Technical background

Electromagnetic field problems can possess three types of planar symmetry: geometric, electric and magnetic. The type of symmetry is defined by the geometric properties of the structure and sources.

Geometric symmetry

In this case the geometry of the structure must be symmetric with respect to the symmetry plane, while the sources may be arbitrarily located. Such a setup generally leads to non-symmetric current distributions on the structure.

Electric symmetry

In the case of an electric symmetry plane, not only must geometric symmetry hold, but additional requirements also have to be met by the sources. Figure 1 shows these requirements. The electric current density must be anti-symmetric and the magnetic current density symmetric. A physical interpretation of an electric symmetry plane is a plane which can be replaced by a PEC wall without changing the field distribution. The tangential component of the electric field and the normal component of the magnetic field thus disappear at such a plane.

Magnetic symmetry

In the case of a magnetic symmetry plane, geometric symmetry must again hold, again with additional requirements on the sources, but different from the electric case. Figure 2 shows these requirements. The electric current density must be symmetric and the magnetic current density anti-symmetric. A physical interpretation of a magnetic symmetry plane is a plane which can be replaced by a PMC wall without changing the field distribution. The normal component of the electric field and the tangential component of the magnetic field thus disappear at such a plane.

Computational benefits of utilizing symmetry in FEKO

When using numerical methods to solve electromagnetic field problems, symmetry may be exploited to reduce computational costs in terms of runtime and memory requirements.

In FEKO, symmetry can be invoked together with the MoM (metallic, dielectric, layered media Green function) and the FEM/MoM, but not together with the MLFMM. In FEKO, the three types of symmetry planes discussed above may be specified. The resulting benefits are as follows.

• Geometric symmetry: since the current/field distribution does not generally possess any symmetric properties in this case, the unknown coefficients on the whole mesh must be solved. Therefore no reduction in memory usage is obtained, as the matrix equation being solved is the same as it would have been, had symmetry not been considered. However, a reduction in the computation time for setting up the matrix equation does result. This reduction is achieved by exploiting the fact that the interaction between any two basis functions is the same as that between their symmetrical counterparts.
  

• Electric symmetry: as with geometric symmetry, less computational time is required to calculate the matrix equation entries. However, the major additional benefit is that the number of unknown coefficients is reduced by a factor of roughly two. Thus the system of linear equations to be solved has dimension half of what it would have been, had symmetry not been considered. The impact for the MoM is a reduction by a factor four (=2*2) in memory requirement, as the MoM leads to fully populated matrices. The impact for the FEM is a reduction by a factor two in memory requirement, as the FEM leads to sparsely populated matrices. The reduction in unknowns also leads to dramatic lowering of matrix equation solution time.
  

• Magnetic symmetry: the same benefits result as in the case of electric symmetry.
  

Clearly, the benefits of symmetry can be significant.

Applying symmetry in CADFEKO

In CADFEKO symmetry is considered a property of the model. Symmetry planes are defined via the Define symmetry planes dialog under Model in the main menu (see Figure 3). The coordinate planes x = 0 (yz plane), y = 0 (zx plane) and z = 0 (xy plane) may be defined as planes of symmetry (geometric, electric or magnetic). There is no restriction on assigning the same type of symmetry to more than one of the coordinate planes, in which case the computational benefits are compounded. CADFEKO indicates the current symmetry of the model in the 3D view, as shown in Figure 3. 

When applying symmetry in CADFEKO, the whole symmetric model should be created, including ports, excitations, loads, losses, etc. The fact that it is not necessary to only create a section of the model makes it very easy to switch between a solution that employs symmetry and one that does not, or adjust the symmetry properties of the model without any geometry or mesh modifications. During meshing CADFEKO will verify that the geometry to be meshed does indeed adhere to the specified model symmetry (both geometric symmetry as well as symmetry of excitation and loads where magnetic or electric symmetry is concerned).

Example of a symmetric antenna structure

In Figure 4 a horn antenna is shown, exited with a dominant mode, rectangular waveguide port (this is Example 14 in the FEKO Suite 5.4 Examples Guide, where a full description of the problem can be found). The plane z=0 is clearly not a geometric symmetry plane, while the other two planes are. The plane y=0 is a plane of electric symmetry, since the incident wave is electrically polarized in the y-direction, and as this is also a geometric symmetry plane. The plane x=0 is a plane of magnetic symmetry, since the incident magnetic field is normal to this plane and the associated electric field is tangential to it, together with the fact that this is also a geometric symmetry plane.

The symmetry properties of the antenna are as follows: