Comparison of Various Full-Wave Methods in Calculating the RCS of Inlet

Dachuan YU, Min ZHANG*

TONGJI UNIVERSITY, Modern Integrated Electromagnetic Simulation R&D Center (MIEMS) 4800 Cao’an Road, Jiading District, Shanghai 201804, China

* Correspondent Author: Prof. Dr.-Ing. Min ZHANG, School of Electronic and Information Engineering, Tongji University, min.zhang@mail.tongji.edu.cn

Abstract—Radar Cross Section (RCS) is a very important

measure in the field of radar stealth and radar anti-stealth. Several full-wave simulation methods are introduced in this paper, and the RCS of an S-shaped inlet is calculated respectively using these methods implanted in the commercial software CST MICROWAVE STUDIO® (CST MWS). The calculated results in CST MWS are in agreement with the measurement. Finally, these methods, which are compared in terms of accuracy, memory requirements and CPU time, are discussed with respect to the advantages and limitations in calculating the RCS of electrically large concave structures.

Key words: RCS, full wave EM algorithm, airplane inlet.

Ⅰ. INTRODUCTION

As the detecting technology developing, the survivability of the military aerial vehicle is threatened during the flight missions. It has become the hotspot in the field of military research to achieve aircrafts’ stealth. At present, it is still mainly by means of lowering the radar cross section (RCS) of the target to develop the stealth aircrafts successfully. The principle of the radar stealth is that the radar signal reflected by the aircraft is lowered or directed to the areas radar receiver can not detect, thus reducing the effective detecting distance of the enemy radar. As in [1], the radar scattering signal from the inlet presents a substantial proportion of that of the whole aircraft, so the research of the inlet and its RCS is significant.

In investigating radar anti-stealth, researchers often need to first establish a scattering model of the target for simulation, and then choose one and more efficient methods to calculate the target’s RCS. Full-wave Computational Electromagnetic (CEM) methods can be divided into two major categories, frequency-domain methods and time-domain methods. Frequency-domain method includes Method of Moments (MoM) and Finite Element Method (FEM), etc.. Time-domain method includes Finite-Difference Time-Domain (FDTD), Finite Integral Time-Domain (FITD), Transmission Line Method (TLM), etc.. The basic idea of frequency-domain methods is to solve the Maxwell’s Equations in harmonic form, while for time-domain methods it is directly associated with the physical and transient characteristics of electromagnetic waves.

Many high-frequency methods and hybrid methods for calculating the RCS of inlets and cavities have been put into

study and applications [2]. For example, in hybrid method for calculating the RCS of cavity, finite-difference time-domain method is used to analyze the complexly shaped terminal of the inlet, and several high-frequency methods are utilized in analyzing the remaining part of the cavity [4].

In this paper, an S-bend inlet is simulated in CST MWS [12] with its four different full-wave methods including Finite Integral Time-Domain (FITD) method, Finite Element Method (FEM), Multi-Level Fast Multipole Method (MLFMM) and Method of Moment (MoM), and the simulation results are compared with the measurement.

As the terminal of the inlet is complex, the model is simplified here to be short ended in order to test the relation among the theoretical analysis, numerical simulation and the measurement. The structure and dimensions of the inlet is shown in Fig.1. The dimension of cross section is 203.2 mm×101.6 mm, and the thickness of the sidewall is 5mm.

Vertical incidence

Direction of

polarization

Fig.1 Construction and dimensions of the whole S-bend inlet in E-plane

Ⅱ. THEORETICAL ANALYSIS

A. RCS When the incident electromagnetic waves come onto certain

metal object, induced current will be generated on the target surface, in accordance with the Maxwell Equations and its corresponding electromagnetic boundary conditions. And the induced current has its own electromagnetic field called the target’s scattering field, which, together with the incident field, constitutes the total field. The space distribution of the scattering power depends on the target’s shape, size and structure, as well as the frequency of the incident wave, polarization mode and some other factors.

The physical quantity representing the intensity of the target scattering field is defined as the effective scattering section towards radar, denoted asσ :

978-1-4244-1880-0/08/$25.00 ©2008 IEEE. ICMMT2008 Proceedings

4σ π= ×Scattering power return to the radar receiver per solid angle

power of incident wave at the target(1)

The precondition of the RCS definition is plane wave incidence to achieve a uniform irradiation. The mathematical expression is:

22

2 2

| | | |lim 4 lim 4| | | |

s s

i iR R

ER RE H

σ π π→∞ →∞

= =2

2 H (2)

In equation (2), sE and s

H represent the scattering electric and magnetic fields, respectively, while and respectively the incident electric and magnetic fields, respectively. The unit of RCS is m

iE

iH

2 or dBsm ( 10logdBsmσ σ= ).

