Finite-difference time-domain (FDTD) or Yee's method (named after the Chinese American applied mathematician Kane S. Yee, born 1934) is a numerical analysis technique used for modeling computational electrodynamics.
History Finite difference schemes for time-dependent partial differential equations (PDEs) have been employed for many years in computational fluid dynamics problems, including the idea of using centered finite difference operators on staggered grids in space and time to achieve second-order accuracy. The novelty of Yee's FDTD scheme, presented in his seminal 1966 paper, was to apply centered finite difference operators on staggered grids in space and time for each electric and magnetic vector field component in Maxwell's curl equations. The descriptor "Finite-difference time-domain" and its corresponding "FDTD" acronym were originated by Allen Taflove in 1980. 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. 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 Popularity). As of 2013, there are at least 25 commercial/proprietary FDTD software vendors; 13 free-software/open-source-software FDTD projects; and 2 freeware/closed-source FDTD projects, some not for commercial use (see External links).
Development of FDTD and Maxwell's equations
An appreciation of the basis, technical development, and possible future of FDTD numerical techniques for Maxwell's equations can be developed by first considering their history. The following lists some of the key publications in this area.
Partial chronology of FDTD techniques and applications for Maxwell's equations.Adapted with permission from Taflove and Hagness (2005). year event 1928 Courant, Friedrichs, and Lewy (CFL) publish seminal paper with the discovery of conditional stability of explicit time-dependent finite difference schemes, as well as the classic FD scheme for solving second-order wave equation in 1-D and 2-D. 1950 First appearance of von Neumann's method of stability analysis for implicit/explicit time-dependent finite difference methods. 1966 Yee described the FDTD numerical technique for solving Maxwell's curl equations on grids staggered in space and time. 1969 Lam reported the correct numerical CFL stability condition for Yee's algorithm by employing von Neumann stability analysis. 1975 Taflove and Brodwin reported 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 and Kunz & Lee applied Yee's algorithm to EMP problems. 1980 Taflove 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 published the first numerically stable, second-order accurate, absorbing boundary condition (ABC) for Yee's grid. 1982–83 Taflove and Umashankar 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 reported an improved ABC based upon space-time extrapolation of the field adjacent to the outer grid boundary. 1985 Gwarek introduced the lumped equivalent circuit formulation of FDTD. 1986 Choi and Hoefer published the first FDTD simulation of waveguide structures. 1987–88 Kriegsmann et al and Moore et al published the first articles on ABC theory in IEEE Transactions on Antennas and Propagation. 1987–88, 1992 Contour-path subcell techniques were introduced by Umashankar et al to permit FDTD modeling of thin wires and wire bundles, by Taflove et al to model penetration through cracks in conducting screens, and by Jurgens et al to conformally model the surface of a smoothly curved scatterer. 1988 Sullivan et al 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. 1990–91 FDTD modeling of frequency-dependent dielectric permittivity was introduced by Kashiwa and Fukai, Luebbers et al, and Joseph et al. 1990–91 FDTD modeling of antennas was introduced by Maloney et al, Katz et al, and Tirkas and Balanis. 1990 FDTD modeling of picosecond optoelectronic switches was introduced by Sano and Shibata, and El-Ghazaly et al. 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; beam self-focusing by Ziolkowski and Judkins; the first temporal solitons in two dimensions by Joseph et al; and the first spatial solitons in two dimensions by Joseph and Taflove. 1992 FDTD modeling of lumped electronic circuit elements was introduced by Sui et al. 1993 Toland et al published the first FDTD models of gain devices (tunnel diodes and Gunn diodes) exciting cavities and antennas. 1993 Aoyagi et al present a hybrid Yee algorithm/scalar-wave equation and demonstrate equivalence of Yee scheme to finite difference scheme for electromagnetic wave equation. 1994 Thomas et al 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 introduced the highly effective, perfectly matched layer (PML) ABC for two-dimensional FDTD grids, which was extended to non-orthogonal meshes by Navarro et al, and three dimensions by Katz et al, and to dispersive waveguide terminations by Reuter et al. 1994 Chew and Weedon introduced the coordinate stretching PML that is easily extended to three dimensions, other coordinate systems and other physical equations. 1995–96 Sacks et al and Gedney introduced a physically realizable, uniaxial perfectly matched layer (UPML) ABC. 1997 Liu introduced the pseudospectral time-domain (PSTD) method, which permits extremely coarse spatial sampling of the electromagnetic field at the Nyquist limit. 1997 Ramahi introduced the complementary operators method (COM) to implement highly effective analytical ABCs. 1998 Maloney and Kesler introduced several novel means to analyze periodic structures in the FDTD space lattice. 1998 Nagra and York introduced a hybrid FDTD-quantum mechanics model of electromagnetic wave interactions with materials having electrons transitioning between multiple energy levels. 1998 Hagness et al introduced FDTD modeling of the detection of breast cancer using ultrawideband radar techniques. 1999 Schneider and Wagner introduced a comprehensive analysis of FDTD grid dispersion based upon complex wavenumbers. 2000–01 Zheng, Chen, and Zhang introduced the first three-dimensional alternating-direction implicit (ADI) FDTD algorithm with provable unconditional numerical stability. 2000 Roden and Gedney introduced the advanced convolutional PML (CPML) ABC. 2000 Rylander and Bondeson introduced a provably stable FDTD - finite-element time-domain hybrid technique. 2002 Hayakawa et al and Simpson and Taflove independently 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. 2004 Soriano and Navarro derived the stability condition for Quantum FDTD technique. 2008 Ahmed, Chua, Li and Chen introduced the three-dimensional locally one-dimensional (LOD)FDTD method and proved unconditional numerical stability. 2008 Taniguchi, Baba, Nagaoka and Ametani introduced a Thin Wire Representation for FDTD Computations for conductive media 2009 Oliveira and Sobrinho applied the FDTD method for simulating lightning strokes in a power substation 2021 Oliveira and Paiva developed the Least Squares Finite-Difference Time-Domain method (LS-FDTD) for using time steps beyond FDTD CFL limit.
FDTD models and methods 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.
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 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.
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. 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. As a result, certain classes of simulations can require many thousands of time-steps for completion.
Using the FDTD method To implement an FDTD solution of Maxwell's equations, a computational domain must first 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.
The permittivity of dispersive materials in tabular form cannot be directly substituted into the FDTD scheme. Instead, it can be approximated using multiple Debye, Drude, Lorentz or critical point terms. This approximation can be obtained using open fitting programs and does not necessarily have physical meaning.
Once the computational domain and the grid materials are established, a source is specified. The source can be current on a wire, applied electric field or impinging plane wave. In the last case FDTD can be used to simulate light scattering from arbitrary shaped objects, planar periodic structures at various incident angles, and photonic band structure of infinite periodic structures.