WHY USE FDTD?

Figure 1 The Yee cell with labeled field components.

Figure 2 A dielectric sphere as meshed in an FDTD grid. The individual cell edges (electric field locations) are displayed as theoverlapping grid lines.

Figure 3 The coordinate system using in XFdtd® for far-zone and incident plane wave directions.

While many electromagnetic simulation techniques are applied in the frequency-domain, FDTD solves Maxwell’s equations in the time domain. This means that the calculation of the electromagnetic field values progresses at discrete steps in time. One benefit of the time domain approach is that it gives broadband output from a single execution of the program. However the main reason for using the FDTD approach is the excellent scaling performance of the method as the problem size grows. As the number of unknowns increases, the FDTD approach quickly outpaces other methods in efficiency. FDTD has also been identified as the preferred method for performing electromagnetic simulations for biological effects from wireless devices [2]). The FDTD method has been shown to be the most efficient approach and provides accurate results of the field penetration into biological tissues.

A BRIEF SUMMARY – FDTD BASICS

In the FDTD approach, both space and time are divided into discrete segments. Space is segmented into box-shaped “cells”, which are small compared to the wavelength. The electric fields are located on the edges of the box and the magnetic fields are positioned on the faces as shown in Figure 1. This orientation of the fields is known as the Yee cell [3] and is the basis for FDTD.Time is quantized into small steps where each step represents the time required for the field to travel from one cell to the next. Given the offset in space of the magnetic fields from the electric fields, the values of the field with respect to time are also offset. The electric and magnetic fields are updated using a leapfrog scheme where first the electric fields, then the magnetic are computed at each step in time.

When many FDTD cells are combined together to form a three-dimensional volume, the result is an FDTD grid or mesh. Each FDTD cell will overlap edges and faces with its neighbors, so by convention each cell will have three electric fields that begin at a common node associated with it. The electric fields at the other nine edges of the FDTD cell will belong to other, adjacent cells. Each cell will also have three magnetic fields originating on the faces of the cell adjacent to the common node of the electric fields as shown in Figure 1.

Within the mesh, materials such as conductors or dielectrics can be added by changing the equations for computing the fields at given locations. For example, to add a perfectly conducting wire segment to a cell edge, the equation for computing the electric field can be replaced by simply setting the field to zero since the electric field in a perfect conductor is identically zero. By joining numerous end-to-end cell edges defined as perfectly conducting material, a wire can be formed. Introducing other materials or other configurations is handled in a similar manner and each may be applied to either the electric or magnetic fields depending on the characteristics of the material. By associating many cell edges with materials, a geometrical structure can be formed within the FDTD grid such as the dielectric sphere shown in Figure 2. Each small box shown in the figure represents one FDTD cell.

The cell size, the dimensions of the box, is the most important constraint in any FDTD simulation since it determines not only the step size in time, but also the upper frequency limit for the calculation. A general rule of thumb sets the minimum resolution, and thus the upper frequency limit, at ten cells per wavelength. In practice the cell size will often be set by dimensions and features of the structure to be simulated such as the thickness of a substrate or the length of a wire.

An excitation may be applied to an FDTD simulation by applying a sampled waveform to the field update equation at one or several locations. At each step in time, the value of the waveform over that time period is added into the field value. The surrounding fields will propagate the introduced waveform throughout the FDTD grid appropriately,depending on the characteristics of each cell. A calculation must continue until a state of convergence has been reached. This typically means that all field values have decayed to essentially zero (at least 60dB down from the peak) or a steady-state condition has been reached.

MATERIALS

FDTD is capable of simulating a wide variety of electric and magnetic materials. The most basic material of course is free space. All FDTD cells are initialized as free space and the fields at all cell edges are updated using the free space equations unless another material is added to replace the free space.

Perfectly conducting electric and magnetic materials are simulated by setting the electric or magnetic field to zero for any cell edges located within these materials. Because of the simplicity of the calculation for these materials, it is better to use a perfect conductor rather than a real conductor whenever feasible. Conductors such as copper can be simulated in FDTD, but since the equations for computing the fields in copper material are more complicated than those for a perfect conductor, the calculation will take longer.Of course for cases where only a small percentage of the FDTD cells are defined as a conductor, the difference in execution time will hardly be noticeable.

Frequency independent dielectric and magnetic materials, considered “normal” materials by XFdtd, are defined by their constitutive parameters of relative permittivity and conductivity for the electrical material, or relative permeability and magnetic conductivity for the magnetic material. In most cases, even when performing a broadband calculation, these materials are appropriate since the parameters do not vary significantly over the frequency range.

In some cases a frequency independent material is not appropriate and instead a frequency dependent, or dispersive, material should be substituted. Some common examples of frequency dependent materials are high water content materials such as human tissues, and metals when excited at optical frequencies. Included in XFdtd is the capability to simulate electric and magnetic Debye and Drude materials such as plasmas, Lorentz materials, and anisotropic magnetic ferrites, as well as frequency independent anisotropic dielectrics, and nonlinear diagonally anisotropic dielectrics.