B. FITD Finite-Difference Time-Domain (FDTD) is a method which

directly discretizes the partial differential form of Maxwell’s equations [3] based on the Yee Grid. In contrast, the FITD algorithm implemented in CST MWS is derived from the integral form of the Maxwell’s Equations rather than the differential form as for FDTD. FITD was first proposed by T. Weiland in 1976 [7]. It is then further developed in combination with the concept of Perfect Boundary Approximation (PBA®) technique [8], which improves the performance of the classic staircase FDTD significantly.

The topology of the system matrix of FITD, only dependant on the structure, dielectric and boundary conditions of the model, is large sparse matrix, which is similar to that of FDTD. With the PBA® technique, the elements of the system matrix of FITD can be accurately evaluated.

A remarkable feature of FITD is that the major mathematic properties of the analytical gradient, curl, and divergence operators are well maintained in the grid space, for example,

and . 0div rot ≡ 0rot grad ≡FITD keeps all the advantages of the FDTD structured

hexahedral grids, e.g. fast mesh generation and explicit formulation, and hence the fast calculation speed, while maintains an accurate modeling of curved structures. Figure 2 shows the comparison of modeling accuracy of a curved object with the hexahedral grid of classic staircase FDTD and FITD (FIT+ PBA®), as well as the tetrahedral grid of FEM.

Fig.2. Comparison in modeling curved structures with classic FDTD、

FIT+PBA® and FEM

The computing time and memory consumption for a zero-order FITD is linearly proportional to the number of grids (N).

C. FEM Finite Element Method (FEM) [9] divides the field domain

into finite small sections each of which is replaced by a chosen approximate function. A group of linear algebraic equations is formulated for solution. In contrast to FITD the final equation for solution is implicit in terms of the unknowns. Therefore

FEM is more computing time intensive and memory intensive. The computing effort and memory goes with . FEM is based on tetrahedral meshes, which show a smooth approximation of curved objects.

2N

D. MoM Method of Moments (MoM) [10] is an integral equation

based algorithm. As integral equations automatically satisfy the radiating boundary conditions, MoM is especially applicable to open domain problems, such as scattering and radiating problems.

Generally, the solving procedure with MoM includes the following four steps [11]: (1) Expanding the unknown quantity into series consisting of basic function; (2) Choosing testing function used in scalar product with basic function; (3) Constructing matrix equations from the scalar product; (4) Solving the matrix equations and getting the solution.

Suppose power series function as the basic function and testing function:

1 1

1 1

M Mi j

iji j

J I x y− −

= =

= ∑∑ (3)

For plane wave incidence, the boundary condition on the surface of perfect conductor is:

s in E n E× = − × (4)

( )s

sE L J= (5) Suppose the th basic function represents the currentl lJ ,

sJ is the surface current of the metal target, which is the source

of scattering field sE and satisfies the equation s l l

lJ I J=∑

)

.

From (5), 1 1

1 1( ) (

M Ms i jl l l ij

l l i iE E L I J L I x y− −

= =

= = =∑ ∑ ∑∑ (6)

Use (6) to expand (4) in accordance with Galerkin – MoM into matrix equations:

[ ][ ] [ ]kl l kZ I V= (7) , ( ) ,

,kl k l k l

ik k

Z J L J J E

V J E

⎧ =< >=< >⎪⎨

=< >⎪⎩ (8)

The formula solving the electric field through potential function is:

l l lE grad V j Aω= − − (9) The magnetic vector potential is expressed as:

0 ( )lgls

A J R dsμ= ∫ (10)

The scalar potential is expressed as:

0

1 ( )ls

V lg Rρε

= ∫ ds (11)

The relation between the surface charge density and surface current is:

( / ) ll j div Jρ ω= (12)

Put (12) into (11), and put (10) into (9), then the scattering field (including the unknown coefficientlE ijI ) generated by the l th basic function is available. The impedance matrix [Z] can be deduced from (8).

The incident field is defined as follows: 0( ) exp( )

iE r E j r kβ= − i (13)

0E , the amplitude of the electric field vector, is complex, and . The phase constant is0| | 1E = β ω με= ; represents the location vector of the equiphase surface; k represents the direction of incidence.

r

Considering different direction of polarization of incident wave, choose the corresponding and put it into (8), the excited matrix [V] can be solved; then solve (12), and the current coefficient matrix [I] is available; from (9) and (13), the scattering electric field

0E

sE is known; finally, put (13) and

sE into (2), the value of RCS can be obtained.

