Off topid: finite element vs finite volume
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Phony Accoun #1 / 21

Off topid: finite element vs finite volume
Hello, I apologize for this offtopic question, but I was hoping to find a few folks that may point me to the right direction to answer this question. I have been dealing with finite X commercial modeling codes for several years now, and noticed that the ones dealing with fluid (and chemistry) modeling are mostly finite volume (I have the lovely book by, ok, the name now escapes me, Pandakar?), while the ones with stress, thermal,and e&m are finite elements (and off course the finite boundary elements). What is the reason for finite volume being favored for fluid type problems? The nature of the pde's, including convecitve effects, or something else. What would be a good reference that would help me understand this better? Thanks, Mirko

Fri, 07 Dec 2007 18:58:40 GMT 


Arjen Marku #2 / 21

Off topid: finite element vs finite volume
Quote:
> Hello, > I apologize for this offtopic question, but I was hoping to find a few > folks that may point me to the right direction to answer this question. > I have been dealing with finite X commercial modeling codes for several > years now, and noticed that the ones dealing with fluid (and chemistry) > modeling are mostly finite volume (I have the lovely book by, ok, the > name now escapes me, Pandakar?), while the ones with stress, thermal,and > e&m are finite elements (and off course the finite boundary elements).
You mean: Patankar? Quote: > What is the reason for finite volume being favored for fluid type > problems? The nature of the pde's, including convecitve effects, or > something else. > What would be a good reference that would help me understand this better?
I do not know of a good reference for this type of question. But I do know something about it:  Finite volumes tend to be less resourceconsuming both on computer memory and programmer agility.  The mathematics supporting finite elements can be regarded as more sophisticated and elegant, but the fact remains it is more involved.  In general FEM results in matrices with no particular structure.  Sometimes, and this is especially so for hydrodynamic problems, setting up the FEM equations from scratch is very tricky indeed (the pressure term is the culprit in hydrodynamics). But:  Good mesh generators are available nowadays and standard packages to help solve the matrix equations that result.  Linux clusters seem to be adequate for solving practical problems in a reasonable time with a reasonable price tag. So, many of the drawbacks of FEM are being conquered. This leaves the cultural and historical differences, I guess. Regards, Arjen

Fri, 07 Dec 2007 19:32:16 GMT 


Tim Princ #3 / 21

Off topid: finite element vs finite volume
Quote:
>>Hello, >>I apologize for this offtopic question, but I was hoping to find a few >>folks that may point me to the right direction to answer this question. >>I have been dealing with finite X commercial modeling codes for several >>years now, and noticed that the ones dealing with fluid (and chemistry) >>modeling are mostly finite volume (I have the lovely book by, ok, the >>name now escapes me, Pandakar?), while the ones with stress, thermal,and >>e&m are finite elements (and off course the finite boundary elements). > You mean: Patankar? >>What is the reason for finite volume being favored for fluid type >>problems? The nature of the pde's, including convecitve effects, or >>something else. >>What would be a good reference that would help me understand this better? > I do not know of a good reference for this type of question. But I do > know > something about it: >  Finite volumes tend to be less resourceconsuming both on computer > memory > and programmer agility. >  The mathematics supporting finite elements can be regarded as more > sophisticated and elegant, but the fact remains it is more involved. >  In general FEM results in matrices with no particular structure. >  Sometimes, and this is especially so for hydrodynamic problems, > setting up the FEM equations from scratch is very tricky indeed > (the pressure term is the culprit in hydrodynamics). > But: >  Good mesh generators are available nowadays and standard packages > to help solve the matrix equations that result. >  Linux clusters seem to be adequate for solving practical problems > in a reasonable time with a reasonable price tag. > So, many of the drawbacks of FEM are being conquered. This leaves > the cultural and historical differences, I guess. > Regards, > Arjen
My knowledge may be somewhat out of date, but you allude to historical differences. FEM is an excellent fit, where Green's theorem and Galerkin method apply with no tricks, so the mathematical formulation is more complete. In many such problems, it would produce a symmetric positive definite set of equations to be solved, for either a static problem, or an implicit transient solution. I don't understand Arjen's comment about "no particular structure." Perhaps he means that arbitrary automatic tetrahedral meshing produces a high bandwidth stiffness matrix. Economy of solution certainly depends on sorting the unknowns to produce a compact skyline matrix. Implicit transient solution in fluids problems may involve an approximate factorization to simplify the matrix, as convection limits the time step anyway. When I was last in that field there was still black magic about balancing all the approximations. Fluids problems may not drive memory requirements per cluster node up to 8GB and more as quickly as implicit structures FEM solutions are in the process of doing.

