research article the mlfma equipped with a hybrid tree ... · a pec rectangular plate with in...

Research ArticleThe MLFMA Equipped with a Hybrid Tree Structure forthe Multiscale EM Scattering

Wei-Bin Kong Hou-Xing Zhou Wei-Dong Li Guang Hua and Wei Hong

State Key Laboratory of Millimeter Waves Southeast University Nanjing 210096 China

Correspondence should be addressed to Hou-Xing Zhou hxzhouemfieldorg

Received 21 October 2013 Revised 8 January 2014 Accepted 8 January 2014 Published 19 February 2014

Academic Editor Ladislau Matekovits

Copyright copy 2014 Wei-Bin Kong et al This 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

We present an efficient strategy for reducing the memory requirement for the near-field matrix in the multilevel fast multipolealgorithm (MLFMA) for solving multiscale electromagnetic (EM) scattering problems A multiscale problem can obviously lowerthe storage efficiency of the MLFMA for the near-field matrix This paper focuses on overcoming this shortcoming to a certainextent A hybrid tree structure for the MLFMA that possesses two kinds of bottom-layer boxes with different edge sizes will bebuilt to significantly reduce the memory requirement for the near-field matrix in the multiscale case compared with the single-tree-structure technique Several numerical examples are provided to demonstrate the efficiency of the proposed scheme in themultiscale EM scattering

1 Introduction

Themethod ofmoments (MoM) [1] has beenwidely used dueto its superior capability to handle arbitrarily shaped objectsin electromagnetic (EM) radiation and scattering problemsThe computational complexity of the MoM is 119874(1198733) for adirect solver like the LU and 119874(1198981198732) for an iterative solverlike the CG where 119873 is the number of unknowns and 119898 isthe iteration count

Because of its higher complexity the traditional MoMis not suitable for solving electrically large EM problemsThe multilevel fast multipole algorithm (MLFMA) [2ndash4]can reduce the computational complexity of a matrix-vectorproduct from 119874(119873

2) to 119874(119873 log119873) as well as the storage

complexity of the MoM matrix from 119874(1198732) to 119874(119873) Such

an advancement thanks to the application of the additiontheorem of the Greenrsquos function and the introduction ofthe octree structure To further improve the computationalefficiency the MLFMA can be combined with parallel com-puting technology [5ndash11]

In EM simulation the Rao-Wilton-Glisson (RWG) basisfunctions [12] are frequently selected to build a MLFMAmodel for an electrically large object For this the surfaceof the electrically large object is generally discretized with

about 10 elements per wavelength In such a case the octreewhose bottom-layer or finest-layer boxes have edge sizeof about 03 wavelength is more suitable for grouping allbasis functions However amultiscale problem can obviouslylower the storage efficiency of the MLFMA for the near-field matrix because some local regions of the object surfaceneed to be overmeshed to capture tiny geometry featuresleading to a very large near-field matrix if the single octreestructure is still used In order to take care of the fine meshparts smaller bottom-layer boxes seem to be consideredHowever the bottom-layer boxes with edge size of 015wavelength are not suitable for grouping those RWG basisfunctions with edge size of about 01 wavelength Besidesthe box edge size of 015 wavelength is already very close tothe subwavelength breakdown of the MLFMA (or the HF-MLFMA) [13ndash19] Introducing some assistive technology canalleviate this problem to a certain extent In [20] Pan etal combined the interpolative decomposition (ID) with theMLFMA to alleviate the multiscale problem In [21] Vikramet al combined the accelerated Cartesian expansion (ACE)with the MLFMA to alleviate the multiscale problem

In this paper we focus on overcoming the multiscaleproblem to a certain extent A hybrid tree structure for theMLFMA that possesses two kinds of bottom-layer boxes

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 281303 6 pageshttpdxdoiorg1011552014281303

2 International Journal of Antennas and Propagation

with different edge sizes will be built to significantly reducethe memory requirement for the near-field matrix in themultiscale case compared with the single-tree-structure tech-nique Compared with the previous methods [20 21] thismethod is particularly simple and easy to implement Theidea of this method germinated in our previous work [22]Several numerical examples are provided to demonstrate theefficiency of the proposed method in the multiscale EMscattering

