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Analysis of periodic structures in FEKO

This page describes the analysis of periodic structures by way of the periodic boundary condition feature, new to FEKO Suite 5.4.

Feature introduction

With FEKO Suite 5.4 comes a new feature, namely the option to analyse infinite periodic structures. The MoM is used in the periodic analysis, employing a periodic version of the free space Green function, such that a PBC is realized. A typical application of this method is to analyse FSS structures, as will be shown in the validation results to follow.

 

Capabilities

The PBC feature has the following capabilities:

  • Periodic boundaries in both one and two dimensions is supported.  
  • In the case of two dimensional periodicity, arbitrary (non-orthogonal) lattice vectors are supported.
  • The unit cell can include conducting surfaces as well as thin dielectric sheets. Wire segments are not supported at present.
  • Since the PBC method is based upon the free space Green function, it can be used together with an infinite PEC or PMC plane, but it cannot be used together with dielectric layers.
  • The geometry of the unit cell may intersect its boundaries (half basis functions are created on the relevant elements in the discretized model), as long as it lines up with the geometry edge on the opposite side of the unit cell.
  • In the case of solving a radiation problem, the periodic phase shift between cell excitations can be specified.
  • In the case of solving a scattering problem, the periodic phase shift follows from the incident plane wave definition and cannot be specified independently.
  • PBC analysis is part of the MoM kernel and can be executed sequentially or in parallel. However, it is not available in conjuction with MLFMM, SEP, VEP, PO, UTD, GO or FEM.
  • Periodicity in three dimensions is not supported.

These features allow the user to analyse a very large class of problems, though there are some restrictions, as noted.

 

Usage

In CADFEKO, a PBC analysis is requested via the Define periodic boundary condition dialog under Model in the main menu. The dialog enables the user to:

  • Choose between one and two dimensional periodicity
  • Define lattice vector(s)
  • Define directional phase shifts for local excitations, if applicable
  • Visualize the PBC's 

Figure 1: PBC analysis dialog in CADFEKO.
  

Schematic unit cell in 1D

Figure 2(a): Schematic depiction of a 1D periodic unit cell.
The phase change can be specified by the user in the direction u1.
The points S1 and S2 define the lattice vector. 
Schematic unit cell in 2D 

Figure 2(b): Schematic depiction of a 2D periodic unit cell.
The phase change can be specified by the user in the directions u1 and u2.
The points S1, S2 and S3 define the lattice vectors. 
 

Validation results

Figure 3 shows the CADFEKO model of a Jerusalem cross FSS unit cell. The geometry is from [1]. To verify the PBC results the MLFMM was used to analyse a large finite array of 51x51 cells. Given a normally incident plane wave excitation at 7 GHz, Figure 4 compares the PBC current distribution to that of the central element in the MLFMM analysis. Excellent agreement can be observed, verifying the validity of the PBC analysis method.

 

Figure 3: Visualization in POSTFEKO of the unit cell geometry of the Jerusalem Cross FSS. Planes of periodicity,
lattice vectors and the incident plane wave source are also shown.

 

 

 

Figure 4(a): current distribution obtained with
PBC method for an infinite array.

Figure 4(b): current distribution obtained with
MLFMM for a very large, finite array. 
 

Further consider the reflection coefficient of this FSS over the frequency range 2-12 GHz. In Figure 5, results calculated with the PBC method is compared to the results published in [1]. For the FEKO results, 328 triangles were used in the discretization of the unit cell. Excellent agreement is again observed.

Figure 5(a): Magnitude of the reflection coefficient.

Figure 5(b): Phase of the reflection coefficient.
 

References

[1]

Ivica Stevanovic, Pedro Crespo-Valero, Katarina Blagovic, Frederic Bongard and Juan R. Mosig, “Integral-Equation Analysis of 3-D Metallic Objects Arranged in 2-D Lattices Using the Ewald Transformation,” IEEE Trans. Microwave Theory and Techniques, vol. 54, no. 10, pp. 3688-3697, October 2006.