# An improved cubic polynomial method for Interpolating/extrapolating MoM matrices over a frequency band.

1. INTRODUCTIONThe method of moments (MoM)-based surface integral equation solver has been a tool in wide use for analyzing time-harmonic electromagnetic (EM) radiation and scattering from perfect electrically conducting (PEC) objects [1,2]. The MoM results in a dense impedance matrix, and the MoM matrix is computationally intensive especially for frequency-sweeping cases. The polynomial interpolation and extrapolation methods [3-12,19], the rational polynomial approaches [13-18,20], and other techniques [21-27] have been proposed for fast generating the MoM matrices, the current distributions, and the radar-cross-section (RCS) results over a frequency band. The quadric polynomial interpolation method originally adopted in [3] was improved by extracting the dominant phase term [4]. A cubic polynomial interpolation scheme with derivative term was proposed in the authors' previous work [11]. In this scheme, the modified matrix element consists of the remaining phase term and the quadratic polynomial term of the normalized frequency, and it is more suitable for polynomial interpolation than the version in [4]. The position of the internal frequency sample is optimized. Hence, this scheme yields more accurate matrices over the frequency band than the schemes in [4] and [10]. However, the positions of the first and third frequency samples still need to be optimized within the frequency band. In [13-16], the rational polynomial approaches such as the Pade approximation have been developed to fast obtain the current distributions or the far field results over a frequency band. Compared with the matrix element, these parameters vary drastically with frequency [4], respectively. Hence, a number of frequency data and frequency-derivative data are needed to obtain the coefficients of the approaches, respectively.

In this paper, a cubic polynomial inter/extrapolation method is investigated for efficiently generating MoM matrices over a frequency band. The method is the improved version of the previous work [11], and its significant difference is that the positions of all three frequency samples are optimized within the frequency band by minimizing the amplitude of the polynomial component and they can be expressed as analytical forms of two terminal points of the frequency band. The lowest and highest frequency samples define one sub-band. The matrices inside and outside the sub-band are generated via interpolation and extrapolation, respectively.

The rest of this paper is organized as follows. In Section 2, a cubic polynomial inter/extrapolation method is described in detail. Section 3 provides numerical examples to validate the proposed method. Some conclusions are given in Section 4.

2. CUBIC POLYNOMIAL INTER/EXTRAPOLATION SCHEME WITH THREE OPTIMAL FREQUENCY SAMPLES

The electric field integral equation (EFIE) or the combined field integral equation (CFIE) is used to analyze the 3D EM problem from PEC objects in a free space [1]. The surface current density J induced by the incident time-harmonic EM field is expanded in terms of the RWG functions [2]. After the Galerkin's procedure, the EFIE-MoM or CFIE-MoM impedance matrix at each frequency f [member of] [[f.sub.l],[f.sub.h]] is yielded. For simplicity, we denote by [Z.sub.mn] the matrix element (10) (or (12)) in the frequency-sweeping case [11]. The modified matrix element employed here is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [f.sub.r] = f / [f.sub.h] is the normalized frequency and varies from [f.sub.l] / [f.sub.h] to 1, and [R.sub.mn] is the distance between the centers of the RWG elements [S.sub.m] and [S.sub.n] in the electrical size at [f.sub.h].

As elucidated in [11], [[??].sub.mn] is a product of a quadratic polynomial in [f.sub.r] and the remaining phase term. Compared with the quadratic polynomial, the cubic polynomial may effectively mitigate the negative effect of the remaining term on the inter/extrapolation accuracy. In [11], the optimal position of the second frequency sample is located, however the positions of the first and third frequency samples still need to be optimized.

The scheme under investigation involves three frequency samples [f.sub.i] (i = 0,1,2) within [[f.sub.l],[f.sub.h]] and the matrices at the normalized frequencies [x.sub.i] = [f.sub.i] / [f.sub.h] (i = 0,1,2) as well as the first derivative at [x.sub.1]. The inter/extrapolation formula for each [f.sub.r] in [[f.sub.l]/[f.sub.h],1] is expressed as

