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Juergen Kertz December 21, 2005 09:10

convective stiffness matrix for FEM

After some more or less successful experiments I came to the "segregated" methods, which would finally meet my needs considering memory consumption.

For previous tries used only diffusive terms (e.g. penalty method), and the stiffness matrix is well done with:

Aij=mue*INTEGRAL{dNi/dx*dNj/dx + dNi/dy*dNj/dy}dOmega

where i and j are matrix indices.

However, as I tried to add the convective term, I found an additional k-index, and I do not know what to do with this. The meaning is not described in any paper I found.

Cij=INTEGRAL{wi*(Nk*uk*dNj/dx + Nk*vk*dNj/dy)}dOmega

I know that the uk and vk are taken from the previous iteration to avoid the nonlinearity,but I do not know where to put the k-index (if it even means an index) into the algorithm and my code.

For further explanation or my set-up: I want to calculate steady viscous flow (plastics extrusion die) in 2D (for the first try) with linear 3-node-triangle-elements.

Can somebody please tell me what the k's mean, or point me to a document where the matrix assembly algorithm is described?

Thank you!


Renato. December 21, 2005 16:54

Re: convective stiffness matrix for FEM
Hi Juergen,

take a look in page 21 of

Although written in portuguese, I think it can help you...



Rami December 22, 2005 03:03

Re: convective stiffness matrix for FEM

According to the Einstein convention a repeated index means summation. Therefore k is a dummy index to be summed on the relevant range (probably k=1,2,3 for 3D).

I hope this helps,


Juergen Kertz December 22, 2005 03:07

Re: convective stiffness matrix for FEM
Hi Renato,

Thank you very much for that document! This is a very good work! The things are described "for the programmer", and this is what I needed. I think I can handle my problem with this now, I will try today.

Did you ever think of translating your work (it is your work, isn't it?) to english (or even better german)? I think A LOT of people could use it very well!! As far as I could read the titles with my little italian knowledge, you describe highly sophiticated concepts in an easy-to-read way.

Thanks again!


Juergen Kertz December 22, 2005 04:22

Re: convective stiffness matrix for FEM
Thank you Rami for you answer!

This is exactly what Renato's document says (which I'm trying to read just in this moment)

Looking at my problem with this knowledge makes my further work much easier! In fact, I feel the answer is so obvious, I must be some kind of stupid that I didn't get it on my own.



Renato. December 23, 2005 06:55

Re: convective stiffness matrix for FEM
Hi Juergen,

When I was stating my CFD studies I had the same kind of doubt about Einstein's notation and after that, I decided to write all FEM matrices to incompressible flow in my MSc's dissertation.

Regarding the translation: unfortunately, here in Brazil, we're not encouraged to write our dissertations and thesis in any other language rather than Portuguese. We only publish articles in English. In fact, most of the Brazilians don't need to know a second language. We are a huge country where everybody speak Portuguese, besides being surrounded by Latin countries.

It brings me to a curiosity: "How many north Americans learn a second language? Maybe Spanish... They are also a huge country speaking an international language" Lucky guys...

Regards and Merry Christmas


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