research article fast direct solution of electromagnetic...

Research ArticleFast Direct Solution of Electromagnetic Scattering fromLeft-Handed Materials Coated Target over Wide Angle

Guo-hua Wang1 and Ying-bao Geng2

1New Star Institute of Applied Technology Hefei 230039 China2School of Medical Information Engineering Anhui University of Chinese Medicine Hefei 230038 China

Correspondence should be addressed to Ying-bao Geng gengyingbao163com

Received 5 February 2016 Revised 6 April 2016 Accepted 9 May 2016

Academic Editor John N Sahalos

Copyright copy 2016 G-h Wang and Y-b Geng This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

When solving the electromagnetic scattering problems over wide angle the traditional method of moments (MoM) needs to repeatthe solving process of dense systems of linear equations using the iteration method at each incident angle which proved to be quiteinefficient To circumvent this problem a fast numerical method based on block LDLT factorization accelerated by adaptive crossapproximation (ACA) algorithm is presented to analyze the electromagnetic scattering of left-handed materials (LHM) coatedtarget The ACA algorithm is applied to impedance matrix filling and all steps of block LDLT factorization process which canaccelerate the computation process and reduce the memory consumptionThe numerical results proved that the proposed methodis efficient in calculatingmonostatic RCS of LHMcoated target withmany required sampling angles Compared with the traditionalMoM computation time and memory consumption are reduced effectively

1 Introduction

Left-handed materials (LHM) [1 2] have electromagneticproperties which cannot be generally found in nature theelectric field direction magnetic field direction and thepropagation vector conform to the laws of the left handwhen electromagnetic wave travels in it LHM are alsocalled double negative materials (DNG) [3] as the dielectricpermittivity and permeability at a certain frequency rangecan be negative together The application of LHM has beenwide in antenna [4] microwave devices and other fields [5]due to their unusual physical properties such as negativerefraction inverse Doppler shift and reversed Cerenkovradiation For the purpose of stealth many conductor targetsurfaces are also coated by LHM this necessitates a fast andefficient solution to analyze the electromagnetic scatteringcharacteristics of LHM coated targets [6ndash9]

Researchers have proposed various methods to analyzethe scattering problem of LHM coated targets In literature[7] the radar cross section of metallic spheres covered byDNGbased onMie series solution is studied but this method

is valid only in certain circumstances when the geometricstructure of the target is simple In literature [8] the physicaloptics (PO) method in combination with the impedanceboundary condition (IBC) is used to analyze the near-fieldelectromagnetic scattering characteristics of LHM coatedtargets In literature [9] the wide band scattering of perfectlyconducting target coated with DNG is computed using finitedifference time domain (FDTD) method Apart from theabove methods this paper mainly discusses the fast methodto compute the RCS of LHM coated target over wide anglebased on MoM [10] framework and hybrid PEC-dielectricformulation However when solving the electromagneticscattering problems withmultiple incident angles traditionalMoMneeds to compute the current coefficients as the changeof incident angle Usually iterationmethod is used to solve thelinear system equations which proved to be a quite effectivemethod for bistatic RCS computing but for monostatic RCSwith many sampling angles the iteration operations mustbe repeated when the incident angle changes each time thisis inefficient when the number of sampling angles is largealso there are often convergence issues when dealing with

Hindawi Publishing CorporationJournal of Electrical and Computer EngineeringVolume 2016 Article ID 2108586 7 pageshttpdxdoiorg10115520162108586

2 Journal of Electrical and Computer Engineering

complex structure targets To circumvent this problem a newblock ACA-LDLT method is proposed in this paper to speedup the filling process of impedance matrix and solve thelinear system equations more effectively which is suitablefor computingmonostatic RCS withmany required samplingangles of complex structure targets

2 Hybrid PEC-Dielectric Formulation forLHM Coated Target

Recently a method [11] based on thin dielectric sheet (TDS)approximation with explicit perfect electric conductor (PEC)boundary conditions at the interfaces of PEC and dielectricswas provided which can get accurate numerical solution ofEM scattering but is only suitable for electrical coating mate-rials In literature [12] a generalized thin coating equivalentmodel was proposed to be suitable for both electrical andmagnetic coatingmaterials but there is no further discussionwhen the coating medium is LHM In this section wemainly discuss the building up of the hybrid PEC-dielectricformulation for LHM coated target

For LHM coated PEC target under the irradiation of idealplane wave shown in Figure 1 119878 stands for the surface ofPEC 119881 is the volume of coating LHM and 120591 is the coatingthickness The total tangential component of electric field iszero following the PEC boundary condition of the electricfield we can obtain

[Einc(r) + Esca

(r)]10038161003816100381610038161003816tan = 0 r isin 119878 (1)

here the subscript ldquotanrdquo stands for the tangential componentr stands for the field point Einc stands for the incident fieldand Esca is the scattering field which could be described as

Esca(r) = Esca

pec (r) + Escadie (r) (2)

The scattering field is produced by conductors and LHMcoating medium together and the scattering field producedby the electric current on 119878 could be described asEscapec (r)