Fri, 07 Dec 2007 21:09:09 GMT 


Arjen Marku #4 / 21

Off topid: finite element vs finite volume
Quote:
> >  In general FEM results in matrices with no particular structure. > >  Sometimes, and this is especially so for hydrodynamic problems, > > setting up the FEM equations from scratch is very tricky indeed > > (the pressure term is the culprit in hydrodynamics). > I don't understand Arjen's comment about "no particular structure." > Perhaps he means that arbitrary automatic tetrahedral meshing produces a > high bandwidth stiffness matrix. Economy of solution certainly depends > on sorting the unknowns to produce a compact skyline matrix.
Yes, the finite volume approach with regular meshes of rectangles or slightly deformed rectangles usually produces band matrices. As finite element methods most often use triangles, the resulting matrices may be sparse but without any regular structure like band matrices or blocked band matrices. It is easier to create grids using triangles (as you need to take certain limits regarding angles into account, AIUI) than using rectangles and still fit with the irregular domains encountered in practice. As for black magic: I recently attended a few presentations about the methods being used nowadays and they sure are not as pretty as a straightforward Galerkin approximation. Regards, Arjen

Fri, 07 Dec 2007 21:19:18 GMT 


beliav.. #5 / 21

Off topid: finite element vs finite volume
I think the best newsgroup for your question is sci.math.numanalysis . A recent book covering finite elements, with accompanying fortran 90 (F77 style but using automatic arrays) and Matlab code, is The Numerical Solution of Ordinary and Partial Differential Equations, 2nd Edition Granville Sewell Wiley (2005) http://www.josseybass.com/WileyCDA/WileyTitle/productCd0471735809,de... . Offtopic, but the same author has another recent book on computational linear algebra, also with accompanying Fortran and Matlab code. Computational Methods of Linear Algebra, 2nd Edition Granville Sewell Wiley (2005) http://www.wiley.com/WileyCDA/WileyTitle/productCd0471735795,descCd...

Fri, 07 Dec 2007 22:39:06 GMT 


Phony Accoun #6 / 21

Off topid: finite element vs finite volume
Quote:
> I think the best newsgroup for your question is sci.math.numanalysis . > A recent book covering finite elements, with accompanying Fortran 90 > (F77 style but using automatic arrays) and Matlab code, is > The Numerical Solution of Ordinary and Partial Differential Equations, > 2nd Edition > Granville Sewell > Wiley (2005) > http://www.josseybass.com/WileyCDA/WileyTitle/productCd0471735809,de... > . > Offtopic, but the same author has another recent book on computational > linear algebra, also with accompanying Fortran and Matlab code. > Computational Methods of Linear Algebra, 2nd Edition > Granville Sewell > Wiley (2005) > http://www.wiley.com/WileyCDA/WileyTitle/productCd0471735795,descCd...
Thanks to all the replies. I will check out the book, the suggested newsgroup, and also try to find some references on application of the methods to the fluid dynamics problems. Mirko

Sun, 09 Dec 2007 19:24:09 GMT 


Arjen Marku #7 / 21

Off topid: finite element vs finite volume
Quote:
> I think, here you identify FEM="unstructured triangular meshes", and > FV="structured, cartesian meshes". > However, nobody will hinder you to use cartesian, regular meshes in a > Finite Element setting, and profit from any resulting advantages. > (but I admit that I haven't heard of any finite volume approach that > uses triangular meshes)
Well, not so much identify them as claiming those are the most common options  at least as far as I have seen them ;) Quote: > >  Good mesh generators are available nowadays and standard packages > > to help solve the matrix equations that result. > >  Linux clusters seem to be adequate for solving practical problems > > in a reasonable time with a reasonable price tag. > > So, many of the drawbacks of FEM are being conquered. This leaves > > the cultural and historical differences, I guess. > The structured/unstructured meshes difference is still a severe one. > If you look at the publications that use unstructured grids, you > notice that the number of unknowns is usually much smaller.
You mean to say, if I understand the stuff correctly, that unstructured meshes result in more unknowns than strucutured meshes  you need consistency relations to get the same number of equations as unknowns ... Quote: > I'd like to add another point of view: > Finite volume starts from conservation laws (mass, energy, momentum), > while in Finite Element setting, conservation is an extra issue to be > dealt with. In certain settings conservation can be a vital question > (turbulence for example), which might explain the bias towards > finite volume.
Having been active in the field of water quality modelling for years, I am rather biased towards methods that obey conservation laws :). Quote: > For literature: > Gresho et. al.: "Incompressible Flow and the Finite Element Method" > might be interesting.
Going to keep that one in mind ... Oh! 1021 pages according to my library. Hm, that will have to wait until after my summer holiday then :D Regards, Arjen

