Institute of Communications and Computer Systems, School of Electrical and Computer Engineering, National Technical. University of Athens, Athens, Greece.
RADIO SCIENCE, VOL. 39, RS5015, doi:10.1029/2004RS003028, 2004
Error estimation and optimization of the method of auxiliary sources (MAS) for scattering from a dielectric circular cylinder Hristos T. Anastassiu and Dimitria I. Kaklamani Institute of Communications and Computer Systems, School of Electrical and Computer Engineering, National Technical University of Athens, Athens, Greece
Received 7 January 2004; revised 21 May 2004; accepted 14 July 2004; published 28 October 2004.
[1] This article presents a rigorous error estimation of the method of auxiliary sources
(MAS) when applied to the solution of the electromagnetic scattering problem involving dielectric objects. The geometry investigated herein is a circular, dielectric cylinder of infinite length. The MAS matrix is inverted analytically, via advanced eigenvalue analysis, and an exact expression for the boundary condition error owing to discretization is derived. Furthermore, an analytical formula for the condition number of the linear system is also extracted, explaining the irregular behavior of the computational error resulting from numerical matrix inversion. Also, the effects of the dielectric parameters on the error are fully investigated. Finally, the optimal location of the auxiliary sources is INDEX TERMS: 0619 Electromagnetics: determined on the grounds of error minimization. Electromagnetic theory; 0644 Electromagnetics: Numerical methods; 0669 Electromagnetics: Scattering and diffraction; 0689 Electromagnetics: Wave propagation (4275); KEYWORDS: method of auxiliary sources, optimization, cylinder Citation: Anastassiu, H. T., and D. I. Kaklamani (2004), Error estimation and optimization of the method of auxiliary sources (MAS) for scattering from a dielectric circular cylinder, Radio Sci., 39, RS5015, doi:10.1029/2004RS003028.
1. Introduction [2] The method of auxiliary sources (MAS) [PopovidiZaridze et al., 1981; Zaridze et al., 1998a, 1998b; Leviatan and Boag, 1987; Leviatan et al., 1988; Leviatan, 1990; Kaklamani and Anastassiu, 2002; Anastassiu et al., 2002, 2003, 2004] is generally considered as a promising alternative to standard integral equation techniques, such as the method of moments (MoM). Its inherent advantages include low computational cost [Anastassiu et al., 2002], simple algorithmic structure (with respect to the matrix elements calculations) and substantial physical insight. Owing to its attractive features, it has successfully been applied to a very large variety of radiation and scattering problems [Kaklamani and Anastassiu, 2002]. Nevertheless, MAS is still not as popular as MoM, since the latter is still considered generally more reliable for the extraction of reference data. The main reason for this is the limited robustness of MAS, which is due to the ambiguity related to the location of the auxiliary
Copyright 2004 by the American Geophysical Union. 0048-6604/04/2004RS003028
sources (AS). In theory, there is no uniquely determinable location for the AS, although its choice affects the solution efficiency, indeed. It has been observed that, poor AS positioning often leads to an inexplicable, irregular behavior of the numerical solution. This behavior usually translates into slow convergence rates or unacceptably high boundary condition errors. [3] Very recently, a rigorous assessment of the MAS accuracy for scattering from a perfectly conducting (PEC) circular cylinder has been carried out [Anastassiu et al., 2003, 2004]. It was shown that for this particular geometry, the MAS matrix can be inverted analytically, via spectral analysis and matrix diagonalization. The eigenvalues and eigenvectors were evaluated using a technique [Warnick and Chew, 2000] based on the addition theorem of cylindrical functions [Hochstadt, 1986, p. 229]. The analysis resulted in the derivation of an exact expression for the boundary condition error, which is due to discretization (from now on merely called ‘‘error’’ for brevity), as well as for the system condition number. Comparisons showed that the theoretical, analytically evaluated error, and the actual, computational error (produced by numerical matrix inversion) were generally identical, except for some range of the auxiliary surface radius, where discrepancies