= 1198951205961205830int119878

(J119878(r1015840) +

1

1198962

0

nablanabla1015840sdot J119878(r1015840))119892 (r r1015840) 119889119878

1015840

(3)

here 119892(r r1015840) = 119890minus119895119896|rminusr1015840|

4120587|r minus r1015840| stands for scalar Greenrsquosfunction in free space J

119878is surface current density on 119878 and

1205830and 119896

0stand for the permeability and wave number in

free space respectivelyThe scattering field produced by LHMcoating medium in 119881 could be described as

Escadie (r) = 119895120596120583

0[int119881

120594 (r1015840) 119892 (r r1015840) 119895120596D (r1015840) 1198891198811015840

minus1

1198962

0

int119878+

119899

120594nabla119892 (r r1015840) 1198951205961015840sdot D (r1015840) 119889119878

1015840

+1

1198962

0

int119878minus

119899

120594nabla119892 (r r1015840) 1198951205961015840sdot D (r1015840) 119889119878

1015840]

minus int119881

nabla119892 (r r1015840) times 119895120596120585 (r1015840)B (r1015840) 1198891198811015840

(4)

where 120594(r1015840) = 1120576119903(r1015840) minus 1 and 120585(r1015840) = 1120583

119903(r1015840) minus 1 are the

dielectric contrast ratios 1015840 is the unit vector normal to theupper surface When the coating thickness 120591 is very smallcompared to the wavelength we can use an approximatemethod named as TDS approximation

D (r1015840) asymp 1015840nabla1015840sdot

J119878(r1015840)119895120596

B (r1015840) asymp minus1205831015840times J119878(r1015840)

r1015840 isin V

(5)

Then the volume integral can be further approximated tosurface integral through conversion 119889119881 asymp 120591119889119878Therefore thescattering field produced by dielectric can be described as

Escadie (r) = 119895120596120583

0[1 minus 120583119903

2J119878120591 + int119878

1015840120591120594119892nabla1015840sdot J1198781198891198781015840

+nabla

1198962

0

int119878

120594 (119892 minus 119892120591) nabla1015840sdot J1198781198891198781015840

+ (1 minus 120583119903) int119878minus1198780

120591nabla119892 times 1015840times J1198781198891198781015840]

(6)

Here the last term in (6) is the principal value integralFinally we substitute (3) and (6) in (2) in combination with(1) we can obtain the hybrid PEC-dielectric formulationwhich can be expressed by the only unknown J

119878

Einc(r) = minus119895120596120583

0[1 minus 120583119903

2J119878120591

+ int119878

119892 (J119878+ 1015840120591120594nabla1015840sdot J119878) 1198891198781015840

+nabla

1198962

0

int119878

(119892 + 120594119892 minus 120594119892120591) nabla1015840sdot J1198781198891198781015840

+ (1 minus 120583119903) int119878

120591nabla119892 times 1015840times J1198781198891198781015840] r isin 119878

(7)

Here 120576119903and 120583

119903are permittivity and permeability of coating

LHM given by Drude model [13 14] 119892120591is scalar Greenrsquos

function defined by

119892120591=

119890minus119895119896|rminus(r1015840+1205911015840)|

412058710038161003816100381610038161003816r minus (r1015840 + 120591

1015840)10038161003816100381610038161003816

(8)

3 Solutions

We will refer to (8) as a hybrid PEC-dielectric formulationwhich can be solved using the MoM or MLFMA with iter-ation method But for monostatic RCS with many requiredsampling angles iteration method must be repeated withmany right-hand sides (RHS) so this part becomes expen-sive The block LDLT factorization method in combinationwith adaptive cross approximation (ACA) is introduced to

Journal of Electrical and Computer Engineering 3

this paper which can reduce time and memory storage andis proven to be more efficient in computing the monostaticRCS than the method in [12]

31 Block LDLT Factorization Method By discretizing thesurface currents using RWG vector basis functions [15 16]the integral equation (8) can be transformed into a densecomplex linear equation

Z sdot J = V (9)

where Z is the impedance matrix of dimension 119873 times 119873 J isinduced current density and V is excitation voltage matrix119873 is the unknowns number of closed scattering mesh WhenusingMoM to solve the equation the storage complexity andmatrix filling time is 119874(119873

2) and the direct LU factorization

complexity is 119874(1198733) both grow rapidly for electrically large

target and bring a heavy burden to computer To overcome

the problem the unknowns in this work have been groupedinto multiple local regions and the rank of the submatricesdecreases with the increase of distance of two blocks then thecompressed algorithm can be used to reduce the storage andoperations count When unknowns are grouped into localspatial regions (9) can be converted into the following form

(

Z11

Z12

sdot sdot sdot Z1119872

Z21

Z22


d

Z1198721

Z1198722


)(

J1

J2

J119872

) = (

V1

V2

V119872

) (10)

where Z119894119895(119894 119895 = 1 2 3 119872) represent the submatrix of

impedance matrix By using RWG vector basis functions asthe test functions the impedance matrix in (10) is a complexcoefficient symmetric matrix which can be expressed asblock LDLT form

(

Z11

Z12


Z21

Z22


d

Z1198721

Z1198722


) = (

1 0 sdot sdot sdot 0

L21

1 sdot sdot sdot 0

d

L1198721

L1198722

sdot sdot sdot 1

)(

D11

0 sdot sdot sdot 0

0 D22

sdot sdot sdot 0

d

0 0 sdot sdot sdot D119872119872

)(

(

1 L11987921

sdot sdot sdot L1198791198721

0 1 sdot sdot sdot L1198791198722

d


)

