Research ArticleSimulation of Airborne Antenna Array Layout Problems UsingParallel Higher-Order MoM

Zhongchao Lin1 Yu Zhang1 Shugang Jiang1 Xunwang Zhao1 and Jingyan Mo2

1 School of Electronic Engineering Xidian University Xirsquoan Shaanxi 710071 China2 School of Communication and Information Engineering Shanghai University Shanghai 200021 China

Correspondence should be addressed to Yu Zhang yuseexidian163com

Received 16 January 2014 Revised 29 March 2014 Accepted 30 June 2014 Published 17 July 2014

Academic Editor Wei Hong

Copyright copy 2014 Zhongchao Lin et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The parallel higher-order Method of Moments based on message passing interface (MPI) has been successfully used to analyzethe changes in radiation patterns of a microstrip patch array antenna mounted on different positions of an airplane The block-partitioned scheme for the large dense MoM matrix and a block-cyclic matrix distribution scheme are designed to achieveexcellent load balance and high parallel efficiency Numerical results demonstrate that the rigorous parallel Method of Momentscan efficiently and accurately solve large complex electromagnetic problems with composite structures

1 Introduction

As the most traditional and widely adopted method themethod of moments (MoM) is a numerically accuratemethod for solving electromagnetic problems [1] Howeverits memory requirement and computational complexity growrapidly with 119874 (1198732) or even 119874 (1198733) where 119873 is thenumber of unknowns This makes the method very diffi-cult to deal with large complex objects such as airborneplatform To overcome this difficulty on one hand higher-order polynomials over wires and quadrilateral plates areused as basis functions over larger subdomain patches toreduce the number of unknowns [2] Polynomial expansionsfor the basis functions over larger subdomains lead to agood approximation of the current distributions over largesurfaces using approximately ten unknowns per wavelengthsquared which is much less than the use of piecewiseRao-Wilton-Glisson basis functions (RWGs) [2 3] On theother hand with the rapid development in computer hard-ware capabilities massively parallel computing on computerclusters and multicore processors has been the method ofchoice for solving modern engineering and science problemsarising from extremely complicated real-life applications Inaddition to improve the computation efficiency of the higher-order MoM the large dense MoM matrix is divided into a

number of smaller block matrices that are nearly equal insize and distributed among all participating processes Thestrategy of distributing the blocks is designed appropriatelyaccording to the parallel lowerupper (LU) decompositionsolver to minimize the communication between processes

The parallel efficiency and scalability of the parallelhigher-orderMoMwere analyzed in our previous studies [2]The accuracy of algorithmwas validated through comparisonwith measurement and MoM code with RWG basis functionwhich can be found in [2 4] In this paper the parallelhigher-orderMoM is used to analyze the changes in radiationpatterns of an antenna array installed on different positionsof a lager airplane which have been simulated with a MoMhybrid with high frequency asymptotic methods [5 6] ratherthan a rigorous MoM

The paper is organized as follows In Section 2 thebasics of integral equations higher-order basis functions andthe parallel strategies are listed respectively Section 3 listssome numerical results Section 31 validates the accuracy ofthis paperrsquos method through the comparison with the finiteelement method (FEM) [7] Section 32 describes the com-putation platforms Section 33 presents radiation patternsof a microstrip patch array antenna with 37 times 9 ElementsSection 34 analyzes the changes in radiation patterns of themicrostrip array installed on different positions of a lager

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 985367 11 pageshttpdxdoiorg1011552014985367

2 International Journal of Antennas and Propagation

airplane Section 35 analyzes another case of airborne arraythat a microstrip array sticks on a lager airplane Finallythe conclusions are presented in Section 4 followed by theacknowledgements

2 Parallel Higher-Order Method of Moments

21 Formulation of the Integral Equations The method isbased on the solution of surface integral equations (SIEs)[8ndash10] in the frequency domain for equivalent currentsover dielectric boundary surfaces and electric currents overperfect electric conductors (PECs) The set of integralequations obtained are solved by using MoM specificallyusing the Galerkin method The method is able to handleinhomogeneous dielectrics categorized by a combination ofvarious homogeneous dielectrics Therefore any compositemetallic and dielectric structure can be represented as anelectromagnetic system consisting of a finite number of finite-size linear homogeneous and isotropic regions situated in anunbounded linear homogeneous isotropic environment

Let us consider an arbitrary region 119894 with a nonzeroelectromagnetic field According to the surface equivalenttheorem the field outside region 119894 becomes zero Hence theregion outside region 119894 can be homogenized with respectthe region 119894 and then a multiple-region problem may bedecomposed into 119899 single-region problems The total electricfield in region 119894 may be expressed as a sum of scattered andincident fields as follows

997888119864

(119894)

=

119899

sum

119895=0119895 =119894

997888119864

(119894)

(997888119869119904119894119895997888119872119904119894119895) +

997888119864

(119894)

inc (1)

where 997888119864

(119894)

(997888119869119904119894119895997888119872119904119894119895) represents the scattered field inside

region 119894 which is calculated by the currents placed on

the boundary surface between regions 119894 and 119895 and 997888119864

(119894)

inc isthe corresponding incident field The electric and magneticscattered fields inside region 119894 are given by

997888119864

(119894)

(997888119869119904119894119895997888119872119904119894119895) = minus119871

(119894)(997888119869119904119894119895) + 119870

(119894)(997888119872119904119894119895) (2)

997888119867

(119894)

(997888119869119904119894119895997888119872119904119894119895) = minus119870

(119894)(997888119869119904119894119895) minus

120576(119894)

120583(119894)119871(119894)(997888119872119904119894119895) (3)

where 119871(119894) and119870(119894) are linear operators given by

119871(119894)(997888119883119904119894119895)

= 119895120596120583(119894)int119878119894119896

(997888119883119904119894119895(997888119903119894119895) 119892(119894)(997888119903

9978881199031015840

) +1

1205962120576(119894)120583(119894)nabla119904119894119895

sdot997888119883119904119894119895(997888119903119894119895) nabla119892(119894)(997888119903

9978881199031015840

))119889119878119894119895

119870(119894)(997888119883119904119894119895) = int119878119894119895

997888119883119904119894119895(997888119903119894119895) times nabla119892

(119894)(997888119903

9978881199031015840

) 119889119878119894119895

(4)

where 997888119883119904119894119895

denotes the electric or magnetic current 9978881199031015840

isthe vector position of the source point 997888119903 is the vectorposition of the field point and 119892

119894(997888119903

9978881199031015840

) is Greenrsquos functionfor the homogeneous medium 119894 Making use of the boundarycondition equations given by

119899119894119895times (

997888119864

(119894)

minus997888119864

(119895)

) = 0 (5)

119899119894119895times (

997888119867

(119894)

minus997888119867

(119895)

) = 0 (6)

and replacing (1) (2) and (3) into (4) and (6) the integralequations for regions 119894 and 119896 are obtained in the form

n119894119896times

119899

sum

119895=0119895 =119894

[119871(119894)(997888119869119904119894119895) minus 119870

(119894)(997888119872119904119894119895)]

minus

119899

sum

119895=0119895 =119896

[119871(119895)

(997888119869119904119896119895) minus 119870

(119895)(997888119872119904119896119895)]

= n119894119896times (

997888119864

(119894)

inc minus997888119864

(119896)

inc)

n119894119896times

119899

sum

119895=0119895 =119894

(120576(119894)

120583(119894)119871(119894)(997888119872119904119894119895) + 119870

(119894)(997888119869119904119894119895))

minus

119899

sum

119895=0119895 =119896

(120576(119895)

120583(119895)119871(119895)

(997888119872119904119896119895) + 119870

(119895)(997888119869119904119896119895))

= n119894119895times (

997888119867

(119894)

inc minus997888119867

(119896)

inc)

(7)

These two sets of equations represent a general form ofthe Poggio-Miller-Chang-Harrington-Wu (PMCHW) [2 11]formulation When the boundary surface of two differentregions is PEC the magnetic currents are equal to zero atthe boundary surface and the equations degenerates into theelectric field integral equation (EFIE) [12 13]

22 Higher-Order Basis Functions The electric and magneticcurrents are approximated by higher-order polynomialswhich reduce the number of unknowns compared with therational piecewise basis functions The code makes use oftruncated cones for wires and bilinear patches to characterizeother surfaces

For bilinear surfaces the surface current is decomposedinto its 119901- and 119904-components 119901 and 119904 being the two para-metric coordinates of the unit quadrangle 119901 119904 isin [minus1 1] Theapproximation for the 119904-component of the electric current is

International Journal of Antennas and Propagation 3

0

2

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1

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1

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11

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91

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48

58

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98

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49

59

69

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89

99

(a)

0

(00)1

(10)2

(20)0

1

2

0

1

2

3

(01)4

(11)5

(21)3

4

5

3

4

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6

(02)7

(12)8

(22)6

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8

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7

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(00)1

(10)2

(20)0

1

2

0

1

2

3

(01)4

(11)5

(21)3

4

5

3

4

5

6

(02)7

(12)8

(22)6

7

8

6

7

8

0

(00)1

(10)2

(20)0

1

2

0

1

2

3

(01)4

(11)5

(21)3

4

5

3

4

5

6

(02)7

(12)8

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

6

7

8

6

7

8

(b)

Figure 1 Block-cyclic distribution of a matrix (a) a matrix consisting of 9 times 9 blocks (b) rank and coordinates of each process owning thecorresponding blocks in (a)

(analogous expressions stand for the 119901-component and forthe magnetic current)

997888119869119904(119901 119904)

=

119873119901

sum

119894=0

[

[

1198881198941

997888119864119894(119901 119904) + 119888

1198942

997888119864119894(119901 minus119904) +

119873119904

sum

119895=2

119886119894119895

997888119875119894119895(119901 119904)]

]

(8)

where 119873119901and 119873

119904are the degrees of the approximations

along the coordinates and 119886119894119895 1198881198941 and 119888

1198942are the unknown

coefficientsThus expression (8) stands for the representation of the

current in terms of edge basis functions 997888119864119894(119901 119904) and interior

or patch basis functions 997888119875119894119895(119901 119904) which can be compactly

expressed as

997888119864119894119896(119901 119904) =

120572119904

10038161003816100381610038161003816120572119901times 120572119904

10038161003816100381610038161003816

119901119894119873(119904) 119896 = 1

119901119894119873(minus119904) 119896 = 2

997888119875119894119895(119901 119904) =

120572119904

10038161003816100381610038161003816120572119901times 120572119904

10038161003816100381610038161003816

119901119894119878119895(119904)

119873 (119904) =1 minus 119904

2 119878119894(119904) =

119904119894minus 1 119894 is even

119904119894minus 119904 119894 is odd

(9)

where symbols 119886119904and 119886119901denote the unitary vectors along the

transformed 119904 and 119901 coordinatesEdge basis functions 997888

1198641198941

and patch basis functions997888119875119894119895are zero along the first edge (119904 = minus1) 997888119864

1198942and 997888

119875119894119895being

Feed

Substrate

Patchz

x

y

o

Figure 2 Model of the microstrip patch antenna element

zero along the second edge (119904 = 1) Thus the continuityequation can easily be imposed on a given mesh made ofpatches

23 Parallel Schemes of Higher-Order MoM The paralleliza-tion of the higher-order MoM solution involves parallelmatrix filling followed by a parallel solution of the densematrix equation To ensure the load-balancing in terms ofdata and CPU operations it is necessary to divide the matrixinto blocks and distribute those blocks to all processorsthrough a block-cyclic distribution procedure The modes ofdistribution of blocks are related to matrix equation solutionmethods and we select the LU factorization as the solutionmethod

4 International Journal of Antennas and Propagation

Parallel higher-order MoMFinite element method

0

4

80

30

60

90

120

150180

210

240

270

300

330

0

4

8

Gai

n (d

B)

minus4

minus8

minus4

(a)

Parallel higher-order MoMFinite element method

02468

0

30

60

90

120

150180

210

240

270

300

330

02468

Gai

n (d

B)

minus2

minus4

minus6

minus4

minus2

(b)

Figure 3 2D gain of microstrip patch antenna (a) yoz plane and (b) xoy plane

As an example consider amatrix [119860] of Figure 1(a) whichis distributed to different processes in the 3 times 3 process gridFigure 1(b) shows to which process the blocks of [119860] aredistributed using the block-cyclic distribution methodology[2 14] In Figure 1(a) the outermost numbers denote therow and column indices of the process coordinates The topand bottom number in any block of Figure 1(b) denotes theprocess ID (rank) and the process coordinates of a certainprocess respectively corresponding to the block of thematrixshown in Figure 1(a) By varying the dimensions of the blocksof [119860] and those of the process grid different mappings canbe obtained

During the procedure of finding the solution to thematrixequation balancing the computational load between all theprocesses is also important However it is less easy to trackcompared with the parallel matrix filling procedure sincean increase in the number of unknowns or the number ofprocesses executing the solution can increase the amountof communication required between the processes Thisincrease in communication will decrease the gains of parallelspeedup However as a rule of thumb more processes typi-cally means less wall clock time for solving large problemseven though the overall parallel efficiency may sometimes beincreased by using fewer processes to solve the same problem

The solution of the matrix equation is essentially thesame regardless of the type of basis functions used for theMoM The parallel LU decomposition solver based on theScaLAPACK library package is described in Chapter 2 in[2] Also one can refer to Chapter 6 in [2] for a detaileddiscussion about the iterative solvers based on the conjugategradient (CG) method The computational complexity of LUdecomposition scales with 119874 (1198733) is much higher than

zx

yo

9 patchelements

37 patch elements (10m)

(25m)

Figure 4 Model of the microstrip patch array antenna with 37 times 9elements

25658

20102

14547

89913

34357

minus21198

minus76754

minus13231

minus18786

minus24342

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

Figure 5 3D gain patterns of the microstrip patch array antenna

that of CG type methods with 119874 (1198732) where 119873 is thenumber of unknowns However in this paper the parallelLU decomposition is utilized as the parallel equation solverrather than the parallel iterative CG method due to the factthat the iterative CG method may encounter a divergence

International Journal of Antennas and Propagation 5

0 60 120 180

0

10

20

30

40G

ain

(dB)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(a)

Gai

n (d

B)

0 60 120 180 240 300 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Figure 6 2D gain patterns of the microstrip patch array antenna (a) yoz plane and (b) xoy plane

problem when dealing with complex antennas composed ofthin structures and various materials

3 Numerical Results

31 Comparison with the Finite Element Method To validatethe accuracy and efficiency of the proposed parallel higher-order MoM a rectangular microstrip patch antenna elementis simulated which is shown in Figure 2 The dimensionsof the patch element are 2056mm times 1548mm and thematerial parameter of the substrate is 120576

119903= 42 The operation

frequency is 440MHz and the simulation is performed ona personal computer with a Quad-core Intel I5 processor(300GHz) and 8GB of RAMThe radiation pattern obtainedusing the parallel higher-order MoM and the finite elementmethod are plotted together in Figure 3 The results agreewith each other very well

32 Description of the Parallel Computational Platform Thecomputing cluster used in this section is the Magic Cubewith 64 nodes at Shanghai Supercomputer Center (SSC) andeach node has four quad-core 19 GHz processors and 64GBmemory The nodes are connected with Infiniband switches

33 A Microstrip Patch Array Antenna with 37 times 9 ElementsIn this section we present the radiation patterns of thealgorithm for a microstrip array Consider a rectangularmicrostrip patch array antenna with 37 times 9 elements Themicrostrip element parameters are the same as the elementanalyzed in Section 31 and the array model is shown inFigure 4 The dimensions of the array are 10m times 25m times

0018m and the distances between any two neighboring ele-ments are 579mm and 833mm along the length and widthdirections The amplitude at the feed of the array is designedby a minus35 dB Taylor distribution and the operation frequencyof the array is 440MHz The three-dimensional (3D) gainpattern is given in Figure 5 and the two-dimensional (2D)

z

xy

o

Microstrip array

Figure 7 Microstrip patch array over an airplane

bull Array center

7 48 5 29 6 3

1

zx o

2m

Figure 8 Nine positions of the array in the xoz plane

gain patterns are given in Figure 6 The side-lobe levels inboth the yoz and xoy planes are below minus35 dB which meetthe design target Note that 120579 coordinate is measured fromxoy plane to 119911 axis and 120593 coordinate is measured from +x axisto 119910 axis in this paper

34 A Microstrip Antenna Array with 37 times 9 Elementsover a Large Airplane Install the 37 times 9 microstrip patcharray that was analyzed in Section 33 over a large airplane(httpwwwscalecraftcom737aewandcaspx) as shown inFigure 7 The dimensions of the airplane are 55m times 476m times

158m The array elements are very thin compared with the

6 International Journal of Antennas and Propagation

26648

21092

15537

99811

44256

minus113

minus66855

minus12241

minus17797

minus23352

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

26401

20846

1529

97347

41792

minus13764

minus69319

minus12487

minus18043

minus23599

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

25902

20346

14791

92352

36796

minus1876

minus74315

minus12987

minus18543

minus24098

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(c)

26452

20897

15341

97854

42299

minus13257

minus68812

minus12437

minus17992

minus23548

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(d)

26458

20902

15347

97913

42357

minus13198

minus68754

minus12431

minus17986

minus23542

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(e)

25794

20239

14683

91277

35722

minus19834

minus7539

minus13095

minus1865

minus24206

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(f)

26527

20972

15416

98608

43053

minus12503

minus68059

minus12361

minus17917

minus23473

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(g)

26307

20752

15196

96406

4085

minus14705

minus70261

minus12582

minus18137

minus23693

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(h)

Figure 9 Continued

International Journal of Antennas and Propagation 7

2576220207146519095635401minus20155minus7571minus13127minus18682minus24238

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(i)

Figure 9 3D gain patterns of the nine installed positions of the array (a) Array at position 1 (b) Array at position 2 (c) Array at position 3(d) Array at position 4 (e) Array at position 5 (f) Array at position 6 (g) Array at position 7 (h) Array at position 8 (i) Array at position 9

47

8 5 2

9 6 3

1

z

x o

(a)

z

x o7 4

2

1

8 5

9 6 3

(b)

Figure 10 Two comparison ways for the 2D gain pattern

airplane platform and the airborne antenna array is a com-posite and multiscale problem Consider a realistic problemthat the microstrip antenna array installed on nine positionsof the airplane with different 119909 coordinates or different 119911coordinates as shown in Figure 8 the mainlobe direction isdirected towards the wing The operation frequency of theairborne array is 440MHz and the number of unknowns is308371

