h-FEM / p-FEM / hp-FEM advantages / disadvantages /your opinion
h-FEM / p-FEM / hp-FEM advantages / disadvantages /your opinion
(OP)
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.





RE: h-FEM / p-FEM / hp-FEM advantages / disadvantages /your opinion
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
RE: h-FEM / p-FEM / hp-FEM advantages / disadvantages /your opinion
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
RE: h-FEM / p-FEM / hp-FEM advantages / disadvantages /your opinion
"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.
RE: h-FEM / p-FEM / hp-FEM advantages / disadvantages /your opinion
RE: h-FEM / p-FEM / hp-FEM advantages / disadvantages /your opinion
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.
RE: h-FEM / p-FEM / hp-FEM advantages / disadvantages /your opinion
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 !
Thanks!
Regards
RE: h-FEM / p-FEM / hp-FEM advantages / disadvantages /your opinion
RE: h-FEM / p-FEM / hp-FEM advantages / disadvantages /your opinion
RE: h-FEM / p-FEM / hp-FEM advantages / disadvantages /your opinion
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
RE: h-FEM / p-FEM / hp-FEM advantages / disadvantages /your opinion
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
RE: h-FEM / p-FEM / hp-FEM advantages / disadvantages /your opinion
Torpen, I'd like to give you five stars if the forum's mechanism allowed me to!!!
Regards