Sun, 09 Dec 2007 21:49:07 GMT 


Michael Bade #8 / 21

Off topid: finite element vs finite volume
Quote:
> I do not know of a good reference for this type of question. But I do > know > something about it: >  Finite volumes tend to be less resourceconsuming both on computer > memory > and programmer agility. >  The mathematics supporting finite elements can be regarded as more > sophisticated and elegant, but the fact remains it is more involved. >  In general FEM results in matrices with no particular structure.
I think, here you identify FEM="unstructured triangular meshes", and FV="structured, cartesian meshes". However, nobody will hinder you to use cartesian, regular meshes in a Finite Element setting, and profit from any resulting advantages. (but I admit that I haven't heard of any finite volume approach that uses triangular meshes) Quote: >  Good mesh generators are available nowadays and standard packages > to help solve the matrix equations that result. >  Linux clusters seem to be adequate for solving practical problems > in a reasonable time with a reasonable price tag. > So, many of the drawbacks of FEM are being conquered. This leaves > the cultural and historical differences, I guess.
The structured/unstructured meshes difference is still a severe one. If you look at the publications that use unstructured grids, you notice that the number of unknowns is usually much smaller. I'd like to add another point of view: Finite volume starts from conservation laws (mass, energy, momentum), while in Finite Element setting, conservation is an extra issue to be dealt with. In certain settings conservation can be a vital question (turbulence for example), which might explain the bias towards finite volume. For literature: Gresho et. al.: "Incompressible Flow and the Finite Element Method" might be interesting. Regards Michael

Sun, 09 Dec 2007 20:37:18 GMT 


Gib Bogl #9 / 21

Off topid: finite element vs finite volume
Quote:
>>The structured/unstructured meshes difference is still a severe one. >>If you look at the publications that use unstructured grids, you >>notice that the number of unknowns is usually much smaller. > You mean to say, if I understand the stuff correctly, that > unstructured meshes result in more unknowns than strucutured > meshes  you need consistency relations to get the same number of > equations as unknowns ...
I wondered about the meaning of that sentence, and decided that he meant people are solving smaller problems (by node count) with FEM than they are with FV. Gib

Mon, 10 Dec 2007 13:43:40 GMT 


Arjen Marku #10 / 21

Off topid: finite element vs finite volume
Quote:
> >>The structured/unstructured meshes difference is still a severe one. > >>If you look at the publications that use unstructured grids, you > >>notice that the number of unknowns is usually much smaller. > > You mean to say, if I understand the stuff correctly, that > > unstructured meshes result in more unknowns than strucutured > > meshes  you need consistency relations to get the same number of > > equations as unknowns ... > I wondered about the meaning of that sentence, and decided that he meant > people are solving smaller problems (by node count) with FEM than they > are with FV. > Gib
Oh, that is an interpretation I had not thought of ;) I am probably biased by my (slightly feeble) attempts to understand FEM ... one publication I read about that discussed the set of unknowns extensively  but from the point of view of getting enough equations that make physical sense. Regards, Arjen

Mon, 10 Dec 2007 14:41:19 GMT 


s.c.kra.. #11 / 21

Off topid: finite element vs finite volume
Arjen Markus schreef: Quote:
> > I think, here you identify FEM="unstructured triangular meshes", and > > FV="structured, cartesian meshes". > > However, nobody will hinder you to use cartesian, regular meshes in a > > Finite Element setting, and profit from any resulting advantages. > > (but I admit that I haven't heard of any finite volume approach that > > uses triangular meshes) > Well, not so much identify them as claiming those are the most > common options  at least as far as I have seen them ;)
If you want to know someone who did use finite volume with unstructured triangular meshes, just ask your colleague (small world, right) Ivo Wenneker, he wrote his PhD thesis about it. A reason why this combination is less common is that it is harder to derive higher order methods for it. So for people who are not satisfied with first order schemes, unstructured becomes kind of synonymous with FEM. Quote: > > The structured/unstructured meshes difference is still a severe one. > > If you look at the publications that use unstructured grids, you > > notice that the number of unknowns is usually much smaller. > You mean to say, if I understand the stuff correctly, that > unstructured meshes result in more unknowns than strucutured > meshes  you need consistency relations to get the same number of > equations as unknowns ...
I think the point is, that in general it is computationally more expensive to solve the equations resulting from unstructured schemes than those from an structured approach with the same number of unknowns. Therefore with a given amount of computational power, we can apply more unknowns using structured grids. However unstructured grids have the big advantage of allowing us to put more gridpoints in the area of interest while keeping low resolution in the rest of the domain. Thus with unstructured grids it is often possible to get better results using less points than structured grids. Quote: > > I'd like to add another point of view: > > Finite volume starts from conservation laws (mass, energy, momentum), > > while in Finite Element setting, conservation is an extra issue to be > > dealt with. In certain settings conservation can be a vital question > > (turbulence for example), which might explain the bias towards > > finite volume. > Having been active in the field of water quality modelling for > years, I am rather biased towards methods that obey conservation laws > :).
Good point indeed. So there is no simple answer which is better. Depends very much on the problem: complicated boundaries for instances are better resolved with unstructured grids, grid refinement, conservation properties. As stated before history also plays a big role: conservative people stick to FDM/FVM on Cartesian meshes (sorry for the poor word play)

