Haotian ZENG, Xunwang ZHAO, Qin SU, Yu ZHANG, Hao LI

Shaanxi Key Laboratory of Large Scale Electromagnetic Computing, Xidian University, Xi'an 710071, China

KEYWORDS Coating modeling;Higher-Order Method of Moments (HOMOM);Radar Absorbing Materials(RAM);Radar Cross Section (RCS);Stealth technology

Abstract In order to fast analyze the aircraft Radar Cross Section(RCS)and accurately reduce it with Radar Absorbing Materials(RAM),a comprehensive analysis method based on Higher-Order Method of Moments (HOMOM), termed Locally Coating Method (LCM), is proposed in this paper. There are two steps to fast analyze coatings for RCS reduction in this method: analyze the RCS of various parts before coating the aircraft; model a coating over the aircraft and analyze the wave absorbing effect of it. The aircraft RCS is calculated as a whole but analyzed in various parts by LCM, and thus the RCS contribution of different parts can be compared without disturbing the current continuity. A model expansion algorithm is also presented in LCM to model absorption coatings on specif ied aircraft parts for later stage RCS calculation of the coated aircraft.

1. Introduction

Radar detection and stealth function are like spear and shield in the f ield of electromagnetic waves.With the development of modern technology, the performance of stealth has become increasingly important, and has become the basic requirement of the fourth generation of f ighters. In order to attain the stealth effect, Radar Absorbing Materials (RAM) are often coated on the surfaces of aircraft, reducing their Radar Cross Sections (RCS).1-2

As a complex target, an aircraft scatters differently at different frequencies. As a result, in order to achieve multiband RCS reduction, it is inappropriate to simply coat the whole target. At a single frequency, it is enough to coat only several specif ied parts. Analysis on coating location and coating effects makes locally coating fast and accurate. The f lowchart of fast coating analysis for RCS reduction by Locally Coating Method (LCM) is shown in Fig. 1.

At present, the mainstream method for locally coating the target is based on surface current distribution. The parts with strong surface currents are considered as those which contribute greatly to the RCS. It is noted that the electric and magnetic f ields generated from the currents are complex vectors in high-frequency situation, and thus the combination of the f ields scattered from different aircraft parts may lead to large or small RCS along different directions. Therefore, the surface currents cannot directly ref lect the RCS contribution of different aircraft parts. Research on the RCS distribution makes it convenient to coat the target properly. This study greatly improves the eff iciency of the later stage reduction of RCS.3-8

Fig. 1 Flowchart of LCM.

The widely used method to calculate the RCS of coated targets is the‘equivalent impedance method'.We take a dielectric layer of 2 mm thick as an example.The wave absorption effect of the dielectric is equivalent to certain impedance in the parallel connection according to the dielectric constant, the permeability and the thickness.9-11The shortcomings, however,are very obvious. As shown in Fig. 2, with the increase of the incident angle, the absorption coating that actually works becomes increasingly thicker than 2 mm.Therefore,the equivalent impedance cannot replace the effect of the coating accurately,and there is large error in the RCS results calculated by this method.12-14This error is evident when it comes to models of f lat shape.

As is known,there are two diff iculties to perform the accurate calculation of complex coatings. One is that the substantially increased unknowns result in a large amount of computation and storage. The enlargement of the model and the introduction of the dielectric coating will generate a large number of unknowns. This change makes it a challenge to solve the problem.15-17The other is that the model accuracy of the coatings is hard to guarantee. It is diff icult to model a coating over a large complex target using a full-wave method in most electromagnetic simulation software. The model editing function ‘Scale' that is available in most software can be useful when dealing with regular and simple models. In this function, the coordinates of all the sampling points are multiplied by a multiplier directly.Thus,there is often a dislocation between the new model and the original one when the model is complex,as shown in Fig.3.To solve this problem,the model expansion algorithm is proposed. In this paper, LCM reduces the RCS by modeling the absorption coating, using the model expansion algorithm.

Fig. 2 Propagation distance in dielectric.

Fig. 3 Dislocation between original and scaled models.

2. Principle introduction

2.1. Theoretical principle

2.1.1.Integral equations for composite metallic-dielectric models

To deal with the problem of a metallic target coated with dielectric coatings, the combination of Electric Field Integral Equation (EFIE) and PMCHW equation are used.

According to the boundary conditions of the dielectric surface, PMCHW integral equation can be established as

where i,j and k are three order counters;s and p are two coordinate axes as depicted in Fig. 4; η represents the wave impedance of the dielectric; ^n represents the normal vector of each surface; Eincand Hincrepresent incident electric f ield and incident magnetic f ield respectively; J and M are the electric and magnetic current density expressed in Eq.(4);L and K are two operators def ined as

where G represents the Green's function and X represents an unknown quantity that can be replaced by either J or M.

Fig. 4 Example of a bilinear surface.