This paper is organized as follows In Section 2 thehybrid tree structure (HTS) is built In Section 3 theMLFMAequipped with a HTS is presented In Section 4 numericalexamples are provided Finally the conclusion is given inSection 5 Besides in this paper 120582 always denotes the wave-length in the free space

2 The Hybrid Tree Structure

In the EM simulation using the RWG basis functions thesurface of an electrically large perfectly conducting objectis discretized with more than or equal to 10 elements perwavelength To build a MLFMA model an octree structurefor grouping all basis functionsmust be constructed For thisthe entire object is enclosed within a large box called 0-layerbox Then the 0-layer box is divided into its eight childrenboxes called 1-layer boxes Each 1-layer box is divided intoits eight children boxes called 2-layer boxes until the edgesize of the finest-layer boxes is about 03 wavelength [2ndash4]All basis functions are grouped into the corresponding finest-layer or bottom-layer boxes Only the nonempty boxes areretained at all levels In this way we build a hierarchy of theboxes for the entire object

The MLFMA splits a matrix-vector product into twoparts near-field interactions and far-field interactions Thenear-field interactions between adjacent boxes at the bottomlayer form the near-field matrix of the MLFMA Now theedge size of the bottom-layer boxes is selected to be 03 120582 Atthis time the number of the RWG basis functions with edgesize of about 01 120582 in a bottom-layer box is usually less than 50However if some local regions of the object are overmeshedfor example if the discrete sizes are less than or equal to005 120582 then the numbers of the RWG basis functions in somebottom-layer boxes can exceed 100 This is why a multiscaleproblem can lead to a very large near-field matrix Thus itis a meaningful work to build a proper tree structure for theMLFMA in the multiscale case

As elucidated in [18] in most cases the numericalerror produced by 02 120582 bottom-layer-box size is acceptableIf the box size is below 02 120582 the MLFMA could sufferfrom subwavelength breakdown [13ndash19] Hence 02 120582 is anappropriate choice for the bottom-layer-box size when thediscrete sizes of some local regions are less than or equalto 005 120582 According to the different surface discretization ahybrid of the octree structure and the 27-tree structure will bea better choice for theMLFMA in themultiscale case In sucha hybrid tree structure there are two kinds of bottom-layerboxes One has the edge size of 03 wavelength and anotherhas the edge size of 02 wavelength as shown in Figure 1

03 12058203 120582

02 120582

03120582

Figure 1 Two kinds of the bottom-layer boxes For example eachpurple or black box is a bottom-layer box in the octree structurewhile each blue box is a bottom-layer box in the 27-tree structure

The formation process of the HTS is closely related tothe surface discretization and difficult to be described witha few formulas but this process is very easy for a computerlanguage to describe Here our purpose is just to show howa hybrid tree structure is applied in the MLFMA Then thesituation in Figure 1 can be used as a simple example In thiscase the formation procedure can be outlined as follows

(1) The entire object is enclosed within a large box called0-layer box

(2) Then the 0-layer box is divided into its eight childrenboxes called 1-layer boxes Each 1-layer box is dividedinto its eight children boxes called 2-layer boxes untilthe edge size of the bottom-layer boxes is about 03 120582

(3) Group all basis functions into the correspondingcurrent bottom-layer boxes and record the number ofthe basis functions in each current bottom-layer box

(4) If one bottom-layer box contains more than 50 basisfunctions then mark its parent box Redivide thisparent box marked into 27 children boxes with edgesize of 02 120582

(5) After the process described by (4) for all bottom-layer boxes is finished some basis functions have tem-porarily lost the bottom-layer boxes containing themRegroup these basis functions into the correspondingnew bottom-layer boxes whose edge size is 02 120582

In this way we build a hierarchy of boxes for all basisfunctions Clearly the finest layer in the above hybrid treestructure is an imprecise concept In fact there are two kindsof the bottom-layer edge sizes that is 03 120582 and 02 120582 Figure 2shows such a tree structure for the MLFMA

