h-FEM / p-FEM / hp-FEM advantages / disadvantages /your opinion 3

[xerf]

Following the interesting aspects debated in the thread "mesh convergence test", I thought it should be better to start a new thread for sharing opinions on the h-/p-/hp- versions of FEM.

 

[cbrn]

Hi,
so let me reply here as a continuation of what I last red in the "mesh convergence" thread.

First of all, I'd be interested in knowing, mathematically, what makes p-elements "more efficient" as regards convergence. Is it because, in case of a stress gradient of - say - 5th order, h-elems-based solution would experience difficulties in reducing the error norm because all they can do is approximating the gradient in terms of 2nd-order functions? OK, I do believe it would be a serious problem with an insufficiently-dense mesh. But then, doesn't the p-refinement (i.e. an increase in the transfer function's order inside the elements) sound quite like the h-refinement (where it's the number of elements which is increased)? Seen like that, the two would only be complementary methods to do the same thing... So, where is the big difference?
"Secondly", is the numerical solution of n equations of order m more or less efficient than the solution of n/2 equations of order 2*m? Given the same model, and given that both p-mesh and h-mesh is converged, do you end up with less integration points in the case of p-method?
"Thirdly", in the test I made, the highest order reached by the p-elems was 3. Probably I didn't produce a case with "extreme" stress singularities. However, it was a "real-life" model, so my questio is: is it so frequent to encounter 4-, 5- or higher-order gradients of stress?

Excuse me if some questions sound naive, but I'm not really experienced with p-method.

Thanks in advance to everybody... I've got plenty more questions ready to be posted here... ;-)

Regards

 

[corus]

I'm not sure what hp-FEM is so can't reply.
The other mathematical techniques I know of for approximating the solution of PDEs are finite difference methods and boundary elements. These have been missed out of the list and should be mentioned.
I believe boundary elements have some advantages in that the solution matrix is smaller (even though it's not banded), though does have the drawback in that only linear material properties can be used (as with p-elements). Generating the mesh only on the boundary also has big adavantages in that you don't have to worry about meshing within the region.
Finite differences seem only to have academic interest these days, as the need to model more complex geometry has arisen, which FD can't cope with. It's good for students to learn though.

corus

 

[xerf]

I found an interesting paper:

"On Efficient Numerical Integration of p-version
Finite Element Stiffness Matrices" by Magda Martins-Wagner

The paper presents an investigation of various schemes for integration of large element stiffness matrices typically arising for higher order finite elements.

 

[xerf]

hp-refinement in context of FEM, is combination of h- and p- mesh refinements.

 

[prost]

This is not correct, p-element software is available to compute engineering quantities for structures with nonlinear materials (StressCheck, for instance). Might not do rubbers, but that's a limitation of the software, not the p-version. More than a decade ago I used p-version to model nonlinear viscoelastic materials, which worked quite well; I don't recall any h-version FE software being able to handle those material types, though I am sure they could, if the FE companies bothered to develop them.

I can imagine the numerical difficulties intrinsic to finite differences contribute significantly to their now being out of favor. Anybody who has ever run an experiment in which an output signal had to be differentiated I am sure is aware that integrating is intrinsically more stable than differentiating.

The 'proof' why 'hp' refinements converge more quickly (that is, for fewer degrees of freedom) can be found in the following references:
[1] Babuska, I, "The p- and hp-versions of the Finite Element Method: the state of the art." Finite Elements: Theory and Applications, ed. D.L. Dwoyer, M.Y. Hussaini, and R.G. Voigt, pp. 199-239, 1988
[2] Gui, W., and Babuska, I, "The h- p- and hp-versions of the finite element method in one dimension. Part I: The error analysis of the p-version. Part II: The error analysis of the h and hp-versions. Part III: The Adaptive hp-version," Numerische Mathematik, Vol. 49, pp. 577-612, 613-657, and 659-683, 1986.
[3] Babuska, I, and Suri, M. "The p and hp-versions of the finite element method. An Overview," Comp. Meth. in Appl. Mech. and Engng, Vol 80, pp. 5-26.

Warning! These are written in the language of a branch of high level mathematics called functional analysis. There'll be many terms unfamiliar to 99% of all engineers: Sobolev spaces, Lebesque integration, etc.

Those are the mathematics based proof of superiority of hp-version over h-version.

 

[cbrn]

Hi,
eh eh, Prost, I agree that hp-method is superior to h-method, but it is superior to p-method as well! ;-)
The references you provide are extremely interesting and Babuska is well-known for his studies in numerical techniques; at the University in Italy they make your head turn into a balloon with purely-theoreticals, but then when you go out and work "for real" all you hope is to forget all these ! ;-) I'll try to catch back my math notions and perhaps try to read the books you mention.
Thanks!

Regards

 

[xerf]

I just want to point out that even if a numerical technique is shown to be better from a mathematical perspective when it comes to coding and software implementation it might end up being not sufficient good and/or feasible.

 

[prost]

xerf, certainly this can be the case. Nevertheless the p-version is as easy (or hard!) to implement as the h-version. I would like to point out that a distinction should be made between what either method can do and what has been implemented in a commercially available code. It is obvious that, considering just the physics that can be modeled (particularly for nonlinear materials such as rubbers), the h-version FE software available commercially is far ahead of comparably available p-version FE software. This would suggest that the approx. 20 year lead the h-version has on the p-version is more of a resources issue than an intrinsic capability issue.

 

[torpen]

Hello,

The following article :

Babuska, I, and Suri, M. "The p and hp-versions of the finite element method. An Overview,"
Comp. Meth. in Appl. Mech. and Engng, Vol 80, pp. 5-26.

Torpen,

Regards

 

[torpen]

Hello,

Frederic Cugnon from the Liege University has written a doctoral thesis about the automatization of computation with p-version finite elements method.
In particular the chapter 2 compares h-version FEM versus p-version FEM.

It is written in french but the first pages of the chapter 2 are clear and quite easy to read.


Regards,

Torpen

 

[cbrn]

Hi,
Torpen, I'd like to give you five stars if the forum's mechanism allowed me to!!!
Regards