The term “electrical size” refers to the largest dimension of the model (for example the length of the ground plane for a patch antenna) measured in wavelengths. For many applications, the electrical size of the model is the main consideration for the solver type. This is due to the fact that computational resources for the MoM grow with respect to electrical size (proportional to the number of mesh elements squared). It will, however, be shown that this not necessarily true for all solvers – for example asymptotic solvers’ suitability actually increases with electrical size while their computational resources do not in most instances.
Popular applications, their solvers and examples
Method of Moments (MoM)
The MoM is the default solver in Feko. It is highly suitable for a variety of applications, provided the electrical size (largest dimension of the model) is not too large. For a laptop computer a few wavelengths would be solvable with MoM. For a large server with tens of cores and hundreds of Gigabytes of memory ten or more wavelengths could be considered.
A good example would be a microstrip patch antenna or a light aircraft at VHF frequencies.
Method of Moments: more on dielectric methods
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The SEP is the default dielectric method. It employs surface meshing for dielectrics that enables the accurate representation of the fields internal to the dielectric with electric and magnetic surface currents on the surface of the dielectric.
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The MoM for planar infinite (2D) dielectrics uses analytical expressions for the dielectric based on the Green function. More specifically, the dielectric is represented in an analytical sense instead creating of a mesh. The formulation is well suited for modelling microstrip and stripline applications such as feed networks or printed antennas.
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The VEP uses tetrahedral meshing to model the volume of the dielectric with electric and magnetic polarization currents. It is well suited for low frequency (electrically small) modelling of dielectrics. The VEP is only dielectric method that also supports low frequency stabilization.
Multilevel fast multipole method (MLFMM):
This method is the method of choice for electrically large problems. For example the MLFMM will be needed if a light aircraft is solved at 1 GHz or higher.
Note that the MLFMM uses an iterative solver compared to the MoM that uses a direct solver. Therefore the MLFMM iterative solution time depends on the number of iterations to achieve convergence.
Finite element method (FEM):
The FEM is well suited for models with multiple dielectrics, such as a multi-layer radome, a stacked patch antenna array, or an inhomogeneous human phantom. In most cases, the actual solver is the hybrid FEM/MoM where the outer surface elements of the FEM are used as boundary elements for the MoM solution.
Typically an enclosing dielectric air layer (permittivity of 1) is used to enclose the model and to fill free space regions with tetrahedra (for example the stacked Sierpinski antenna shown bottom right). The reason for the air layer is that mesh elements on the outer surface are computationally expensive to solve compared to the FEM tetrahedra internal to the mesh. For the stacked Sierpinski the mesh on the right will consume a lower amount of memory and runtime.
The dielectric resonator (DRA) example in the Feko Example Guide can also be studied as an example of FEM/MoM. When the FEM is completely enclosed by PEC elements such as a waveguide (see the waveguide step example in the Feko Example Guide), then there are no boundary elements for the MoM and the solution is a pure FEM solution. The two path cutoff waveguide filter (Shigesawa et. al.) shown below with the top face hidden is well suited to the FEM solver. The default FEM solver is an iterative solver, but a direct solution, albeit more memory intensive, can be selected.
Finite Element Method / Multilevel Fast Multipole Method (FEM/MLFMM):
For electrically large models where some part of the model is solved with the FEM, then the FEM/MLFMM can be used. In this case the surfaces of the FEM and the rest of the model is solved with the MLFMM instead of the MoM. A good example of this would be an antenna placement scenario where the stacked Sierpinski (above) is placed on an aircraft such as a helicopter.
Asymptotic methods
An asymptotic method makes specific assumptions about the electrical size and the model construction. For example, it assumes that a smooth current distribution will exist over the surfaces and that sources are electrically far away. An asymptotic method is typically chosen when the computational resources for the MLFMM are too large. In most cases the asymptotic method is coupled with a small MoM part, which comprises the source (excitation) such as an antenna.
Physical optics (PO)
The physical optics employs a mesh identical to that used for the MoM. The difference is in the solution where the asymptotic assumptions are applied. A reflector antenna is well suited to this solver, where the reflector is solved with PO and the horn antenna with MoM. Note the smooth and electrically large surface of the reflector. Also note the source that is several wavelengths away from the reflector. To reduce computational resources, often an equivalent source can be used instead of the horn. The Feko Example Guide contains an example of a reflector antenna with multiple feed options.
If the reflector becomes electrically much larger, the number of mesh elements could be in the millions. Then the Large Element Physical Optics (LE-PO) solution could be considered. For the LE-PO the meshed triangles are much larger (size typically around 1 to 2 wavelengths) than for PO, reducing the number of required meshed elements and consequently the computational resources.
Uniform theory of diffraction (UTD)
The UTD should be selected when the computational resources and the number of meshed triangles for the MLFMM and for the PO (LE-PO) are too high. An example would be a ship at a few GHz. The benefit of the UTD is that computational resources are not dependent on frequency (electrical size). If the model contains curved surfaces the faceted UTD (fUTD) should be selected. In the fUTD a triangular mesh is created allowing for diffraction to be computed for curved surfaces. Note that due to caustics in the UTD, the calculation of RCS is not supported for the UTD. For the ship below, to compute RCS, the RL-GO could be employed instead of the (f)UTD.
Ray-launching geometrical optics (RL-GO):
This method is used for electrically huge dielectric and metallic problems. Examples would be a lens antenna or a radome. Also for electrically huge RCS applications, the RL-GO should be used.
Finite difference time domain (FDTD):
The FDTD is well suited for wideband problems of small to medium size. Examples are a printed LPDA or a Vivaldi antenna. Note for curved structures the cuboidal voxel mesh will approximate the geometry with a stair-cased mesh. 
Multiconductor transmission line (MTL)
A shielded or unshielded cable can be solved using the Multiconductor Transmission Line method. Furthermore, the MTL is hybridized with the MoM, MLFMM or FDTD. For example, a cable harness inside a vehicle could be solved with the MTL: the MTL is used to efficiently model the cable harness, while the MoM or MLFMM is used for the everything else.
The MTL is limited to scenarios where cables harnesses run close to a ground plane. Where cables are further separated from the nearest metallic surfaces, Feko employs a hybrid MTL/MoM or MTL/MLFMM solution.
Conclusion
It may not always be clear which technique to use, especially when more than one method is suitable.
In general it is recommended to first consider the default solver (MoM) and if the electrical size is very large, consider MLFMM. Then if MLFMM is still too computationally intensive, consider an asymptotic technique. For accuracy users may want to start with the default solver.
The following table summarizes each solver in terms of complexity (how easy it is to create a model for the solver), computational resources (memory and solution speed) and accuracy of results. Although the table indicates the accuracy of the asymptotic solvers as lower than for full wave solvers, if used for the right application they can give just as accurate results as full wave techniques. The table below should be seen as a very general guideline only.
| Method | Setup Complexity | Speed | Memory | |
| MoM with SEP | easy | medium | medium to high | |
| MoM with infinite substrates | easy | fast | low to medium | |
| VEP | easy | medium | medium to high | |
| FEM/MoM | medium | medium | medium | |
| FEM/MLFMM | medium | medium | medium | |
| MLFMM | easy | fast | medium to low | |
| MoM/PO | medium | medium to fast | low | |
| MoM/UTD | medium to high | fast | low | |
| RL-GO | medium | fast | low | |
| FDTD | medium to high | fast | low | |
Conclusion
When a model is solved Feko will sometimes give a message indicating that a better technique could be used. But this is not true in all cases. If there is doubt about which solver to use before starting a new project, users are more than welcome to contact support for some tips.
