Research on wide-band electromagnetic scattering characteristics of random distributed objects using Monte Carlo method

Abstract

In the engineering applications, the distribution of objects is mostly random. Therefore, scattering analysis of randomly distributed objects has been one of the important problems in broadband electromagnetic calculation field. To resolve the problem, the Asymptotic Waveform Evaluation technique in conjunction with Monte Carlo Method is presented. First, the stochastic distribution is modeled by the Monte Carlo Method, and then the Asymptotic Waveform Evaluation technique using Padé approximation is utilized to achieve the Radar Cross Section at a wide frequency band. Numerical results show that the Asymptotic Waveform Evaluation technique can solve the random distributed object problems efficiently and accurately.

Keywords

Scattering characteristics Asymptotic Waveform Evaluation Monte Carlo

1. Introduction

The broadband scattering characteristics of the objects contain a lot of information. In the low frequency region, it includes the volume information of objects. In the resonance region, it involves the outline information. In the high frequency region, it contains the details information of objects. The solutions of Electric-Field Integral Equation (EFIE) via the Method of Moments (MOM) have been an available method to study the scattering characteristics [1]. Using this kind of method to solve broadband scattering characteristics, we must select the small frequency interval, and the computational process has to be repeated for each frequency. It will inevitably consume a large amount of memory space and computing time, especially for multi-objective stochastic distribution. Thus, this method can not be very effective.

To solve this problem, a very efficient extrapolation technique based on Asymptotic Waveform Evaluation (AWE) Technology [2, 3, 4] is applied. The application of AWE technique can decrease the unnecessary repeated computation process. This will save memory space and computational time. In the literature [5, 6], the AWE technique in conjunction with AIM/PO can be generalized to broad-band analysis of antennas installed in the large complex conductive platforms. In the literature [7], the AWE technology is used to deal with wideband electromagnetic question of surface-wire junction structures. In the literature [8], the AWE technology is utilized to Mixed Potential Integral Equation for fast frequency scanning in half-space scattering problems. And in the literature [9], the AWE technology is proposed to analyze broad-band scattering features of thin dielectric layers. However, AWE technique in these literature is applied to study the scattering characteristics of regular arrays of objects. In these applications, the distribution of objects are mostly random, such as mineral distribution, snow distribution [10] et al. Due to the distribution of the materials on earth is not uniform, the material particle is modeled as distribution objectives randomly, and the positions of them are calculated by Monte Carlo (MC) [11].

In this paper, AWE technique combined with the MC method is generalized to wide-band electromagnetic scattering analysis of randomly distributed objects. First, MC method is used to construct the theoretical modeling of random distribution targets. Second, the broad-band Radar Cross Section (RCS) of the model is analyzed by adopting the AWE technology. Numerical results are certified the correctness and efficiency of this combined technique.

2. Integral equations of conduct-dielectric-mixing objects

In the space area $V$ , the boundary of $\sum=S+S_{1}$ is closed. The Maxwell Equations are:

$\displaystyle\nabla\times\vec{E}=-\vec{M}-j\omega\mu\vec{H}.$ (1) $\displaystyle\nabla\times\vec{E}=\vec{J}+j\omega\varepsilon\vec{E}.$ (2) $\displaystyle\nabla\cdot\vec{E}=\frac{\rho}{\varepsilon}.$ (3) $\displaystyle\nabla\cdot\vec{H}=\frac{\rho_{m}}{\mu}.$ (4)

According to the above equations, nonhomogeneous vector wave equations can be solved.

$\displaystyle\nabla\times\nabla\times\vec{E}-k^{2}\vec{E}=-j\omega\mu\vec{J}-% \nabla\times\vec{M}.$ (5) $\displaystyle\nabla\times\nabla\times\vec{H}-k^{2}\vec{H}=-j\omega\mu\vec{M}+% \nabla\times\vec{J}$ (6)

According to the Vector Green Theorem

$\displaystyle\iiint\nolimits_{v}(\vec{P}\cdot\nabla\times\nabla\times\vec{Q}-% \vec{Q}\cdot\nabla\times\nabla\times\vec{P})dv=\oiint\nolimits_{\Sigma}(\vec{Q% }\times\nabla\times\vec{P}-\vec{P}\times\nabla\times\vec{Q})\cdot d\vec{s}.$ (7)