International Journal of Antennas and Propagation 3

Figure 2 The hybrid tree structure in the 2D case

nnb

nnb

nnb nnb

nnb

nnb nnb nnb nnb nnb nnb

nnb

nnb nnb

nnb

nnb nnb

nnbnnb

nnb

nnb

nnb

nnb nnb

nnb nnb

nnb nnb nnb nnb

nnb

nb

nbob

nbnbnb

nb

nb

nnbnnbnnbnnb

nnb

nnb

nnb nnb nnb

nnb nnb

nnb

Figure 3 An observed box (ob) at the bottom layer and all its near-neighbours (nb) and all next near-neighbours (nnb)

3 The MLFMA Equipped with a HTS

Two boxes at the same layer are called near-neighbours ifthey have at least one common point In the following allboxes mentioned are not empty For convenience we alsoneed some notations

Definition 1 Let ldquoobrdquo be an observed box then 119873119899(ob) is a

collection that consists of ldquoobrdquo and all its near-neighboursHere the subscript 119899 in119873

119899(ob)means ldquonear-neighboursrdquo

Definition 2 Let ldquoobrdquo be an observed box then 119873119899119899(ob)

is a collection that consists of those children of all near-neighbours of ob parent box that are well separated fromob Here the subscript 119899119899 in 119873

119899119899(ob) means ldquonext near-

neighboursrdquo Figure 3 shows the members of 119873119899(ob) and

119873119899119899(ob) related to a bottom-layer box ob in the HTS

As usual the near-field matrix formed by the interactionsbetween the near-neighbours at the bottom layer is directly

calculated and stored in advance The product of the near-field matrix with a vector is always directly calculated

Assume that the RWG functions 997888119869119894and 997888119869

119895belong to the

next near-neighbours 119861119898and 119861

119899 respectively The location

vectors of the centers of 119861119898

and 119861119899are denoted by 997888119903

119898

and 997888119903119899 respectively The location vectors of any points on

supports of997888119869119894and997888119869

119895are denoted by997888119903

119894and997888119903

119895 respectively

Furthermore define 997888119903119894119898=997888119903119894minus997888119903119898 997888119903119899119895=997888119903119899minus997888119903119895 and

997888119903119898119899

=997888119903119898minus997888119903119899 Then the far-matrix elements formed by

the interactions between these two next near-neighbours forthe combined field integral equation (CFIE) are expressed as

119885C119894119895= intΩ

1198892997888119881119891119894119898

() sdot 120572119898119899() sdot

997888119881119904119899119895() (1)

where997888119881119891119894119898

() = 120574int119878119894

[119868 minus ] sdot997888119869119894(997888119903119894119898) 119890minus119895997888119896 sdot997888119903 119894119898119889119904

+ (1 minus 120574)int119878119894

[997888119869119894(997888119903119894119898) times 119899] times 119890

minus119895997888119896 sdot997888119903 119894119898119889119904

997888119881119904119899119895() = int

119878119895


(2)

120572119898119899() =

1198951205780

2119879119871(997888119896 997888119903119898119899) (3)

where 120574 is the combination factor and 0 le 120574 le 1 In this paper120574 is selected to be 05

The formula (1) can be decomposed into three processesthe aggregation process the translation process and thedisaggregation process For the sake of clarity an examplein Figure 4 is considered Compared with the conventionalMLFMA the MLFMA equipped with a HTS contains moredetails about the aggregation translation and disaggregationat the bottom layer

In Figure 4 the solid purple arrow line illustratesthe aggregation translation and disaggregation processesbetween two next near-neighbours at the bottom layer ofdifferent sizes whose centers are marked with 997888119903

119898and 997888119903

119899

respectively There is an indirect transmission of informa-tion between two basis functions marked with 997888119903

119894and 997888119903

119895

respectively That is 997888119903119895rarr

997888119903119899rarr

997888119903119898rarr

997888119903119894 Two solid

black arrow lines illustrate the aggregation translation anddisaggregation processes between two next near-neighboursat the bottom layer of the same sizes It should be noted thatthe higher layer calculation is the same as the conventionalMLFMA