)

(11)

where L119894119895are the lower triangular entities and D

119894119894are the

diagonal entities thenmaking the substitutionsU119894119894= D119894119894and

U119894119895

= D119894119894L119879119895119894 we can obtain the standard block LU form

(

Z11

Z12


Z21

Z22


d

Z1198721

Z1198722


)

= (


L21

1 sdot sdot sdot 0

d

L1198721

L1198722

sdot sdot sdot 1

)(

U11

U12

sdot sdot sdot U1119872

0 U22


d

0 0 sdot sdot sdot U119872119872

)

(12)

The iterative formula of U119894119895is

U119894119895

= Z119894119895minus

119894minus1

sum

119901=1

U119879119901119894Dminus1119901119901U119901119895 (13)

The surface currents can be obtained through the forwarditerative step

X119894= V119894minus

119894minus1

sum

119901=1

U119879119901119894Dminus1119901119901X119901 (14)

and back iterative step

J119894= Dminus1119894119894

(X119894minus

119899

sum

119901=119894+1

U119894119901J119901) (15)

32 ACA Acceleration The impedance matrix in (10) iscomposed of multiple submatrices and the off-diagonalblocks are of low rank and can be compressed In literature[17 18] Shaeffer points out that the upper triangular matrixU and multiple plane wave excitation RHS voltage matrixV also have low rank characteristics and can be compressedtoo so the compression operation can be used for all steps ofthe solutions including impedance filling LU factorizationand LU solving In numerous matrix compression methodsthe ACA algorithm in literature [19] is widely used [20ndash22]The ACA algorithm operates by scanning a row followed bya column of the matrix at each iteration and progressivelybuilds up a low rank estimate of the matrix based on the rowsand columns that have been scannedThe algorithm operatesas follows

(1) Initialize the approximate matrix as A119898times119899 asymp 0119898times119899 and

the iteration count as 119895 = 1 arbitrarily choose a rowof the matrix

(2) Scan the 119895th row of the matrix(3) Find the error of the previous field approximation at

the 119895th row R = A119898times119899(119895th row) minus A119898times119899(119895th row) andchoose the 119895th column to be the one containing themaximum element of |R|


(4) Assign V1times119899119895

= RR (119895th column)(5) Scan the 119895th column and find the error of the previous

estimate at this column C = A119898times119899(119895th column) minus

A119898times119899(119895th column) Choose the (119895 + 1)th row to bethe one containing the maximum element of |C|

(ensuring that the 119895th row is not chosen again)(6) Assign 119906

119898times1

119895= C

(7) Update the field approximation A119898times119899 = A119898times119899 +

119906119898times1

119895V1times119899119895

(8) If 119906119898times1

119895V1times119899119895

lt 120576A119898times119899 stop scanning elseincrement 119895 and repeat steps (2) to (8)

A low rank matrix can be well approximated by theproduct of two full rank matrices using ACA method

A119898times119899 asymp A119898times119899 = A119898times119903119880

A119903times119899119881

(16)

where 119903 is the effective rank of matrix and the memoryrequirement decreased from119898times119899 entries to 119903(119898+119899) entriesWe can get the block decomposition expression by using theACA to the iterative process of U

119894119895

U119894119895

= [U119880U119881]119894119895

= [Z119880Z119881]119894119895minus

119894minus1

sum

119901=1

[U119879119881U119879119880]119901119894Dminus1119901119901

[U119880U119881]119901119895

(17)

By using the ACA to the LU solving solution the inducedcurrent density J

119894on 119878 can be obtained by the forward

iterative steps

X119894= V119894minus

119894minus1

sum

119901=1

[U119879119881U119879119880]119901119894Dminus1119901119901

[X119880X119881]119901

(18)

and back iterative steps

J119894= Dminus1119894119894

(X119894minus

119899

sum

119901=119894+1

[U119880U119881]119894119901

[J119880J119881]119901) (19)

4 Numerical Results

In this section we simulated three numerical examplesto demonstrate the accuracy and validity of the proposedmethod All the results are computed on an Intel Core2Duo PC with 340GHz processor and 16GB RAM only onecore is used the ACA iteration error threshold is 1119890 minus 3 Thepercentage error of advised algorithm is defined as

radicV minus Z sdot J

Vtimes 100 (20)

(A) A Dielectric Coated PEC Sphere We start by consideringthe scattering problem from a dielectric coated PEC spherewith radius 1 120582 Incident frequency is 300MHz from 120579 = 0

∘

PEC

S

S+n

Sminusn

120591

V

Einc

Figure 1 A PEC target coated by LHM

0 30 60 90 120 150 180

0

5

10

15

20

25

RCS

(dBs

m)

minus5

minus10

minus15

FEKOTDS-LDLT-ACA

∘) (120601 = 0∘)120579 (

Figure 2 LHM coated PEC sphere

(120593 = 0∘) The thickness of the coating material is 120591 = 001 120582

The relative permittivity of LHM is 120576119903

= 90 and relativepermeability is 120583

119903= 177 minus 119895406 The geometry is divided

into 2220 triangular patches with an average length of 12058210and this gives rise to 3330 unknowns In Figure 2 the bistaticradar cross section (RCS) is shown from 120579 = 0

