Finite-difference time-domain method

Finite-difference time-domain (FDTD) is a popular computational electrodynamics modeling technique. It is considered easy to understand and easy to implement in software. Since it is a time-domain method, solutions can cover a wide frequency range with a single simulation run.

The FDTD method belongs in the general class of grid-based differential time-domain numerical modeling methods. The time-dependent Maxwell's equations (in partial differential form) are discretized using central-difference approximations to the space and time partial derivatives. The resulting finite-difference equations are solved in either software or hardware in a leapfrog manner: the electric field vector components in a volume of space are solved at a given instant in time; then the magnetic field vector components in the same spatial volume are solved at the next instant in time; and the process is repeated over and over again until the desired transient or steady-state electromagnetic field behavior is fully evolved.

The basic FDTD space grid and time-stepping algorithm trace back to a seminal 1966 paper by Kane Yee in IEEE Transactions on Antennas and Propagation (Yee 1966). The descriptor "Finite-difference time-domain" and its corresponding "FDTD" acronym were originated by Allen Taflove in a 1980 paper in IEEE Transactions on Electromagnetic Compatibility (Taflove 1980). See "References" for these and other important journal papers in the development of FDTD techniques, as well as relevant textbooks and research monographs.

Since about 1990, FDTD techniques have emerged as primary means to computationally model many scientific and engineering problems dealing with electromagnetic wave interactions with material structures. As summarized in Taflove & Hagness (2005), current FDTD modeling applications range from near-DC (ultralow-frequency geophysics involving the entire Earth-ionosphere waveguide) through microwaves (radar signature technology, antennas, wireless communications devices, digital interconnects, biomedical imaging/treatment) to visible light (photonic crystals, nanoplasmonics, solitons, and biophotonics). In 2006, an estimated 2,000 FDTD-related publications appeared in the science and engineering literature (see "Growth of FDTD publications"). At present, there are at least 27 commercial/proprietary FDTD software vendors; 8 free-software/open-source-software FDTD projects; and 2 freeware/closed-source FDTD projects, some not for commercial use (see "External links").

Workings of the FDTD method

When Maxwell's differential equations are examined, it can be seen that the change in the E-field in time (the time derivative) is dependent on the change in the H-field across space (the curl). This results in the basic FDTD time-stepping relation that, at any point in space, the updated value of the E-field in time is dependent on the stored value of the E-field and the numerical curl of the local distribution of the H-field in space (Yee 1966).

The H-field is time-stepped in a similar manner. At any point in space, the updated value of the H-field in time is dependent on the stored value of the H-field and the numerical curl of the local distribution of the E-field in space. Iterating the E-field and H-field updates results in a marching-in-time process wherein sampled-data analogs of the continuous electromagnetic waves under consideration propagate in a numerical grid stored in the computer memory.

This description holds true for 1-D, 2-D, and 3-D FDTD techniques. When multiple dimensions are considered, calculating the numerical curl can become complicated. Kane Yee's seminal 1966 paper in IEEE Transactions on Antennas and Propagation proposed spatially staggering the vector components of the E-field and H-field about rectangular unit cells of a Cartesian computational grid so that each E-field vector component is located midway between a pair of H-field vector components, and conversely. This scheme, now known as a Yee lattice, has proven to be very robust, and remains at the core of many current FDTD software constructs (Yee 1966).

Furthermore, Yee proposed a leapfrog scheme for marching in time wherein the E-field and H-field updates are staggered so that E-field updates are conducted midway during each time-step between successive H-field updates, and conversely (Yee 1966). On the plus side, this explicit time-stepping scheme avoids the need to solve simultaneous equations, and furthermore yields dissipation-free numerical wave propagation. On the minus side, this scheme mandates an upper bound on the time-step to ensure numerical stability (Taflove & Brodwin 1975). As a result, certain classes of simulations can require many thousands of time-steps for completion.

Using the FDTD method

In order to use FDTD a computational domain must be established. The computational domain is simply the physical region over which the simulation will be performed. The E and H fields are determined at every point in space within that computational domain. The material of each cell within the computational domain must be specified. Typically, the material is either free-space (air), metal, or dielectric. Any material can be used as long as the permeability, permittivity, and conductivity are specified.