Supposing

$\displaystyle\vec{Q}=\vec{E},\vec{P}=\vec{a}g(\vec{r},\vec{r}^{\prime})$ (8) $\displaystyle g(\vec{r},\vec{r}^{\prime})=\frac{e^{-jk|\vec{r}-\vec{r}^{\prime% }|}}{|\vec{r}-\vec{r}^{\prime}|}.$

Finally, the equations can be described as follow:

$\displaystyle\iiint\nolimits_{v}\left[-j\omega\mu g\vec{J}-\vec{M}\times\nabla g% +\frac{\rho}{\varepsilon}\nabla g\right]dv=$ (9) $\displaystyle\quad\oiint\nolimits_{\Sigma}[-j\omega\mu g(\texttoptiebar{n}% \times\vec{H})+(\texttoptiebar{n}\times\vec{E})\times\nabla g+(\texttoptiebar{% n}\cdot\vec{E})\nabla g]ds$

In the free space, there are M dielectric objects and N fully conductive objects, which their intersecting surface is arbitrary shapes. As showing in Figs 1 and 2. S means the outsides of the objects, the subscript cn means the conductive objects, the subscript dm means the dielectric objects. Guessing TM wave is the incident field, the equations are as following:

$\displaystyle\vec{E}=\vec{E}^{i}+\vec{E}^{s}(\vec{J}_{c},\vec{J}_{d},\vec{M}_{% d}).$ (10) $\displaystyle\vec{H}=\vec{H}^{i}+\vec{H}^{s}(\vec{J}_{c},\vec{J}_{d},\vec{M}_{% d}).$ (11)

Where $\vec{E}$ and $\vec{H}$ are the total fields, $\vec{E}^{i}$ and $\vec{H}^{i}$ are the incident fields, $\vec{E}^{s}$ and $\vec{H}^{s}$ are the scattering fields. $\vec{J}_{c}$ is an equivalent current of the conducting surface, $\vec{J}_{d}$ is an equivalent current of the dielectric surface, $\vec{M}_{d}$ is an equivalent magnetic current of the dielectric surface.

Figure 1.

Scattering of conduct and dielectric objects.

Figure 2.

External equivalent figure.

Further, $\vec{E}^{s}$ and $\vec{H}^{s}$ are calculated by

$\displaystyle\vec{E}^{s}=\sum^{N}_{n=1}[-j\omega\vec{A}(\vec{J}_{cn})-\nabla% \Phi(\vec{J}_{cn})]+\sum^{M}_{m=1}[-j\omega\vec{A}(\vec{J}_{dm})-\nabla\Phi(% \vec{J}_{dm})-\frac{1}{\varepsilon}\nabla\times\vec{F}(\vec{M}_{dm})].$ (12) $\displaystyle\vec{H}^{s}=\sum^{N}_{n=1}\frac{1}{\mu}\nabla\times\vec{A}(\vec{J% }_{cn})+\sum^{M}_{m=1}[-j\omega\vec{F}(\vec{M}_{dm})-\nabla\Phi(\vec{M}_{dm})+% \frac{1}{\mu}\nabla\times\vec{A}(\vec{J}_{dm})].$ (13)

Where the subscript m means the M-th dielectric objects and the subscript n means the N-th conducting objects.

3. Asymptotic waveform evaluation

The basic idea of Padé approximation is using rational fraction function to approximate the function expressed by the power series. The coefficients of the rational fraction function can be resolved by the coefficients of the power series based on the Padé approximation method. That is to say, from the first item, the Taylor expansion of the rational fraction function is consistent with power series. Compared with the Taylor expansion, Padé approximation has the faster convergence speed and the larger convergence radius, so it can extrapolate in the large scale. And the Padé approximation can well reflect the zero and pole characteristic of the function, so it is good to approach the function of singularity.