[[??].sup.A.sub.mn] ([f.sub.r]) = [2.summation over (i=0)][y.sub.i][[PHI].sub.i] ([f.sub.r]) + [y'.sub.1][[phi].sub.1] ([f.sub.r]), (2)

where [y.sub.i] is [[??].sub.mn] at [x.sub.i] for i = 0,1,2, and [y'.sub.1] is [[??]'.sub.mn] at [x.sub.i]; [[PHI].sub.i] ([f.sub.r]) (i = 0,1,2) and [[phi].sub.i] ([f.sub.r]) have been defined in [11]. Its remainder term is given by

[[??].sub.mn] - [[??].sup.A.sub.mn] (fr) = [[??].sup.(4).sub.mn](xi) / 4! W ([x.sub.0],[x.sub.1],[x.sub.2],[f.sub.r]), (3)

where [xi] lies in ([f.sub.l]/[f.sub.h],1), and

W ([x.sub.0],[x.sub.1],[x.sub.2],[f.sub.r]) = [([f.sub.r] - [x.sub.o])([f.sub.r] - [x.sub.i]).sup.2] ([f.sub.r] - [x.sub.2]). (4)

Let

t = [f.sub.r] - [f.sub.l]/[f.sub.h] / 1 - [f.sub.l]/[f.sub.h],

[t.sub.i] is obtained by substituting [x.sub.i] into (5) for i = 0, 1, 2. Then,

W([x.sub.0],[x.sub.1],[x.sub.2],[f.sub.r]) = [(1 - [f.sub.l]/[f.sub.h]).sup.4] U([t.sub.0],[t.sub.l],[t.sub.2],t), (6)

where t varies within [0,1], and

U([t.sub.0],[t.sub.1],[t.sub.2],t) = (t - [t.sub.0]) [(t - [t.sub.1]).sup.2] (t - [t.sub.2]). (7)

The error norm is defined, for indicating the inter/extrapolation error of matrix, as

error ([f.sub.r]) = [[parallel][??] - [[??].sup.A][parallel].sub.F] / [[parallel][??][parallel].sub.F], (8)

where [[parallel]*[parallel].sub.F] denotes the Frobenius norm of matrix. By using (3) and (6), (8) is rewritten as

error ([f.sub.r]) = coeff([t.sub.0],[t.sub.1],[t.sub.2],[f.sub.r]) [(1 - [f.sub.l]/[f.sub.h]).sup.4][absolute value of U ([t.sub.0],[t.sub.1],[t.sub.2],t)], (9)

where the error coefficient is

coeff([t.sub.0],[t.sub.1],[t.sub.2],[f.sub.r]) = [[parallel][[??].sup.(4)] ({[xi]})[parallel].sub.F] / 4![[parallel][??][parallel].sub.F] (10)

and {[xi]} denotes the set of [xi] in (3).

We have investigated extensive numerical examples for analyzing the frequency response of (8) and (10), respectively. Without loss of generality, a PEC sphere of radius 1 cm is considered. Seen from Figure 1, the error norm varies drastically with the positions of the frequency samples and the operating frequency, and the error coefficient is not sensitive to ([t.sub.0],[t.sub.1],[t.sub.2]) and [f.sub.r]. It is practical to minimize [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by optimizing ([t.sub.0],[t.sub.1],[t.sub.2]).

Section 3 in [11] shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] attains the minimum when [x.sub.i] = ([x.sub.0] + [x.sub.2]) / 2, which means that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] attains the minimum when [t.sub.1] = ([t.sub.0] + [t.sub.2]) / 2. Hence, we minimize [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by optimizing ([t.sub.0],[t.sub.2]) over [ohm] = [0,1] x [0,1].

[FIGURE 1 OMITTED]

Since the stationary points of [ansolute value of U ([t.sub.0],[t.sub.1],[t.sub.2],t)] with t over [0,1] are [t.sup.[+ or -].sub.1] = [t.sub.1] [+ or -] ([t.sub.2] - [t.sub.0])[square root of 2] / 4 and [t.sub.i] (i = 0,1,2), the minimum of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with ([t.sub.0],[t.sub.2]) over n can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

After solving (11) (see the Appendix), the optimal values of [t.sub.0] and [t.sub.2] can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] attains the minimal value of [([square root of 2] - 1).sup.2] / 16. As a result, (9) is less than or equal to

coeff([t.sub.0],[t.sub.1],[t.sub.2],[f.sub.r]) [(1 - [f.sub.l]/[f.sub.h]).sup.4] [([square root of 2] - 1).sup.2] / 16. (13)

