Forum for Electromagnetic Research Methods and Application Technologies (FERMAT)
The Precise Integration Time Domain (PITD) Method - A Supplement to the Computational Electromagnetics
Xikui Ma1 and Tianyu Dong 1,2
1 State Key Laboratory of Electrical Insulation and Power Equipment, School of Electrical
Engineering, Xi'an Jiaotong University, Xi'an 710049, China
2 State Key Lab for Strength and Vibration of Mechanical Structures, School of Aerospace,
Xi'an Jiaotong University, Xi'an 710049, China
Abstract: The finite-difference time-domain (FDTD) method [1] is one of the most widely used full-wave
electromagnetic simulation tools for solving Maxwell’s curl equations due to its simplicity,
straightforwardness and easy implementation. However, the Courant-Friedrich-Levy (CFL) stability
condition, i.e., Δ𝑡FDTD ≤ 1/ (𝑣√1
Δ𝑥2+
1
Δ𝑦2+
1
Δ𝑧2), and the increasing numerical dispersion error have
limited the intense utilization of the FDTD. Thus, the development of accurate and efficient numerical
algorithms for solving Maxwell’s equations is still an attractive topic in our computational electromagnetics
(CEM) society.
The precise integration time domain (PITD) method [2] is proposed for solving Maxwell’s equations,
which breaks through the CFL limit and improves the computational efficiency. The fundamental idea of
the PITD method consists of reducing Maxwell’s curl equations to a set of ordinary differential equations
(ODEs) by approximating the spatial derivative with difference only, and solving the ODEs by using the
precise integration (PI) technique with an accuracy of machinery precision. Compared to the
convectional FDTD method, the PITD method posseses some striking features, which are summerized
hereinafter.
First, different with that of the conventional FDTD method, the stability condition of the PITD
method can be expressed as Δ𝑡PITD ≤ √2𝑙/ (𝑣√1
Δ𝑥2+
1
Δ𝑦2+
1
Δ𝑧2) , where 2Nl and N is a preselected
integer. It can be clearly seen that the upper limit of the time-step size of the PITD method is determined by
the preselected integer N and the spatial mesh size. Hence, the requirement of the time-step size can be
satisfied by the alternative choices of the integer N . Meanwhile, it should be emphasized that the time-step
size of the PITD method is much larger than that of the FDTD method.
Second, although the numerical dispersion error of the PITD method is slightly larger than that of the
FDTD method, the numerical dispersion performance of the PITD method is much better than other
unconditionally stable algorithms, such as alternating direction implicit FDTD (ADI-FDTD) method.
Furthermore, the numerical dispersion errors can be made nearly independent of the time-step size [3, 4].
Finally, the PITD method has advantages over the conventional Yee’s FDTD method in using a large
time step, and over the ADI-FDTD method in having high computational accuracy, respectively, and
numerical dispersion errors of the PITD scheme can be made nearly independent of the time-step size.
Therefore, we can use large time-step size to solve Maxwell’s equations without increasing the
computational error.
Due to its excellent performance on computational electromagnetics, the PITD method attracts more
and more attentions [2-7]. Recently, various low-dispersion and low-memory-requirement PITD methods
have been proposed, such as the fourth-order accurate PITD [PITD (4)] method [3], the wavelet Galerkin
scheme-based PITD (WG-PITD) method [4], the leapfrog scheme-based PITD (L-PITD) method [5], the
unified split-step PITD (SS-PITD) method [6], the compact PITD (CPITD) method [7] and so on. With the
rapid development of the computer hardware technology, the PITD method still has
considerable development potential in the future.
Keywords: computational electromagnetics, matrix-exponential methods, time-domain analysis, large time-steps, and precise integration technique.
References
[1] K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE
Trans. Antennas Propag., 3 (1966) 302–307.
[2] X. K. Ma, X. T. Zhao, and Y. Z. Zhao, A 3-D precise integration time-domain method without the restrains of the Courant-
Friedrich-Levy stability condition for the numerical solution of Maxwell’s equations, IEEE Trans. Microw. Theory Tech.
7 (2006) 3026-3037.
[3] Z. M. Bai, X. K. Ma, and G. Sun, A low-dispersion realization of precise integration time-domain method using a fourth-
order accurate finite difference scheme, IEEE Trans. Antennas Propag., 4 (2011) 1311-1320.