∘ to 120579 = 180∘

The dashed lines by our proposed TDS-LDLT-ACA methodagree well with the solid lines by the commercial EM softwareFEKO

(B) LHM Coated PEC Cylinder The second example is theproblem of scattering from LHM coated PEC cylinder withradius of 05 120582 and height of 2 120582 as shown in Figure 3 theplane wave is incident from 120579 = 0

∘ (120593 = 0∘) to 120579 = 180

∘

(120593 = 0∘) the relative permittivity of LHM is 120576

119903= minus90 and

relative permeability is 120583119903= minus177 minus 119895406 As a comparison

we have examined the RHM coated case with the relativepermittivity 120576

119903= 90 and relative permeability 120583

119903= 177 minus

119895406 The geometry is divided into 1456 triangular patcheswith an average length of 12058210 and this gives rise to 2184unknowns In ACA operation the geometry is subdividedinto 8 spatial regions as shown in Figure 3 The monostatic


x

y

z

Einc

Figure 3 LHM coated PEC cylinder

0 30 60 90 120 150 180

0

10

20

30

RCS

(dBs

m)

minus10

minus20

minus30

minus40

minus10

minus20

minus30

minus40

LHM_FEKORHM_FEKO

LHM_TDSRHM_TDS

∘) (120601 = 0∘)120579 (

(a) Coating thickness 0005120582

0 30 60 90 120 150 180

0

10

20

30

RCS

(dBs

m)

minus10

minus20

minus30

minus40

LHM_FEKORHM_FEKO

LHM_TDSRHM_TDS

∘) (120601 = 0∘)120579 (

(b) Coating thickness 001 120582

Figure 4 VV polarization monostatic RCS of coated cylinder

x

y

z

Einc

Figure 5 LHM coated PEC missile model

RCS calculated forVVpolarization is shown in Figure 3 alongwith the reference solution provided by the FEKO

In Figure 4(a) the coating thickness is 001 120582 a quitegood agreement between the reference FEKO solution andthe results of advised method can be observed Also we cansee that the monostatic RCS value of coated cylinder withRHM is less than that of LHM coated circumstance in mostof angle range

The monostatic RCS is shown in Figure 4(b) when coat-ing thickness 120591 reduced to 0005 120582 We can find that theRCS change rule with incident plane wave angle is similar toFigure 4(a) When coating thickness reduced the RCS valueof RHM drops about 3 dB in most of incident angle areawhile the RCS values of LHM increase about 2 dB in most ofincident angle area The phenomenon is also consistent withthe conclusion in literature [23] that the reflectivity and RCSwill vary with the coating thickness

(C) LHM Coated PEC Missile Model The third example isLHM coated PEC missile model shown in Figure 5 whichhas a length of 991m a wingspan of 628m and a height of219mThe geometry is divided into 11300 triangular patcheswith an average length of 12058210 and this gives rise to 16950unknowns

The thickness of the material is 120591 = 001 120582 The relativepermittivity of LHM is 120576

119903= minus15 minus 119895015 and relative per-

meability is 120583119903= minus10minus11989501Themissile model is illuminated

by an incident plane wave with the incident direction of


Table 1 Time and RAM requirements

The targets and the unknowns Number of incident plane waves Method RAM requirementsMB Times

Coated sphere 3330 1 FEKO 146 56TDS-LDLT-ACA 132 46

LHM coated cylinder 2184 180 FEKO 118 948TDS-LDLT-ACA 28 385

LHM coated missile model 16950 1 FEKO 1347 924TDS-LDLT-ACA 1355 816

0 30 60 90 120 150 180

0

10

20

30

40

RCS

(dBs

m)

minus10

FEKOTDS-LDLT-ACA

∘) (120601 = 0∘)120579 (

Figure 6 Bistatic RCS of LHM coated PEC missile model

120579 = 90∘ and 120593 = 270

∘ Bistatic RCS is shown in Figure 6along with the reference solution provided by FEKO Weobserve that the agreement between the FEKO results andthose obtainedwith theTDS-LDLT-ACAmethod is excellentThere exists difference between the incident angles 110

∘lt

120579 lt 160∘ however the difference of scattering results is rela-

tively smallerThe total CPU time and RAM requirements of above

three examples are shown in Table 1 It can be seen that theproposedmethod has compressedmuchmore CPU time andrandom memory demand than conventional MoM

5 Conclusion

ATDS approach together with explicit PEC boundary condi-tions has been proposed to handle the EM scattering prob-lems of thin LHM coating target efficiently The modelingprocess has been greatly simplified comparedwith the surfaceor volume integral equation approach The ACA methodis incorporated into block LDLT factorization algorithmto reduce the filling time of impedance matrix and speedup the factorization and solving processes The numericalresults demonstrate that the backscattering monostatic RCSvalues in specific angle area can be reduced by selecting theappropriate LHM coating thicknessThemethod can be usedfor electromagnetic scattering problems of complex structure

LHM coated PEC target and has an important referencevalue in practical engineering applications of arbitrary shapestargets