4 Numerical Examples

In this section the conductor surface is meshed with trian-gular elements and RWG functions [12] are chosen as bothbasis and test functions where the edge sizes of the triangularelements are from 005 120582 to 01 120582 The singularity of the near-field matrix elements is dealt with through the approach in[23]


rm

ri

rj

rn

Figure 4 The process of aggregation translation and disaggrega-tion where each ldquo∙rdquo represents the center of a bottom-layer box

Figure 5 Mesh of a PEC sphere with radius 2 120582

41 A PEC Sphere with Radius 20 120582 A perfectly electricconducting (PEC) sphere with radius size 20 120582 is consideredin Figure 5The upper hemisphere surface is discretized withedge size of about 005 120582 and the lower hemisphere surfaceis discretized with edge size of about 01 120582 producing 47148unknowns

Figure 6 depicts the bistatic RCS curves obtained bythe Mie solution [24] the conventional MLFMA and theMLFMA equipped with a HTS The numbers of the bottom-layer boxes in the conventional MLFMA and the MLFMAequipped with a HTS are 784 and 1273 respectively asshown in Table 1 The memory requirement for both the fastmethods is 896796MB and 450876MB respectively andhence the new method leads to 5028 memory reduction

42 A PEC Rectangular Plate with Thin Slots Next weconsider the EM scattering from a PEC rectangular plate

0 20 40 60 80 100 120 140 160 180

0

5

10

15

20

25

30

35

Mie solutionMLFMAMLFMA equipped with a HTS

minus10

minus5

RCS120582

2(d

B)

120601in = 00 120579in = 00

120601s = 00

120579 (deg)

Figure 6 Bistatic RCS curves of the PEC sphere

Table 1 Comparison between the two MLFMAs in Section 41

MLFMA MLFMA equippedwith a HTS

Number of levels 4 4Number ofbottom-layer boxes 784 1273

Memory for thenear-field matrix 896796MB 450876MB





with thin slots as shown in Figure 7 The plate surface isdiscretized into a lot of triangles yielding 15153 unknownsBecause there are tiny geometric details the resultingmesh isa multiscale mesh and also an extremely nonuniform meshThe longest edge size is about 01 120582 while the shortest edgesize is about 005 120582

Figure 8 depicts the bistatic RCS curves obtained by theMoM the conventional MLFMA and theMLFMA equippedwith a HTS The numbers of the bottom-layer boxes in theconventional MLFMA and in the MLFMA equipped with aHTS are 378 and 473 respectively as shown in Table 2 Thememory requirement for both the fast methods is 10476MBand 8185MB respectively and hence the new method leadsto 219 memory reduction


8

4

x

y

x

y10

10

Figure 7 Mesh of a PEC rectangular plate with thin slots The dimension of the plate is 8 120582 by 4 120582 and that of each slot is 4 120582 by 005 120582

0 20 40 60 80 100 120 140 160 180

0

10

20

30

40

50

MoMMLFMAMLFMA equipped with a HTS

RCS120582

2(d

B)

minus50

minus40

minus30

minus20

minus10

120601in = 00 120579in = 00

120601s = 00

120579 (deg)

Figure 8 Bistatic RCS curves of the PEC plate with thin slots

43 A PEC Aircraft-Like Model The last structure is a PECaircraft-like model in Figure 9 The width is 13 120582 and thelength is 185 120582The head part of flyer is discretized with edgesize of about 005 120582 and the remaining part is discretizedwithedge size of about 01 120582 The number of unknowns is 54159

In Figure 10 the bistatic RCS results are plotted whichis obtained by the conventional MLFMA and the MLFMAequippedwith aHTSThenumbers of the bottom-layer boxesin the conventionalMLFMA and theMLFMA equipped withaHTS are 1793 and 1902 respectively as shown inTable 3Thememory requirement for both the fast methods is 49483MBand 40997MB respectively and hence the newmethod leadsto 1715 memory reduction