[4] G. Sun, X. K. Ma, and Z. M. Bai, A low dispersion precise integration time domain method based on wavelet Galerkin
scheme, IEEE Microw. Wireless Compon. Lett., 12 (2010) 651-653.
[5] G. Sun, X. K. Ma, and Z. M. Bai, A low-memory-requirement realization of precise integration time domain method using
a leapfrog scheme, IEEE Microw. Wireless Compon. Lett., 6 (2012) 294-296.
[6] Q. Liu, X. K. Ma, and F. Chen, Unified split-step precise integration time-domain method for dispersive media, Electron.
lett. 18 (2013) 1135-1136.
[7] Z. Kang, X. K. Ma, and X. Zhuansun, An efficient 2-D compact precise-integration time-domain method for longitudinally
invariant waveguiding structures, IEEE Trans. Microw. Theory Tech. 7 (2013) 2535-2544.
Xikui Ma was born in Shaanxi, China, in 1958. He received the B.Sc. and M.Sc. degrees
in electrical engineering from Xi'an Jiaotong University, Shaanxi, China, in 1982 and
1985, respectively. In 1985, he joined the School of Electrical Engineering, Xi'an
Jiaotong University, as a Lecturer, and became a Professor in 1992. From 1994 to 1995,
he was a Visiting Scientist with the Department of Electrical Engineering and Computer,
University of Toronto. He has authored or coauthored over 140 scientific and technical
papers. He has authored five books in electromagnetic fields. His main areas of research
interests include electromagnetic field theory and its applications, analytical and
numerical methods in solving electromagnetic problems, chaotic dynamics and their
applications in power electronics, and the applications of digital control to power
electronics.
Tianyu Dong is an Assistant Professor in the School of Electrical Engineering at Xi’an
Jiaotong University, China. He is affiliated with the Group for Advanced Electrical
Technologies (GAET), an interdisciplinary center for research and education with an
emphasis on electromagnetics, circuits, power electronics, nonlinear phenomenon and
renewable energies. He is also affiliated with the State Key Laboratory of Electrical
Insulation and Power Equipment (SKLEI), and the State Key Laboratory for Strength and
Vibration of Mechanical Structures (MSSV) as a postdoctoral researcher. Prior to joining
XJTU as a tenure-tracked faculty at the beginning of the year of 2015, he received the BS
and PhD degrees from Xi'an Jiaotong University, China, respectively in 2008 and 2014.
During the PhD, he spent two years from 2011 to 2013 in the Electromagnetic
Communication (EMC) Lab of Electrical Engineering Department at The Pennsylvania
State University (Penn State) in the United States, where he was hosted by Prof. Dr. Raj Mittra. His current
research interests include graphene plasmonics, metasurfaces, scattering of nanoparticles, and wireless power
transfer.
*This use of this work is restricted solely for academic purposes. The author of this work owns the copyright and no reproduction in any form is permitted without written permission by the author. *
The Precise Integration Time Domain
(PITD) Method— A Supplement to the Computational Electromagnetics
Xikui Ma1 and Tianyu Dong1,2
[email protected] [email protected]
September 19, 20161 SKLEI, School of Electrical Engineering, XJTU
2 MSSV, School of Aerospace, XJTU
Overview Methods Examples Outlooks
Outline
1 Overview
2 MethodsIntroduction to PITDNumerical dispersion and stabilitySource and boundary conditionsImproved methods
3 ExamplesResonant cavityMicrostrip LPFSmall antenna
4 Outlooks
ICEF2016 Xi’an — PITD 2/35
Overview Methods Examples Outlooks
CEM map ˚
˚source: http://feko.info/product-detail/numerical methods
ICEF2016 Xi’an — PITD 3/35
Overview Methods Examples Outlooks
FDTD timeline
birth of F
DTD (Yee)
Engquist-M
ajda ABC
1965 1970 1975 1980
1985199019952000
20052010 2015
Coining of FDTD
Mur ABC
NTFF/RCS
Liao ABCPML (Berenger)
UPML (Sack
s)
CPML (Gedney)
ADI(Namiki; Z
heng et al)
CFDTD (Mitt
ra);
PSTD (Liu)
PI tech
nique to CEM (M
a)
Nonlinear m
edia/circu
its
Recursi
ve convolutio
n
ADE Dispersi
ve dielectric
Unconstr
uctured grid
Contour p
ath
subce
ll model
3D-PITD (Ma)
High-ord
er FDTD
FDTD “Bible” (T
a�ove)
LOD-FDTDMetamateria
l modelin
g
(Hao & M
ittra)
Photonics
& Nanotech
.
