Finite-difference time-domain method

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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").


[edit] 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.

Illustration of a standard Cartesian Yee cell used for FDTD, about which electric and magnetic field vector components are distributed (Yee 1966). Visualized as a cubic voxel, the electric field components form the edges of the cube, and the magnetic field components form the normals to the faces of the cube. A three-dimensional space lattice is comprised of a multiplicity of such Yee cells. An electromagnetic wave interaction structure is mapped into the space lattice by assigning appropriate values of permittivity to each electric field component, and permeability to each magnetic field component.

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.

[edit] 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).

[edit] 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.

[edit] 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.

[edit] 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.

[edit] History of FDTD Techniques and Applications for Maxwell's Equations

We can begin to develop an appreciation of the basis, technical development, and possible future of FDTD numerical techniques for Maxwell’s equations by first considering their history. The following lists some of the key publications in this area, starting with Yee's seminal Paper #1 (1966), which has over 4000 citations according to the ISI Web of Science.

Partial Chronology of FDTD Techniques and Applications for Maxwell's Equations. (Adapted with permission from Taflove and Hagness (2005))

1966 — Yee (1966) described the basis of the FDTD numerical technique for solving Maxwell’s curl equations directly in the time domain on a space grid.

1975 — Taflove and Brodwin (1975a, 1975b) reported the correct numerical stability criterion for Yee’s algorithm; the first sinusoidal steady-state FDTD solutions of two- and three-dimensional electromagnetic wave interactions with material structures; and the first bioelectromagnetics models.

1977 — Holland (1977), and Kunz and Lee (1977) applied Yee’s algorithm to EMP problems.

1980 — Taflove (1980) coined the FDTD acronym and published the first validated FDTD models of sinusoidal steady-state electromagnetic wave penetration into a three-dimensional metal cavity.

1981 — Mur (1981) published the first numerically stable, second-order accurate, absorbing boundary condition (ABC) for Yee’s grid.

1982, '83 — Taflove and Umashankar (1982, 1983) developed the first FDTD electromagnetic wave scattering models computing sinusoidal steady-state near-fields, far-fields, and radar cross-section for two- and three-dimensional structures.

1984 — Liao et al. (1984) reported an improved ABC based upon space-time extrapolation of the field adjacent to the outer grid boundary.

1985 — Gwarek (1985) introduced the lumped equivalent circuit formulation of FDTD.

1986 — Choi and Hoefer (1986) published the first FDTD simulation of waveguide structures.

1987, '88 — Kriegsmann et al. (1987) and Moore et al. (1988) published the first articles on ABC theory in IEEE Trans. Antennas and Propagation.

1987, '88, '92 — Contour-path subcell techniques were introduced by Umashankar et al. (1987) to permit FDTD modeling of thin wires and wire bundles, by Taflove et al. (1988) to model penetration through cracks in conducting screens, and by Jurgens et al. (1992) to conformally model the surface of a smoothly curved scatterer.

1988 — Sullivan et al. (1988) published the first 3-D FDTD model of sinusoidal steady-state electromagnetic wave absorption by a complete human body.

1988 — FDTD modeling of microstrips was introduced by Zhang et al. (1988).

1990, '91 — FDTD modeling of frequency-dependent dielectric permittivity was introduced by Kashiwa and Fukai (1990), Luebbers et al. (1990), and Joseph et al. (1991).

1990, '91 — FDTD modeling of antennas was introduced by Maloney et al. (1990), Katz et al. (1991), and Tirkas and Balanis (1991).

1990 — FDTD modeling of picosecond optoelectronic switches was introduced by Sano and Shibata (1990), and El-Ghazaly et al. (1990).

1992–94 — FDTD modeling of the propagation of optical pulses in nonlinear dispersive media was introduced, including the first temporal solitons in one dimension by Goorjian and Taflove (1992); beam self-focusing by Ziolkowski and Judkins (1993); the first temporal solitons in two dimensions by Joseph et al. (1993); and the first spatial solitons in two dimensions by Joseph and Taflove (1994).

1992 — FDTD modeling of lumped electronic circuit elements was introduced by Sui et al. (1992).

1993 — Toland et al. (1993) published the first FDTD models of gain devices (tunnel diodes and Gunn diodes) exciting cavities and antennas.

1994 — Thomas et al. (1994) introduced a Norton’s equivalent circuit for the FDTD space lattice, which permits the SPICE circuit analysis tool to implement accurate subgrid models of nonlinear electronic components or complete circuits embedded within the lattice.

1994 — Berenger (1994) introduced the highly effective, perfectly matched layer (PML) ABC for two-dimensional FDTD grids, which was extended to three dimensions by Katz et al. (1994), and to dispersive waveguide terminations by Reuter et al. (1994).

1995, '96 — Sacks et al. (1995) and Gedney (1996) introduced a physically realizable, uniaxial perfectly matched layer (UPML) ABC.

1997 — Liu (1997) introduced the pseudospectral time-domain (PSTD) method, which permits extremely coarse spatial sampling of the electromagnetic field at the Nyquist limit.

1997 — Ramahi (1997) introduced the complementary operators method (COM) to implement highly effective analytical ABCs.

1998 — Maloney and Kesler (1998) introduced several novel means to analyze periodic structures in the FDTD space lattice.

