A Summary Review on 25 Years of Progress and Future Challenges in FDTD and FETD Techniques
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A Summary Review on 25 Years of Progress and Future Challenges in FDTD and FETD TechniquesAbstract
The finite-difference time-domain (FDTD) method has established itself among the most popular methods for the numerical solution of Maxwell equations. Reasons for its popularity include its versatility, matrix-free characteristic, ease for parallelization, and low computational complexity. In recent years, the finite-element time-domain (FETD) has also become another very popular algorithm for solving time-domain Maxwell equations due to its geometrical flexibility and the steady growth in hardware computing power. In this review, we succinctly recollect some of the milestones in the development of FDTD and FETD over the last 25 years, and briefly discuss some challenges for the future development of these two algorithms.
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References
K. S. Yee, “Numerical solution of initial
boundary value problems involving
Maxwell ́s equations in isotropic media,”
IEEE Trans. Antennas Propag., vol. 14, pp.
-307, 1966.
A. Taflove and S. Hagness, Computational
Electrodynamics: The Finite-Difference
Time-Domain Method, 3 rd ed., Artech
House, 2005.
J. F. Lee, R. Lee, and A. C. Cangellaris,
“Time domain finite element methods,”
IEEE Trans. Antennas Propag., vol. 45, no.
, pp. 430-442, 1997.
J. M. Jin, The Finite Element Method in
Electromagnetics, 2 nd ed., John Wiley, 2003.
B. Donderici and F. L. Teixeira, “Mixed
finite-element time-domain method for
transient Maxwell equations in doubly
dispersive media,” IEEE Trans. Microwave
Theory Tech. , vol. 56, no. 1, pp. 113-120,
K. L. Shlager and J. B. Schneider, “A
selective survey of the finite-difference
time-domain literature,” IEEE Antennas
Propag. Mag., vol. 37, no. 4, pp. 39-56,
F. L. Teixeira, “Time-domain finite-
difference and finite-element methods for
Maxwell equations in complex media,”
IEEE Trans. Antennas Propag., vol. 56, no.
, pp. 2150-2166, 2008.
A. Taflove and M. E. Brodwin, “Numerical
solution of steady-state electromagnetic
scattering problems using the time-
dependent Maxwel l's equations,” IEEE
Trans. Microwave Theory Tech., vol. 23, pp.
–630, 1975.
R. Holland, “Threde: A free-field EMP
coupling and scattering code,” IEEE Trans.
Nuclear Sci., vol. 24, pp. 2416–2421, 1977.
K. S. Kunz and K. M. Lee, “A three-
dimensional finite-difference solution of the
external response of an aircraft to a complex
transient EM environment,” IEEE Trans.
Electromagn. Compat., vol. 20, pp. 328–
, 1978.
A. Taflove, “Application of the finite-
difference time-domain method to sinusoidal
steady state electromagnetic penetration
problems,” IEEE Trans. on Electromagn.
Compat., vol. 22, pp. 191–202, 1980.
T. Weiland, “A discretization method for the
solution of Maxwell’s equations for six-
component fields,” Electronics and
Communications AEU, vol. 31, no.3, pp.
–120, 1977.
G. Mur, “Absorbing boundary conditions for
the finite-difference approximation of the
time-domain electromagnetic field
equations,” IEEE Trans. Electromagn.
Compat., vol. 23, pp. 377–382., 1981.
Z. P. Liao, H. L. Wong, B. P. Yang, and Y.
F. Yuan, “A transmitting boundary for
transient wave analysis,” Scientia Sinica A
vol. 27, pp. 1063–1076, 1984.
G. A. Kriegsmann, A. Taflove, and K. R.
Umashankar, “A new formulation of
electromagnetic wave scattering using an
on-surface radiation boundary condition
approach,” IEEE Trans. Antennas Propag.,
vol. 35, pp. 153–161, 1987.
T. G. Moore, J. G. Blaschak, A. Taflove,
and G. A. Kriegsmann, “Theory and
application of radiation boundary
operators,” IEEE Trans. Antennas Propag .,
vol. 36, pp. 1797–1812, 1988.
K. R. Umashankar and A. Taflove, “A novel
method to analyze electromagnetic
scattering of complex objects,” IEEE Trans.
Electromagn. Compat., vol. 24, pp. 397–
, 1982.
