Integration by parts-I can't reproduce a given answer

[bzz77]

Integration by parts--I can't reproduce a given answer

I am working through an example problem that involves integration by parts. The answer is given, but the answer I get is different.

Can anyone help me identify the problem? Is it me or the given answer?

Question:
∫ [ (N_i) (d² N_j / dx²) ] dx

Given answer:
- ∫ [ (dN_i / dx) (dN_j / dx) ] dx + boundary terms

Answer I get (additional term):
N_i * dN_i/dx - ∫ [ (dN_i / dx) (dN_j / dx) ] dx + boundary terms

Thanks a lot for any help!

 

[tiny-tim]

hi bzz77!

you have an extra [N_i dN_j/dx]₀^∞, which is correct

what is N ?

I assume it's defined so that [N_i dN_j/dx]₀^∞ = 0

 

[bzz77]

Hi tiny-tim:

Thanks a lot for your help!

Ni = (1 - x/L); Nj = x/L

Am I making a silly mistake? I still can't see where I'm going wrong! Thanks for any assistance!

 

[tiny-tim]

hi bzz77!

(i don't know where i got those limits from )

what problem does this come from?

are the limits from 0 to L ?

i suspect that at the boundaries either N or N' has to be zero

 

[bzz77]

Hi tiny-tim:

Thanks again! It comes from an example problem I'm working through in finite element analysis. Sorry, I should have mentioned that the limits are 0 to L! I'm an old fart and I have totally forgotten my integral calculus! Sorry for the silly question!

P is a continuous variable that we are approximating within an element in terms of the P at two nodes P1 and P2.

So: N1 = 1 - x/L and N2 = x/L
L is the length of an element and x is the spatial variable that varies from 0 at node 1 to L at node 2.

N1=1 at node 1 and N1=0 at node 2.
N2=0 at node 1 while N2=1 at node 2.
N1 + N2 = 1 (over the entire element).

 

[tiny-tim]

hmm …

it looks like N₁ = 0 at one limit, and 1 at the other, but dN₂/dx is constant

are you integrating just between one pair of nodes, or over a large number of nodes?

 

[bzz77]

Well, I'm integrating it between one pair of nodes... Then doing it over another pair of nodes until I have covered my whole mesh.

I think I have an inkling about what the issue is now... It seems like that extra term has to do with the boundary conditions. And that they somehow cancel out.

 

[bzz77]

Hey tiny-tim:

Just letting you know that the boundary terms (my extra terms) are often ignored in finite element analysis. So maybe that's why that term is not included in my example. Anyway, thank you so much for all your help!