The FDTD Method for Electromagnetic Simulation

A Brief Summary - FDTD Simulation 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 [2, p. 37] 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.

Figure 1:  The Yee cell with labeled field components

Figure 2:  A dielectric sphere as meshed in a traditional FDTD grid with staircasing. The individual cell edges (electric field locations) are displayed as overlapping grid lines.

Cell Staircasing Advancements

Traditionally, staircased FDTD cells are considered cumbersome and inefficient at resolving curved surfaces or highly varying fields. Advancements in FDTD allow for conformal meshes and singularity correction.

The cubicle Yee cell of Figure 1 can in general be rectangular. The spacings between cells in the x, y, and z-directions can vary throughout the problem space. This allows more cell edges to be placed in regions of strong fields, such as around small features of highly conductive material. Within a cell, the standard update equations of FDTD may be refined in many ways, for example to allow for wires that are thinner than a cell size or to calculate the strong fields on the edges of conductors such as microstrip lines. Another refinement can allow for objects whose surface intersects the cell at arbitrary angles with respect to the principle axes. These “conformal” cells can be further refined to account for curvature of the object surface within the volume of the cell.

Figure 3 shows the geometry of a portion of a mobile phone. For clarity, the visibility of many of the parts, including the outer case, has been turned off to show the region near the antenna. A small portion of the FDTD mesh of the phone is shown in Figure 4a using simple rectangular cells. In Figure 4b the same part of the phone mesh is shown, this time using a type of conformal treatment for cells which contain portions of the surface of an object. Figure 5 shows a larger view of the surfaces of the conformal mesh.

FDTD simulation software 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 are defined by their constitutive parameters of relative permittivity and conductivity or loss tangent for the electrical material, or relative permeability and magnetic conductivity for the magnetic material. In many 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, metals when excited at optical frequencies, and dielectrics over wide bandwidths. 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. These materials may have permittivities or permeabilities that are negative at some frequencies, making them effective for simulating metamaterials macroscopically. FDTD is also particularly effective at simulating nonlinear materials, several of which are included in XFdtd.