Once the computational domain and the grid materials are established, a source is specified. The source can be an impinging plane wave, a current on a wire, or an applied electric field, depending on the application.

Since the E and H fields are determined directly, the output of the simulation is usually the E or H field at a point or a series of points within the computational domain. The simulation evolves the E and H fields forward in time.

Processing may be done on the E and H fields returned by the simulation. Data processing may also occur while the simulation is ongoing.

While the FDTD technique computes electromagnetic fields within a compact spatial region, scattered and/or radiated far fields can be obtained via near-to-far-field transformations, as reported originally by Umashankar and Taflove (1982).

Strengths of FDTD modeling

Every modeling technique has strengths and weaknesses, and the FDTD method is no different.

FDTD is a versatile modeling technique used to solve Maxwell's equations. It is intuitive, so users can easily understand how to use it and know what to expect from a given model.

FDTD is a time-domain technique, and when a broadband pulse (such as a Gaussian pulse) is used as the source, then the response of the system over a wide range of frequencies can be obtained with a single simulation. This is useful in applications where resonant frequencies are not exactly known, or anytime that a broadband result is desired.

Since FDTD calculates the E and H fields everywhere in the computational domain as they evolve in time, it lends itself to providing animated displays of the electromagnetic field movement through the model. This type of display is useful in understanding what is going on in the model, and to help ensure that the model is working correctly.

The FDTD technique allows the user to specify the material at all points within the computational domain. A wide variety of linear and nonlinear dielectric and magnetic materials can be naturally and easily modeled.

FDTD allows the effects of apertures to be determined directly. Shielding effects can be found, and the fields both inside and outside a structure can be found directly or indirectly.

FDTD uses the E and H fields directly. Since most EMI/EMC modeling applications are interested in the E and H fields, it is convenient that no conversions must be made after the simulation has run to get these values.

Weaknesses of FDTD modeling

Since FDTD requires that the entire computational domain be gridded, and the grid spatial discretization must be sufficiently fine to resolve both the smallest electromagnetic wavelength and the smallest geometrical feature in the model, very large computational domains can be developed, which results in very long solution times. Models with long, thin features, (like wires) are difficult to model in FDTD because of the excessively large computational domain required.

FDTD finds the E/H fields directly everywhere in the computational domain. If the field values at some distance are desired, it is likely that this distance will force the computational domain to be excessively large. Far-field extensions are available for FDTD, but require some amount of postprocessing (Taflove & Hagness 2005).

Since FDTD simulations calculate the E and H fields at all points within the computational domain, the computational domain must be finite to permit its residence in the computer memory. In many cases this is achieved by inserting artificial boundaries into the simulation space. Care must be taken to minimize errors introduced by such boundaries. There are a number of available highly effective absorbing boundary conditions (ABCs) to simulate an infinite unbounded computational domain (Taflove & Hagness 2005). Most modern FDTD implementations instead use a special absorbing "material", called a perfectly matched layer (PML) to implement absorbing boundaries (Berenger 1994, Gedney 1996).

Because FDTD is solved by propagating the fields forward in the time domain, the electromagnetic time response of the medium must be modeled explicitly. For an arbitrary response, this involves a computationally expensive time convolution, although in most cases the time response of the medium (or Dispersion (optics)) can be adequately and simply modeled using either the recursive convolution (RC) technique, the auxiliary differential equation (ADE) technique, or the Z-transform technique. An alternative way of solving Maxwell's equations that can treat arbitrary dispersion easily is the Pseudospectral Spatial-Domain method (PSSD), which instead propagates the fields forward in space.

Grid truncation techniques for open-region FDTD modeling problems

The most commonly used grid truncation techniques for open-region FDTD modeling problems are the Mur absorbing boundary condition (ABC) (Mur 1981), the Liao ABC (Liao "et al." 1984), and various perfectly matched layer (PML) formulations (Berenger 1994, Gedney 1996, Taflove & Hagness 2005). The Mur and Liao techniques are simpler than PML. However, PML (which is technically an absorbing region rather than a boundary condition "per se") can provide orders-of-magnitude lower reflections. The PML concept was introduced by J.-P. Berenger in a seminal 1994 paper in the Journal of Computational Physics (Berenger 1994). Since 1994, Berenger's original split-field implementation has been modified and extended to the uniaxial PML (UPML), the convolutional PML (CPML), and the higher-order PML. The latter two PML formulations have increased ability to absorb evanescent waves, and therefore can in principle be placed closer to a simulated scattering or radiating structure than Berenger's original formulation.

