# Basis Functions

Basis functions are elementary functions for the modelling of the unknown quantity on a mesh element.

## Categories of Basis Functions

There are two main categories of basis functions:
• entire-domain basis functions
• sub-domain (sub-sectional) basis functions

Entire-domain basis functions are defined over the entire surface of the scatterer - they are non-zero over the entire domain. The formulation of these functions is deemed rather trivial, provided the shape of the scatterer is regular. For most practical applications, the shape of the scatterer is irregular and the formulation of such basis functions is near impossible. This requires the usage of sub-domain basis functions.

In the application of sub-domain basis functions the entire surface of the scatterer is subdivided into small surfaces. On each subdivided surface a simple function is employed to represent the unknown quantity (such as charge or current). Sub-domain basis functions are non-zero on only a small part of the entire domain.
Note: For FEM and VEP, the volume is subdivided and on each volumetric element a simple function is employed to represent the field.

## Types of Sub-Domain Basis Functions

The different types of basis functions are distinguished from each other based on their spatial variations. A few well-known ones are as follows:
• constant (also known as pulse or stair-step)
• linear
• polynomial
• piecewise sinusoidal

## The Rao-Wilton-Glisson (RWG) element

The MoM in Feko is based on a triangular mesh. Triangular meshes can approximate surfaces much better than for example, rectangular patches. Feko makes use of linear roof-top basis functions introduced by Rao, Wilton and Glisson in 1982. 1 These basis functions enforces current continuity over a common edge of a triangle pair.

In Figure 1, only two triangles are shown sharing a common edge. Each triangle also has two other edges. If these edges are connected to triangles, then additional basis functions would be required. Therefore for a triangle connected on all three sides, a total of three basis functions would be defined. Within the triangle element the total current would then be the sum of these three basis functions.

In Figure 5, the Yagi-Uda was modelled with wire segments. Similar to triangle pairs, linear roof-top basis functions are used across vertices between wire segment pairs.
1 S.M. Rao, D.R. Wilton and A.W. Glisson. "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propagation, 30, 409-418, May 1982.