Competing Interests

The authors declare that they have no competing interests

References

[1] H Chen B-I Wu and J A Kong ldquoReview of electromagnetictheory in left-handed materialsrdquo Journal of ElectromagneticWaves and Applications vol 20 no 15 pp 2137ndash2151 2006

[2] V G Veselago ldquoThe electrodynamics of substances with simul-taneously negative valueS OF 120598 and 120583rdquo Soviet Physics Uspekhivol 10 no 4 pp 509ndash514 1968

[3] N Engheta and RW Ziolkowski ldquoA positive future for double-negative metamaterialsrdquo IEEE Transactions on Microwave The-ory and Techniques vol 53 no 4 pp 1535ndash1556 2005

[4] K Z Rajab R Mittra and M T Lanagan ldquoSize reductionof microstrip antennas using metamaterialsrdquo in Proceedings ofthe IEEE Antennas and Propagation Society International Sym-posium pp 296ndash299 Washington DC USA July 2005

[5] C J Zapata-Rodrıguez D Pastor L E Martınez and J J MiretldquoLeft-handed metamaterial coatings for subwavelength-resolu-tion imagingrdquo Journal of the Optical Society of America A Opticsand Image Science andVision vol 29 no 9 pp 1992ndash1998 2012

[6] L K Hady and A A Kishk ldquoElectromagnetic scattering fromconducting circular cylinder coated by metamaterials andloaded with helical strips under oblique incidencerdquo Progress InElectromagnetics Research B vol 3 pp 189ndash206 2008

[7] M-Y Wang J Xu J Wu andW-F Wu ldquoElectromagnetic scat-tering ofmetallic sphere covered by double-negativemetamate-rials based on Mie seriesrdquo Systems Engineering and Electronicsvol 30 no 11 pp 2082ndash2086 2008

[8] R Zhao H-L Yang N Wu and D-G Xie ldquoNear-field EMscattering by coated objects with left-handed materialsrdquo inProceedings of the Cross Strait Quad-Regional Radio Science andWireless Technology Conference (CSQRWC rsquo11) vol 1 pp 44ndash47Harbin China July 2011

[9] B Wei D-B Ge and M-Y Wang ldquoWide band scatteringof perfectly conducting object coated with DNMrdquo SystemsEngineering and Electronics vol 28 no 6 pp 840ndash843 2006

[10] R F Harrington Field Computation by Moment MethodsMacMillan New York NY USA 1968

[11] C P Davis and W C Chew ldquoAn alternative to impedanceboundary conditions for dielectric-coated PEC surfacesrdquo inProceedings of the IEEE Antennas and Propagation Society Inter-national Symposium pp 2785ndash2788 IEEE Honolulu HawaiiUSA June 2007


[12] S HeTheoretical Modeling and Efficient Algorithm Research onElectromagnetic Scattering from Inhomogeneous and ComplexStructures University of Electronic Science and Technology ofChina (UESTC) Chengdu China 2011

[13] Z Liu J Cao and Y Geng ldquoSimulation study on the EMscattering by left-hand-materialrdquo Chinese Journal of ElectronDevices vol 34 no 2 pp 154ndash158 2011

[14] M S Wheeler J S Aitchison and M Mojahedi ldquoCoated non-magnetic spheres with a negative index of refraction at infraredfrequenciesrdquo Physical Review BmdashCondensed Matter and Mate-rials Physics vol 73 no 4 Article ID 045105 2006

[15] S M Rao D R Wilton and A W Glisson ldquoElectromagneticscattering by surfaces of arbitrary shaperdquo IEEE Transactions onAntennas and Propagation vol 30 no 3 pp 409ndash418 1982

[16] A F Peterson S L Ray and R Mittra Computational Methodsfor Electromagnetics IEEE New York NY USA 1998

[17] J Shaeffer ldquoDirect solve of electrically large integral equationsfor problem sizes to 1M unknownsrdquo IEEE Transactions onAntennas and Propagation vol 56 no 8 pp 2306ndash2313 2008

[18] J Shaeffer and F Canning ldquoAdaptive cross approximation forMoM matrix fill for PC problem sizes to 157000 unknownsrdquoin Proceedings of the IEEEACES International Conference onWireless Communications and Applied Computational Electro-magnetics Society (ACES rsquo05) pp 748ndash753 Honolulu HawaiiUSA April 2005

[19] M Bebendorf ldquoApproximation of boundary element matricesrdquoNumerische Mathematik vol 86 no 4 pp 565ndash589 2000

[20] K Zhao M Vouvakis and J-F Lee ldquoThe adaptive crossapproximation algorithm for accelerated method of momentscomputations of EMC problemsrdquo IEEE Transactions on Electro-magnetic Compatibility vol 47 no 4 pp 763ndash773 2005

[21] R Maaskant R M Mittra and A Tijhuis ldquoFast analysis oflarge antenna arrays using the characteristic basis functionmethod and the adaptive cross approximation algorithmrdquo IEEETransactions on Antennas and Propagation vol 56 no 11 pp3440ndash3451 2008

