Improvement on the Forward-backward Iterative PhysicalOptics Algorithm Applied to Computing the RCS of Large

Open-ended Cavities

Zhang Peng-fei Gong Shu-xiState Key Lab OfAntennas and Microwave Technology, Xidian Univ., Xi'an 710071, China

Email: shan1ii218@sina.com

Abstract: The method of initial value succession isintroduced to the forward-backward iterative physicaloptics(FBIPO) algorithm. The initial residual error isreduced by inheriting the current values. Combined witha relaxation parameter, the method improves theconvergence of the FBIPO algorithm for cavityscattering problems. The accuracy of RCS calculation isalso enhanced by using a more precise mixed-face model.Then, the RCS of a jet inlet model is calculated, with itsresults in good agreement with the experimental ones.The method is also used to calculate the RCS reductionof a circular wave guide by coating the inner wall with athin absorber layer.Key words: Large open-ended cavities; forward-backward iterative physical optics(FBIPO); RCScalculation

FDTD[9-1 1]. The efficiency and the calculation speed ofIPO have been improved in several ways[5-8]. It has alsobeen extended to deal with cavities with lossy walls byusing an equivalent impedance boundary condition[5-6].In this paper, the forward-backward propagationmethodology of [5] is improved by introducing the initialvalue succession into the iterative process. Numericalresults show that the initial error of the FBIPO method ismuch smaller than that of the original FBIPO. Combinedwith the relaxation parameter, the method improves theconvergence of the algorithm for cavity scatteringproblems. Section two reviews the principle of IPOmethod. Section three explains the forward-backwarditerative algorithms and the improvement of FBIPOalgorithm. Numerical results and discussion arepresented in Sections four.

2. IPO Method1. Instruction

The problem of EM scattering of large open-endedcavities, such as jet engine inlets, is of great importancein the RCS forecast of aircrafts [1-6]. Study on EMscattering of large open cavities leads to several RCScalculating methods, such as model match method[1],shooting and bouncing ray method[1,2], Gaussian Beammethod[1], complex ray expansion method[2], iterativephysical optics method [4-6], etc.. Model match methodis limited to a relatively small set of canonicalgeometries for which waveguide modes can be found inclose form. The ray-based methods are limited by theshape and depth of cavities. IPO overcomes thelimitations of these methods[4-8]. The principles ofphysical optics are applied iteratively to evaluate theequivalent surface currents on the inner walls of thecavity. The iterative re-radiation of the wall currentsaccounts for the multiple reflections inside thecavities[5]. On the other hand, IPO is based onhigh-frequency asymptotic principles of physical opticsand uses a much coarser discretization density(4-9 testpoints per square of wavelength), thus being much moreefficient than purely numerical methods (about 100 testpoints per square of wavelength)such as MoM, FEM and

The scattering fields of the outside of the cavitiesstructure are varied with the different ways the cavitiesembedded in the aircraft. The paper deals with the fieldscoupled into the cavities through the aperture of cavities.Fig. 1(a) shows the geometry of arbitrarily shapedopen-ended cavities illuminated by a plane wave in the

aperture S.' The inner cavity wall sc is assumed to beelectrically large and smooth. In the following analysisthe harmonic time convention is assumed to be ejwt.z0 and A present for the impendence and wavelength inthe free space respectively. In a similar way as in theoriginal IPO[1], the original problem of Fig.l(a) can bereplaced by a equivalent one depicted in Fig.l(b) bysubstitute the inner wall by the equivalent electric and

s, -

A(EH.J M

t

(a)Cavity geometry (b) Equivalent current model

Figure . Open-ended cavity and its equivalent modelmagnetic surface currents( J, IMV ) radiate in free space.The currents are found by enforcing the magneticintegral equation (MFIE) of (1) on sc [5].

Jc(r) = n X Ha + n x (pv. fs J(rj ) x VG0(r - rj )dsc

+ 1 Vxpv. MQ'(rc )xVGO(r -r_c)dsc<)} (1)jkzo

where "pv" denotes the principal value integral, which

does not include the area of ri - r'.

e-jkRG0 (i -- ) = R is the Green function.

Ha = f JQ<) x VGo (r-r<a)dSa

+ Vx YaMQ )XVGo(Q- -)dSa (2)

Ha is the magnetic field radiated from the electric and

magnetic aperture currents Ja,Ma on Sa .The unit

back scattering fields E',H' on Sa radiated by

Jc andMc can be calculated by the Green function. The

scattering fields of far region and RCS are calculated bycombining the Kirchhoff equivalent currents

Js andMs of (6) with aperture integral. It should be

noted that unit vector n of (6) points to the outside ofcavity, which is in the opposite direction ofn in (3).