In this Section 1024 CPU cores are used and the directsolver is observed in solving this kind of problems to avoid theslow convergence of iterative CG type methods The matrixfilling time is about 4700 seconds and the matrix equationsolving time is about 18500 seconds The computed three-dimensional (3D) gain patterns of the antenna array installedon different positions of the airplane are shown in Figure 9To make the comparison more clearly the computed two-dimensional (2D) gain patterns of the antenna array installedon different positions of the airplane are compared in twoways as shown in Figure 10 One is the same 119909 coordinatesand different 119911 coordinates as shown in Figure 11 One is thedifferent 119909 coordinates and same 119911 coordinates as shown inFigure 12 Furthermore the patterns of the array alone arealso given for comparison

Form the comparison it is clearly seen that the gainpatterns of the airborne antenna array in the yoz planes(elevation planes) seriously deteriorate compared with theantenna array alone especially in the mainlobe regionAnd in contrast the patterns in the xoy plane (azimuth

plane) show slight changes From the comparisons resultsof the array installed on same 119909 coordinates and different 119911coordinates shown in Figure 11 one can see that with the 119911coordinates of the installation positions increasing (ie formposition 1 to 2 and 3 form position 4 to 5 and 6 and formposition 7 to 8 and 9) the mainlobe region of gain patterns(ie minus30∘ sim30∘ in the yoz plane) gets better mainly due to thereflection and diffraction effects of airplane wing becomingweak Furthermore the sidelobe region of gain patterns (ieminus150∘ sim minus30∘ in the yoz plane 115∘ sim120∘ in the xoy plane)becomes low with the installation height increasing Formthe comparisons results of the array installed on different 119909coordinates and same 119911 coordinates shown in Figure 12 wecan see that gain patterns (ie minus150∘ sim30∘ in the yoz plane)get better with the 119909 coordinates of the installation positionsincreasing (ie form position 1 to 4 and 7 form position 2 to5 and 8 and form position 3 to 6 and 9) However the overalldesign of airplane is a composite and multiscale problem itis necessary to take many factors into account such as theaerodynamics Therefore the array is installed not too highand not close to the head of airplane taking these factors intoconsideration the better installation position is position 5 inthis paper

Moreover the model of airborne array is a whole inpractical engineeringThe array is connected to airplane witha bracket which is installed below the array When the mainbeam of the array is direct to the wings of the airplane theeffect of the bracket can be neglected So in the simulations

8 International Journal of Antennas and Propagation

Array aloneArray at position 1

Array at position 2Array at position 3

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(a)

Array aloneArray at position 1

Array at position 2Array at position 3

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 4

Array at position 5Array at position 6

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(c)

Array aloneArray at position 4

Array at position 5Array at position 6

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 7

Array at position 8Array at position 9

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(e)

Array aloneArray at position 7

Array at position 8Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 11 2D gain patterns of the array installed on same 119909 coordinates and different 119911 coordinates (a) yoz plane of positions 1 2 and 3 (b)xoy plane of positions 1 2 and 3 (c) yoz plane of positions 4 5 and 6 (d) xoy plane of positions 4 5 and 6 (e) yoz plane of positions 7 8and 9 (f) xoy plane of positions 7 8 and 9

International Journal of Antennas and Propagation 9

Array aloneArray at position 1

Array at position 4Array at position 7

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(a)

Array aloneArray at position 1

Array at position 4Array at position 7

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 2

Array at position 5Array at position 8

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(c)

Array aloneArray at position 2

Array at position 5Array at position 8

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 3

Array at position 6Array at position 9

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(e)

Array aloneArray at position 3

Array at position 6Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 12 2D gain patterns of the array installed on different 119909 coordinates and same 119911 coordinates (a) yoz plane of positions 1 4 and 7 (b)xoy plane of positions 1 4 and 7 (c) yoz plane of positions 2 5 and 8 (d) xoy plane of positions 2 5 and 8 (e) yoz plane of positions 3 6 and9 (f) xoy plane of position 3 6 and 9

10 International Journal of Antennas and Propagation

zx

yo

(a)

Microstrip array

x

y

o

(b)

x

y

o

(c)

Figure 13 Simulation models (a) microstrip array (b) airborne array (c) the position of the array

z

x yo

1039

48344

minus62767

minus072116

minus17388

minus11832

minus22943

minus28499

minus34054

minus3961

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

z

x yo

93514

37959

minus17597

minus73152

minus12871

minus18426

minus23982

minus29537

minus35093

minus40649

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

Figure 14 3D gain patterns of the array alone and airborne array (a) array alone (b) airborne array

of the airborne array in this section the bracket is removedfrom the simulation model

35 A Microstrip Antenna Array with 9 times 1 Elements stick onanAirplane To demonstrate the ability of themethod to dealwith the problem of an array connected to the airplane amicrostrip antenna array with 9 times 1 elements sticks on thesurface of a large airplane is considered in this section [15]The model of the array and the airborne array are shown inFigures 13(a) and 13(b) and the position of the array is given

in Figure 13(c) The parameters of the array and the airplaneare same to the previous sections The computed 3D gainpatterns of the array alone and airborne array are shown inFigure 14 The computed 2D gain patterns of the array aloneand airborne array are compared in Figure 15

Form the comparisons we can see that when the arraysticks on the airplane the back gain of the array is signifi-cantly reduced while the front gain shows slight changes Itis mainly due to that when the array is stuck on the airplaneit can be considered as enlarging the ground of the array sothe result in Figure 15 is reasonable

International Journal of Antennas and Propagation 11

0

10

0

30

60

90

120

150180

210

240

270

300

330

0

10

Array aloneAirborne array

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus40

minus30

minus20

minus10

Figure 15 Comparison result of 2D gain patterns in xoz plane

4 Conclusion

An efficient parallel higher-order MoM is developed to solveextremely large and complex electromagnetics problemsTheparallel strategies based on block matrices and the higher-order basis functions make it possible to solve airborneantenna array problems using rigorousMoM and the simula-tion time of one installed position is about seven hours using1024 CPU cores Numerical results confirm the accuracy andefficiency of the method and its applicability to the analysisof radiation from airborne platform systems with complexantennas

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)(2012AA01A308) the NSFC (61301069 and 61072019) andthe Program for New Century Excellent Talents in Universityof China (NCET-13-0949) the Project with contract no2013KJXX-67 The computational resources utilized in thisresearch are provided by Shanghai Supercomputer Center

References

[1] R F Harrington ldquoField computation by moment methodsrdquo inIEEE Series on Electromagnetic Waves IEEE New York NYUSA 1993

[2] Y Zhang and T K Sarkar Parallel Solution of Integral EquationBased EM Problems in the Frequency Domain John Wiley ampSons Hoboken NJ USA 2009

[3] 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

[4] Y Zhang T K Sarkar X Zhao et al Higher Order Basis BasedIntegral Equation Solver (HOBBIES) John Wiley Hoboken NJUSA 2012

[5] Y Zhang X Zhao D G Donoro and T K Sarkar ldquoParallelizedhybrid method with higher-order MoM And PO for analysisof phased array antennas on electrically large platformsrdquo IEEETransactions on Antennas and Propagation vol 58 no 12 pp4110ndash4115 2010

[6] X-W Zhao Y Zhang H-W Zhang et al ldquoParallel MoM-POmethod with out-of-core technique for analysis of complexarrays on electrically large platformsrdquo Progress in Electromag-netics Research vol 108 pp 1ndash21 2010

[7] G Hennigan and S Castillo ldquoOpen region electromagneticfinite-element scattering calculations in anisotropic media onparallel computersrdquo in IEEE Antennas and Propagation SocietyInternational Symposium vol 1 pp 484ndash487 July 1999

[8] P Yla-OijalaM Taskinen and S Jarvenpaa ldquoAnalysis of surfaceintegral equations in electromagnetic scattering and radiationproblemsrdquo Engineering Analysis with Boundary Elements vol32 no 3 pp 196ndash209 2008

[9] P Yla-Oijala M Taskinen and J Sarvas ldquoSurface integralequation method for general composite metallic and dielec-tric structures with junctionsrdquo in Progress in ElectromagneticsResearch vol 52 pp 81ndash108 2005

[10] K Yang W T Sheng Z Y Zhu and M S Tong ldquoElectromag-netic analysis for inhomogeneous interconnect and packagingstructures based on volume-surface integral equationsrdquo IEEETransactions on Components Packaging and ManufacturingTechnology vol 3 no 8 pp 1364ndash1371 2013

[11] R F Harrington ldquoBoundary integral formulations for homoge-nous material bodiesrdquo Journal of Electromagnetic Waves andApplications vol 3 no 1 pp 1ndash15 1989

[12] D Z Ding Y Shi Z N Jiang and R S Chen ldquoAugmentedEFIE with adaptive cross approximation algorithm for analysisof electromagnetic problemsrdquo International Journal of Antennasand Propagation vol 2013 Article ID 487276 9 pages 2013

[13] J L Volakis and K Sertel Integral Equation Methods forElectromagnetics SciTech Raleigh NC USA 2012

[14] Y Yan Y Zhang C-H Liang H Zhao and D Garcıa-DonoroldquoRCS computation by parallel MoM using higher-order basisfunctionsrdquo International Journal of Antennas and Propagationvol 2012 Article ID 745893 9 pages 2012

[15] Z Yang S Jiang X Zhao Y Zhang and H Qu ldquoRadiationof an airborne patch array antennardquo in Proceedings of the IETInternational Radar Conference Institution of Engineering andTechnology Xirsquoan China April 2013

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VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

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DistributedSensor Networks

International Journal of

2 International Journal of Antennas and Propagation

airplane Section 35 analyzes another case of airborne arraythat a microstrip array sticks on a lager airplane Finallythe conclusions are presented in Section 4 followed by theacknowledgements

2 Parallel Higher-Order Method of Moments

21 Formulation of the Integral Equations The method isbased on the solution of surface integral equations (SIEs)[8ndash10] in the frequency domain for equivalent currentsover dielectric boundary surfaces and electric currents overperfect electric conductors (PECs) The set of integralequations obtained are solved by using MoM specificallyusing the Galerkin method The method is able to handleinhomogeneous dielectrics categorized by a combination ofvarious homogeneous dielectrics Therefore any compositemetallic and dielectric structure can be represented as anelectromagnetic system consisting of a finite number of finite-size linear homogeneous and isotropic regions situated in anunbounded linear homogeneous isotropic environment

Let us consider an arbitrary region 119894 with a nonzeroelectromagnetic field According to the surface equivalenttheorem the field outside region 119894 becomes zero Hence theregion outside region 119894 can be homogenized with respectthe region 119894 and then a multiple-region problem may bedecomposed into 119899 single-region problems The total electricfield in region 119894 may be expressed as a sum of scattered andincident fields as follows

997888119864

(119894)

=

119899

sum

119895=0119895 =119894

997888119864

(119894)

(997888119869119904119894119895997888119872119904119894119895) +

997888119864

(119894)

inc (1)

where 997888119864

(119894)

(997888119869119904119894119895997888119872119904119894119895) represents the scattered field inside

region 119894 which is calculated by the currents placed on

the boundary surface between regions 119894 and 119895 and 997888119864

(119894)

inc isthe corresponding incident field The electric and magneticscattered fields inside region 119894 are given by

997888119864

(119894)

(997888119869119904119894119895997888119872119904119894119895) = minus119871

(119894)(997888119869119904119894119895) + 119870

(119894)(997888119872119904119894119895) (2)

997888119867

(119894)

(997888119869119904119894119895997888119872119904119894119895) = minus119870

(119894)(997888119869119904119894119895) minus

120576(119894)

120583(119894)119871(119894)(997888119872119904119894119895) (3)

where 119871(119894) and119870(119894) are linear operators given by

119871(119894)(997888119883119904119894119895)

= 119895120596120583(119894)int119878119894119896

(997888119883119904119894119895(997888119903119894119895) 119892(119894)(997888119903

9978881199031015840

) +1

1205962120576(119894)120583(119894)nabla119904119894119895

sdot997888119883119904119894119895(997888119903119894119895) nabla119892(119894)(997888119903

9978881199031015840

))119889119878119894119895

119870(119894)(997888119883119904119894119895) = int119878119894119895

997888119883119904119894119895(997888119903119894119895) times nabla119892

(119894)(997888119903

9978881199031015840

) 119889119878119894119895

(4)

where 997888119883119904119894119895

denotes the electric or magnetic current 9978881199031015840

isthe vector position of the source point 997888119903 is the vectorposition of the field point and 119892

119894(997888119903

9978881199031015840

) is Greenrsquos functionfor the homogeneous medium 119894 Making use of the boundarycondition equations given by

119899119894119895times (

997888119864

(119894)

minus997888119864

(119895)

) = 0 (5)

119899119894119895times (

997888119867

(119894)

minus997888119867

(119895)

) = 0 (6)

and replacing (1) (2) and (3) into (4) and (6) the integralequations for regions 119894 and 119896 are obtained in the form

n119894119896times

119899

sum

119895=0119895 =119894

[119871(119894)(997888119869119904119894119895) minus 119870

(119894)(997888119872119904119894119895)]

minus

119899

sum

119895=0119895 =119896

[119871(119895)

(997888119869119904119896119895) minus 119870

(119895)(997888119872119904119896119895)]

= n119894119896times (

997888119864

(119894)

inc minus997888119864

(119896)

inc)

n119894119896times

119899

sum

119895=0119895 =119894

(120576(119894)

120583(119894)119871(119894)(997888119872119904119894119895) + 119870

(119894)(997888119869119904119894119895))

minus

119899

sum

119895=0119895 =119896

(120576(119895)

120583(119895)119871(119895)

(997888119872119904119896119895) + 119870

(119895)(997888119869119904119896119895))

= n119894119895times (

997888119867

(119894)

inc minus997888119867

(119896)

inc)

(7)

These two sets of equations represent a general form ofthe Poggio-Miller-Chang-Harrington-Wu (PMCHW) [2 11]formulation When the boundary surface of two differentregions is PEC the magnetic currents are equal to zero atthe boundary surface and the equations degenerates into theelectric field integral equation (EFIE) [12 13]

22 Higher-Order Basis Functions The electric and magneticcurrents are approximated by higher-order polynomialswhich reduce the number of unknowns compared with therational piecewise basis functions The code makes use oftruncated cones for wires and bilinear patches to characterizeother surfaces

For bilinear surfaces the surface current is decomposedinto its 119901- and 119904-components 119901 and 119904 being the two para-metric coordinates of the unit quadrangle 119901 119904 isin [minus1 1] Theapproximation for the 119904-component of the electric current is

International Journal of Antennas and Propagation 3

0

2

1

1

0

2

1

0

2

0 1 2 0 1 2 0 1 2

11

21

31

41

51

61

71

81

91

12

22

32

42

52

62

72

82

92

13

23

33

43

53

63

73

83

93

14

24

34

44

54

64

74

84

94

15

25

35

45

55

65

75

85

95

16

26

36

46

56

66

76

86

96

17

27

37

47

57

67

77

87

97

18

28

38

48

58

68

78

88

98

19

29

39

49

59

69

79

89

99

(a)

0

(00)1

(10)2

(20)0

1

2

0

1

2

3

(01)4

(11)5

(21)3

4

5

3

4

5

6

(02)7

(12)8

(22)6

7

8

6

7

8

0

(00)1

(10)2

(20)0

1

2

0

1

2

3

(01)4

(11)5

(21)3

4

5

3

4

5

6

(02)7

(12)8

(22)6

7

8

6

7

8

0

(00)1

(10)2

(20)0

1

2

0

1

2

3

(01)4

(11)5

(21)3

4

5

3

4

5

6

(02)7

(12)8

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

6

7

8

6

7

8

(b)

Figure 1 Block-cyclic distribution of a matrix (a) a matrix consisting of 9 times 9 blocks (b) rank and coordinates of each process owning thecorresponding blocks in (a)

(analogous expressions stand for the 119901-component and forthe magnetic current)

997888119869119904(119901 119904)

=

119873119901

sum

119894=0

[

[

1198881198941

997888119864119894(119901 119904) + 119888

1198942

997888119864119894(119901 minus119904) +

119873119904

sum

119895=2

119886119894119895

997888119875119894119895(119901 119904)]

]

(8)

where 119873119901and 119873

119904are the degrees of the approximations

along the coordinates and 119886119894119895 1198881198941 and 119888

1198942are the unknown

coefficientsThus expression (8) stands for the representation of the

current in terms of edge basis functions 997888119864119894(119901 119904) and interior

or patch basis functions 997888119875119894119895(119901 119904) which can be compactly

expressed as

997888119864119894119896(119901 119904) =

120572119904

10038161003816100381610038161003816120572119901times 120572119904

10038161003816100381610038161003816

119901119894119873(119904) 119896 = 1

119901119894119873(minus119904) 119896 = 2

997888119875119894119895(119901 119904) =

120572119904

10038161003816100381610038161003816120572119901times 120572119904

10038161003816100381610038161003816

119901119894119878119895(119904)

119873 (119904) =1 minus 119904

2 119878119894(119904) =

119904119894minus 1 119894 is even

119904119894minus 119904 119894 is odd

(9)

where symbols 119886119904and 119886119901denote the unitary vectors along the

transformed 119904 and 119901 coordinatesEdge basis functions 997888

1198641198941

and patch basis functions997888119875119894119895are zero along the first edge (119904 = minus1) 997888119864

1198942and 997888

119875119894119895being

Feed

Substrate

Patchz

x

y

o

Figure 2 Model of the microstrip patch antenna element

zero along the second edge (119904 = 1) Thus the continuityequation can easily be imposed on a given mesh made ofpatches

23 Parallel Schemes of Higher-Order MoM The paralleliza-tion of the higher-order MoM solution involves parallelmatrix filling followed by a parallel solution of the densematrix equation To ensure the load-balancing in terms ofdata and CPU operations it is necessary to divide the matrixinto blocks and distribute those blocks to all processorsthrough a block-cyclic distribution procedure The modes ofdistribution of blocks are related to matrix equation solutionmethods and we select the LU factorization as the solutionmethod

4 International Journal of Antennas and Propagation

Parallel higher-order MoMFinite element method

0

4

80

30

60

90

120

150180

210

240

270

300

330

0

4

8

Gai

n (d

B)

minus4

minus8

minus4

(a)

Parallel higher-order MoMFinite element method

02468

0

30

60

90

120

150180

210

240

270

300

330

02468

Gai

n (d

B)

minus2

minus4

minus6

minus4

minus2

(b)