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occurred. These discrepancies revealed an irregular behavior of the computational error, and were fully explained on the grounds of noise smearing, introduced by poor matrix conditioning. It was also demonstrated that a few specific locations of the AS yielded particularly high boundary errors. It was shown that these locations were related to the zeros of Bessel functions of integer order, and were physically interpreted as resonance effects. Finally, optimization of the MAS solution was achieved, by choosing the AS location so that the boundary error was minimized. [4] Although the results in the work of Anastassiu et al. [2003, 2004] explained the MAS behavior to a great extent for scattering from a PEC, cylindrical surface, they are not directly applicable to nonmetallic or arbitrarily shaped boundaries. However, it is well known that MAS capabilities can be exploited in many more cases [Kaklamani and Anastassiu, 2002], including dielectric circular [Leviatan and Boag, 1987], coated circular [Leviatan et al., 1988] and impedance square cylinders [Anastassiu et al., 2002]. In the references above it was shown that the far field results could be highly accurate, since the error was usually possible to reduce by moving the AS on the basis of experience and physical insight, but there was no investigation into the mathematics of the error-producing mechanisms. However, the error skyrocketed abruptly and unexpectedly, for special AS locations, just like in the PEC case. These locations were merely avoided in the solution process, but it remained unclear what exactly the cause of this phenomenon was. [5] Given the potential of MAS to become a very useful and efficient technique for a wide range of electromagnetic problems in the near future, deep understanding of its accuracy characteristics is absolutely necessary, for the most generic scatterers possible. Since the MAS optimization issue has been largely unresolved so far, it is natural that relatively simple, canonical geometries must be investigated, as a first step toward this direction, setting the basis for future analysis of more complicated configurations. The purpose of this work is the MAS error estimation for scattering from dielectric circular cylinders. The method invoked is closely related to Anastassiu et al. [2003, 2004], involving eigenvalue analysis and matrix diagonalization, although significant modifications are necessary, since the boundary conditions are now different. Knowledge of an analytical expression for the error, as a function of the geometry parameters, allows MAS optimization with respect to the location of the AS, exactly like in the PEC case [Anastassiu et al., 2003, 2004]. [6] The paper is organized as follows: Section 2 presents the analytical inversion of the MAS linear system, in addition to the derivation of the analytical error expressions for transverse magnetic (TM) polarization. Section 3 presents the derivation of the relevant
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Figure 1. Geometry of the problem. Black, white, and gray bullets represent auxiliary sources (AS), collocation points (CP), and midpoints (MP), respectively. The gray disk corresponds to the dielectric cylinder. matrix condition number. Section 4 presents numerical results whereas optimization issues are discussed. Finally, section 5 summarizes and concludes the article. An ejwt time convention is assumed and suppressed throughout the paper.
