**New Feature in Feko 2023: Characteristic Basis Functions Method (CBFM)**

In the recent Feko 2023 release a new Simulation Method **CBFM** has been introduced. CBFM stands for **Characteristic Basis Functions Method**, and it is based on the use of Macro Basis Functions. The CBFM can be seen as a modification of the conventional Method of Moments that **reduces the number of unknowns** of the problem. In other words, in the CBFM the original RWG (Rao-Wilton-Glisson) basis/testing functions are replaced by CBFs.

The CBFM is activated in the solver settings:

CBFM can be applied for MoM and MLFMM simulations. There are two options available to compute the characteristic base functions (CBF) using a *MoM-based* or a faster but for low field levels less accurate *PO-based* approach. The MoM-based approach is the default choice and you can find more details about that under "Tips and Tricks" below.

**How does CBFM work? **

Each CBF is internally expressed as an aggregation of low-level basis functions with specific weights. Inside each block the CBFs are orthonormal.

In the first step of the simulation process the CBFs are generated using a plane wave spectrum in order to approximate the current distribution over a subdomain on the structure. Because of this step, setting up the matrix elements takes longer with characteristic basis functions than with standard RWG basis functions of MoM/MLFMM. In return, the effort to solve the system of equations will be lower since the problem size is reduced. For the combination CBFM with MoM, the memory requirements are also significantly lower than for standard MoM.

**What are typical application cases for CBFM?**

The CBFM proves its advantages in problems with many variants of the right-hand side of the linear system. This is the case especially for computation of **monostatic radar cross section (RCS**), where the structure is excited by multiple plane EM waves from different directions.

**Tips and Tricks**

In the use cases of monostatic RCS, structures with closed surfaces may appear. In this situation, the MLFMM computation time can be reduced by switching the type of integral equation in Feko from **EFIE (Electric Field Integral Equation)** to **CFIE (Combined Field Integral Equation)** for such closed metallic surfaces. This can be defined in the *Modify Face menu* of the relevant faces.

In this case the PO-based CBFM has same accuracy as the MoM-based CBFM and therefore it is recommended to switch to the faster PO-based approach. And "PO-based" can be used (also for EFIE) whenever high RCS values are to be detected accurately, but lower accuracy is sufficient for low RCS levels.

**Examples **

Figure 4 shows an aircraft with its monostatic RCS at f = 350 MHz in the theta = 90° plane.

The angle increment is 0.5° for the multiple plane wave excitation and at this frequency the simulation mesh contains 109.918 triangle elements. For this problem size the MLFMM is more suitable than MoM, but in order to demo the CBFM properties we compare for both methods the solutions with CBFM and standard RWG basis functions. All simulations are done using 24 parallel processes on 2 CPUs.

All four variants show the same result. As expected, the MLFMM solution has significantly better performance parameters than MoM.

- For MLFMM, the main advantage of CBFM is the shorter computation time.
- For MoM, on the other hand, the advantage of CBFM is the greatly reduced memory requirement by factor 18.5!

This nice CBFM property offers new simulation options in cases where MLFMM shows bad convergence and shifts the threshold when MLFMM should be used instead of MoM to much larger models (or higher frequencies).

If we increase the frequency to 1 GHz the model has 895.234 triangle elements and MLFMM will be the preferred method instead of MoM.

Here, the advantage of switching from Electric Field Integral Equation (EFIE) to Combined Field Integral Equation (CFIE) is well seen with huge improvements regarding simulation runtime for the simulation with 361 plane wave excitations. In this case, the advantage of CBFM over MLFMM is the lower memory requirement. However, the example also shows that reducing the matrix size with CBFM does not always guarantee a shorter computation time, since the problem-specific convergence behavior also plays a role.

Finally, we look at a ship with 114 m length at 300 MHz. The model has 1.752.408 triangular elements and the RCS is calculated for 181 incidence angles. Since the MoM solution would require more than 51 TB of memory, we compare MLFMM (with classical RGW base functions) and MLFMM-CBFM. For both solutions the CFIE approach is used and both approaches provide very similar results.

If one looks more closely at the allocation of the computation times, then one recognizes that the (PO-based) CBFM needs more time for the computation of the matrix elements and for this the solution of the linear system is clearly faster. This advantage becomes larger the more angles of incidence are considered in the RCS analysis.

**Requirements for CBFM**

CBFM is available for models with metallic triangles only. Wire segments and dielectric media are not supported yet. These restrictions will be gradually resolved in future Feko releases.