Higher order basis functions vs low order basis functions

Hello All,

I have a question that I have been searching for an answer to it for over a year :). I was trying to find out whether using higher order basis functions (in MoM or FEM) with small mesh is better or lower order basis functions with larger mesh ?

I noticed that some programs uses higher order basis functions (e.g. HFSS in FEM, WIPL-D in MoM) and some programs don't (e.g FEKO in MoM).

I was wondering what is there effect in terms of computational complexity when requiring the same accuracy.

If anybody have any idea, I would be glad if he could share it,

Thanks in advance,
Adel

Hi Adel,

it depends on the structure you are trying to solve. The Mesh in any 3D field simulator has to fulfill two tasks. It has to represent the geometry AND it has to represent field inside or on the surface of the geometry.

Higher order Elements allow you to represent the "fields" with equally accuracy with a coarse mesh with less computational effort. Each higher order element has certainly more unknowns then a first order element, but since you can use a coarser mesh the total effort is reduced.

The problem is: At some parts in your structure the mesh might already be very dense just because of the resolved geometry. Think about the fields inside a plate capacitor. In between the plates, the field is pretty much homogeneous and can be represent almost perfectly with single order elements. Using higher order elements at this location would just increase the computational effort without giving you any extra accuracy.


If your mesh has to be extremely dense in terms of the wavelength (because of the geometry sampling) first order elements will therefore be more efficient, if your mesh can be coarse -> second order elements are better.


F.

I agree.

This is not a problem! Software codes that use higher order basis functions can also use low order elements where they are enough / where the meshing elements are small. For instance, if you use a hierarchical set of basis functions it is very easy to adaptively apply different orders of basis functions to different elements in the mesh according to electrical size of those elements or some other criterion. That way, the code is equally efficient like low-order codes in the parts where small mesh elements are needed but it is much more efficient in parts where you can apply larger mesh elements.

And what about third, fourth order? Up to the fifth order there is a significant decrease of the number of unknowns which is needed to represent surface currents on a wavelength squared surface. In total, in most models, higher order basis results in 5 times lower number of unknowns than low-order basis.