Defining Outer Boundary Conditions
In this section, you will learn about how to:
- Distinguish absorbing and reflecting boundaries in EMPro
- Choose which boundary type to use for your project calculation
Specifying an outer radiation boundary is necessary to indicate how the calculation treats the boundaries of the problem space. During an EMPro calculation, the fields updated at every cell location are dependent upon the neighboring fields. However, due to memory limitations, the fields on the outer edges of the grid cannot be updated correctly because the grid must be a finite size. To correct this situation, outer radiation boundary conditions are applied at the edges of the EMPro grid. Thus, the performance of the outer boundaries is a significant factor in the accuracy of the calculation, and it is important to use them correctly. This chapter details several available options for defining the outer boundaries of an EMPro project.
Outer boundaries are defined in the Outer Boundary Editor, located in the Fdtd: Outer Boundary branch of the Project Tree, as shown below.
Outer Boundary Branch
Absorbing Boundaries vs Reflecting Boundaries
The outer radiation boundary is a method for absorbing fields propagating from the EMPro grid toward the boundary. By absorbing these fields, the grid appears to extend infinitely; however, it is actually finite in order to fall within reasonable memory usage. There are two numerical absorbers designed to allow electromagnetic fields radiated or scattered by the FDTD geometry to be absorbed with very little reflection from the boundary. These include a Uni-Axial Perfectly Matched Layer (PML) and a second-order, stabilized Liao radiation boundary.
In some cases a reflecting boundary rather than an absorbing one is preferred. A perfectly conducting boundary (either electric, PEC, or magnetic, PMC) may be used in these cases, for example, to provide a ground plane, or to image the fields in an EMPro calculation.
The Liao and PML boundaries may not be mixed together in the same calculation. Furthermore, PML may not be used with the PMC boundary. The Liao boundary may be used with both PEC and PMC boundaries.
The default boundary condition for EMPro is PML.
In addition, EMPro has Periodic boundary conditions that enable periodic structures to be modeled. These boundary conditions equate the corresponding outer surfaces of the mesh.
The figure below shows the Outer Boundary Editor.
The Outer Boundary Editor
The Liao outer boundary condition is an estimation method, which is makes it fundamentally different from PML boundary conditions. By looking into the FDTD space and back in time, it estimates the electric fields just outside the limits of the FDTD mesh. These estimated values are then used in the FDTD equations inside the space. The Liao estimation assumes that waves are allowed to travel outward from the space but not reflect back in. This method works well provided that there is enough space between the radiating geometry and the outer boundary. Typical limits are at least 10 cells of spacing to ensure that instability does not occur.
For more on calculation instability, refer to the section on Calculation Stability.
A homogeneous dielectric may be located against the Liao boundary. For example, in a lossy earth or strip line calculation, the earth or dielectric layer may touch the outer boundary. Liao will usually function well in this situation provided that there are no air gaps within five cells of the Liao boundary. Liao assumes homogeneous material within five cells, and if this is not the case then the EMPro calculation will usually be unstable with rapidly rising field amplitudes.
Since Liao is an estimation method, the size of the FDTD mesh is not increased by using it. Some storage is needed for saving electric values at previous timesteps, but this is usually negligible in a typical calculation.
The Perfectly Matched Layer (PML) boundary condition is offered as an alternative to Liao. PML is an artificial absorbing material that absorbs the incident energy as it propagates through the PML layers. Better absorption, that is, smaller reflection, is obtained by adding more layers at the expense of increasing the size of the FDTD mesh. For example, consider an EMPro calculation on a mesh using the Liao absorber that is 50 x 60 x 70 cells or a total of 210,000 cells. There is a 15 cell free space border all around the geometry so that the Liao boundaries can provide small reflections. If the Liao is changed to eight PML layers, the geometry mesh will not change. However, outside of this defined mesh region, eight additional FDTD mesh layers are added on each side of the geometry. This means that the actual number of FDTD cells that must be calculated grows to 66 x 76 x 86 or 431,000 cells, more than double. Since PML cells require more arithmetic operations than normal cells, the time penalty is actually greater.
This time penalty for PML is also increased because the PML cells have special equations for both electric and magnetic fields. For an EMPro calculation with no magnetic materials present, the magnetic fields are computed very quickly. However, when PML is added, the magnetic field update equations are more complicated even when no actual magnetic fields are present and this adds to the time penalty.
The benefit of using the PML layers is that they provide better absorption than Liao even with only a five-cell border of free space, and perhaps only six PML layers would provide this. In such a situation, calculation time would be saved. Making this comparison would require meshing the object again with a smaller free space margin to the outer boundary. This can be done easily in EMPro using the mesh tab and choosing a smaller padding around the geometry.
Both PML and Liao boundary conditions are offered to provide flexibility. Both methods should provide similar results when properly used although in some cases, particularly when low frequencies (compared to the cell size) are used, PML is superior. It is also recommended that PML boundary conditions are used wherever possible when using the adaptive meshing feature.
In some situations, terminating one or more faces of the FDTD geometry space with a Perfect Electric Conductor (PEC) outer boundary is advantageous. For example, the conducting ground plane of a microstrip could be located on one face of the FDTD space.
If all of the outer boundaries of the calculation are not absorbing, a plane wave should not be used to excite the calculation and the far-zone transformations will not provide correct results for far-zone fields. The sole exception is in the case of one PEC boundary and five absorbing boundaries, which will compute far zone over infinite PEC ground.
An edge of the FDTD space should be set to PEC using the PEC Boundary Condition. Do not set FDTD cells to PEC material in the geometry and set the outer boundary to absorbing, as this will cause instabilities in the calculation.
The Perfect Magnetic Conductor (PMC) outer boundary condition may be useful in reducing the size of the FDTD mesh, memory requirements, and calculation time by taking advantage of symmetries in the geometry. For example, this condition would be a good choice in a symmetric problem space where magnetic fields are strictly normal to a plane.
Similar to the PMC boundary condition, the Periodic boundary condition may be useful in taking advantage of geometry/field symmetry to reduce the size of the FDTD mesh and therefore the memory and calculation time required. In this case the upper and lower edges of the mesh are forced to be equal during the analysis. This may be useful for cases when small geometries are repeated over and over (i.e., optics examples).
For more information about using the periodic boundary with Plane Wave excitations, refer to Creating a New Simulation.