(Ta�ove et. a
l.)
ICEF2016 Xi’an — PITD 4/35
Overview Methods Examples Outlooks
FDTD major technical paths
4 Absorbing boundary conditions• Mur; Engquist-Majda; Berenger PML, UPML, CPML
4 Numerical dispersion• High-order space differences; MRTD; PSTD
4 Numerical stability• ADI techniques; PITD; One-step Chebyshev method
4 Conforming grids• Locally/globally conforming; Stable hybrid FETD/FDTD
4 Digital signal processing• Near-to-far-field transformation
4 Dispersive and nonlinear materials• Isotropic/anisotropic dispersions; Nonlinear dispersions
4 Multiphysics
ICEF2016 Xi’an — PITD 5/35
Overview Methods Examples Outlooks
PITD timeline
PI techniqe forLTI ODEs (Zhong)
1990
1995
2000
2005
2010
2015
PI techniqe for CEM (Ma)
3D-PITD (Ma) Engquist-Madja & PML ABCs for PITD (Ma)
PITD in cylin
drical c
oord. (M
a);
Numerical d
ispersi
on analysis
& stabilit
y conditio
n (Chan)
Closed-su
rface
crite
rion (M
a) Sub-domain PITD (Ma);Wavelet Galerkin PITD (Ma)
High order PITD (Ma)
Leap-frog scheme (Ma);Lossy media (Ma)
Compact PITD (Ma);Split-step-scheme-based PITD (Ma)
Uni�ed split-step PITD (Ma)
Hybrid PITD-FDTDKrylov subspace-based PITD
PITD monograph (Sci. Press, Ma)
ICEF2016 Xi’an — PITD 6/35
Overview Methods Examples Outlooks
Maxwell’s equations
And God said:
BE
Bt“ ε´1 ¨∇ˆH,
BH
Bt“ ´µ´1 ¨∇ˆ E,
and then there was light.
J. C. Maxwell
“From a long view of the history of mankind the most significant event of
the nineteenth century will be judged as Maxwell’s discovery of the laws
of electrodynamics.” — Richard P. Feynman
ICEF2016 Xi’an — PITD 7/35
Overview Methods Examples Outlooks
Discretization and Yee cells
FDTD• Finite difference in space
• Finite difference in time
PITD
• Finite difference in space
• ODEs in time
ICEF2016 Xi’an — PITD 8/35
Overview Methods Examples Outlooks
Updating equations / ODEs
FDTD
pR` FqXn`1 “ pR´ FqXn ` fn`1
where
Xn “
„
En
Hn`1{2
• Dε|µ|σe |σm — diagonal matricescontaining ε, µ, σe , σm for each cell
• K — arises from the discretization ofthe curl operators
• fn`1 — sources
PITD
dXptq
dt“MXptq ` fptq
where
Xptq “
„
EptqHptq
• M — matrix containing materialproperties and the discretization of thecurl operators
• f — sources
R “1
2
«
2∆t
Dε ´K
´KT 2∆t
Dµ
ff
F “1
2
„
Dσe K
´KT Dσm
E and H are staggered in time. E and H are non-staggered in time.
ICEF2016 Xi’an — PITD 9/35
Overview Methods Examples Outlooks
Formal solution to dXdt “MX` fptq
• Analytical form
Xptq “ exppMtqX p0q `
ż t
0exprMpt ´ sqsfpsqds
• Recursive form
Xn`1 “ TXn ` Tn`1
ż tn`1
tn
expp´sMqfpsqds
T “ eM∆t1
2
Key points!
ICEF2016 Xi’an — PITD 10/35
Overview Methods Examples Outlooks
Ways to compute eM∆t
• Series methods
• ODE methods
• Polynomial methods
• Matrix decomposition methods
• Splitting methods
• Krylov space methods
• ...
ICEF2016 Xi’an — PITD 11/35
Overview Methods Examples Outlooks
PI technique to compute T “ eM∆t
1 scaling and squaring
T “ eM∆t “
˜
eM ∆t{`
¸`
“
´
eMτ¯`
2 series expansion of eMτ
eMτ “ I` Ta « I` pMτq `pMτq2
2!`pMτq3
3!`pMτq4
4!
3 compute T “`
eMτ˘`“ pI` Taq
` — to be contd.
