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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 15 — Jul. 16, 2012
  • pp: 17237–17249
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Improving condition number and convergence of the surface integral-equation method of moments for penetrable bodies

Luis Landesa, Marta Gómez Araújo, José Manuel Taboada, Luis Bote, and Fernando Obelleiro  »View Author Affiliations


Optics Express, Vol. 20, Issue 15, pp. 17237-17249 (2012)
http://dx.doi.org/10.1364/OE.20.017237


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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

Most practical electromagnetic scattering and radiation problems can be defined as a combination of conducting and penetrable objects. The surface integral equation (SIE) method [1

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

] is perhaps the most powerful numerical method in the electromagnetic analysis of this kind of problems. It has demonstrated to be very accurate and efficient for the analysis of composite objects with real conductors and dielectrics in radio frequency and microwave regimes [2

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

10

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]

]. Recently, this approach was successfully extended to the analysis of homogenized metamaterials and plasmonic problems [11

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

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]

] in the context of nanoscience applications. However, traditional SIE methods are still restricted to electrically small problems (small in terms of the wavelength) because the implicit discretization process of the SIE results in a dense matrix equation which is very expensive to solve and to store. In order to solve electrically large problems, advanced methods like the fast multipole method (FMM) [19

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]

] and its multilevel version (MLFMA) [20

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

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]

] are required in combination with iterative solutions of the matrix equation [22

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

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]

]. Nevertheless, many of the traditional SIE formulations lead to ill-conditioned matrix equations. A high condition number may cause problems in the numerical solution, such as slow convergence of iterative solvers (also depending on the clustering of eigenvalues) and even inaccuracy of the solution when the condition number is really high.

An alternative way to improve the condition of the matrix is a proper choice of the SIE to be solved. The main reason for the poor conditioning of the SIE matrix is that the unknowns (the electric and magnetic surface current densities), and the integral equations (the electric and magnetic field integral equations), are of the very different scales. Usually the balance between the unknowns and between the equations is improved, for example, by multiplying the magnetic field integral equation (MFIE) with the wave impedance [33

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]

]; or scaling the magnetic, or electric, surface current density by the wave impedance [34

34. 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).

36

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]

] or both [37

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

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]

]. In these cases, the conditioning of the matrix is improved by using proper weighting coefficients and combination of the equations [8

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]

]. In fact, the widespread combined tangential formulation (CTF) can be seen as a weighted version of the original Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) formulation [34

34. 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).

], in which the balance between the diagonal blocks of the matrix has been improved. And the same goes for the combined normal formulation (CNF), which can be seen as a variation of the original Müller formulation [2

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

,39

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]

]. Anyway, the resultant matrices in all these SIE formulations are still unbalanced because, for example, when balancing the diagonal blocks of the matrices, the off-diagonal blocks become unbalanced. Otherwise, modifying the formulation through the appropriate selection of combination coefficients to improve block balance may affect other desirable properties. This is the case of CTF and PMCHWT when dealing with plasmonic problems. While CTF has a much better convergence than PMCHWT, this is at the expense of a lower accuracy, as has been demonstrated in [18

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]

].

In a recent paper [40

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]

], an efficient method to improve the balance between the unknowns and the equations and, hence, also between the matrix elements has been presented. The method is based on the use of normalized field quantities and unknowns together with carefully chosen scaling factors, achieving a reduction in the condition number of the SIE matrices of several orders of magnitude.

In this paper we propose a similar solution, although in this case the problem is addressed from novel and a strictly numerical point of view. The SIE matrix formulation is not modified anymore; namely, the balance between matrix elements is straightforwardly addressed during the iterative solution process by means of a left and right (L&R) diagonal matrix preconditioner. This allows improving the matrix balance independently of the formulation, which can now be selected solely in terms of accuracy for a given application. The proposed preconditioning formulation is developed in a general way, so that particular expressions can be derived for any particular SIE formulation.

Numerical results show that, owing to the good balance between the matrix elements, the proposed preconditioner provides a significant reduction in the condition number. In this paper, the proposed preconditioner has been tested for the conventional SIE formulations previously mentioned: PMCHWT formulation, CTF, the normal Müller formulation and CNF, and for the electric and magnetic current combined field integral equation (JMCFIE) [9

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]

], although it can be straight forwardly extended to any other formulation.

