Do you know the difference between triangle and quadrilateral elements?
Theoretical solution will give a von mises stress value of 372.7 MPa
Now let’s try FEA with different number of elements and both with Triangle and quad.
(There is no picture for quad, but I just did the same…with quad)
When you do the calculation, it is actually funny to see how much the difference is actually huge! Same number of elements will give a totally different error in function of triangle and quad.
Here are the same results with pictures:
Do you realize what it means??
To get the same less than 5% error, you will need 16 times more triangle elements than quad elements.
Now let’s assume the speed to solve the problem is proportional to the number of elements (it’s not totally true, but let’s just assume that)
if it takes 1 sec to solve the model for 6346 tetra elements, it will take only 0.06 sec to solve the same problem with quad with less elements but the same accuracy.
So… Do you still want to use triangle/tetra elements??
Let me know what you think about that
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Very nice comparison. It is well known that degenerate linear elements lead to innacurate solutions because they are subjected in structural analysis to several effects (shear/volumetric locking…) that makes solutions be too stiff in general. However this is not true when using quadratic interpolation elements… It would be very interesting if you could add this variable to your comparative analysis as well.
Some observations:
1- In FEA, usually total DOF is used for benchmark/error comparison purpose and not number of elements. If total DOF is considered, the difference in error between event types will become less significant.
2- Linear Traingular elements are called CST ( Constant Strain Triangle). Therefore, they should either not be used where there is rapid change in stress/strain or much larger number must be used. On the other hand, linear (1st order) Quad elements are based on linear strain shape function.
3- It is not clear what integration order has been used (reduced or full).
4- it is not clear whether nodal raw nodal data or average has been used. They affect the comparison.
5- The analytical solution does not reprsent Von Mises Stress; it is the normal stress in L direction. Therefore, it’s comparison with VM Stress from FEA is not correct.
6- Given that linear elements are used, more elements must be used around the curve regardless of element type (triangular or Quad). If the analyst follows this fact, the cases with very course mesh should be excluded from comparison as they don’t satisfy the fundamental rules of linear elements.
Thank you Mehdi, I am glad that you spotted those problems! I will take them in account to improve the article.
The title is “The Accuracy Secret” but the article is about efficiency, not accuracy. Solutions with both element types converge to the exact solution, indicating that both provide accurate answers if enough elements are used. Accuracy is not the issue, speed is.
That said, if you use 2nd order triangle or quad elements (or higher order) the issue becomes a non-issue. You can obtain a solution with much less than 5% error using only 5 quad elements or 10 tri element if you use, for example, 8th order shape functions.
I agree with Watkins here. This is more a problem of “convergence” than accuracy. The study here shows that hexa elements converge faster to the solution than tets do.
Also tets are not always avoidable. Sometimes we have to resort to using tets may be with higher order elements.
Hi Cyprien,
First of all, i appreciate your effort about teaching all about FEA.
An observation is that at first picture, F = 100 N/mm2 is written. It should be force unit not pressure. Except this, everything is seems good.
Hello Cyprien.
Thank you very much for sharing useful FEA knowledge through demonstration. Very inspired by this blog. I will be happy to contribute to your blog with limited solving resources I have. I am positive this is only the beginning!
Best Regards,
Syed
Hi Syed, Thank you very much, I am glad you like the blog and the articles !
Just a question. with four nodes I can get only one quadrilateral element or two triangular elements. means, the same number of nodes the triangular elements are double the quadrilateral elements. So, why the results for quadrilateral elements is better?
Good Question Salaheldin, This is related to the finite element method itself. If you read about the theory of finite elements, you will notice that the algorithm is much more accurate when your have a 4-node approximation rather than a 3-nodes approximation…I know this is not the best answer, so if someone has it with more details, please explain here 😉
An interesting article Cyprien which opens up many questions:
1) I am first curious as to whether or not your ‘theoretical’ solution is actually correct. There is a closed-form solution for a plate with a hole in it subject to uniform tractions but these tractions are applied at infinity so that for a finite plate such as yours the tractions are not uniform. I dare say that the result will not be too different but caution should be exercised. The closed-form solution can be found on page 116 of the following:
http://www.ramsay-maunder.co.uk/downloads/Exeter_Thesis.pdf
Actually if you look at the results you will see that the FE solution is not converging to your ‘theoretical’ solution – the quadrilateral element starts off with a positive error which then goes negative with mesh refinement.
2) You have used lower-order elements so there is a further approximation to account for namely that the FE mesh is not modelling the geometry correctly – piecewise linear to model a circular arc. Even higher-order elements will not model this accurately as a circular arc is not contained within a parabolic description. Again this probably has little effect but it should be noted.
3) I would say that your article does not ‘explain’ the difference but it does ‘demonstrate’ a difference in the results.
4) The results presented are somewhat misleading in that the stress field is only ‘complicated’ local to the stress concentration. Away from this the stresses are more or less constant and could be modelled accurately with lower-order triangular elements. The results as presented use global refinement and if a local refinement scheme had been used then the number of elements required to achieve the specified level of accuracy would have been less than suggested.
5) Not sure what form of quadrilateral element you have used but for a comparison it was presumably the four-noded quad? However, even with this element there are a number of options in terms of integration schemes and additional bubble functions. These can make a significant difference to the results.
6) Whilst the time to formulate the global stiffness matrix might well be proportional to the number of elements, the solution of this system will not be – I don’t think that solvers are yet available that are capable of this.
7) Although most FE systems have available the lower-order elements, I would suspect that most people would adopt the higher-order elements for any real analysis. The advantage of using higher-order elements over the lower-order elements is demonstrated in the following technical note which illustrates the value of p-type refinement.
http://www.ramsay-maunder.co.uk/downloads/P-H-REFINEMENT.pdf
It would be good to see the results of a p-type refinement on the problem in your article.
I think your initiative is good one and I hope that these points might be of value to you as you develop other articles.
s
Hello Angus,
Thank you very much for your comments.
My purpose is actually more to raise the awareness on the possibility of such problem.
I agree that the method I used may not have been totally correct and that a lot of hypothesis have to be verified..but I suppose that if I did, I would never have written this post (and Blog) and I would have rather written a PhD Thesis like you that very few people would have read probably…
PhD thesis? hahaha this is pretty basic stuff to be aware of, man.