NEAR-ZONE VERSUS FAR-ZONE

For any given calculation the geometry of the structure being simulated is defined by setting the cell edges at specific locations to certain materials. The entire FDTD geometry space, commonly called the grid or the mesh, is composed of a three dimensional block of these cells. This three-dimensional volume is considered to be the near-zone region in XFdtd in terms of the data storage. The field value at any edge in the FDTD grid may be observed as a function of time by saving a “near-zone” point in XFdtd. Other types of data such as steady-state field magnitudes, specific absorption rates, S-parameters, or impedance may be stored as well “near-zone” (within the grid)values.

It is possible to make an FDTD grid that is large enough to allow sampling of points inthe far-field of a geometry. In general this will be extremely costly in terms of computer memory and calculation time since the number of unknowns (cells) will most likely be large. Note that each FDTD cell has a maximum size of one-tenth of a wavelength, so moving several wavelengths away from a structure will require many cells. In most cases this is not an appropriate method of monitoring far-field results.

A more practical method for transforming field values to the far-zone and for calculations of radiation gain or radar scattering patterns is to use a transformation to convert the near-zone values in the FDTD grid into a far-field value at some location away from the grid. In XFdtd this is done by enclosing the geometry in a box and storing the fields onthe six faces of this box. The faces of the box are located five FDTD cells from each outer edge of the FDTD grid. For the transformation to be valid, all parts of the XFdtd geometry must be contained within the box.

The coordinate system used in XFdtd is defined with the azimuthal (phi) angle referenced from the X axis and the elevation (theta) angle referenced from the Z axis, as shown in Figure 3. This coordinate system is used for locating far-zone positions andfor defining the incident plane wave direction in XFdtd.

BROADBAND AND STEADY-STATE CALCULATIONS

Generally a broadband calculation is performed with FDTD since a single calculation can yield results for a wide range of frequencies. In some cases, in particular when antenna patterns are desired at a single frequency or for biological applications such as the Specific Absorption Rate, a sinusoidal input is preferable or necessary.

A sinusoidal source must be used whenever a steady-state output is desired. The most common example is the Specific Absorption Rate (SAR) calculation, but XFdtd can also store electric field magnitudes, magnetic flux magnitudes, and conduction currents as steady-state quantities.

A sinusoidal source can be useful for generating gain patterns due to the method used to compute the far-field data. With the steady-state far-zone transformation, the calculation overhead is minimal and any pattern may be computed in post-processing. The transient far-zone calculation requires extra calculation time for each far-zone angle specified and all far-zone angles must be defined before the calculation. The benefit of the transient far-zone calculation is that broadband data is available in the output while the steady-state transform is for a single frequency.

OUTER RADIATION BOUNDARIES

A three-dimensional grid of cells forms the XFdtd geometry and the fields updated atevery cell location are dependent on the neighboring fields. However, due to memory limitations the grid must end at some point and because of this, the fields on the outer edges of the grid cannot be updated correctly. To correct this situation, outer radiation boundary conditions are applied at the edges of the XFdtd grid.

The outer radiation boundary is a method for absorbing fields propagating from the XFdtd grid toward the boundary. By absorbing these fields, the grid appears to extend forever. The performance of the outer boundaries is an important factor in the accuracy of an FDTD calculation and care should be taken to correctly use them.

In some cases a reflecting boundary rather than an absorbing one is preferred. A perfectly conducting boundary (either electric or magnetic) may be used to image the fields in an FDTD calculation.

COMPUTER RESOURCES

FDTD is a computationally intensive method and most reasonable calculations will need a fast computer and several hundred megabytes of computer memory. For most applications it is fairly simple to estimate the amount of computer memory required for a calculation. The most important factor for the memory usage, and in large part the run time, is the number of FDTD cells used to represent the structure under test. Each FDTD cell has six field values associated with it: three electric fields and three magnetic fields. Additionally each cell has six flags associated with it to indicate the material type present at each of the six field locations. The field values are real numbers, each four bytes in length, while the flags are each one byte. This gives a memory usage per FDTD cell of 24 bytes for fields and 6 bytes for flags for a total of 30 bytes. To estimate the total memory required, in bytes, simply multiply the number of FDTD cells by the 30 bytes per cell value. There is some overhead in the calculation, but it is generally quite small. One notable exception is the transient far-zone angles that allocate six one dimensional real value arrays per direction.

Estimating the execution time of an FDTD calculation is more complicated since performance of computer processors varies. One method of estimating is to compute the total number of operations to be performed. There are about 80 operations per cell, per time step during the FDTD calculations. The total number of operations is found from the product of the number of cells, the number of time steps, and the factor of 80 operations per cell, per time step. If a value of the floating-point performance of the processor is known, a value for execution time can be computed. In general though, a better estimating method is to determine the execution time of a simple problem on a given computer, and the scale the time by the ratio of the number of operations between the desired calculation and the simple one. Note: The XFdtd software will estimate both the computer memory resources needed for simulation. The information in this section is presented to explain the basis for the estimates.

References

1. K. Kunz and R. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, 1993, CRC Press Catalog Number 8657, 496 pages.

2. C95.3.2002, “Recommended Practice for Measurements and Computations with Respect to Human Exposure to Radio Frequency Electromagnetic Fields, 100kHz to 300GHz”, IEEE Standards and Coordinating Committee 28 on Non-Ionizing Radiation Hazards, April 2002.