Figure 3 2D gain of microstrip patch antenna (a) yoz plane and (b) xoy plane

As an example consider amatrix [119860] of Figure 1(a) whichis distributed to different processes in the 3 times 3 process gridFigure 1(b) shows to which process the blocks of [119860] aredistributed using the block-cyclic distribution methodology[2 14] In Figure 1(a) the outermost numbers denote therow and column indices of the process coordinates The topand bottom number in any block of Figure 1(b) denotes theprocess ID (rank) and the process coordinates of a certainprocess respectively corresponding to the block of thematrixshown in Figure 1(a) By varying the dimensions of the blocksof [119860] and those of the process grid different mappings canbe obtained

During the procedure of finding the solution to thematrixequation balancing the computational load between all theprocesses is also important However it is less easy to trackcompared with the parallel matrix filling procedure sincean increase in the number of unknowns or the number ofprocesses executing the solution can increase the amountof communication required between the processes Thisincrease in communication will decrease the gains of parallelspeedup However as a rule of thumb more processes typi-cally means less wall clock time for solving large problemseven though the overall parallel efficiency may sometimes beincreased by using fewer processes to solve the same problem

The solution of the matrix equation is essentially thesame regardless of the type of basis functions used for theMoM The parallel LU decomposition solver based on theScaLAPACK library package is described in Chapter 2 in[2] Also one can refer to Chapter 6 in [2] for a detaileddiscussion about the iterative solvers based on the conjugategradient (CG) method The computational complexity of LUdecomposition scales with 119874 (1198733) is much higher than

zx

yo

9 patchelements

37 patch elements (10m)

(25m)

Figure 4 Model of the microstrip patch array antenna with 37 times 9elements

25658

20102

14547

89913

34357

minus21198

minus76754

minus13231

minus18786

minus24342

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

Figure 5 3D gain patterns of the microstrip patch array antenna

that of CG type methods with 119874 (1198732) where 119873 is thenumber of unknowns However in this paper the parallelLU decomposition is utilized as the parallel equation solverrather than the parallel iterative CG method due to the factthat the iterative CG method may encounter a divergence

International Journal of Antennas and Propagation 5

0 60 120 180

0

10

20

30

40G

ain

(dB)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(a)

Gai

n (d

B)

0 60 120 180 240 300 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Figure 6 2D gain patterns of the microstrip patch array antenna (a) yoz plane and (b) xoy plane

problem when dealing with complex antennas composed ofthin structures and various materials

3 Numerical Results

31 Comparison with the Finite Element Method To validatethe accuracy and efficiency of the proposed parallel higher-order MoM a rectangular microstrip patch antenna elementis simulated which is shown in Figure 2 The dimensionsof the patch element are 2056mm times 1548mm and thematerial parameter of the substrate is 120576

119903= 42 The operation

frequency is 440MHz and the simulation is performed ona personal computer with a Quad-core Intel I5 processor(300GHz) and 8GB of RAMThe radiation pattern obtainedusing the parallel higher-order MoM and the finite elementmethod are plotted together in Figure 3 The results agreewith each other very well

32 Description of the Parallel Computational Platform Thecomputing cluster used in this section is the Magic Cubewith 64 nodes at Shanghai Supercomputer Center (SSC) andeach node has four quad-core 19 GHz processors and 64GBmemory The nodes are connected with Infiniband switches

33 A Microstrip Patch Array Antenna with 37 times 9 ElementsIn this section we present the radiation patterns of thealgorithm for a microstrip array Consider a rectangularmicrostrip patch array antenna with 37 times 9 elements Themicrostrip element parameters are the same as the elementanalyzed in Section 31 and the array model is shown inFigure 4 The dimensions of the array are 10m times 25m times

0018m and the distances between any two neighboring ele-ments are 579mm and 833mm along the length and widthdirections The amplitude at the feed of the array is designedby a minus35 dB Taylor distribution and the operation frequencyof the array is 440MHz The three-dimensional (3D) gainpattern is given in Figure 5 and the two-dimensional (2D)

z

xy

o

Microstrip array

Figure 7 Microstrip patch array over an airplane

bull Array center

7 48 5 29 6 3

1

zx o

2m

Figure 8 Nine positions of the array in the xoz plane

gain patterns are given in Figure 6 The side-lobe levels inboth the yoz and xoy planes are below minus35 dB which meetthe design target Note that 120579 coordinate is measured fromxoy plane to 119911 axis and 120593 coordinate is measured from +x axisto 119910 axis in this paper

34 A Microstrip Antenna Array with 37 times 9 Elementsover a Large Airplane Install the 37 times 9 microstrip patcharray that was analyzed in Section 33 over a large airplane(httpwwwscalecraftcom737aewandcaspx) as shown inFigure 7 The dimensions of the airplane are 55m times 476m times

158m The array elements are very thin compared with the

6 International Journal of Antennas and Propagation

26648

21092

15537

99811

44256

minus113

minus66855

minus12241

minus17797

minus23352

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

26401

20846

1529

97347

41792

minus13764

minus69319

minus12487

minus18043

minus23599

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

25902

20346

14791

92352

36796

minus1876

minus74315

minus12987

minus18543

minus24098

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(c)

26452

20897

15341

97854

42299

minus13257

minus68812

minus12437

minus17992

minus23548

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(d)

26458

20902

15347

97913

42357

minus13198

minus68754

minus12431

minus17986

minus23542

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(e)

25794

20239

14683

91277

35722

minus19834

minus7539

minus13095

minus1865

minus24206

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(f)

26527

20972

15416

98608

43053

minus12503

minus68059

minus12361

minus17917

minus23473

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(g)

26307

20752

15196

96406

4085

minus14705

minus70261

minus12582

minus18137

minus23693

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(h)

Figure 9 Continued

International Journal of Antennas and Propagation 7

2576220207146519095635401minus20155minus7571minus13127minus18682minus24238

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(i)

Figure 9 3D gain patterns of the nine installed positions of the array (a) Array at position 1 (b) Array at position 2 (c) Array at position 3(d) Array at position 4 (e) Array at position 5 (f) Array at position 6 (g) Array at position 7 (h) Array at position 8 (i) Array at position 9

47

8 5 2

9 6 3

1

z

x o

(a)

z

x o7 4

2

1

8 5

9 6 3

(b)

Figure 10 Two comparison ways for the 2D gain pattern

airplane platform and the airborne antenna array is a com-posite and multiscale problem Consider a realistic problemthat the microstrip antenna array installed on nine positionsof the airplane with different 119909 coordinates or different 119911coordinates as shown in Figure 8 the mainlobe direction isdirected towards the wing The operation frequency of theairborne array is 440MHz and the number of unknowns is308371

In this Section 1024 CPU cores are used and the directsolver is observed in solving this kind of problems to avoid theslow convergence of iterative CG type methods The matrixfilling time is about 4700 seconds and the matrix equationsolving time is about 18500 seconds The computed three-dimensional (3D) gain patterns of the antenna array installedon different positions of the airplane are shown in Figure 9To make the comparison more clearly the computed two-dimensional (2D) gain patterns of the antenna array installedon different positions of the airplane are compared in twoways as shown in Figure 10 One is the same 119909 coordinatesand different 119911 coordinates as shown in Figure 11 One is thedifferent 119909 coordinates and same 119911 coordinates as shown inFigure 12 Furthermore the patterns of the array alone arealso given for comparison

Form the comparison it is clearly seen that the gainpatterns of the airborne antenna array in the yoz planes(elevation planes) seriously deteriorate compared with theantenna array alone especially in the mainlobe regionAnd in contrast the patterns in the xoy plane (azimuth

plane) show slight changes From the comparisons resultsof the array installed on same 119909 coordinates and different 119911coordinates shown in Figure 11 one can see that with the 119911coordinates of the installation positions increasing (ie formposition 1 to 2 and 3 form position 4 to 5 and 6 and formposition 7 to 8 and 9) the mainlobe region of gain patterns(ie minus30∘ sim30∘ in the yoz plane) gets better mainly due to thereflection and diffraction effects of airplane wing becomingweak Furthermore the sidelobe region of gain patterns (ieminus150∘ sim minus30∘ in the yoz plane 115∘ sim120∘ in the xoy plane)becomes low with the installation height increasing Formthe comparisons results of the array installed on different 119909coordinates and same 119911 coordinates shown in Figure 12 wecan see that gain patterns (ie minus150∘ sim30∘ in the yoz plane)get better with the 119909 coordinates of the installation positionsincreasing (ie form position 1 to 4 and 7 form position 2 to5 and 8 and form position 3 to 6 and 9) However the overalldesign of airplane is a composite and multiscale problem itis necessary to take many factors into account such as theaerodynamics Therefore the array is installed not too highand not close to the head of airplane taking these factors intoconsideration the better installation position is position 5 inthis paper

Moreover the model of airborne array is a whole inpractical engineeringThe array is connected to airplane witha bracket which is installed below the array When the mainbeam of the array is direct to the wings of the airplane theeffect of the bracket can be neglected So in the simulations

8 International Journal of Antennas and Propagation

Array aloneArray at position 1

Array at position 2Array at position 3

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(a)

Array aloneArray at position 1

Array at position 2Array at position 3

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 4

Array at position 5Array at position 6

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(c)

Array aloneArray at position 4

Array at position 5Array at position 6

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 7

Array at position 8Array at position 9

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(e)

Array aloneArray at position 7

Array at position 8Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 11 2D gain patterns of the array installed on same 119909 coordinates and different 119911 coordinates (a) yoz plane of positions 1 2 and 3 (b)xoy plane of positions 1 2 and 3 (c) yoz plane of positions 4 5 and 6 (d) xoy plane of positions 4 5 and 6 (e) yoz plane of positions 7 8and 9 (f) xoy plane of positions 7 8 and 9

International Journal of Antennas and Propagation 9

Array aloneArray at position 1

Array at position 4Array at position 7

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(a)

Array aloneArray at position 1

Array at position 4Array at position 7

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 2

Array at position 5Array at position 8

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(c)

Array aloneArray at position 2

Array at position 5Array at position 8

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 3

Array at position 6Array at position 9

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(e)

Array aloneArray at position 3

Array at position 6Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 12 2D gain patterns of the array installed on different 119909 coordinates and same 119911 coordinates (a) yoz plane of positions 1 4 and 7 (b)xoy plane of positions 1 4 and 7 (c) yoz plane of positions 2 5 and 8 (d) xoy plane of positions 2 5 and 8 (e) yoz plane of positions 3 6 and9 (f) xoy plane of position 3 6 and 9

10 International Journal of Antennas and Propagation

zx

yo

(a)

Microstrip array

x

y

o

(b)

x

y

o

(c)

Figure 13 Simulation models (a) microstrip array (b) airborne array (c) the position of the array

z

x yo

1039

48344

minus62767

minus072116

minus17388

minus11832

minus22943

minus28499

minus34054

minus3961

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

z

x yo

93514

37959

minus17597

minus73152

minus12871

minus18426

minus23982

minus29537

minus35093

minus40649

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

Figure 14 3D gain patterns of the array alone and airborne array (a) array alone (b) airborne array

of the airborne array in this section the bracket is removedfrom the simulation model

35 A Microstrip Antenna Array with 9 times 1 Elements stick onanAirplane To demonstrate the ability of themethod to dealwith the problem of an array connected to the airplane amicrostrip antenna array with 9 times 1 elements sticks on thesurface of a large airplane is considered in this section [15]The model of the array and the airborne array are shown inFigures 13(a) and 13(b) and the position of the array is given

in Figure 13(c) The parameters of the array and the airplaneare same to the previous sections The computed 3D gainpatterns of the array alone and airborne array are shown inFigure 14 The computed 2D gain patterns of the array aloneand airborne array are compared in Figure 15

Form the comparisons we can see that when the arraysticks on the airplane the back gain of the array is signifi-cantly reduced while the front gain shows slight changes Itis mainly due to that when the array is stuck on the airplaneit can be considered as enlarging the ground of the array sothe result in Figure 15 is reasonable

International Journal of Antennas and Propagation 11

0

10

0

30

60

90

120

150180

210

240

270

300

330

0

10

Array aloneAirborne array

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus40

minus30

minus20

minus10

Figure 15 Comparison result of 2D gain patterns in xoz plane

4 Conclusion

An efficient parallel higher-order MoM is developed to solveextremely large and complex electromagnetics problemsTheparallel strategies based on block matrices and the higher-order basis functions make it possible to solve airborneantenna array problems using rigorousMoM and the simula-tion time of one installed position is about seven hours using1024 CPU cores Numerical results confirm the accuracy andefficiency of the method and its applicability to the analysisof radiation from airborne platform systems with complexantennas

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)(2012AA01A308) the NSFC (61301069 and 61072019) andthe Program for New Century Excellent Talents in Universityof China (NCET-13-0949) the Project with contract no2013KJXX-67 The computational resources utilized in thisresearch are provided by Shanghai Supercomputer Center

References

[1] R F Harrington ldquoField computation by moment methodsrdquo inIEEE Series on Electromagnetic Waves IEEE New York NYUSA 1993

[2] Y Zhang and T K Sarkar Parallel Solution of Integral EquationBased EM Problems in the Frequency Domain John Wiley ampSons Hoboken NJ USA 2009

[3] 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

[4] Y Zhang T K Sarkar X Zhao et al Higher Order Basis BasedIntegral Equation Solver (HOBBIES) John Wiley Hoboken NJUSA 2012

[5] Y Zhang X Zhao D G Donoro and T K Sarkar ldquoParallelizedhybrid method with higher-order MoM And PO for analysisof phased array antennas on electrically large platformsrdquo IEEETransactions on Antennas and Propagation vol 58 no 12 pp4110ndash4115 2010

[6] X-W Zhao Y Zhang H-W Zhang et al ldquoParallel MoM-POmethod with out-of-core technique for analysis of complexarrays on electrically large platformsrdquo Progress in Electromag-netics Research vol 108 pp 1ndash21 2010

[7] G Hennigan and S Castillo ldquoOpen region electromagneticfinite-element scattering calculations in anisotropic media onparallel computersrdquo in IEEE Antennas and Propagation SocietyInternational Symposium vol 1 pp 484ndash487 July 1999

[8] P Yla-OijalaM Taskinen and S Jarvenpaa ldquoAnalysis of surfaceintegral equations in electromagnetic scattering and radiationproblemsrdquo Engineering Analysis with Boundary Elements vol32 no 3 pp 196ndash209 2008

[9] P Yla-Oijala M Taskinen and J Sarvas ldquoSurface integralequation method for general composite metallic and dielec-tric structures with junctionsrdquo in Progress in ElectromagneticsResearch vol 52 pp 81ndash108 2005

[10] K Yang W T Sheng Z Y Zhu and M S Tong ldquoElectromag-netic analysis for inhomogeneous interconnect and packagingstructures based on volume-surface integral equationsrdquo IEEETransactions on Components Packaging and ManufacturingTechnology vol 3 no 8 pp 1364ndash1371 2013

[11] R F Harrington ldquoBoundary integral formulations for homoge-nous material bodiesrdquo Journal of Electromagnetic Waves andApplications vol 3 no 1 pp 1ndash15 1989

[12] D Z Ding Y Shi Z N Jiang and R S Chen ldquoAugmentedEFIE with adaptive cross approximation algorithm for analysisof electromagnetic problemsrdquo International Journal of Antennasand Propagation vol 2013 Article ID 487276 9 pages 2013

[13] J L Volakis and K Sertel Integral Equation Methods forElectromagnetics SciTech Raleigh NC USA 2012

[14] Y Yan Y Zhang C-H Liang H Zhao and D Garcıa-DonoroldquoRCS computation by parallel MoM using higher-order basisfunctionsrdquo International Journal of Antennas and Propagationvol 2012 Article ID 745893 9 pages 2012

[15] Z Yang S Jiang X Zhao Y Zhang and H Qu ldquoRadiationof an airborne patch array antennardquo in Proceedings of the IETInternational Radar Conference Institution of Engineering andTechnology Xirsquoan China April 2013

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 3

0

2

1

1

0

2

1

0

2

0 1 2 0 1 2 0 1 2

11

21

31

41

51

61

71

81

91

12

22

32

42

52

62

72

82

92

13

23

33

43

53

63

73

83

93

14

24

34

44

54

64

74

84

94

15

25

35

45

55

65

75

85

95

16

26

36

46

56

66

76

86

96

17

27

37

47

57

67

77

87

97

18

28

38

48

58

68

78

88

98

19

29

39

49

59

69

79

89

99

(a)

0

(00)1

(10)2

(20)0

1

2

0

1

2

3

(01)4

(11)5

(21)3

4

5

3

4

5

6

(02)7

(12)8

(22)6

7

8

6

7

8

0

(00)1

(10)2

(20)0

1

2

0

1

2

3

(01)4

(11)5

(21)3

4

5

3

4

5

6

(02)7

(12)8

(22)6

7

8

6

7

8

0

(00)1

(10)2

(20)0

1

2

0

1

2

3

(01)4

(11)5

(21)3

4

5

3

4

5

6

(02)7

(12)8

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

(00)

(10)

(20)

(01)

(11)

(21)

(02)

(12)

(22)

6

7

8

6

7

8

(b)

Figure 1 Block-cyclic distribution of a matrix (a) a matrix consisting of 9 times 9 blocks (b) rank and coordinates of each process owning thecorresponding blocks in (a)

(analogous expressions stand for the 119901-component and forthe magnetic current)

997888119869119904(119901 119904)

=

119873119901

sum

119894=0

[

[

1198881198941

997888119864119894(119901 119904) + 119888

1198942

997888119864119894(119901 minus119904) +

119873119904

sum

119895=2

119886119894119895

997888119875119894119895(119901 119904)]

]

(8)

where 119873119901and 119873

119904are the degrees of the approximations

along the coordinates and 119886119894119895 1198881198941 and 119888

1198942are the unknown

coefficientsThus expression (8) stands for the representation of the

current in terms of edge basis functions 997888119864119894(119901 119904) and interior

or patch basis functions 997888119875119894119895(119901 119904) which can be compactly

expressed as

997888119864119894119896(119901 119904) =

120572119904

10038161003816100381610038161003816120572119901times 120572119904

10038161003816100381610038161003816

119901119894119873(119904) 119896 = 1

119901119894119873(minus119904) 119896 = 2

997888119875119894119895(119901 119904) =

120572119904

10038161003816100381610038161003816120572119901times 120572119904

10038161003816100381610038161003816

119901119894119878119895(119904)