[22] C Li S-Y He J Yang Z Zhang B-X Xiao and G-Q ZhuldquoMonostatic scattering from two-dimensional two-layer roughsurfaces using hybrid 3DMLUV-ACA methodrdquo InternationalJournal of Applied Electromagnetics and Mechanics vol 42 no1 pp 1ndash11 2013

[23] D Lu and C Tong ldquoRCS simulation of target coated with chiralmediumrdquo Fire Control amp Command Control vol 36 no 6 pp67ndash70 2011

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complex structure targets To circumvent this problem a newblock ACA-LDLT method is proposed in this paper to speedup the filling process of impedance matrix and solve thelinear system equations more effectively which is suitablefor computingmonostatic RCS withmany required samplingangles of complex structure targets

2 Hybrid PEC-Dielectric Formulation forLHM Coated Target

Recently a method [11] based on thin dielectric sheet (TDS)approximation with explicit perfect electric conductor (PEC)boundary conditions at the interfaces of PEC and dielectricswas provided which can get accurate numerical solution ofEM scattering but is only suitable for electrical coating mate-rials In literature [12] a generalized thin coating equivalentmodel was proposed to be suitable for both electrical andmagnetic coatingmaterials but there is no further discussionwhen the coating medium is LHM In this section wemainly discuss the building up of the hybrid PEC-dielectricformulation for LHM coated target

For LHM coated PEC target under the irradiation of idealplane wave shown in Figure 1 119878 stands for the surface ofPEC 119881 is the volume of coating LHM and 120591 is the coatingthickness The total tangential component of electric field iszero following the PEC boundary condition of the electricfield we can obtain

[Einc(r) + Esca

(r)]10038161003816100381610038161003816tan = 0 r isin 119878 (1)

here the subscript ldquotanrdquo stands for the tangential componentr stands for the field point Einc stands for the incident fieldand Esca is the scattering field which could be described as

Esca(r) = Esca

pec (r) + Escadie (r) (2)

The scattering field is produced by conductors and LHMcoating medium together and the scattering field producedby the electric current on 119878 could be described asEscapec (r)

= 1198951205961205830int119878

(J119878(r1015840) +

1

1198962

0

nablanabla1015840sdot J119878(r1015840))119892 (r r1015840) 119889119878

1015840

(3)

here 119892(r r1015840) = 119890minus119895119896|rminusr1015840|

4120587|r minus r1015840| stands for scalar Greenrsquosfunction in free space J

119878is surface current density on 119878 and

1205830and 119896

0stand for the permeability and wave number in

free space respectivelyThe scattering field produced by LHMcoating medium in 119881 could be described as

Escadie (r) = 119895120596120583

0[int119881

120594 (r1015840) 119892 (r r1015840) 119895120596D (r1015840) 1198891198811015840

minus1

1198962

0

int119878+

119899

120594nabla119892 (r r1015840) 1198951205961015840sdot D (r1015840) 119889119878

1015840

+1

1198962

0

int119878minus

119899

120594nabla119892 (r r1015840) 1198951205961015840sdot D (r1015840) 119889119878

1015840]

minus int119881

nabla119892 (r r1015840) times 119895120596120585 (r1015840)B (r1015840) 1198891198811015840

(4)

where 120594(r1015840) = 1120576119903(r1015840) minus 1 and 120585(r1015840) = 1120583

119903(r1015840) minus 1 are the

dielectric contrast ratios 1015840 is the unit vector normal to theupper surface When the coating thickness 120591 is very smallcompared to the wavelength we can use an approximatemethod named as TDS approximation

D (r1015840) asymp 1015840nabla1015840sdot

J119878(r1015840)119895120596

B (r1015840) asymp minus1205831015840times J119878(r1015840)

r1015840 isin V

(5)

Then the volume integral can be further approximated tosurface integral through conversion 119889119881 asymp 120591119889119878Therefore thescattering field produced by dielectric can be described as

Escadie (r) = 119895120596120583

0[1 minus 120583119903

2J119878120591 + int119878

1015840120591120594119892nabla1015840sdot J1198781198891198781015840

+nabla

1198962

0

int119878

120594 (119892 minus 119892120591) nabla1015840sdot J1198781198891198781015840

+ (1 minus 120583119903) int119878minus1198780

120591nabla119892 times 1015840times J1198781198891198781015840]

(6)

Here the last term in (6) is the principal value integralFinally we substitute (3) and (6) in (2) in combination with(1) we can obtain the hybrid PEC-dielectric formulationwhich can be expressed by the only unknown J

119878

Einc(r) = minus119895120596120583

0[1 minus 120583119903

2J119878120591

+ int119878

119892 (J119878+ 1015840120591120594nabla1015840sdot J119878) 1198891198781015840

+nabla

1198962

0

int119878

(119892 + 120594119892 minus 120594119892120591) nabla1015840sdot J1198781198891198781015840

+ (1 minus 120583119903) int119878

120591nabla119892 times 1015840times J1198781198891198781015840] r isin 119878

(7)

Here 120576119903and 120583

119903are permittivity and permeability of coating

LHM given by Drude model [13 14] 119892120591is scalar Greenrsquos

function defined by

119892120591=

119890minus119895119896|rminus(r1015840+1205911015840)|

412058710038161003816100381610038161003816r minus (r1015840 + 120591

1015840)10038161003816100381610038161003816

(8)