When [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] attains the minimal value of 1 / 64, which is the case in [11]. Then, (9) is less than or equal to

coeff(0,0.5,1,[f.sub.r]) [(1 - [f.sub.l]/[f.sub.h]).sup.4] / 64. (14)

[FIGURE 2 OMITTED]

The Equations (5) and (12) show that the optimal frequency samples can be expressed as analytical forms of [f.sub.l] and [f.sub.h], and they are all located inside [[f.sub.l],[f.sub.h]]. In this way, the range between the lowest and highest frequency samples is one sub-band of [[f.sub.l],[f.sub.h]]. By using (2), the matrices inside and outside the sub-band are generated via interpolation and extrapolation, respectively.

Figures 2 and 3 plot the error norms and the error coefficients for the optimized scheme and the interpolation scheme with ([t.sub.0],[t.sub.1],[t.sub.2]) = (0,0.5,1), respectively, in the EFIE and CFIE cases. The error coefficients for the two schemes are almost identical to each other, and the error norm for the optimized scheme is less than that for the interpolation scheme. From (13) and (14), it is easily known that the error norm for the optimized scheme is reduced to about 68.63% of that for the interpolation scheme.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

A model of bowtie with edge 7.5 cm is investigated as shown in Figure 4(a), where 422 unknowns are involved. In Figure 5, the error norm for the optimized scheme is less than that for the interpolation scheme. The reason is the same as in the above case.

In the optimized scheme, the inter/extrapolation error of matrix is a product of the error coefficient and the polynomial component. The error coefficient is not sensitive to the positions of the frequency samples and the operating frequency, and the frequency behavior of the inter/extrapolation error of matrix is closely related to the polynomial component. Hence, it is practical to minimize the amplitude of the polynomial component rather than the inter/extrapolation error of matrix by optimizing the frequency samples. The above approach offers a practical guideline to locate the optimal frequency samples.

[FIGURE 6 OMITTED]

3. NUMERICAL EXAMPLES

In this section, the inter/extrapolation scheme optimized in (12) is tested on the two objects of Figures 4(a) and (b) through comparison with the MoM and the Pade approximation [14]. The incident wave is a [??]-polarized plane wave propagating along [??]-axis. The EFIE and CFIE equations are applied to the open and close objects, respectively. All the calculations are carried out on a Xeron computer with 2.5 GHz.

The first example is the model of bowtie as elucidated in Section 2. The current distributions and the monostatic RCS are calculated through the optimized scheme, the Pade approximation (with the expanded points of 1.875 and 4.625 GHz), and the MoM, respectively. Figure 6 shows that the optimized scheme yields more accurate current distributions than the Pade approximation. Here, the error is defined as the ratio of the difference between the results obtained from the MoM and one of the other methods to the results from the MoM. The Pade approximation yields the results deviating from those from the MoM at the low and middle frequency ranges. In Figure 7, the monostatic RCS results from the optimized scheme are in good agreement with those from the MoM and the results from the Pade approximation are not.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

The second example is a hemisphere with cone cavity as shown in Figure 4(b), where the radius of the hemisphere is 5 cm and the radius and depth of the cone cavity are 3.5 cm, respectively. The surface is discretized into 2426 triangles with 3639 unknowns. The errors of the current distributions from the optimized scheme and the Pade approximation are plotted in Figure 8, respectively, and the monostatic RCS results plotted in Figure 9, respectively. The Pade approximation (with the expanded points of 1.875 and 4.625 GHz) gives less accurate results than the optimized scheme.

Listed in Table 1 is the CPU time for solving the current distributions at 56 frequencies used in the optimized scheme, the Pade approximation, and the MoM, respectively. The CPU time for calculating and inter/extrapolating the matrices used in the optimized scheme is 22.53% of that used in the Pade approximation and 7.24% of that used in the MoM. The total CPU time used in the optimized scheme is much less than that used in the other methods, respectively, though the iteration time used in the optimized scheme is the largest.

[FIGURE 9 OMITTED]

4. CONCLUSION

In this paper, a cubic polynomial inter/extrapolation method is improved for efficiently generating the MoM matrices over a frequency band. The frequency samples employed are optimized by minimizing the amplitude of the polynomial component of the error norm. The optimized scheme yields more accurate current distributions and far-field results over the frequency band than the Pade approximation.