2. Surface integral-equation formulations for composite penetrable bodies

Let us consider a homogeneous penetrable object in a homogeneous (unbounded) medium. The exterior region is denoted byR1and the interior region is denoted byR2. Let us define with εi=εriε0 the complex permittivity and with μi=μriμ0 the complex permeability of region Ri with i = 1,2. εri and μri are the complex valued relative permittivity and permeability constants of region Ri and ε0,μ0 are the constitutive parameters of vacuum. A time harmonic variation exp(jωt) is assumed and suppressed from the formulation. Let us denote with S the interface surface between regions R1 and R2. n^i denotes the unit vector normal to S and pointing toward Ri, hence n^2=n^1.

According to the equivalence principle [41

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

] the electric field integral equation (EFIE) and the magnetic field integral equation (MFIE) can be formulated separately in each regionRi. Two alternative formulations can be derived depending on how the fields are projected onto the surfaces surrounding this region. The tangential (T) equations in Ri are given by the following expressions [1

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

,8

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]

]:

T-EFIEi:(ηii(Ji)Ki(Mi))tan+12n^i×Mi=(Eiinc)tan
(1)
T-MFIEi:(Ki(Ji)+ηi1i(Mi))tan12n^i×Ji=(Hiinc)tan
(2)

In a similar way, the normal (N) equations in Ri can be written as

N-EFIEi:n^i×(ηii(Ji)Ki(Mi))12Mi=n^i×Eiinc
(3)
N-MFIEi:n^i×(Ki(Ji)+ηi1i(Mi))+12Ji=n^i×Hiinc
(4)

Here ηi is the intrinsic impedance in medium Ri. The integro-differential i and Ki operators are defined as
i(Xi)=jki[SXi(r')Gi(r,r')dS'+1ki2S'XiGi(r,r')dS']
(5)
Ki(Xi)=S,PVXi(r')×Gi(r,r')dS'
(6)
where r denotes the observation points approaching to S from the inner of region Ri and r'S denotes the source points. ' is the divergence in the primed (source) coordinates, PV denotes the principal value of the integral in Eq. (6), ki is the wavenumber in Ri, and
Gi(r,r')=exp(jki|rr'|)4π|rr'|
(7)
is the homogeneous Green’s function in Ri. Eiinc(r') and Hiinc(r') are the incident fields due to the sources located inside region Ri, and Ji(r') and Mi(r') are the equivalent electric and magnetic currents placed over the boundary surface S, as given by the equivalence principle for each region Ri. According to the boundary conditions (continuity of tangential field components across the boundary surface S), the surface current densities in both sides of S should satisfy J1=J2 and M1=M2, thus the problem can be described using only one set of equivalent electric and magnetic currents on S, namely J=J1=J2 and M=M1=M2.

Among the various possibilities of combination of the integral equations Eqs. (1)(4), we have chosen the procedure of [7

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]

] to derive two stable and well tested SIEs. The 1/ηT-EFIE is combined with the N-MFIE, and the N-EFIE is combined with the ηT-MFIE leading to the electric current combined field integral equation (JCFIE) and the magnetic current combined field integral equation (MCFIE) in regionRi:

JCFIEi=aiηi1T-EFIEi+biN-MFIEi
(8)
MCFIEi=ciN-EFIEi+diηiT-MFIEi
(9)

The JCFIE and MCFIE equations for regions R1 and R2 are then combined respectively to obtain two independent integral equations for the two unknown functions J and M:

JCFIE:a1(1(J)η11K1(M))tan+a2(2(J)η21K2(M))tan+b1n^×(K1(J)+η111(M))b2n^×(K2(J)+η212(M))+12(a1η11a2η21)n^×M+12(b1+b2)J=a1η11(E1inc)tana2η21(E2inc)tan+b1n^×H1inc+b2n^×H2inc,rS
(10)
MCFIE:c1n^×(η11(J)K1(M))+c2n^×(η22(J)K2(M))+d1(η1K1(J)+1(M))tan+d2(η2K2(J)+2(M))tan+12(c1+c2)M12(d1η1d2η2)n^×J=c1n^×E1incc2n^×E2inc+d1η1(H1inc)tand2η2(H2inc)tan,rS
(11)

3. Discretization of the integral equations

Amni=Δmfmi(fn)dS
(15)
Bmni=ΔmfmKi(fn)dS
(16)
A'mni=Δmfmn^m×i(fn)dS
(17)
B'mni=Δmfmn^m×Ki(fn)dS
(18)
Imn=ΔmfmfndS
(19)
I'mn=Δmfmn^m×fndS.
(20)
Em=Δmfm[a1(ηi1Eiinc)tana2(η21Ejinc)tan]dS
(21)
Hm=Δmfm[d1(η1H1inc)tand2(η2H2inc)tan]dS
(22)
E'm=Δmfm[c1n^m×E1inc+c2n^m×E2inc]dS
(23)
H'm=Δmfm[b1n^m×H1inc+b2n^m×H2inc]dS
(24)

