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Research ArticleIntegral Equation Analysis of EM Scattering fromMultilayered Metallic Photonic Crystal Accelerated withAdaptive Cross Approximation
Jianxun Su1 Zengrui Li1 Xujin Yuan2 Yaoqing (Lamar) Yang3 and Junhong Wang4
1Electromagnetic Laboratory Communication University of China Beijing 100024 China2Science and Technology on Electromagnetic Scattering Laboratory Beijing 100854 China3Department of Computer and Electronics Engineering University of Nebraska-Lincoln Lincoln NE 68182 USA4Institute of Lightwave Technology Beijing Jiaotong University Beijing 100044 China
Correspondence should be addressed to Jianxun Su sujianxun jlgx163com
Received 6 March 2015 Accepted 8 July 2015
Academic Editor Luis Landesa
Copyright copy 2015 Jianxun Su 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
A space-domain integral equation method accelerated with adaptive cross approximation (ACA) is presented for the fast andaccurate analysis of electromagnetic (EM) scattering from multilayered metallic photonic crystal (MPC) The method directlysolves for the electric field in order to easily enable the periodic boundary condition (PBC) in the spatial domain The ACA isa purely algebraic method allowing the compression of fully populated matrices hence its formulation and implementation areindependent of integral equation kernel (Greenrsquos function)Therefore the ACA is very well suited for accelerating integral equationanalysis of periodic structurewith the integral kernel of the periodicGreenrsquos function (PGF)The computation of the spatial-domainperiodic Greenrsquos function (PGF) is accelerated by the modified Ewald transformation such that the multilayered periodic structurecan be analyzed efficiently and accurately An effective interpolation method is also proposed to fast compute the periodic Greenrsquosfunction which can greatly reduce the time of matrix filling Numerical examples show that the proposed method can greatly savethe frequency sweep time for multilayered periodic structure
1 Introduction
Photonic crystals [1] are periodic structures of great interestfor their applications both in themicrowave region and in theoptical range The main feature is the presence of frequencybands wherein the waves are highly attenuated and cannotpropagate It results from the removal of degeneracies ofthe free-photon states at the Bragg planes provoked by theperiodicity which produces forbidden frequency gaps so-called photonic band gaps (PBGs) This property is exploitedin the electromagnetic and optical applications
Many numerical methods for computing such bandstructures have been adapted from solid-state physics Theyinclude but are not restricted to plane-wave expansions[2] the Korringa-Kohn-Rostoker method [3] and shellmethodologies [4] Over the past two decades finite dif-ference time domain (FDTD) [5] has been widely used in
computational optics and photonics Recent developmentsinclude specialized versions of the finite element method(FEM) [6] and the flexible local approximation method[7] FDTD and FEM can cope with arbitrary complicatedshapes andmaterials However the radiation into unboundedregions requires either absorbing boundary conditions orperfectly matched layers and special measures need to betaken into account when employing these conditions tothe scattered field formulation For open-region problemsintegral equationmethod of moments (IEMoM) has a greatadvantage
The ACA is a purely algebraic method which is inde-pendent of the kernel The ACA was originally introduced in2000 by Bebendorf [8] and since then has been successfullyapplied for integral formulations involved electromagneticscattering [9 10] The integral kernel is a free space Greenrsquosfunction In addition it is noted that most photonic crystal
Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 274307 7 pageshttpdxdoiorg1011552015274307
2 International Journal of Antennas and Propagation
applications deal with two-dimensional (2D) structures thatare invariant along a longitudinal axis and periodic in thetransverse plane [11] The manufacture of a 2D photoniccrystal structure is easier than that of three-dimensional (3D)one [12]
In this paper a space-domain integral equation acceler-ated with ACA is presented for the fast and accurate analysisof electromagnetic scattering from 2D and 3D multilayeredperiodic structures The ACA is first introduced for surfaceintegral equation with the kernel of periodic Greenrsquos function(PGF) The ACA can be applied to compress impedancematrix blocks and accelerate matrix-vector multiplication(MVP) for reducing computational complexity A modifiedEwald transformation is proposed to efficiently computethe PGFs Compared with traditional Ewald method it hasless computational cost and can eliminate the imbalance oftwo Ewald sequences for multilayered periodic structureAn efficient interpolation method is also proposed to fastcompute the PGFs and then accelerate impedance matrixfilling Our proposed method can effectively reduce thefrequency sweep time for multilayered periodic structureinvolving a lot of unknowns
2 Formulation
Figure 1 shows the multilayered double periodic structurewith identical metallic objects of arbitrary shape periodicallyrepeated in the 119909119910-plane 997888119886 1 and
997888119886 2 are the primitive lattice
vectors 119860 = |997888119886 1 times
997888119886 2| is the cross-sectional area of the unit
cell 997888119864119894
= (120579 cos120572 + 120593 sin120572)1198640 exp(minus119895120573119896 sdot
997888119903 ) is the incident
electric field 120572 is the polarization angle The wave number is120573 = 120596radic120583120576 and 119896 = minus(119909 sin 120579119894 cos120593119894+119910 sin 120579119894 sin120593119894+ cos 120579119894)The wave vector is
997888
119896 = 120573119896 and phase shift
997888
11989611990500 =
997888
119896 |119905 The
array is assumed to be periodic in the 119909119910-plane and the cell ofthe array is obtained by shifting the cell through the relation997888120588119898119899= 119898sdot
997888119886 1+119899sdot
997888119886 2
997888120588119898119899
is defined as the translation vectorof the lattice
21 Integral Equation In this paper we consider the metallicphotonic crystal Let 119878 be the surface of a metallic object Byenforcing the boundary conditions that the total tangentialelectric field is zero on the perfect electric conductor (PEC)surface 119878 the electric field integral equation (EFIE) is givenas [13]
119895120596120583 [int[
997888
119869119904(997888119903
1015840
) +
11205732nabla1015840sdot
997888
119869119904(997888119903
1015840
)nabla]
sdot 119866119901(997888119903 997888119903
1015840
) 1198891199041015840]
119905
=
997888
119864
inc119905
(1)
where 119866119901is a doubly periodic Greenrsquos function The Rao-
Wilton-Glisson (RWG) basis functions on triangular ele-ments are employed to discretize the electric current [14] Inaddition the special basis and test functions derived fromRWG basis function are introduced to ensure current con-tinuity across the periodic boundaries [15] The applicationof the Floquet-Bloch theorem reduces the computational
x
y
zPlane wave120579
120593
a1
a2
Figure 1 Multilayered periodic structure with a general skewedlattice
domain of infinite periodic structures to a single unit cell butleads to the numerical evaluation of very slowly convergingseries
22 Modified Ewald Transformation We propose the follow-ing strategy to compute the spatial periodic Greenrsquos functionin an arbitrary point 997888119906 = 997888119903 minus 997888119903
1015840
If 119906119911= |119911 minus 119911
1015840| gt 05radic119860 PGF is computed by its spectral
representation with exponential convergence and thus noacceleration technique is required [16] Consider
119866119901(997888119903 997888119903
1015840
) =
infin
sum
119898=minusinfin
infin
sum
119899=minusinfin
119890minus119895997888119896119905119898119899sdot(997888120588minus9978881205881015840
)
2119895119860119896119911119898119899
119890minus119895119896119911119898119899|119911minus1199111015840| (2)
where
997888119903 =
997888120588 +119911 sdot
997888
119896119905119898119899
=
997888
11989611990500 +
2120587119860
[119898 (997888119886 2 times ) + 119899 ( times
997888119886 1)]
(3)
is the reciprocal lattice vector Consider
119896119911119898119899
=radic11989620 minus
997888
119896119905119898119899
sdot
997888
119896119905119898119899
(4)
where Re(119896119911119898119899
) ge 0 Im(119896119911119898119899
) le 0If 119906119911= |119911 minus 119911
1015840| lt 05radic119860 PGF is computed by the
representation of Ewald transformation [17] which are twoexponential converging series Consider
119866119901(997888119903 997888119903
1015840
) = 1198661199011 (
997888119903 997888119903
1015840
) +1198661199012 (
997888119903 997888119903
1015840
) (5)
International Journal of Antennas and Propagation 3
Primary source
Field
Image sources
Image sources
(a)
(minus1 0) (0 0) (1 0)
(minus1 minus1) (0 minus1) (1 minus1)
(minus1 1) (0 1) (1 1)
(b)
Figure 2 (a) Periodic image sources in triangular boundary mesh (b) Corresponding lattice numbers (0 0) lattice is the primary unit cell
where
1198661199011 (
997888119903 997888119903
1015840
) =
infin
sum
119898=minusinfin
infin
sum
119899=minusinfin
119890minus119895997888119896119905119898119899sdot(997888120588minus9978881205881015840
)
4119895119860119896119911119898119899
sdot [119890minus119895119896119911119898119899(119911minus1199111015840) erfc(
119895119896119911119898119899
2119864minus (119911 minus 119911
1015840) 119864)
+ 119890119895119896119911119898119899(119911minus1199111015840) erfc(
119895119896119911119898119899
2119864+ (119911 minus 119911
1015840) 119864)]
(6)
1198661199012 (
997888119903 997888119903
1015840
) =
infin
sum
119898=minusinfin
infin
sum
119899=minusinfin
119890minus11989599788811989611990500 sdot997888120588119898119899
4120587119877119898119899
sdot real [1198901198951198960119877119898119899 erfc(119877119898119899119864+
119895119896
2119864)]
(7)
where erfc(119911) is the complementary error function of com-plex argument and 119877
119898119899= |997888119903 minus
997888119903
1015840
minus997888120588119898119899|
By analyzing the asymptotic behavior of the series termsthe approximation to the optimal value for general skewedlattices is given by
119864 = max(radic 120587
119860
119896
2119867) (8)
where1198672 is the maximum exponent permitted In almost allpractical cases it is sufficient to include only nine terms in (6)and (7) (ie the summation limits are fromminus1 to +1) inwhichthe error is usually less than 01
For multilayered periodic structure when 119906119911= |119911 minus 119911
1015840| gt
05radic119860 the PGFs are computed by their spectral represen-tation rather than direct Ewald transformation In doing sothere are two advantages
(1) The spectral representation just requires evaluation ofthe exponential function which is much simpler than
the computation of complex-argument complemen-tary error function (erfc) Compared with erfc(119911)the computational complexity of the exponentialfunction exp(119911) can be ignored
(2) In addition it can eliminate the imbalance of twoEwald sequences when 119906
119911is too large For the multi-
layered structure the complementary error functionsin the first terms (119898 = 119899 = 0) of both Ewald serieswill take large imaginary arguments when 119906
119911is too
largeThese two terms will have very large values thatare approximately equal in amplitude but of oppositesigns so that summing them up will lead to a severeloss of accuracy due to the finite machine precision
23 Singularity Extraction When the distance between theobservation point 997888119903 and the source point 997888119903
1015840
or one of itsperiodic extensions is electrically small the contribution toGreenrsquos function is essentially quasistatic Usually the centerdistance between source and field triangles is less than 01120582then both are considered to be ldquonear interactionrdquo Singularterms in (5) and (7) often arise for example from one ofnine spatial terms 119877
119898119899with119898 119899 = minus1 0 or +1 depending on
which term represents the source nearest to the observer 997888119903 By extracting its singularity [18] the PGF can be expressedas 119866119901= 1198661199011+ [1198661199012minus 1(4120587119877)] + 1(4120587119877) The singularity
subtraction technique [19] used in this paper can compute thesingularity of |997888119903 minus 997888119903
1015840
|119899(119899 ge minus3) type
Supposing the source point nearest to the observer pointappears at (1 0) term rather than (0 0) term as shown inFigure 2 the primary source point can be transferred to (1 0)lattice Then the singular term always appears at (0 0) term11986600
1199012 The singularity of 11986600
1199012can be extracted as
119866001199012 (
997888119903 997888119903
1015840
) = 119865001199012 (
997888119903 997888119903
1015840
) +
14120587119877
(9)
4 International Journal of Antennas and Propagation
o
a2
a1
Gn119909n119910n119911Gn119909+1n119910n119911
Gn119909+1n119910+1n119911Gn119909n119910+1n119911
Gn119909+1n119910n119911+1
Gn119909n119910+1n119911+1
Gn119909n119910n119911+1
Gn119909+1n119910+1n119911+1
dx
dydz
G(u)
Figure 3 Depiction of trilinear interpolation of PGF
where 119865001199012
is the nonsingular part
119865001199012 (
997888119903 997888119903
1015840
) =
14120587119877
sdot real119890minus1198951198960119877 sdot erfc(119877119864minus11989511989602119864
)minus erfc(minus11989511989602119864
)
(10)
As 119877 rarr 0 the result of applying Lrsquo Hopitalrsquos rule on (10) is asfollows
lim119877rarr 0
119865001199012 (
997888119903 997888119903
1015840
)
=
12120587
sdot real [minus1198951198960 erfc(minus11989511989602119864
)minus
2119864radic120587
119890119896204119864
2]
(11)
24 Interpolation of PGF PGFs are very smooth andamenable to interpolation Fast interpolation of PGFs cangreatly reduce the time of matrix filling The reason is thatinterpolating the PGF values from the table is much fasterthan directly computing the Ewald sums (since this has tobe done for every pair of source and field triangles in thenumerical integration using cubature formulas) Trilinearinterpolation [20] suited for the calculation of complex-valued function with very high efficiency is applied hereAs shown in Figure 3 the cuboid grouping the unit cell isequally divided intomany subgrids with side length 119871 Yellowpoints are the precomputed PGFs at the corner of the subgridThen the three-dimensional PGF values can be obtained bytrilinear interpolation
119866(997888119906) = (1minus120572
119909) (1minus120572
119910) (1minus120572
119911) 119866119899119909+1119899119910+1119899119911+1
+ (1minus120572119909) (1minus120572
119910) 120572119911119866119899119909+1119899119910+1119899119911+2
+ (1minus120572119909) 120572119910(1minus120572
119911) 119866119899119909+1119899119910+2119899119911+1
