acceleration of augmented efie using multilevel complex...

Research ArticleAcceleration of Augmented EFIE Using Multilevel ComplexSource Beam Method

Lianning Song Yongpin Chen Ming Jiang Jun Hu and Zaiping Nie

Department of Microwave Engineering University of Electronic Science and Technology of China Chengdu Sichuan 610054 China

Correspondence should be addressed to Jun Hu hujunuestceducn

Received 20 April 2017 Accepted 12 June 2017 Published 13 July 2017

Academic Editor Song Guo

Copyright copy 2017 Lianning Song 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

The computation of the augmented electric field integral equation (A-EFIE) is accelerated by using the multilevel complex sourcebeam (MLCSB) method As an effective solution of the low-frequency problem A-EFIE includes both current and charge asunknowns to avoid the imbalance between the vector potentials and the scalar potentials in the conventional EFIE However denseimpedance submatrices are involved in the A-EFIE system and the computational cost becomes extremely high for problems witha large number of unknowns As an exact solution to Maxwellrsquos equations the complex source beam (CSB) method can be welltailored for A-EFIE to accelerate thematrix-vector products in an iterative solver Different from the commonly usedmultilevel fastmultipole algorithm (MLFMA) the CSBmethod is free from the problem of low-frequency breakdown In our implementation theexpansion operators of CSB are first derived for the vector potentials and the scalar potentials Consequently the aggregation anddisaggregation operators are introduced to form a multilevel algorithm to reduce the computational complexity The accuracy andefficiency of the proposedmethod are discussed in detail through a variety of numerical examples It is observed that the numericalerror of theMLCSB-AEFIE keeps constant for a broad frequency range indicating the good stability and scalability of the proposedmethod

1 Introduction

The method of moments (MoM) [1] for solving electric fieldintegral equation (EFIE) has received intensive study in theanalysis of electromagnetic (EM) radiation scattering andcircuit problems in recent years However the commonlyused EFIE with RWG basis function suffers from a ldquolow-frequency breakdownrdquo problem [2] when the frequencydecreases andor when the mesh is refined In this situa-tion EFIE is dominated by the scalar potentials while thecontribution from the vector potentials is overwhelmed dueto the finite machine precision Since the scalar potentialterm is singular the EFIE matrix system becomes extremelyill-conditioned and results in convergence issue when theiterative solver is applied

To address this difficulty several methods have beenproposed in the past years A widely used remedy is theloop-tree or loop-star decomposition [3] which decouplesthe magneto- and electrostatic physics However the extrac-tion of global loops is difficult for complex interconnectingstructures and the systemmatrix becomes ill-conditioned as

the frequency increases On the other hand the convergenceissue can be improved by a Calderon preconditioner [4]when iterative solvers are applied To avoid the involvedloop searching process alternative formulations have beenproposed by separating current and charge such as the cur-rent and charge integral equation (CCIE) [5] split potentialintegral equation (SPIE) [6 7] and augmented electric fieldintegral equation (A-EFIE) [8ndash11] In the A-EFIE methodthe contributions of the vector potentials and the scalarpotentials are separated by adding charge as extra unknownswhere the current continuity equation is explicitly enforcedFrequency scaling is then implemented to stabilize the systemequation The resulting impendence matrix of A-EFIE hasthe characteristics of a saddle point matrix and is well-conditioned at low frequencies Recently the perturbationmethod is introduced to enhance the accuracy of A-EFIE atextremely low frequencies [12] The augmented equivalenceprinciple algorithm (A-EPA) [13] and discontinuousGalerkin(DG) method [14] are combined with the A-EFIE for domaindecomposition problems

HindawiInternational Journal of Antennas and PropagationVolume 2017 Article ID 9640136 8 pageshttpsdoiorg10115520179640136

2 International Journal of Antennas and Propagation

As the number of unknowns increases a fast algorithmhas to be incorporated into the iterative solver to reduce theoperation complexity of the matrix-vector product (MVP)Such fast algorithms should be low-frequency stable forinstance the low-frequency multilevel fast multipole algo-rithm (LF-MLFMA) [15] the multilevel accelerated Carte-sian expansion algorithm (MLACEA) [16] the multileveladaptive cross-approximation (MLACA) algorithm [17] andfast Fourier transform (FFT) [18] are eligible candidates forsuch purpose Recently a complex source beam-method ofmoment (CSB-MoM) is proposed to accelerate the far-fieldinteractions of MoM at midfrequencies [19] The object isfirst divided into groups and complex source beams (CSBs)are used to expand the fields of the basis functions residingin each group [20 21] A multilevel version of this methodis developed in [22] To further improve the computationalefficiency a nested complex source beam (NCSB) methodis proposed by utilizing an equivalent relationship betweenadjacent levels [23] Since the CSBs are exact solutions ofMaxwellrsquos equations any arbitrary EMfields can be expandedin terms of a set of CSBs [20 24] Therefore this method canbe extended to solve the low-frequency problemswithout anytheoretical barriers

This paper is organized as followsThe basis formulationsof CSB-MoM and A-EFIE are briefly reviewed in Section 2In Section 3 the CSB-MoM is integrated into the A-EFIEsystem to remedy the low-frequency breakdown problemThe detailed derivation of the CSB expansions for A-EFIEis first presentedThe aggregationdisaggregation translationoperators for a multilevel CSB method are then discussedFinally numerical examples are summarized in Section 4to demonstrate the validity and efficiency of the proposedmethod

2 Theory Background

Given a 3D perfectly electrical conducting (PEC) bodydefined by its surface the conventional MoM formulationcan be applied to the electric field integral equation (EFIE)leading to a matrix equation of the form

Zj = b (1)

where j is the unknown vector for the surface current densityb is the excitation vector and Z is the dense impedancematrix The impedance matrix can be expressed as

Z119894119895 = int119878119894

f119894 (r) sdot int119878119895

119866(r r1015840) f119895 (r1015840) 1198891198781015840 (2)

where

119866(r r1015840) = [119868 + nablanabla11989620 ]119892 (r r1015840) (3)

is the dyadicGreen function and 119892(rr1015840)= 1198901198941198960|rminusr1015840|4120587|rminusr1015840| isthe scalar Green function Moreover f119895(r1015840) and f119894(r) denotethe RWG basis functions for expanding surface current andGalerkin testing respectively This equation suffers from the

low-frequency breakdown problem because the contributionof the vector potential is swamped by that of the scalar poten-tial at low frequencies due to the finite machine precision

21 Formulation of CSB-MoM for EFIE In CSB-MoMmeth-od the MVP acceleration of Zj = b is achieved througha FMM-like near-field and far-field decomposition For thefar-field interactions Z119865j between well separated groups theMVP are carried out by a series of CSBs launched on acomplex equivalence surface enclosing each group

Z119865j = Z1198981198981015840 j1198981015840 = sum1205761205761015840=120579120601

(W120576119898)119879 T12057612057610158401198711198981198981015840W12057610158401198981015840 j1198981015840 (4)

where 119898 and 1198981015840 denote the observation and source groupsand 119871 indicates the finest level (single level in CSB-MoM)W1205761198981015840 and W120576