119873 (119904) =1 minus 119904

2 119878119894(119904) =

119904119894minus 1 119894 is even

119904119894minus 119904 119894 is odd

(9)

where symbols 119886119904and 119886119901denote the unitary vectors along the

transformed 119904 and 119901 coordinatesEdge basis functions 997888

1198641198941

and patch basis functions997888119875119894119895are zero along the first edge (119904 = minus1) 997888119864

1198942and 997888

119875119894119895being

Feed

Substrate

Patchz

x

y

o

Figure 2 Model of the microstrip patch antenna element

zero along the second edge (119904 = 1) Thus the continuityequation can easily be imposed on a given mesh made ofpatches

23 Parallel Schemes of Higher-Order MoM The paralleliza-tion of the higher-order MoM solution involves parallelmatrix filling followed by a parallel solution of the densematrix equation To ensure the load-balancing in terms ofdata and CPU operations it is necessary to divide the matrixinto blocks and distribute those blocks to all processorsthrough a block-cyclic distribution procedure The modes ofdistribution of blocks are related to matrix equation solutionmethods and we select the LU factorization as the solutionmethod

4 International Journal of Antennas and Propagation

Parallel higher-order MoMFinite element method

0

4

80

30

60

90

120

150180

210

240

270

300

330

0

4

8

Gai

n (d

B)

minus4

minus8

minus4

(a)

Parallel higher-order MoMFinite element method

02468

0

30

60

90

120

150180

210

240

270

300

330

02468

Gai

n (d

B)

minus2

minus4

minus6

minus4

minus2

(b)

Figure 3 2D gain of microstrip patch antenna (a) yoz plane and (b) xoy plane

As an example consider amatrix [119860] of Figure 1(a) whichis distributed to different processes in the 3 times 3 process gridFigure 1(b) shows to which process the blocks of [119860] aredistributed using the block-cyclic distribution methodology[2 14] In Figure 1(a) the outermost numbers denote therow and column indices of the process coordinates The topand bottom number in any block of Figure 1(b) denotes theprocess ID (rank) and the process coordinates of a certainprocess respectively corresponding to the block of thematrixshown in Figure 1(a) By varying the dimensions of the blocksof [119860] and those of the process grid different mappings canbe obtained

During the procedure of finding the solution to thematrixequation balancing the computational load between all theprocesses is also important However it is less easy to trackcompared with the parallel matrix filling procedure sincean increase in the number of unknowns or the number ofprocesses executing the solution can increase the amountof communication required between the processes Thisincrease in communication will decrease the gains of parallelspeedup However as a rule of thumb more processes typi-cally means less wall clock time for solving large problemseven though the overall parallel efficiency may sometimes beincreased by using fewer processes to solve the same problem

The solution of the matrix equation is essentially thesame regardless of the type of basis functions used for theMoM The parallel LU decomposition solver based on theScaLAPACK library package is described in Chapter 2 in[2] Also one can refer to Chapter 6 in [2] for a detaileddiscussion about the iterative solvers based on the conjugategradient (CG) method The computational complexity of LUdecomposition scales with 119874 (1198733) is much higher than

zx

yo

9 patchelements

37 patch elements (10m)

(25m)

Figure 4 Model of the microstrip patch array antenna with 37 times 9elements

25658

20102

14547

89913

34357

minus21198

minus76754

minus13231

minus18786

minus24342

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

Figure 5 3D gain patterns of the microstrip patch array antenna

that of CG type methods with 119874 (1198732) where 119873 is thenumber of unknowns However in this paper the parallelLU decomposition is utilized as the parallel equation solverrather than the parallel iterative CG method due to the factthat the iterative CG method may encounter a divergence

International Journal of Antennas and Propagation 5

0 60 120 180

0

10

20

30

40G

ain

(dB)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(a)

Gai

n (d

B)

0 60 120 180 240 300 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Figure 6 2D gain patterns of the microstrip patch array antenna (a) yoz plane and (b) xoy plane

problem when dealing with complex antennas composed ofthin structures and various materials

3 Numerical Results

31 Comparison with the Finite Element Method To validatethe accuracy and efficiency of the proposed parallel higher-order MoM a rectangular microstrip patch antenna elementis simulated which is shown in Figure 2 The dimensionsof the patch element are 2056mm times 1548mm and thematerial parameter of the substrate is 120576

119903= 42 The operation

frequency is 440MHz and the simulation is performed ona personal computer with a Quad-core Intel I5 processor(300GHz) and 8GB of RAMThe radiation pattern obtainedusing the parallel higher-order MoM and the finite elementmethod are plotted together in Figure 3 The results agreewith each other very well

32 Description of the Parallel Computational Platform Thecomputing cluster used in this section is the Magic Cubewith 64 nodes at Shanghai Supercomputer Center (SSC) andeach node has four quad-core 19 GHz processors and 64GBmemory The nodes are connected with Infiniband switches

33 A Microstrip Patch Array Antenna with 37 times 9 ElementsIn this section we present the radiation patterns of thealgorithm for a microstrip array Consider a rectangularmicrostrip patch array antenna with 37 times 9 elements Themicrostrip element parameters are the same as the elementanalyzed in Section 31 and the array model is shown inFigure 4 The dimensions of the array are 10m times 25m times

0018m and the distances between any two neighboring ele-ments are 579mm and 833mm along the length and widthdirections The amplitude at the feed of the array is designedby a minus35 dB Taylor distribution and the operation frequencyof the array is 440MHz The three-dimensional (3D) gainpattern is given in Figure 5 and the two-dimensional (2D)

z

xy

o

Microstrip array

Figure 7 Microstrip patch array over an airplane

bull Array center

7 48 5 29 6 3

1

zx o

2m

Figure 8 Nine positions of the array in the xoz plane

gain patterns are given in Figure 6 The side-lobe levels inboth the yoz and xoy planes are below minus35 dB which meetthe design target Note that 120579 coordinate is measured fromxoy plane to 119911 axis and 120593 coordinate is measured from +x axisto 119910 axis in this paper

34 A Microstrip Antenna Array with 37 times 9 Elementsover a Large Airplane Install the 37 times 9 microstrip patcharray that was analyzed in Section 33 over a large airplane(httpwwwscalecraftcom737aewandcaspx) as shown inFigure 7 The dimensions of the airplane are 55m times 476m times

158m The array elements are very thin compared with the

6 International Journal of Antennas and Propagation

26648

21092

15537

99811

44256

minus113

minus66855

minus12241

minus17797

minus23352

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

26401

20846

1529

97347

41792

minus13764

minus69319

minus12487

minus18043

minus23599

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

25902

20346

14791

92352

36796

minus1876

minus74315

minus12987

minus18543

minus24098

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(c)

26452

20897

15341

97854

42299

minus13257

minus68812

minus12437

minus17992

minus23548

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(d)

26458

20902

15347

97913

42357

minus13198

minus68754

minus12431

minus17986

minus23542

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(e)

25794

20239

14683

91277

35722

minus19834

minus7539

minus13095

minus1865

minus24206

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(f)

26527

20972

15416

98608

43053

minus12503

minus68059

minus12361

minus17917

minus23473

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(g)

26307

20752

15196

96406

4085

minus14705

minus70261

minus12582

minus18137

minus23693

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(h)

Figure 9 Continued

International Journal of Antennas and Propagation 7

2576220207146519095635401minus20155minus7571minus13127minus18682minus24238

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(i)

Figure 9 3D gain patterns of the nine installed positions of the array (a) Array at position 1 (b) Array at position 2 (c) Array at position 3(d) Array at position 4 (e) Array at position 5 (f) Array at position 6 (g) Array at position 7 (h) Array at position 8 (i) Array at position 9

47

8 5 2

9 6 3

1

z

x o

(a)

z

x o7 4

2

1

8 5

9 6 3

(b)

Figure 10 Two comparison ways for the 2D gain pattern

airplane platform and the airborne antenna array is a com-posite and multiscale problem Consider a realistic problemthat the microstrip antenna array installed on nine positionsof the airplane with different 119909 coordinates or different 119911coordinates as shown in Figure 8 the mainlobe direction isdirected towards the wing The operation frequency of theairborne array is 440MHz and the number of unknowns is308371

In this Section 1024 CPU cores are used and the directsolver is observed in solving this kind of problems to avoid theslow convergence of iterative CG type methods The matrixfilling time is about 4700 seconds and the matrix equationsolving time is about 18500 seconds The computed three-dimensional (3D) gain patterns of the antenna array installedon different positions of the airplane are shown in Figure 9To make the comparison more clearly the computed two-dimensional (2D) gain patterns of the antenna array installedon different positions of the airplane are compared in twoways as shown in Figure 10 One is the same 119909 coordinatesand different 119911 coordinates as shown in Figure 11 One is thedifferent 119909 coordinates and same 119911 coordinates as shown inFigure 12 Furthermore the patterns of the array alone arealso given for comparison

Form the comparison it is clearly seen that the gainpatterns of the airborne antenna array in the yoz planes(elevation planes) seriously deteriorate compared with theantenna array alone especially in the mainlobe regionAnd in contrast the patterns in the xoy plane (azimuth

plane) show slight changes From the comparisons resultsof the array installed on same 119909 coordinates and different 119911coordinates shown in Figure 11 one can see that with the 119911coordinates of the installation positions increasing (ie formposition 1 to 2 and 3 form position 4 to 5 and 6 and formposition 7 to 8 and 9) the mainlobe region of gain patterns(ie minus30∘ sim30∘ in the yoz plane) gets better mainly due to thereflection and diffraction effects of airplane wing becomingweak Furthermore the sidelobe region of gain patterns (ieminus150∘ sim minus30∘ in the yoz plane 115∘ sim120∘ in the xoy plane)becomes low with the installation height increasing Formthe comparisons results of the array installed on different 119909coordinates and same 119911 coordinates shown in Figure 12 wecan see that gain patterns (ie minus150∘ sim30∘ in the yoz plane)get better with the 119909 coordinates of the installation positionsincreasing (ie form position 1 to 4 and 7 form position 2 to5 and 8 and form position 3 to 6 and 9) However the overalldesign of airplane is a composite and multiscale problem itis necessary to take many factors into account such as theaerodynamics Therefore the array is installed not too highand not close to the head of airplane taking these factors intoconsideration the better installation position is position 5 inthis paper

Moreover the model of airborne array is a whole inpractical engineeringThe array is connected to airplane witha bracket which is installed below the array When the mainbeam of the array is direct to the wings of the airplane theeffect of the bracket can be neglected So in the simulations

8 International Journal of Antennas and Propagation

Array aloneArray at position 1

Array at position 2Array at position 3

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(a)

Array aloneArray at position 1

Array at position 2Array at position 3

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 4

Array at position 5Array at position 6

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(c)

Array aloneArray at position 4

Array at position 5Array at position 6

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 7

Array at position 8Array at position 9

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(e)

Array aloneArray at position 7

Array at position 8Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 11 2D gain patterns of the array installed on same 119909 coordinates and different 119911 coordinates (a) yoz plane of positions 1 2 and 3 (b)xoy plane of positions 1 2 and 3 (c) yoz plane of positions 4 5 and 6 (d) xoy plane of positions 4 5 and 6 (e) yoz plane of positions 7 8and 9 (f) xoy plane of positions 7 8 and 9

International Journal of Antennas and Propagation 9

Array aloneArray at position 1

Array at position 4Array at position 7

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(a)

Array aloneArray at position 1

Array at position 4Array at position 7

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 2

Array at position 5Array at position 8

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(c)

Array aloneArray at position 2

Array at position 5Array at position 8

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 3

Array at position 6Array at position 9

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(e)

Array aloneArray at position 3

Array at position 6Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 12 2D gain patterns of the array installed on different 119909 coordinates and same 119911 coordinates (a) yoz plane of positions 1 4 and 7 (b)xoy plane of positions 1 4 and 7 (c) yoz plane of positions 2 5 and 8 (d) xoy plane of positions 2 5 and 8 (e) yoz plane of positions 3 6 and9 (f) xoy plane of position 3 6 and 9

10 International Journal of Antennas and Propagation

zx

yo

(a)

Microstrip array

x

y

o

(b)

x

y

o

(c)

Figure 13 Simulation models (a) microstrip array (b) airborne array (c) the position of the array

z

x yo

1039

48344

minus62767

minus072116

minus17388

minus11832

minus22943

minus28499

minus34054

minus3961

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

z

x yo

93514

37959

minus17597

minus73152

minus12871

minus18426

minus23982

minus29537

minus35093

minus40649

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

Figure 14 3D gain patterns of the array alone and airborne array (a) array alone (b) airborne array

of the airborne array in this section the bracket is removedfrom the simulation model

35 A Microstrip Antenna Array with 9 times 1 Elements stick onanAirplane To demonstrate the ability of themethod to dealwith the problem of an array connected to the airplane amicrostrip antenna array with 9 times 1 elements sticks on thesurface of a large airplane is considered in this section [15]The model of the array and the airborne array are shown inFigures 13(a) and 13(b) and the position of the array is given

in Figure 13(c) The parameters of the array and the airplaneare same to the previous sections The computed 3D gainpatterns of the array alone and airborne array are shown inFigure 14 The computed 2D gain patterns of the array aloneand airborne array are compared in Figure 15

Form the comparisons we can see that when the arraysticks on the airplane the back gain of the array is signifi-cantly reduced while the front gain shows slight changes Itis mainly due to that when the array is stuck on the airplaneit can be considered as enlarging the ground of the array sothe result in Figure 15 is reasonable

International Journal of Antennas and Propagation 11

0

10

0

30

60

90

120

150180

210

240

270

300

330

0

10

Array aloneAirborne array

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus40

minus30

minus20

minus10

Figure 15 Comparison result of 2D gain patterns in xoz plane

4 Conclusion

An efficient parallel higher-order MoM is developed to solveextremely large and complex electromagnetics problemsTheparallel strategies based on block matrices and the higher-order basis functions make it possible to solve airborneantenna array problems using rigorousMoM and the simula-tion time of one installed position is about seven hours using1024 CPU cores Numerical results confirm the accuracy andefficiency of the method and its applicability to the analysisof radiation from airborne platform systems with complexantennas

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)(2012AA01A308) the NSFC (61301069 and 61072019) andthe Program for New Century Excellent Talents in Universityof China (NCET-13-0949) the Project with contract no2013KJXX-67 The computational resources utilized in thisresearch are provided by Shanghai Supercomputer Center

References

[1] R F Harrington ldquoField computation by moment methodsrdquo inIEEE Series on Electromagnetic Waves IEEE New York NYUSA 1993

[2] Y Zhang and T K Sarkar Parallel Solution of Integral EquationBased EM Problems in the Frequency Domain John Wiley ampSons Hoboken NJ USA 2009

[3] 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

[4] Y Zhang T K Sarkar X Zhao et al Higher Order Basis BasedIntegral Equation Solver (HOBBIES) John Wiley Hoboken NJUSA 2012

[5] Y Zhang X Zhao D G Donoro and T K Sarkar ldquoParallelizedhybrid method with higher-order MoM And PO for analysisof phased array antennas on electrically large platformsrdquo IEEETransactions on Antennas and Propagation vol 58 no 12 pp4110ndash4115 2010

[6] X-W Zhao Y Zhang H-W Zhang et al ldquoParallel MoM-POmethod with out-of-core technique for analysis of complexarrays on electrically large platformsrdquo Progress in Electromag-netics Research vol 108 pp 1ndash21 2010

[7] G Hennigan and S Castillo ldquoOpen region electromagneticfinite-element scattering calculations in anisotropic media onparallel computersrdquo in IEEE Antennas and Propagation SocietyInternational Symposium vol 1 pp 484ndash487 July 1999

[8] P Yla-OijalaM Taskinen and S Jarvenpaa ldquoAnalysis of surfaceintegral equations in electromagnetic scattering and radiationproblemsrdquo Engineering Analysis with Boundary Elements vol32 no 3 pp 196ndash209 2008

[9] P Yla-Oijala M Taskinen and J Sarvas ldquoSurface integralequation method for general composite metallic and dielec-tric structures with junctionsrdquo in Progress in ElectromagneticsResearch vol 52 pp 81ndash108 2005

[10] K Yang W T Sheng Z Y Zhu and M S Tong ldquoElectromag-netic analysis for inhomogeneous interconnect and packagingstructures based on volume-surface integral equationsrdquo IEEETransactions on Components Packaging and ManufacturingTechnology vol 3 no 8 pp 1364ndash1371 2013

[11] R F Harrington ldquoBoundary integral formulations for homoge-nous material bodiesrdquo Journal of Electromagnetic Waves andApplications vol 3 no 1 pp 1ndash15 1989

[12] D Z Ding Y Shi Z N Jiang and R S Chen ldquoAugmentedEFIE with adaptive cross approximation algorithm for analysisof electromagnetic problemsrdquo International Journal of Antennasand Propagation vol 2013 Article ID 487276 9 pages 2013

[13] J L Volakis and K Sertel Integral Equation Methods forElectromagnetics SciTech Raleigh NC USA 2012

[14] Y Yan Y Zhang C-H Liang H Zhao and D Garcıa-DonoroldquoRCS computation by parallel MoM using higher-order basisfunctionsrdquo International Journal of Antennas and Propagationvol 2012 Article ID 745893 9 pages 2012

[15] Z Yang S Jiang X Zhao Y Zhang and H Qu ldquoRadiationof an airborne patch array antennardquo in Proceedings of the IETInternational Radar Conference Institution of Engineering andTechnology Xirsquoan China April 2013

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

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DistributedSensor Networks

International Journal of

4 International Journal of Antennas and Propagation

Parallel higher-order MoMFinite element method

0

4

80

30

60

90

120

150180

210

240

270

300

330

0

4

8

Gai

n (d

B)

minus4

minus8

minus4

(a)

Parallel higher-order MoMFinite element method

02468

0

30

60

90

120

150180

210

240

270

300

330

02468

Gai

n (d

B)

minus2

minus4

minus6

minus4

minus2

(b)

Figure 3 2D gain of microstrip patch antenna (a) yoz plane and (b) xoy plane

As an example consider amatrix [119860] of Figure 1(a) whichis distributed to different processes in the 3 times 3 process gridFigure 1(b) shows to which process the blocks of [119860] aredistributed using the block-cyclic distribution methodology[2 14] In Figure 1(a) the outermost numbers denote therow and column indices of the process coordinates The topand bottom number in any block of Figure 1(b) denotes theprocess ID (rank) and the process coordinates of a certainprocess respectively corresponding to the block of thematrixshown in Figure 1(a) By varying the dimensions of the blocksof [119860] and those of the process grid different mappings canbe obtained