ACES JOURNAL, VOL. 25, NO. 1, JANUARY 2010
A. Taflove and K. R. Umashankar, “Radar
cross section of general three-dimensional
scatterers,” IEEE Trans. Electromagn.
Compat., vol. 25, pp. 433–440, 1983.
D. H. Choi and W. J. Hoefer, “The finite-
difference time-domain method and its
application to eigenvalue problems,” IEEE
Trans. Microwave Theory Tech., vol. 34, pp.
–1470, 1986.
X. Zhang, J. Fang, K. K. Mei, and Y. Liu,
“Calculation of the dispersive characteristics
of microstrips by the time-domain finite-
difference method,” IEEE Trans. Microwave
Theory Tech., vol. 36, pp. 263–267, 1988.
D. M. Sullivan, O. P. Gandhi, and A.
Taflove, “Use of the finite-difference time-
domain method in calculating EM
absorption in man models,” IEEE Trans.
Biomed. Eng., vol. 35, pp. 179–186, 1988.
K. R. Umashankar, A. Taflove, and B.
Beker, “Calculation and experimental
validation of induced currents on coupled
wires in an arbitrary shaped cavity,” IEEE
Trans. Antennas Propag., vol. 35, pp. 1248–
, 1987.
A. Taflove, K. R. Umashankar, B. Beker, F.
A. Harfoush, and K. S. Yee, “Detailed
FDTD analysis of electromagnetic fields
penetrating narrow slots and lapped joints in
thick conducting screens,” IEEE Trans.
Antennas Propag., vol. 36, pp. 247–257,
C. J. Railton, I. J. Craddock, and J. B.
Schneider, "An improved locally distorted
CPFDTD algorithm with provable stability,"
Electron. Lett., vol. 31, no. 18, pp. 1585-
, 1995.
S. Dey and R. Mittra, “A locally
conformal finite-difference time-domain
(FDTD) algorithm for modeling three-
dimensional perfectly conducting objects,”
IEEE Microwave Guided Wave Lett ., vol.
, pp. 273-275, 1997.
C. J. Railton and J. B. Schneider, "An
analytical and numerical analysis of several
locally-conformal FDTD Schemes," IEEE
Trans. Microwave Theory Tech., vol. 47, no.
, pp. 56-66, 1999
W. Gwarek, “Analysis of an arbitrarily
shaped planar circuit — A time-domain
approach,” IEEE Trans. Microwave Theory
Tech., vol. 33, pp. 1067–1072, 1985.
G. Maloney, G. S. Smith, and W. R. Scott,
Jr., “Accurate computation of the radiation
from simple antennas using the finite-
difference time-domain method,” IEEE
Trans. Antennas Propag., vol. 38, pp. 1059–
, 1990
D. S. Katz, A. Taflove, M. J. Piket-May, and
K. R. Umashankar, “FDTD analysis of
electromagnetic wave radiation from
systems containing horn antennas,” IEEE
Trans. Antennas Propag., vol. 39, pp. 1203–
, 1991.
K. Li, C. F. Lee, S. Y. Poh, R. T. Shin, and
J. A. Kong, “Application of FDTD method
to analysis of electromagnetic radiation from
VLSI heatsink configurations,” IEEE Trans.
Electromagn. Compat., vol. 35, no. 2, pp.
– 214, 1993.
R. Lee and T.T. Chia, “Analysis of
electromagnetic scattering from a cavity
with a complex termination by means of a
hybrid ray-FDTD method,” IEEE Trans.
Antennas Propag., vol. 41, pp. 1560-1569,
V. A. Thomas, M. E. Jones, M. J. Piket-
May, A. Taflove, and E. Harrigan, “The use
of SPICE lumped circuits as sub-grid
models for FDTD high-speed electronic
circuit design,” IEEE Microwave Guided
Wave Lett., vol. 4, pp. 141–143, 1994.
A. Bataineh, R. Lee, and F. Ozguner,
“Electrical characterization of high-speed
interconnects with a parallel three-
dimensional finite difference time-domain
algorithm,” Simulation, vol. 64, pp. 289-
, 1995.
E. Sano and T. Shibata, “Full-wave analysis
of picosecond photoconductive switches,”
IEEE J. Quantum Electron., vol. 26, pp.
–377, 1990.