And on the Perfect Electrical Conductor (PEC) surface,EFIE can be established as

2.1.2. Bilinear surface

Metallic and dielectric surfaces are modeled by bilinear surfaces in Higher-Order Method of Moments (HOMOM). A bilinear surface is,in general,a nonplanar quadrilateral,which is def ined uniquely by its four arbitrarily spaced vertices, as shown in Fig. 4.

The parametric equation can be written in the form:

where r11, r12, r21and r22are the position vectors of the four vertices, and p and s are the local coordinates.18-20

2.1.3. Current expansion

The surface currents over a bilinear surface are decomposed into its p and s components. Without loss of generality, we take the s component as an example. The s component of the electric and magnetic current density over a bilinear surface is respectively expressed as

where Npand Nsare the degrees of approximations along the coordinates;aijand bijare the unknown coeff icients;Fij(p,s)are the basis functions. They are mathematically characterized by

where αpand αsare the unitary vectors;^ipand^isare the corresponding unit vectors;fi(p)represents a polynomial of order i;hj(s) represents a polynomial of order j and r(p,s) is given by Eq. (3).

2.1.4. Matrix equations

The f irst step to reduce aircraft RCS in LCM is to f ind the appropriate location that contributes most to the RCS. The aircraft is divided into various parts as shown in Fig.5.θ represents the elevation and φ represents the azimuth. θ of the Z axis is 0° and φ of the X axis is 0°.

The integral equations are discretized into an N×N dense matrix equation in a general form of

where Z denotes the complex dense matrix, I is the unknown vector to be determined, and V denotes the given source vector.

In consideration of the multiple aircraft parts divided by the proposed method, the matrix Eq. (6) can be written in a block form. Take the ten parts in Fig. 5 as an example, and the matrix equation can be written as

where Zaais the self-impedance of Part a and Zbais the mutual impedance of Part a to Part b. Iais the current coeff icients on Part a. The current coeff icients are obtained by solving the above matrix equation. In this paper, we use a parallel LU decomposition based direct solver to solve the equation. Substitute the current coeff icients in Eq. (4) and use auxiliary potentials, and the electric and magnetic f ields can be computed.

2.2. RCS contribution of each part

The core idea of analyzing the RCS contribution of the concerned part is to take the currents on surfaces of that part into consideration and use the related currents to compute the corresponding RCS.21-25Because the coupling among all the parts is rigorously computed in Eq. (7), the currents involve the coupling effects from all other parts, as shown in Fig. 6.

2.3. Model expansion algorithm

The core idea of creating a coating over the given model depends on two facts:

Fig. 5 Various parts of aircraft.

Fig. 6 Analysis on RCS of one part.

(1) If the distances between four non-collinear points and a surface are all equal to the given distance d, then the plane containing these four points will be parallel to the surface.The distance between the plane and the surface will also be d, as shown in Fig. 7;

(2) The parallel planes of three non-coplanar surfaces are Planes A,B and C respectively,and their common point is Point p′′, as shown in Fig. 8. If the distances between Planes A, B, C and their respective original surfaces are all equal to d,then the distances between p′′and the surfaces will also be d. Create the coating surface with all the new points like p′′, and then the distance between the coating surface and the original model surface will right be equal to the given distance d.It can be said that the process of creating the coating model is mainly that of creating the common points.

2.3.1. Creation of common points

In LCM, the model expansion algorithm reads quadrilateral meshing information of the model and creates new surfaces as the coating.It f igures out the normal vector n with the equation written as

where raband racare two vectors specif ied by the two adjacent edges of a surface as Fig. 8 shows.

According to the normal vector n, the coordinate origin o,the original Point p and the given distance d, a new point named p′that belongs to the new Plane A is created by the equation below:

where rop′represents the vector from the coordinate origin to Point p and roprepresents the vector from the coordinate origin to Point p′.Plane A can be specif ied by the equation written as

Fig. 7 Locating coating surface by points.

Fig. 8 Creation of new points.

where rprandp′represents the vector from a random point that belongs to Plane A to Point p′.

Repeat the previous steps and when the other two Planes B and C are specif ied(see Eq. (10)), the common solution of the coordinates of Point prandcan be f igured out.This solution represents the coordinates of Point p′′, the common point of the three planes.When all common points are specif ied,the model of coating surface can be created.

2.3.2. Different cases of original surfaces

Depending on the number of non-collinear normal vectors,the process of f iguring out Point p′′is divided into four cases:

(1) When there is only one non-collinear normal vector, it means that all surfaces containing Point p are coplanar and there would be only one Point p′. Locate Point p′and it would exactly be Point p′′, as shown in Fig. 9.

(2) When there are two non-collinear normal vectors,Plane A and Plane B can be specif ied. The two vectors are taken as raband racrespectively to f igure out the third normal vector n.Then Point p is taken as the third Point p′to specify Plane C. Subsequently, Point p′′can be located, as shown in Fig. 10.

Fig. 9 Surfaces on one non-collinear plane.

Fig. 10 Surfaces on two non-coplanar planes.