According to the Padé approximation method, The unknown current density distribution at any wave numbers is expanded into a Taylor series at the given wave number. The steps of using AWE technique to achieve the wideband RCS are:

$\displaystyle I(k)=\sum^{\infty}_{n=0}m_{n}(k-k_{0})^{n}.$ (14) $\displaystyle m_{n}=Z^{-1}(k_{0})\left[\frac{V^{n}(k_{0})}{n!}-\sum^{n}_{i=1}% \frac{Z^{(i)}(k_{0})m_{n-i}}{i!}\right].$ (15)

Where $k_{0}$ is the central wave number, $Z^{(i)}(k_{0})$ is the $i$ -th derivative in respect of $Z(k)$ at $k=k_{0}$ ; $V^{(n)}(k_{0})$ is the n-th derivative in respect of $V(k)$ at $k=k_{0}$ .

If the surface current distribution is approximated by the Taylor series shown in the Eq. (14), the higher order moments must be calculated at the given wave number. So the Taylor series is transformed into rational fraction function by using Padé approximation:

$\displaystyle I(k)=\frac{\sum\limits^{L}_{i=0}a_{i}(k-k_{0})^{i}}{1+\sum% \limits^{M}_{j=1}b_{j}(k-k_{0})^{j}}.$ (16)

Generally, $b_{0}=$ 1, $L=(M|M+1)$ .

The unknown coefficients of $a_{i}$ and $b_{j}$ in the Eq. (16) are calculated as follows:

$\displaystyle\left[\begin{array}[]{ccccc}m_{L}&m_{L-1}&m_{L-2}&\cdots&m_{L-M+1% }\\ m_{L+1}&m_{L}&m_{L-1}&\cdots&m_{L-M+2}\\ m_{L+2}&m_{L+1}&m_{L}&\cdots&m_{L-M+3}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ m_{L+M-1}&m_{L+M-2}&m_{L+M-3}&\cdots&m_{L}\end{array}\right]∼{}\left[\begin{% array}[]{c}b_{1}\\ b_{2}\\ b_{3}\\ \vdots\\ b_{M}\end{array}\right]=-\left[\begin{array}[]{c}m_{L+1}\\ m_{L+2}\\ m_{L+3}\\ \vdots\\ m_{L+M}\end{array}\right].$ (17)

$\displaystyle\left[\begin{array}[]{c}a_{0}\\ a_{1}\\ a_{2}\\ \vdots\\ a_{L}\end{array}\right]=\left[\begin{array}[]{ccccc}m_{0}&0&0&\cdots&0\\ m_{1}&m_{0}&0&\cdots&0\\ m_{2}&m_{1}&m_{0}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ m_{L}&m_{L-1}&m_{L-2}&\cdots&m_{L-M}\end{array}\right]∼{}\left[\begin{array}[]% {c}1\\ b_{1}\\ b_{2}\\ \vdots\\ b_{M}\end{array}\right].$ (18)

4. MC simulation

MC method is known as the stochastic simulation technique, the random sampling technology and so on. The basic idea of MC is to establish random process, and then determine the random value that is equal to the solution of the required questions. Then calculate the statistical characteristics of the parameters by meaning of observing the process. Finally, use the statistical characteristics of parameters instead of the solutions of the problem. The processes of using MC to simulate the random distribution are as follows:

Step1: Supposing a unit square, which has $N$ targets in this area:

$\displaystyle N=N_{1}^{2}.$ (19) $\displaystyle b=\sqrt{\frac{4f}{\pi N}}.$ (20)

Where $N_{1}$ is an integer. The size of each target is the same. $b$ is the diameter of each target. $f$ is the representation of the duty cycle.

Step2: In the above area, the target is set regularly in the direction and direction. The distance between the target is

$\displaystyle s=\frac{1}{N_{1}}$ (21)

Thus the initial position of each target is determined.

For example, the first line of targets’ coordinates is respectively:

$\displaystyle\left(\left(\frac{s}{2},\frac{s}{2}\right)，\left(\frac{3s}{2},% \frac{s}{2}\right)，\left(\frac{5s}{2},\frac{s}{2}\right)，\ldots ，\left(N% _{1}-\frac{1}{2}\right)s,\frac{s}{2}\right).$ (22)

The coordinates of the second lines are respectively:

$\displaystyle\left(\left(\frac{s}{2},\frac{3s}{2}\right)，\left(\frac{3s}{2},% \frac{3s}{2}\right)，\left(\frac{5s}{2},\frac{3s}{2}\right)，\ldots ，\left% (N_{1}-\frac{1}{2}\right)s,\frac{3s}{2}\right).$ (23)

Step3: Moving targets.