1015840

119898 are the expansion matrices for both 120579 and 120601components which can be obtained by the far-fieldmatchingof basis functions in groups 119898 and 1198981015840 respectively [19]Superscript 119879 stands for the transpose of the correspondingmatrix It needs to be emphasized that only 120579 and 120601 com-ponents are considered in CSBs since the far-field radiationfields contain only 120579 and 120601 components

By using the expansion matrices the CSB expansioncoefficients for the source group 1198981015840 can be expanded fromthe surface currents as

s1198711198981015840 = [[s1205791198711198981015840

s1206011198711198981015840

]] = [[W1205791198981015840

W1206011198981015840

]] j1198981015840 (5)

T1205761205761015840

1198711198981198981015840 in (4) is the translation matrix of which theelements are expressed as

[T12057612057610158401198711198981198981015840]1199021199021015840 = 120576119902 sdot 119866 (r119902119871119898 r101584011990210158401198711198981015840) sdot 12057610158401199021015840 (6)

where 120576119902 and 12057610158401199021015840 denote the unit vectors of 119902th and 1199021015840thCSB respectivelyThe complex position vectors r119902119871119898 and r

10158401199021015840

1198711198981015840

are the launch points of CSBs associated with groups 119898 and1198981015840 119866(r119902119871119898 r101584011990210158401198711198981015840) is the dyadic Green function with complexarguments The directional property of CSBs can be used toreduce the computational cost in the translation procedure

22 Formulation of A-EFIE In the augmented electric fieldintegral equation the surface current is discretized by thenormalized RWG basis function which is modified byremoving the length of the common edge

120588119894 (r) =

r minus r+1198942119860+119894 r isin 119879+119894rminus119894 minus r2119860minus119894 r isin 119879minus1198940 otherwise

(7)

where 119860plusmn119894 are the area of triangles 119879plusmn119894 rplusmn119894 are the free verticesof the two triangles On the other hand the surface charge

International Journal of Antennas and Propagation 3

density is approximated by the pulse function defined on eachtriangle which is expressed as

ℎ119894 (r) = 1119860 119894 r isin 1198791198940 otherwise (8)

Combining EFIE in (1) and the current continuity condi-tion between current and charge we can arrive at the A-EFIEequation

[[V D119879 sdot PD 11989620I ]] sdot [1198941198960j1198880120601] = [120578minus10 b0 ] (9)

with

V119894119895 = int119878119894

120588119894 (r) sdot int119878119895

119892 (r r1015840)120588119895 (r1015840) 1198891198781015840 119889119878 (10)

P119894119895 = int119878119894

ℎ119894 (r) sdot int119878119895

119892 (r r1015840) ℎ119895 (r1015840) 1198891198781015840 119889119878 (11)

b119894 = int119878119894

f119894 (r) sdot E119894119889119878 (12)

D119894119895 = 1 119879119894 isin 119878119895 the positive partminus1 119879119894 isin 119878119895 the negative part0 119879119894 notin 119878119895

(13)

where j and 120601 denote the unknown coefficients for currentand charge 1198960 and 1205780 are the wave number and the waveimpedance in free space and 1198880 is the light speed in freespace The dense matrix V represents the vector potentialsand depicts the current interactions between inner edgesOn the other hand P denotes the scalar potentials whichdescribes charges interactions between triangles The sparsematrix D represents the relationship between edge and thetriangles patch and I is the identity matrix In the matrixsystem (9) the vector potential and scalar potential arebalanced by using a proper frequency scaling which is criticalfor low-frequency problems

3 Implementation of MLCSB for A-EFIE

31 CSB Expansions for A-EFIE Motivated by the idea of CSBexpansionmethod for EFIE we can derive the expansions forA-EFIE in a similar way The inner integrals of (10) and (11)can be written as

V119894 (r) = int119878119894

119892 (r r1015840)120588119894 (r1015840) 1198891198781015840 (14)

P119895 (r) = int119878119895

119892 (r r1015840) ℎ119895 (r1015840) 1198891198781015840 (15)

where 119894 and 119895 represent the 119894th normalized RWG basisfunction and the 119895th pulse function in group 1198981015840 The above

integral equation is equivalent to the summation of CSBvector and scalar potentials

V119894 (r) = 119876sum119902=1

119892 (r r1015840119902) (119909119908119909119902119894 + 119910119908119910119902119894 + 119908119911119902119894) (16)

P119895 (r) = 119876sum119902=1

119892 (r r1015840119902)119908119901119902119895 (17)

where 119909119908119909119902119894+119910119908119910119902119894+119908119911119902119894 is the vector CSBweight for current119894 119908119901119902119895 is the weight for charge 119895 and 119876 represents the totalnumber of beams in this discretization Combining (16) with(14) and (17) with (15) and testing on a far-field matchingpoint we get

119909 sdot 119876sum119902=1

119892 (r119905 r1015840119902) 119909119908119909119902119894 = 119909 sdot int119878119894

119892 (r119905 r1015840)120588119894 (r1015840) 1198891198781015840119910 sdot 119876sum119902=1

119892 (r119905 r1015840119902) 119910119908119910119902119894 = 119910 sdot int119878119894

119892 (r119905 r1015840)120588119894 (r1015840) 1198891198781015840 sdot 119876sum119902=1

119892 (r119905 r1015840119902) 119908119911119902119894 = sdot int119878119894

119892 (r119905 r1015840)120588119894 (r1015840) 1198891198781015840119876sum119902=1

119892 (r119905 r1015840119902)119908119901119902119895 = int119878119895

119892 (r119905 r1015840) ℎ119895 (r1015840) 1198891198781015840

(18)

where r119905 (119905 = 1 2 119876) is the matching point Those linearequations can be converted into a matrix form and combinedas a multiple right-hand side problem by all the current andcharge basis in the group1198981015840

G [W1199091198981015840 W1199101198981015840 W1199111198981015840 W1199011198981015840] = [V1199091198981015840 V1199101198981015840 V1199111198981015840 P1198981015840] (19)

where[G]119905119902 = 119892 (r119905 r1015840119902)[WV1198981015840]119902119894 = 119908V

119902119894 (V = 119909 119910 119911)[W1199011198981015840

]119902119895

= 119908119901119902119895[VV1198981015840]119905119894 = V119905 sdot int

119878119894

119892 (r119905 r1015840)120588119894 (r1015840) 1198891198781015840 (V = 119909 119910 119911)[P1198981015840]119905119895 = int

119878119895

119892 (r119905 r1015840) ℎ119895 (r1015840) 1198891198781015840(20)

By using the expansion matrices W1199091198981015840 W1199101198981015840 W1199111198981015840 andW1199011198981015840 the CSB expansion coefficients for vector potentials of

group1198981015840 can be expanded from j1198981015840 as

sV1198711198981015840 = [s1199091198711198981015840 s1199101198711198981015840 s1199111198711198981015840] = [W1199091198981015840 W1199101198981015840 W1199111198981015840] j1198981015840 (21)