P. M. Goorjian and A. Taflove, “Direct time
integration of Maxwell’s equations in
nonlinear dispersive media for propagation
and scattering of femtosecond
electromagnetic solitons,” Opt. Lett., vol.
, pp. 180–182, 1992.
R. W. Ziolkowski and J. B. Judkins, “Full-
wave vector Maxwell’s equations modeling
TEIXERIA: SUMMARY REVIEW ON 25 YEARS IN FDTD AND FETD
of self-focusing of ultra-short optical pulses
in a nonlinear Kerr medium exhibiting a
finite response time,” J. Opt. Soc. Am. B,
vol. 10, pp. 186–198, 1983.
R. Luebbers, F. Hunsberger, K. Kunz, R.
Standler, and M. Schneider, “A frequency-
dependent finite-difference time-domain
formulation for dispersive materials,” IEEE
Trans. Electromagn. Compat., vol. 32, pp.
–227, 1990.
R. M. Joseph, S. C. Hagness, and A.
Taflove, “Direct time integration of
Maxwell’s equations in linear dispersive
media with absorption for scattering and
propagation of femtosecond electromagnetic
pulses,” Opt. Lett., vol. 16, pp. 1412–1414,
A. C. Cangellaris, M. Gribbons, and G.
Sohos, “A hybrid spectral/FDTD method for
the electromagnetic analysis of guided
waves in periodic structures,” IEEE
Microwave Guided Wave Lett., vol. 3, no.
, pp. 375–377, 1993.
P. Harms, R. Mittra, and W. Ko,
“Implementation of the periodic boundary
condition in the finite-difference time-
domain algorithm for FSS structures,” IEEE
Trans. Antennas Propag. , vol. 42, no. 9,
pp.1317–1324, 1994.
A. Taflove and S. Hagness, “Analysis of
Periodic Structures,” in Computational
Electrodynamics: The Finite Difference
Time Domain Method. Boston, MA: Artech
House, 1995, ch. 13.
M. Celuch-Marcysiak and W. K. Gwarek,
“Spatially looped algorithms for time-
domain analysis of periodic structures,”
IEEE Trans. Microwave Theory Tech. , vol.
, no. 4, pp. 860–865, 1995.
S. Wang and F. L. Teixeira, “Dispersion-
relation-preserving FDTD schemes of large-
scale three-dimensional problems,” IEEE
Trans. Antennas Propag., vol. 51, no. 8, pp.
-1828, 2003.
B. Finkelstein and R. Kastner, “A
comprehensive new methodology for
formulating FDTD schemes with controlled
order of accuracy and dispersion,” IEEE
Trans. Antennas Propag., vol. 56, no.
, pp. 3516-3525, 2008.
Q.H. Liu, “The PSTD algorithm: A time-
domain method requiring only two cells per
wavelength,” Microwave Opt. Technol.
Lett., vol. 15, no. 3, pp. 357-361, 1997.
J. Berenger, “A perfectly matched layer for
the absorption of electromagnetic waves,” J.
Comput. Phys., vol. 114, pp. 185–200, 1994.
D. S. Katz, E. T. Thiele, and A. Taflove,
“Validation and extension to three
dimensions of the Berenger PML absorbing
boundary condition for FDTD meshes,”
IEEE Microwave Guided Wave Lett., vol. 4,
pp. 268–270, 1994.
Z. S. Sacks, D. M. Kingsland, R. Lee, and J.
F. Lee, “A perfectly matched anisotropic
absorber for use as an absorbing boundary
condition,” IEEE Trans . Antennas Propag .,
vol. 43, pp. 1460–1463, 1995.
F. L. Teixeira and W. C. Chew, “Complex
space approach to perfectly matched layers:
A review and some new developments,” Int.
J. Num. Model., vol. 13, no. 5, pp. 441-455,
F. L. Teixeira, K. P. Hwang, W. C. Chew,
and J. M. Jin, “Conformal PML-FDTD
schemes for electromagnetic field
simulations: A dynamic stability study,”
IEEE Trans. Antennas Propag., vol. 49, no.
, pp. 902-907, 2001.
C. D. Moss, F. L. Teixeira, Y.E. Yang, and
J. A. Kong, “Finite-difference time-domain
simulation of scattering from objects in
continuous random media,” IEEE Trans.
Geosci. Remote Sens. , vol. 40, no. 1, pp.
-186, 2002.