5 Conclusions

In the multiscale case a single octree structure for theMLFMA can produce a very large near-field matrix In thispaper in order to overcome this shortcoming to a certain

z

x

y

Figure 9 Mesh of the PEC aircraft-like model

0 20 40 60 80 100 120 140 160 180

05

101520253035

MLFMA MLFMA equipped with a HTS

RCS120582

2(d

B)

120601in = 00 120579in = 00

120601s = 00

120579 (deg)

minus15

minus10

minus5

Figure 10 Bistatic RCS curves of the PEC aircraft-like model

extent a hybrid tree structure scheme has been establishedIn this scheme there are two kinds of bottom-layer boxeswith different edge sizes With the hybrid tree structurethe memory requirement for the near-field matrix can besignificantly reduced in the multiscale case compared withthe MLFMA equipped with a single octree structure

It should be pointed out that to what extent the totalamount of storage is reduced by the proposed method is




Number of levels 7 7Number of thebottom-layer boxes 1793 1902


closely related to the proportion of the local regions with thetiny geometry features to the whole surface domainThis facthas been revealed by the three numerical examples providedin this paper Therefore in the general case to what exactextent the proposed method can reduce the total amount ofstorage can not be determined in advance

Conflict of Interests

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

Acknowledgment

This work was supported by the National Basic ResearchProgram of China (2013CB329002 2010CB327400 and2009CB320203)

References

[1] R F Harrington Field Computation by Moment Methods TheMacMillian New York NY USA 1968

[2] JM Song andWC Chew ldquoMultilevel fast-multipole algorithmfor solving combined field integral equations of electromagneticscatteringrdquo Microwave and Optical Technology Letters vol 10no 1 pp 14ndash19 1995

[3] J-M Song C-C Lu andW C Chew ldquoMultilevel fast multipolealgorithm for electromagnetic scattering by large complexobjectsrdquo IEEE Transactions on Antennas and Propagation vol45 no 10 pp 1488ndash1493 1997

[4] W C Chew J M Jin E Michielssen and J M Song Fast andEfficient Algorithms in Computational Electro-Magnetics ArtechHouse Boston Mass USA 2001

[5] S Velamparambil and W C Chew ldquoAnalysis and performanceof a distributed memory multilevel fast multipole algorithmrdquoIEEE Transactions on Antennas and Propagation vol 53 no 8pp 2719ndash2727 2005

[6] O Ergul and L Gurel ldquoEfficient parallelization of themultilevelfast multipole algorithm for the solution of large-scale scatter-ing problemsrdquo IEEE Transactions on Antennas and Propagationvol 56 no 8 pp 2335ndash2345 2008

[7] L Gurel and O Ergul ldquoHierarchical parallelization of themultilevel fast multipole algorithm (MLFMA)rdquo Proceedings ofthe IEEE vol 101 no 2 pp 332ndash341 2013

[8] X M Pan W C Pi M L Yang Z Peng and X Q ShengldquoSolving problems with over one billion unknowns by theMLFMArdquo IEEE Transaction on Antennas and Propagation vol60 no 5 pp 2571ndash2574 2012

[9] V Melapudi B Shanker S Seal and S Aluru ldquoA scalable paral-lel wideband MLFMA for efficient electromagnetic simulationson large scale clustersrdquo IEEE Transactions on Antennas andPropagation vol 59 no 7 pp 2565ndash2577 2011

[10] B Michiels J Fostier I Bogaert and D De Zutter ldquoWeak scal-ability analysis of the distributed-memory parallel MLFMArdquoIEEE Transaction on Antennas and Propagation vol 61 no 11pp 5567ndash5574 2013

[11] J Guan Y Su and J M Jin ldquoAn openMP-CUDA implementa-tion of multilevel fast multipole algorithm for electromagneticsimulation on multi-GPU computing systemsrdquo IEEE Transac-tion on Antennas and Propagation vol 61 no 7 pp 3607ndash36162013

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

[13] J S Zhao and W C Chew ldquoThree-dimensional multilevel fastmultipole algorithm from static to electrodynamicrdquoMicrowaveand Optical Technology Letters vol 26 no 1 pp 43ndash48 2000