Similarly the coefficients for scalar ones are

s1199011198711198981015840

= W1199011198981015840

1206011198981015840 (22)


Hence the far-field MVP procedure for (10) and (11) canbe obtained with a similar process in (4)

V1198981198981015840 j1198981015840 = sumV=119909119910119911

(WV119898)119879 T1198711198981198981015840WV

1198981015840 j1198981015840

P11989811989810158401206011198981015840 = (W119889119898)119879 T1198711198981198981015840W11990111989810158401206011198981015840 (23)

where the translation operator T1198711198981198981015840 is expressed as

[T1198711198981198981015840]1199021199021015840 = 119892 (r119902119871119898 r101584011990210158401198711198981015840) (24)

It is noticed that the scalar Green function with complexarguments is used here and the translation for the vector andscalar expansion coefficients shares the same operators

32 Multilevel Algorithm of CSB for A-EFIE So far wehave derived the CSB expansion method for A-EFIE andhave presented the far-group MVP process with the helpof translation operators in a single level algorithm In thefollowing we will obtain the aggregation and disaggregationoperators of CSB expansion coefficients for both (10) and (11)to realize a multilevel algorithm Firstly the vector and scalarpotentials by the 119902119897+1th CSB in a group is

V119902119897+1 (r) = 119892 (r r1015840119902119897+1) (119909119904119909119902119897+1 + 119910119904119910119902119897+1 + 119904z119902119897+1)P119902119897+1 (r) = 119892 (r r1015840119902119897+1) 119904119901119902119897+1 (25)

where 119897 + 1 means the (119897 + 1)th level of the octree in themultilevel algorithm By applying the summation and testingprocedures similar to (16)ndash(18) a linear system can be set upto obtain the equivalent relationship of CSBs between twoadjacent levels

G119897A119897 = P119897+1 (26)

where [G119897]119905119897119902119897 = 119892(r119905119897 r1015840119902119897) is the matching matrix in parentlevel 119897 This matching matrix connects the equivalent CSBsources to the scalar potentials P119897+1 Different from the right-hand side in (19) the right-hand side here for each vectorpotential component is the same as the scalar one which is[P119897+1]119902119897119902119897+1 = 119892 (r119902119897 | r1015840119902119897+1) (27)

Once G119897 and P119897+1 are assembled the aggregation matrixA119897 for level 119897 can be numerically solved By using theaggregation matrix the CSB expansion coefficients of theparent group in level 119897 can be obtained efficiently from itschild groups in level 119897 + 1

sV1198971198981015840 = [s1199091198971198981015840 s1199101198971198981015840 s1199111198971198981015840] = sum1198991015840isinchild(1198981015840)

A119897sV119897+11198991015840

s1199011198971198981015840

= sum1198991015840isinchild(1198981015840)

A119897s119901

119897+11198991015840 (28)

Similar to the multilevel fast multipole algorithm(MLFMA) [25] the CSB expansion coefficients of a receivinggroup in level 119897 + 1 are obtained from the translation in thesame level as well as the disaggregation from its parent level119897 The disaggregation matrix can be easily obtained from thetranspose of the aggregation matrices

MieMLCSB-AEFIE

8040 60 100 120 180140 160200

휃 (∘)

minus320

minus300

minus280

minus260

minus240

minus220

minus200

BiRC

S (d

Bsm

)

Figure 1 Bistatic RCS of the PEC sphere at 100Hz validated by theMie series

4 Numerical Results

In this section the accuracy error analysis computationalcomplexity and the efficiency of the method are investigatedthrough several numerical examples All the examples wererun on a computer of 2 processors each with 14 cores at26GHz 512GB memory and OpenMP parallelization

41 Small Sphere To show the accuracy of MLCSB-AEFIEat low frequencies the electromagnetic scattering by a PECsphere of 1m radius is analyzed at 100Hz The sphere isdiscretized with 1764 triangular patches which correspondsto 2646 inner edges A three levelsrsquo MLCSB algorithm is usedwith a group size of 83times10minus8120582 at the finest levelThe incidentangle of a plane wave is 120579119894 = 0∘ 120593119894 = 0∘ and the observedazimuth angle is fixed at 120579119894 = 0∘ The residual error thresholdis set to be 10minus15 for GMRES-30 It takes 80 iteration stepsto converge with the help of the saddle point preconditionerin [9] A good agreement of the bistatic RCS is observed inFigure 1 as compared with the analytical solution of Mieseries

42 Computational Complexity To demonstrate the com-putational complexity of the proposed method the planewave scattering of a PEC cube with a side length of 01mis calculated at 300MHz The electric size of the cube is0173120582The surface of the cube is discretized into six differentmeshesMeshAMesh BMeshCMeshDMesh E andMeshF The coarsest Mesh A has 1262 planar triangles and 1893interior edges and the average edge length is 105 times 10minus2120582Then we refineMesh A by halving the edge length recursivelyuntil Mesh F Mesh F comprises 1534536 planar trianglesand 2301804 interior edges and the average edge length is3 times 10minus4120582


Table 1 Computational statistics of the PEC cube with different meshes at 300MHz

Mesh Number ofRWG

Number oftriangles

Near-fieldMem(MB)

ExpansionMem(MB)

TranslationMem (MB)

InterpolationMem (MB)

TotalMem(MB)

MVP time(s)

Iterationnumber

Total time(s)

A 1893 1262 27 215 591 42 875 015 25 41B 7992 5328 102 907 886 84 1979 06 36 228C 33330 22220 435 3783 1182 126 5526 22 52 1184D 139002 92668 178 1541 1478 168 18836 87 74 6558E 563292 375528 7279 6245 17736 21 71704 312 110 3480F 2301804 1534536 29329 25518 2069 252 28683 1209 174 212326

101

102

103

104

105

Mem

ory

(MB)

104

105

106

107

103

Number of unknowns

10minus1

100

101

102

103

Tim

e (s)

Memory usageMVP time

O(13e minus 2 lowast N)


Figure 2 CPU time for each MVP and memory usage in theexample of the PEC cube

The CPU times for each MVP and the memory usage isplotted in Figure 2 It can be found that the computationalcomplexity and the memory requirement of MLCSB-AEFIEboth scale as 119874(119873) where 119873 are the unknown numbersThe detailed computational statistics are summarized inTable 1 and the iteration histories are shown in Figure 3 forcomparison By using the preconditioner in [9] the iterationconverges quickly to 10minus4 A good agreement of the bistaticradar cross sections (RCS) of the five cases is demonstratedin Figure 4

43 Error Analysis In the proposed MLCSB-AEFIE thenumerical error is mainly from expansion and aggregationSince the expansion and aggregation processes are similarfor the vector and scalar potentials we only show the scalarpotential case in the following For the expansion the relativeerrors related to a group with the size of 0005m and averagediscretization length of 0001m are studiedThe relative errorof expansion is defined as

119890 = 10038161003816100381610038161003816119875 (r) minus 119875CSB (r)10038161003816100381610038161003816|119875 (r)| (29)