+ (1minus120572119909) 120572119910120572119911119866119899119909+1119899119910+2119899119911+2
+120572119909(1minus120572
119910) (1minus120572
119911) 119866119899119909+2119899119910+1119899119911+1
+120572119909(1minus120572
119910) 120572119911119866119899119909+2119899119910+1119899119911+2
+120572119909120572119910(1minus120572
119911) 119866119899119909+2119899119910+2119899119911+1
+120572119909120572119910120572119911119866119899119909+2119899119910+2119899119911+2
(12)
where 120572119909= 119889119909119871 120572119910= 119889119910119871 120572119911= 119889119911119871
The relative error 120576 obtained by interpolating the PGFs isdefined as
120576 =
100381610038161003816100381610038161003816100381610038161003816
119866intminus 119866
ex
119866ex
100381610038161003816100381610038161003816100381610038161003816
(13)
where 119866int and 119866
ex are the interpolated and the exactdata respectively The exact data are obtained by the aboveaccelerating method with an accuracy 120576 = 10minus7 The relativeerrors are less than 120576 = 10minus7 and have therefore beenneglected
Take the multilayered structure with orthogonal latticesas an example The relative error is shown in Figure 4 forthe interpolation of PGF involving a series of interpolationpoints through the trilinear interpolation method withvarious values of 119873int In Figure 4 positions (A D) arethe sampling points for interpolation A very high level ofaccuracy is easily obtained with a relatively small number ofinterpolation points The side length of the subgrid is usuallyfixed to 002120582 and the higher interpolation accuracy (120576 lt
10minus4) can be obtained
3 ACA Technique
Let the119898times119899 rectangular matrix119885119898times119899
represent the couplingbetween twowell-separated groups in theMoMcomputationThe ACA algorithm aims to accurately approximate 119885
119898times119899by
a low-rank matrix 119885119898times119899
In particular the ACA algorithm
International Journal of Antennas and Propagation 5
o
B
C
D
AB
CD
A
10 20 30 40 50 60 70 80 90 100 1101E minus 7
1E minus 6
1E minus 5
1E minus 4
1E minus 3
Rela
tive e
rror
120576Nint
a1
a2
Figure 4 The relative error of a series of points Reference structure primitive vector |9978881198861| = |
9978881198862| = 055120582 ℎ = 22120582 phase shifts 119896
119909=
119896 cos(45∘) 119896119910= 119896 sin(30∘) Accuracy reached interpolatingPGFs through trilinear interpolationwith a 3Dgrid of119873inttimes119873inttimes2119873int equispaced
points in minus055120582 lt Δ119909 lt 055120582 minus055120582 lt Δ119910 lt 055120582 0 lt Δ119911 lt 22120582 The data plotted refers to the straight line from (015120582 015120582 0) to(015120582 015120582 22120582) The number of equally spaced sampling points is 4
constructs the approximate matrix 119885119898times119899
through a productform Namely
119885119898times119899
= 119880119898times119903
119881119903times119899
=
119903
sum
119894=1119906119894
119898times1V119894
119899times1 (14)
where 119903 is the rank of 119885119898times119899
or the effective rank of the matrix119885119898times119899
119880119898times119903
and 119881119903times119899
are two dense rectangular matrices119906119894
119898times1 is the column 119894th of 119880 and V119894119899times1 is the row 119894th of 119881
respectively The goal of ACA is to achieve10038171003817100381710038171003817119885119898times119899
minus119885119898times119899
10038171003817100381710038171003817119865le 120576
1003817100381710038171003817119885119898times119899
1003817100381710038171003817119865
(15)
for a given tolerance 120576 Note that in this paper sdot refers tothe matrix Frobenus norm [9]
The ACA technique is numerically based on the principleof the approximation above but does not need to knowall terms of 119885
119898times119899 Indeed only 119903 rows and 119903 columns
are calculated during the algorithm allowing a significantreduction in the number of integral computations to performTherefore not only is the memory requirement reduced butthe assembly time is also saved Yet if we agree to deal withtheMVP119880
119898times119903119881119903times119899119887119899times1 instead of the exact value of119885119898times119899119887119899times1
the number of operations needed is also remarkably reducedby performing 119880
119898times119903(119881119903times119899119887119899times1) Thus if an iterative solver is
used one can accelerate matrix vector multiply and save onceagain the computation time [10]
The compression rate 120578 of matrix block 119885119898times119899
whichdenotes the memory saved by using ACA can be expressedas
120578 () = (1minus 119903 sdot (119898 + 119899)
119898 sdot 119899
) sdot 100 (16)
Obviously the approximation byACA approach above is use-ful only if 119903 lt (12) sdotmin119898 119899 where 119885
119898times119899presents similar
values For the electromagnetic application the effective rank119903 lt min(119898 119899) Therefore instead of storing entire 119898 times 119899
entries the algorithm only requires storing (119898+119899)times119903 entries
4 Numerical Results
The EFIE and ACA approaches to PEC are implementedusing Fortran-95 language Numerical experiments run ona PC with Intel i5-2400 31 GHz CPU and 16GB RAMThe resulting impedance matrices are iteratively solved byusing the transpose-free quasi-minimal residual (TFQMR)solver with diagonal preconditioning [21] the relative errortolerance is set to be 10minus3
41 Infinite Metallic Rod As a 2D example we study thescattering properties of a photonic band gap (PBG) structureconsisting of infinite metallic rod [17] Each unit cell consistsof two infinitely long metallic rods The inset of Figure 5 istwo meshed cylinders in a unit cell The axis of each cylinderis along 119909-direction 997888119886
1= 119909 6mm and 997888119886
2= 119910 6mm Each
cylinder has the radius 06mm and length 6mm The spacebetween two cylinders is 6mm Each cylinder was discretizedinto 494 triangle facets with 741 unknowns to ensure accurateresults obtained in the entire frequency band Figure 5 showsthe magnitude of the transmission and reflection coefficientsfor the Floquet TM
00mode with the electric field oriented
in the 119909-direction Our results agree well with the referenceresults obtained using 2D IE approach [15] The resultsconverged for 9 terms in both Ewald sums and there are 1482unknowns for the geometry The time for matrix filling andsolving is 058 s and 021 s per frequency point respectivelyInACA the box size is 6times6times6mm3 that is each box includes
6 International Journal of Antennas and Propagation
minus50
minus40
minus30
minus20
minus10
0
Refl
and
Tran
s co
effici
ents
(dB)
Frequency (GHz)
Refl Trans Refl [15]
o
1
2
Unit cell
10 50403020
a1
a2
Figure 5 Reflection and transmission coefficient of two-layer rodPBG
20 4540353025minus80
minus60
minus40
minus20
0
Tran
simiss
ion
(dB)
Frequency (GHz)
5-layer10-layer21-layer
Layer
1
2
3
N minus 1
N
Figure 6 Transmission coefficient of multilayered PBG
one cylinder If the trilinear interpolation method is notused the time ofmatrix filling would increase dramatically to1286 s Therefore trilinear interpolation of PGFs can reducethe time of matrix filling by more than 995
Next the number of layers increases from 2 to 5 10 and21 respectively Figure 6 shows the transmission coefficientofmultilayered structureThe transmission coefficient curvesexhibit the typical resonances of photonic band gapmaterialsIt can be seen that the null of transmission becomes deeperas the layer number increasesThe total solution times for perfrequency point are 51 s 124 s and 325 s respectively
The 21-layer structure is meshed into 10374 triangle facetswith 15561 unknownsThe compression rate 120578 ofmatrix blockis shown in Table 1 The times for matrix filling and matrixsolving are 238 s and 87 s per frequency point respectively
10 201816141200
02
04
06
08
10
Refle
ctio
n
4-layer 8-layer
16-layer 16-layer [Ansys HFSS]
w
d
h
120582 (120583m)
Figure 7 Reflection spectra for multilayered woodpile MPC
Table 1 Compression rate 120578 () of matrix block
11988511
11988512
11988513
11988514
11988515
11988516
11988517
mdash 2078 864 540 594 567 48611988518
11988519
119885110
119885111
119885112
119885113
119885114
621 567 405 351 378 459 432119885115
119885116
119885117
119885118
119885119
119885120
119885121
486 540 432 378 405 324 351mdash denotes the matrix block calculated directly by MoM
The corresponding times of direct solution by MoM withoutACA acceleration are 1876 s and 752 s per frequency pointrespectively Obviously ACA can greatly reduce the totalsolution time especially for frequency sweeping
42 Woodpile A 3D example showing the inset in Figure 7is a woodpile MPC with 119908 = 200 nm ℎ = 300 nm andin-layer lattice constant 119889 = 1 120583m The incident wave is aTE polarized (ie 119864 parallel to the top metallic layer) planewave through the stacking direction The number of layersare 4 8 and 16 respectively Two intersecting metallic pipesare considered to be fully contacting with each other innumerical modelingTheACA box size is 1times1times03 120583m3Thereflection coefficients are shown in Figure 7 Almost the sameresult is obtained for TM polarization Our numerical resultsagree well with that of Ansys HFSS Previous results reportedby other studies [12 22] for a similar kind of the structuralgeometry also validate our calculated results For the 16-layerMPC with 13149 unknowns (8766 triangle facets) the timesformatrix filling and linear system solving are 403 s and 152 sper frequency point respectivelyThe corresponding times ofdirect solution by MoMwithout ACA acceleration are 1323 sand 925 s respectively
5 Conclusion
An integral equation accelerated with adaptive cross approx-imation (ACA) is proposed for the analysis of multilayeredmetallic photonic crystal The ACA is a purely algebraic and
International Journal of Antennas and Propagation 7
kernel independent algorithm It is verified to be suited foraccelerating integral equation analysis of periodic structureThe ACA can be applied to compress impedance matrixblocks and accelerate MVP for reducing computational com-plexity A modified Ewald transformation and an efficientinterpolation method are also proposed to fast compute theperiodic Greenrsquos function hence that can greatly reducethe time of matrix filling Numerical results confirm thevalidity and accuracy of the proposed method In our nextstage of research ACA will be applied to accelerate hybridfield integral equation for arbitrarily complex multilayeredperiodic structure composed ofmetal and dielectricmaterialswith more unknowns
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was supported in part by the National ScienceFoundation of China (NSFC) under Grants no 61331002no 61102011 and no 61201082 and in part by ExcellentInnovation Team of CUC under Grant no yxtd201303
References
[1] J D Joannopoulos R D Meade and J N Winn PhotonicCrystals Molding the Flow of Light Princeton University PressPrinceton NJ USA 1995
[2] R D Meade A M Rappe K D Brommer J D Joannopoulosand O L Alerhand ldquoAccurate theoretical analysis of photonicband-gapmaterialsrdquo Physical Review B vol 48 no 11 pp 8434ndash8437 1993
[3] X Wang X-G Zhang Q Yu and B N Harmon ldquoMultiple-scattering theory for electro-magnetic wavesrdquo Physical ReviewB vol 47 no 8 pp 4161ndash4167 1993
[4] J B Pendry ldquoPhotonic structuresrdquo Journal of Modern Opticsvol 41 pp 209ndash229 1994
[5] J A Roden S D Gedney M P Kesler J G Maloney andP H Harms ldquoTime-domain analysis of periodic structuresat oblique incidence orthogonal and nonorthogonal FDTDimplementationsrdquo IEEE Transactions on MicrowaveTheory andTechniques vol 46 no 4 pp 420ndash427 1998
[6] A A Tavallaee and J P Webb ldquoFinite-element modeling ofevanescent modes in the stopband of periodic structuresrdquo IEEETransactions on Magnetics vol 44 no 6 pp 1358ndash1361 2008
[7] I Tsukerman and F Cajko ldquoPhotonic band structure compu-tation using FLAMErdquo IEEE Transactions on Magnetics vol 44no 6 pp 1382ndash1385 2008
[8] M Bebendorf ldquoApproximation of boundary element matricesrdquoNumerische Mathematik vol 86 no 4 pp 565ndash589 2000
[9] K ZhaoM Vouvakis and J-F Lee ldquoThe adaptive cross approx-imation algorithm for accelerated method of moments compu-tations of EMC problemsrdquo IEEE Transactions on Electromag-netic Compatibility vol 47 no 4 pp 763ndash773 2005
[10] A Heldring J M Tamayo C Simon E Ubeda and J M RiusldquoSparsified adaptive cross approximation algorithm for acceler-ated method of moments computationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 1 pp 240ndash246 2013
[11] M Sarnowski T Vaupel V Hansen E Kreysa and H PGemuend ldquoCharacterization of diffraction anomalies in 2-Dphotonic bandgap structuresrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 49 no 10 pp 1868ndash1872 2001
[12] S Y Lin G Arjavalingam andWM Robertson ldquoInvestigationof absolute photonic bandgaps in 2-dimensional dielectricstructuresrdquo Journal of Modern Optics vol 41 pp 385ndash393 1994
[13] W C Chew J M Jin E Michielssen and J Song Fast andEfficient Algorithm in Computational Electromagnetics chapter3 Artech House Boston Mass USA 2001
[14] 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
[15] F-G Hu and J Song ldquoIntegral-equation analysis of scatteringfrom doubly periodic array of 3-D conducting objectsrdquo IEEETransactions on Antennas and Propagation vol 59 no 12 pp4569ndash4578 2011
[16] M G Silveirinha and C A Fernandes ldquoA new accelerationtechnique with exponential convergence rate to evaluate peri-odic Green functionsrdquo IEEE Transactions on Antennas andPropagation vol 53 no 1 pp 347ndash355 2005
[17] K E Jordan G R Richter and P Sheng ldquoAn efficient numericalevaluation of the Greenrsquos function for the Helmholtz operatoron periodic structuresrdquo Journal of Computational Physics vol63 no 1 pp 222ndash235 1986
[18] R D Graglia ldquoOn the numerical integration of the linear shapefunctions times the 3-D Greenrsquos function or its gradient on aplane trianglerdquo IEEETransactions onAntennas andPropagationvol 41 no 10 pp 1448ndash1455 1993
[19] S Jarvenpaa M Taskinen and P Yla-Oijala ldquoSingularitysubtraction technique for high-order polynomial vector basisfunctions on planar trianglesrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 42ndash49 2006
[20] J F Hughes and A V Dam Computer Graphics Principles andPractice Addison-Wesley 3rd edition 2013
[21] Y Saad Iterative Methods for Sparse Linear Systems SIAMSociety for Industrial amp Applied Mathematics 2nd edition2003
[22] J G Fleming S-Y Lin I El-Kady R Biswas andKMHo ldquoAll-metallic three-dimensional photonic crytal with a large infraredbandgaprdquo Nature vol 417 no 6884 pp 52ndash55 2002
International Journal of
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Journal ofEngineeringVolume 2014
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Shock and Vibration