(i) Random movement of regular targets.

Supposing the original coordinates of the regular targets are ( $x_{j},y_{j}$ ). After random movement, its coordinates are:

$\displaystyle x^{\prime}_{j}=x_{j}+r_{1}\Delta.$ (24) $\displaystyle y^{\prime}_{j}=y_{j}+r_{2}\Delta.$ (25)

Where $r_{1}$ and $r_{2}$ are the average distribution of random number in the range of [ $-$ 1, 1]. $\Delta=C_{st}b$ , where $C_{st}$ is the displacement calibration constant.

(ii) Check the rationality of the coordinates after the target displacement.

Rationality means that one target cannot be coincident with another target, and the target cannot be moved beyond the square area. In order to check that one target is coincident with another target. First:

$\displaystyle r_{x}=x_{l}-x^{\prime}_{j}\quad(l=1,2,\cdots N)\quad(l\neq j)$ (26)

According to the periodic boundary conditions:

$\displaystyle\left\{\begin{array}[]{ll}r_{x}=r_{x}-1&r_{x}>0.5\\ r_{x}=r_{x}+1&r_{x}<-0.5.\end{array}\right.$ (27)

Secondly:

$\displaystyle r_{d}=\sqrt{r_{x}^{2}+r^{2}_{y}}$ (28) $\displaystyle\left\{\begin{array}[]{ll}r_{d}\geqslant b&(x^{\prime}_{j},y^{% \prime}_{j})\\ r_{d}<b&(x_{j},y_{j})\\ \end{array}\right.$ (29)

Using the above method, the determined coordinates are called new coordinates.

Finally, examine whether the coordinates are outside of the square.

$\displaystyle\left\{\begin{array}[]{ll}xn_{j}<0&xn_{j}=xn_{j}+1\\ xn_{j}>1&xn_{j}=xn_{j}-1.\end{array}\right.$ (30)

The analysis on the $\hat{y}$ direction is the same of the $\hat{x}$ direction.

5. Numerical analysis

According to the MC method and AWE technology, the derivation process of the target broadband electromagnetic scattering characteristics is as follows:

Step1:Accroding to the coordinate value calculated by MC method, $l_{mn}$ can be calculated.

$\displaystyle l_{mn}=\left\{\begin{array}[]{ll}\frac{\eta}{4}k\Delta C_{n}H_{o% }^{(2)}k\sqrt{(x_{n}-x_{m})^{2}+(y_{n}-y_{m})^{2}}&m\neq n\\ \frac{\eta}{4}k\Delta C_{n}\left[1-j\frac{2}{\pi}\ln\left(\frac{\gamma k\Delta C% _{n}}{4e}\right)\right]&m=n\end{array}\right.$ (31)

Where $k$ is wave number in free space, $\eta$ is the wave impedance, $\gamma=$ 1.781, $H_{o}^{(2)}$ is zero order Hankel function of the first kind, $H^{(2)}_{1}$ is the first order Hankel function.

Step 2: the derivative of $l_{mn}$ can be calculated.

When $m=n$ , there is

$\displaystyle l_{mn}^{(1)}=\frac{\eta\Delta C_{n}}{4}\left[1-j\frac{2}{\pi}% \left(\gamma+\ln\frac{k\Delta C_{n}}{4}\right)\right]$ $\displaystyle l_{mn}^{(2)}=\frac{\eta\Delta C_{n}}{4}\left[-j\frac{2}{\pi}k^{-% 1}\right]$ $\displaystyle l_{mn}^{(3)}=\frac{\eta\Delta C_{n}}{4}\left[-j\frac{2}{\pi}(-1)% ^{3}k^{-2}\right]$ $\displaystyle l_{mn}^{(4)}=\frac{\eta\Delta C_{n}}{4}\left[-j\frac{2}{\pi}(-1)% ^{4}2!k^{-3}\right]$ (32) $\displaystyle l_{mn}^{(5)}=\frac{\eta\Delta C_{n}}{4}\left[-j\frac{2}{\pi}(-1)% ^{5}3!k^{-4}\right]$ $\displaystyle l_{mn}^{(6)}=\frac{\eta\Delta C_{n}}{4}\left[-j\frac{2}{\pi}(-1)% ^{6}4!k^{-5}\right]$ $\displaystyle l_{mn}^{(7)}=\frac{\eta\Delta C_{n}}{4}\left[-j\frac{2}{\pi}(-1)% ^{7}5!k^{-6}\right]$