8040 60 100 12020 140 160 1800

Number of iterations

10minus4

10minus3

10minus2

10minus1

100

Resid

ual e

rror

s

Mesh A

Mesh FMesh BMesh C

Mesh DMesh E

Figure 3 Iteration history of the PEC cube for different meshes

where 119875CSB(r) is the scalar potential obtained via the expan-sion process in (17) 119875(r) is the exact data calculated by (15)and the observation point r is fixed at the center of thenearest cousin group Figure 5 shows the relative errors 119890 fordifferent numbers of CSBs 119876 = 30 42 58 and 82 withrespect to the frequencies from 30Hz to 300MHz It isnoticed that the expansion errors are almost constant forthe same 119876 in a very wide frequency range indicating anexcellent stability and scalability of the proposed method forbroadband computations

To investigate the interpolation error in the aggregationprocess we compare the scalar potentials related to a groupwith the size of 0005m and the parent group with the sizeof 001m The scalar potentials are calculated by CSBs whilethe CSB coefficients of the parent group are aggregated fromthe child group The RMS relative error is shown in Figure 6which is defined as

RMS (119875) = radic 1119873 119873sum119894=1

10038171003817100381710038171003817119875CSB119897 (r119894) minus 119875CSB

119897+1 (r119894)10038171003817100381710038171003817210038171003817100381710038171003817119875CSB119897+1

(r119894)100381710038171003817100381710038172 (30)


0 8020 40 60 120 180100 140 160

휃 (∘)

minus60

minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

BiRC

S (d

Bsm

)

Mesh A

Mesh F

Mesh DMesh E

Mesh CMesh B

Figure 4 Bistatic RCS of the PEC cube with different meshes

10minus6

10minus5

10minus4

10minus3

10minus2

Relat

ive e

rror

s of e

xpan

sion

102

103

104

105

106

107

108

109

101

Frequency (Hz)

Q = 30

Q = 42

Q = 58

Q = 82

Figure 5 Relative errors of expansion

and here 119875CSB119897 (r119894) 119875CSB

119897+1 (r119894) are the scalar potentials of theparent group and the child group respectively and r119894 is seton azimuth circle centered at the parent group Figure 6 alsoshows a constant accuracy level for different frequencies withthe same 11987644 Scattering of Multiscale Cone The performance of theMLCSB-AEFIE for multiscale structures is evaluated by anonuniformly meshed cone illustrated in Figure 7 Theradius of the cone is 5m at the bottom and the heightis 10m We mesh the cone with a length of 01m at thebottom and gradually reduce it to 0001m at the sharp

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

RMS

erro

rs o

f int

erpo

latio

n

102

103

104

105

106

107

108

109

101

Frequency (Hz)

Q = 30

Q = 42

Q = 58

Q = 82

Figure 6 RMS errors of interpolation

001 m

01 m

Z

YX

Figure 7 A cone of radius 5m and height 10m which is meshedwith average length 01m at the bottom and 0001m at the sharppoint Mesh details are shown with reference lengths 01m and001m

point Finally the cone is discretized with 191976 planarpatches and hence 287964 interior edges Figure 8 showsthe bistatic RCS of the cone excited by a 119910-polarized planewave incident from the 119909 direction at 10 KHz and 10MHzIn this exampleMLCSB-AEFIE converges to relative residualerror of 10minus3 within 73 and 67 iterations for 10 KHz and10MHz respectively


10 KHz10 MHz

20 40 60 80 100 120 140 160 1800

120579 (∘)

minus140

minus130

minus120

minus110

minus100

minus90

0

10

20

30

BiRC

S (d

Bsm

)

Figure 8 Bistatic RCS of the PEC cone at 10 KHz and 10Mhz

5 Conclusion

In this paper we have proposed a MLCSB-AEFIE methodfor the well-known low-frequency problem The vector andscalar potentials from the current and charge unknowns areexpended with CSBs An aggregation matrix is obtained forthe CSB expansion coefficients to form an efficient multilevelalgorithm Numerical examples have validated the goodaccuracy efficiency and scalability of the proposed methodfor low-frequency problems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported partly by the National ExcellentYouth Fund by NSFC no 61425010 Programme of Intro-ducing Talents of Discipline to Universities under Grant nob07046 the Chang Jiang Scholar Project of MOE and theNational Natural Science Foundation of China under Grantno 61490695

References

[1] R F Harrington and J L Harrington Field Computation byMoment Methods Oxford University Press Oxford UK 1996

[2] Z G Qian and W C Chew ldquoA quantitative study on thelow frequency breakdown of EFIErdquo Microwave and OpticalTechnology Letters vol 50 no 5 pp 1159ndash1162 2008

[3] F P Andriulli ldquoLoop-star and loop-tree decompositions analy-sis and efficient algorithmsrdquo IEEETransactions onAntennas andPropagation vol 60 no 5 pp 2347ndash2356 2012

[4] S Yan J-M Jin and Z Nie ldquoEFIE analysis of low-frequencyproblems with loop-star decomposition and Calderon multi-plicative preconditionerrdquo IEEE Transactions on Antennas andPropagation vol 58 no 3 pp 857ndash867 2010

[5] M Taskinen and P Yla-Oijala ldquoCurrent and charge integralequation formulationrdquo IEEE Transactions on Antennas andPropagation vol 54 no 1 pp 58ndash67 2006

[6] D Gope A Ruehli and V Jandhyala ldquoSolving low-frequencyEM-CKTproblems using the PEECmethodrdquo IEEETransactionson Advanced Packaging vol 30 no 2 pp 313ndash320 2007

[7] A Das and D Gope ldquoModified SPIE formulation for low-frequency stability of electric field integral equationrdquo in Pro-ceedings of the 5th IEEE Applied Electromagnetics Conference(AEMC rsquo15) Guwahati India December 2015

[8] ZGQian andWCChew ldquoAn augmented electric field integralequation for high-speed interconnect analysisrdquoMicrowave andOptical Technology Letters vol 50 no 10 pp 2658ndash2662 2008

[9] Z-G Qian and W C Chew ldquoFast full-wave surface integralequation solver for multiscale structure modelingrdquo IEEE Trans-actions on Antennas and Propagation vol 57 no 11 pp 3594ndash3601 2009

[10] Y P Chen L Jiang Z-G Qian and W C Chew ldquoModelingelectrically small structures in layeredmediumwith augmentedEFIE methodrdquo in Proceedings of the IEEE International Sympo-sium on Antennas and Propagation and USNCURSI NationalRadio Science Meeting APSURSI 2011 pp 3218ndash3221 SpokaneWash USA July 2011

[11] Y G Liu W C Chew L Jiang and Z Qian ldquoA memorysaving fast A-EFIE solver for modeling low-frequency large-scale problemsrdquo Applied Numerical Mathematics vol 62 no 6pp 682ndash698 2012

[12] Z-G Qian andW C Chew ldquoEnhanced A-EFIE with perturba-tion methodrdquo IEEE Transactions on Antennas and Propagationvol 58 no 10 pp 3256ndash3264 2010