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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DistributedSensor Networks
International Journal of
2 International Journal of Antennas and Propagation
applications deal with two-dimensional (2D) structures thatare invariant along a longitudinal axis and periodic in thetransverse plane [11] The manufacture of a 2D photoniccrystal structure is easier than that of three-dimensional (3D)one [12]
In this paper a space-domain integral equation acceler-ated with ACA is presented for the fast and accurate analysisof electromagnetic scattering from 2D and 3D multilayeredperiodic structures The ACA is first introduced for surfaceintegral equation with the kernel of periodic Greenrsquos function(PGF) The ACA can be applied to compress impedancematrix blocks and accelerate matrix-vector multiplication(MVP) for reducing computational complexity A modifiedEwald transformation is proposed to efficiently computethe PGFs Compared with traditional Ewald method it hasless computational cost and can eliminate the imbalance oftwo Ewald sequences for multilayered periodic structureAn efficient interpolation method is also proposed to fastcompute the PGFs and then accelerate impedance matrixfilling Our proposed method can effectively reduce thefrequency sweep time for multilayered periodic structureinvolving a lot of unknowns
2 Formulation
Figure 1 shows the multilayered double periodic structurewith identical metallic objects of arbitrary shape periodicallyrepeated in the 119909119910-plane 997888119886 1 and
997888119886 2 are the primitive lattice
vectors 119860 = |997888119886 1 times
997888119886 2| is the cross-sectional area of the unit
cell 997888119864119894
= (120579 cos120572 + 120593 sin120572)1198640 exp(minus119895120573119896 sdot
997888119903 ) is the incident
electric field 120572 is the polarization angle The wave number is120573 = 120596radic120583120576 and 119896 = minus(119909 sin 120579119894 cos120593119894+119910 sin 120579119894 sin120593119894+ cos 120579119894)The wave vector is
997888
119896 = 120573119896 and phase shift
997888
11989611990500 =
997888
119896 |119905 The
array is assumed to be periodic in the 119909119910-plane and the cell ofthe array is obtained by shifting the cell through the relation997888120588119898119899= 119898sdot
997888119886 1+119899sdot
997888119886 2
997888120588119898119899
is defined as the translation vectorof the lattice
21 Integral Equation In this paper we consider the metallicphotonic crystal Let 119878 be the surface of a metallic object Byenforcing the boundary conditions that the total tangentialelectric field is zero on the perfect electric conductor (PEC)surface 119878 the electric field integral equation (EFIE) is givenas [13]
119895120596120583 [int[
997888
119869119904(997888119903
1015840
) +
11205732nabla1015840sdot
997888
119869119904(997888119903
1015840
)nabla]
sdot 119866119901(997888119903 997888119903
1015840
) 1198891199041015840]
119905
=
997888
119864
inc119905
(1)
where 119866119901is a doubly periodic Greenrsquos function The Rao-
Wilton-Glisson (RWG) basis functions on triangular ele-ments are employed to discretize the electric current [14] Inaddition the special basis and test functions derived fromRWG basis function are introduced to ensure current con-tinuity across the periodic boundaries [15] The applicationof the Floquet-Bloch theorem reduces the computational
x
y
zPlane wave120579
120593
a1
a2
Figure 1 Multilayered periodic structure with a general skewedlattice
domain of infinite periodic structures to a single unit cell butleads to the numerical evaluation of very slowly convergingseries
22 Modified Ewald Transformation We propose the follow-ing strategy to compute the spatial periodic Greenrsquos functionin an arbitrary point 997888119906 = 997888119903 minus 997888119903
1015840
If 119906119911= |119911 minus 119911
1015840| gt 05radic119860 PGF is computed by its spectral
representation with exponential convergence and thus noacceleration technique is required [16] Consider
119866119901(997888119903 997888119903
1015840
) =
infin
sum
119898=minusinfin
infin
sum
119899=minusinfin
119890minus119895997888119896119905119898119899sdot(997888120588minus9978881205881015840
)
2119895119860119896119911119898119899
119890minus119895119896119911119898119899|119911minus1199111015840| (2)
where
997888119903 =
997888120588 +119911 sdot
997888
119896119905119898119899
=
997888
11989611990500 +
2120587119860
[119898 (997888119886 2 times ) + 119899 ( times
997888119886 1)]
(3)
is the reciprocal lattice vector Consider
119896119911119898119899
=radic11989620 minus
997888
119896119905119898119899
sdot
997888
119896119905119898119899
(4)
where Re(119896119911119898119899
) ge 0 Im(119896119911119898119899
) le 0If 119906119911= |119911 minus 119911
1015840| lt 05radic119860 PGF is computed by the
representation of Ewald transformation [17] which are twoexponential converging series Consider
119866119901(997888119903 997888119903
1015840
) = 1198661199011 (
997888119903 997888119903
1015840
) +1198661199012 (
997888119903 997888119903
1015840
) (5)
International Journal of Antennas and Propagation 3
Primary source
Field
Image sources
Image sources
(a)
(minus1 0) (0 0) (1 0)
(minus1 minus1) (0 minus1) (1 minus1)
(minus1 1) (0 1) (1 1)
(b)
Figure 2 (a) Periodic image sources in triangular boundary mesh (b) Corresponding lattice numbers (0 0) lattice is the primary unit cell
where
1198661199011 (
997888119903 997888119903
1015840
) =
infin
sum
119898=minusinfin
infin
sum
119899=minusinfin
119890minus119895997888119896119905119898119899sdot(997888120588minus9978881205881015840
)
4119895119860119896119911119898119899
sdot [119890minus119895119896119911119898119899(119911minus1199111015840) erfc(
119895119896119911119898119899
2119864minus (119911 minus 119911
1015840) 119864)
+ 119890119895119896119911119898119899(119911minus1199111015840) erfc(
119895119896119911119898119899
2119864+ (119911 minus 119911
1015840) 119864)]
(6)
1198661199012 (
997888119903 997888119903
1015840
) =
infin
sum
119898=minusinfin
infin
sum
119899=minusinfin
119890minus11989599788811989611990500 sdot997888120588119898119899
4120587119877119898119899
sdot real [1198901198951198960119877119898119899 erfc(119877119898119899119864+
119895119896
2119864)]
(7)
where erfc(119911) is the complementary error function of com-plex argument and 119877
119898119899= |997888119903 minus
997888119903
1015840
minus997888120588119898119899|
By analyzing the asymptotic behavior of the series termsthe approximation to the optimal value for general skewedlattices is given by
119864 = max(radic 120587
119860
119896
2119867) (8)
where1198672 is the maximum exponent permitted In almost allpractical cases it is sufficient to include only nine terms in (6)and (7) (ie the summation limits are fromminus1 to +1) inwhichthe error is usually less than 01
For multilayered periodic structure when 119906119911= |119911 minus 119911
1015840| gt
05radic119860 the PGFs are computed by their spectral represen-tation rather than direct Ewald transformation In doing sothere are two advantages
(1) The spectral representation just requires evaluation ofthe exponential function which is much simpler than
the computation of complex-argument complemen-tary error function (erfc) Compared with erfc(119911)the computational complexity of the exponentialfunction exp(119911) can be ignored
(2) In addition it can eliminate the imbalance of twoEwald sequences when 119906
119911is too large For the multi-
layered structure the complementary error functionsin the first terms (119898 = 119899 = 0) of both Ewald serieswill take large imaginary arguments when 119906
119911is too
largeThese two terms will have very large values thatare approximately equal in amplitude but of oppositesigns so that summing them up will lead to a severeloss of accuracy due to the finite machine precision
23 Singularity Extraction When the distance between theobservation point 997888119903 and the source point 997888119903
1015840
or one of itsperiodic extensions is electrically small the contribution toGreenrsquos function is essentially quasistatic Usually the centerdistance between source and field triangles is less than 01120582then both are considered to be ldquonear interactionrdquo Singularterms in (5) and (7) often arise for example from one ofnine spatial terms 119877
119898119899with119898 119899 = minus1 0 or +1 depending on
which term represents the source nearest to the observer 997888119903 By extracting its singularity [18] the PGF can be expressedas 119866119901= 1198661199011+ [1198661199012minus 1(4120587119877)] + 1(4120587119877) The singularity
subtraction technique [19] used in this paper can compute thesingularity of |997888119903 minus 997888119903
1015840
|119899(119899 ge minus3) type
Supposing the source point nearest to the observer pointappears at (1 0) term rather than (0 0) term as shown inFigure 2 the primary source point can be transferred to (1 0)lattice Then the singular term always appears at (0 0) term11986600
1199012 The singularity of 11986600
1199012can be extracted as
119866001199012 (
997888119903 997888119903
1015840
) = 119865001199012 (
997888119903 997888119903
1015840
) +
14120587119877
(9)
4 International Journal of Antennas and Propagation
o
a2
a1
Gn119909n119910n119911Gn119909+1n119910n119911
Gn119909+1n119910+1n119911Gn119909n119910+1n119911
Gn119909+1n119910n119911+1
Gn119909n119910+1n119911+1
Gn119909n119910n119911+1
Gn119909+1n119910+1n119911+1
dx
dydz
G(u)
Figure 3 Depiction of trilinear interpolation of PGF
where 119865001199012
is the nonsingular part
119865001199012 (
997888119903 997888119903
1015840
) =
14120587119877
sdot real119890minus1198951198960119877 sdot erfc(119877119864minus11989511989602119864
)minus erfc(minus11989511989602119864
)
(10)
As 119877 rarr 0 the result of applying Lrsquo Hopitalrsquos rule on (10) is asfollows
lim119877rarr 0
119865001199012 (
997888119903 997888119903
1015840
)
=
12120587
sdot real [minus1198951198960 erfc(minus11989511989602119864
)minus
2119864radic120587
119890119896204119864
2]
(11)
24 Interpolation of PGF PGFs are very smooth andamenable to interpolation Fast interpolation of PGFs cangreatly reduce the time of matrix filling The reason is thatinterpolating the PGF values from the table is much fasterthan directly computing the Ewald sums (since this has tobe done for every pair of source and field triangles in thenumerical integration using cubature formulas) Trilinearinterpolation [20] suited for the calculation of complex-valued function with very high efficiency is applied hereAs shown in Figure 3 the cuboid grouping the unit cell isequally divided intomany subgrids with side length 119871 Yellowpoints are the precomputed PGFs at the corner of the subgridThen the three-dimensional PGF values can be obtained bytrilinear interpolation
119866(997888119906) = (1minus120572
119909) (1minus120572
119910) (1minus120572
119911) 119866119899119909+1119899119910+1119899119911+1
+ (1minus120572119909) (1minus120572
119910) 120572119911119866119899119909+1119899119910+1119899119911+2
+ (1minus120572119909) 120572119910(1minus120572
119911) 119866119899119909+1119899119910+2119899119911+1
+ (1minus120572119909) 120572119910120572119911119866119899119909+1119899119910+2119899119911+2
+120572119909(1minus120572
119910) (1minus120572
119911) 119866119899119909+2119899119910+1119899119911+1
+120572119909(1minus120572
119910) 120572119911119866119899119909+2119899119910+1119899119911+2
+120572119909120572119910(1minus120572
119911) 119866119899119909+2119899119910+2119899119911+1
+120572119909120572119910120572119911119866119899119909+2119899119910+2119899119911+2
(12)
where 120572119909= 119889119909119871 120572119910= 119889119910119871 120572119911= 119889119911119871
The relative error 120576 obtained by interpolating the PGFs isdefined as
120576 =
100381610038161003816100381610038161003816100381610038161003816
119866intminus 119866
ex
119866ex
100381610038161003816100381610038161003816100381610038161003816
(13)
where 119866int and 119866
ex are the interpolated and the exactdata respectively The exact data are obtained by the aboveaccelerating method with an accuracy 120576 = 10minus7 The relativeerrors are less than 120576 = 10minus7 and have therefore beenneglected
Take the multilayered structure with orthogonal latticesas an example The relative error is shown in Figure 4 forthe interpolation of PGF involving a series of interpolationpoints through the trilinear interpolation method withvarious values of 119873int In Figure 4 positions (A D) arethe sampling points for interpolation A very high level ofaccuracy is easily obtained with a relatively small number ofinterpolation points The side length of the subgrid is usuallyfixed to 002120582 and the higher interpolation accuracy (120576 lt
10minus4) can be obtained
3 ACA Technique
Let the119898times119899 rectangular matrix119885119898times119899
represent the couplingbetween twowell-separated groups in theMoMcomputationThe ACA algorithm aims to accurately approximate 119885
119898times119899by
a low-rank matrix 119885119898times119899
In particular the ACA algorithm
International Journal of Antennas and Propagation 5
o
B
C
D
AB
CD
A
10 20 30 40 50 60 70 80 90 100 1101E minus 7
1E minus 6
1E minus 5
1E minus 4
1E minus 3
Rela
tive e
rror
120576Nint
a1
a2
Figure 4 The relative error of a series of points Reference structure primitive vector |9978881198861| = |
9978881198862| = 055120582 ℎ = 22120582 phase shifts 119896
119909=
119896 cos(45∘) 119896119910= 119896 sin(30∘) Accuracy reached interpolatingPGFs through trilinear interpolationwith a 3Dgrid of119873inttimes119873inttimes2119873int equispaced
points in minus055120582 lt Δ119909 lt 055120582 minus055120582 lt Δ119910 lt 055120582 0 lt Δ119911 lt 22120582 The data plotted refers to the straight line from (015120582 015120582 0) to(015120582 015120582 22120582) The number of equally spaced sampling points is 4
constructs the approximate matrix 119885119898times119899
through a productform Namely
119885119898times119899
= 119880119898times119903
119881119903times119899
=
119903
sum
119894=1119906119894
119898times1V119894
119899times1 (14)
where 119903 is the rank of 119885119898times119899
or the effective rank of the matrix119885119898times119899
119880119898times119903
and 119881119903times119899
are two dense rectangular matrices119906119894
119898times1 is the column 119894th of 119880 and V119894119899times1 is the row 119894th of 119881
respectively The goal of ACA is to achieve10038171003817100381710038171003817119885119898times119899
minus119885119898times119899
10038171003817100381710038171003817119865le 120576
1003817100381710038171003817119885119898times119899
1003817100381710038171003817119865
(15)
for a given tolerance 120576 Note that in this paper sdot refers tothe matrix Frobenus norm [9]
