XF's meshing algorithm utilizes dielectric volume averaging (DVA) to consider the volume around each cell edge and set material properties based on the average of those findings. This affects the mesh of a part's surface where it meets other materials rather than the internal, homogeneous area of the part.

## Background

The finite-difference time-domain (FDTD) method discretizes CAD geometry into cell edges. It is common to think of discretization with regard to cells, but users should take care not to let this common association abstract their understanding of the FDTD update equations. Each cell is associated with an electric field but it is the cell's edges, not the cell itself, that carry the electromagnetic information during timestepping. Therefore, each cell edge represents the volume around it, which is roughly one third the size of the cell.

This leads to accuracy degeneration at dielectric material interfaces ([1], [2], [3], [4], [5]). For example, the uncertainty of an Ey cell edge surrounded by different dielectric materials is emphasized by the FDTD formulation, which assigns the edge only a single value of permittivity and conductivity. XF typically assigns the permittivity and conductivity from the material with the highest meshing priority to the Ey edge, which is a reasonable approximation in many cases.

One way to increase accuracy in such cases is calculating an effective permittivity and conductivity for each cell edge, which is known in XF as dielectric volume averaging. Enabling the dielectric volume averaging option causes XF's meshing algorithm to consider the volume around each cell edge rather than looking solely at the geometry with the highest meshing priority. By using the volume, the algorithm determines the averaged material properties and applies it to that cell edge. This smoothes the transition between dielectrics at material interfaces and results in a more accurate representation.

Users should note that using DVA to treat material interfaces in the FDTD space requires additional memory and computational resources.

#### References

1. A. Christ, J. F. S. Benkler, and N. Kuster, "Analysis of the accuracy of the numerical reflection coefficient of the finite-difference time-domain method at planar material interfaces," IEEE Trans. Electromagnetic Compatibility, vol. 48, p. 264, May 2006.
2. T. Hirono, Y. Shibata, W. W. Lui, S. Seki, and Y. Yoshikuni, "The second-order condition for the dielectric interface orthogonal to the yee-lattice axis in the fdtd scheme," IEEE Microwave and Guided Wave Letters, vol. 10, p. 359, September 2000.
3. K.-P. Hwang and A. Cangellaris, "Effective permittivities for second-order accurate fdtd equations at dielectric interfaces," IEEE Microwave and Wireless Comp. Letters, vol. 11, p. 158, April 2001.
4. B. Yang and C. A. Balanis, "Dielectric interface conditions for general fourth-order finite difference," IEEE Microwave and Wireless Comp. Letters, vol. 17, p. 559, August 2007.
5. T. T. Zygiridis, T. K. Katsibas, C. S. Antonopoulos, and T. D. Tsiboukis, "Treatment of grid-conforming dielectric interfaces in fdtd methods," IEEE Trans. Magnetics, vol. 45, p. 1396, March 2009.

## Enabling and Visualization

Users can enable DVA for a part, multiple parts, or an assembly by right-clicking on the desired object in the Project Tree, and selecting Gridding/MeshingEnable dielectric volume averaging. The option to disable DVA is also available through this right-click menu.

Users can also access the DVA option by right-clicking on a part in the Project Tree and selecting Gridding/MeshingMeshing Properties to open the Meshing Properties Editor. Checking and unchecking the Use dielectric volume averaging setting enables and disables DVA, respectively.

To verify that DVA is enabled for a part, right-click on Parts in the Project Tree and select View Parts List (All Parts) to open the Parts List - Parts editor. A percent icon appears in the row of a part for which DVA is enabled.

Additional information is available through the meshing controls, which are opened by double-clicking on Mesh in the Project Tree. When viewing a slice of the mesh, users can check the View Mesh Information: Material option to display the averaged material properties per cell edge.

The meshing controls also provide an Averaged Materials Visibility option for visualizing averaged material properties. This setting's computational requirements may decrease XF's speed when selected with a large number of averaged cells. Users can de-select this option to avoid a slow interface when slicing through the mesh.

Averaged material properties per cell edge are formulated using the weighted average. To conserve memory, the number of allowable weights is limited and like materials are grouped together for consideration in the computation engine.

Users can control how many subdivisions are considered for an averaged material by right-clicking on Mesh in the Project Tree and selecting Properties to open the Mesh Properties Editor. Then, enter the desired number of subdivisions in the Average Material Subdivisions field. Higher numbers of specified subdivisions result in greater resolution of averaged materials generated, as well as increased requirements for computational resources.

## Recommendations

DVA improves simulation results for microstrip antennas and similar applications in which antennas are mounted on dielectric substrates. This is due to the averaging of the substrate material and air, or free space, for cell edges in the antenna plane, which increases the accuracy of the edge field computations. The importance of DVA for these applications also increases with the permittivity of the substrate, so the higher the permittivity of the substrate, the greater the accuracy improvement due to proper material averaging.

Activating DVA for SAR calculations involving tissue materials ensures accurate SAR averaging results that are in compliance with the latest SAR standard.

The following recommendations are based on DVA's characteristics:

• Enable DVA for substrates and other dielectric structures requiring one or more cells to resolve its cross section.
• Enable DVA for tissue materials used in SAR calculations.
• Do not enable DVA for metals or other good conductors.

It is also recommended to run simulations with, as well as without, DVA in order to ensure it has the expected and desired effect on the simulation results.

## Limitations

DVA is not appropriate for all situations:

• DVA is only suitable on parts with either a Nondispersive or Debye-Drude material definition.
• When a dielectric part is adjacent to a part with XACT mesh enabled, averaging is not applied to cell edges near the XACT part.