[13] Z-H Ma L J Jiang and W C Chew ldquoLoop-tree freeaugmented equivalence principle algorithm for low-frequencyproblemsrdquo Microwave and Optical Technology Letters vol 55no 10 pp 2475ndash2479 2013

[14] K-J Xu X-M Pan andX-Q Sheng ldquoAn augmented EFIEwithdiscontinuous Galerkin discretizationrdquo in Proceedings of theIEEE International Conference on Computational Electromag-netics ICCEM 2016 pp 106ndash108 Guangzhou China February2016

[15] J S Zhao and W C Chew ldquoApplying LF-MLFMA to solvecomplex PEC structuresrdquo Microwave and Optical TechnologyLetters vol 28 no 3 pp 155ndash160 2001

[16] Y Zheng Y Zhao Z Nie and Q Cai ldquoFull-wave fast solver forcircuit devices modelingrdquo Applied Computational Electromag-netics Society Journal vol 30 no 10 pp 1115ndash1121 2015

[17] 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

[18] M M Jia S Sun and W C Chew ldquoAccelerated A-EFIE withperturbation method using fast fourier transformrdquo in Proceed-ings of the IEEE Antennas and Propagation Society InternationalSymposium (APSURSI rsquo14) pp 2148-2149Memphis TennUSAJuly 2014

[19] K Tap P H Pathak and R J Burkholder ldquoComplex sourcebeam-moment method procedure for accelerating numerical


integral equation solutions of radiation and scattering prob-lemsrdquo IEEE Transactions on Antennas and Propagation vol 62no 4 part 2 pp 2052ndash2062 2014

[20] K Tap Complex source point beam expansions for some electro-magnetic radiation and scattering problems [PhD thesis] TheOhio State University Columbus Ohio USA 2007

[21] E Martini and S Maci ldquoGeneration of complex source pointexpansions from radiation integralsrdquo Progress in Electromagnet-ics Research vol 152 no 3 pp 17ndash31 2015

[22] Z H Fan X Hu and R S Chen ldquoMultilevel complex sourcebeam method for electromagnetic scattering problemsrdquo IEEEAntennas andWireless Propagation Letters vol 14 pp 843ndash8462015

[23] K C Wang Z H Fan M M Li and R S Chen ldquoAn effectiveMoM Solution with nested complex source beam method forelectromagnetic scattering problemsrdquo IEEE Transactions onAntennas and Propagation vol 64 no 6 pp 2546ndash2551 2016

[24] T B Hansen and G Kaiser ldquoHuygensrsquo principle for complexspheresrdquo IEEE Transactions on Antennas and Propagation vol59 no 10 pp 3835ndash3847 2011

[25] O Ergul and L Gurel The Multilevel Fast Multipole Algorithm(MLFMA) for Solving Large-Scale Computational Electromag-netics Problems John Wiley amp Sons Hoboken NJ USA 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



As the number of unknowns increases a fast algorithmhas to be incorporated into the iterative solver to reduce theoperation complexity of the matrix-vector product (MVP)Such fast algorithms should be low-frequency stable forinstance the low-frequency multilevel fast multipole algo-rithm (LF-MLFMA) [15] the multilevel accelerated Carte-sian expansion algorithm (MLACEA) [16] the multileveladaptive cross-approximation (MLACA) algorithm [17] andfast Fourier transform (FFT) [18] are eligible candidates forsuch purpose Recently a complex source beam-method ofmoment (CSB-MoM) is proposed to accelerate the far-fieldinteractions of MoM at midfrequencies [19] The object isfirst divided into groups and complex source beams (CSBs)are used to expand the fields of the basis functions residingin each group [20 21] A multilevel version of this methodis developed in [22] To further improve the computationalefficiency a nested complex source beam (NCSB) methodis proposed by utilizing an equivalent relationship betweenadjacent levels [23] Since the CSBs are exact solutions ofMaxwellrsquos equations any arbitrary EMfields can be expandedin terms of a set of CSBs [20 24] Therefore this method canbe extended to solve the low-frequency problemswithout anytheoretical barriers

This paper is organized as followsThe basis formulationsof CSB-MoM and A-EFIE are briefly reviewed in Section 2In Section 3 the CSB-MoM is integrated into the A-EFIEsystem to remedy the low-frequency breakdown problemThe detailed derivation of the CSB expansions for A-EFIEis first presentedThe aggregationdisaggregation translationoperators for a multilevel CSB method are then discussedFinally numerical examples are summarized in Section 4to demonstrate the validity and efficiency of the proposedmethod

2 Theory Background

Given a 3D perfectly electrical conducting (PEC) bodydefined by its surface the conventional MoM formulationcan be applied to the electric field integral equation (EFIE)leading to a matrix equation of the form

Zj = b (1)

where j is the unknown vector for the surface current densityb is the excitation vector and Z is the dense impedancematrix The impedance matrix can be expressed as

Z119894119895 = int119878119894

f119894 (r) sdot int119878119895

119866(r r1015840) f119895 (r1015840) 1198891198781015840 (2)

where

119866(r r1015840) = [119868 + nablanabla11989620 ]119892 (r r1015840) (3)

is the dyadicGreen function and 119892(rr1015840)= 1198901198941198960|rminusr1015840|4120587|rminusr1015840| isthe scalar Green function Moreover f119895(r1015840) and f119894(r) denotethe RWG basis functions for expanding surface current andGalerkin testing respectively This equation suffers from the

low-frequency breakdown problem because the contributionof the vector potential is swamped by that of the scalar poten-tial at low frequencies due to the finite machine precision

21 Formulation of CSB-MoM for EFIE In CSB-MoMmeth-od the MVP acceleration of Zj = b is achieved througha FMM-like near-field and far-field decomposition For thefar-field interactions Z119865j between well separated groups theMVP are carried out by a series of CSBs launched on acomplex equivalence surface enclosing each group

Z119865j = Z1198981198981015840 j1198981015840 = sum1205761205761015840=120579120601

(W120576119898)119879 T12057612057610158401198711198981198981015840W12057610158401198981015840 j1198981015840 (4)

where 119898 and 1198981015840 denote the observation and source groupsand 119871 indicates the finest level (single level in CSB-MoM)W1205761198981015840 and W120576

1015840

119898 are the expansion matrices for both 120579 and 120601components which can be obtained by the far-fieldmatchingof basis functions in groups 119898 and 1198981015840 respectively [19]Superscript 119879 stands for the transpose of the correspondingmatrix It needs to be emphasized that only 120579 and 120601 com-ponents are considered in CSBs since the far-field radiationfields contain only 120579 and 120601 components

By using the expansion matrices the CSB expansioncoefficients for the source group 1198981015840 can be expanded fromthe surface currents as

s1198711198981015840 = [[s1205791198711198981015840

s1206011198711198981015840

]] = [[W1205791198981015840

W1206011198981015840

]] j1198981015840 (5)

T1205761205761015840

1198711198981198981015840 in (4) is the translation matrix of which theelements are expressed as