The ACA technique is numerically based on the principleof the approximation above but does not need to knowall terms of 119885
119898times119899 Indeed only 119903 rows and 119903 columns
are calculated during the algorithm allowing a significantreduction in the number of integral computations to performTherefore not only is the memory requirement reduced butthe assembly time is also saved Yet if we agree to deal withtheMVP119880
119898times119903119881119903times119899119887119899times1 instead of the exact value of119885119898times119899119887119899times1
the number of operations needed is also remarkably reducedby performing 119880
119898times119903(119881119903times119899119887119899times1) Thus if an iterative solver is
used one can accelerate matrix vector multiply and save onceagain the computation time [10]
The compression rate 120578 of matrix block 119885119898times119899
whichdenotes the memory saved by using ACA can be expressedas
120578 () = (1minus 119903 sdot (119898 + 119899)
119898 sdot 119899
) sdot 100 (16)
Obviously the approximation byACA approach above is use-ful only if 119903 lt (12) sdotmin119898 119899 where 119885
119898times119899presents similar
values For the electromagnetic application the effective rank119903 lt min(119898 119899) Therefore instead of storing entire 119898 times 119899
entries the algorithm only requires storing (119898+119899)times119903 entries
4 Numerical Results
The EFIE and ACA approaches to PEC are implementedusing Fortran-95 language Numerical experiments run ona PC with Intel i5-2400 31 GHz CPU and 16GB RAMThe resulting impedance matrices are iteratively solved byusing the transpose-free quasi-minimal residual (TFQMR)solver with diagonal preconditioning [21] the relative errortolerance is set to be 10minus3
41 Infinite Metallic Rod As a 2D example we study thescattering properties of a photonic band gap (PBG) structureconsisting of infinite metallic rod [17] Each unit cell consistsof two infinitely long metallic rods The inset of Figure 5 istwo meshed cylinders in a unit cell The axis of each cylinderis along 119909-direction 997888119886
1= 119909 6mm and 997888119886
2= 119910 6mm Each
cylinder has the radius 06mm and length 6mm The spacebetween two cylinders is 6mm Each cylinder was discretizedinto 494 triangle facets with 741 unknowns to ensure accurateresults obtained in the entire frequency band Figure 5 showsthe magnitude of the transmission and reflection coefficientsfor the Floquet TM
00mode with the electric field oriented
in the 119909-direction Our results agree well with the referenceresults obtained using 2D IE approach [15] The resultsconverged for 9 terms in both Ewald sums and there are 1482unknowns for the geometry The time for matrix filling andsolving is 058 s and 021 s per frequency point respectivelyInACA the box size is 6times6times6mm3 that is each box includes
6 International Journal of Antennas and Propagation
minus50
minus40
minus30
minus20
minus10
0
Refl
and
Tran
s co
effici
ents
(dB)
Frequency (GHz)
Refl Trans Refl [15]
o
1
2
Unit cell
10 50403020
a1
a2
Figure 5 Reflection and transmission coefficient of two-layer rodPBG
20 4540353025minus80
minus60
minus40
minus20
0
Tran
simiss
ion
(dB)
Frequency (GHz)
5-layer10-layer21-layer
Layer
1
2
3
N minus 1
N
Figure 6 Transmission coefficient of multilayered PBG
one cylinder If the trilinear interpolation method is notused the time ofmatrix filling would increase dramatically to1286 s Therefore trilinear interpolation of PGFs can reducethe time of matrix filling by more than 995
Next the number of layers increases from 2 to 5 10 and21 respectively Figure 6 shows the transmission coefficientofmultilayered structureThe transmission coefficient curvesexhibit the typical resonances of photonic band gapmaterialsIt can be seen that the null of transmission becomes deeperas the layer number increasesThe total solution times for perfrequency point are 51 s 124 s and 325 s respectively
The 21-layer structure is meshed into 10374 triangle facetswith 15561 unknownsThe compression rate 120578 ofmatrix blockis shown in Table 1 The times for matrix filling and matrixsolving are 238 s and 87 s per frequency point respectively
10 201816141200
02
04
06
08
10
Refle
ctio
n
4-layer 8-layer
16-layer 16-layer [Ansys HFSS]
w
d
h
120582 (120583m)
Figure 7 Reflection spectra for multilayered woodpile MPC
Table 1 Compression rate 120578 () of matrix block
11988511
11988512
11988513
11988514
11988515
11988516
11988517
mdash 2078 864 540 594 567 48611988518
11988519
119885110
119885111
119885112
119885113
119885114
621 567 405 351 378 459 432119885115
119885116
119885117
119885118
119885119
119885120
119885121
486 540 432 378 405 324 351mdash denotes the matrix block calculated directly by MoM
The corresponding times of direct solution by MoM withoutACA acceleration are 1876 s and 752 s per frequency pointrespectively Obviously ACA can greatly reduce the totalsolution time especially for frequency sweeping
42 Woodpile A 3D example showing the inset in Figure 7is a woodpile MPC with 119908 = 200 nm ℎ = 300 nm andin-layer lattice constant 119889 = 1 120583m The incident wave is aTE polarized (ie 119864 parallel to the top metallic layer) planewave through the stacking direction The number of layersare 4 8 and 16 respectively Two intersecting metallic pipesare considered to be fully contacting with each other innumerical modelingTheACA box size is 1times1times03 120583m3Thereflection coefficients are shown in Figure 7 Almost the sameresult is obtained for TM polarization Our numerical resultsagree well with that of Ansys HFSS Previous results reportedby other studies [12 22] for a similar kind of the structuralgeometry also validate our calculated results For the 16-layerMPC with 13149 unknowns (8766 triangle facets) the timesformatrix filling and linear system solving are 403 s and 152 sper frequency point respectivelyThe corresponding times ofdirect solution by MoMwithout ACA acceleration are 1323 sand 925 s respectively
5 Conclusion
An integral equation accelerated with adaptive cross approx-imation (ACA) is proposed for the analysis of multilayeredmetallic photonic crystal The ACA is a purely algebraic and
International Journal of Antennas and Propagation 7
kernel independent algorithm It is verified to be suited foraccelerating integral equation analysis of periodic structureThe ACA can be applied to compress impedance matrixblocks and accelerate MVP for reducing computational com-plexity A modified Ewald transformation and an efficientinterpolation method are also proposed to fast compute theperiodic Greenrsquos function hence that can greatly reducethe time of matrix filling Numerical results confirm thevalidity and accuracy of the proposed method In our nextstage of research ACA will be applied to accelerate hybridfield integral equation for arbitrarily complex multilayeredperiodic structure composed ofmetal and dielectricmaterialswith more unknowns
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was supported in part by the National ScienceFoundation of China (NSFC) under Grants no 61331002no 61102011 and no 61201082 and in part by ExcellentInnovation Team of CUC under Grant no yxtd201303
References
[1] J D Joannopoulos R D Meade and J N Winn PhotonicCrystals Molding the Flow of Light Princeton University PressPrinceton NJ USA 1995
[2] R D Meade A M Rappe K D Brommer J D Joannopoulosand O L Alerhand ldquoAccurate theoretical analysis of photonicband-gapmaterialsrdquo Physical Review B vol 48 no 11 pp 8434ndash8437 1993
[3] X Wang X-G Zhang Q Yu and B N Harmon ldquoMultiple-scattering theory for electro-magnetic wavesrdquo Physical ReviewB vol 47 no 8 pp 4161ndash4167 1993
[4] J B Pendry ldquoPhotonic structuresrdquo Journal of Modern Opticsvol 41 pp 209ndash229 1994
[5] J A Roden S D Gedney M P Kesler J G Maloney andP H Harms ldquoTime-domain analysis of periodic structuresat oblique incidence orthogonal and nonorthogonal FDTDimplementationsrdquo IEEE Transactions on MicrowaveTheory andTechniques vol 46 no 4 pp 420ndash427 1998
[6] A A Tavallaee and J P Webb ldquoFinite-element modeling ofevanescent modes in the stopband of periodic structuresrdquo IEEETransactions on Magnetics vol 44 no 6 pp 1358ndash1361 2008
[7] I Tsukerman and F Cajko ldquoPhotonic band structure compu-tation using FLAMErdquo IEEE Transactions on Magnetics vol 44no 6 pp 1382ndash1385 2008
[8] M Bebendorf ldquoApproximation of boundary element matricesrdquoNumerische Mathematik vol 86 no 4 pp 565ndash589 2000
[9] K ZhaoM Vouvakis and J-F Lee ldquoThe adaptive cross approx-imation algorithm for accelerated method of moments compu-tations of EMC problemsrdquo IEEE Transactions on Electromag-netic Compatibility vol 47 no 4 pp 763ndash773 2005
[10] A Heldring J M Tamayo C Simon E Ubeda and J M RiusldquoSparsified adaptive cross approximation algorithm for acceler-ated method of moments computationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 1 pp 240ndash246 2013
[11] M Sarnowski T Vaupel V Hansen E Kreysa and H PGemuend ldquoCharacterization of diffraction anomalies in 2-Dphotonic bandgap structuresrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 49 no 10 pp 1868ndash1872 2001
[12] S Y Lin G Arjavalingam andWM Robertson ldquoInvestigationof absolute photonic bandgaps in 2-dimensional dielectricstructuresrdquo Journal of Modern Optics vol 41 pp 385ndash393 1994
[13] W C Chew J M Jin E Michielssen and J Song Fast andEfficient Algorithm in Computational Electromagnetics chapter3 Artech House Boston Mass USA 2001
[14] 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
[15] F-G Hu and J Song ldquoIntegral-equation analysis of scatteringfrom doubly periodic array of 3-D conducting objectsrdquo IEEETransactions on Antennas and Propagation vol 59 no 12 pp4569ndash4578 2011
[16] M G Silveirinha and C A Fernandes ldquoA new accelerationtechnique with exponential convergence rate to evaluate peri-odic Green functionsrdquo IEEE Transactions on Antennas andPropagation vol 53 no 1 pp 347ndash355 2005
[17] K E Jordan G R Richter and P Sheng ldquoAn efficient numericalevaluation of the Greenrsquos function for the Helmholtz operatoron periodic structuresrdquo Journal of Computational Physics vol63 no 1 pp 222ndash235 1986
[18] R D Graglia ldquoOn the numerical integration of the linear shapefunctions times the 3-D Greenrsquos function or its gradient on aplane trianglerdquo IEEETransactions onAntennas andPropagationvol 41 no 10 pp 1448ndash1455 1993
[19] S Jarvenpaa M Taskinen and P Yla-Oijala ldquoSingularitysubtraction technique for high-order polynomial vector basisfunctions on planar trianglesrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 42ndash49 2006
[20] J F Hughes and A V Dam Computer Graphics Principles andPractice Addison-Wesley 3rd edition 2013
[21] Y Saad Iterative Methods for Sparse Linear Systems SIAMSociety for Industrial amp Applied Mathematics 2nd edition2003
[22] J G Fleming S-Y Lin I El-Kady R Biswas andKMHo ldquoAll-metallic three-dimensional photonic crytal with a large infraredbandgaprdquo Nature vol 417 no 6884 pp 52ndash55 2002
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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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 3
Primary source
Field
Image sources
Image sources
(a)
(minus1 0) (0 0) (1 0)
(minus1 minus1) (0 minus1) (1 minus1)
(minus1 1) (0 1) (1 1)
(b)
Figure 2 (a) Periodic image sources in triangular boundary mesh (b) Corresponding lattice numbers (0 0) lattice is the primary unit cell
where
1198661199011 (
997888119903 997888119903
1015840
) =
infin
sum
119898=minusinfin
infin
sum
119899=minusinfin
119890minus119895997888119896119905119898119899sdot(997888120588minus9978881205881015840
)
4119895119860119896119911119898119899
sdot [119890minus119895119896119911119898119899(119911minus1199111015840) erfc(
119895119896119911119898119899
2119864minus (119911 minus 119911
1015840) 119864)
+ 119890119895119896119911119898119899(119911minus1199111015840) erfc(
119895119896119911119898119899
2119864+ (119911 minus 119911
1015840) 119864)]
(6)
1198661199012 (
997888119903 997888119903
1015840
) =
infin
sum
119898=minusinfin
infin
sum
119899=minusinfin
119890minus11989599788811989611990500 sdot997888120588119898119899
4120587119877119898119899
sdot real [1198901198951198960119877119898119899 erfc(119877119898119899119864+
119895119896
2119864)]
(7)
where erfc(119911) is the complementary error function of com-plex argument and 119877
119898119899= |997888119903 minus
997888119903
1015840
minus997888120588119898119899|
By analyzing the asymptotic behavior of the series termsthe approximation to the optimal value for general skewedlattices is given by
119864 = max(radic 120587
119860
119896
2119867) (8)
where1198672 is the maximum exponent permitted In almost allpractical cases it is sufficient to include only nine terms in (6)and (7) (ie the summation limits are fromminus1 to +1) inwhichthe error is usually less than 01
For multilayered periodic structure when 119906119911= |119911 minus 119911
1015840| gt
05radic119860 the PGFs are computed by their spectral represen-tation rather than direct Ewald transformation In doing sothere are two advantages
(1) The spectral representation just requires evaluation ofthe exponential function which is much simpler than
the computation of complex-argument complemen-tary error function (erfc) Compared with erfc(119911)the computational complexity of the exponentialfunction exp(119911) can be ignored
(2) In addition it can eliminate the imbalance of twoEwald sequences when 119906
119911is too large For the multi-
layered structure the complementary error functionsin the first terms (119898 = 119899 = 0) of both Ewald serieswill take large imaginary arguments when 119906
119911is too
largeThese two terms will have very large values thatare approximately equal in amplitude but of oppositesigns so that summing them up will lead to a severeloss of accuracy due to the finite machine precision
23 Singularity Extraction When the distance between theobservation point 997888119903 and the source point 997888119903
1015840
or one of itsperiodic extensions is electrically small the contribution toGreenrsquos function is essentially quasistatic Usually the centerdistance between source and field triangles is less than 01120582then both are considered to be ldquonear interactionrdquo Singularterms in (5) and (7) often arise for example from one ofnine spatial terms 119877
119898119899with119898 119899 = minus1 0 or +1 depending on
which term represents the source nearest to the observer 997888119903 By extracting its singularity [18] the PGF can be expressedas 119866119901= 1198661199011+ [1198661199012minus 1(4120587119877)] + 1(4120587119877) The singularity
subtraction technique [19] used in this paper can compute thesingularity of |997888119903 minus 997888119903
1015840
|119899(119899 ge minus3) type