Regrouping terms in Eq. (13) and Eq. (14), the system of linear equations can be expressed in the usual dense matrix system form:
Z¯I=V
(25)
where the impedance matrix Z¯can be written as
Z¯=[Z¯1JZ¯1MZ¯2JZ¯2M]
(26)
with the elements given by

Z¯mn1J=a1Amn1+a2Amn2+b1B'mn1b2B'mn2+12(b1+b2)Imn
(27)
Z¯mn1M=a1η11Bmn1a2η21Bmn2+b1η11A'mn1b2η21A'mn2+12(a1η11a2η21)I'mn
(28)
Z¯mn2J=d1η1Bmn1+d2η2Bmn2c1η1A'mn1+c2η2A'mn212(d1η1d2η2)I'mn
(29)
Z¯mn2M=d1Amn1+d2Amn2+c1B'mn1c2B'mn2+12(c1+c2)Imn
(30)

Similarly we define I=[JM]T and V=[V1V2]T, with superscript T indicating vector transposition, and where

Vm1=Em+H'm
(31)
Vm2=E'm+Hm.
(32)

Different known formulations can be obtained depending on the selection of the complex combination parameters ai,bi,ci,di for i=1,2 (see Table 1

Table 1. Combination parameters ai,bi,ci,di and Onorm(Z¯) for the five formulations considered

table-icon
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).

4. Numerical balance of the impedance matrix

The impedance matrix in Eq. (26) has a four block structure. At this point, we will address the balance between the elements inside this matrix. Looking at the definition of the elements given in Eqs. (27)(30), and considering that the quantities given in Eqs. (15)(20) are the same order of magnitude, namely, O(Amni)O(Amn'i)O(Bmni)O(Bmn'i)O(Imni)O(Imn'i)Θ for i=1,2, the order of the four blocks of Z¯ becomes
O(Z¯)=O([Z¯1JZ¯1MZ¯2JZ¯2M])(Θf(a1,a2,b1,b2)Θg(a1,a2,b1,b2,η1,η2)Θh(c1,c2,d1,d2,η1,η2)Θf(c1,c2,d1,d2))
(33)
where 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).

Thereafter, since the combination parameters ai,bi,ci,di are defined in the same way inside each region (although they might actually have different values when they depend on the constitutive parameters of the media), we can assume that they are the same order of magnitude. So, let us assume that O(a1)O(a2)a, O(b1)O(b2)b, O(c1)O(c2)c, and O(d1)O(d2)d. Similarly, we can assume that O(η1)O(η2)η.

O([Z¯1JZ¯1MZ¯2JZ¯2M])(Θ(a+b)Θ(a+b)ηΘ(c+d)ηΘ(c+d))
(34)

In order to appreciate/study the imbalance in the order of magnitude between the different blocks of matrix Z¯, let us consider a normalized version of the above expression, in which the order of magnitude has been normalized with respect to the order of magnitude of Z¯1J:

Onorm(Z¯)=O(Z¯)O(Z¯1J)(11η(c+d)(a+b)η(c+d)(a+b))
(35)

Looking at Table 1, it becomes clear that in the PMCHWT formulation the diagonal blocks differ by a factor of η2, while the off-diagonal blocks are the same order and differ by a factor of 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 η2, and 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 η2and 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.

Z˜=[Z¯1JZ¯1MZ¯2JZ¯2M](1η(a+b)(c+d)1η(a+b)(c+d))
(36)

Different designations can be made for a, b, c, d and η. In this work, we have selected for these parameters the outer medium values: a = a1, b = b1, c = c1, d = d1, and η = η1, 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

Z˜=M¯LZ¯M¯R
(42)

The diagonal values of the diagonal matrices in Eq. (39) and Eq. (40), α11,α22,β11 and β22, can be obtained by comparing Eq. (42) to Eq. (36). For simplicity in the derivation of the wanted elements, we will take α11=β11=1. So, the coefficients for the left and right diagonal matrices that balance the impedance matrix blocks are given by

α11=β11=1β22=ηα22=a+bc+d1η
(43)

Using the above definitions, Eq. (41) is a well-balanced system with low condition number which can be easily solved using any iterative scheme. Once Eq. (41) is solved, the solution to the original electromagnetic problem is finally obtained from Eq. (37) as I=M¯RI˜.