Ja n Has (ra) Ala Ea (ra)n(6To set up the cavity model and get the messages of thetest points, the traditional way is decomposing the modelinto triangle or rectangle faces which present the currentelements, and placing the test points in the center of thefaces. However, the limitations are that the rectanglefaces cannot simulate the cavity precisely, and thatsimulation of the cavity with triangle faces will lead to agreat number of faces and test points. In this paper, themodels of cavities are mixed-face model which includetriangle faces and rectangle faces as shown in Fig.2. Themain part of the circular surface is simulated byrectangle faces which ensure the fitness of the modelused in the IPO algorithm. The triangle faces areemployed to simulate the small part, such as edge, toensure the precision of the model.

surface vector nI points at the inside of the cavity.

Ja n x H' (r) ,M E' (ra)n (3)

The electric and magnetic surface currents Jc , Mc on (1)

are related by the impendence boundary condition [5]:

MJr)=-zj?)hxJc(?) (4)

Incifield- For-a-i'Forward,

Figure2. Mixed-face modelof a circular surface

O<p<q4 0

Figure3. Forward andbackward substitution

where zs = jz0 tg(kt cr/'r) (5)

zs is the surface impendence.

The MFIE of (1) is solved for the unknown cavity

surface currents J, (using the relation of(4) for MA )by

iterative numerical integration. The detail iterative

process will be discussed in section three. Once the

stable values of equivalent currents are achieved, the

3. FBIPO and Improvement

The most important work in IPO is to solve the unknown

cavity surface currents J, , M, by the iteration of

equation(1). The cavity with a PEC inner wall as shownin Fig.3 is taken here as an example to show the iterativeprocess. N is the total number of faces in sc and k is the

iterative number. In the PEC case, the equation of (1)reduces to the form of (11):

Nk+l (r) =2hXfI +2hx k( I)xVG0 -j)AK

p-i(11)

The PO fields on the inner wall are used to calculate the

q 0

q<p<x

initial values in the iteration.

J0(r ) snhaoCO shadow

(12)

To solve Jc , Mc, the original way is directly iterativecalculation of Equation (11). The FBIPO developed fromthe original IPO is demonstrated to provide a veryrapidly convergent solution. The principle of FBIPO isshown in Fig 3. The discrete current elements arenumerical consecutively in the direction of propagation.The currents are updated by using forward and backwardsubstitution for each iteration[5].The forwardpass forq= 1,2,.....N:

-k+112 ~~~q-1l k1 2jk+l (r) Ji + 2n xEj (r' )xVGo (r - rC )Ascp=

N

+ E Jk (r') xVGO (r - rj)Asjp=q+l

(13)a

The backward pass for q=N,N-1,... 1:

-k+1 ~~~~q-1l +1J$+l (r ) =J' +2nxZj, (r1 ) xVGo (r )-rA)Asp=1

N

+ E Jp+ (ri )xVGO(r- <r )Asc (12-b)p=q+l

Analysis shows that the FBIPO can update currentstwice in a whole forward-backward iteration. But theintegration calculation work is just the same as that ofone interaction in the original IPO[5].It is demonstrated that the adequate number of iterationsis straight forwardly related to the number of expectedinternal reflection of the cavity fields. It can be foreseenthat the iterative time will increase very quickly with theinternal reflection of the cavity fields when the incidentangle departs the propagation axial. Thus the calculationspeed will slower with the increase of the incident angle.To solve this problem, the paper proposes a new methodthat combines the initial value succession with FBIPO. Itis obvious that the convergent currents in the iteration donot vary sharply with the incident angle. Thus, if theconvergent currents in the iteration of 0 were adoptedas the initial values of currents in the iteration of0 + 1 as in (14), it could be found that the initial valueswould be very close to the convergent ones, and that afew times of iteration could match the convergentrequirement.

-+r ) Js r )(4J~~~A (/~~~~~ (14)As in the original FBIPO method, relaxation parameterco= 0.5 is employed to improve convergence andreduce the chance of divergence of iterative solution.