2. Analytical Considerations (TM Polarization) [7] Assume a dielectric, infinite, circular cylinder of radius b characterized by complex relative permittivity er and relative permeability equal to 1. The dielectric is assumed to be linear, homogeneous and isotropic. The structure is illuminated by a plane wave impinging from a direction with polar angle fi (see Figure 1). The polarization of the plane wave is assumed to be transverse magnetic (TM) with respect to the cylinder axis z. The incident electric field at an arbitrary point (r, f) is therefore expressed by Ezi ¼ E0 exp jk0 r cos fi f ; ð1Þ where k0 is the free space wave number and E0 is the amplitude of the incident electric field, whereas the corresponding incident magnetic field, tangential to the cylinder, is given by Hfi ¼
k0 E0 cos fi f exp jk0 r cos fi f : ð2Þ wm0
To construct the MAS solution [Leviatan and Boag, 1987], two fictitious auxiliary surfaces Sin and Sout are
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defined, both conformal to the actual boundary S. The first surface Sin is located inside the dielectric scatterer, and hence has a circular cross section of radius ain < b. The second surface Sout is located outside the dielectric scatterer, and hence has a circular cross section of radius aout > b (see Figure 1). A number N of AS, in the form of elementary electric currents, are located on each one of Sin or Sout, radiating elementary electric fields, proportional to the two-dimensional Green’s function. The AS on Sin radiate outside the scatterer, whereas AS on Sout radiate inside it. Matching the boundary condition (equality of the electric and magnetic tangential fields) at M = N collocation points (CP) on the z plane projection of the scatterer surface yields the MAS square linear system. [8] Assume that the AS on Sin are located at azimuth angles f0p p 2p/N, p = 1,. . ., N, radiating an electric field equal to j ð2Þ out Epz ðRÞ ¼ dp H0 k0 R R0p 4 1 X j ð2Þ ¼ dp Jl ðk0 ain ÞHl ðk0 rÞ 4 l¼1 n o exp jl f f0p ; jRj b;
ð3Þ
of the latter remains intact. The boundary condition for the electric field yields N X
in Eqz ðRm Þ
q¼1
j ð2Þ in Eqz ðRÞ ¼ cq H0 k R R0q 4 1 X j ð2Þ ¼ cq Jl ðkrÞHl ðkaout Þ 4 l¼1 n o exp jl f f0q ; jRj b;
N X
out Epz ðRm Þ ¼ Ezi ðRm Þ;
ð5Þ
whereas the boundary condition for the magnetic field yields N X
in Hqf ðRm Þ
q¼1
N X
out Hpf ðRm Þ ¼ Hfi ðRm Þ;
m ¼ 1; . . . ; N :
p¼1
ð6Þ
After evaluating the magnetic fields corresponding to equations (3) and (4), the combination of equations (5) and (6) can be written in a compact, block matrix form as 9 8 9 2 38 ½ U ½ V < fC g = < A i = 4 5 ; ð7Þ ¼ : ; : i ; ½W ½Y fD g B where [U], [V], [W], [Y] are N N square matrices with elements given by 1 j X ð2Þ Jl ðkbÞHl ðkaout Þ expfjl ðfm fn Þg; 4 l¼1
ð8Þ
Vmn
1 j X ð2Þ Jl ðk0 ain ÞHl ðk0 bÞ expfjl ðfm fn Þg; 4 l¼1
ð9Þ
Wmn
pffiffiffiffi 1 j er X ð2Þ J_l ðkbÞHl ðkaout Þ 4 l¼1 expfjl ðfm fn Þg;
ð4Þ
cq being the unknown weight of the corresponding AS, and k being the wave number inside the dielectric. Finally, the CP lie at azimuth angles fm m 2p/N, m = out,in corre1,. . ., N. Obviously, the magnetic fields Hp,q,f sponding to equations (3) and (4) are trivially derived via the Maxwell’s equations, but their expressions are not given here for brevity. The weights cq and dp are determined by imposing the boundary condition for all CP located at the points Rm on the cylindrical boundary. The j/4 term in equations (3) and (4) has not been incorporated to the weights, so that the physical meaning
m ¼ 1; . . . ; N ;
p¼1
Umn
where H(2) l (.) is the Hankel function of l order and second kind, and dp is the unknown weight of the AS. The second expression in equation (3) has been derived via use of the addition theorem for cylindrical functions [Hochstadt, 1986, p. 229], in a manner similar to Anastassiu et al. [2003, 2004]. Similarly, the AS on Sout are located at azimuth angles fq0 q 2p/N, q = 1,. . ., N, radiating an electric field equal to
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Ymn
ð10Þ
1 j X ð2Þ Jl ðk0 ain ÞH_ l ðk0 bÞ expfjl ðfm fn Þg; 4 l¼1
ð11Þ