3 Solutions

We will refer to (8) as a hybrid PEC-dielectric formulationwhich can be solved using the MoM or MLFMA with iter-ation method But for monostatic RCS with many requiredsampling angles iteration method must be repeated withmany right-hand sides (RHS) so this part becomes expen-sive The block LDLT factorization method in combinationwith adaptive cross approximation (ACA) is introduced to




Z sdot J = V (9)






(

Z11

Z12


Z21

Z22


d

Z1198721

Z1198722


)(

J1

J2

J119872

) = (

V1

V2

V119872

) (10)



(

Z11

Z12


Z21

Z22


d

Z1198721

Z1198722


) = (


L21

1 sdot sdot sdot 0

d

L1198721

L1198722

sdot sdot sdot 1

)(

D11

0 sdot sdot sdot 0

0 D22

sdot sdot sdot 0

d


)(

(

1 L11987921


0 1 sdot sdot sdot L1198791198722

d


)

)

(11)


119894119894are the


U119894119895


(

Z11

Z12


Z21

Z22


d

Z1198721

Z1198722


)

= (


L21

1 sdot sdot sdot 0

d

L1198721

L1198722

sdot sdot sdot 1

)(

U11

U12


0 U22


d


)

(12)


U119894119895

= Z119894119895minus

119894minus1

sum

119901=1

U119879119901119894Dminus1119901119901U119901119895 (13)


X119894= V119894minus

119894minus1

sum

119901=1

U119879119901119894Dminus1119901119901X119901 (14)


J119894= Dminus1119894119894

(X119894minus

119899

sum

119901=119894+1

U119894119901J119901) (15)












119898times1

119895= C


119906119898times1

119895V1times119899119895

(8) If 119906119898times1

119895V1times119899119895




A119903times119899119881

(16)


119894119895

U119894119895

= [U119880U119881]119894119895

= [Z119880Z119881]119894119895minus

119894minus1

sum

119901=1

[U119879119881U119879119880]119901119894Dminus1119901119901

[U119880U119881]119901119895

(17)



iterative steps

X119894= V119894minus

119894minus1

sum

119901=1

[U119879119881U119879119880]119901119894Dminus1119901119901

[X119880X119881]119901

(18)


J119894= Dminus1119894119894

(X119894minus

119899

sum

119901=119894+1

[U119880U119881]119894119901

[J119880J119881]119901) (19)

4 Numerical Results



Vtimes 100 (20)


∘

PEC

S

S+n

Sminusn

120591

V

Einc


0 30 60 90 120 150 180

0

5

10

15

20

25

RCS

(dBs

m)

minus5

minus10

minus15

FEKOTDS-LDLT-ACA

∘) (120601 = 0∘)120579 (







∘ to 120579 = 180∘



∘ (120593 = 0∘) to 120579 = 180

∘


119903= minus90 and




119903= 177 minus



x

y

z

Einc


0 30 60 90 120 150 180

0

10

20

30

RCS

(dBs

m)

minus10

minus20

minus30

minus40

minus10

minus20

minus30

minus40

LHM_FEKORHM_FEKO

LHM_TDSRHM_TDS

∘) (120601 = 0∘)120579 (


0 30 60 90 120 150 180

0

10

20

30

RCS

(dBs

m)

minus10

minus20

minus30

minus40

LHM_FEKORHM_FEKO

LHM_TDSRHM_TDS

∘) (120601 = 0∘)120579 (



x

y

z

Einc
















0 30 60 90 120 150 180

0

10

20

30

40

RCS

(dBs

m)

minus10

FEKOTDS-LDLT-ACA

∘) (120601 = 0∘)120579 (


120579 = 90∘ and 120593 = 270


∘lt




5 Conclusion



Competing Interests


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Z sdot J = V (9)






(

Z11

Z12


Z21

Z22


d

Z1198721

Z1198722


)(

J1

J2

J119872

) = (

V1

V2

V119872

) (10)



(

Z11

Z12


Z21

Z22


d

Z1198721

Z1198722


) = (


L21

1 sdot sdot sdot 0

d

L1198721

L1198722

sdot sdot sdot 1

)(

D11

0 sdot sdot sdot 0

0 D22

sdot sdot sdot 0

d


)(

(

1 L11987921


0 1 sdot sdot sdot L1198791198722

d


)

)

(11)


119894119894are the


U119894119895


(

Z11

Z12


Z21

Z22


d

Z1198721

Z1198722


)

= (


L21

1 sdot sdot sdot 0

d

L1198721

L1198722

sdot sdot sdot 1

)(

U11

U12


0 U22


d


)

(12)


U119894119895

= Z119894119895minus

119894minus1

sum

119901=1

U119879119901119894Dminus1119901119901U119901119895 (13)


X119894= V119894minus

119894minus1

sum

119901=1

U119879119901119894Dminus1119901119901X119901 (14)


J119894= Dminus1119894119894

(X119894minus

119899

sum

119901=119894+1

U119894119901J119901) (15)












119898times1

119895= C


119906119898times1

119895V1times119899119895

(8) If 119906119898times1

119895V1times119899119895




A119903times119899119881

(16)