Fig. 11 Surfaces on three non-coplanar planes.

Fig. 12 Surfaces on multiple non-coplanar planes.

(3) When there are three non-collinear normal vectors, as shown in Fig. 11, the process of locating Point p′′is introduced previously as a typical case.

(4) When there are four or more non-collinear normal vectors, as shown in Fig. 12, any three of them would be enough to be used as in case (3).

3. Results and discussion

3.1. Comparison between coating calculation methods

In order to compare the RCS of coatings calculated by the equivalent impedance method and by LCM, a PEC plate coated with RAM and another PEC plate assigned with equivalent impedance are calculated respectively.The parameters of this comparison are as follows: The square plate has an edge length of 0.1 m; the thickness of the RAM is 1 mm; the frequency of the incident wave is 10 GHz;the relative permittivity of the RAM is 22-j8 and the relative permeability is 1. The equivalent impedance is obtained according to the RAM parameters by FEKO. Fig. 13 shows the calculation of the monostatic RCS in the range of θ from 0° to 90°.

Through the comparison in Fig.14,it is clear that there is a relatively large error in the calculation of monostatic RCS by using the equivalent impedance method. In detail, the calculated results are in good agreement when the elevation is less than 10°, but the discrepancy occurs when the angle is greater than 45°. In comparison, the results of LCM calculated by Method of Moments (MOM) and HOMOM are consistent in Fig. 14.

Fig. 13 Calculation of monostatic RCS.

Fig.14 Comparison between equivalent impedance method and LCM.

3.2. Application of LCM

According to the division in Fig. 5, the RCS contribution of each aircraft part is calculated.Fig.15 show the results of total RCS.

In these examples,the frequency of the incident plane-wave is 1 GHz and the polarization direction is horizontal polarization. The proposed method requires 21.5 GB memory and about 880 s to compute the current coeff icients at an incident angle. The RCS of each part can be computed within 50 s.Ten dodeca-core Intel Xeon 2690 V2 2.2 GHz CPUs are used in the calculation.

Fig. 15(a) shows that, when the elevation θ=95°and azimuth -15°＜φ ＜15°, Part c, which is the rear cockpit of the aircraft,contributes most to the monostatic RCS. Besides,Part b and Part e, which are the front cockpit and the inlet respectively, also contribute a lot to the RCS. When θ=90°,as shown in Fig. 15(b), the RCS contributions are similar to those in Fig. 15(a).

As a comparison, the surface currents are calculated under the same condition when θ=90°and the result is shown in Fig. 16.

It is clear to see in Fig. 16 that the surface currents are strong in Part a, Part b and Part d. As discussed previously,the scattering f ields are complex vectors. The surface currents cannot directly ref lect the RCS contribution of different aircraft parts. This phenomenon is worth paying attention to because it is not accurate to coat the target by analyzing the current distribution.

Fig. 15 Monostatic RCS contribution of various parts when azimuth -15° ＜φ ＜15°.

Fig. 16 Surface current distribution of aircraft.

According to the RCS in Fig. 15, a RAM of 1 mm thick is coated on Part b,Part c and Part e.The relative permittivity of the RAM is 14.8-j1.5 and the relative permeability is 4.2-j1.1. As a comparison, the RAM is coated on the whole aircraft, as shown in Fig. 17.

The results in Fig.17 prove that it is effective to reduce the RCS by coating on several parts according to the analysis on RCS of various parts.

Fig. 17 RCS reduction by partly coating and totally coating.

4. Conclusions

In the research of aircraft RCS reduction, LCM saves much time to f ind the suitable coating location, improves the accuracy of the coating RCS calculation and overcomes the diff iculty in irregular coating modeling. The conclusions and details are summarized as follows:

(1) Analysis on the RCS contribution of various parts,as a preliminary work, makes it easy to coat the aircraft.With the analysis, it is no longer necessary to coat the whole target.

(2) Compared with the analysis on current distribution,it is more reliable to locally coat the target according to the analysis on RCS by LCM.

(3) Coatings on surfaces of different shapes can be modeled by the method proposed in this paper.In most cases,no more modif ication is needed to be made on the new model.

(4) Obvious differences in the RCS results between two modeling methods are observed in this paper, which demonstrate that the traditional equivalent impedance method is not accurate as LCM.

(5) There is an urgent demand for the accurate calculation of the stealth aircraft RCS. The feasibility of the presented method is verif ied by a variety of examples.

Acknowledgements

This study was supported by the National Key Research and Development Program of China (No. 2017YFB0202102), the China Postdoctoral Science Foundation funded project (No.2017M 613068), the National Key Research and Development Program of China(No.2016YFE0121600),the National High Technology Research and Development Program of China(863 Program) (No. 2014AA01A302), the Key Research and Development Program of Shandong Province, China (No.2015GGX101028), and the Special Program for Applied Research on Super Computation of the NSFC(National Natural Science Foundation of China)-Guangdong Joint Fund,China (the second phase) (No. U1501501).