Finite-Difference Time-Domain (FDTD) is a popular electromagnetic modeling technique. It is considered easy to understand and easy to implement in software. Since it is a time-___domain technique, solutions can cover a wide frequency range with a single simulation run.
The FDTD method belongs in the general class of differential time-___domain numerical modeling methods. Maxwell's equations (in partial differential form) are modified to central-difference equations, discretized, and implemented in software. The equations are solved in a leapfrog manner: the electric field is solved at a given instant in time, then the magnetic field is solved at the next instant in time, and the process is repeated over and over again.
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.
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. 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 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
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.
Strengths of the FDTD technique
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 broad-band 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.
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.
What are the weaknesses of the FDTD Technique?
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.
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.
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 or the auxiliary differential equation (ADE) 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.
References
Journal Articles
- Kane Yee (1966). "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media". Antennas and Propagation, IEEE Transactions on. 14: 302–307.