Error estimation and optimization of the method of auxiliary sources (MAS) for scattering from a dielectric circular cylinder

Authors


Abstract

[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 determined on the grounds of error minimization.

1. Introduction

[2] The method of auxiliary sources (MAS) [Popovidi-Zaridze 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 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 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 matrix condition number. Section 4 presents numerical results whereas optimization issues are discussed. Finally, section 5 summarizes and concludes the article. An ejωt 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 ɛr 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 ϕi (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 (ρ, ϕ) is therefore expressed by

equation image

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

equation image

To construct the MAS solution [Leviatan and Boag, 1987], two fictitious auxiliary surfaces Sin and Sout are 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.

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.

[8] Assume that the AS on Sin are located at azimuth angles ϕ′pp · 2π/N, p = 1,…, N, radiating an electric field equal to

equation image

where Hl(2)(•) 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 ϕ′qq · 2π/N, q = 1,…, N, radiating an electric field equal to

equation image

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 ϕmm · 2π/N, m = 1,…, N. Obviously, the magnetic fields Hp,qout,in corresponding 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 ρm 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 of the latter remains intact. The boundary condition for the electric field yields

equation image

whereas the boundary condition for the magnetic field yields

equation image

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

equation image

where [U], [V], [W], [Y] are N × N square matrices with elements given by

equation image
equation image
equation image
equation image

where the dot over the cylindrical functions denotes differentiation with respect to the entire argument, {Ai}, {Bi} are N × 1 column vectors, with

equation image
equation image

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

equation image
equation image

where [Du], [Dv], [Dw], [Dy] are diagonal matrices containing the eigenvalues of [U], [V], [W], [Y] respectively, given by (q = 1,…, N)

equation image
equation image
equation image
equation image

and [G] is the eigenvector square matrix (common for all [U], [V], [W], [Y]), defined by

equation image

where

equation image

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 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

equation image

where [0] is the N × N null matrix, and therefore its inverse is

equation image

where

equation image

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 surface with azimuth angles equal to ϕm + equation image, where 0 ≤ equation image ≤ 2π/N. Obviously the choice equation image = π/N corresponds to 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

equation image

where

equation image
equation image

and [equation imageu], [equation imagev], [equation imagew], [equation imagey] are diagonal matrices with respective elements given by (q = 1,…, N)

equation image
equation image
equation image
equation image

The MP normalized error in the boundary condition can be defined, in a manner analogous to Anastassiu et al. [2003, 2004] by

equation image

where {equation imagei}, {equation imagei} are the incident fields of equations (12) and (13) evaluated at the MP, and ∥•∥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., when equation image = 0, the error in equation (32) vanishes, as expected. Also, for N → ∞, it follows that (q + sN)equation image = (q + sN)equation imagesπ ⇒ equation image → (−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 → ∞, 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

equation image

where

equation image
equation image

whereas ϕpp · 2π/N and

equation image
equation image

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

equation image

Substituting equations (16)–(19) into equation (38), and assuming very large N, so that only the s = 0 terms are significant, we obtain

equation image

Equation (39) implies that a resonance may occur whenever any of the three factors in the product vanishes. To begin with, the equation Hq(2)(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 ejωt 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 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 only difference being that now in equation (10) the equation image 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 (round-off) 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 κ2 cannot be determined by the ratio of its eigenvalues, but only through its singular values μq, i.e.,

equation image

where λq 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

equation image

where [Γ], [Δ], [Ξ], [Ψ] are N × N diagonal matrices with respective elements (q = 1,…, N)

equation image
equation image

Since equation (41) is a similarity transformation, the eigenvalues of [Z]* [Z] are equal to the eigenvalues of

equation image

which are evidently determined by setting det([Ω] − λ[I]) = 0. Since all blocks of [Ω] − λ[I] are diagonal, the latter matrix can be inverted via the technique explained in equations (23) and (24). Therefore the determinant vanishes if ([Γ] − λ[I])([Ψ] − λ[I]) − [Ξ][Δ] is not invertible, i.e., if

equation image

Since all matrices are diagonal, equation (45) is equivalently written as

equation image

where

equation image

From equation (46) it follows that the eigenvalues of [Ω] are equal to λ1+, λ2+,…, λN+, λ1, λ2,…, λN and the condition number of [Z] is finally equal to

equation image

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 condition number approaches infinity for very large N, as expected.

4. Numerical Results and Discussion

[14] To validate the expressions derived, direct comparisons were performed between the analytical error given in equations (33)–(37) and the computational error calculated by a LU decomposition of the MAS matrix, in a manner similar to Anastassiu et al. [2003, 2004]. Figure 2 presents the comparison for a geometry with b = 0.5λ, aout = λ, ɛr = 5, TM incidence and ain varying from 0 to b. The vertical axis maps the base 10 logarithm of the error. Three sets of curves are plotted, for N = 10, 20 and 40. Although not identical in values, the overall behavior of both the analytical and the computational plots are very similar to the PEC case [Anastassiu et al., 2003, 2004]. The geometrical parameters and the numbers of unknowns were deliberately chosen to be equal to one of the PEC case studied in the work of Anastassiu et al. [2003, 2004], allowing immediate comparisons and assessment of the material effects. The irregular behavior of the computational error for small values of ain is explained by the high values of the condition number, plotted in Figure 3, which also strikingly resembles the PEC case. Discrepancies for large ain are due to the coincidence of the AS with the CP [Anastassiu et al., 2003, 2004]. It is emphasized that the MAS solution, as a discrete approximation of a continuous problem, is only indirectly responsible for the irregular behavior of the computational error for small radii ain, since the algorithm structure, which 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).