τ “ ∆t{`
` “ 2N
N – preselected integer
ICEF2016 Xi’an — PITD 12/35
Overview Methods Examples Outlooks
PI technique to compute T “ eM∆t (contd.)
3 compute T “`
eMτ˘`“ pI` Taq
`
T “ pI` Taq2N“ pI` Taq
2N´1
ˆ pI` Taq2N´1
“ ...
with pI` Taq2“ I` 2Ta ` Ta ˆ Ta.
Pseudo code
do i “ 1, ...,NTa Ð 2Ta ` Ta ˆ Ta
end doTÐ I` Ta
The algorithm holds thecomputational precision.
ICEF2016 Xi’an — PITD 13/35
Overview Methods Examples Outlooks
Evaluating T
ż tn`1
tn
e´sMfpsqds & the recursive formula
(1) Analytical solution under the linear representation of fptq
• 1st order Taylor approximation of fptq, i.e.,
f “ f0 ` pt ´ tnqf1, t P ptn, tn`1q
• integrate T
ż tn`1
tn
e´sMfpsqds analytically
• recursive form for Xn`1:
Xn`1 “T“
Xn `M´1pf0 `M´1f1q‰
´M´1“
f0 `M´1f1 ` f1∆t‰
compute M´1? It is noninvertible generally!
ICEF2016 Xi’an — PITD 14/35
Overview Methods Examples Outlooks
Evaluating T
ż tn`1
tn
e´sMfpsqds & the recursive formula (contd.)
(2) Recursive formula by using the Gaussian quadrature rule
• three-point Gaussian quadrature
• recursive form for Xn`1:
Xn`1 “ TXn `5
18e
´
1`?
3{5¯
M∆t{2f”
tn `´
1´a
3{5¯
∆t{2ı
`5
18e
´
1´?
3{5¯
M∆t{2f”
tn `´
1`a
3{5¯
∆t{2ı
`8
18eM∆t{2f ptn `∆t{2q
ICEF2016 Xi’an — PITD 15/35
Overview Methods Examples Outlooks
Remarks
dX
dt“MX` f, X “ rE,HsT
1 finite difference in space, but differential in time
2 scaling and squaring for exppM∆tq
3 Ta Ð 2Ta ` Ta ˆ Ta to guarantee the computationalprecision
4 Gaussian quadrature for the excitation term
5 non-staggered E and H in time
Precise Integration
ICEF2016 Xi’an — PITD 16/35
Overview Methods Examples Outlooks
Numerical stability condition
FDTD CFL criteria: ∆tFDTDupperbound
“1
cb
1p∆xq2
` 1p∆yq2
` 1p∆zq2
PITD
Stability conditions vary for different orders of Taylor approximation:
• 1st/2nd-order: unstable;
• 3rd-order: ∆t ă?
3{2 ` ∆tFDTDupperbound
“?
3{2 2N ∆tFDTDupperbound
• 4th-order: ∆t ă?
2`∆tFDTDupperbound
“ 2N`1{2∆tFDTDupperbound
• 5th-order:?
2p15´?
65q
4?
15´?
65`∆tFDTD
upperbound
ă ∆t ă?
2p15`?
65q
4?
15`?
65`∆tFDTD
upperbound
Almost unconditionally
stable for large N.
ICEF2016 Xi’an — PITD 17/35
Overview Methods Examples Outlooks
Numerical dispersion analysis
FDTD numerical dispersion relation:
W 2t {c
2 “W 2x `W 2
y `W 2z ,
where Wx |y |z “sinpk̃x|y |z∆x |y |z{2q
∆x |y |z{2 , Wt “sinpω∆t{2q
∆t{2 .
PITD
tan2
ˆ
ω∆t
`
˙
“
`
ΛPITD ´ Λ3PITD{3!
˘2
1` Λ2PITD{2!´ Λ4
PITD{4!
where ΛPITD “c∆t`
b
W 2x `W 2
y `W 2z .
ICEF2016 Xi’an — PITD 18/35
Overview Methods Examples Outlooks
Numerical dispersion analysis (contd.)
• numerical dispersion is slightlyworse than that of FDTD
• independent of the time step
• dense griding improves theaccuracy
ICEF2016 Xi’an — PITD 19/35
Overview Methods Examples Outlooks
Source and boundary conditions
Source conditions
• Hard sources
• Plane waves & TS/SF technique
• ...