The computational time and memory for 2nd order MoM is proportional to . 3N

E. MLFMM Despite high computational accuracy of MoM, it requires so

long computational time and so large memory, that it is hard to apply MoM in scattering analysis of three-dimensional electrically large objects like the inlet in this paper..

Multi-Level Fast Multipole Method (MLFMM) [5-6] is deduced based on the MoM method. The basic idea of MLFMM is to group the discrete grids of traditional MoM into far-zone and near-zone according to the distance. The electromagnetic coupling among unit elements in the near-zone group still uses the MoM formulation, while electromagnetic coupling to all the elements in the far-zone group is decomposed into three steps of aggregation, transmission and disaggregation. It speeds up the solution of matrix through simplifying the coupling of induced currents. In scattering problems, Lagrange's interpolation formula technique is applied.

The computing effort and memory consumption for MLFMM is proportional to NlogN. It is dramatically reduced compared to MoM.

Ⅲ. NUMERICAL SIMULATION

A. FITD, FEM, MLFMM All simulations presented in this paper are the converged

ones. By convergence we mean that multiple calculations are performed with different number of meshes until the results converge with one another.

The bi-static RCS results in H-plane calculated with FITD, FEM, and MLFMM methods in CST MWS are shown in Fig.3 and the monostatic RCS values are -3.135, -8.277, and -8.933 dBsm, respectively. It can be seen that FITD and MLFMM show a converged result in the forward scattering direction while FEM and MLFMM coincide well in the backward direction. The strong oscillating behavior of FEM in the forward direction differs rather large compared to physical

understanding. Figure 4 shows the surface current distribution calculated

with MLFMM.

Fig.3. Comparison of bistatic RCS results of the inlet

with FITD、FEM and MLFMM

Fig.4. Surface current distribution on the inlet

B. MoM As we pointed above that MoM is a memory intensive algorithm, it is hard to simulate the whole inlet. In this test we are only able to simulate a degenerated inlet as shown in Fig.5. This is the biggest structure that MoM can handle with the given hardware power. It is a straight open cavity with depth of 300mm. The bistatic H-plane RCS curves simulated with FITD, FEM, and MoM are shown in Fig.6, with the monostatic RCS being 5.132dBsm, 5.837dBsm, and 6.034dBsm, respectively. It is found that FITD and MoM curves overlap with each other very well, but FEM deviates greatly from the other two, showing again a clear oscillating behavior near the forward solid angle.

Vertical incidence Direction of

polarization

Fig.5. Geometry of the straight cavity used for MoM（Length: 300mm）

Fig.6. Comparison of bistatic RCS of straight cavity

with FITD、FEM, and MoM

C. Simulation Summary The computational time, memory spent, RCS values, and the

measurement are shown in Table 1. Table.1. Simulation Summary

FDTD FEM MLFMMMesh Quantity 6,519,240 1,427,792 67,618

CPU Time 4h 44m 35s 6h 52m 49s 5h 20m 07sMemory 886MB 21.63GB 4.51GB

Measurement -1.12 dBsm Simulation Results -3.14 dBsm -8.23 dBsm -8.93 dBsm

From Table 1, it is obvious that FDTD is superior over the other two methods in memory consumption, CPU time and accuracy. MLFMM is not very efficient as long as cavity or concave structure exist. The performance of FEM is the worst or least efficient method here for such inlet structure in terms of accuracy, memory, and simulation time. As for MoM, it is too harsh for the memory requirements to be applied in scattering analysis of electrically large model.

Ⅳ. CONCLUSION

Four full-wave CEM algorithms including FITD, FEM, MoM and MLFMM are investigated for monostatic RCS simulation of an S shaped inlet in terms of their accuracy, memory, and CPU time on a PC workstation with 2 CPUs and 24 GB RAM. By comparing the calculated results of each algorithm with the measurement, the following conclusion can be drawn: MoM has the highest accuracy in calculating the RCS of electrically small objects. As the electrical dimension gets larger, CPU time and memory requirements increase tremendously, make it less and less efficient. MLFMM is the most promising method for scattering analysis of electrically large convex structures with metal surface. However, for concave and/or electrically large structures like the inlet in this paper, FITD is the most superior over all the other algorithm in terms of accuracy, CPU time, and memory consumption.

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