where the dot over the cylindrical functions denotes differentiation with respect to the entire argument, {Ai}, {Bi} are N 1 column vectors, with Ain E0 exp jk0 b cos fi fn ; ð12Þ
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Bin jE0 cos fi fn exp jk0 b cos fi fn ; ð13Þ
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representing the samples of the incident fields, and finally, {C}, {D} are N 1 column vectors containing the unknown AS weights. [9] To derive an expression for the boundary condition error, in a way similar to Anastassiu et al. [2003, 2004], equation (7) must be inverted analytically. Such an inversion is feasible, given that each one of [U], [V], [W], [Y] is exactly diagonalizable, on the basis of the method described in the work of Anastassiu et al. [2003, 2004] and Warnick and Chew [2000]. Indeed, invoking the diagonalization scheme of Anastassiu et al. [2003, 2004], it can be shown that ½U ¼ ½G ½Du ½G 1 ;
½V ¼ ½G ½Dv ½G 1 ;
ð14Þ
½W ¼ ½G ½Dw ½G 1 ;
½Y ¼ ½G Dy ½G 1 ;
ð15Þ
where [Du], [Dv], [Dw], [Dy] are diagonal matrices containing the eigenvalues of [U], [V], [W], [Y] respectively, given by (q = 1,. . ., N)
uq ¼
vq ¼
1 jN X ð2Þ JqþsN ðkbÞHqþsN ðkaout Þ; 4 s¼1
jN 4
1 X
ð2Þ
JqþsN ðk0 ain ÞHqþsN ðk0 bÞ;
yq ¼
1 jN X ð2Þ JqþsN ðk0 ain ÞH_ qþsN ðk0 bÞ; 4 s¼1
that [U], [V], [W], [Y] are all circulant, and therefore possess the properties described in Appendix A. Using equations (14) – (21) the square matrix in equation (7) can be written as 2 ½Z 4
½U ½V
3
2
5¼4
½W ½Y
½ G ½ 0
32 54
½0 ½G
½Du ½Dv
½Dw Dy
32 54
½G 1 ½0 ½0 ½G
1
3 5;
ð22Þ
where [0] is the N N null matrix, and therefore its inverse is 2 32 3 ½ G ½ 0 Dy ½Dv 6 76 7 ½Z 1 ¼4 54 5 ½0 ½G ½Dw ½Du 2 1 32 1 3 ½G ½0 ½P ½0 6 76 7 ð23Þ 4 54 5; ½0 ½P 1
½0 ½G 1
where ½P ½Du Dy ½Dw ½Dv
ð16Þ
ð17Þ
s¼1
pffiffiffiffi 1 j er N X ð2Þ wq ¼ J_qþsN ðkbÞHqþsN ðkaout Þ; 4 s¼1
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ð18Þ
ð19Þ
and [G] is the eigenvector square matrix (common for all [U], [V], [W], [Y]), defined by ð20Þ ½ G fg g1 ; fg g2 ; . . . ; fg gN ;
is the block matrix ‘‘determinant’’ (the proof of equation (23) is straightforward, by showing directly that [Z]1 [Z] = [Z][Z]1 = [I], i.e., that both products equal the identity matrix). Hence inversion of equation (7) in view of equation (23) yields an analytic expression for the unknown weights. [10] Suppose, now, that we are interested in calculating the boundary condition error at points of the boundary e where 0 surface with azimuth angles equal to fm + f, e 2p/N. Obviously the choice f e = p/N corresponds to f the midpoints (MP) between the CP (see Figure 1). The net fields at the MP, i.e., the inner minus outer field differences occurring as left-hand sides in equations (5) and (6), are given, in a way similar to Anastassiu et al. [2003, 2004] by 8n o 9 2 h ih i 38 9 e rad > > e < fC g = e V U = < A 6 7 ; n o ¼ 4 h ih i 5 > > : B e : fD g ; f Y e rad ; W
where 1 fggq pffiffiffiffi N ½expfjqf1 g; expfjqf2 g; . . . ; expfjqfN g T ;
ð24Þ
ð21Þ
where h i h i e ¼ ½G D e u ½G 1 ; U
are the normalized eigenvectors, identical to the PEC case [Anastassiu et al., 2003, 2004]. Alternatively, the results in equations (14) – (21) can be derived in view of the fact
h i h i f ¼ ½G D e w ½G 1 ; W
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h i h i e ¼ ½G D e v ½G 1 ; V
ð25Þ
ð26Þ
h i h i e ¼ ½G D e y ½G 1 ; ð27Þ Y
ANASTASSIU AND KAKLAMANI: ERROR OF METHOD OF AUXILIARY SOURCES
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e u], [D e v], [D e w], [D e y] are diagonal matrices with and [D respective elements given by (q = 1,. . ., N) 1 jN X ~ ð2Þ e JqþsN ðkbÞHqþsN ðkaout ÞejðqþsN Þf ; ð28Þ uq ¼ 4 s¼1
where em1
ð30Þ eyq ¼
em2
k
k
k
ei
ei
: i ; B
k
2