Boundary conditions
• Engquist-Majda ABC
• PMLs
• ...
“它山之石,可以攻玉。” —《诗经·小雅·鹤鸣》
“Stones from other hills may serve topolish the jade of this one.”
— Classic of Poetry ‚ Lesser Court Hymns‚ Singing of Cranes
ICEF2016 Xi’an — PITD 20/35
Overview Methods Examples Outlooks
Remarks
Characteristics of the PITD method:
3 preselected N determines the upper bound of ∆tPITD
3 ∆tPITDupperbound
ąą ∆tFDTDupperbound
3 slight worse numerical dispersion compared with that of theFDTD method
3 numerical dispersion can be independent of ∆t
3 technique paths of the FDTD method can be learned
“Stones from other hills may serve to polish the jade of this one.”— Classic of Poetry ‚ Lesser Court Hymns ‚ Singing of Cranes
ICEF2016 Xi’an — PITD 21/35
Overview Methods Examples Outlooks
Improved methods
• Fourth-order PITD [PITD(4)] method
• Wavelet Galerkin PITD (WG-PITD) method
• Leapfrog PITD (L-PITD) method
• Compact PITD (CPITD) method
• Hybrid PITD-FDTD method
• Krylov subspace method
• ...
ICEF2016 Xi’an — PITD 22/35
Overview Methods Examples Outlooks
Improved methods — PITD(4) 1
4th-order spatial difference scheme is used as:
BuiBx
“1
∆x
„
1
24
`
ui´3{2 ´ ui`3{2
˘
´27
24
`
ui´1{2 ´ ui`1{2
˘
`O”
p∆xq4ı
Numerical dispersion
1IEEE T-AP, 59(4), 2011: 1311-1320.
ICEF2016 Xi’an — PITD 23/35
Overview Methods Examples Outlooks
Improved methods — WG-PITD method 2
Discretization form of Maxwell equation(s) in space:
d Ex |l`1{2,m,n
dt“ ´
σ|l`1{2,m,n
ε|l`1{2,m,n
Ex |l`1{2,m,n `ÿ
i
ai
«
Hz |l`1{2,m`i`1{2,n
ε|l`1{2,m,n ∆y´
Hyˇ
ˇ
l`1{2,m,n`i`1{2
ε|l`1{2,m,n ∆z
ff
where ai “ş8
´8
dφ1{2pxq
dx φ´i pxqdx .
Numerical dispersion S11-parameter
2IEEE MWCL, 20(12), 2010: 651-653
ICEF2016 Xi’an — PITD 24/35
Overview Methods Examples Outlooks
Improved methods — Hybrid PITD-FDTD method 3
How to handle multiscale problems with fine geometrical features?
• Subgriding in FDTD — ∆t depends on ∆xmin
• PITD — need to compute eM∆t , but ∆t can be relaxed
Hybriding
3Please refer to PA-11 (Mon. 14:00-15:30, Function Room 2, 2F)
ICEF2016 Xi’an — PITD 25/35
Overview Methods Examples Outlooks
Improved methods — Krylov space method 4
Recursive form of the PITD method:
Xn`1 “ eM∆tXn `ÿ
i
αieMβi∆tfptn ` γi∆tq
number of nonzeroelements
Denseness Memory cost (MB)
M 1558 0.0039 0.003eM∆t 370482 0.9334 0.76
Evaluate eM∆t explicitly? 7 ÝÑ Estimate eM∆tv directly. 3
4Please refer to OC2-6 (Tue. 08:30-10:00, Function Room 3, 3F)
ICEF2016 Xi’an — PITD 26/35
Overview Methods Examples Outlooks
Improved methods — Krylov space method 4 (contd.)
Direct estimation of eM∆tv
1 mth-order Krylov subspace: KmpM, vq “ spanpv,Mv, . . . ,Mm´1vq
2 Arnoldi process
• Vm “ rv1, v2, . . . , vmsT – orthogonal basis of Km
• Hm « VTmMVm – matrix generated during the Arnoldi process
3 eM∆tv « VmeHm∆tVT
mv “ VmeHm∆te1, e1 “ r1, 0, 0, . . . , 0s
TP Rmˆ1
CPU time (s)Memory
cost (MB)Krylov-PITD 23.78 0.30
FDTD 137.28 1.34
4Please refer to OC2-6 (Tue. 08:30-10:00, Function Room 3, 3F)
ICEF2016 Xi’an — PITD 27/35
Overview Methods Examples Outlooks
Rectangular cavity
fana. (GHz)FDTD scheme ADI-FDTD scheme PITD scheme
∆t = 1 ps ∆t = 60 ps ∆t = 60 psf (GHz) rel. err. f (GHz) rel. err. f (GHz) rel. err.