Figure 2.

Midpoint (boundary condition) error plots as a function of ain/b, for b = 0.5λ, aout = λ, ɛr = 5, TM incidence, and various numbers of unknowns N.

Figure 3.

Matrix condition number as a function of ain/b for b = 0.5λ, aout = λ, ɛr = 5, TM incidence, and various numbers of unknowns N.

[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 j0,1 ≅ 2.405 [Abramowitz and Stegun, 1972, p. 409]. The argument of J0(k0ain) equals j0,1 when ain/b ≅ 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

equation image

[16] What is particular to the dielectric cylinder, though, is the existence of an additional parameter, namely the radius of the outer auxiliary surface aout. A plot of the computational and analytical error as a function of aout for b = 0.5λ, ain = 0.325λ, ɛr = 5 and TM polarization is depicted in Figure 4. It is concluded that the MAS error is practically independent of the Sout location (unless aout is too close to b), meaning that Sout can be arbitrarily chosen without risking degradation of the method's accuracy. This result was partially expected after the analysis in section 2, where it was proven that there are no resonances associated with Sout. Also, there is some physical insight into this observation. As discussed in [Zaridze et al., 1998a, 1998b], the boundary condition error generally depends on the relative position and the proximity between the AS and the caustics of the scattered field. Since the analytic continuation of the interior modes outside the scatterer has no singularities [Harrington, 1961, p. 261], it should be expected that the exterior AS locations do not affect significantly the accuracy of the algorithm, because they cannot possibly associate with any caustics. These results on the error behavior for varying radii of Sin and Sout completely agree with the empirical observations in the work of Leviatan and Boag [1987], where no mathematical proofs were given.

Figure 4.

Midpoint (boundary condition) error plots as a function of aout/b for b = 0.5λ, ain = 0.325λ, ɛr = 5, TM incidence, and various numbers of unknowns N.

[17] Finally, the effect of the dielectric permittivity on the error is shown in Figure 5, where the logarithm of the error is plotted as a function of ɛr, for b = 0.5λ, ain = 0.325λ, aout = 10λ, and TM polarization. It is observed that the variation is not very smooth, however the occurring protrusions are due to the rapidly oscillating behavior of the functions involved in equations (32)–(37), and not to any vanishing denominators which would possibly characterize a resonance.

Figure 5.

Midpoint (boundary condition) error plot for varying dielectric permittivity, b = 0.5λ, ain = 0.325λ, aout = 10λ, TM incidence, and various numbers of unknowns N.

[18] Given the properties of the error examined in this analysis, it is interesting to address the fundamental question of MAS, i.e., what is the choice of the AS that produces the most accurate numerical solution. From the above discussion it is concluded that with respect to Sout, almost any choice, not too close to the boundary, would perform equally well. However, with respect to Sin, the inner radius should be chosen in a manner identical to the PEC [Anastassiu et al., 2003, 2004] case, i.e., the ratio ain/b should be chosen as small as possible, provided that the condition number of the system is not exceedingly large. The allowable condition number levels are determined by the arithmetic precision of the calculations. Also, the product k0ain should not lie in the vicinity of any root of any Bessel function of integer order, to avoid resonance effects. Although these results are finally proven to be the same with the PEC case, they cannot be assumed to be valid without the mathematical analysis presented herein, specifically for the dielectric cylinder, which is significantly more complicated than in the PEC configuration.

5. Conclusions

[19] In this paper, a thorough analysis of the MAS accuracy for plane wave scattering from an infinite, circular dielectric cylinder was presented. The MAS linear system was inverted analytically through eigenvalue evaluation and block matrix diagonalization, and an exact expression for the midpoint, boundary condition error due to discretization was derived. Furthermore, an exact formula for the condition number of the system was also developed. Comparative plots of the analytical and the computational error showed that the accuracy behavior of the method for a dielectric scatterer is analogous to the PEC case. Again, the analytical error approaches 0 for N → ∞, verifying the convergence properties of the technique, whereas the condition number of the system approaches infinity, implying that exceedingly large numbers of unknowns in a MAS formulation of the problem may yield unreliable results. Finally, several criteria for the optimal choice of the auxiliary surfaces' location were presented. The location of the outer auxiliary surface is practically irrelevant to the solution accuracy, whereas the criterion for the inner auxiliary surface location is identical to the one developed earlier in the literature for the PEC cylindrical scatterer.

Appendix A:: Theorem From Davies [1979]

[20] Let [A] be a N × N circulant complex matrix, i.e., [A] = circ(a1, a2,…, aN). Then [A] is normal, and thus diagonalizable. Its eigenvalues are given by

equation image

and the corresponding eigenvectors by

equation image

Appendix B

B1. Preposition

[21] The quantity

equation image

cannot vanish for any cylinder radius b, any integer order q and any, possibly lossy dielectric material, filling the cylinder.

B2. Proof

[22] By suitably rearranging equation (B1) we obtain

equation image

Expanding the Hankel functions into sums of Bessel and Neumann functions, and using the expression for their Wronskian [Abramowitz and Stegun, 1972, p. 360]

equation image

we conclude that the imaginary part of the right-hand side of equation (B2) is

equation image

However, making use of the equality [Hochstadt, 1986, p. 256]

equation image

for any complex z, where jq,p is the pth zero of Jq(•) (and hence real and positive). Since for the ejωt time convention Im{k} ≤ 0, some tedious but elementary calculation shows that

equation image

Therefore equations (B2), (B4), and (B6) clearly prove that equation (B1) cannot vanish under any circumstances.

Ancillary