T. Namiki, “A new FDTD algorithm based
on alternating-direction implicit method,”
IEEE Trans. Microwave Theory Tech. , vol.
, no. 10, pp. 2003–2007, Oct. 1999.
F. Zheng, Z. Chen, and J. Zhang, “Toward
the development of a three-dimensional
unconditionally stable finite-difference time-
domain method,” IEEE Trans. Microw.
Theory Tech., vol. 48, no. 9, pp. 1550–1558,
Sep. 2000.
V. E. do Nascimento, B.-H. V. Borges, and
F. L. Teixeira, “Split-field PML
implementations for the unconditionally
stable LOD-FDTD method,” IEEE
ACES JOURNAL, VOL. 25, NO. 1, JANUARY 2010
Microwave Wireless Components Lett. , vol.
, no. 7, pp. 398-400, 2006.
K.-Y. Jung and F. L. Teixeira, “An iterative
unconditionally stable LOD-FDTD
method,” IEEE Microwave Wireless
Components Lett., vol. 18, no. 2, pp. 76-78,
S. S. Zivanovic, K. S.Yee, and K. K. Mei,
“A subgridding method for the time-domain
finite-difference method to solve Maxwell’s
equations,” IEEE Trans. Microwave Theory
Tech., vol. 39, no. 3, pp. 471–479, 1991.
S. Kapoor, “Sub-cellular technique for
finite-difference time-domain method,”
IEEE Trans. Microwave Theory Tech. , vol.
, no. 5, pp. 673–677, 1997.
M. Okoniewski, E. Okoniewska, and M. A.
Stuchly, “Three-dimensional subgridding
algorithm for FDTD,” IEEE Trans.
Antennas Propag., vol. 45, no. 3, pp. 422–
, 1997.
K. M. Krishnaiah and C. J. Railton,
‘‘Passive equivalent circuit of FDTD: an
application to subgridding,’’ Electron. Lett.,
vol. 33, 1277–1278, 1997.
P. Thoma and T. Weiland, ‘‘A consistent
subgridding scheme for the finite difference
time domain method,’’ Int. J. Num. Model.
, 359–374, 1996.
K. M. Krishnaiah and C. J. Railton, "A
stable subgridding algorithm and its
application to eigenvalue problems," IEEE
Trans. Microwave Theory Tech., vol. 47, pp.
-628, 1999.
S. Wang, F. L. Teixeira, R. Lee, and J.-F.
Lee, `Optimization of subgridding schemes
for FDTD,' IEEE Microwave Wireless
Components Lett. , vol. 12, no. 6, pp. 223-
, 2002.
B. Donderici and F. L. Teixeira, “Improved
FDTD subgridding algorithms via digital
filtering and domain overriding,” IEEE
Trans. Antennas Propag., vol. 53, no. 9, pp.
-2951, 2005.
J. P. Berenger, “A Huygens subgridding for
the FDTD method,” IEEE Trans. Antennas
Propag., vol. 54, no. 12, pp. 3797 – 3804,
R. Schuhmann and T. Weiland, “A stable
interpolation technique for FDTD on non-
orthogonal grids,” Int. J. Num. Model., vol.
, no. 6, pp. 299 – 306, 1999.
S. D. Gedney and J. A. Roden, “Numerical
stability of nonorthogonal FDTD methods,”
IEEE Trans. Antennas Propag., vol. 48,
issue 2, pp. 231-239, 2000.
A. Bossavit, “Whitney forms: A new class
of finite elements for three-dimensional
computations in electromagnetics,” IEE
Proc. A , vol. 135, pp. 493–500, 1988.
F. L. Teixeira and W. C. Chew, “Analytical
derivation of a conformal perfectly matched
absorber for electromagnetic waves,”
Microwave and Optical Technology Letters,
vol. 17, no. 4, pp. 231-236, 1998.
T. Rylander and J.-M. Jin, “Conformal
perfectly matched layers for the time domain
finite element method,” in IEEE AP-S Int.
Symp. Digest, vol. 1, pp. 698–701, 2003.
B. Donderici and F. L. Teixeira, “Conformal
perfectly matched layer for the mixed finite-
element time-domain method,” IEEE Trans.
Antennas Propag., vol. 56, no. 4, pp. 1017-
, 2008.
M. M. Ilic and B. M. Notaros, “Higher
order hierarchical curved hexahedral vector
finite elements for electromagnetic
modeling,” IEEE Trans. Microwave Theory
Tech., vol. 51, no. 3, pp. 1026-103, 2003.