1998 — Nagra and York (1998) introduced a hybrid FDTD-quantum mechanics model of electromagnetic wave interactions with materials having electrons transitioning between multiple energy levels.

1998 — Hagness et al. (1998) introduced FDTD modeling of the detection of breast cancer using ultrawideband radar techniques.

1999 — Schneider and Wagner (1999) introduced a comprehensive analysis of FDTD grid dispersion based upon complex wavenumbers.

2000, '01 — Zheng, Chen, and Zhang (2000, 2001) introduced the first three-dimensional alternating-direction implicit (ADI) FDTD algorithm with provable unconditional numerical stability.

2000 — Roden and Gedney (2000) introduced the advanced convolutional PML (CPML) ABC.

2000 — Rylander and Bondeson (2000) introduced a provably stable FDTD - finite-element time-domain hybrid technique.

2002 , '06 — Hayakawa et al. (2002) and Simpson and Taflove (2006) introduced FDTD modeling of the global Earth-ionosphere waveguide for extremely low-frequency geophysical phenomena.

2003 — DeRaedt introduced the unconditionally stable, “one-step” FDTD technique (2003).

Interest in FDTD Maxwell’s equations solvers has increased nearly exponentially over the past 20 years. Increasingly, engineers and scientists in nontraditional electromagnetics-related areas such as photonics and nanotechnology have become aware of the power of FDTD techniques. As shown in the figure on the left, an estimated 2,000 FDTD-related publications appeared in the science and engineering literature in 2006, as opposed to fewer than 10 as recently as 1985. 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.

There are seven primary reasons for the tremendous expansion of interest in FDTD computational solution approaches for Maxwell’s equations:

1. FDTD uses no linear algebra. Being a fully explicit computation, FDTD avoids the difficulties with linear algebra that limit the size of frequency-domain integral-equation and finite-element electromagnetics models to generally fewer than 1e7 electromagnetic field unknowns. FDTD models with as many as 1e9 field unknowns have been run; there is no intrinsic upper bound to this number.

2. FDTD is accurate and robust. The sources of error in FDTD calculations are well understood, and can be bounded to permit accurate models for a very large variety of electromagnetic wave interaction problems.

3. FDTD treats impulsive behavior naturally. Being a time-domain technique, FDTD directly calculates the impulse response of an electromagnetic system. Therefore, a single FDTD simulation can provide either ultrawideband temporal waveforms or the sinusoidal steady-state response at any frequency within the excitation spectrum.

4. FDTD treats nonlinear behavior naturally. Being a time-domain technique, FDTD directly calculates the nonlinear response of an electromagnetic system. This allows natural hybriding of FDTD with sets of auxiliary differential equations that describe nonlinearities from either the classical or semi-classical standpoint. An exciting research frontier here is the development of hybrid algorithms which join FDTD classical electrodynamics models with phenomena arising from quantum electrodynamics, especially vacuum fluctuations.

5. FDTD is a systematic approach. With FDTD, specifying a new structure to be modeled is reduced to a problem of mesh generation rather than the potentially complex reformulation of an integral equation. For example, FDTD requires no calculation of structure-dependent Green functions.

6. Parallel-processing computer architectures have come to dominate supercomputing. FDTD scales with high efficiency on parallel-processing CPU-based computers, and extremely well on recently developed GPU-based accelerator technology.

7. Computer visualization capabilities are increasing rapidly. While this trend positively influences all numerical techniques, it is of particular advantage to FDTD methods, which generate time-marched arrays of field quantities suitable for use in color videos to illustrate the field dynamics.

These factors combine to indicate that FDTD will likely remain one of the dominant computational electrodynamics techniques; and indeed may emerge as the dominant technique for mid-21st-century problems of surpassing volumetric complexity and/or multiphysics.

[edit] See also

[edit] References

[edit] Journal articles

  • R. Holland (1977). "Threde: A free-field EMP coupling and scattering code". Nuclear Science, IEEE Transactions on 24: 2416-2421. doi:10.1109/TNS.1977.4329229. 
  • K. S. Kunz and K. M. Lee (1978). "A three-dimensional finite-difference solution of the external response of an aircraft to a complex transient EM environment". Electromagnetic Compatibility, IEEE Transactions on 20: 328-341. doi:10.1109/TEMC.1978.303727. 
  • Z. P. Liao, H. L. Wong, B. P. Yang, and Y. F. Yuan (1984). "A transmitting boundary for transient wave analysis". Scientia Sinica a 27: 1063–1076. 
  • T. Kashiwa and I. Fukai (1990). "A treatment by FDTD method of dispersive characteristics associated with electronic polarization". Microwave and Optics Technology Letters 3: 203-205. 
  • R. W. Ziolkowski and J. B. Judkins (1993). "Full-wave vector Maxwell’s equations modeling of self-focusing of ultra-short optical pulses in a nonlinear Kerr medium exhibiting a finite response time". Optical Society of America B, Journal of 10: 186-198. 
  • J. G. Maloney and M. P. Kesler (1998). "Analysis of Periodic Structures". Chap. 6 in Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, A. Taflove, ed., Artech House, publishers. 

[edit] University-level textbooks

[edit] External links

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