Supposing the source point nearest to the observer pointappears at (1 0) term rather than (0 0) term as shown inFigure 2 the primary source point can be transferred to (1 0)lattice Then the singular term always appears at (0 0) term11986600
1199012 The singularity of 11986600
1199012can be extracted as
119866001199012 (
997888119903 997888119903
1015840
) = 119865001199012 (
997888119903 997888119903
1015840
) +
14120587119877
(9)
4 International Journal of Antennas and Propagation
o
a2
a1
Gn119909n119910n119911Gn119909+1n119910n119911
Gn119909+1n119910+1n119911Gn119909n119910+1n119911
Gn119909+1n119910n119911+1
Gn119909n119910+1n119911+1
Gn119909n119910n119911+1
Gn119909+1n119910+1n119911+1
dx
dydz
G(u)
Figure 3 Depiction of trilinear interpolation of PGF
where 119865001199012
is the nonsingular part
119865001199012 (
997888119903 997888119903
1015840
) =
14120587119877
sdot real119890minus1198951198960119877 sdot erfc(119877119864minus11989511989602119864
)minus erfc(minus11989511989602119864
)
(10)
As 119877 rarr 0 the result of applying Lrsquo Hopitalrsquos rule on (10) is asfollows
lim119877rarr 0
119865001199012 (
997888119903 997888119903
1015840
)
=
12120587
sdot real [minus1198951198960 erfc(minus11989511989602119864
)minus
2119864radic120587
119890119896204119864
2]
(11)
24 Interpolation of PGF PGFs are very smooth andamenable to interpolation Fast interpolation of PGFs cangreatly reduce the time of matrix filling The reason is thatinterpolating the PGF values from the table is much fasterthan directly computing the Ewald sums (since this has tobe done for every pair of source and field triangles in thenumerical integration using cubature formulas) Trilinearinterpolation [20] suited for the calculation of complex-valued function with very high efficiency is applied hereAs shown in Figure 3 the cuboid grouping the unit cell isequally divided intomany subgrids with side length 119871 Yellowpoints are the precomputed PGFs at the corner of the subgridThen the three-dimensional PGF values can be obtained bytrilinear interpolation
119866(997888119906) = (1minus120572
119909) (1minus120572
119910) (1minus120572
119911) 119866119899119909+1119899119910+1119899119911+1
+ (1minus120572119909) (1minus120572
119910) 120572119911119866119899119909+1119899119910+1119899119911+2
+ (1minus120572119909) 120572119910(1minus120572
119911) 119866119899119909+1119899119910+2119899119911+1
+ (1minus120572119909) 120572119910120572119911119866119899119909+1119899119910+2119899119911+2
+120572119909(1minus120572
119910) (1minus120572
119911) 119866119899119909+2119899119910+1119899119911+1
+120572119909(1minus120572
119910) 120572119911119866119899119909+2119899119910+1119899119911+2
+120572119909120572119910(1minus120572
119911) 119866119899119909+2119899119910+2119899119911+1
+120572119909120572119910120572119911119866119899119909+2119899119910+2119899119911+2
(12)
where 120572119909= 119889119909119871 120572119910= 119889119910119871 120572119911= 119889119911119871
The relative error 120576 obtained by interpolating the PGFs isdefined as
120576 =
100381610038161003816100381610038161003816100381610038161003816
119866intminus 119866
ex
119866ex
100381610038161003816100381610038161003816100381610038161003816
(13)
where 119866int and 119866
ex are the interpolated and the exactdata respectively The exact data are obtained by the aboveaccelerating method with an accuracy 120576 = 10minus7 The relativeerrors are less than 120576 = 10minus7 and have therefore beenneglected
Take the multilayered structure with orthogonal latticesas an example The relative error is shown in Figure 4 forthe interpolation of PGF involving a series of interpolationpoints through the trilinear interpolation method withvarious values of 119873int In Figure 4 positions (A D) arethe sampling points for interpolation A very high level ofaccuracy is easily obtained with a relatively small number ofinterpolation points The side length of the subgrid is usuallyfixed to 002120582 and the higher interpolation accuracy (120576 lt
10minus4) can be obtained
3 ACA Technique
Let the119898times119899 rectangular matrix119885119898times119899
represent the couplingbetween twowell-separated groups in theMoMcomputationThe ACA algorithm aims to accurately approximate 119885
119898times119899by
a low-rank matrix 119885119898times119899
In particular the ACA algorithm
International Journal of Antennas and Propagation 5
o
B
C
D
AB
CD
A
10 20 30 40 50 60 70 80 90 100 1101E minus 7
1E minus 6
1E minus 5
1E minus 4
1E minus 3
Rela
tive e
rror
120576Nint
a1
a2
Figure 4 The relative error of a series of points Reference structure primitive vector |9978881198861| = |
9978881198862| = 055120582 ℎ = 22120582 phase shifts 119896
119909=
119896 cos(45∘) 119896119910= 119896 sin(30∘) Accuracy reached interpolatingPGFs through trilinear interpolationwith a 3Dgrid of119873inttimes119873inttimes2119873int equispaced
points in minus055120582 lt Δ119909 lt 055120582 minus055120582 lt Δ119910 lt 055120582 0 lt Δ119911 lt 22120582 The data plotted refers to the straight line from (015120582 015120582 0) to(015120582 015120582 22120582) The number of equally spaced sampling points is 4
constructs the approximate matrix 119885119898times119899
through a productform Namely
119885119898times119899
= 119880119898times119903
119881119903times119899
=
119903
sum
119894=1119906119894
119898times1V119894
119899times1 (14)
where 119903 is the rank of 119885119898times119899
or the effective rank of the matrix119885119898times119899
119880119898times119903
and 119881119903times119899
are two dense rectangular matrices119906119894
119898times1 is the column 119894th of 119880 and V119894119899times1 is the row 119894th of 119881
respectively The goal of ACA is to achieve10038171003817100381710038171003817119885119898times119899
minus119885119898times119899
10038171003817100381710038171003817119865le 120576
1003817100381710038171003817119885119898times119899
1003817100381710038171003817119865
(15)
for a given tolerance 120576 Note that in this paper sdot refers tothe matrix Frobenus norm [9]
The ACA technique is numerically based on the principleof the approximation above but does not need to knowall terms of 119885
119898times119899 Indeed only 119903 rows and 119903 columns
are calculated during the algorithm allowing a significantreduction in the number of integral computations to performTherefore not only is the memory requirement reduced butthe assembly time is also saved Yet if we agree to deal withtheMVP119880
119898times119903119881119903times119899119887119899times1 instead of the exact value of119885119898times119899119887119899times1
the number of operations needed is also remarkably reducedby performing 119880
119898times119903(119881119903times119899119887119899times1) Thus if an iterative solver is
used one can accelerate matrix vector multiply and save onceagain the computation time [10]
The compression rate 120578 of matrix block 119885119898times119899
whichdenotes the memory saved by using ACA can be expressedas
120578 () = (1minus 119903 sdot (119898 + 119899)
119898 sdot 119899
) sdot 100 (16)
Obviously the approximation byACA approach above is use-ful only if 119903 lt (12) sdotmin119898 119899 where 119885
119898times119899presents similar
values For the electromagnetic application the effective rank119903 lt min(119898 119899) Therefore instead of storing entire 119898 times 119899
entries the algorithm only requires storing (119898+119899)times119903 entries
4 Numerical Results
The EFIE and ACA approaches to PEC are implementedusing Fortran-95 language Numerical experiments run ona PC with Intel i5-2400 31 GHz CPU and 16GB RAMThe resulting impedance matrices are iteratively solved byusing the transpose-free quasi-minimal residual (TFQMR)solver with diagonal preconditioning [21] the relative errortolerance is set to be 10minus3
41 Infinite Metallic Rod As a 2D example we study thescattering properties of a photonic band gap (PBG) structureconsisting of infinite metallic rod [17] Each unit cell consistsof two infinitely long metallic rods The inset of Figure 5 istwo meshed cylinders in a unit cell The axis of each cylinderis along 119909-direction 997888119886
1= 119909 6mm and 997888119886
2= 119910 6mm Each
cylinder has the radius 06mm and length 6mm The spacebetween two cylinders is 6mm Each cylinder was discretizedinto 494 triangle facets with 741 unknowns to ensure accurateresults obtained in the entire frequency band Figure 5 showsthe magnitude of the transmission and reflection coefficientsfor the Floquet TM
00mode with the electric field oriented
in the 119909-direction Our results agree well with the referenceresults obtained using 2D IE approach [15] The resultsconverged for 9 terms in both Ewald sums and there are 1482unknowns for the geometry The time for matrix filling andsolving is 058 s and 021 s per frequency point respectivelyInACA the box size is 6times6times6mm3 that is each box includes
6 International Journal of Antennas and Propagation
minus50
minus40
minus30
minus20
minus10
0
Refl
and
Tran
s co
effici
ents
(dB)
Frequency (GHz)
Refl Trans Refl [15]
o
1
2
Unit cell
10 50403020
a1
a2
Figure 5 Reflection and transmission coefficient of two-layer rodPBG
20 4540353025minus80
minus60
minus40
minus20
0
Tran
simiss
ion
(dB)
Frequency (GHz)
5-layer10-layer21-layer
Layer
1
2
3
N minus 1
N
Figure 6 Transmission coefficient of multilayered PBG
one cylinder If the trilinear interpolation method is notused the time ofmatrix filling would increase dramatically to1286 s Therefore trilinear interpolation of PGFs can reducethe time of matrix filling by more than 995
Next the number of layers increases from 2 to 5 10 and21 respectively Figure 6 shows the transmission coefficientofmultilayered structureThe transmission coefficient curvesexhibit the typical resonances of photonic band gapmaterialsIt can be seen that the null of transmission becomes deeperas the layer number increasesThe total solution times for perfrequency point are 51 s 124 s and 325 s respectively
The 21-layer structure is meshed into 10374 triangle facetswith 15561 unknownsThe compression rate 120578 ofmatrix blockis shown in Table 1 The times for matrix filling and matrixsolving are 238 s and 87 s per frequency point respectively
10 201816141200
02
04
06
08
10
Refle
ctio
n
4-layer 8-layer
16-layer 16-layer [Ansys HFSS]
w
d
h
120582 (120583m)
Figure 7 Reflection spectra for multilayered woodpile MPC
Table 1 Compression rate 120578 () of matrix block
11988511
11988512
11988513
11988514
11988515
11988516
11988517
mdash 2078 864 540 594 567 48611988518
11988519
119885110
119885111
119885112
119885113
119885114
621 567 405 351 378 459 432119885115
119885116
119885117
119885118
119885119
119885120
119885121
486 540 432 378 405 324 351mdash denotes the matrix block calculated directly by MoM
The corresponding times of direct solution by MoM withoutACA acceleration are 1876 s and 752 s per frequency pointrespectively Obviously ACA can greatly reduce the totalsolution time especially for frequency sweeping
42 Woodpile A 3D example showing the inset in Figure 7is a woodpile MPC with 119908 = 200 nm ℎ = 300 nm andin-layer lattice constant 119889 = 1 120583m The incident wave is aTE polarized (ie 119864 parallel to the top metallic layer) planewave through the stacking direction The number of layersare 4 8 and 16 respectively Two intersecting metallic pipesare considered to be fully contacting with each other innumerical modelingTheACA box size is 1times1times03 120583m3Thereflection coefficients are shown in Figure 7 Almost the sameresult is obtained for TM polarization Our numerical resultsagree well with that of Ansys HFSS Previous results reportedby other studies [12 22] for a similar kind of the structuralgeometry also validate our calculated results For the 16-layerMPC with 13149 unknowns (8766 triangle facets) the timesformatrix filling and linear system solving are 403 s and 152 sper frequency point respectivelyThe corresponding times ofdirect solution by MoMwithout ACA acceleration are 1323 sand 925 s respectively
5 Conclusion
An integral equation accelerated with adaptive cross approx-imation (ACA) is proposed for the analysis of multilayeredmetallic photonic crystal The ACA is a purely algebraic and
International Journal of Antennas and Propagation 7
kernel independent algorithm It is verified to be suited foraccelerating integral equation analysis of periodic structureThe ACA can be applied to compress impedance matrixblocks and accelerate MVP for reducing computational com-plexity A modified Ewald transformation and an efficientinterpolation method are also proposed to fast compute theperiodic Greenrsquos function hence that can greatly reducethe time of matrix filling Numerical results confirm thevalidity and accuracy of the proposed method In our nextstage of research ACA will be applied to accelerate hybridfield integral equation for arbitrarily complex multilayeredperiodic structure composed ofmetal and dielectricmaterialswith more unknowns
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was supported in part by the National ScienceFoundation of China (NSFC) under Grants no 61331002no 61102011 and no 61201082 and in part by ExcellentInnovation Team of CUC under Grant no yxtd201303
References
[1] J D Joannopoulos R D Meade and J N Winn PhotonicCrystals Molding the Flow of Light Princeton University PressPrinceton NJ USA 1995
[2] R D Meade A M Rappe K D Brommer J D Joannopoulosand O L Alerhand ldquoAccurate theoretical analysis of photonicband-gapmaterialsrdquo Physical Review B vol 48 no 11 pp 8434ndash8437 1993
[3] X Wang X-G Zhang Q Yu and B N Harmon ldquoMultiple-scattering theory for electro-magnetic wavesrdquo Physical ReviewB vol 47 no 8 pp 4161ndash4167 1993
[4] J B Pendry ldquoPhotonic structuresrdquo Journal of Modern Opticsvol 41 pp 209ndash229 1994
[5] J A Roden S D Gedney M P Kesler J G Maloney andP H Harms ldquoTime-domain analysis of periodic structuresat oblique incidence orthogonal and nonorthogonal FDTDimplementationsrdquo IEEE Transactions on MicrowaveTheory andTechniques vol 46 no 4 pp 420ndash427 1998
[6] A A Tavallaee and J P Webb ldquoFinite-element modeling ofevanescent modes in the stopband of periodic structuresrdquo IEEETransactions on Magnetics vol 44 no 6 pp 1358ndash1361 2008
[7] I Tsukerman and F Cajko ldquoPhotonic band structure compu-tation using FLAMErdquo IEEE Transactions on Magnetics vol 44no 6 pp 1382ndash1385 2008
[8] M Bebendorf ldquoApproximation of boundary element matricesrdquoNumerische Mathematik vol 86 no 4 pp 565ndash589 2000
[9] K ZhaoM Vouvakis and J-F Lee ldquoThe adaptive cross approx-imation algorithm for accelerated method of moments compu-tations of EMC problemsrdquo IEEE Transactions on Electromag-netic Compatibility vol 47 no 4 pp 763ndash773 2005
[10] A Heldring J M Tamayo C Simon E Ubeda and J M RiusldquoSparsified adaptive cross approximation algorithm for acceler-ated method of moments computationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 1 pp 240ndash246 2013