During the procedure of finding the solution to thematrixequation balancing the computational load between all theprocesses is also important However it is less easy to trackcompared with the parallel matrix filling procedure sincean increase in the number of unknowns or the number ofprocesses executing the solution can increase the amountof communication required between the processes Thisincrease in communication will decrease the gains of parallelspeedup However as a rule of thumb more processes typi-cally means less wall clock time for solving large problemseven though the overall parallel efficiency may sometimes beincreased by using fewer processes to solve the same problem

The solution of the matrix equation is essentially thesame regardless of the type of basis functions used for theMoM The parallel LU decomposition solver based on theScaLAPACK library package is described in Chapter 2 in[2] Also one can refer to Chapter 6 in [2] for a detaileddiscussion about the iterative solvers based on the conjugategradient (CG) method The computational complexity of LUdecomposition scales with 119874 (1198733) is much higher than

zx

yo

9 patchelements

37 patch elements (10m)

(25m)

Figure 4 Model of the microstrip patch array antenna with 37 times 9elements

25658

20102

14547

89913

34357

minus21198

minus76754

minus13231

minus18786

minus24342

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

Figure 5 3D gain patterns of the microstrip patch array antenna

that of CG type methods with 119874 (1198732) where 119873 is thenumber of unknowns However in this paper the parallelLU decomposition is utilized as the parallel equation solverrather than the parallel iterative CG method due to the factthat the iterative CG method may encounter a divergence

International Journal of Antennas and Propagation 5

0 60 120 180

0

10

20

30

40G

ain

(dB)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(a)

Gai

n (d

B)

0 60 120 180 240 300 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Figure 6 2D gain patterns of the microstrip patch array antenna (a) yoz plane and (b) xoy plane

problem when dealing with complex antennas composed ofthin structures and various materials

3 Numerical Results

31 Comparison with the Finite Element Method To validatethe accuracy and efficiency of the proposed parallel higher-order MoM a rectangular microstrip patch antenna elementis simulated which is shown in Figure 2 The dimensionsof the patch element are 2056mm times 1548mm and thematerial parameter of the substrate is 120576

119903= 42 The operation

frequency is 440MHz and the simulation is performed ona personal computer with a Quad-core Intel I5 processor(300GHz) and 8GB of RAMThe radiation pattern obtainedusing the parallel higher-order MoM and the finite elementmethod are plotted together in Figure 3 The results agreewith each other very well

32 Description of the Parallel Computational Platform Thecomputing cluster used in this section is the Magic Cubewith 64 nodes at Shanghai Supercomputer Center (SSC) andeach node has four quad-core 19 GHz processors and 64GBmemory The nodes are connected with Infiniband switches

33 A Microstrip Patch Array Antenna with 37 times 9 ElementsIn this section we present the radiation patterns of thealgorithm for a microstrip array Consider a rectangularmicrostrip patch array antenna with 37 times 9 elements Themicrostrip element parameters are the same as the elementanalyzed in Section 31 and the array model is shown inFigure 4 The dimensions of the array are 10m times 25m times

0018m and the distances between any two neighboring ele-ments are 579mm and 833mm along the length and widthdirections The amplitude at the feed of the array is designedby a minus35 dB Taylor distribution and the operation frequencyof the array is 440MHz The three-dimensional (3D) gainpattern is given in Figure 5 and the two-dimensional (2D)

z

xy

o

Microstrip array

Figure 7 Microstrip patch array over an airplane

bull Array center

7 48 5 29 6 3

1

zx o

2m

Figure 8 Nine positions of the array in the xoz plane

gain patterns are given in Figure 6 The side-lobe levels inboth the yoz and xoy planes are below minus35 dB which meetthe design target Note that 120579 coordinate is measured fromxoy plane to 119911 axis and 120593 coordinate is measured from +x axisto 119910 axis in this paper

34 A Microstrip Antenna Array with 37 times 9 Elementsover a Large Airplane Install the 37 times 9 microstrip patcharray that was analyzed in Section 33 over a large airplane(httpwwwscalecraftcom737aewandcaspx) as shown inFigure 7 The dimensions of the airplane are 55m times 476m times

158m The array elements are very thin compared with the

6 International Journal of Antennas and Propagation

26648

21092

15537

99811

44256

minus113

minus66855

minus12241

minus17797

minus23352

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

26401

20846

1529

97347

41792

minus13764

minus69319

minus12487

minus18043

minus23599

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

25902

20346

14791

92352

36796

minus1876

minus74315

minus12987

minus18543

minus24098

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(c)

26452

20897

15341

97854

42299

minus13257

minus68812

minus12437

minus17992

minus23548

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(d)

26458

20902

15347

97913

42357

minus13198

minus68754

minus12431

minus17986

minus23542

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(e)

25794

20239

14683

91277

35722

minus19834

minus7539

minus13095

minus1865

minus24206

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(f)

26527

20972

15416

98608

43053

minus12503

minus68059

minus12361

minus17917

minus23473

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(g)

26307

20752

15196

96406

4085

minus14705

minus70261

minus12582

minus18137

minus23693

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(h)

Figure 9 Continued

International Journal of Antennas and Propagation 7

2576220207146519095635401minus20155minus7571minus13127minus18682minus24238

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(i)

Figure 9 3D gain patterns of the nine installed positions of the array (a) Array at position 1 (b) Array at position 2 (c) Array at position 3(d) Array at position 4 (e) Array at position 5 (f) Array at position 6 (g) Array at position 7 (h) Array at position 8 (i) Array at position 9

47

8 5 2

9 6 3

1

z

x o

(a)

z

x o7 4

2

1

8 5

9 6 3

(b)

Figure 10 Two comparison ways for the 2D gain pattern

airplane platform and the airborne antenna array is a com-posite and multiscale problem Consider a realistic problemthat the microstrip antenna array installed on nine positionsof the airplane with different 119909 coordinates or different 119911coordinates as shown in Figure 8 the mainlobe direction isdirected towards the wing The operation frequency of theairborne array is 440MHz and the number of unknowns is308371

In this Section 1024 CPU cores are used and the directsolver is observed in solving this kind of problems to avoid theslow convergence of iterative CG type methods The matrixfilling time is about 4700 seconds and the matrix equationsolving time is about 18500 seconds The computed three-dimensional (3D) gain patterns of the antenna array installedon different positions of the airplane are shown in Figure 9To make the comparison more clearly the computed two-dimensional (2D) gain patterns of the antenna array installedon different positions of the airplane are compared in twoways as shown in Figure 10 One is the same 119909 coordinatesand different 119911 coordinates as shown in Figure 11 One is thedifferent 119909 coordinates and same 119911 coordinates as shown inFigure 12 Furthermore the patterns of the array alone arealso given for comparison

Form the comparison it is clearly seen that the gainpatterns of the airborne antenna array in the yoz planes(elevation planes) seriously deteriorate compared with theantenna array alone especially in the mainlobe regionAnd in contrast the patterns in the xoy plane (azimuth

plane) show slight changes From the comparisons resultsof the array installed on same 119909 coordinates and different 119911coordinates shown in Figure 11 one can see that with the 119911coordinates of the installation positions increasing (ie formposition 1 to 2 and 3 form position 4 to 5 and 6 and formposition 7 to 8 and 9) the mainlobe region of gain patterns(ie minus30∘ sim30∘ in the yoz plane) gets better mainly due to thereflection and diffraction effects of airplane wing becomingweak Furthermore the sidelobe region of gain patterns (ieminus150∘ sim minus30∘ in the yoz plane 115∘ sim120∘ in the xoy plane)becomes low with the installation height increasing Formthe comparisons results of the array installed on different 119909coordinates and same 119911 coordinates shown in Figure 12 wecan see that gain patterns (ie minus150∘ sim30∘ in the yoz plane)get better with the 119909 coordinates of the installation positionsincreasing (ie form position 1 to 4 and 7 form position 2 to5 and 8 and form position 3 to 6 and 9) However the overalldesign of airplane is a composite and multiscale problem itis necessary to take many factors into account such as theaerodynamics Therefore the array is installed not too highand not close to the head of airplane taking these factors intoconsideration the better installation position is position 5 inthis paper

Moreover the model of airborne array is a whole inpractical engineeringThe array is connected to airplane witha bracket which is installed below the array When the mainbeam of the array is direct to the wings of the airplane theeffect of the bracket can be neglected So in the simulations

8 International Journal of Antennas and Propagation

Array aloneArray at position 1

Array at position 2Array at position 3

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(a)

Array aloneArray at position 1

Array at position 2Array at position 3

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 4

Array at position 5Array at position 6

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(c)

Array aloneArray at position 4

Array at position 5Array at position 6

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 7

Array at position 8Array at position 9

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(e)

Array aloneArray at position 7

Array at position 8Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 11 2D gain patterns of the array installed on same 119909 coordinates and different 119911 coordinates (a) yoz plane of positions 1 2 and 3 (b)xoy plane of positions 1 2 and 3 (c) yoz plane of positions 4 5 and 6 (d) xoy plane of positions 4 5 and 6 (e) yoz plane of positions 7 8and 9 (f) xoy plane of positions 7 8 and 9

International Journal of Antennas and Propagation 9

Array aloneArray at position 1

Array at position 4Array at position 7

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(a)

Array aloneArray at position 1

Array at position 4Array at position 7

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 2

Array at position 5Array at position 8

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(c)

Array aloneArray at position 2

Array at position 5Array at position 8

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 3

Array at position 6Array at position 9

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(e)

Array aloneArray at position 3

Array at position 6Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 12 2D gain patterns of the array installed on different 119909 coordinates and same 119911 coordinates (a) yoz plane of positions 1 4 and 7 (b)xoy plane of positions 1 4 and 7 (c) yoz plane of positions 2 5 and 8 (d) xoy plane of positions 2 5 and 8 (e) yoz plane of positions 3 6 and9 (f) xoy plane of position 3 6 and 9

10 International Journal of Antennas and Propagation

zx

yo

(a)

Microstrip array

x

y

o

(b)

x

y

o

(c)

Figure 13 Simulation models (a) microstrip array (b) airborne array (c) the position of the array

z

x yo

1039

48344

minus62767

minus072116

minus17388

minus11832

minus22943

minus28499

minus34054

minus3961

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

z

x yo

93514

37959

minus17597

minus73152

minus12871

minus18426

minus23982

minus29537

minus35093

minus40649

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

Figure 14 3D gain patterns of the array alone and airborne array (a) array alone (b) airborne array

of the airborne array in this section the bracket is removedfrom the simulation model

35 A Microstrip Antenna Array with 9 times 1 Elements stick onanAirplane To demonstrate the ability of themethod to dealwith the problem of an array connected to the airplane amicrostrip antenna array with 9 times 1 elements sticks on thesurface of a large airplane is considered in this section [15]The model of the array and the airborne array are shown inFigures 13(a) and 13(b) and the position of the array is given

in Figure 13(c) The parameters of the array and the airplaneare same to the previous sections The computed 3D gainpatterns of the array alone and airborne array are shown inFigure 14 The computed 2D gain patterns of the array aloneand airborne array are compared in Figure 15

Form the comparisons we can see that when the arraysticks on the airplane the back gain of the array is signifi-cantly reduced while the front gain shows slight changes Itis mainly due to that when the array is stuck on the airplaneit can be considered as enlarging the ground of the array sothe result in Figure 15 is reasonable

International Journal of Antennas and Propagation 11

0

10

0

30

60

90

120

150180

210

240

270

300

330

0

10

Array aloneAirborne array

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus40

minus30

minus20

minus10

Figure 15 Comparison result of 2D gain patterns in xoz plane

4 Conclusion

An efficient parallel higher-order MoM is developed to solveextremely large and complex electromagnetics problemsTheparallel strategies based on block matrices and the higher-order basis functions make it possible to solve airborneantenna array problems using rigorousMoM and the simula-tion time of one installed position is about seven hours using1024 CPU cores Numerical results confirm the accuracy andefficiency of the method and its applicability to the analysisof radiation from airborne platform systems with complexantennas

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)(2012AA01A308) the NSFC (61301069 and 61072019) andthe Program for New Century Excellent Talents in Universityof China (NCET-13-0949) the Project with contract no2013KJXX-67 The computational resources utilized in thisresearch are provided by Shanghai Supercomputer Center

References

[1] R F Harrington ldquoField computation by moment methodsrdquo inIEEE Series on Electromagnetic Waves IEEE New York NYUSA 1993

[2] Y Zhang and T K Sarkar Parallel Solution of Integral EquationBased EM Problems in the Frequency Domain John Wiley ampSons Hoboken NJ USA 2009

[3] 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

[4] Y Zhang T K Sarkar X Zhao et al Higher Order Basis BasedIntegral Equation Solver (HOBBIES) John Wiley Hoboken NJUSA 2012

[5] Y Zhang X Zhao D G Donoro and T K Sarkar ldquoParallelizedhybrid method with higher-order MoM And PO for analysisof phased array antennas on electrically large platformsrdquo IEEETransactions on Antennas and Propagation vol 58 no 12 pp4110ndash4115 2010

[6] X-W Zhao Y Zhang H-W Zhang et al ldquoParallel MoM-POmethod with out-of-core technique for analysis of complexarrays on electrically large platformsrdquo Progress in Electromag-netics Research vol 108 pp 1ndash21 2010

[7] G Hennigan and S Castillo ldquoOpen region electromagneticfinite-element scattering calculations in anisotropic media onparallel computersrdquo in IEEE Antennas and Propagation SocietyInternational Symposium vol 1 pp 484ndash487 July 1999

[8] P Yla-OijalaM Taskinen and S Jarvenpaa ldquoAnalysis of surfaceintegral equations in electromagnetic scattering and radiationproblemsrdquo Engineering Analysis with Boundary Elements vol32 no 3 pp 196ndash209 2008

[9] P Yla-Oijala M Taskinen and J Sarvas ldquoSurface integralequation method for general composite metallic and dielec-tric structures with junctionsrdquo in Progress in ElectromagneticsResearch vol 52 pp 81ndash108 2005

[10] K Yang W T Sheng Z Y Zhu and M S Tong ldquoElectromag-netic analysis for inhomogeneous interconnect and packagingstructures based on volume-surface integral equationsrdquo IEEETransactions on Components Packaging and ManufacturingTechnology vol 3 no 8 pp 1364ndash1371 2013

[11] R F Harrington ldquoBoundary integral formulations for homoge-nous material bodiesrdquo Journal of Electromagnetic Waves andApplications vol 3 no 1 pp 1ndash15 1989

[12] D Z Ding Y Shi Z N Jiang and R S Chen ldquoAugmentedEFIE with adaptive cross approximation algorithm for analysisof electromagnetic problemsrdquo International Journal of Antennasand Propagation vol 2013 Article ID 487276 9 pages 2013

[13] J L Volakis and K Sertel Integral Equation Methods forElectromagnetics SciTech Raleigh NC USA 2012

[14] Y Yan Y Zhang C-H Liang H Zhao and D Garcıa-DonoroldquoRCS computation by parallel MoM using higher-order basisfunctionsrdquo International Journal of Antennas and Propagationvol 2012 Article ID 745893 9 pages 2012

[15] Z Yang S Jiang X Zhao Y Zhang and H Qu ldquoRadiationof an airborne patch array antennardquo in Proceedings of the IETInternational Radar Conference Institution of Engineering andTechnology Xirsquoan China April 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

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International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 5

0 60 120 180

0

10

20

30

40G

ain

(dB)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(a)

Gai

n (d

B)

0 60 120 180 240 300 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Figure 6 2D gain patterns of the microstrip patch array antenna (a) yoz plane and (b) xoy plane

problem when dealing with complex antennas composed ofthin structures and various materials

3 Numerical Results

31 Comparison with the Finite Element Method To validatethe accuracy and efficiency of the proposed parallel higher-order MoM a rectangular microstrip patch antenna elementis simulated which is shown in Figure 2 The dimensionsof the patch element are 2056mm times 1548mm and thematerial parameter of the substrate is 120576

119903= 42 The operation

frequency is 440MHz and the simulation is performed ona personal computer with a Quad-core Intel I5 processor(300GHz) and 8GB of RAMThe radiation pattern obtainedusing the parallel higher-order MoM and the finite elementmethod are plotted together in Figure 3 The results agreewith each other very well

32 Description of the Parallel Computational Platform Thecomputing cluster used in this section is the Magic Cubewith 64 nodes at Shanghai Supercomputer Center (SSC) andeach node has four quad-core 19 GHz processors and 64GBmemory The nodes are connected with Infiniband switches

33 A Microstrip Patch Array Antenna with 37 times 9 ElementsIn this section we present the radiation patterns of thealgorithm for a microstrip array Consider a rectangularmicrostrip patch array antenna with 37 times 9 elements Themicrostrip element parameters are the same as the elementanalyzed in Section 31 and the array model is shown inFigure 4 The dimensions of the array are 10m times 25m times

0018m and the distances between any two neighboring ele-ments are 579mm and 833mm along the length and widthdirections The amplitude at the feed of the array is designedby a minus35 dB Taylor distribution and the operation frequencyof the array is 440MHz The three-dimensional (3D) gainpattern is given in Figure 5 and the two-dimensional (2D)

z

xy

o

Microstrip array

Figure 7 Microstrip patch array over an airplane

bull Array center

7 48 5 29 6 3

1

zx o

2m

Figure 8 Nine positions of the array in the xoz plane

gain patterns are given in Figure 6 The side-lobe levels inboth the yoz and xoy planes are below minus35 dB which meetthe design target Note that 120579 coordinate is measured fromxoy plane to 119911 axis and 120593 coordinate is measured from +x axisto 119910 axis in this paper

34 A Microstrip Antenna Array with 37 times 9 Elementsover a Large Airplane Install the 37 times 9 microstrip patcharray that was analyzed in Section 33 over a large airplane(httpwwwscalecraftcom737aewandcaspx) as shown inFigure 7 The dimensions of the airplane are 55m times 476m times

158m The array elements are very thin compared with the

6 International Journal of Antennas and Propagation

26648

21092

15537

99811

44256

minus113

minus66855

minus12241

minus17797

minus23352

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

26401

20846

1529

97347

41792

minus13764

minus69319

minus12487

minus18043

minus23599

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

25902

20346

14791

92352

36796

minus1876

minus74315

minus12987

minus18543

minus24098

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(c)

26452

20897

15341

97854

42299

minus13257

minus68812

minus12437

minus17992

minus23548

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(d)

26458

20902

15347

97913

42357

minus13198

minus68754

minus12431

minus17986

minus23542

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(e)

25794

20239

14683

91277

35722

minus19834

minus7539

minus13095

minus1865

minus24206

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(f)