APPENDIX A.

Let

[[PSI].sub.I] (x,y) = [(y - x).sup.4] / 64, (A1)

[[PSI].sub.II] (x,y) = (1 - x) [(2 - x - y).sup.2] (1 - y) / 4, (A2)

[[PSI].sub.III] (x,y) = x[(x + y).sup.2]y / 4, (A3)

and then we define over the domain [OMEGA] = [0,1] x [0,1]

[bar.[omega]] (x,y) = max {[[PSI].sub.I] (x,y), [[PSI].sub.II] (x,y), [[PSI].sub.III] (x,y)}. (A4)

The focus of the appendix is to minimize [bar.[omega]] (x,y) by optimizing the point (x,y) over [OMEGA].

[FIGURE A1 OMITTED]

The function [bar.[omega]] (x,y) is continuous in [OMEGA] and can be further expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A5)

where [[omega].sub.I], [[omega].sub.II], and [[omega].sub.III] are separated by the following lines [l.sub.i] (i = 1,2,...,5) in Figure A1:

[l.sub.1] : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[l.sub.2] : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[l.sub.3] : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[l.sub.4] : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[l.sub.5] : x + y = 1

In fact, [l.sub.2] and [l.sub.3] are solutions to [[PSI].sub.I] (x,y) = [[PSI].sub.II] (x,y), and [l.sub.1] and [l.sub.4] are solutions to [[PSI].sub.I] (x,y) = [[PSI].sub.III] (x,y); [l.sub.5] is a solution to [[PSI].sub.II] (x,y) = [[PSI].sub.III] (x,y).

Since [nabla][[PSI].sub.I] (x,y) is not equal to zero inside [[OMEGA].sub.i] for i = I, II, III, no stationary point of [bar.[omega]] (x,y) exists inside these sub- domains. Consequently, [bar.[omega]](x,y) attains the global minimum only on the lines [l.sub.i] (i = 1,2,...,5) and the boundary of [OMEGA]. By comparing the values of [bar.omega]](x,y) on the lines and the boundary, two optimal points in [OMEGA], i.e., ([x.sup.*.sub.0],[y.sup.*.sub.0]) intersected by [l.sub.1] and [l.sub.2], and ([x.sup.*.sub.1],[y.sup.*.sub.1]) by [l.sub.3] and [l.sub.4], can be expressed as

[x.sup.*.sub.0] = [y.sup.*.sub.1] = 1/2 (1 - [square root of 2] [[square root of [[square root of 2] - 1]]]), (A6)

[x.sup.*.sub.1] = [y.sup.*.sub.0] = 1/2 (1 + [square root of 2] [[square root of [[square root of 2] - 1]]]), (A7)

In such case, the minimum of [bar.[omega]] (x,y) is [([square root of 2] - 1).sup.2] / 16.

ACKNOWLEDGMENT

This work was supported in part by NSFC under Grant 60901013 and 60921063, in part by National 973 projects under Grant 2010CB327400 and 2009CB320203, in part by National High-Tech Research Plan of China under Grant 2008AA01Z223 and 2009AA011503, in part by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and in part by Innovation Project Program of State Key Laboratory of Millimeter Waves under Grant Z201006.

Received 10 May 2011, Accepted 1 June 2011, Scheduled 9 June 2011

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W.-D. Li (1), *, J.-X. Miao (2), J. Hu (1), Z. Song (1), and H.-X. Zhou (1)

(1) State Key Laboratory of Millimeter Waves, School of Information Science and Engineering, Southeast University, Nanjing 210096, P. R. China

(2) School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210046, P. R. China

* Corresponding author: Wei-Dong Li (wdli@emfield.org).

Table 1. CPU time for solving the current distributions at 56 frequencies for the hemisphere with cone cavity used in the different methods (in minutes). Optimized Pade MoM Matrices at frequency samples 133.64 601.36 1870.09 Inter/extrapolation 1.84 0.029 / of matrices or currents Iteration solution 20.54 6.60 20.47 Total time 156.02 607.99 1890.56

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Author: | Li, W.-D.; Miao, J.-X.; Hu, J.; Song, Z.; Zhou, H.-X. |
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Publication: | Progress In Electromagnetics Research |

Article Type: | Report |

Geographic Code: | 9CHIN |

Date: | Jul 1, 2011 |

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