[T12057612057610158401198711198981198981015840]1199021199021015840 = 120576119902 sdot 119866 (r119902119871119898 r101584011990210158401198711198981015840) sdot 12057610158401199021015840 (6)

where 120576119902 and 12057610158401199021015840 denote the unit vectors of 119902th and 1199021015840thCSB respectivelyThe complex position vectors r119902119871119898 and r

10158401199021015840

1198711198981015840

are the launch points of CSBs associated with groups 119898 and1198981015840 119866(r119902119871119898 r101584011990210158401198711198981015840) is the dyadic Green function with complexarguments The directional property of CSBs can be used toreduce the computational cost in the translation procedure

22 Formulation of A-EFIE In the augmented electric fieldintegral equation the surface current is discretized by thenormalized RWG basis function which is modified byremoving the length of the common edge

120588119894 (r) =

r minus r+1198942119860+119894 r isin 119879+119894rminus119894 minus r2119860minus119894 r isin 119879minus1198940 otherwise

(7)

where 119860plusmn119894 are the area of triangles 119879plusmn119894 rplusmn119894 are the free verticesof the two triangles On the other hand the surface charge






with

V119894119895 = int119878119894

120588119894 (r) sdot int119878119895

119892 (r r1015840)120588119895 (r1015840) 1198891198781015840 119889119878 (10)

P119894119895 = int119878119894

ℎ119894 (r) sdot int119878119895

119892 (r r1015840) ℎ119895 (r1015840) 1198891198781015840 119889119878 (11)

b119894 = int119878119894

f119894 (r) sdot E119894119889119878 (12)


(13)




V119894 (r) = int119878119894

119892 (r r1015840)120588119894 (r1015840) 1198891198781015840 (14)

P119895 (r) = int119878119895

119892 (r r1015840) ℎ119895 (r1015840) 1198891198781015840 (15)



V119894 (r) = 119876sum119902=1

119892 (r r1015840119902) (119909119908119909119902119894 + 119910119908119910119902119894 + 119908119911119902119894) (16)

P119895 (r) = 119876sum119902=1

119892 (r r1015840119902)119908119901119902119895 (17)


119909 sdot 119876sum119902=1

119892 (r119905 r1015840119902) 119909119908119909119902119894 = 119909 sdot int119878119894

119892 (r119905 r1015840)120588119894 (r1015840) 1198891198781015840119910 sdot 119876sum119902=1

119892 (r119905 r1015840119902) 119910119908119910119902119894 = 119910 sdot int119878119894

119892 (r119905 r1015840)120588119894 (r1015840) 1198891198781015840 sdot 119876sum119902=1

119892 (r119905 r1015840119902) 119908119911119902119894 = sdot int119878119894

119892 (r119905 r1015840)120588119894 (r1015840) 1198891198781015840119876sum119902=1

119892 (r119905 r1015840119902)119908119901119902119895 = int119878119895

119892 (r119905 r1015840) ℎ119895 (r1015840) 1198891198781015840

(18)


G [W1199091198981015840 W1199101198981015840 W1199111198981015840 W1199011198981015840] = [V1199091198981015840 V1199101198981015840 V1199111198981015840 P1198981015840] (19)

where[G]119905119902 = 119892 (r119905 r1015840119902)[WV1198981015840]119902119894 = 119908V

119902119894 (V = 119909 119910 119911)[W1199011198981015840

]119902119895

= 119908119901119902119895[VV1198981015840]119905119894 = V119905 sdot int

119878119894

119892 (r119905 r1015840)120588119894 (r1015840) 1198891198781015840 (V = 119909 119910 119911)[P1198981015840]119905119895 = int

119878119895

119892 (r119905 r1015840) ℎ119895 (r1015840) 1198891198781015840(20)



sV1198711198981015840 = [s1199091198711198981015840 s1199101198711198981015840 s1199111198711198981015840] = [W1199091198981015840 W1199101198981015840 W1199111198981015840] j1198981015840 (21)


s1199011198711198981015840

= W1199011198981015840

1206011198981015840 (22)



V1198981198981015840 j1198981015840 = sumV=119909119910119911

(WV119898)119879 T1198711198981198981015840WV

1198981015840 j1198981015840

P11989811989810158401206011198981015840 = (W119889119898)119879 T1198711198981198981015840W11990111989810158401206011198981015840 (23)


[T1198711198981198981015840]1199021199021015840 = 119892 (r119902119871119898 r101584011990210158401198711198981015840) (24)



V119902119897+1 (r) = 119892 (r r1015840119902119897+1) (119909119904119909119902119897+1 + 119910119904119910119902119897+1 + 119904z119902119897+1)P119902119897+1 (r) = 119892 (r r1015840119902119897+1) 119904119901119902119897+1 (25)


G119897A119897 = P119897+1 (26)




A119897sV119897+11198991015840

s1199011198971198981015840

= sum1198991015840isinchild(1198981015840)

A119897s119901

119897+11198991015840 (28)


MieMLCSB-AEFIE

8040 60 100 120 180140 160200

휃 (∘)

minus320

minus300

minus280

minus260

minus240

minus220

minus200

BiRC

S (d

Bsm

)


4 Numerical Results






Mesh Number ofRWG

Number oftriangles

Near-fieldMem(MB)

ExpansionMem(MB)

TranslationMem (MB)


TotalMem(MB)

MVP time(s)

Iterationnumber

Total time(s)

A 1893 1262 27 215 591 42 875 015 25 41B 7992 5328 102 907 886 84 1979 06 36 228C 33330 22220 435 3783 1182 126 5526 22 52 1184D 139002 92668 178 1541 1478 168 18836 87 74 6558E 563292 375528 7279 6245 17736 21 71704 312 110 3480F 2301804 1534536 29329 25518 2069 252 28683 1209 174 212326

101

102

103

104

105

Mem

ory

(MB)

104

105

106

107

103

Number of unknowns

10minus1

100

101

102

103

Tim

e (s)







119890 = 10038161003816100381610038161003816119875 (r) minus 119875CSB (r)10038161003816100381610038161003816|119875 (r)| (29)

8040 60 100 12020 140 160 1800


10minus4

10minus3

10minus2

10minus1

100

Resid

ual e

rror

s

Mesh A

Mesh FMesh BMesh C

Mesh DMesh E




RMS (119875) = radic 1119873 119873sum119894=1

10038171003817100381710038171003817119875CSB119897 (r119894) minus 119875CSB

119897+1 (r119894)10038171003817100381710038171003817210038171003817100381710038171003817119875CSB119897+1

(r119894)100381710038171003817100381710038172 (30)


0 8020 40 60 120 180100 140 160

휃 (∘)

minus60

minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

BiRC

S (d

Bsm

)

Mesh A

Mesh F

Mesh DMesh E

Mesh CMesh B


10minus6

10minus5

10minus4

10minus3

10minus2

Relat

ive e

rror

s of e

xpan

sion

102

103

104

105

106

107

108

109

101

Frequency (Hz)