26527

20972

15416

98608

43053

minus12503

minus68059

minus12361

minus17917

minus23473

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(g)

26307

20752

15196

96406

4085

minus14705

minus70261

minus12582

minus18137

minus23693

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(h)

Figure 9 Continued

International Journal of Antennas and Propagation 7

2576220207146519095635401minus20155minus7571minus13127minus18682minus24238

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(i)

Figure 9 3D gain patterns of the nine installed positions of the array (a) Array at position 1 (b) Array at position 2 (c) Array at position 3(d) Array at position 4 (e) Array at position 5 (f) Array at position 6 (g) Array at position 7 (h) Array at position 8 (i) Array at position 9

47

8 5 2

9 6 3

1

z

x o

(a)

z

x o7 4

2

1

8 5

9 6 3

(b)

Figure 10 Two comparison ways for the 2D gain pattern

airplane platform and the airborne antenna array is a com-posite and multiscale problem Consider a realistic problemthat the microstrip antenna array installed on nine positionsof the airplane with different 119909 coordinates or different 119911coordinates as shown in Figure 8 the mainlobe direction isdirected towards the wing The operation frequency of theairborne array is 440MHz and the number of unknowns is308371

In this Section 1024 CPU cores are used and the directsolver is observed in solving this kind of problems to avoid theslow convergence of iterative CG type methods The matrixfilling time is about 4700 seconds and the matrix equationsolving time is about 18500 seconds The computed three-dimensional (3D) gain patterns of the antenna array installedon different positions of the airplane are shown in Figure 9To make the comparison more clearly the computed two-dimensional (2D) gain patterns of the antenna array installedon different positions of the airplane are compared in twoways as shown in Figure 10 One is the same 119909 coordinatesand different 119911 coordinates as shown in Figure 11 One is thedifferent 119909 coordinates and same 119911 coordinates as shown inFigure 12 Furthermore the patterns of the array alone arealso given for comparison

Form the comparison it is clearly seen that the gainpatterns of the airborne antenna array in the yoz planes(elevation planes) seriously deteriorate compared with theantenna array alone especially in the mainlobe regionAnd in contrast the patterns in the xoy plane (azimuth

plane) show slight changes From the comparisons resultsof the array installed on same 119909 coordinates and different 119911coordinates shown in Figure 11 one can see that with the 119911coordinates of the installation positions increasing (ie formposition 1 to 2 and 3 form position 4 to 5 and 6 and formposition 7 to 8 and 9) the mainlobe region of gain patterns(ie minus30∘ sim30∘ in the yoz plane) gets better mainly due to thereflection and diffraction effects of airplane wing becomingweak Furthermore the sidelobe region of gain patterns (ieminus150∘ sim minus30∘ in the yoz plane 115∘ sim120∘ in the xoy plane)becomes low with the installation height increasing Formthe comparisons results of the array installed on different 119909coordinates and same 119911 coordinates shown in Figure 12 wecan see that gain patterns (ie minus150∘ sim30∘ in the yoz plane)get better with the 119909 coordinates of the installation positionsincreasing (ie form position 1 to 4 and 7 form position 2 to5 and 8 and form position 3 to 6 and 9) However the overalldesign of airplane is a composite and multiscale problem itis necessary to take many factors into account such as theaerodynamics Therefore the array is installed not too highand not close to the head of airplane taking these factors intoconsideration the better installation position is position 5 inthis paper

Moreover the model of airborne array is a whole inpractical engineeringThe array is connected to airplane witha bracket which is installed below the array When the mainbeam of the array is direct to the wings of the airplane theeffect of the bracket can be neglected So in the simulations

8 International Journal of Antennas and Propagation

Array aloneArray at position 1

Array at position 2Array at position 3

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(a)

Array aloneArray at position 1

Array at position 2Array at position 3

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 4

Array at position 5Array at position 6

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(c)

Array aloneArray at position 4

Array at position 5Array at position 6

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 7

Array at position 8Array at position 9

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(e)

Array aloneArray at position 7

Array at position 8Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 11 2D gain patterns of the array installed on same 119909 coordinates and different 119911 coordinates (a) yoz plane of positions 1 2 and 3 (b)xoy plane of positions 1 2 and 3 (c) yoz plane of positions 4 5 and 6 (d) xoy plane of positions 4 5 and 6 (e) yoz plane of positions 7 8and 9 (f) xoy plane of positions 7 8 and 9

International Journal of Antennas and Propagation 9

Array aloneArray at position 1

Array at position 4Array at position 7

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(a)

Array aloneArray at position 1

Array at position 4Array at position 7

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 2

Array at position 5Array at position 8

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(c)

Array aloneArray at position 2

Array at position 5Array at position 8

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 3

Array at position 6Array at position 9

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(e)

Array aloneArray at position 3

Array at position 6Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 12 2D gain patterns of the array installed on different 119909 coordinates and same 119911 coordinates (a) yoz plane of positions 1 4 and 7 (b)xoy plane of positions 1 4 and 7 (c) yoz plane of positions 2 5 and 8 (d) xoy plane of positions 2 5 and 8 (e) yoz plane of positions 3 6 and9 (f) xoy plane of position 3 6 and 9

10 International Journal of Antennas and Propagation

zx

yo

(a)

Microstrip array

x

y

o

(b)

x

y

o

(c)

Figure 13 Simulation models (a) microstrip array (b) airborne array (c) the position of the array

z

x yo

1039

48344

minus62767

minus072116

minus17388

minus11832

minus22943

minus28499

minus34054

minus3961

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

z

x yo

93514

37959

minus17597

minus73152

minus12871

minus18426

minus23982

minus29537

minus35093

minus40649

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

Figure 14 3D gain patterns of the array alone and airborne array (a) array alone (b) airborne array

of the airborne array in this section the bracket is removedfrom the simulation model

35 A Microstrip Antenna Array with 9 times 1 Elements stick onanAirplane To demonstrate the ability of themethod to dealwith the problem of an array connected to the airplane amicrostrip antenna array with 9 times 1 elements sticks on thesurface of a large airplane is considered in this section [15]The model of the array and the airborne array are shown inFigures 13(a) and 13(b) and the position of the array is given

in Figure 13(c) The parameters of the array and the airplaneare same to the previous sections The computed 3D gainpatterns of the array alone and airborne array are shown inFigure 14 The computed 2D gain patterns of the array aloneand airborne array are compared in Figure 15

Form the comparisons we can see that when the arraysticks on the airplane the back gain of the array is signifi-cantly reduced while the front gain shows slight changes Itis mainly due to that when the array is stuck on the airplaneit can be considered as enlarging the ground of the array sothe result in Figure 15 is reasonable

International Journal of Antennas and Propagation 11

0

10

0

30

60

90

120

150180

210

240

270

300

330

0

10

Array aloneAirborne array

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus40

minus30

minus20

minus10

Figure 15 Comparison result of 2D gain patterns in xoz plane

4 Conclusion

An efficient parallel higher-order MoM is developed to solveextremely large and complex electromagnetics problemsTheparallel strategies based on block matrices and the higher-order basis functions make it possible to solve airborneantenna array problems using rigorousMoM and the simula-tion time of one installed position is about seven hours using1024 CPU cores Numerical results confirm the accuracy andefficiency of the method and its applicability to the analysisof radiation from airborne platform systems with complexantennas

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)(2012AA01A308) the NSFC (61301069 and 61072019) andthe Program for New Century Excellent Talents in Universityof China (NCET-13-0949) the Project with contract no2013KJXX-67 The computational resources utilized in thisresearch are provided by Shanghai Supercomputer Center

References

[1] R F Harrington ldquoField computation by moment methodsrdquo inIEEE Series on Electromagnetic Waves IEEE New York NYUSA 1993

[2] Y Zhang and T K Sarkar Parallel Solution of Integral EquationBased EM Problems in the Frequency Domain John Wiley ampSons Hoboken NJ USA 2009

[3] 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

[4] Y Zhang T K Sarkar X Zhao et al Higher Order Basis BasedIntegral Equation Solver (HOBBIES) John Wiley Hoboken NJUSA 2012

[5] Y Zhang X Zhao D G Donoro and T K Sarkar ldquoParallelizedhybrid method with higher-order MoM And PO for analysisof phased array antennas on electrically large platformsrdquo IEEETransactions on Antennas and Propagation vol 58 no 12 pp4110ndash4115 2010

[6] X-W Zhao Y Zhang H-W Zhang et al ldquoParallel MoM-POmethod with out-of-core technique for analysis of complexarrays on electrically large platformsrdquo Progress in Electromag-netics Research vol 108 pp 1ndash21 2010

[7] G Hennigan and S Castillo ldquoOpen region electromagneticfinite-element scattering calculations in anisotropic media onparallel computersrdquo in IEEE Antennas and Propagation SocietyInternational Symposium vol 1 pp 484ndash487 July 1999

[8] P Yla-OijalaM Taskinen and S Jarvenpaa ldquoAnalysis of surfaceintegral equations in electromagnetic scattering and radiationproblemsrdquo Engineering Analysis with Boundary Elements vol32 no 3 pp 196ndash209 2008

[9] P Yla-Oijala M Taskinen and J Sarvas ldquoSurface integralequation method for general composite metallic and dielec-tric structures with junctionsrdquo in Progress in ElectromagneticsResearch vol 52 pp 81ndash108 2005

[10] K Yang W T Sheng Z Y Zhu and M S Tong ldquoElectromag-netic analysis for inhomogeneous interconnect and packagingstructures based on volume-surface integral equationsrdquo IEEETransactions on Components Packaging and ManufacturingTechnology vol 3 no 8 pp 1364ndash1371 2013

[11] R F Harrington ldquoBoundary integral formulations for homoge-nous material bodiesrdquo Journal of Electromagnetic Waves andApplications vol 3 no 1 pp 1ndash15 1989

[12] D Z Ding Y Shi Z N Jiang and R S Chen ldquoAugmentedEFIE with adaptive cross approximation algorithm for analysisof electromagnetic problemsrdquo International Journal of Antennasand Propagation vol 2013 Article ID 487276 9 pages 2013

[13] J L Volakis and K Sertel Integral Equation Methods forElectromagnetics SciTech Raleigh NC USA 2012

[14] Y Yan Y Zhang C-H Liang H Zhao and D Garcıa-DonoroldquoRCS computation by parallel MoM using higher-order basisfunctionsrdquo International Journal of Antennas and Propagationvol 2012 Article ID 745893 9 pages 2012

[15] Z Yang S Jiang X Zhao Y Zhang and H Qu ldquoRadiationof an airborne patch array antennardquo in Proceedings of the IETInternational Radar Conference Institution of Engineering andTechnology Xirsquoan China April 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 International Journal of Antennas and Propagation

26648

21092

15537

99811

44256

minus113

minus66855

minus12241

minus17797

minus23352

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

26401

20846

1529

97347

41792

minus13764

minus69319

minus12487

minus18043

minus23599

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

25902

20346

14791

92352

36796

minus1876

minus74315

minus12987

minus18543

minus24098

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(c)

26452

20897

15341

97854

42299

minus13257

minus68812

minus12437

minus17992

minus23548

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(d)

26458

20902

15347

97913

42357

minus13198

minus68754

minus12431

minus17986

minus23542

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(e)

25794

20239

14683

91277

35722

minus19834

minus7539

minus13095

minus1865

minus24206

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(f)

26527

20972

15416

98608

43053

minus12503

minus68059

minus12361

minus17917

minus23473

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(g)

26307

20752

15196

96406

4085

minus14705

minus70261

minus12582

minus18137

minus23693

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(h)

Figure 9 Continued

International Journal of Antennas and Propagation 7

2576220207146519095635401minus20155minus7571minus13127minus18682minus24238

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(i)

Figure 9 3D gain patterns of the nine installed positions of the array (a) Array at position 1 (b) Array at position 2 (c) Array at position 3(d) Array at position 4 (e) Array at position 5 (f) Array at position 6 (g) Array at position 7 (h) Array at position 8 (i) Array at position 9

47

8 5 2

9 6 3

1

z

x o

(a)

z

x o7 4

2

1

8 5

9 6 3

(b)

Figure 10 Two comparison ways for the 2D gain pattern

airplane platform and the airborne antenna array is a com-posite and multiscale problem Consider a realistic problemthat the microstrip antenna array installed on nine positionsof the airplane with different 119909 coordinates or different 119911coordinates as shown in Figure 8 the mainlobe direction isdirected towards the wing The operation frequency of theairborne array is 440MHz and the number of unknowns is308371

In this Section 1024 CPU cores are used and the directsolver is observed in solving this kind of problems to avoid theslow convergence of iterative CG type methods The matrixfilling time is about 4700 seconds and the matrix equationsolving time is about 18500 seconds The computed three-dimensional (3D) gain patterns of the antenna array installedon different positions of the airplane are shown in Figure 9To make the comparison more clearly the computed two-dimensional (2D) gain patterns of the antenna array installedon different positions of the airplane are compared in twoways as shown in Figure 10 One is the same 119909 coordinatesand different 119911 coordinates as shown in Figure 11 One is thedifferent 119909 coordinates and same 119911 coordinates as shown inFigure 12 Furthermore the patterns of the array alone arealso given for comparison

Form the comparison it is clearly seen that the gainpatterns of the airborne antenna array in the yoz planes(elevation planes) seriously deteriorate compared with theantenna array alone especially in the mainlobe regionAnd in contrast the patterns in the xoy plane (azimuth

plane) show slight changes From the comparisons resultsof the array installed on same 119909 coordinates and different 119911coordinates shown in Figure 11 one can see that with the 119911coordinates of the installation positions increasing (ie formposition 1 to 2 and 3 form position 4 to 5 and 6 and formposition 7 to 8 and 9) the mainlobe region of gain patterns(ie minus30∘ sim30∘ in the yoz plane) gets better mainly due to thereflection and diffraction effects of airplane wing becomingweak Furthermore the sidelobe region of gain patterns (ieminus150∘ sim minus30∘ in the yoz plane 115∘ sim120∘ in the xoy plane)becomes low with the installation height increasing Formthe comparisons results of the array installed on different 119909coordinates and same 119911 coordinates shown in Figure 12 wecan see that gain patterns (ie minus150∘ sim30∘ in the yoz plane)get better with the 119909 coordinates of the installation positionsincreasing (ie form position 1 to 4 and 7 form position 2 to5 and 8 and form position 3 to 6 and 9) However the overalldesign of airplane is a composite and multiscale problem itis necessary to take many factors into account such as theaerodynamics Therefore the array is installed not too highand not close to the head of airplane taking these factors intoconsideration the better installation position is position 5 inthis paper

Moreover the model of airborne array is a whole inpractical engineeringThe array is connected to airplane witha bracket which is installed below the array When the mainbeam of the array is direct to the wings of the airplane theeffect of the bracket can be neglected So in the simulations

8 International Journal of Antennas and Propagation

Array aloneArray at position 1

Array at position 2Array at position 3

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(a)

Array aloneArray at position 1

Array at position 2Array at position 3

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 4

Array at position 5Array at position 6

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(c)

Array aloneArray at position 4

Array at position 5Array at position 6

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 7

Array at position 8Array at position 9

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(e)

Array aloneArray at position 7

Array at position 8Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 11 2D gain patterns of the array installed on same 119909 coordinates and different 119911 coordinates (a) yoz plane of positions 1 2 and 3 (b)xoy plane of positions 1 2 and 3 (c) yoz plane of positions 4 5 and 6 (d) xoy plane of positions 4 5 and 6 (e) yoz plane of positions 7 8and 9 (f) xoy plane of positions 7 8 and 9

International Journal of Antennas and Propagation 9

Array aloneArray at position 1

Array at position 4Array at position 7

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(a)

Array aloneArray at position 1

Array at position 4Array at position 7

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 2

Array at position 5Array at position 8

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(c)

Array aloneArray at position 2

Array at position 5Array at position 8

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 3

Array at position 6Array at position 9

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(e)

Array aloneArray at position 3

Array at position 6Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 12 2D gain patterns of the array installed on different 119909 coordinates and same 119911 coordinates (a) yoz plane of positions 1 4 and 7 (b)xoy plane of positions 1 4 and 7 (c) yoz plane of positions 2 5 and 8 (d) xoy plane of positions 2 5 and 8 (e) yoz plane of positions 3 6 and9 (f) xoy plane of position 3 6 and 9

10 International Journal of Antennas and Propagation

zx

yo

(a)

Microstrip array

x

y

o

(b)

x

y

o

(c)

Figure 13 Simulation models (a) microstrip array (b) airborne array (c) the position of the array

z

x yo

1039

48344

minus62767

minus072116

minus17388

minus11832

minus22943

minus28499

minus34054

minus3961

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

z

x yo

93514

37959

minus17597

minus73152

minus12871

minus18426

minus23982

minus29537

minus35093

minus40649

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

Figure 14 3D gain patterns of the array alone and airborne array (a) array alone (b) airborne array

of the airborne array in this section the bracket is removedfrom the simulation model

35 A Microstrip Antenna Array with 9 times 1 Elements stick onanAirplane To demonstrate the ability of themethod to dealwith the problem of an array connected to the airplane amicrostrip antenna array with 9 times 1 elements sticks on thesurface of a large airplane is considered in this section [15]The model of the array and the airborne array are shown inFigures 13(a) and 13(b) and the position of the array is given

in Figure 13(c) The parameters of the array and the airplaneare same to the previous sections The computed 3D gainpatterns of the array alone and airborne array are shown inFigure 14 The computed 2D gain patterns of the array aloneand airborne array are compared in Figure 15

Form the comparisons we can see that when the arraysticks on the airplane the back gain of the array is signifi-cantly reduced while the front gain shows slight changes Itis mainly due to that when the array is stuck on the airplaneit can be considered as enlarging the ground of the array sothe result in Figure 15 is reasonable

International Journal of Antennas and Propagation 11

0

10

0

30

60

90

120

150180

210

240

270

300

330

0

10

Array aloneAirborne array

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus40

minus30

minus20

minus10

Figure 15 Comparison result of 2D gain patterns in xoz plane

4 Conclusion

An efficient parallel higher-order MoM is developed to solveextremely large and complex electromagnetics problemsTheparallel strategies based on block matrices and the higher-order basis functions make it possible to solve airborneantenna array problems using rigorousMoM and the simula-tion time of one installed position is about seven hours using1024 CPU cores Numerical results confirm the accuracy andefficiency of the method and its applicability to the analysisof radiation from airborne platform systems with complexantennas

