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

In summary, the conversation involves a user seeking help with an example problem involving integration by parts. The given answer is different from the answer they get, and they are trying to identify the mistake. The user is using finite element analysis and the boundaries are from 0 to L. It is pointed out that there is an extra term which is correct, and that N1 = 0 at one limit and 1 at the other, while dN2/dx is constant. The user is integrating between one pair of nodes and then over other pairs until the whole mesh is covered. They suspect the issue may be related to boundary conditions. The extra terms are often ignored in finite element analysis.
  • #1
bzz77
34
0
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:
∫ [ (Ni) (d2 Nj / dx2) ] dx

Given answer:
- ∫ [ (dNi / dx) (dNj / dx) ] dx + boundary terms

Answer I get (additional term):
Ni * dNi/dx - ∫ [ (dNi / dx) (dNj / dx) ] dx + boundary terms

Thanks a lot for any help!
 
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  • #2
hi bzz77! :smile:

you have an extra [Ni dNj/dx]0∞, which is correct

what is N ?

I assume it's defined so that [Ni dNj/dx]0∞ = 0
 
  • #3


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!
 
  • #4
hi bzz77! :smile:

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

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
 
  • #5


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).
 
  • #6
hmm …

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

are you integrating just between one pair of nodes, or over a large number of nodes?
 
  • #7


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.
 
  • #8


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!
 

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

1. How does integration by parts work?

Integration by parts is a technique used to integrate the product of two functions. It involves using the product rule of differentiation to rewrite the integral in a different form, making it easier to solve. The formula for integration by parts is ∫u dv = uv - ∫v du.

2. What do I do if I can't reproduce a given answer using integration by parts?

If you are having trouble reproducing a given answer using integration by parts, the first step is to double check your calculations. Make sure you have correctly identified which function to designate as u and which to designate as dv. You should also check your integration and any algebraic simplifications you may have made.

3. Can I use integration by parts for any type of integral?

No, integration by parts is most useful for integrals involving products of functions. It is not always the most efficient method for solving integrals, so it is important to consider other integration techniques such as substitution or partial fractions.

4. Is there a specific order in which I should choose u and dv for integration by parts?

There is no specific order in which u and dv should be chosen. However, it is generally recommended to choose u as the function that becomes simpler when differentiated, and dv as the function that becomes easier to integrate.

5. Are there any common mistakes to avoid when using integration by parts?

One common mistake when using integration by parts is to choose u as the entire integral, which leads to an infinitely repeating process. It is important to choose u and dv as specific functions within the integral. Other mistakes include incorrect application of the formula and forgetting to include the constant of integration.

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