[11] M Sarnowski T Vaupel V Hansen E Kreysa and H PGemuend ldquoCharacterization of diffraction anomalies in 2-Dphotonic bandgap structuresrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 49 no 10 pp 1868ndash1872 2001
[12] S Y Lin G Arjavalingam andWM Robertson ldquoInvestigationof absolute photonic bandgaps in 2-dimensional dielectricstructuresrdquo Journal of Modern Optics vol 41 pp 385ndash393 1994
[13] W C Chew J M Jin E Michielssen and J Song Fast andEfficient Algorithm in Computational Electromagnetics chapter3 Artech House Boston Mass USA 2001
[14] 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
[15] F-G Hu and J Song ldquoIntegral-equation analysis of scatteringfrom doubly periodic array of 3-D conducting objectsrdquo IEEETransactions on Antennas and Propagation vol 59 no 12 pp4569ndash4578 2011
[16] M G Silveirinha and C A Fernandes ldquoA new accelerationtechnique with exponential convergence rate to evaluate peri-odic Green functionsrdquo IEEE Transactions on Antennas andPropagation vol 53 no 1 pp 347ndash355 2005
[17] K E Jordan G R Richter and P Sheng ldquoAn efficient numericalevaluation of the Greenrsquos function for the Helmholtz operatoron periodic structuresrdquo Journal of Computational Physics vol63 no 1 pp 222ndash235 1986
[18] R D Graglia ldquoOn the numerical integration of the linear shapefunctions times the 3-D Greenrsquos function or its gradient on aplane trianglerdquo IEEETransactions onAntennas andPropagationvol 41 no 10 pp 1448ndash1455 1993
[19] S Jarvenpaa M Taskinen and P Yla-Oijala ldquoSingularitysubtraction technique for high-order polynomial vector basisfunctions on planar trianglesrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 42ndash49 2006
[20] J F Hughes and A V Dam Computer Graphics Principles andPractice Addison-Wesley 3rd edition 2013
[21] Y Saad Iterative Methods for Sparse Linear Systems SIAMSociety for Industrial amp Applied Mathematics 2nd edition2003
[22] J G Fleming S-Y Lin I El-Kady R Biswas andKMHo ldquoAll-metallic three-dimensional photonic crytal with a large infraredbandgaprdquo Nature vol 417 no 6884 pp 52ndash55 2002
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
4 International Journal of Antennas and Propagation
o
a2
a1
Gn119909n119910n119911Gn119909+1n119910n119911
Gn119909+1n119910+1n119911Gn119909n119910+1n119911
Gn119909+1n119910n119911+1
Gn119909n119910+1n119911+1
Gn119909n119910n119911+1
Gn119909+1n119910+1n119911+1
dx
dydz
G(u)
Figure 3 Depiction of trilinear interpolation of PGF
where 119865001199012
is the nonsingular part
119865001199012 (
997888119903 997888119903
1015840
) =
14120587119877
sdot real119890minus1198951198960119877 sdot erfc(119877119864minus11989511989602119864
)minus erfc(minus11989511989602119864
)
(10)
As 119877 rarr 0 the result of applying Lrsquo Hopitalrsquos rule on (10) is asfollows
lim119877rarr 0
119865001199012 (
997888119903 997888119903
1015840
)
=
12120587
sdot real [minus1198951198960 erfc(minus11989511989602119864
)minus
2119864radic120587
119890119896204119864
2]
(11)
24 Interpolation of PGF PGFs are very smooth andamenable to interpolation Fast interpolation of PGFs cangreatly reduce the time of matrix filling The reason is thatinterpolating the PGF values from the table is much fasterthan directly computing the Ewald sums (since this has tobe done for every pair of source and field triangles in thenumerical integration using cubature formulas) Trilinearinterpolation [20] suited for the calculation of complex-valued function with very high efficiency is applied hereAs shown in Figure 3 the cuboid grouping the unit cell isequally divided intomany subgrids with side length 119871 Yellowpoints are the precomputed PGFs at the corner of the subgridThen the three-dimensional PGF values can be obtained bytrilinear interpolation
119866(997888119906) = (1minus120572
119909) (1minus120572
119910) (1minus120572
119911) 119866119899119909+1119899119910+1119899119911+1
+ (1minus120572119909) (1minus120572
119910) 120572119911119866119899119909+1119899119910+1119899119911+2
+ (1minus120572119909) 120572119910(1minus120572
119911) 119866119899119909+1119899119910+2119899119911+1
+ (1minus120572119909) 120572119910120572119911119866119899119909+1119899119910+2119899119911+2
+120572119909(1minus120572
119910) (1minus120572
119911) 119866119899119909+2119899119910+1119899119911+1
+120572119909(1minus120572
119910) 120572119911119866119899119909+2119899119910+1119899119911+2
+120572119909120572119910(1minus120572
119911) 119866119899119909+2119899119910+2119899119911+1
+120572119909120572119910120572119911119866119899119909+2119899119910+2119899119911+2
(12)
where 120572119909= 119889119909119871 120572119910= 119889119910119871 120572119911= 119889119911119871
The relative error 120576 obtained by interpolating the PGFs isdefined as
120576 =
100381610038161003816100381610038161003816100381610038161003816
119866intminus 119866
ex
119866ex
100381610038161003816100381610038161003816100381610038161003816
(13)
where 119866int and 119866
ex are the interpolated and the exactdata respectively The exact data are obtained by the aboveaccelerating method with an accuracy 120576 = 10minus7 The relativeerrors are less than 120576 = 10minus7 and have therefore beenneglected
Take the multilayered structure with orthogonal latticesas an example The relative error is shown in Figure 4 forthe interpolation of PGF involving a series of interpolationpoints through the trilinear interpolation method withvarious values of 119873int In Figure 4 positions (A D) arethe sampling points for interpolation A very high level ofaccuracy is easily obtained with a relatively small number ofinterpolation points The side length of the subgrid is usuallyfixed to 002120582 and the higher interpolation accuracy (120576 lt
10minus4) can be obtained
3 ACA Technique
Let the119898times119899 rectangular matrix119885119898times119899
represent the couplingbetween twowell-separated groups in theMoMcomputationThe ACA algorithm aims to accurately approximate 119885
119898times119899by
a low-rank matrix 119885119898times119899
In particular the ACA algorithm
International Journal of Antennas and Propagation 5
o
B
C
D
AB
CD
A
10 20 30 40 50 60 70 80 90 100 1101E minus 7
1E minus 6
1E minus 5
1E minus 4
1E minus 3
Rela
tive e
rror
120576Nint
a1
a2
Figure 4 The relative error of a series of points Reference structure primitive vector |9978881198861| = |
9978881198862| = 055120582 ℎ = 22120582 phase shifts 119896
119909=
119896 cos(45∘) 119896119910= 119896 sin(30∘) Accuracy reached interpolatingPGFs through trilinear interpolationwith a 3Dgrid of119873inttimes119873inttimes2119873int equispaced
points in minus055120582 lt Δ119909 lt 055120582 minus055120582 lt Δ119910 lt 055120582 0 lt Δ119911 lt 22120582 The data plotted refers to the straight line from (015120582 015120582 0) to(015120582 015120582 22120582) The number of equally spaced sampling points is 4
constructs the approximate matrix 119885119898times119899
through a productform Namely
119885119898times119899
= 119880119898times119903
119881119903times119899
=
119903
sum
119894=1119906119894
119898times1V119894
119899times1 (14)
where 119903 is the rank of 119885119898times119899
or the effective rank of the matrix119885119898times119899
119880119898times119903
and 119881119903times119899
are two dense rectangular matrices119906119894
119898times1 is the column 119894th of 119880 and V119894119899times1 is the row 119894th of 119881
respectively The goal of ACA is to achieve10038171003817100381710038171003817119885119898times119899
minus119885119898times119899
10038171003817100381710038171003817119865le 120576
1003817100381710038171003817119885119898times119899
1003817100381710038171003817119865
(15)
for a given tolerance 120576 Note that in this paper sdot refers tothe matrix Frobenus norm [9]
The ACA technique is numerically based on the principleof the approximation above but does not need to knowall terms of 119885
119898times119899 Indeed only 119903 rows and 119903 columns
are calculated during the algorithm allowing a significantreduction in the number of integral computations to performTherefore not only is the memory requirement reduced butthe assembly time is also saved Yet if we agree to deal withtheMVP119880
119898times119903119881119903times119899119887119899times1 instead of the exact value of119885119898times119899119887119899times1
the number of operations needed is also remarkably reducedby performing 119880
119898times119903(119881119903times119899119887119899times1) Thus if an iterative solver is
used one can accelerate matrix vector multiply and save onceagain the computation time [10]
The compression rate 120578 of matrix block 119885119898times119899
whichdenotes the memory saved by using ACA can be expressedas
120578 () = (1minus 119903 sdot (119898 + 119899)
119898 sdot 119899
) sdot 100 (16)
Obviously the approximation byACA approach above is use-ful only if 119903 lt (12) sdotmin119898 119899 where 119885
119898times119899presents similar
values For the electromagnetic application the effective rank119903 lt min(119898 119899) Therefore instead of storing entire 119898 times 119899
entries the algorithm only requires storing (119898+119899)times119903 entries
4 Numerical Results
The EFIE and ACA approaches to PEC are implementedusing Fortran-95 language Numerical experiments run ona PC with Intel i5-2400 31 GHz CPU and 16GB RAMThe resulting impedance matrices are iteratively solved byusing the transpose-free quasi-minimal residual (TFQMR)solver with diagonal preconditioning [21] the relative errortolerance is set to be 10minus3
41 Infinite Metallic Rod As a 2D example we study thescattering properties of a photonic band gap (PBG) structureconsisting of infinite metallic rod [17] Each unit cell consistsof two infinitely long metallic rods The inset of Figure 5 istwo meshed cylinders in a unit cell The axis of each cylinderis along 119909-direction 997888119886
1= 119909 6mm and 997888119886
2= 119910 6mm Each
cylinder has the radius 06mm and length 6mm The spacebetween two cylinders is 6mm Each cylinder was discretizedinto 494 triangle facets with 741 unknowns to ensure accurateresults obtained in the entire frequency band Figure 5 showsthe magnitude of the transmission and reflection coefficientsfor the Floquet TM
00mode with the electric field oriented
in the 119909-direction Our results agree well with the referenceresults obtained using 2D IE approach [15] The resultsconverged for 9 terms in both Ewald sums and there are 1482unknowns for the geometry The time for matrix filling andsolving is 058 s and 021 s per frequency point respectivelyInACA the box size is 6times6times6mm3 that is each box includes
6 International Journal of Antennas and Propagation
minus50
minus40
minus30
minus20
minus10
0
Refl
and
Tran
s co
effici
ents
(dB)
Frequency (GHz)
Refl Trans Refl [15]
o
1
2
Unit cell
10 50403020
a1
a2
Figure 5 Reflection and transmission coefficient of two-layer rodPBG
20 4540353025minus80
minus60
minus40
minus20
0
Tran
simiss
ion
(dB)
Frequency (GHz)
5-layer10-layer21-layer
Layer
1
2
3
N minus 1
N
Figure 6 Transmission coefficient of multilayered PBG
one cylinder If the trilinear interpolation method is notused the time ofmatrix filling would increase dramatically to1286 s Therefore trilinear interpolation of PGFs can reducethe time of matrix filling by more than 995
Next the number of layers increases from 2 to 5 10 and21 respectively Figure 6 shows the transmission coefficientofmultilayered structureThe transmission coefficient curvesexhibit the typical resonances of photonic band gapmaterialsIt can be seen that the null of transmission becomes deeperas the layer number increasesThe total solution times for perfrequency point are 51 s 124 s and 325 s respectively
The 21-layer structure is meshed into 10374 triangle facetswith 15561 unknownsThe compression rate 120578 ofmatrix blockis shown in Table 1 The times for matrix filling and matrixsolving are 238 s and 87 s per frequency point respectively
10 201816141200
02
04
06
08
10
Refle
ctio
n
4-layer 8-layer
16-layer 16-layer [Ansys HFSS]
w
d
h
120582 (120583m)
Figure 7 Reflection spectra for multilayered woodpile MPC
Table 1 Compression rate 120578 () of matrix block
11988511
11988512
11988513
11988514
11988515
11988516
11988517
mdash 2078 864 540 594 567 48611988518
11988519
119885110
119885111
119885112
119885113
119885114
621 567 405 351 378 459 432119885115
119885116
119885117
119885118
119885119
119885120
119885121
486 540 432 378 405 324 351mdash denotes the matrix block calculated directly by MoM
The corresponding times of direct solution by MoM withoutACA acceleration are 1876 s and 752 s per frequency pointrespectively Obviously ACA can greatly reduce the totalsolution time especially for frequency sweeping
42 Woodpile A 3D example showing the inset in Figure 7is a woodpile MPC with 119908 = 200 nm ℎ = 300 nm andin-layer lattice constant 119889 = 1 120583m The incident wave is aTE polarized (ie 119864 parallel to the top metallic layer) planewave through the stacking direction The number of layersare 4 8 and 16 respectively Two intersecting metallic pipesare considered to be fully contacting with each other innumerical modelingTheACA box size is 1times1times03 120583m3Thereflection coefficients are shown in Figure 7 Almost the sameresult is obtained for TM polarization Our numerical resultsagree well with that of Ansys HFSS Previous results reportedby other studies [12 22] for a similar kind of the structuralgeometry also validate our calculated results For the 16-layerMPC with 13149 unknowns (8766 triangle facets) the timesformatrix filling and linear system solving are 403 s and 152 sper frequency point respectivelyThe corresponding times ofdirect solution by MoMwithout ACA acceleration are 1323 sand 925 s respectively
5 Conclusion
An integral equation accelerated with adaptive cross approx-imation (ACA) is proposed for the analysis of multilayeredmetallic photonic crystal The ACA is a purely algebraic and
International Journal of Antennas and Propagation 7
kernel independent algorithm It is verified to be suited foraccelerating integral equation analysis of periodic structureThe ACA can be applied to compress impedance matrixblocks and accelerate MVP for reducing computational com-plexity A modified Ewald transformation and an efficientinterpolation method are also proposed to fast compute theperiodic Greenrsquos function hence that can greatly reducethe time of matrix filling Numerical results confirm thevalidity and accuracy of the proposed method In our nextstage of research ACA will be applied to accelerate hybridfield integral equation for arbitrarily complex multilayeredperiodic structure composed ofmetal and dielectricmaterialswith more unknowns
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was supported in part by the National ScienceFoundation of China (NSFC) under Grants no 61331002no 61102011 and no 61201082 and in part by ExcellentInnovation Team of CUC under Grant no yxtd201303