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)(2012AA01A308) the NSFC (61301069 and 61072019) andthe Program for New Century Excellent Talents in Universityof China (NCET-13-0949) the Project with contract no2013KJXX-67 The computational resources utilized in thisresearch are provided by Shanghai Supercomputer Center

References

[1] R F Harrington ldquoField computation by moment methodsrdquo inIEEE Series on Electromagnetic Waves IEEE New York NYUSA 1993

[2] Y Zhang and T K Sarkar Parallel Solution of Integral EquationBased EM Problems in the Frequency Domain John Wiley ampSons Hoboken NJ USA 2009

[3] 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

[4] Y Zhang T K Sarkar X Zhao et al Higher Order Basis BasedIntegral Equation Solver (HOBBIES) John Wiley Hoboken NJUSA 2012

[5] Y Zhang X Zhao D G Donoro and T K Sarkar ldquoParallelizedhybrid method with higher-order MoM And PO for analysisof phased array antennas on electrically large platformsrdquo IEEETransactions on Antennas and Propagation vol 58 no 12 pp4110ndash4115 2010

[6] X-W Zhao Y Zhang H-W Zhang et al ldquoParallel MoM-POmethod with out-of-core technique for analysis of complexarrays on electrically large platformsrdquo Progress in Electromag-netics Research vol 108 pp 1ndash21 2010

[7] G Hennigan and S Castillo ldquoOpen region electromagneticfinite-element scattering calculations in anisotropic media onparallel computersrdquo in IEEE Antennas and Propagation SocietyInternational Symposium vol 1 pp 484ndash487 July 1999

[8] P Yla-OijalaM Taskinen and S Jarvenpaa ldquoAnalysis of surfaceintegral equations in electromagnetic scattering and radiationproblemsrdquo Engineering Analysis with Boundary Elements vol32 no 3 pp 196ndash209 2008

[9] P Yla-Oijala M Taskinen and J Sarvas ldquoSurface integralequation method for general composite metallic and dielec-tric structures with junctionsrdquo in Progress in ElectromagneticsResearch vol 52 pp 81ndash108 2005

[10] K Yang W T Sheng Z Y Zhu and M S Tong ldquoElectromag-netic analysis for inhomogeneous interconnect and packagingstructures based on volume-surface integral equationsrdquo IEEETransactions on Components Packaging and ManufacturingTechnology vol 3 no 8 pp 1364ndash1371 2013

[11] R F Harrington ldquoBoundary integral formulations for homoge-nous material bodiesrdquo Journal of Electromagnetic Waves andApplications vol 3 no 1 pp 1ndash15 1989

[12] D Z Ding Y Shi Z N Jiang and R S Chen ldquoAugmentedEFIE with adaptive cross approximation algorithm for analysisof electromagnetic problemsrdquo International Journal of Antennasand Propagation vol 2013 Article ID 487276 9 pages 2013

[13] J L Volakis and K Sertel Integral Equation Methods forElectromagnetics SciTech Raleigh NC USA 2012

[14] Y Yan Y Zhang C-H Liang H Zhao and D Garcıa-DonoroldquoRCS computation by parallel MoM using higher-order basisfunctionsrdquo International Journal of Antennas and Propagationvol 2012 Article ID 745893 9 pages 2012

[15] Z Yang S Jiang X Zhao Y Zhang and H Qu ldquoRadiationof an airborne patch array antennardquo in Proceedings of the IETInternational Radar Conference Institution of Engineering andTechnology Xirsquoan China April 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 7

2576220207146519095635401minus20155minus7571minus13127minus18682minus24238

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(i)

Figure 9 3D gain patterns of the nine installed positions of the array (a) Array at position 1 (b) Array at position 2 (c) Array at position 3(d) Array at position 4 (e) Array at position 5 (f) Array at position 6 (g) Array at position 7 (h) Array at position 8 (i) Array at position 9

47

8 5 2

9 6 3

1

z

x o

(a)

z

x o7 4

2

1

8 5

9 6 3

(b)

Figure 10 Two comparison ways for the 2D gain pattern

airplane platform and the airborne antenna array is a com-posite and multiscale problem Consider a realistic problemthat the microstrip antenna array installed on nine positionsof the airplane with different 119909 coordinates or different 119911coordinates as shown in Figure 8 the mainlobe direction isdirected towards the wing The operation frequency of theairborne array is 440MHz and the number of unknowns is308371

In this Section 1024 CPU cores are used and the directsolver is observed in solving this kind of problems to avoid theslow convergence of iterative CG type methods The matrixfilling time is about 4700 seconds and the matrix equationsolving time is about 18500 seconds The computed three-dimensional (3D) gain patterns of the antenna array installedon different positions of the airplane are shown in Figure 9To make the comparison more clearly the computed two-dimensional (2D) gain patterns of the antenna array installedon different positions of the airplane are compared in twoways as shown in Figure 10 One is the same 119909 coordinatesand different 119911 coordinates as shown in Figure 11 One is thedifferent 119909 coordinates and same 119911 coordinates as shown inFigure 12 Furthermore the patterns of the array alone arealso given for comparison

Form the comparison it is clearly seen that the gainpatterns of the airborne antenna array in the yoz planes(elevation planes) seriously deteriorate compared with theantenna array alone especially in the mainlobe regionAnd in contrast the patterns in the xoy plane (azimuth

plane) show slight changes From the comparisons resultsof the array installed on same 119909 coordinates and different 119911coordinates shown in Figure 11 one can see that with the 119911coordinates of the installation positions increasing (ie formposition 1 to 2 and 3 form position 4 to 5 and 6 and formposition 7 to 8 and 9) the mainlobe region of gain patterns(ie minus30∘ sim30∘ in the yoz plane) gets better mainly due to thereflection and diffraction effects of airplane wing becomingweak Furthermore the sidelobe region of gain patterns (ieminus150∘ sim minus30∘ in the yoz plane 115∘ sim120∘ in the xoy plane)becomes low with the installation height increasing Formthe comparisons results of the array installed on different 119909coordinates and same 119911 coordinates shown in Figure 12 wecan see that gain patterns (ie minus150∘ sim30∘ in the yoz plane)get better with the 119909 coordinates of the installation positionsincreasing (ie form position 1 to 4 and 7 form position 2 to5 and 8 and form position 3 to 6 and 9) However the overalldesign of airplane is a composite and multiscale problem itis necessary to take many factors into account such as theaerodynamics Therefore the array is installed not too highand not close to the head of airplane taking these factors intoconsideration the better installation position is position 5 inthis paper

Moreover the model of airborne array is a whole inpractical engineeringThe array is connected to airplane witha bracket which is installed below the array When the mainbeam of the array is direct to the wings of the airplane theeffect of the bracket can be neglected So in the simulations

8 International Journal of Antennas and Propagation

Array aloneArray at position 1

Array at position 2Array at position 3

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(a)

Array aloneArray at position 1

Array at position 2Array at position 3

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 4

Array at position 5Array at position 6

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(c)

Array aloneArray at position 4

Array at position 5Array at position 6

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 7

Array at position 8Array at position 9

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(e)

Array aloneArray at position 7

Array at position 8Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 11 2D gain patterns of the array installed on same 119909 coordinates and different 119911 coordinates (a) yoz plane of positions 1 2 and 3 (b)xoy plane of positions 1 2 and 3 (c) yoz plane of positions 4 5 and 6 (d) xoy plane of positions 4 5 and 6 (e) yoz plane of positions 7 8and 9 (f) xoy plane of positions 7 8 and 9

International Journal of Antennas and Propagation 9

Array aloneArray at position 1

Array at position 4Array at position 7

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(a)

Array aloneArray at position 1

Array at position 4Array at position 7

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 2

Array at position 5Array at position 8

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(c)

Array aloneArray at position 2

Array at position 5Array at position 8

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 3

Array at position 6Array at position 9

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(e)

Array aloneArray at position 3

Array at position 6Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 12 2D gain patterns of the array installed on different 119909 coordinates and same 119911 coordinates (a) yoz plane of positions 1 4 and 7 (b)xoy plane of positions 1 4 and 7 (c) yoz plane of positions 2 5 and 8 (d) xoy plane of positions 2 5 and 8 (e) yoz plane of positions 3 6 and9 (f) xoy plane of position 3 6 and 9

10 International Journal of Antennas and Propagation

zx

yo

(a)

Microstrip array

x

y

o

(b)

x

y

o

(c)

Figure 13 Simulation models (a) microstrip array (b) airborne array (c) the position of the array

z

x yo

1039

48344

minus62767

minus072116

minus17388

minus11832

minus22943

minus28499

minus34054

minus3961

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

z

x yo

93514

37959

minus17597

minus73152

minus12871

minus18426

minus23982

minus29537

minus35093

minus40649

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

Figure 14 3D gain patterns of the array alone and airborne array (a) array alone (b) airborne array

of the airborne array in this section the bracket is removedfrom the simulation model

35 A Microstrip Antenna Array with 9 times 1 Elements stick onanAirplane To demonstrate the ability of themethod to dealwith the problem of an array connected to the airplane amicrostrip antenna array with 9 times 1 elements sticks on thesurface of a large airplane is considered in this section [15]The model of the array and the airborne array are shown inFigures 13(a) and 13(b) and the position of the array is given

in Figure 13(c) The parameters of the array and the airplaneare same to the previous sections The computed 3D gainpatterns of the array alone and airborne array are shown inFigure 14 The computed 2D gain patterns of the array aloneand airborne array are compared in Figure 15

Form the comparisons we can see that when the arraysticks on the airplane the back gain of the array is signifi-cantly reduced while the front gain shows slight changes Itis mainly due to that when the array is stuck on the airplaneit can be considered as enlarging the ground of the array sothe result in Figure 15 is reasonable

International Journal of Antennas and Propagation 11

0

10

0

30

60

90

120

150180

210

240

270

300

330

0

10

Array aloneAirborne array

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus40

minus30

minus20

minus10

Figure 15 Comparison result of 2D gain patterns in xoz plane

4 Conclusion

An efficient parallel higher-order MoM is developed to solveextremely large and complex electromagnetics problemsTheparallel strategies based on block matrices and the higher-order basis functions make it possible to solve airborneantenna array problems using rigorousMoM and the simula-tion time of one installed position is about seven hours using1024 CPU cores Numerical results confirm the accuracy andefficiency of the method and its applicability to the analysisof radiation from airborne platform systems with complexantennas

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)(2012AA01A308) the NSFC (61301069 and 61072019) andthe Program for New Century Excellent Talents in Universityof China (NCET-13-0949) the Project with contract no2013KJXX-67 The computational resources utilized in thisresearch are provided by Shanghai Supercomputer Center

References

[1] R F Harrington ldquoField computation by moment methodsrdquo inIEEE Series on Electromagnetic Waves IEEE New York NYUSA 1993

[2] Y Zhang and T K Sarkar Parallel Solution of Integral EquationBased EM Problems in the Frequency Domain John Wiley ampSons Hoboken NJ USA 2009

[3] 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

[4] Y Zhang T K Sarkar X Zhao et al Higher Order Basis BasedIntegral Equation Solver (HOBBIES) John Wiley Hoboken NJUSA 2012

[5] Y Zhang X Zhao D G Donoro and T K Sarkar ldquoParallelizedhybrid method with higher-order MoM And PO for analysisof phased array antennas on electrically large platformsrdquo IEEETransactions on Antennas and Propagation vol 58 no 12 pp4110ndash4115 2010

[6] X-W Zhao Y Zhang H-W Zhang et al ldquoParallel MoM-POmethod with out-of-core technique for analysis of complexarrays on electrically large platformsrdquo Progress in Electromag-netics Research vol 108 pp 1ndash21 2010

[7] G Hennigan and S Castillo ldquoOpen region electromagneticfinite-element scattering calculations in anisotropic media onparallel computersrdquo in IEEE Antennas and Propagation SocietyInternational Symposium vol 1 pp 484ndash487 July 1999

[8] P Yla-OijalaM Taskinen and S Jarvenpaa ldquoAnalysis of surfaceintegral equations in electromagnetic scattering and radiationproblemsrdquo Engineering Analysis with Boundary Elements vol32 no 3 pp 196ndash209 2008

[9] P Yla-Oijala M Taskinen and J Sarvas ldquoSurface integralequation method for general composite metallic and dielec-tric structures with junctionsrdquo in Progress in ElectromagneticsResearch vol 52 pp 81ndash108 2005

[10] K Yang W T Sheng Z Y Zhu and M S Tong ldquoElectromag-netic analysis for inhomogeneous interconnect and packagingstructures based on volume-surface integral equationsrdquo IEEETransactions on Components Packaging and ManufacturingTechnology vol 3 no 8 pp 1364ndash1371 2013

[11] R F Harrington ldquoBoundary integral formulations for homoge-nous material bodiesrdquo Journal of Electromagnetic Waves andApplications vol 3 no 1 pp 1ndash15 1989

[12] D Z Ding Y Shi Z N Jiang and R S Chen ldquoAugmentedEFIE with adaptive cross approximation algorithm for analysisof electromagnetic problemsrdquo International Journal of Antennasand Propagation vol 2013 Article ID 487276 9 pages 2013

[13] J L Volakis and K Sertel Integral Equation Methods forElectromagnetics SciTech Raleigh NC USA 2012

[14] Y Yan Y Zhang C-H Liang H Zhao and D Garcıa-DonoroldquoRCS computation by parallel MoM using higher-order basisfunctionsrdquo International Journal of Antennas and Propagationvol 2012 Article ID 745893 9 pages 2012

[15] Z Yang S Jiang X Zhao Y Zhang and H Qu ldquoRadiationof an airborne patch array antennardquo in Proceedings of the IETInternational Radar Conference Institution of Engineering andTechnology Xirsquoan China April 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 International Journal of Antennas and Propagation

Array aloneArray at position 1

Array at position 2Array at position 3

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(a)

Array aloneArray at position 1

Array at position 2Array at position 3

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 4

Array at position 5Array at position 6

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(c)

Array aloneArray at position 4

Array at position 5Array at position 6

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 7

Array at position 8Array at position 9

0 60 120 180

010203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

(e)

Array aloneArray at position 7

Array at position 8Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 11 2D gain patterns of the array installed on same 119909 coordinates and different 119911 coordinates (a) yoz plane of positions 1 2 and 3 (b)xoy plane of positions 1 2 and 3 (c) yoz plane of positions 4 5 and 6 (d) xoy plane of positions 4 5 and 6 (e) yoz plane of positions 7 8and 9 (f) xoy plane of positions 7 8 and 9

International Journal of Antennas and Propagation 9

Array aloneArray at position 1

Array at position 4Array at position 7

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(a)

Array aloneArray at position 1

Array at position 4Array at position 7

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 2

Array at position 5Array at position 8

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(c)

Array aloneArray at position 2

Array at position 5Array at position 8

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 3

Array at position 6Array at position 9

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(e)

Array aloneArray at position 3

Array at position 6Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 12 2D gain patterns of the array installed on different 119909 coordinates and same 119911 coordinates (a) yoz plane of positions 1 4 and 7 (b)xoy plane of positions 1 4 and 7 (c) yoz plane of positions 2 5 and 8 (d) xoy plane of positions 2 5 and 8 (e) yoz plane of positions 3 6 and9 (f) xoy plane of position 3 6 and 9

10 International Journal of Antennas and Propagation

zx

yo

(a)

Microstrip array

x

y

o

(b)

x

y

o

(c)

Figure 13 Simulation models (a) microstrip array (b) airborne array (c) the position of the array

z

x yo

1039

48344

minus62767

minus072116

minus17388

minus11832

minus22943

minus28499

minus34054

minus3961

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

z

x yo

93514

37959

minus17597

minus73152

minus12871

minus18426

minus23982

minus29537

minus35093

minus40649

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

Figure 14 3D gain patterns of the array alone and airborne array (a) array alone (b) airborne array

of the airborne array in this section the bracket is removedfrom the simulation model

35 A Microstrip Antenna Array with 9 times 1 Elements stick onanAirplane To demonstrate the ability of themethod to dealwith the problem of an array connected to the airplane amicrostrip antenna array with 9 times 1 elements sticks on thesurface of a large airplane is considered in this section [15]The model of the array and the airborne array are shown inFigures 13(a) and 13(b) and the position of the array is given

in Figure 13(c) The parameters of the array and the airplaneare same to the previous sections The computed 3D gainpatterns of the array alone and airborne array are shown inFigure 14 The computed 2D gain patterns of the array aloneand airborne array are compared in Figure 15

Form the comparisons we can see that when the arraysticks on the airplane the back gain of the array is signifi-cantly reduced while the front gain shows slight changes Itis mainly due to that when the array is stuck on the airplaneit can be considered as enlarging the ground of the array sothe result in Figure 15 is reasonable

International Journal of Antennas and Propagation 11

0

10

0

30

60

90

120

150180

210

240

270

300

330

0

10

Array aloneAirborne array

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus40

minus30

minus20

minus10

Figure 15 Comparison result of 2D gain patterns in xoz plane

4 Conclusion

An efficient parallel higher-order MoM is developed to solveextremely large and complex electromagnetics problemsTheparallel strategies based on block matrices and the higher-order basis functions make it possible to solve airborneantenna array problems using rigorousMoM and the simula-tion time of one installed position is about seven hours using1024 CPU cores Numerical results confirm the accuracy andefficiency of the method and its applicability to the analysisof radiation from airborne platform systems with complexantennas

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)(2012AA01A308) the NSFC (61301069 and 61072019) andthe Program for New Century Excellent Talents in Universityof China (NCET-13-0949) the Project with contract no2013KJXX-67 The computational resources utilized in thisresearch are provided by Shanghai Supercomputer Center

References

[1] R F Harrington ldquoField computation by moment methodsrdquo inIEEE Series on Electromagnetic Waves IEEE New York NYUSA 1993

[2] Y Zhang and T K Sarkar Parallel Solution of Integral EquationBased EM Problems in the Frequency Domain John Wiley ampSons Hoboken NJ USA 2009

[3] 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

[4] Y Zhang T K Sarkar X Zhao et al Higher Order Basis BasedIntegral Equation Solver (HOBBIES) John Wiley Hoboken NJUSA 2012

[5] Y Zhang X Zhao D G Donoro and T K Sarkar ldquoParallelizedhybrid method with higher-order MoM And PO for analysisof phased array antennas on electrically large platformsrdquo IEEETransactions on Antennas and Propagation vol 58 no 12 pp4110ndash4115 2010

[6] X-W Zhao Y Zhang H-W Zhang et al ldquoParallel MoM-POmethod with out-of-core technique for analysis of complexarrays on electrically large platformsrdquo Progress in Electromag-netics Research vol 108 pp 1ndash21 2010

