Re: BIM versus Finite Element

[c.]

On 7 Sep 2011, at 17:34, J Stasko wrote:

> I have been looking at Anastasis C. Polycarpou's book, Introduction to the 
> Finite Element Method in Electromagnetics, and I understand it well so far, 
> I'm just starting the second half.
> 
> I have also been looking at Johann Cervenka's dissertation which can be found 
> here, http://www.iue.tuwien.ac.at/phd/cervenka/diss.html .  
> 
> He mentions that there are three different techniques, finite element, finite 
> difference, and box integration.  He does mention that finite elements use 
> Ansatzfunktionen,
the use of Shape/Test Functions is partially apparent in the mesh structure:

>> msh = msh2m_structured_mesh (linspace(0,1,3), linspace(0,1,3), 1, 1:4);
>> msh = bim2c_mesh_properties (msh)
msh =

  scalar structure containing the fields:

    p: 2x9 matrix
    e: 7x8 matrix
    t: 4x8 matrix
    wjacdet: 3x8 matrix
    area: 8x1 matrix
    shg: 2x3x8 matrix

The field shg represents the gradient of the shape functions computed at each 
element vertices
(as the trapezoidal rule is used for quadrature triangle vertices coincide with 
quadrature nodes).


> which is covered in Polycarpou's work, and I don't see much mention of it in 
> the BIM software.  I'm beginning to suspect that BIM is a special case of 
> FEM, where the Ansatzfunktionen chosen are the set with a polynomial of 
> degree one (as Polycarpou describes in section 1.5).

Yes, BIM uses piece-wise linear continuous Finite Elements with Exponential 
Fitting 
stabilization for convection dominated problems and Mass-Matrix Lumping.
The resulting algebraic system is equivalent to that of the Box Integration 
Method (hence the name).

> Does anyone has a reference which explicitly compares the three?

The equivalence of many methods with Exponential Fitting is shown here:
http://10.1007/s007910050012

a comparison of even more discretization methods is in my thesis:
http://mox.polimi.it/it/progetti/pubblicazioni/viewtesi.php?id=159&en=en

c.