Compute derivatives from a FE model 1

[Eddy83]

Maybe somebody can help me out with this:

When from a Finite Element model you want to calculate derivatives of a nodal displacement w.r.t. a parameter of your model (e.g. you have a simple beam and you want to calculate the derivative of the nodal displacement of the node at the free tip w.r.t. the length of the beam: dui/dL) then it turns out that the more elements you use for your model, the worse your results become.
This is of course quite peculiar since normally more elements make your results improve. When I use 10 elements the derivatives computed from the FE model correspond quite well with the exact results, but when I use 100 elements the computed derivatives get further off from the exact results.
I remember from a lecture that this "phenomenon" has been explained times ago by some person, but I don't know the explanation itself. Maybe anybody else knows?

 

[JStephen]

If you have a simple beam with uniform load, won't the FEA model be an exact solution with 1-element? In that case, you'd never gain anything by increasing the number of elements, and have to start losing accuracy at some point due to numerical problems. With a 100 elements, I wouldn't expect that to happen unless maybe it was a low-quality solution method.

 

[Eddy83]

Thanks JStephen

Yes indeed, the results of my FE analysis are indeed exact (apart from small numerical errors) and only 1 element should be enough. When using 1 element the derivatives I compute (using Global Finite Differences or the Discrete Semi-Analytical method or the Discrete Analytical method) are very close to the exact analytical derivatives, which are easy to compute for this simple beam. However I don't gain anything when I increase the number of elements for this simple beam, the question is interesting since for larger and more complicated problems, where the FE solution won't be exact, it's good to understand the influence the number of elements.
Anyhow, I also think starts to get the supposition that the numerical quality of the finite element problem somehow starts to deteriorate, so I think you're right. Yet if this is true I would like to know somewhat more about this numerical 'problem'.

 

[xerf]

The mathematical theory based on which the common FE elements are developed ensures the convergences of FE solution as the size of the elements decreases.

 

[prost]

What scale are you talking about here? The difference between 1e-8 and 1e-10?

 

[JStephen]

If you're using a decent canned software, I would expect the results to be very accurate even with a 100 elements. If you're programming it up yourself, it's easy to come up with an equation-solving routine that loses accuracy in various conditions, even with a lot less than 100 elements.

 

[123ash]

same problem I found in structure analysis using catia v5
i analyzed with linear element and then with parabolic element and found 10 times diff in stress so what actually i have to select ? linear element ? or parabolic?

 

[electricpete]

A note alittle bit out in left field - If you were solving Laplace or Poisson's equation using an iterative technique, more elements means longer to converge, but when it does converge it comes closer to the exact solution. Do iterative techniques apply to mechanical (stress / displacement) FEM?

Also, aren't there varying degrees of exactness depending on what math model you are comparing your solution to? Euler element is exact solution of Euler beam (gives same analytical result). Timoshenko element is exact soltion of Timoshenko beam which may be closer to reality. Just for my edification, what is meant by an exact element?



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