[7] G Hennigan and S Castillo ldquoOpen region electromagneticfinite-element scattering calculations in anisotropic media onparallel computersrdquo in IEEE Antennas and Propagation SocietyInternational Symposium vol 1 pp 484ndash487 July 1999

[8] P Yla-OijalaM Taskinen and S Jarvenpaa ldquoAnalysis of surfaceintegral equations in electromagnetic scattering and radiationproblemsrdquo Engineering Analysis with Boundary Elements vol32 no 3 pp 196ndash209 2008

[9] P Yla-Oijala M Taskinen and J Sarvas ldquoSurface integralequation method for general composite metallic and dielec-tric structures with junctionsrdquo in Progress in ElectromagneticsResearch vol 52 pp 81ndash108 2005

[10] K Yang W T Sheng Z Y Zhu and M S Tong ldquoElectromag-netic analysis for inhomogeneous interconnect and packagingstructures based on volume-surface integral equationsrdquo IEEETransactions on Components Packaging and ManufacturingTechnology vol 3 no 8 pp 1364ndash1371 2013

[11] R F Harrington ldquoBoundary integral formulations for homoge-nous material bodiesrdquo Journal of Electromagnetic Waves andApplications vol 3 no 1 pp 1ndash15 1989

[12] D Z Ding Y Shi Z N Jiang and R S Chen ldquoAugmentedEFIE with adaptive cross approximation algorithm for analysisof electromagnetic problemsrdquo International Journal of Antennasand Propagation vol 2013 Article ID 487276 9 pages 2013

[13] J L Volakis and K Sertel Integral Equation Methods forElectromagnetics SciTech Raleigh NC USA 2012

[14] Y Yan Y Zhang C-H Liang H Zhao and D Garcıa-DonoroldquoRCS computation by parallel MoM using higher-order basisfunctionsrdquo International Journal of Antennas and Propagationvol 2012 Article ID 745893 9 pages 2012

[15] Z Yang S Jiang X Zhao Y Zhang and H Qu ldquoRadiationof an airborne patch array antennardquo in Proceedings of the IETInternational Radar Conference Institution of Engineering andTechnology Xirsquoan China April 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 9

Array aloneArray at position 1

Array at position 4Array at position 7

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(a)

Array aloneArray at position 1

Array at position 4Array at position 7

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(b)

Array aloneArray at position 2

Array at position 5Array at position 8

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(c)

Array aloneArray at position 2

Array at position 5Array at position 8

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(d)

Array aloneArray at position 3

Array at position 6Array at position 9

0 60 120 180

10203040

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus50

minus60minus60minus180 minus120

120579 (deg)

0

(e)

Array aloneArray at position 3

Array at position 6Array at position 9

Gai

n (d

B)

0 6030 90 120 150 180 240210 270 300 330 360

0

10

20

30

minus10

minus20

minus30

minus40

minus50

Φ (deg)

(f)

Figure 12 2D gain patterns of the array installed on different 119909 coordinates and same 119911 coordinates (a) yoz plane of positions 1 4 and 7 (b)xoy plane of positions 1 4 and 7 (c) yoz plane of positions 2 5 and 8 (d) xoy plane of positions 2 5 and 8 (e) yoz plane of positions 3 6 and9 (f) xoy plane of position 3 6 and 9

10 International Journal of Antennas and Propagation

zx

yo

(a)

Microstrip array

x

y

o

(b)

x

y

o

(c)

Figure 13 Simulation models (a) microstrip array (b) airborne array (c) the position of the array

z

x yo

1039

48344

minus62767

minus072116

minus17388

minus11832

minus22943

minus28499

minus34054

minus3961

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

z

x yo

93514

37959

minus17597

minus73152

minus12871

minus18426

minus23982

minus29537

minus35093

minus40649

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

Figure 14 3D gain patterns of the array alone and airborne array (a) array alone (b) airborne array

of the airborne array in this section the bracket is removedfrom the simulation model

35 A Microstrip Antenna Array with 9 times 1 Elements stick onanAirplane To demonstrate the ability of themethod to dealwith the problem of an array connected to the airplane amicrostrip antenna array with 9 times 1 elements sticks on thesurface of a large airplane is considered in this section [15]The model of the array and the airborne array are shown inFigures 13(a) and 13(b) and the position of the array is given

in Figure 13(c) The parameters of the array and the airplaneare same to the previous sections The computed 3D gainpatterns of the array alone and airborne array are shown inFigure 14 The computed 2D gain patterns of the array aloneand airborne array are compared in Figure 15

Form the comparisons we can see that when the arraysticks on the airplane the back gain of the array is signifi-cantly reduced while the front gain shows slight changes Itis mainly due to that when the array is stuck on the airplaneit can be considered as enlarging the ground of the array sothe result in Figure 15 is reasonable

International Journal of Antennas and Propagation 11

0

10

0

30

60

90

120

150180

210

240

270

300

330

0

10

Array aloneAirborne array

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus40

minus30

minus20

minus10

Figure 15 Comparison result of 2D gain patterns in xoz plane

4 Conclusion

An efficient parallel higher-order MoM is developed to solveextremely large and complex electromagnetics problemsTheparallel strategies based on block matrices and the higher-order basis functions make it possible to solve airborneantenna array problems using rigorousMoM and the simula-tion time of one installed position is about seven hours using1024 CPU cores Numerical results confirm the accuracy andefficiency of the method and its applicability to the analysisof radiation from airborne platform systems with complexantennas

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)(2012AA01A308) the NSFC (61301069 and 61072019) andthe Program for New Century Excellent Talents in Universityof China (NCET-13-0949) the Project with contract no2013KJXX-67 The computational resources utilized in thisresearch are provided by Shanghai Supercomputer Center

References

[1] R F Harrington ldquoField computation by moment methodsrdquo inIEEE Series on Electromagnetic Waves IEEE New York NYUSA 1993

[2] Y Zhang and T K Sarkar Parallel Solution of Integral EquationBased EM Problems in the Frequency Domain John Wiley ampSons Hoboken NJ USA 2009

[3] 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

[4] Y Zhang T K Sarkar X Zhao et al Higher Order Basis BasedIntegral Equation Solver (HOBBIES) John Wiley Hoboken NJUSA 2012

[5] Y Zhang X Zhao D G Donoro and T K Sarkar ldquoParallelizedhybrid method with higher-order MoM And PO for analysisof phased array antennas on electrically large platformsrdquo IEEETransactions on Antennas and Propagation vol 58 no 12 pp4110ndash4115 2010

[6] X-W Zhao Y Zhang H-W Zhang et al ldquoParallel MoM-POmethod with out-of-core technique for analysis of complexarrays on electrically large platformsrdquo Progress in Electromag-netics Research vol 108 pp 1ndash21 2010

[7] G Hennigan and S Castillo ldquoOpen region electromagneticfinite-element scattering calculations in anisotropic media onparallel computersrdquo in IEEE Antennas and Propagation SocietyInternational Symposium vol 1 pp 484ndash487 July 1999

[8] P Yla-OijalaM Taskinen and S Jarvenpaa ldquoAnalysis of surfaceintegral equations in electromagnetic scattering and radiationproblemsrdquo Engineering Analysis with Boundary Elements vol32 no 3 pp 196ndash209 2008

[9] P Yla-Oijala M Taskinen and J Sarvas ldquoSurface integralequation method for general composite metallic and dielec-tric structures with junctionsrdquo in Progress in ElectromagneticsResearch vol 52 pp 81ndash108 2005

[10] K Yang W T Sheng Z Y Zhu and M S Tong ldquoElectromag-netic analysis for inhomogeneous interconnect and packagingstructures based on volume-surface integral equationsrdquo IEEETransactions on Components Packaging and ManufacturingTechnology vol 3 no 8 pp 1364ndash1371 2013

[11] R F Harrington ldquoBoundary integral formulations for homoge-nous material bodiesrdquo Journal of Electromagnetic Waves andApplications vol 3 no 1 pp 1ndash15 1989

[12] D Z Ding Y Shi Z N Jiang and R S Chen ldquoAugmentedEFIE with adaptive cross approximation algorithm for analysisof electromagnetic problemsrdquo International Journal of Antennasand Propagation vol 2013 Article ID 487276 9 pages 2013

[13] J L Volakis and K Sertel Integral Equation Methods forElectromagnetics SciTech Raleigh NC USA 2012

[14] Y Yan Y Zhang C-H Liang H Zhao and D Garcıa-DonoroldquoRCS computation by parallel MoM using higher-order basisfunctionsrdquo International Journal of Antennas and Propagationvol 2012 Article ID 745893 9 pages 2012

[15] Z Yang S Jiang X Zhao Y Zhang and H Qu ldquoRadiationof an airborne patch array antennardquo in Proceedings of the IETInternational Radar Conference Institution of Engineering andTechnology Xirsquoan China April 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 International Journal of Antennas and Propagation

zx

yo

(a)

Microstrip array

x

y

o

(b)

x

y

o

(c)

Figure 13 Simulation models (a) microstrip array (b) airborne array (c) the position of the array

z

x yo

1039

48344

minus62767

minus072116

minus17388

minus11832

minus22943

minus28499

minus34054

minus3961

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(a)

z

x yo

93514

37959

minus17597

minus73152

minus12871

minus18426

minus23982

minus29537

minus35093

minus40649

Contour fill of total gain (dB) gt Genno1 44000000000 MHz (dB)

(b)

Figure 14 3D gain patterns of the array alone and airborne array (a) array alone (b) airborne array

of the airborne array in this section the bracket is removedfrom the simulation model

35 A Microstrip Antenna Array with 9 times 1 Elements stick onanAirplane To demonstrate the ability of themethod to dealwith the problem of an array connected to the airplane amicrostrip antenna array with 9 times 1 elements sticks on thesurface of a large airplane is considered in this section [15]The model of the array and the airborne array are shown inFigures 13(a) and 13(b) and the position of the array is given

in Figure 13(c) The parameters of the array and the airplaneare same to the previous sections The computed 3D gainpatterns of the array alone and airborne array are shown inFigure 14 The computed 2D gain patterns of the array aloneand airborne array are compared in Figure 15

Form the comparisons we can see that when the arraysticks on the airplane the back gain of the array is signifi-cantly reduced while the front gain shows slight changes Itis mainly due to that when the array is stuck on the airplaneit can be considered as enlarging the ground of the array sothe result in Figure 15 is reasonable

International Journal of Antennas and Propagation 11

0

10

0

30

60

90

120

150180

210

240

270

300

330

0

10

Array aloneAirborne array

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus40

minus30

minus20

minus10

Figure 15 Comparison result of 2D gain patterns in xoz plane

4 Conclusion

An efficient parallel higher-order MoM is developed to solveextremely large and complex electromagnetics problemsTheparallel strategies based on block matrices and the higher-order basis functions make it possible to solve airborneantenna array problems using rigorousMoM and the simula-tion time of one installed position is about seven hours using1024 CPU cores Numerical results confirm the accuracy andefficiency of the method and its applicability to the analysisof radiation from airborne platform systems with complexantennas

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)(2012AA01A308) the NSFC (61301069 and 61072019) andthe Program for New Century Excellent Talents in Universityof China (NCET-13-0949) the Project with contract no2013KJXX-67 The computational resources utilized in thisresearch are provided by Shanghai Supercomputer Center

References

[1] R F Harrington ldquoField computation by moment methodsrdquo inIEEE Series on Electromagnetic Waves IEEE New York NYUSA 1993

[2] Y Zhang and T K Sarkar Parallel Solution of Integral EquationBased EM Problems in the Frequency Domain John Wiley ampSons Hoboken NJ USA 2009

[3] 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

[4] Y Zhang T K Sarkar X Zhao et al Higher Order Basis BasedIntegral Equation Solver (HOBBIES) John Wiley Hoboken NJUSA 2012

[5] Y Zhang X Zhao D G Donoro and T K Sarkar ldquoParallelizedhybrid method with higher-order MoM And PO for analysisof phased array antennas on electrically large platformsrdquo IEEETransactions on Antennas and Propagation vol 58 no 12 pp4110ndash4115 2010

[6] X-W Zhao Y Zhang H-W Zhang et al ldquoParallel MoM-POmethod with out-of-core technique for analysis of complexarrays on electrically large platformsrdquo Progress in Electromag-netics Research vol 108 pp 1ndash21 2010

[7] G Hennigan and S Castillo ldquoOpen region electromagneticfinite-element scattering calculations in anisotropic media onparallel computersrdquo in IEEE Antennas and Propagation SocietyInternational Symposium vol 1 pp 484ndash487 July 1999

[8] P Yla-OijalaM Taskinen and S Jarvenpaa ldquoAnalysis of surfaceintegral equations in electromagnetic scattering and radiationproblemsrdquo Engineering Analysis with Boundary Elements vol32 no 3 pp 196ndash209 2008

[9] P Yla-Oijala M Taskinen and J Sarvas ldquoSurface integralequation method for general composite metallic and dielec-tric structures with junctionsrdquo in Progress in ElectromagneticsResearch vol 52 pp 81ndash108 2005

[10] K Yang W T Sheng Z Y Zhu and M S Tong ldquoElectromag-netic analysis for inhomogeneous interconnect and packagingstructures based on volume-surface integral equationsrdquo IEEETransactions on Components Packaging and ManufacturingTechnology vol 3 no 8 pp 1364ndash1371 2013

[11] R F Harrington ldquoBoundary integral formulations for homoge-nous material bodiesrdquo Journal of Electromagnetic Waves andApplications vol 3 no 1 pp 1ndash15 1989

[12] D Z Ding Y Shi Z N Jiang and R S Chen ldquoAugmentedEFIE with adaptive cross approximation algorithm for analysisof electromagnetic problemsrdquo International Journal of Antennasand Propagation vol 2013 Article ID 487276 9 pages 2013

[13] J L Volakis and K Sertel Integral Equation Methods forElectromagnetics SciTech Raleigh NC USA 2012

[14] Y Yan Y Zhang C-H Liang H Zhao and D Garcıa-DonoroldquoRCS computation by parallel MoM using higher-order basisfunctionsrdquo International Journal of Antennas and Propagationvol 2012 Article ID 745893 9 pages 2012

[15] Z Yang S Jiang X Zhao Y Zhang and H Qu ldquoRadiationof an airborne patch array antennardquo in Proceedings of the IETInternational Radar Conference Institution of Engineering andTechnology Xirsquoan China April 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 11

0

10

0

30

60

90

120

150180

210

240

270

300

330

0

10

Array aloneAirborne array

Gai

n (d

B)

minus10

minus20

minus30

minus40

minus40

minus30

minus20

minus10

Figure 15 Comparison result of 2D gain patterns in xoz plane

4 Conclusion

An efficient parallel higher-order MoM is developed to solveextremely large and complex electromagnetics problemsTheparallel strategies based on block matrices and the higher-order basis functions make it possible to solve airborneantenna array problems using rigorousMoM and the simula-tion time of one installed position is about seven hours using1024 CPU cores Numerical results confirm the accuracy andefficiency of the method and its applicability to the analysisof radiation from airborne platform systems with complexantennas

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)(2012AA01A308) the NSFC (61301069 and 61072019) andthe Program for New Century Excellent Talents in Universityof China (NCET-13-0949) the Project with contract no2013KJXX-67 The computational resources utilized in thisresearch are provided by Shanghai Supercomputer Center

References

[1] R F Harrington ldquoField computation by moment methodsrdquo inIEEE Series on Electromagnetic Waves IEEE New York NYUSA 1993

[2] Y Zhang and T K Sarkar Parallel Solution of Integral EquationBased EM Problems in the Frequency Domain John Wiley ampSons Hoboken NJ USA 2009

[3] 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

[4] Y Zhang T K Sarkar X Zhao et al Higher Order Basis BasedIntegral Equation Solver (HOBBIES) John Wiley Hoboken NJUSA 2012

[5] Y Zhang X Zhao D G Donoro and T K Sarkar ldquoParallelizedhybrid method with higher-order MoM And PO for analysisof phased array antennas on electrically large platformsrdquo IEEETransactions on Antennas and Propagation vol 58 no 12 pp4110ndash4115 2010

[6] X-W Zhao Y Zhang H-W Zhang et al ldquoParallel MoM-POmethod with out-of-core technique for analysis of complexarrays on electrically large platformsrdquo Progress in Electromag-netics Research vol 108 pp 1ndash21 2010

[7] G Hennigan and S Castillo ldquoOpen region electromagneticfinite-element scattering calculations in anisotropic media onparallel computersrdquo in IEEE Antennas and Propagation SocietyInternational Symposium vol 1 pp 484ndash487 July 1999

[8] P Yla-OijalaM Taskinen and S Jarvenpaa ldquoAnalysis of surfaceintegral equations in electromagnetic scattering and radiationproblemsrdquo Engineering Analysis with Boundary Elements vol32 no 3 pp 196ndash209 2008

[9] P Yla-Oijala M Taskinen and J Sarvas ldquoSurface integralequation method for general composite metallic and dielec-tric structures with junctionsrdquo in Progress in ElectromagneticsResearch vol 52 pp 81ndash108 2005

[10] K Yang W T Sheng Z Y Zhu and M S Tong ldquoElectromag-netic analysis for inhomogeneous interconnect and packagingstructures based on volume-surface integral equationsrdquo IEEETransactions on Components Packaging and ManufacturingTechnology vol 3 no 8 pp 1364ndash1371 2013

[11] R F Harrington ldquoBoundary integral formulations for homoge-nous material bodiesrdquo Journal of Electromagnetic Waves andApplications vol 3 no 1 pp 1ndash15 1989

[12] D Z Ding Y Shi Z N Jiang and R S Chen ldquoAugmentedEFIE with adaptive cross approximation algorithm for analysisof electromagnetic problemsrdquo International Journal of Antennasand Propagation vol 2013 Article ID 487276 9 pages 2013

[13] J L Volakis and K Sertel Integral Equation Methods forElectromagnetics SciTech Raleigh NC USA 2012

[14] Y Yan Y Zhang C-H Liang H Zhao and D Garcıa-DonoroldquoRCS computation by parallel MoM using higher-order basisfunctionsrdquo International Journal of Antennas and Propagationvol 2012 Article ID 745893 9 pages 2012

[15] Z Yang S Jiang X Zhao Y Zhang and H Qu ldquoRadiationof an airborne patch array antennardquo in Proceedings of the IETInternational Radar Conference Institution of Engineering andTechnology Xirsquoan China April 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of