Q = 30

Q = 42

Q = 58

Q = 82


and here 119875CSB119897 (r119894) 119875CSB


10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

RMS

erro

rs o

f int

erpo

latio

n

102

103

104

105

106

107

108

109

101

Frequency (Hz)

Q = 30

Q = 42

Q = 58

Q = 82


001 m

01 m

Z

YX




10 KHz10 MHz

20 40 60 80 100 120 140 160 1800

120579 (∘)

minus140

minus130

minus120

minus110

minus100

minus90

0

10

20

30

BiRC

S (d

Bsm

)


5 Conclusion




Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














with

V119894119895 = int119878119894

120588119894 (r) sdot int119878119895

119892 (r r1015840)120588119895 (r1015840) 1198891198781015840 119889119878 (10)

P119894119895 = int119878119894

ℎ119894 (r) sdot int119878119895

119892 (r r1015840) ℎ119895 (r1015840) 1198891198781015840 119889119878 (11)

b119894 = int119878119894

f119894 (r) sdot E119894119889119878 (12)


(13)




V119894 (r) = int119878119894

119892 (r r1015840)120588119894 (r1015840) 1198891198781015840 (14)

P119895 (r) = int119878119895

119892 (r r1015840) ℎ119895 (r1015840) 1198891198781015840 (15)



V119894 (r) = 119876sum119902=1

119892 (r r1015840119902) (119909119908119909119902119894 + 119910119908119910119902119894 + 119908119911119902119894) (16)

P119895 (r) = 119876sum119902=1

119892 (r r1015840119902)119908119901119902119895 (17)


119909 sdot 119876sum119902=1

119892 (r119905 r1015840119902) 119909119908119909119902119894 = 119909 sdot int119878119894

119892 (r119905 r1015840)120588119894 (r1015840) 1198891198781015840119910 sdot 119876sum119902=1

119892 (r119905 r1015840119902) 119910119908119910119902119894 = 119910 sdot int119878119894

119892 (r119905 r1015840)120588119894 (r1015840) 1198891198781015840 sdot 119876sum119902=1

119892 (r119905 r1015840119902) 119908119911119902119894 = sdot int119878119894

119892 (r119905 r1015840)120588119894 (r1015840) 1198891198781015840119876sum119902=1

119892 (r119905 r1015840119902)119908119901119902119895 = int119878119895

119892 (r119905 r1015840) ℎ119895 (r1015840) 1198891198781015840

(18)


G [W1199091198981015840 W1199101198981015840 W1199111198981015840 W1199011198981015840] = [V1199091198981015840 V1199101198981015840 V1199111198981015840 P1198981015840] (19)

where[G]119905119902 = 119892 (r119905 r1015840119902)[WV1198981015840]119902119894 = 119908V

119902119894 (V = 119909 119910 119911)[W1199011198981015840

]119902119895

= 119908119901119902119895[VV1198981015840]119905119894 = V119905 sdot int

119878119894

119892 (r119905 r1015840)120588119894 (r1015840) 1198891198781015840 (V = 119909 119910 119911)[P1198981015840]119905119895 = int

119878119895

119892 (r119905 r1015840) ℎ119895 (r1015840) 1198891198781015840(20)



sV1198711198981015840 = [s1199091198711198981015840 s1199101198711198981015840 s1199111198711198981015840] = [W1199091198981015840 W1199101198981015840 W1199111198981015840] j1198981015840 (21)


s1199011198711198981015840

= W1199011198981015840

1206011198981015840 (22)



V1198981198981015840 j1198981015840 = sumV=119909119910119911

(WV119898)119879 T1198711198981198981015840WV

1198981015840 j1198981015840

P11989811989810158401206011198981015840 = (W119889119898)119879 T1198711198981198981015840W11990111989810158401206011198981015840 (23)


[T1198711198981198981015840]1199021199021015840 = 119892 (r119902119871119898 r101584011990210158401198711198981015840) (24)



V119902119897+1 (r) = 119892 (r r1015840119902119897+1) (119909119904119909119902119897+1 + 119910119904119910119902119897+1 + 119904z119902119897+1)P119902119897+1 (r) = 119892 (r r1015840119902119897+1) 119904119901119902119897+1 (25)


G119897A119897 = P119897+1 (26)




A119897sV119897+11198991015840

s1199011198971198981015840

= sum1198991015840isinchild(1198981015840)

A119897s119901

119897+11198991015840 (28)


MieMLCSB-AEFIE

8040 60 100 120 180140 160200

휃 (∘)

minus320

minus300

minus280

minus260

minus240

minus220

minus200

BiRC

S (d

Bsm

)


4 Numerical Results






Mesh Number ofRWG

Number oftriangles

Near-fieldMem(MB)

ExpansionMem(MB)

TranslationMem (MB)


TotalMem(MB)

MVP time(s)

Iterationnumber

Total time(s)

A 1893 1262 27 215 591 42 875 015 25 41B 7992 5328 102 907 886 84 1979 06 36 228C 33330 22220 435 3783 1182 126 5526 22 52 1184D 139002 92668 178 1541 1478 168 18836 87 74 6558E 563292 375528 7279 6245 17736 21 71704 312 110 3480F 2301804 1534536 29329 25518 2069 252 28683 1209 174 212326

101

102

103

104

105

Mem

ory

(MB)

104

105

106

107

103

Number of unknowns

10minus1

100

101

102

103

Tim

e (s)







119890 = 10038161003816100381610038161003816119875 (r) minus 119875CSB (r)10038161003816100381610038161003816|119875 (r)| (29)

8040 60 100 12020 140 160 1800


10minus4

10minus3

10minus2

10minus1

100

Resid

ual e

rror

s

Mesh A

Mesh FMesh BMesh C

Mesh DMesh E




RMS (119875) = radic 1119873 119873sum119894=1

10038171003817100381710038171003817119875CSB119897 (r119894) minus 119875CSB

119897+1 (r119894)10038171003817100381710038171003817210038171003817100381710038171003817119875CSB119897+1

(r119894)100381710038171003817100381710038172 (30)


0 8020 40 60 120 180100 140 160

휃 (∘)

minus60

minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

BiRC

S (d

Bsm

)

Mesh A

Mesh F

Mesh DMesh E

Mesh CMesh B


10minus6

10minus5

10minus4

10minus3

10minus2

Relat

ive e

rror

s of e

xpan

sion

102

103

104

105

106

107

108

109

101

Frequency (Hz)

Q = 30

Q = 42

Q = 58

Q = 82


and here 119875CSB119897 (r119894) 119875CSB


10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

RMS

erro

rs o

f int

erpo

latio

n

102

103

104

105

106

107

108

109

101

Frequency (Hz)

Q = 30

Q = 42

Q = 58

Q = 82


001 m

01 m

Z

YX




10 KHz10 MHz

20 40 60 80 100 120 140 160 1800

120579 (∘)

minus140

minus130

minus120

minus110

minus100

minus90

0

10

20

30

BiRC

S (d

Bsm

)


5 Conclusion




Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











V1198981198981015840 j1198981015840 = sumV=119909119910119911

(WV119898)119879 T1198711198981198981015840WV

1198981015840 j1198981015840

P11989811989810158401206011198981015840 = (W119889119898)119879 T1198711198981198981015840W11990111989810158401206011198981015840 (23)