References
[1] J D Joannopoulos R D Meade and J N Winn PhotonicCrystals Molding the Flow of Light Princeton University PressPrinceton NJ USA 1995
[2] R D Meade A M Rappe K D Brommer J D Joannopoulosand O L Alerhand ldquoAccurate theoretical analysis of photonicband-gapmaterialsrdquo Physical Review B vol 48 no 11 pp 8434ndash8437 1993
[3] X Wang X-G Zhang Q Yu and B N Harmon ldquoMultiple-scattering theory for electro-magnetic wavesrdquo Physical ReviewB vol 47 no 8 pp 4161ndash4167 1993
[4] J B Pendry ldquoPhotonic structuresrdquo Journal of Modern Opticsvol 41 pp 209ndash229 1994
[5] J A Roden S D Gedney M P Kesler J G Maloney andP H Harms ldquoTime-domain analysis of periodic structuresat oblique incidence orthogonal and nonorthogonal FDTDimplementationsrdquo IEEE Transactions on MicrowaveTheory andTechniques vol 46 no 4 pp 420ndash427 1998
[6] A A Tavallaee and J P Webb ldquoFinite-element modeling ofevanescent modes in the stopband of periodic structuresrdquo IEEETransactions on Magnetics vol 44 no 6 pp 1358ndash1361 2008
[7] I Tsukerman and F Cajko ldquoPhotonic band structure compu-tation using FLAMErdquo IEEE Transactions on Magnetics vol 44no 6 pp 1382ndash1385 2008
[8] M Bebendorf ldquoApproximation of boundary element matricesrdquoNumerische Mathematik vol 86 no 4 pp 565ndash589 2000
[9] K ZhaoM Vouvakis and J-F Lee ldquoThe adaptive cross approx-imation algorithm for accelerated method of moments compu-tations of EMC problemsrdquo IEEE Transactions on Electromag-netic Compatibility vol 47 no 4 pp 763ndash773 2005
[10] A Heldring J M Tamayo C Simon E Ubeda and J M RiusldquoSparsified adaptive cross approximation algorithm for acceler-ated method of moments computationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 1 pp 240ndash246 2013
[11] M Sarnowski T Vaupel V Hansen E Kreysa and H PGemuend ldquoCharacterization of diffraction anomalies in 2-Dphotonic bandgap structuresrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 49 no 10 pp 1868ndash1872 2001
[12] S Y Lin G Arjavalingam andWM Robertson ldquoInvestigationof absolute photonic bandgaps in 2-dimensional dielectricstructuresrdquo Journal of Modern Optics vol 41 pp 385ndash393 1994
[13] W C Chew J M Jin E Michielssen and J Song Fast andEfficient Algorithm in Computational Electromagnetics chapter3 Artech House Boston Mass USA 2001
[14] 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
[15] F-G Hu and J Song ldquoIntegral-equation analysis of scatteringfrom doubly periodic array of 3-D conducting objectsrdquo IEEETransactions on Antennas and Propagation vol 59 no 12 pp4569ndash4578 2011
[16] M G Silveirinha and C A Fernandes ldquoA new accelerationtechnique with exponential convergence rate to evaluate peri-odic Green functionsrdquo IEEE Transactions on Antennas andPropagation vol 53 no 1 pp 347ndash355 2005
[17] K E Jordan G R Richter and P Sheng ldquoAn efficient numericalevaluation of the Greenrsquos function for the Helmholtz operatoron periodic structuresrdquo Journal of Computational Physics vol63 no 1 pp 222ndash235 1986
[18] R D Graglia ldquoOn the numerical integration of the linear shapefunctions times the 3-D Greenrsquos function or its gradient on aplane trianglerdquo IEEETransactions onAntennas andPropagationvol 41 no 10 pp 1448ndash1455 1993
[19] S Jarvenpaa M Taskinen and P Yla-Oijala ldquoSingularitysubtraction technique for high-order polynomial vector basisfunctions on planar trianglesrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 42ndash49 2006
[20] J F Hughes and A V Dam Computer Graphics Principles andPractice Addison-Wesley 3rd edition 2013
[21] Y Saad Iterative Methods for Sparse Linear Systems SIAMSociety for Industrial amp Applied Mathematics 2nd edition2003
[22] J G Fleming S-Y Lin I El-Kady R Biswas andKMHo ldquoAll-metallic three-dimensional photonic crytal with a large infraredbandgaprdquo Nature vol 417 no 6884 pp 52ndash55 2002
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 5
o
B
C
D
AB
CD
A
10 20 30 40 50 60 70 80 90 100 1101E minus 7
1E minus 6
1E minus 5
1E minus 4
1E minus 3
Rela
tive e
rror
120576Nint
a1
a2
Figure 4 The relative error of a series of points Reference structure primitive vector |9978881198861| = |
9978881198862| = 055120582 ℎ = 22120582 phase shifts 119896
119909=
119896 cos(45∘) 119896119910= 119896 sin(30∘) Accuracy reached interpolatingPGFs through trilinear interpolationwith a 3Dgrid of119873inttimes119873inttimes2119873int equispaced
points in minus055120582 lt Δ119909 lt 055120582 minus055120582 lt Δ119910 lt 055120582 0 lt Δ119911 lt 22120582 The data plotted refers to the straight line from (015120582 015120582 0) to(015120582 015120582 22120582) The number of equally spaced sampling points is 4
constructs the approximate matrix 119885119898times119899
through a productform Namely
119885119898times119899
= 119880119898times119903
119881119903times119899
=
119903
sum
119894=1119906119894
119898times1V119894
119899times1 (14)
where 119903 is the rank of 119885119898times119899
or the effective rank of the matrix119885119898times119899
119880119898times119903
and 119881119903times119899
are two dense rectangular matrices119906119894
119898times1 is the column 119894th of 119880 and V119894119899times1 is the row 119894th of 119881
respectively The goal of ACA is to achieve10038171003817100381710038171003817119885119898times119899
minus119885119898times119899
10038171003817100381710038171003817119865le 120576
1003817100381710038171003817119885119898times119899
1003817100381710038171003817119865
(15)
for a given tolerance 120576 Note that in this paper sdot refers tothe matrix Frobenus norm [9]
The ACA technique is numerically based on the principleof the approximation above but does not need to knowall terms of 119885
119898times119899 Indeed only 119903 rows and 119903 columns
are calculated during the algorithm allowing a significantreduction in the number of integral computations to performTherefore not only is the memory requirement reduced butthe assembly time is also saved Yet if we agree to deal withtheMVP119880
119898times119903119881119903times119899119887119899times1 instead of the exact value of119885119898times119899119887119899times1
the number of operations needed is also remarkably reducedby performing 119880
119898times119903(119881119903times119899119887119899times1) Thus if an iterative solver is
used one can accelerate matrix vector multiply and save onceagain the computation time [10]
The compression rate 120578 of matrix block 119885119898times119899
whichdenotes the memory saved by using ACA can be expressedas
120578 () = (1minus 119903 sdot (119898 + 119899)
119898 sdot 119899
) sdot 100 (16)
Obviously the approximation byACA approach above is use-ful only if 119903 lt (12) sdotmin119898 119899 where 119885
119898times119899presents similar
values For the electromagnetic application the effective rank119903 lt min(119898 119899) Therefore instead of storing entire 119898 times 119899
entries the algorithm only requires storing (119898+119899)times119903 entries
4 Numerical Results
The EFIE and ACA approaches to PEC are implementedusing Fortran-95 language Numerical experiments run ona PC with Intel i5-2400 31 GHz CPU and 16GB RAMThe resulting impedance matrices are iteratively solved byusing the transpose-free quasi-minimal residual (TFQMR)solver with diagonal preconditioning [21] the relative errortolerance is set to be 10minus3
41 Infinite Metallic Rod As a 2D example we study thescattering properties of a photonic band gap (PBG) structureconsisting of infinite metallic rod [17] Each unit cell consistsof two infinitely long metallic rods The inset of Figure 5 istwo meshed cylinders in a unit cell The axis of each cylinderis along 119909-direction 997888119886
1= 119909 6mm and 997888119886
2= 119910 6mm Each
cylinder has the radius 06mm and length 6mm The spacebetween two cylinders is 6mm Each cylinder was discretizedinto 494 triangle facets with 741 unknowns to ensure accurateresults obtained in the entire frequency band Figure 5 showsthe magnitude of the transmission and reflection coefficientsfor the Floquet TM
00mode with the electric field oriented
in the 119909-direction Our results agree well with the referenceresults obtained using 2D IE approach [15] The resultsconverged for 9 terms in both Ewald sums and there are 1482unknowns for the geometry The time for matrix filling andsolving is 058 s and 021 s per frequency point respectivelyInACA the box size is 6times6times6mm3 that is each box includes
6 International Journal of Antennas and Propagation
minus50
minus40
minus30
minus20
minus10
0
Refl
and
Tran
s co
effici
ents
(dB)
Frequency (GHz)
Refl Trans Refl [15]
o
1
2
Unit cell
10 50403020
a1
a2
Figure 5 Reflection and transmission coefficient of two-layer rodPBG
20 4540353025minus80
minus60
minus40
minus20
0
Tran
simiss
ion
(dB)
Frequency (GHz)
5-layer10-layer21-layer
Layer
1
2
3
N minus 1
N
Figure 6 Transmission coefficient of multilayered PBG
one cylinder If the trilinear interpolation method is notused the time ofmatrix filling would increase dramatically to1286 s Therefore trilinear interpolation of PGFs can reducethe time of matrix filling by more than 995
Next the number of layers increases from 2 to 5 10 and21 respectively Figure 6 shows the transmission coefficientofmultilayered structureThe transmission coefficient curvesexhibit the typical resonances of photonic band gapmaterialsIt can be seen that the null of transmission becomes deeperas the layer number increasesThe total solution times for perfrequency point are 51 s 124 s and 325 s respectively
The 21-layer structure is meshed into 10374 triangle facetswith 15561 unknownsThe compression rate 120578 ofmatrix blockis shown in Table 1 The times for matrix filling and matrixsolving are 238 s and 87 s per frequency point respectively
10 201816141200
02
04
06
08
10
Refle
ctio
n
4-layer 8-layer
16-layer 16-layer [Ansys HFSS]
w
d
h
120582 (120583m)
Figure 7 Reflection spectra for multilayered woodpile MPC
Table 1 Compression rate 120578 () of matrix block
11988511
11988512
11988513
11988514
11988515
11988516
11988517
mdash 2078 864 540 594 567 48611988518
11988519
119885110
119885111
119885112
119885113
119885114
621 567 405 351 378 459 432119885115
119885116
119885117
119885118
119885119
119885120
119885121
486 540 432 378 405 324 351mdash denotes the matrix block calculated directly by MoM
The corresponding times of direct solution by MoM withoutACA acceleration are 1876 s and 752 s per frequency pointrespectively Obviously ACA can greatly reduce the totalsolution time especially for frequency sweeping
42 Woodpile A 3D example showing the inset in Figure 7is a woodpile MPC with 119908 = 200 nm ℎ = 300 nm andin-layer lattice constant 119889 = 1 120583m The incident wave is aTE polarized (ie 119864 parallel to the top metallic layer) planewave through the stacking direction The number of layersare 4 8 and 16 respectively Two intersecting metallic pipesare considered to be fully contacting with each other innumerical modelingTheACA box size is 1times1times03 120583m3Thereflection coefficients are shown in Figure 7 Almost the sameresult is obtained for TM polarization Our numerical resultsagree well with that of Ansys HFSS Previous results reportedby other studies [12 22] for a similar kind of the structuralgeometry also validate our calculated results For the 16-layerMPC with 13149 unknowns (8766 triangle facets) the timesformatrix filling and linear system solving are 403 s and 152 sper frequency point respectivelyThe corresponding times ofdirect solution by MoMwithout ACA acceleration are 1323 sand 925 s respectively
5 Conclusion
An integral equation accelerated with adaptive cross approx-imation (ACA) is proposed for the analysis of multilayeredmetallic photonic crystal The ACA is a purely algebraic and
International Journal of Antennas and Propagation 7
kernel independent algorithm It is verified to be suited foraccelerating integral equation analysis of periodic structureThe ACA can be applied to compress impedance matrixblocks and accelerate MVP for reducing computational com-plexity A modified Ewald transformation and an efficientinterpolation method are also proposed to fast compute theperiodic Greenrsquos function hence that can greatly reducethe time of matrix filling Numerical results confirm thevalidity and accuracy of the proposed method In our nextstage of research ACA will be applied to accelerate hybridfield integral equation for arbitrarily complex multilayeredperiodic structure composed ofmetal and dielectricmaterialswith more unknowns
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was supported in part by the National ScienceFoundation of China (NSFC) under Grants no 61331002no 61102011 and no 61201082 and in part by ExcellentInnovation Team of CUC under Grant no yxtd201303
References
[1] J D Joannopoulos R D Meade and J N Winn PhotonicCrystals Molding the Flow of Light Princeton University PressPrinceton NJ USA 1995
[2] R D Meade A M Rappe K D Brommer J D Joannopoulosand O L Alerhand ldquoAccurate theoretical analysis of photonicband-gapmaterialsrdquo Physical Review B vol 48 no 11 pp 8434ndash8437 1993
[3] X Wang X-G Zhang Q Yu and B N Harmon ldquoMultiple-scattering theory for electro-magnetic wavesrdquo Physical ReviewB vol 47 no 8 pp 4161ndash4167 1993
[4] J B Pendry ldquoPhotonic structuresrdquo Journal of Modern Opticsvol 41 pp 209ndash229 1994
[5] J A Roden S D Gedney M P Kesler J G Maloney andP H Harms ldquoTime-domain analysis of periodic structuresat oblique incidence orthogonal and nonorthogonal FDTDimplementationsrdquo IEEE Transactions on MicrowaveTheory andTechniques vol 46 no 4 pp 420ndash427 1998
[6] A A Tavallaee and J P Webb ldquoFinite-element modeling ofevanescent modes in the stopband of periodic structuresrdquo IEEETransactions on Magnetics vol 44 no 6 pp 1358ndash1361 2008
[7] I Tsukerman and F Cajko ldquoPhotonic band structure compu-tation using FLAMErdquo IEEE Transactions on Magnetics vol 44no 6 pp 1382ndash1385 2008
[8] M Bebendorf ldquoApproximation of boundary element matricesrdquoNumerische Mathematik vol 86 no 4 pp 565ndash589 2000
[9] K ZhaoM Vouvakis and J-F Lee ldquoThe adaptive cross approx-imation algorithm for accelerated method of moments compu-tations of EMC problemsrdquo IEEE Transactions on Electromag-netic Compatibility vol 47 no 4 pp 763ndash773 2005
[10] A Heldring J M Tamayo C Simon E Ubeda and J M RiusldquoSparsified adaptive cross approximation algorithm for acceler-ated method of moments computationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 1 pp 240ndash246 2013
[11] M Sarnowski T Vaupel V Hansen E Kreysa and H PGemuend ldquoCharacterization of diffraction anomalies in 2-Dphotonic bandgap structuresrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 49 no 10 pp 1868ndash1872 2001
[12] S Y Lin G Arjavalingam andWM Robertson ldquoInvestigationof absolute photonic bandgaps in 2-dimensional dielectricstructuresrdquo Journal of Modern Optics vol 41 pp 385ndash393 1994