4. Results and Discussion

Fig.4 shows the convergence of the residual error for theoriginal IPO, IPO with a relaxation parameter, FBIPOand FBIPO with initial value succession. The residualerror here presents for the difference of the currentsbetween two iteration. The Geometry is a circular PECwave guide Fig.8(a). A 1OGHz plane wave incident from450 off axial in the horizontal plane illuminates the cavity.As shown in Fig4, the relaxation parameter improves theconvergence of the iteration. The FBIPO is remarkablyfaster than the traditional IPO. Succession of initialvalues reduces the initial residual error greatly. Therefore,the times of iteration to match the convergencerequirement is reduced greatly. The agreement betweenthe RCS calculation results calculated from themixed-face model and the reference results shown inFig.5 are quite good. The RCS calculation results basedon the rectangle faces are also shown in Fig.5 todemonstrate the advantage of the mixed-face model.Fig.7 gives the RCS calculation results of a jet inletmodel shown in Fig.6. A 1OGHz plane wave incident inthe horizontal plane illuminates the cavity. The RCScalculation results in the horizontal plane are in goodagreement with the experimental ones. In Fig.8(b), thecircular wave guide is coated with a thin layer ofabsorber material between z= -4 -82 , theequivalent impedance of the absorber materialsiszs = Zo(0.11537- jO.18369). RCS calculation results

shown in Fig.9 present an obvious reduction.

5. Summarry

The initial value succession applied to the FBIPOalgorithm has been shown to reduce the initial residualerror of iteration, hence increasing the convergencespeed of FBIPO greatly. It is also shown that themixed-face model includes triangle and rectangle facescan simulate the cavity more precisely and that it canlead to more precise RCS calculation results. The RCSreduction by coating the inner wall of cavity withabsorber layer is also shown in the calculation example.

E

z

LLFU

u)7au

IPO------ IPO+Relax parameter----FBIPO+Relax parameter

-FBIPO+lnitial value succession+relax parameter

N 20 30

N u m be r of Ite ratio n

Figure4. Residual error curves for four iterative methods

~~- -_

1 .0 -

O .8 -

O0.6 -

O0.4 -

0.2 -

Degree

ire5. RCS patterns of a circular wave guide

Figure6.Ajet inlet model

degr 0

Figure7. RCS patterns of the jet inlet model

08 8

12A

Figure8. (a) Circularwave guide

I-

(b) Circular wave guidewith absorber lining

10-id ~ --Sarroevu 'fabobrl,it la - RCSof .del1 ith b d.,Mling

Fiue.RCSSfmcd to-rdcin ofmdircularwave guideibyaddingan absorber lining~~~~~~~~~~~~~1

References

[1] P. H. Pathak and R. J. Burkholder, "Modal, ray, andbeam techniques for analyzing the EM scattering byopen-ended waveguide cavities", IEEE Trans. Antennas

Propagat, 37 , p. 635-647( 1989).[2] H. Shirai and L. B. Felsen, "Rays and modes forplane wave coupling into a large open-ended circularwaveguide" , Wuiv Motion, 9, pp. 461-482, (1987).[3] P. D. Einziger, Y. Harmaty, and L. B. Felsen,"Complex rays for radiation from discretized aperture

distribution" IEEE Trans. Antennas Propagat,35,p.1031.(1987)[4] 1. F. Obelleiro, J.L. Rodriguez, and R.J. Burkholder,"An iterative physical optics approach for analyzing theelectromagnetic scattering by large open-ended cavities",IEEE Trans Antennas Propagat, 43, p.356_361,(1995).[5] Burkholder, R.J.; Lundin, T.; "Forward-backwarditerative physical optics algorithm for computing theRCS of open-ended cavities Antennas and Propagation",IEEE Transactions, 53, P.793 - 799,(2005).[6] F. Obelleiro, M. G. Araujo, and J. L. Rodriguez,"Iterative physical-optics formulation for analyzing largewaveguides with lossy walls," Micro. Opt. Tech. Lett.,28, p. 21-26,(2001).[7]R. J. Burkholder, "A fast and rapidly convergentiterative physical optics algorithm for computing theRCS of open-ended cavities," Appl. ComputationalElectromagn. Soc. J, 16, p. 53-60,(2001).[8]C. C. Lu and W. C. Chew, "Fast far-fieldapproximation for calculating the RCS of large objects,"Microw. Opt. Tech. Lett., 8, p.238-241,(1995).[9]H. Ammari, G. G. Bao, and A. W. Wood, "An integralequation method for the electromagnetic scattering from

cavities," Math. Methods Appl. Sci, 23, p. 1057-1072,(2000).

[Io]J. Liu and J. M. Jin, "A special higher order

finite-element method for scattering by deep cavities,"IEEE Trans. Antennas Propag, 48, p. 694-703,(2000).[1 I]T. Chia, R. J. Burkholder, and R. Lee, "The

application ofFDTD in hybrid methods for cavityscattering analysis," IEEE Trans. Antennas Propag, 43,,p.1082-1090,( 1995).

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