[T1198711198981198981015840]1199021199021015840 = 119892 (r119902119871119898 r101584011990210158401198711198981015840) (24)



V119902119897+1 (r) = 119892 (r r1015840119902119897+1) (119909119904119909119902119897+1 + 119910119904119910119902119897+1 + 119904z119902119897+1)P119902119897+1 (r) = 119892 (r r1015840119902119897+1) 119904119901119902119897+1 (25)


G119897A119897 = P119897+1 (26)




A119897sV119897+11198991015840

s1199011198971198981015840

= sum1198991015840isinchild(1198981015840)

A119897s119901

119897+11198991015840 (28)


MieMLCSB-AEFIE

8040 60 100 120 180140 160200

휃 (∘)

minus320

minus300

minus280

minus260

minus240

minus220

minus200

BiRC

S (d

Bsm

)


4 Numerical Results






Mesh Number ofRWG

Number oftriangles

Near-fieldMem(MB)

ExpansionMem(MB)

TranslationMem (MB)


TotalMem(MB)

MVP time(s)

Iterationnumber

Total time(s)

A 1893 1262 27 215 591 42 875 015 25 41B 7992 5328 102 907 886 84 1979 06 36 228C 33330 22220 435 3783 1182 126 5526 22 52 1184D 139002 92668 178 1541 1478 168 18836 87 74 6558E 563292 375528 7279 6245 17736 21 71704 312 110 3480F 2301804 1534536 29329 25518 2069 252 28683 1209 174 212326

101

102

103

104

105

Mem

ory

(MB)

104

105

106

107

103

Number of unknowns

10minus1

100

101

102

103

Tim

e (s)







119890 = 10038161003816100381610038161003816119875 (r) minus 119875CSB (r)10038161003816100381610038161003816|119875 (r)| (29)

8040 60 100 12020 140 160 1800


10minus4

10minus3

10minus2

10minus1

100

Resid

ual e

rror

s

Mesh A

Mesh FMesh BMesh C

Mesh DMesh E




RMS (119875) = radic 1119873 119873sum119894=1

10038171003817100381710038171003817119875CSB119897 (r119894) minus 119875CSB

119897+1 (r119894)10038171003817100381710038171003817210038171003817100381710038171003817119875CSB119897+1

(r119894)100381710038171003817100381710038172 (30)


0 8020 40 60 120 180100 140 160

휃 (∘)

minus60

minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

BiRC

S (d

Bsm

)

Mesh A

Mesh F

Mesh DMesh E

Mesh CMesh B


10minus6

10minus5

10minus4

10minus3

10minus2

Relat

ive e

rror

s of e

xpan

sion

102

103

104

105

106

107

108

109

101

Frequency (Hz)

Q = 30

Q = 42

Q = 58

Q = 82


and here 119875CSB119897 (r119894) 119875CSB


10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

RMS

erro

rs o

f int

erpo

latio

n

102

103

104

105

106

107

108

109

101

Frequency (Hz)

Q = 30

Q = 42

Q = 58

Q = 82


001 m

01 m

Z

YX




10 KHz10 MHz

20 40 60 80 100 120 140 160 1800

120579 (∘)

minus140

minus130

minus120

minus110

minus100

minus90

0

10

20

30

BiRC

S (d

Bsm

)


5 Conclusion




Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Mesh Number ofRWG

Number oftriangles

Near-fieldMem(MB)

ExpansionMem(MB)

TranslationMem (MB)


TotalMem(MB)

MVP time(s)

Iterationnumber

Total time(s)

A 1893 1262 27 215 591 42 875 015 25 41B 7992 5328 102 907 886 84 1979 06 36 228C 33330 22220 435 3783 1182 126 5526 22 52 1184D 139002 92668 178 1541 1478 168 18836 87 74 6558E 563292 375528 7279 6245 17736 21 71704 312 110 3480F 2301804 1534536 29329 25518 2069 252 28683 1209 174 212326

101

102

103

104

105

Mem

ory

(MB)

104

105

106

107

103

Number of unknowns

10minus1

100

101

102

103

Tim

e (s)







119890 = 10038161003816100381610038161003816119875 (r) minus 119875CSB (r)10038161003816100381610038161003816|119875 (r)| (29)

8040 60 100 12020 140 160 1800


10minus4

10minus3

10minus2

10minus1

100

Resid

ual e

rror

s

Mesh A

Mesh FMesh BMesh C

Mesh DMesh E




RMS (119875) = radic 1119873 119873sum119894=1

10038171003817100381710038171003817119875CSB119897 (r119894) minus 119875CSB

119897+1 (r119894)10038171003817100381710038171003817210038171003817100381710038171003817119875CSB119897+1

(r119894)100381710038171003817100381710038172 (30)


0 8020 40 60 120 180100 140 160

휃 (∘)

minus60

minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

BiRC

S (d

Bsm

)

Mesh A

Mesh F

Mesh DMesh E

Mesh CMesh B


10minus6

10minus5

10minus4

10minus3

10minus2

Relat

ive e

rror

s of e

xpan

sion

102

103

104

105

106

107

108

109

101

Frequency (Hz)

Q = 30

Q = 42

Q = 58

Q = 82


and here 119875CSB119897 (r119894) 119875CSB


10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

RMS

erro

rs o

f int

erpo

latio

n

102

103

104

105

106

107

108

109

101

Frequency (Hz)

Q = 30

Q = 42

Q = 58

Q = 82


001 m

01 m

Z

YX




10 KHz10 MHz

20 40 60 80 100 120 140 160 1800

120579 (∘)

minus140

minus130

minus120

minus110

minus100

minus90

0

10

20

30

BiRC

S (d

Bsm

)


5 Conclusion




Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 8020 40 60 120 180100 140 160

휃 (∘)

minus60

minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

BiRC

S (d

Bsm

)

Mesh A

Mesh F

Mesh DMesh E

Mesh CMesh B


10minus6

10minus5

10minus4

10minus3

10minus2

Relat

ive e

rror

s of e

xpan

sion

102

103

104

105

106

107

108

109

101

Frequency (Hz)

Q = 30

Q = 42

Q = 58

Q = 82


and here 119875CSB119897 (r119894) 119875CSB


10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

RMS

erro

rs o

f int

erpo

latio

n

102

103

104

105

106

107

108

109

101

Frequency (Hz)

Q = 30

Q = 42

Q = 58

Q = 82


001 m

01 m

Z

YX




10 KHz10 MHz

20 40 60 80 100 120 140 160 1800

120579 (∘)

minus140

minus130

minus120

minus110

minus100

minus90

0

10

20

30

BiRC

S (d

Bsm

)


5 Conclusion




Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10 KHz10 MHz

20 40 60 80 100 120 140 160 1800

120579 (∘)

minus140

minus130

minus120

minus110

minus100

minus90

0

10

20

30

BiRC

S (d

Bsm

)


5 Conclusion




Acknowledgments


References




























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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