[14] H Wallen S Jarvenpaa and P Yla-Oijala ldquoBroadband multi-level fast multipole algorithm for acoustic scattering problemsrdquoJournal of Computational Acoustics vol 14 no 4 pp 507ndash5262006

[15] H-W Wallen and J Sarvas ldquoTranslation procedures for broad-band MLFMArdquo Progress in Electromagnetics Research vol 55pp 47ndash78 2005

[16] H-W ChengW Y Crutchfield Z Gimbutas et al ldquoAwidebandfast multipole method for the Helmholtz equation in threedimensionsrdquo Journal of Computational Physics vol 216 no 1pp 300ndash325 2006

[17] L-J Jiang and W C Chew ldquoLow-frequency fast inhomo-geneous plane-wave algorithm (LF-FIPWA)rdquo Microwave andOptical Technology Letters vol 40 no 2 pp 117ndash122 2004

[18] L-J Jiang and W C Chew ldquoA mixed-form fast multipolealgorithmrdquo IEEE Transactions on Antennas and Propagationvol 53 no 12 pp 4145ndash4156 2005

[19] HWallen S Jarvenpaa P Yla-Oijala and J Sarvas ldquoBroadbandMuller-MLFMA for electromagnetic scattering by dielectricobjectsrdquo IEEE Transactions on Antennas and Propagation vol55 no 5 pp 1423ndash1430 2007

[20] X-M Pan J-G Wei Z Peng and X-Q Sheng ldquoA fast algo-rithm for multiscale electromagnetic problems using interpola-tive decomposition and multilevel fast multipole algorithmrdquoRadio Science vol 47 no 1 Article ID RS1011 2012

[21] M Vikram H Huang B Shanker and T Van ldquoA novelwideband FMM for fast integral equation solution of multiscaleproblems in electromagneticsrdquo IEEE Transactions on Antennasand Propagation vol 57 no 7 pp 2094ndash2104 2009

[22] W-B KongH-X ZhouW-D Li GHua andWHong ldquoUsinga hybrid tree structure in the MLFMA in the multiresolutioncaserdquo in International Conference on Computational Problem-Solving (ICCP rsquo12) pp 2ndash8 2012

[23] P Yla-Oijala andM Taskinen ldquoCalculation of CFIE impedancematrix elements with RWG and n times RWG functionsrdquo IEEETransactions on Antennas and Propagation vol 51 no 8 pp1837ndash1846 2003

[24] J J Bowman T B A Senior and P L E Uslenghi Electromag-netic Scattering by Simple Shapes Revised Hemisphere NewYork NY USA 1987

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Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



with different edge sizes will be built to significantly reducethe memory requirement for the near-field matrix in themultiscale case compared with the single-tree-structure tech-nique Compared with the previous methods [20 21] thismethod is particularly simple and easy to implement Theidea of this method germinated in our previous work [22]Several numerical examples are provided to demonstrate theefficiency of the proposed method in the multiscale EMscattering

This paper is organized as follows In Section 2 thehybrid tree structure (HTS) is built In Section 3 theMLFMAequipped with a HTS is presented In Section 4 numericalexamples are provided Finally the conclusion is given inSection 5 Besides in this paper 120582 always denotes the wave-length in the free space

2 The Hybrid Tree Structure

In the EM simulation using the RWG basis functions thesurface of an electrically large perfectly conducting objectis discretized with more than or equal to 10 elements perwavelength To build a MLFMA model an octree structurefor grouping all basis functionsmust be constructed For thisthe entire object is enclosed within a large box called 0-layerbox Then the 0-layer box is divided into its eight childrenboxes called 1-layer boxes Each 1-layer box is divided intoits eight children boxes called 2-layer boxes until the edgesize of the finest-layer boxes is about 03 wavelength [2ndash4]All basis functions are grouped into the corresponding finest-layer or bottom-layer boxes Only the nonempty boxes areretained at all levels In this way we build a hierarchy of theboxes for the entire object