Growth of FDTD publications

An estimated 2,000 FDTD-related publications appeared in the science and engineering literature in 2006. The current rate of growth (based upon a study of ISI Web of Science data) is approximately 5:1 over the period 1995 to 2006.

ee also

* Computational electromagnetics
* Maxwell's equations

References

Journal articles

*cite journal | author= Kane Yee | title= Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media | journal= Antennas and Propagation, IEEE Transactions on | year= 1966 | volume= 14 | pages= 302–307 | url=http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1138693 | doi= 10.1109/TAP.1966.1138693 (Seminal Paper #1 in the FDTD technology area)

*cite journal | author= A. Taflove and M. E. Brodwin | title= Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations | journal= Microwave Theory and Techniques, IEEE Transactions on | year= 1975 | volume= 23 | pages= 623–630| url=http://www.ece.northwestern.edu/ecefaculty/taflove/Paper2.pdf | doi= 10.1109/TMTT.1975.1128640 (Correct numerical stability criterion for Yee's algorithm)

*cite journal | author= A. Taflove | title= Application of the finite-difference time-domain method to sinusoidal steady state electromagnetic penetration problems | journal= Electromagnetic Compatibility, IEEE Transactions on | year= 1980 | volume= 22 | pages= 191–202| url=http://www.ece.northwestern.edu/ecefaculty/taflove/Paper7.pdf | doi= 10.1109/TEMC.1980.303879 ("Finite difference time domain" first used to describe Yee's algorithm; "FDTD" acronym first appears; first rigorous comparison of FDTD, experimental, and phasor-domain moment-method results for electromagnetic wave penetration into a cavity)

*cite journal | author= G. Mur | title= Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations | journal= Electromagnetic Compatibility, IEEE Transactions on | year= 1981 | volume= 23 | pages= 377–382 | url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=4091487&arnumber=4091495 | doi= 10.1109/TEMC.1981.303970 | format= abstract (First effective absorbing boundary condition for FDTD)

*cite journal | author= K. R. Umashankar and A. Taflove | title= A novel method to analyze electromagnetic scattering of complex objects | journal= Electromagnetic Compatibility, IEEE Transactions on | year= 1982 | volume= 24 | pages= 397–405| url=http://www.ece.northwestern.edu/ecefaculty/taflove/Paper9.pdf | doi= 10.1109/TEMC.1982.304054 (First near-to-far-field transformation for FDTD; first rigorous comparison of FDTD and phasor-domain moment-method results for electromagnetic wave scattering problems)

*cite journal | author= Z. P. Liao, H. L. Wong, B. P. Yang, and Y. F. Yuan| title= A transmitting boundary for transient wave analysis| journal= Scientia Sinica a | year= 1984 | volume= 27 | pages= 1063–1076 | url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1325804| format= abstract (Improved absorbing boundary condition)

*cite journal | author= J. Berenger | title= A perfectly matched layer for the absorption of electromagnetic waves | journal= Journal of Computational Physics | year= 1994 | volume= 114 | pages= 185–200 | url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHY-45P0TJR-1P&_coverDate=10%2F31%2F1994&_alid=375031649&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=6863&_sort=d&view=c&_acct=C000051292&_version=1&_urlVersion=0&_userid=1072239&md5=9afa9291d22a44614de567dd9427c63d | doi= 10.1006/jcph.1994.1159 (Seminal paper which revolutionized absorbing boundary condition theory, reducing outer-boundary reflection errors by orders of magnitude)

*cite journal | author= S.D. Gedney | title= An anisotropic perfectly matched layer absorbing media for the truncation of FDTD latices| journal= Antennas and Propagation, IEEE Transactions on | year= 1996 | volume= 44 | pages= 1630–1639 | url=http://www.opticsinfobase.org/abstract.cfm?id=70065 | doi= 10.1109/8.546249 | format= abstract (Placed Berenger's theory in a context consistent with the historic Maxwell's equations)

