Off topid: finite element vs finite volume 
Author Message
 Off topid: finite element vs finite volume

Hello,

I apologize for this off-topic 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  
 Off topid: finite element vs finite volume

Quote:

> Hello,

> I apologize for this off-topic 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 resource-consuming 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  
 Off topid: finite element vs finite volume
Quote:


>>Hello,

>>I apologize for this off-topic 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 resource-consuming 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  
 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  
 Off topid: finite element vs finite volume
I think the best newsgroup for your question is sci.math.num-analysis .

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/productCd-0471735809,de...
.

Off-topic, 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/productCd-0471735795,descCd-...



Fri, 07 Dec 2007 22:39:06 GMT  
 Off topid: finite element vs finite volume
Quote:

> I think the best newsgroup for your question is sci.math.num-analysis .

> 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/productCd-0471735809,de...
> .

> Off-topic, 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/productCd-0471735795,descCd-...

Thanks to all the replies.  I will check out the book, the suggested
news-group, 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  
 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  
 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 resource-consuming 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  
 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  
 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  
 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  
 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, non-magnetic, 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  
 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  
 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 CFD-ACE multi-physics solver for fluid coupled with chemistry
problems and electro-magnetic fields.  It uses the finite volume with
both structured and un-structured 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  
 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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