3.125 2.983 4.54% 2.900 7.20% 2.983 4.54%4.881 4.750 2.68% 4.650 4.73% 4.750 2.68%5.340 5.450 2.06% 5.580 4.49% 5.450 2.06%7.289 7.333 0.60% 6.817 6.92% 7.333 0.60%7.529 7.567 0.51% 7.000 7.03% 7.567 0.51%
• relative error increases as the time-stepincreases for the ADI-FDTD method
• relative error is independent of the timestep for the PITD method
8 cm
4 cm
6 cm
ICEF2016 Xi’an — PITD 28/35
Overview Methods Examples Outlooks
Microstrip low pass filter 5
Comparison between the FDTD method and the L-PITD (Leapfrog PITD) method :
∆x ∆y ∆z ∆t memory CPU time
FDTD 0.41 mm 0.26 mm 0.42 mm 0.441 ps 24.44 MB 1024 sL-PITD 0.41 mm 0.26 mm 0.42 mm 0.884 ps 248.4 MB 851 s: Simulations were performed on Intel® CoreTM Duo CPU T8100 2.10 GHz PC.
5IEEE MWCL, 22(6), 2012: 294 – 296.
ICEF2016 Xi’an — PITD 29/35
Overview Methods Examples Outlooks
Small antenna
Comparison between the FDTD method and the Krylov-PITD method
CPU time (s) Memory cost (MB)
Krylov-PITD 23.78 0.30FDTD 137.28 1.34
ICEF2016 Xi’an — PITD 30/35
Overview Methods Examples Outlooks
Remarks
4 almost unconditionally stable
4 relative large time step size can be used
4 PI technique maintains the computational precision
4 relative error independent of the time-step
4 hybrid PITD-FDTD technique suitable for multiscale problems
4 memory cost can be relaxed by using the Kyrlov space method
“瑕不掩瑜。” —《礼记·聘义》
“One flaw cannot obscure the splendor of the jade.”— Book of Rites ‚ Meaning of Interchange of Missions twixt Different Courts
ICEF2016 Xi’an — PITD 31/35
Overview Methods Examples Outlooks
Outlook
Future work :
• Sub-domain technique
• Parallel computing technique
• Extend to complex materials
Future prospects :
• Nanophotonics and nanoplasmonics. Ultimately, combinationof quantum and classical electrodynamics
• Multiphysics
ICEF2016 Xi’an — PITD 32/35
Overview Methods Examples Outlooks
Acknowledgements & References
Acknowledgements
Dr. Jinquan Zhao, Mr. Min Tang, Ms. Mei Yang, Dr. Xintai Zhao, Dr. Gang Sun,
Dr. Zhongming Bai, Dr. Qi Liu and Dr. Zhen Kang are kindly acknowledged.
References• IEEE T-MTT, 2006, 54(7): 3026
• IEEE MWCL, 2007, 17(7): 471
• PIER, 2007, 69: 201
• IEEE T-MTT, 2008, 56(12): 2859
• IEEE MWCL, 2010, 20(12): 651
• IEEE T-AP, 2011, 59(4): 1311
• IEEE T-MTT, 2012, 60(9): 2723
• IEEE MWCL, 2012, 22(6): 294
• Electron. Lett., 2013, 49(18): 1135
• COMPEL, 2013, 33(1/2): 85
• IEEE T-MTT, 2013, 61(7): 2535
• Electron. Lett., 2014, 50(18): 1297
• IEEE MWCL, 2016, 26(2): 83
• Ma, Xikui. “The precise integrationtime domain method.”(in Chinese)Beijing: Science Press (2015).
ICEF2016 Xi’an — PITD 33/35
Recruitment Ad
Our Group for Advanced Electrical Technologies (GAET) isseeking for highly self-motivated students with the solid scientificstrength in one of the following areas:
• Electromagnetics (theory and computation)
• Metamaterials and plasmonics (graphene plasmonics)
• Wireless power tansfer
• Power electronics
Please visit http://tydong.gr.xjtu.edu.cn/ for details.