University-level textbooks

*cite book | author=Karl S. Kunz and Raymond J. Luebbers | title=The Finite Difference Time Domain Method for Electromagnetics | publisher=CRC Press | year=1993 | id=ISBN 0-8493-8657-8 | url=http://www.crcpress.com/shopping_cart/products/product_detail.asp?sku=8657&af=W1129
*cite book | author=Allen Taflove and Susan C. Hagness | title=Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. | publisher=Artech House Publishers | year=2005 | id=ISBN 1-58053-832-0 | url=http://www.artechhouse.com/Detail.aspx?strBookId=1123

*cite book | author=Wenhua Yu, Raj Mittra, Tao Su, Yongjun Liu, and Xiaoling Yang | title=Parallel Finite-Difference Time-Domain Method | publisher=Artech House Publishers | year=2006 | id=ISBN 1-59693-085-3 | url=http://www.artechhouse.com/default.asp?frame=book.asp&book=1-59693-085-3&Country=US&Continent=NO&State=

External links

* Free software/Open-source software FDTD projects:
** [http://www.ece.ncsu.edu/oleg/wiki/WOLFSIM WOLFSIM] (NCSU) (2-D)
** [http://ab-initio.mit.edu/meep/ Meep] (MIT)
** [http://www.welcomes-you.com/radarfdtd/ (Geo-) Radar FDTD]
** [http://sourceforge.net/projects/bigboy bigboy] (unmaintained, no release files. must get source from cvs)
** [http://www.cemtach.com/reference/software/toyFDTD/ toyFDTD]
** [http://www.2comu.com/products/OpenGEMS.html OpenGEMS] (3-D Parallel FDTD Package)
** [http://www.evtsz.bme.hu/web/staff/szabo/web_fdtd/FDTD_main.html FDTD codes in C++] (developed by Zs. Szabó)
** [http://cs.tu-berlin.de/~peutetre/sfdtd/ FDTD code in Fortran 90]
* Freeware/Closed source FDTD projects (some not for commercial use):
** [http://www.emexplorer.net/ EM Explorer]
** [http://www.gprmax.org GprMax]
* Commercial/proprietary FDTD software vendors:
** [http://www.2comu.com/index.html 2COMU]
** [http://www.acceleware.com/ Acceleware Inc.]
** [http://www.aplac.hut.fi/aplac/general.html APLAC ]
** [http://www.apollophoton.com/apollo/page.php?id=4 Apollo Photonics]
** [http://www.apsimtech.com/ Applied Simulation Technology]
** [http://www.cfdrc.com/bizareas/microelec/rf_dev_inconct/rf-devices.html CFDRC]
** [http://lc.cray.com/ Cray LC]
** [http://www.cst.com/ CST - Computer Simulation Technology]
** [http://www.electromagneticapplications.com/software/ema3d.htm Electro Magnetic Applications Inc.]
** [http://www.emagware.com Emagware.com]
** [http://www.emphotonics.com/ EM Photonics]
** [http://www.empire.de/ Empire]
** [http://www.ems-plus.com/ezfdtd.html EMS Plus]
** [http://www.iis.ee.ethz.ch/research/bioemc/em_simulation_platform.en.html ETHZ]
** [http://www.gdfidl.de/ GdfidL]
** [http://www.lumerical.com/ Lumerical Solutions]
** [http://www.nlcstr.com/sim3d.htm Nonlinear Control Strategies]
** [http://www.optiwave.com/ Optiwave]
** [http://www.photond.com Photon Design]
** [http://www.qwed.eu QuickWave]
** [http://www.remcom.com/ Remcom]
** [http://www.rm-associates.biz/ RM Associates]
** [http://rsoftdesign.com/products.php?sub=Component+Design&itm=FullWAVE Rsoft]
** [http://www.speag.com/ SPEAG]
** [http://www.artechhouse.com/default.asp?Frame=Book.asp&Book=1-58053-832-0&Country=US&Continent=NO&State= Taflove-Hagness book software]
** [http://www.vectorfields.com/content/view/47/70/ Vector Fields]
** [http://www.zeland.com/ Zeland]


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