The MLFMA splits a matrix-vector product into twoparts near-field interactions and far-field interactions Thenear-field interactions between adjacent boxes at the bottomlayer form the near-field matrix of the MLFMA Now theedge size of the bottom-layer boxes is selected to be 03 120582 Atthis time the number of the RWG basis functions with edgesize of about 01 120582 in a bottom-layer box is usually less than 50However if some local regions of the object are overmeshedfor example if the discrete sizes are less than or equal to005 120582 then the numbers of the RWG basis functions in somebottom-layer boxes can exceed 100 This is why a multiscaleproblem can lead to a very large near-field matrix Thus itis a meaningful work to build a proper tree structure for theMLFMA in the multiscale case

As elucidated in [18] in most cases the numericalerror produced by 02 120582 bottom-layer-box size is acceptableIf the box size is below 02 120582 the MLFMA could sufferfrom subwavelength breakdown [13ndash19] Hence 02 120582 is anappropriate choice for the bottom-layer-box size when thediscrete sizes of some local regions are less than or equalto 005 120582 According to the different surface discretization ahybrid of the octree structure and the 27-tree structure will bea better choice for theMLFMA in themultiscale case In sucha hybrid tree structure there are two kinds of bottom-layerboxes One has the edge size of 03 wavelength and anotherhas the edge size of 02 wavelength as shown in Figure 1

03 12058203 120582

02 120582

03120582

Figure 1 Two kinds of the bottom-layer boxes For example eachpurple or black box is a bottom-layer box in the octree structurewhile each blue box is a bottom-layer box in the 27-tree structure

The formation process of the HTS is closely related tothe surface discretization and difficult to be described witha few formulas but this process is very easy for a computerlanguage to describe Here our purpose is just to show howa hybrid tree structure is applied in the MLFMA Then thesituation in Figure 1 can be used as a simple example In thiscase the formation procedure can be outlined as follows

(1) The entire object is enclosed within a large box called0-layer box

(2) Then the 0-layer box is divided into its eight childrenboxes called 1-layer boxes Each 1-layer box is dividedinto its eight children boxes called 2-layer boxes untilthe edge size of the bottom-layer boxes is about 03 120582

(3) Group all basis functions into the correspondingcurrent bottom-layer boxes and record the number ofthe basis functions in each current bottom-layer box

(4) If one bottom-layer box contains more than 50 basisfunctions then mark its parent box Redivide thisparent box marked into 27 children boxes with edgesize of 02 120582

(5) After the process described by (4) for all bottom-layer boxes is finished some basis functions have tem-porarily lost the bottom-layer boxes containing themRegroup these basis functions into the correspondingnew bottom-layer boxes whose edge size is 02 120582

In this way we build a hierarchy of boxes for all basisfunctions Clearly the finest layer in the above hybrid treestructure is an imprecise concept In fact there are two kindsof the bottom-layer edge sizes that is 03 120582 and 02 120582 Figure 2shows such a tree structure for the MLFMA



nnb

nnb

nnb nnb

nnb


nnb

nnb nnb

nnb

nnb nnb

nnbnnb

nnb

nnb

nnb

nnb nnb

nnb nnb

nnb nnb nnb nnb

nnb

nb

nbob

nbnbnb

nb

nb

nnbnnbnnbnnb

nnb

nnb

nnb nnb nnb

nnb nnb

nnb















119895belong to the





119898


supports of997888119869119894and997888119869

119895are denoted by997888119903

119894and997888119903

119895 respectively


997888119903119898119899



119885C119894119895= intΩ

1198892997888119881119891119894119898

() sdot 120572119898119899() sdot

997888119881119904119899119895() (1)

where997888119881119891119894119898

() = 120574int119878119894


+ (1 minus 120574)int119878119894

[997888119869119894(997888119903119894119898) times 119899] times 119890

minus119895997888119896 sdot997888119903 119894119898119889119904

997888119881119904119899119895() = int

119878119895


(2)

120572119898119899() =

1198951205780

2119879119871(997888119896 997888119903119898119899) (3)




119898and 997888119903

119899


119894and 997888119903

119895


997888119903119899rarr

997888119903119898rarr

997888119903119894 Two solid





rm

ri

rj

rn






0 20 40 60 80 100 120 140 160 180

0

5

10

15

20

25

30

35


minus10

minus5

RCS120582

2(d

B)