where {A }, {B } are the incident fields of equations (12) and (13) evaluated at the MP, and k .k 2 is the standard 2-norm. It is evident, in view of equations (7) and (25) – (31), that when the MP coincide with the CP, i.e., e = 0, the error in equation (32) vanishes, as when f e= expected. Also, for N ! 1, it follows that (q + sN)f e (q + sN)Np ! sp ) ejðqþsN Þf ! (1)s, thus the sums in equations (16) – (19) and (28) – (31) become one by one identical, in view of the rapidly decreasing behavior of the cylindrical functions products involved, which effectively obliterates all terms except the s = 0 one. On the basis of this observation, equations (7), (25), and (32) imply that the error vanishes in the limit as N ! 1, verifying the convergence properties of MAS, just like in the PEC cylinder [Anastassiu et al., 2003, 2004]. In the general case, equation (32) can be evaluated explicitly using equations (7) and (25) – (31), after a considerable amount of tedious algebra. The final result for the normalized error can be written as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N N uX X u jem1 j2 þ jem2 j2 u um¼1 m¼1 ; ð33Þ eðain ; aout ; b; N Þ ¼ u t 3 2 NE0 2
ð34Þ
N X N 1 X exp jðm nÞfp lpð21Þ Ain þ lpð22Þ Bin N n¼1 p¼1
ei ; B m
ð35Þ
whereas fp p 2p/N and
1 jN X e ð2Þ JqþsN ðk0 ain ÞH_ qþsN ðk0 bÞejðqþsN Þf : ð31Þ 4 s¼1
The MP normalized error in the boundary condition can be defined, in a manner analogous to Anastassiu et al. [2003, 2004] by 8n o9 8n o9 ei > e rad > > = > = < A < A n o n o > ; > ; : B : B e rad > ei > 2 eðain ; aout ; b; N Þ ; ð32Þ 8 i 9 < A =
N X N 1 X exp jðm nÞfp lpð11Þ Ain þ lpð12Þ Bin N n¼1 p¼1
ei ; A m
1 jN X e ð2Þ evq ¼ JqþsN ðk0 ain ÞHqþsN ðk0 bÞejðqþsN Þf ; ð29Þ 4 s¼1
pffiffiffiffi 1 j er N X e ð2Þ eq ¼ J_qþsN ðkbÞHqþsN ðkaout ÞejðqþsN Þf ; w 4 s¼1
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lqð11Þ
e uq v q uq yq evq wq ð12Þ evq uq e ; l ; uq yq vq wq q uq yq vq wq
ð36Þ
lqð21Þ
e q vq e q yq eyq wq ð22Þ eyq uq w w ; l : uq yq vq wq q uq y q v q w q
ð37Þ
To achieve the highest possible accuracy for the MAS solution, e in equation (33), must be minimized by choosing the most appropriate ain and aout for given b, and N. [11] Like in the PEC case [Anastassiu et al., 2003, 2004], the analytical expression for the boundary condition error reveals the occurrence of resonance effects, i.e., situations where a poor choice of the AS location may cause very high errors. In the dielectric case, it follows from equations (33) – (37) that a resonance occurs when uq yq vq wq ¼ 0:
ð38Þ
Substituting equations (16) – (19) into equation (38), and assuming very large N, so that only the s = 0 terms are significant, we obtain Jq ðk0 ain ÞHqð2Þ ðkaout Þ " # k _ ð2Þ ð2Þ _ Jq ðkbÞHq ðk0 bÞ Jq ðkbÞHq ðk0 bÞ ffi 0: ð39Þ k0
Equation (39) implies that a resonance may occur whenever any of the three factors in the product vanishes. To begin with, the equation H(2) q (kaout) = 0 does not have any roots with Im{kaout} 0 [Abramowitz and Stegun, 1972, p. 373], and therefore no location of the AS on Sout may cause any resonance effects (for all physically realizable dielectrics, Im{k} 0 due to the ejwt time dependence convention). Moreover, it is rigorously proven in Appendix B that the square brackets of equation (39) cannot vanish for any, possibly lossy dielectric, and any scatterer radius b. Finally, the situation
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Jq(k0ain) = 0 is identical with the PEC case [Anastassiu et al., 2003, 2004], and is the only one that may cause resonance effects. Hence in the dielelectric cylinder case, like in the PEC situation, it is only the location of the interior AS that should be carefully chosen to avoid any poor behavior of the method. [12] To simulate the same scattering geometry for the TE polarization, it is assumed that the incident magnetic field is polarized along the z direction, and the AS are represented by elementary magnetic currents. Performing the same analysis described earlier, it turns out that the expression for the resulting MAS linear system is almost identical to equations (7) – (13), the pffiffiffiffionly difference being that now in equation (10) the er term occurs in the denominator. Since the expressions are shown to be so similar for the two polarizations, there is no need to repeat the analysis for the TE incidence.