119894119895

U119894119895

= [U119880U119881]119894119895

= [Z119880Z119881]119894119895minus

119894minus1

sum

119901=1

[U119879119881U119879119880]119901119894Dminus1119901119901

[U119880U119881]119901119895

(17)



iterative steps

X119894= V119894minus

119894minus1

sum

119901=1

[U119879119881U119879119880]119901119894Dminus1119901119901

[X119880X119881]119901

(18)


J119894= Dminus1119894119894

(X119894minus

119899

sum

119901=119894+1

[U119880U119881]119894119901

[J119880J119881]119901) (19)

4 Numerical Results



Vtimes 100 (20)


∘

PEC

S

S+n

Sminusn

120591

V

Einc


0 30 60 90 120 150 180

0

5

10

15

20

25

RCS

(dBs

m)

minus5

minus10

minus15

FEKOTDS-LDLT-ACA

∘) (120601 = 0∘)120579 (







∘ to 120579 = 180∘



∘ (120593 = 0∘) to 120579 = 180

∘


119903= minus90 and




119903= 177 minus



x

y

z

Einc


0 30 60 90 120 150 180

0

10

20

30

RCS

(dBs

m)

minus10

minus20

minus30

minus40

minus10

minus20

minus30

minus40

LHM_FEKORHM_FEKO

LHM_TDSRHM_TDS

∘) (120601 = 0∘)120579 (


0 30 60 90 120 150 180

0

10

20

30

RCS

(dBs

m)

minus10

minus20

minus30

minus40

LHM_FEKORHM_FEKO

LHM_TDSRHM_TDS

∘) (120601 = 0∘)120579 (



x

y

z

Einc
















0 30 60 90 120 150 180

0

10

20

30

40

RCS

(dBs

m)

minus10

FEKOTDS-LDLT-ACA

∘) (120601 = 0∘)120579 (


120579 = 90∘ and 120593 = 270


∘lt




5 Conclusion



Competing Interests


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















119898times1

119895= C


119906119898times1

119895V1times119899119895

(8) If 119906119898times1

119895V1times119899119895




A119903times119899119881

(16)


119894119895

U119894119895

= [U119880U119881]119894119895

= [Z119880Z119881]119894119895minus

119894minus1

sum

119901=1

[U119879119881U119879119880]119901119894Dminus1119901119901

[U119880U119881]119901119895

(17)



iterative steps

X119894= V119894minus

119894minus1

sum

119901=1

[U119879119881U119879119880]119901119894Dminus1119901119901

[X119880X119881]119901

(18)


J119894= Dminus1119894119894

(X119894minus

119899

sum

119901=119894+1

[U119880U119881]119894119901

[J119880J119881]119901) (19)

4 Numerical Results



Vtimes 100 (20)


∘

PEC

S

S+n

Sminusn

120591

V

Einc


0 30 60 90 120 150 180

0

5

10

15

20

25

RCS

(dBs

m)

minus5

minus10

minus15

FEKOTDS-LDLT-ACA

∘) (120601 = 0∘)120579 (







∘ to 120579 = 180∘



∘ (120593 = 0∘) to 120579 = 180

∘


119903= minus90 and




119903= 177 minus



x

y

z

Einc


0 30 60 90 120 150 180

0

10

20

30

RCS

(dBs

m)

minus10

minus20

minus30

minus40

minus10

minus20

minus30

minus40

LHM_FEKORHM_FEKO

LHM_TDSRHM_TDS

∘) (120601 = 0∘)120579 (


0 30 60 90 120 150 180

0

10

20

30

RCS

(dBs

m)

minus10

minus20

minus30

minus40

LHM_FEKORHM_FEKO

LHM_TDSRHM_TDS

∘) (120601 = 0∘)120579 (



x

y

z

Einc
















0 30 60 90 120 150 180

0

10

20

30

40

RCS

(dBs

m)

minus10

FEKOTDS-LDLT-ACA

∘) (120601 = 0∘)120579 (


120579 = 90∘ and 120593 = 270


∘lt




5 Conclusion



Competing Interests


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x

y

z

Einc


0 30 60 90 120 150 180

0

10

20

30

RCS

(dBs

m)

minus10

minus20

minus30

minus40

minus10

minus20

minus30

minus40

LHM_FEKORHM_FEKO

LHM_TDSRHM_TDS

∘) (120601 = 0∘)120579 (


0 30 60 90 120 150 180

0

10

20

30

RCS

(dBs

m)

minus10

minus20

minus30

minus40

LHM_FEKORHM_FEKO

LHM_TDSRHM_TDS

∘) (120601 = 0∘)120579 (



x

y

z

Einc
















0 30 60 90 120 150 180

0

10

20

30

40

RCS

(dBs

m)

minus10

FEKOTDS-LDLT-ACA

∘) (120601 = 0∘)120579 (


120579 = 90∘ and 120593 = 270


∘lt




5 Conclusion



Competing Interests


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















0 30 60 90 120 150 180

0

10

20

30

40

RCS

(dBs

m)

minus10

FEKOTDS-LDLT-ACA

∘) (120601 = 0∘)120579 (


120579 = 90∘ and 120593 = 270


∘lt




5 Conclusion



Competing Interests


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article fast direct solution of electromagnetic...

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