When $m\neq n$ , there is:

$\displaystyle l_{mn}^{(1)}=\frac{\eta\Delta C_{n}}{4}\left[1-j\frac{2}{\pi}% \left(\gamma+\ln\frac{k\Delta C_{n}}{4}\right)\right]$ $\displaystyle l_{mn}^{(2)}=\frac{\eta\Delta C_{n}}{4}\left[-j\frac{2}{\pi}k^{-% 1}\right]$ $\displaystyle l_{mn}^{(3)}=\frac{\eta\Delta C_{n}}{4}\left[-j\frac{2}{\pi}(-1)% ^{3}k^{-2}\right]$ $\displaystyle l_{mn}^{(4)}=\frac{\eta\Delta C_{n}}{4}\left[-j\frac{2}{\pi}(-1)% ^{4}2!k^{-3}\right]$ (33) $\displaystyle l_{mn}^{(5)}=\frac{\eta\Delta C_{n}}{4}\left[-j\frac{2}{\pi}(-1)% ^{5}3!k^{-4}\right]$ $\displaystyle l_{mn}^{(6)}=\frac{\eta\Delta C_{n}}{4}\left[-j\frac{2}{\pi}(-1)% ^{6}4!k^{-5}\right]$ $\displaystyle l_{mn}^{(7)}=\frac{\eta\Delta C_{n}}{4}\left[-j\frac{2}{\pi}(-1)% ^{7}5!k^{-6}\right]$

Figure 3.

Random distribution of 16 conduct objects.

Step 3: Supposing the incident wave $V_{m}=e^{-jk(x_{m}\cos\phi_{1}+y_{m}\sin\phi_{1})}$ .

Step4: According to the equations 17 and 18, $b_{i}(i=1,2,3\cdots)$ and $a_{i}(i=1,2,3\cdots)$ can be calculated.

Step5: Substituting the values of $a_{i}$ and $b_{i}$ into equation 16, $I(k)$ can be obtained.

Figure 4.

Broadband RCS corresponding to Fig. 3.

Figure 5.

Random distribution of 16 dielectric objects.

6. Numerical demonstrations

In order to test and verify the correctness and the efficiency of the proposed method, we give three numerical experiments: the random distribution of 16 conductor objects, the random distribution of 16 dielectric objects, and the random distribution of 16 conductor and dielectric mixing objects. These objects are two-dimensional. All experiments are performed under the same conditions, and all calculations are computed on a personal computer with the Intel ${}^{\@setsize{\scriptsize}{9.5pt}{\viiipt}{\@viiipt}\textregistered}$ Core™ 2 Duo CPU with 2.1 GHz and 2.0 GB RAM.

Figure 6.

Broadband RCS corresponding to Fig. 5.

Figure 7.

Random distribution of 16 mixing objects.

6.1 Example A

The TM plane wave is given by $\vec{E}^{i}_{z}=e^{-jkx}$ . The center frequency is 30 GHz, and the frequency range is from 20 GHz to 40 GHz. The size of each object is $\frac{\lambda}{20}\times\frac{\lambda}{20}$ , where $\lambda$ is the corresponding wavelength for the center frequency. The shape is square. The relative permittivity $\varepsilon_{r}$ of the dielectric is 4.

The broad-band electromagnetic scattering of the random distribution model of 16 conductor objects is considered. According to the MC simulation procedure described in the last section, the random distribution graph is shown in Fig. 3. The RCS of this model in Fig. 4 is computed by the proposed method and conventional MoM, respectively. It is noted that the proposed method agrees very well with the conventional MoM at the range of 20 GHz to 40 GHz.

6.2 Example B

The experimental conditions are the same as Example A. The broad-band electromagnetic scattering of the random distribution model of 16 dielectric objects is calculated. In this example, the distribution of random dielectric objects is simulated by MC method firstly. As shown in Fig. 5. To obtain the RCS of this model, Padé approximation with (L,M) $=$ (4,3) has been used. The result of this model using the proposed method matches the consequence of conventional MoM at the range of 20 GHz to 40 GHz. As is shown in Fig. 6.