Regarding the computational cost, the matrices involved in the preconditioner are diagonal, so the required number of multiplications become O(N), N being the number of unknowns.

6. Numerical examples

The condition number of the matrix corresponding to each conventional SIE formulation under consideration is studied next. Spheres of different size and different permittivity values have been considered. The Rao-Wilton-Glisson (RWG) [42

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]

] basis functions have been used for the discretization of the problems. The analyses have been carried out at an exterior (vacuum) wavelength of λ0 = 546 nm and a mean mesh size of λ0/15 has been considered in all cases except for the smallest sphere, where a mesh refinement was required in order to guarantee the geometrical sphericity.

Figure 1
Fig. 1 Condition number without/with preconditioner vs. sphere size for dielectric spheres of εr = 2.1 analyzed with the five formulations.
shows the matrix condition number with and without preconditioner for several dielectric spheres characterized by εr = 2.1 that have been analyzed using the five formulations. The condition number is represented as a function of k0r, where k0 is the wave number in vacuum and r the radius of the sphere. The results of 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.
and Fig. 3
Fig. 3 Condition number without/with preconditioner vs. sphere size for dielectric spheres of εr = 25 analyzed with the five formulations.
correspond to a gold plasmonic sphere with εr = −5.84−j2.11 [43

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

] and a high-contrast dielectric sphere of εr = 25, respectively.

Under the above conditions, the use of a low-cost L&R preconditioner that improves the convergence without affecting the nature of the SIE formulation constitutes the best option. Figure 4 shows that the pursued improvement of the PMCHWT convergence is attained with the preconditioner proposed in this work, enabling the accurate analysis of the problem. In fact, even taking the result after only 8 outer iterations (relative residue below 10−3) 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

An efficient left and right (L&R) diagonal preconditioner has been presented to balance the different-scale blocks of the impedance matrix in the context of SIE-MoM formulations. The good balance between matrix elements and between electric and magnetic unknowns is straightforwardly obtained during the iterative solution, without altering the formulation at all. This allows fixing the condition number and convergence issues regardless the applied formulation, which can now be selected solely in terms of accuracy for a given application, without worrying at all about its convergence properties. The results demonstrate that the preconditioner leads to significantly better conditioned systems, making even possible to reach fast convergence with typically very slowly converging formulations. Indeed, this is the case with PMCHWT. Although this formulation has demonstrated to be very accurate in plasmonics, its lack of convergence prevented its use in the case of large problems where the direct solution is not feasible. The proposed preconditioner overcomes this difficulty achieving the desirable combination of fast convergence and high accuracy with the PMCHWT formulation.

Finally, the proposed preconditioning scheme is completely general. Proper expressions can be easily derived for any particular SIE formulation. Besides, due to its simplicity and low cost of O(N), it can be easily and efficiently integrated into any existing MoM implementation.

Acknowledgments

This work was supported by the Spanish Government and FEDER (projects TEC2011-28784-C02-01, TEC2011-28784-C02-02, CONSOLIDER-INGENIO2010 CSD2008-00068).

References and links

1.

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

2.

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

3.

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

4.

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5.

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

6.

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]

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]

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]

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]

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]

12.

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]

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 27(10), 2261–2271 (2010). [CrossRef] [PubMed]

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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]

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 18(15), 15876–15886 (2010).

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 28(7), 1341–1348 (2011). [CrossRef] [PubMed]

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]

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]

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]

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]

25.

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

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]

34.

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).

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 11(4), 1383–1398 (1994). [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]

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]

41.

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

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]

43.

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

45.

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

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

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  2. C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer, 1969).
  3. A. J. Poggio and E. K. Miller, Computer Techniques for Electromagnetics (Permagon, 1973), Chap. 4.
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  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]
  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]
  12. 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]
  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 27(10), 2261–2271 (2010). [CrossRef] [PubMed]
  14. 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]
  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 18(15), 15876–15886 (2010).
  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 28(7), 1341–1348 (2011). [CrossRef] [PubMed]
  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]
  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]
  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]
  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]
  25. Y. Saad, Iterative Methods for Sparse Linear Systems (PWS Publishing, 1996).
  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]
  34. 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).
  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 11(4), 1383–1398 (1994). [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]
  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]
  41. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).
  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]
  43. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [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]
  45. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

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