*,

*d*and

*η*.

*O*(

*N*),

*N*being the number of unknowns.

## 6. Numerical examples

42. 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]

*λ*

_{0}= 546 nm and a mean mesh size of

*λ*

_{0}/15 has been considered in all cases except for the smallest sphere, where a mesh refinement was required in order to guarantee the geometrical sphericity.

*ε*= 2.1 that have been analyzed using the five formulations. The condition number is represented as a function of

_{r}*k*, where

_{0}r*k*is the wave number in vacuum and

_{0}*r*the radius of the sphere. The results of Fig. 2 and Fig. 3 correspond to a gold plasmonic sphere with

*ε*= −5.84−

_{r}*j*2.11 [43

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

*ε*= 25, respectively.

_{r}8. 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]

17. 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]

18. M. G. Araújo, J. M. Taboada, D. M. Solís, J. Rivero, L. Landesa, and F. Obelleiro, “Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers,” Opt. Express **20**(8), 9161–9171 (2012). [CrossRef] [PubMed]

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). [CrossRef]

39. 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]

40. 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]

44. 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]

**20**(8), 9161–9171 (2012). [CrossRef] [PubMed]

**40**(6), RS6002 (2005). [CrossRef]

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). [CrossRef]

*σ*) with respect to the analytical Mie’s series result [45] has been considered:with

*N*the number of RCS samples. An error of 2.01⋅10

^{−4}has been obtained for PMCHWT with both direct and precorrected iterative solutions, while an error of 0.013 has been obtained for CTF. These values show that, unlike in conventional problems, CTF leads to an important lack of accuracy in the context of plasmonic problems despite its good iterative convergence. This is in accordance with the behavior reported in the thorough comparison of [18

**20**(8), 9161–9171 (2012). [CrossRef] [PubMed]

^{−3}) an error level of 2.06⋅10

^{−4}is already obtained. As expected, improvement on the iteration convergence is not observed with the preconditioned CTF since convergence essentially depends on the good balance of the diagonal blocks, which is already satisfied by CTF without applying the preconditioner.

## 7. Conclusion

*O*(

*N*), it can be easily and efficiently integrated into any existing MoM implementation.

## Acknowledgments

## References and links

1. | B. M. Kolundzija and A. R. Djordjevic, |

2. | C. Müller, |

3. | A. J. Poggio and E. K. Miller, |

4. | Y. Chang and R. F. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antenn. Propag. |

5. | T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. |

6. | S. M. Rao and D. R. Wilton, “E-field, H-field, and combined field solution for arbitrarily shaped three-dimensional dielectric bodies,” Electromagetics |

7. | M. S. Yeung, “Single integral equation for electromagnetic scattering by three-dimensional homogeneous dielectric objects,” IEEE Trans. Antenn. Propag. |

8. | P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. |

9. | P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antenn. Propag. |

10. | 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. |

11. | D. L. Smith, L. N. Medgyesi-Mitschang, and D. W. Forester, “Surface integral equation formulations for left-handed materials,” Prog. Electromagn. Res. |

12. | Y. A. Liu and W. C. Chew, “Stability of surface integral equation for left-handed materials,” IET Microwaves Antenn. Propag. |

13. | B. Gallinet, A. M. Kern, and O. J. F. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A |

14. | Ö. Ergül and L. Gürel, “Efficient solutions of metamaterial problems using a low-frequency multilevel fast multipole algorithm,” Prog. Electromagn. Res. |

15. | J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express |

16. | 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 |

17. | M. G. Araújo, J. M. Taboada, J. Rivero, and F. Obelleiro, “Comparison of surface integral equations for left-handed materials,” Prog. Electromagn. Res. |

18. | M. G. Araújo, J. M. Taboada, D. M. Solís, J. Rivero, L. Landesa, and F. Obelleiro, “Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers,” Opt. Express |

19. | R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: a pedestrian prescription,” IEEE Antenn. Propag. Mag. |

20. | J. M. Song and W. C. Chew, “Multilevel fast multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. |

21. | J. M. Song, C. C. Lu, W. C. Chew, and S. Lee, “Fast Illinois solver code (FISC),” IEEE Antenn. Propag. Mag. |

22. | 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. |

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. |

24. | M. G. Araújo, J. M. Taboada, J. Rivero, D. M. Solís, and F. Obelleiro, “Solution of large-scale plasmonic problems with the multilevel fast multipole algorithm,” Opt. Lett. |

25. | Y. Saad, |

26. | 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. |

27. | K. Sertel and J. L. Volakis, “Incomplete LU preconditioner for FMM implementation,” Microw. Opt. Technol. Lett. |

28. | 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. |

29. | R. J. Adams, “Physical and analytical properties of a stabilized electric field integral equation,” IEEE Trans. Antenn. Propag. |

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. |

31. | Y. Saad and M. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. |

32. | 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. |

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. |

34. | J. R. Mautz and R. F. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektron. Ubertragungstechn. (Electron. Commun.) |

35. | 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 |

36. | 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. |

37. | 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. |

38. | S. Chen, J.-S. Zhao, and W. C. Chew, “Analyzing low-frequency electromagnetic scattering from a composite object,” IEEE Trans. Geosci. Rem. Sens. |

39. | P. Ylä-Oijala and M. Taskinen, “Well-conditioned Müller formulation for electromagnetic scattering by dielectric objects,” IEEE Trans. Antenn. Propag. |

40. | P. Ylä-Oijala and M. Taskinen, “Improving conditioning of electromagnetic surface integral equations using normalized field quantities,” IEEE Trans. Antenn. Propag. |

41. | R. F. Harrington, |

42. | S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. |

43. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

44. | T. W. Lloyd, J. M. Song, and M. Yang, “Numerical study of surface integral formulations for low-contrast objects,” IEEE Antennas Wirel. Propag. Lett. |

45. | C. F. Bohren and D. R. Huffman, |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(160.2100) Materials : Electro-optical materials

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: May 15, 2012

Revised Manuscript: June 26, 2012

Manuscript Accepted: June 26, 2012

Published: July 13, 2012

**Citation**

Luis Landesa, Marta Gómez Araújo, José Manuel Taboada, Luis Bote, and Fernando Obelleiro, "Improving condition number and convergence of the surface integral-equation method of moments for penetrable bodies," Opt. Express **20**, 17237-17249 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-17237

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### References

- 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. 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]
- 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]
- 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, 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]
- 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]
- B. Gallinet, A. M. Kern, and O. J. F. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27(10), 2261–2271 (2010). [CrossRef] [PubMed]
- 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]
- J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express 18(15), 15876–15886 (2010).
- 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]
- M. G. Araújo, J. M. Taboada, D. M. Solís, J. Rivero, L. Landesa, and F. Obelleiro, “Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers,” Opt. Express 20(8), 9161–9171 (2012). [CrossRef] [PubMed]
- 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 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]
- 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]
- 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. G. Araújo, J. M. Taboada, J. Rivero, D. M. Solís, and F. Obelleiro, “Solution of large-scale plasmonic problems with the multilevel fast multipole algorithm,” Opt. Lett. 37(3), 416–418 (2012). [CrossRef] [PubMed]
- Y. Saad, Iterative Methods for Sparse Linear Systems (PWS Publishing, 1996).
- 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]
- K. Sertel and J. L. Volakis, “Incomplete LU preconditioner for FMM implementation,” Microw. Opt. Technol. Lett. 26(4), 265–267 (2000). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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).
- 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]
- 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]
- 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]
- 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]
- R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).
- 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. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [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]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

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