Figure 8.

Broadband RCS corresponding to Fig. 7.

6.3 Example C

The experimental conditions are the same as Example A. The broad-band electromagnetic scattering of the random distribution model of 16 dielectric and conduct mixing objects is analyzed. Grounded on the MC simulation procedure, the graph is shown in Fig. 7. In the figure, the unfilled districts are the conductor objects, and the filled districts are the dielectric objects. Figure 8 shows the RCS of this model. To obtain the RCS, the AWE technique using Padé approximation is applied. By comparison, the RCS acquired by the conventional MoM is shown in the same figure. It can be seen that the agreement in the frequency range is from 29 GHz to 35 GHz. The frequency band width is smaller than that of the single material objects. This is mainly caused by the influence of the coupling relationship between the conductor and the dielectric.

7. Conclusions

In this paper, AWE technology combined with the MC method is successfully utilized in the broad-band electromagnetic scattering characteristics of the random objects. First, we use the MC method to model the distribution of the random objects, and then the AWE technology is applied to analyse the broad-band RCS characteristics of the random objects. The numerical results have been certified the correctness of the proposed methods. Meanwhile, outcomes using this proposed method are identical with the conventional MoM.

Footnotes

Acknowledgments

This work was supported by the Natural Science Research Project of Anhui Province under Grant No. KJ2016A608, the Project of Anhui Province under Grant No.2019kfkc124, the Project of Hefei university teaching research under Grant No.2018hfkf03 and No 2018xk03, the Project of Young Core teacher visiting Foundation from Anhui Province under Grant No. gxgnfx2018031, the Natural Science Research Project of Hefei University under Grant No. 18ZR08ZDA and No. 18-19RC37.

References

Dault

and Shanker

, A mixed potential MLFMA for higher order moment methods with application to the generalized method of moments, IEEE Trans. Antennas Propag 64 (2016), 650–662.

Hassan

M.A.M.

and Kishk

A.A.

, A combined asymptotic waveform evaluation and random auxiliary sources method for wideband solutions of general-purpose EM problems, IEEE Transactions on Antennas and Propagation 67 (2019), 4010–4021.

Jeong

Hong

Lee

and Yook

, Fast frequency sweep using asymptotic waveform evaluation technique and thin dielectric sheet approximation, IEEE Transactions on Antennas and Propagation 64 (2016), 1800–1806.

Jemai

and Kouki

A.B.

, New adaptive multi-expansion frequencies approach for sp-more techniques with application to the well-conditioned asymptotic waveform evaluation, IEEE Transactions on Microwave Theory and Techniques 65 (2017), 3709–3719.

Wang

et al., Fast and accurate wide-band analysis of antennas mounted on conducting platform using AIM and asymptotic waveform evaluation technique, IEEE Trans Antennas Propag 59 (2011), 4624–4633.

et al., Efficient wide-band analysis of antennas around a conducting platform using MoM-PO hybrid method and asymptotic waveform evaluation technique, IEEE Trans Antennas Propag 60 (2012), 6048–6052.

Dong

H.-L.

et al., A hybrid broadband analysis approach for surface-wire junctions structures by applying AIM and asymptotic waveform evaluation technique, Journal of Electromagnetic Waves and Applications 30 (2016), 141–153.

B.-Y.

et al., Frequency sweep computation of half-space scattering using asymptotic waveform evaluation, Microwave Opt Technol Lett 58 (2016), 495–498.

Jeong

Y.-R.

et al., Fast frequency sweep using asymptotic waveform evaluation technique and thin dielectric sheet approximation, IEEE Trans Antennas Propag 64 (2016), 1800–1806.

10.

Shi

J.C.

et al., Review of snow water equivalent microwave remote sensing, Sci China Earth Sci 59 (2016), 731–745.

11.

Napolitano

Tossani

Borghetti

and Nucci

C.A.

, Lightning performance assessment of power distribution lines by means of stratified sampling monte carlo method, IEEE Transactions on Power Delivery 33 (2018), 2571–2577.