[13] W C Chew J M Jin E Michielssen and J Song Fast andEfficient Algorithm in Computational Electromagnetics chapter3 Artech House Boston Mass USA 2001
[14] 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
[15] F-G Hu and J Song ldquoIntegral-equation analysis of scatteringfrom doubly periodic array of 3-D conducting objectsrdquo IEEETransactions on Antennas and Propagation vol 59 no 12 pp4569ndash4578 2011
[16] M G Silveirinha and C A Fernandes ldquoA new accelerationtechnique with exponential convergence rate to evaluate peri-odic Green functionsrdquo IEEE Transactions on Antennas andPropagation vol 53 no 1 pp 347ndash355 2005
[17] K E Jordan G R Richter and P Sheng ldquoAn efficient numericalevaluation of the Greenrsquos function for the Helmholtz operatoron periodic structuresrdquo Journal of Computational Physics vol63 no 1 pp 222ndash235 1986
[18] R D Graglia ldquoOn the numerical integration of the linear shapefunctions times the 3-D Greenrsquos function or its gradient on aplane trianglerdquo IEEETransactions onAntennas andPropagationvol 41 no 10 pp 1448ndash1455 1993
[19] S Jarvenpaa M Taskinen and P Yla-Oijala ldquoSingularitysubtraction technique for high-order polynomial vector basisfunctions on planar trianglesrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 42ndash49 2006
[20] J F Hughes and A V Dam Computer Graphics Principles andPractice Addison-Wesley 3rd edition 2013
[21] Y Saad Iterative Methods for Sparse Linear Systems SIAMSociety for Industrial amp Applied Mathematics 2nd edition2003
[22] J G Fleming S-Y Lin I El-Kady R Biswas andKMHo ldquoAll-metallic three-dimensional photonic crytal with a large infraredbandgaprdquo Nature vol 417 no 6884 pp 52ndash55 2002
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
minus50
minus40
minus30
minus20
minus10
0
Refl
and
Tran
s co
effici
ents
(dB)
Frequency (GHz)
Refl Trans Refl [15]
o
1
2
Unit cell
10 50403020
a1
a2
Figure 5 Reflection and transmission coefficient of two-layer rodPBG
20 4540353025minus80
minus60
minus40
minus20
0
Tran
simiss
ion
(dB)
Frequency (GHz)
5-layer10-layer21-layer
Layer
1
2
3
N minus 1
N
Figure 6 Transmission coefficient of multilayered PBG
one cylinder If the trilinear interpolation method is notused the time ofmatrix filling would increase dramatically to1286 s Therefore trilinear interpolation of PGFs can reducethe time of matrix filling by more than 995
Next the number of layers increases from 2 to 5 10 and21 respectively Figure 6 shows the transmission coefficientofmultilayered structureThe transmission coefficient curvesexhibit the typical resonances of photonic band gapmaterialsIt can be seen that the null of transmission becomes deeperas the layer number increasesThe total solution times for perfrequency point are 51 s 124 s and 325 s respectively
The 21-layer structure is meshed into 10374 triangle facetswith 15561 unknownsThe compression rate 120578 ofmatrix blockis shown in Table 1 The times for matrix filling and matrixsolving are 238 s and 87 s per frequency point respectively
10 201816141200
02
04
06
08
10
Refle
ctio
n
4-layer 8-layer
16-layer 16-layer [Ansys HFSS]
w
d
h
120582 (120583m)
Figure 7 Reflection spectra for multilayered woodpile MPC
Table 1 Compression rate 120578 () of matrix block
11988511
11988512
11988513
11988514
11988515
11988516
11988517
mdash 2078 864 540 594 567 48611988518
11988519
119885110
119885111
119885112
119885113
119885114
621 567 405 351 378 459 432119885115
119885116
119885117
119885118
119885119
119885120
119885121
486 540 432 378 405 324 351mdash denotes the matrix block calculated directly by MoM
The corresponding times of direct solution by MoM withoutACA acceleration are 1876 s and 752 s per frequency pointrespectively Obviously ACA can greatly reduce the totalsolution time especially for frequency sweeping
42 Woodpile A 3D example showing the inset in Figure 7is a woodpile MPC with 119908 = 200 nm ℎ = 300 nm andin-layer lattice constant 119889 = 1 120583m The incident wave is aTE polarized (ie 119864 parallel to the top metallic layer) planewave through the stacking direction The number of layersare 4 8 and 16 respectively Two intersecting metallic pipesare considered to be fully contacting with each other innumerical modelingTheACA box size is 1times1times03 120583m3Thereflection coefficients are shown in Figure 7 Almost the sameresult is obtained for TM polarization Our numerical resultsagree well with that of Ansys HFSS Previous results reportedby other studies [12 22] for a similar kind of the structuralgeometry also validate our calculated results For the 16-layerMPC with 13149 unknowns (8766 triangle facets) the timesformatrix filling and linear system solving are 403 s and 152 sper frequency point respectivelyThe corresponding times ofdirect solution by MoMwithout ACA acceleration are 1323 sand 925 s respectively
5 Conclusion
An integral equation accelerated with adaptive cross approx-imation (ACA) is proposed for the analysis of multilayeredmetallic photonic crystal The ACA is a purely algebraic and
International Journal of Antennas and Propagation 7
kernel independent algorithm It is verified to be suited foraccelerating integral equation analysis of periodic structureThe ACA can be applied to compress impedance matrixblocks and accelerate MVP for reducing computational com-plexity A modified Ewald transformation and an efficientinterpolation method are also proposed to fast compute theperiodic Greenrsquos function hence that can greatly reducethe time of matrix filling Numerical results confirm thevalidity and accuracy of the proposed method In our nextstage of research ACA will be applied to accelerate hybridfield integral equation for arbitrarily complex multilayeredperiodic structure composed ofmetal and dielectricmaterialswith more unknowns
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was supported in part by the National ScienceFoundation of China (NSFC) under Grants no 61331002no 61102011 and no 61201082 and in part by ExcellentInnovation Team of CUC under Grant no yxtd201303
References
[1] J D Joannopoulos R D Meade and J N Winn PhotonicCrystals Molding the Flow of Light Princeton University PressPrinceton NJ USA 1995
[2] R D Meade A M Rappe K D Brommer J D Joannopoulosand O L Alerhand ldquoAccurate theoretical analysis of photonicband-gapmaterialsrdquo Physical Review B vol 48 no 11 pp 8434ndash8437 1993
[3] X Wang X-G Zhang Q Yu and B N Harmon ldquoMultiple-scattering theory for electro-magnetic wavesrdquo Physical ReviewB vol 47 no 8 pp 4161ndash4167 1993
[4] J B Pendry ldquoPhotonic structuresrdquo Journal of Modern Opticsvol 41 pp 209ndash229 1994
[5] J A Roden S D Gedney M P Kesler J G Maloney andP H Harms ldquoTime-domain analysis of periodic structuresat oblique incidence orthogonal and nonorthogonal FDTDimplementationsrdquo IEEE Transactions on MicrowaveTheory andTechniques vol 46 no 4 pp 420ndash427 1998
[6] A A Tavallaee and J P Webb ldquoFinite-element modeling ofevanescent modes in the stopband of periodic structuresrdquo IEEETransactions on Magnetics vol 44 no 6 pp 1358ndash1361 2008
[7] I Tsukerman and F Cajko ldquoPhotonic band structure compu-tation using FLAMErdquo IEEE Transactions on Magnetics vol 44no 6 pp 1382ndash1385 2008
[8] M Bebendorf ldquoApproximation of boundary element matricesrdquoNumerische Mathematik vol 86 no 4 pp 565ndash589 2000
[9] K ZhaoM Vouvakis and J-F Lee ldquoThe adaptive cross approx-imation algorithm for accelerated method of moments compu-tations of EMC problemsrdquo IEEE Transactions on Electromag-netic Compatibility vol 47 no 4 pp 763ndash773 2005
[10] A Heldring J M Tamayo C Simon E Ubeda and J M RiusldquoSparsified adaptive cross approximation algorithm for acceler-ated method of moments computationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 1 pp 240ndash246 2013
[11] M Sarnowski T Vaupel V Hansen E Kreysa and H PGemuend ldquoCharacterization of diffraction anomalies in 2-Dphotonic bandgap structuresrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 49 no 10 pp 1868ndash1872 2001
[12] S Y Lin G Arjavalingam andWM Robertson ldquoInvestigationof absolute photonic bandgaps in 2-dimensional dielectricstructuresrdquo Journal of Modern Optics vol 41 pp 385ndash393 1994
[13] W C Chew J M Jin E Michielssen and J Song Fast andEfficient Algorithm in Computational Electromagnetics chapter3 Artech House Boston Mass USA 2001
[14] 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
[15] F-G Hu and J Song ldquoIntegral-equation analysis of scatteringfrom doubly periodic array of 3-D conducting objectsrdquo IEEETransactions on Antennas and Propagation vol 59 no 12 pp4569ndash4578 2011
[16] M G Silveirinha and C A Fernandes ldquoA new accelerationtechnique with exponential convergence rate to evaluate peri-odic Green functionsrdquo IEEE Transactions on Antennas andPropagation vol 53 no 1 pp 347ndash355 2005
[17] K E Jordan G R Richter and P Sheng ldquoAn efficient numericalevaluation of the Greenrsquos function for the Helmholtz operatoron periodic structuresrdquo Journal of Computational Physics vol63 no 1 pp 222ndash235 1986
[18] R D Graglia ldquoOn the numerical integration of the linear shapefunctions times the 3-D Greenrsquos function or its gradient on aplane trianglerdquo IEEETransactions onAntennas andPropagationvol 41 no 10 pp 1448ndash1455 1993
[19] S Jarvenpaa M Taskinen and P Yla-Oijala ldquoSingularitysubtraction technique for high-order polynomial vector basisfunctions on planar trianglesrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 42ndash49 2006
[20] J F Hughes and A V Dam Computer Graphics Principles andPractice Addison-Wesley 3rd edition 2013
[21] Y Saad Iterative Methods for Sparse Linear Systems SIAMSociety for Industrial amp Applied Mathematics 2nd edition2003
[22] J G Fleming S-Y Lin I El-Kady R Biswas andKMHo ldquoAll-metallic three-dimensional photonic crytal with a large infraredbandgaprdquo Nature vol 417 no 6884 pp 52ndash55 2002
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
kernel independent algorithm It is verified to be suited foraccelerating integral equation analysis of periodic structureThe ACA can be applied to compress impedance matrixblocks and accelerate MVP for reducing computational com-plexity A modified Ewald transformation and an efficientinterpolation method are also proposed to fast compute theperiodic Greenrsquos function hence that can greatly reducethe time of matrix filling Numerical results confirm thevalidity and accuracy of the proposed method In our nextstage of research ACA will be applied to accelerate hybridfield integral equation for arbitrarily complex multilayeredperiodic structure composed ofmetal and dielectricmaterialswith more unknowns
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was supported in part by the National ScienceFoundation of China (NSFC) under Grants no 61331002no 61102011 and no 61201082 and in part by ExcellentInnovation Team of CUC under Grant no yxtd201303
References
[1] J D Joannopoulos R D Meade and J N Winn PhotonicCrystals Molding the Flow of Light Princeton University PressPrinceton NJ USA 1995
[2] R D Meade A M Rappe K D Brommer J D Joannopoulosand O L Alerhand ldquoAccurate theoretical analysis of photonicband-gapmaterialsrdquo Physical Review B vol 48 no 11 pp 8434ndash8437 1993
[3] X Wang X-G Zhang Q Yu and B N Harmon ldquoMultiple-scattering theory for electro-magnetic wavesrdquo Physical ReviewB vol 47 no 8 pp 4161ndash4167 1993
[4] J B Pendry ldquoPhotonic structuresrdquo Journal of Modern Opticsvol 41 pp 209ndash229 1994
[5] J A Roden S D Gedney M P Kesler J G Maloney andP H Harms ldquoTime-domain analysis of periodic structuresat oblique incidence orthogonal and nonorthogonal FDTDimplementationsrdquo IEEE Transactions on MicrowaveTheory andTechniques vol 46 no 4 pp 420ndash427 1998
[6] A A Tavallaee and J P Webb ldquoFinite-element modeling ofevanescent modes in the stopband of periodic structuresrdquo IEEETransactions on Magnetics vol 44 no 6 pp 1358ndash1361 2008
[7] I Tsukerman and F Cajko ldquoPhotonic band structure compu-tation using FLAMErdquo IEEE Transactions on Magnetics vol 44no 6 pp 1382ndash1385 2008
[8] M Bebendorf ldquoApproximation of boundary element matricesrdquoNumerische Mathematik vol 86 no 4 pp 565ndash589 2000
[9] K ZhaoM Vouvakis and J-F Lee ldquoThe adaptive cross approx-imation algorithm for accelerated method of moments compu-tations of EMC problemsrdquo IEEE Transactions on Electromag-netic Compatibility vol 47 no 4 pp 763ndash773 2005
[10] A Heldring J M Tamayo C Simon E Ubeda and J M RiusldquoSparsified adaptive cross approximation algorithm for acceler-ated method of moments computationsrdquo IEEE Transactions onAntennas and Propagation vol 61 no 1 pp 240ndash246 2013
[11] M Sarnowski T Vaupel V Hansen E Kreysa and H PGemuend ldquoCharacterization of diffraction anomalies in 2-Dphotonic bandgap structuresrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 49 no 10 pp 1868ndash1872 2001
[12] S Y Lin G Arjavalingam andWM Robertson ldquoInvestigationof absolute photonic bandgaps in 2-dimensional dielectricstructuresrdquo Journal of Modern Optics vol 41 pp 385ndash393 1994
[13] W C Chew J M Jin E Michielssen and J Song Fast andEfficient Algorithm in Computational Electromagnetics chapter3 Artech House Boston Mass USA 2001
[14] 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
[15] F-G Hu and J Song ldquoIntegral-equation analysis of scatteringfrom doubly periodic array of 3-D conducting objectsrdquo IEEETransactions on Antennas and Propagation vol 59 no 12 pp4569ndash4578 2011
[16] M G Silveirinha and C A Fernandes ldquoA new accelerationtechnique with exponential convergence rate to evaluate peri-odic Green functionsrdquo IEEE Transactions on Antennas andPropagation vol 53 no 1 pp 347ndash355 2005
[17] K E Jordan G R Richter and P Sheng ldquoAn efficient numericalevaluation of the Greenrsquos function for the Helmholtz operatoron periodic structuresrdquo Journal of Computational Physics vol63 no 1 pp 222ndash235 1986
[18] R D Graglia ldquoOn the numerical integration of the linear shapefunctions times the 3-D Greenrsquos function or its gradient on aplane trianglerdquo IEEETransactions onAntennas andPropagationvol 41 no 10 pp 1448ndash1455 1993
[19] S Jarvenpaa M Taskinen and P Yla-Oijala ldquoSingularitysubtraction technique for high-order polynomial vector basisfunctions on planar trianglesrdquo IEEE Transactions on Antennasand Propagation vol 54 no 1 pp 42ndash49 2006
[20] J F Hughes and A V Dam Computer Graphics Principles andPractice Addison-Wesley 3rd edition 2013
[21] Y Saad Iterative Methods for Sparse Linear Systems SIAMSociety for Industrial amp Applied Mathematics 2nd edition2003
[22] J G Fleming S-Y Lin I El-Kady R Biswas andKMHo ldquoAll-metallic three-dimensional photonic crytal with a large infraredbandgaprdquo Nature vol 417 no 6884 pp 52ndash55 2002
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
top related