3. Matrix Condition Number (TM Polarization) [13] To fully assess the accuracy of the numerical MAS solution, it is important to investigate the behavior of the matrix condition number, since the latter largely determines the significance of the computational (roundoff) errors. Unlike in the PEC case [Anastassiu et al., 2003, 2004], the MAS square matrix for the dielectric cylinder, given in the left-hand side of equation (7), is not normal. Therefore the condition number k2 cannot be determined by the ratio of its eigenvalues, but only through its singular values mq, i.e., n o pffiffiffiffiffi max mq max lq q q n o¼ pffiffiffiffiffi ; k2 ½Z ¼ min lq min mq q
ð40Þ
Since equation (41) is a similarity transformation, the eigenvalues of [Z]* [Z] are equal to the eigenvalues of 2 ½6 4
2 ½Z * ½Z 4
½G ½0
32 54
½0 ½G
½G ½D ½7 ½Y
32 54
½G 1 ½0 1
3 5;
ð41Þ
½0 ½G
where [&], [#], [7], [Y Y] are N N diagonal matrices with respective elements (q = 1,. . ., N) 2 2 gq uq þwq ; dq u*v ð42Þ q q þ w*y q q ¼ x*; q xq vq*uq þ yq*wq ¼ dq*;
2 2 yq vq þyq : ð43Þ
½& ½# ½7 ½Y
3 5;
ð44Þ
which are evidently determined by setting det([6] l[I]) = 0. Since all blocks of [6] l[I] are diagonal, the latter matrix can be inverted via the technique explained in equations (23) and (24). Therefore the determinant vanishes if ([&] l[I])([Y Y] l[I]) [7][#] is not invertible, i.e., if detfð½& l½I Þð½Y l½I Þ ½7 ½# g ¼ 0:
ð45Þ
Since all matrices are diagonal, equation (45) is equivalently written as N h N i Y Y gq l yq l xq dq ¼ 0 , l lþ q q¼1
q¼1
l l ¼ 0; q
ð46Þ
where l q
1 gq þ yq 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi! gq yq
2
þ4xq dq :
ð47Þ
From equation (46) it follows that the eigenvalues of [6] are equal to l+1 , l+2 ,. . ., l+N, l 1 , l2 ,. . ., lN and the condition number of [Z] is finally equal to nqffiffiffiffiffiffio max lþ q q *qffiffiffiffiffiffi+ : k2 ½Z ¼ min l q
q
where lq are the eigenvalues of the matrix [Z]* [Z] (the asterisk denotes the complex transpose). Using equation (22), it can be shown after some elementary algebra that
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ð48Þ
q
Owing to the intricacy of the expression for the condition number, any asymptotic estimates that were feasible in the work of Anastassiu et al. [2003, 2004] and Warnick and Chew [2000] for other types of boundary conditions, are not obviously derivable in the dielectric case. However, it is possible to determine the situations when equation (48) can have a vanishing denominator. Indeed, setting equation (47) equal to 0, some tedious algebra finally yields that equivalently Jq(k0ain) = 0, which is exactly the situation when the boundary condition error approaches infinity (see equation (39) and the discussion following it). This property is strongly reminiscent of the PEC case [Anastassiu et al., 2003, 2004]. Finally, using the appropriate asymptotics, it can be shown that the
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yields the actual values of the matrix elements, introduces bad system conditioning. However, the discrete nature of the method gives rise only to the analytical error, given by equations (32) – (37). [15] The resonance effects, related to the zeros of the Bessel function, are represented by a protrusion between ain/b = 0.7 and 0.8. This ‘‘bump’’ corresponds to the first zero of J0(.), which is equal to j0,1ffi 2.405 [Abramowitz and Stegun, 1972, p. 409]. The argument of J0(k0ain) equals j0,1 when ain/b ffi 0.7655, which is the precise location of the bump in the plot. In general, if jq,n is the nth root of Jq(.), error protrusions are expected for
0