Mon, 10 Dec 2007 22:14:40 GMT 


glen herrmannsfeld #12 / 21

Off topid: finite element vs finite volume
(snip) Quote: > I think the point is, that in general it is computationally more > expensive to solve the equations resulting from unstructured schemes > than those from an structured approach with the same number of > unknowns. Therefore with a given amount of computational power, we can > apply more unknowns using structured grids. However unstructured grids > have the big advantage of allowing us to put more gridpoints in the > area of interest while keeping low resolution in the rest of the > domain. Thus with unstructured grids it is often possible to get better > results using less points than structured grids.
They also allow following the boundary between different materials. The first unstructured mesh program I knew is a magnet design program. Different magnetic, nonmagnetic, and current carrying regions can be described, and it will determine the magnetic field (or vector potential) in some region of space. (POISSON, and it is, I believe, still written in Fortran.)  glen

Tue, 11 Dec 2007 01:33:11 GMT 


Arjen Marku #13 / 21

Off topid: finite element vs finite volume
Quote:
> Arjen Markus schreef:
> > > I think, here you identify FEM="unstructured triangular meshes", and > > > FV="structured, cartesian meshes". > > > However, nobody will hinder you to use cartesian, regular meshes in a > > > Finite Element setting, and profit from any resulting advantages. > > > (but I admit that I haven't heard of any finite volume approach that > > > uses triangular meshes) > > Well, not so much identify them as claiming those are the most > > common options  at least as far as I have seen them ;) > If you want to know someone who did use finite volume with unstructured > triangular meshes, just ask your colleague (small world, right) Ivo > Wenneker, he wrote his PhD thesis about it. A reason why this > combination is less common is that it is harder to derive higher order > methods for it. So for people who are not satisfied with first order > schemes, unstructured becomes kind of synonymous with FEM.
Ah, I know _of_ his work, I do not know it that well. Yes, he has been promoting this kind of approach within our organisation, as you know. It is one reason I have been looking into the background of FEM (and I did indeed confuse it with unstructured meshes). Quote: > Good point indeed. So there is no simple answer which is better. > Depends very much on the problem: complicated boundaries for instances > are better resolved with unstructured grids, grid refinement, > conservation properties. As stated before history also plays a big > role: conservative people stick to FDM/FVM on Cartesian meshes (sorry > for the poor word play)
Perhaps, those people are conservative, but I also think that the physical interpretation of FDM/FVM is easier than that of FEM, which tends to be the domain of mathematicians: just look at the average text book on FEM and compare that to a text book of FDM/FVM :) Regards, Arjen

Tue, 11 Dec 2007 14:41:34 GMT 


Phony Accoun #14 / 21

Off topid: finite element vs finite volume
Quote:
> I think, here you identify FEM="unstructured triangular meshes", and > FV="structured, cartesian meshes". > However, nobody will hinder you to use cartesian, regular meshes in a > Finite Element setting, and profit from any resulting advantages. > (but I admit that I haven't heard of any finite volume approach that > uses triangular meshes)
I use CFDACE multiphysics solver for fluid coupled with chemistry problems and electromagnetic fields. It uses the finite volume with both structured and unstructured meshes. But for problems involving electrostatic component, they recommend a structured mesh. Quote: > I'd like to add another point of view: > Finite volume starts from conservation laws (mass, energy, momentum), > while in Finite Element setting, conservation is an extra issue to be > dealt with. In certain settings conservation can be a vital question > (turbulence for example), which might explain the bias towards > finite volume.
Excellent point. Quote: > For literature: > Gresho et. al.: "Incompressible Flow and the Finite Element Method" > might be interesting. > Regards > Michael
A naive question (I don't have a FEM textbook handy): Does FEM reduce to FV if we used constant elements? Thanks, Mirko

Tue, 11 Dec 2007 18:56:27 GMT 


Tim Princ #15 / 21

Off topid: finite element vs finite volume
Quote:
>>Good point indeed. So there is no simple answer which is better. >>Depends very much on the problem: complicated boundaries for instances >>are better resolved with unstructured grids, grid refinement, >>conservation properties. As stated before history also plays a big >>role: conservative people stick to FDM/FVM on Cartesian meshes (sorry >>for the poor word play) > Perhaps, those people are conservative, but I also think that the > physical interpretation of FDM/FVM is easier than that of FEM, which > tends to be the domain of mathematicians: just look at the average > text book on FEM and compare that to a text book of FDM/FVM :) > Regards, > Arjen
Someone is missing the opportunity here. Has the physical interpretation of FEM truly been lost sight of?

Tue, 11 Dec 2007 21:16:10 GMT 


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