R. N. Rieben , G. H. Rodrigue , and D. A.
White, “A high order mixed vector finite
element method for solving the time
dependent Maxwell equations on
unstructured grids,” J. Comp. Phys ., vol.
, no. 2, pp. 490-519, 2005.
G. Cohen, Higher Order Numerical
Methods for Transient Wave Equations ,
Springer-Verlag, 2001.
N. Marais and D. B. Davidson, “Numerical
evaluation of high-order finite element time
domain formulations in electromagnetics,”
IEEE Trans. Antennas Propag., vol. 56, no.
, pp. 3743-3751, 2008.
B. He and F. L. Teixeira, “A sparse and
explicit FETD via approximate inverse
Hodge (mass) matrix,” IEEE Microwave
Wireless Components Lett. , vol. 16, no. 6,
pp. 348-350, 2006.
M. Wong, O. Picon, and V. F. Hanna, “A
finite element method based on Whitney
TEIXERIA: SUMMARY REVIEW ON 25 YEARS IN FDTD AND FETD
forms to solve Maxwell equations in the
time domain,” IEEE Trans. Magn. , vol. 31,
no. 3, pp. 1618–1621, 1995.
F. L. Teixeira and W. C. Chew, “Lattice
electromagnetic theory from a topological
viewpoint,” J. Math. Phys., vol. 40, no. 1,
pp. 169–187, 1999.
G. Cohen and P. Monk, “Gauss point mass
lumping schemes for Maxwell’s equations,”
Numer. Meth. Partial Diff. Eq., vol. 14 , pp.
–88, 1998.
J. R. Munkres, Elements of Algebraic
Topology, Addison-Wesley, Reading, Mass.,
F. L. Teixeira (ed.), Geometrical Methods
for Computational Electromagnetics,
Progress In Electromagnetics Research
Series 32, EMW Publishing, Cambridge,
Mass., 2001.
B. He and F. L. Teixeira, “On the degrees of
freedom of lattice electrodynamics,” Phys.
Lett. A, vol. 336, pp. 1-7, 2005.
P. Monk, Finite Element Methods for
Maxwell's Equations, Oxford Univ. Press,
S. Adams, J. Payne, and R. Boppana, “Finite
Difference Time Domain (FDTD)
Simulations Using Graphics Processors,”
Proc. 2007 DoD High Performance
Computing Modernization Program Users
Group Conf., pp. 334-338, 2007.
Y. Liu and C.D. Sarris, “Fast Time-Domain
Simulation of Optical Waveguide Structures
with a Multilevel Dynamically Adaptive
Mesh Refinement FDTD Approach,” J.
Lightwave Technol., vol. 24, no. 8, pp. 3235-
, 2006.
R. A. Abd-Alhameed and P. S. Excell,
Complex computational electromagnetic
using hybridization techniques,” in
Advances in Information Technologies for
Electromagnetics (L. Tarricone, A. Espostio,
eds.), Springer-Verlag, pp.69-145, 2006.
B. Donderici and F. L. Teixeira, “Accurate
interfacing of heterogeneous structured
FDTD grid components,” IEEE Trans.
Antennas Propag., vol. 54, no. 6. pp. 1826-
, 2006.
A. Lew, A., J.E. Mardsen, M. Ortiz, and M.
West, “Asynchronous variational
integrators,” Arch. Rational Mech. Anal. ,
vol. 167, p. 85, 2003.
D. J. Riley and C. D. Turner, “VOLMAX: a
solid-model-based, transient volumetric
Maxwell solver using hybrid grids,” IEEE
Antennas Propag. Mag., vol. 39, no. 1, pp.
-33, 1997.
C.-T. Hwang and R.-B. Wu, “Treating late-
time instabilit y of hybrid finite-
element/finite-difference time-domain
method,” IEEE Trans. Antennas Propag. ,
vol. 47, no. 2. pp. 227-232, 1999.
T. Rylander and A. Bondeson, “Stable
FDTD-FEM hybrid method for Maxwell’s
equations,” Comp. Phys. Comm., vol. 125,
pp. 75–82, 2000.
J. Jin and D. J. Riley, Finite Element
Analysis of Antennas and Arrays , Wiley-
IEEE Press, 2009