120601in = 00 120579in = 00

120601s = 00

120579 (deg)













8

4

x

y

x

y10

10


0 20 40 60 80 100 120 140 160 180

0

10

20

30

40

50


RCS120582

2(d

B)

minus50

minus40

minus30

minus20

minus10

120601in = 00 120579in = 00

120601s = 00

120579 (deg)




5 Conclusions


z

x

y


0 20 40 60 80 100 120 140 160 180

05

101520253035


RCS120582

2(d

B)

120601in = 00 120579in = 00

120601s = 00

120579 (deg)

minus15

minus10

minus5












Acknowledgment


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











nnb

nnb

nnb nnb

nnb


nnb

nnb nnb

nnb

nnb nnb

nnbnnb

nnb

nnb

nnb

nnb nnb

nnb nnb

nnb nnb nnb nnb

nnb

nb

nbob

nbnbnb

nb

nb

nnbnnbnnbnnb

nnb

nnb

nnb nnb nnb

nnb nnb

nnb















119895belong to the





119898


supports of997888119869119894and997888119869

119895are denoted by997888119903

119894and997888119903

119895 respectively


997888119903119898119899



119885C119894119895= intΩ

1198892997888119881119891119894119898

() sdot 120572119898119899() sdot

997888119881119904119899119895() (1)

where997888119881119891119894119898

() = 120574int119878119894


+ (1 minus 120574)int119878119894

[997888119869119894(997888119903119894119898) times 119899] times 119890

minus119895997888119896 sdot997888119903 119894119898119889119904

997888119881119904119899119895() = int

119878119895


(2)

120572119898119899() =

1198951205780

2119879119871(997888119896 997888119903119898119899) (3)




119898and 997888119903

119899


119894and 997888119903

119895


997888119903119899rarr

997888119903119898rarr

997888119903119894 Two solid





rm

ri

rj

rn






0 20 40 60 80 100 120 140 160 180

0

5

10

15

20

25

30

35


minus10

minus5

RCS120582

2(d

B)

120601in = 00 120579in = 00

120601s = 00

120579 (deg)













8

4

x

y

x

y10

10


0 20 40 60 80 100 120 140 160 180

0

10

20

30

40

50


RCS120582

2(d

B)

minus50

minus40

minus30

minus20

minus10

120601in = 00 120579in = 00

120601s = 00

120579 (deg)




5 Conclusions


z

x

y


0 20 40 60 80 100 120 140 160 180

05

101520253035


RCS120582

2(d

B)

120601in = 00 120579in = 00

120601s = 00

120579 (deg)

minus15

minus10

minus5












Acknowledgment


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










rm

ri

rj

rn






0 20 40 60 80 100 120 140 160 180

0

5

10

15

20

25

30

35


minus10

minus5

RCS120582

2(d

B)

120601in = 00 120579in = 00

120601s = 00

120579 (deg)













8

4

x

y

x

y10

10


0 20 40 60 80 100 120 140 160 180

0

10

20

30

40

50


RCS120582

2(d

B)

minus50

minus40

minus30

minus20

minus10

120601in = 00 120579in = 00

120601s = 00

120579 (deg)




5 Conclusions


z

x

y


0 20 40 60 80 100 120 140 160 180

05

101520253035


RCS120582

2(d

B)

120601in = 00 120579in = 00

120601s = 00

120579 (deg)

minus15

minus10

minus5












Acknowledgment


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










8

4

x

y

x

y10

10


0 20 40 60 80 100 120 140 160 180

0

10

20

30

40

50


RCS120582

2(d

B)

minus50

minus40

minus30

minus20

minus10

120601in = 00 120579in = 00

120601s = 00

120579 (deg)




5 Conclusions


z

x

y


0 20 40 60 80 100 120 140 160 180

05

101520253035


RCS120582

2(d

B)

120601in = 00 120579in = 00

120601s = 00

120579 (deg)

minus15

minus10

minus5












Acknowledgment


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















Acknowledgment


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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