[Kiffer]

hello, may I ask what the advantadges and disadvantadges of FEM and FVM in CFD? And why the mainstream commercial CFD codes are based on FVM? Thanks a lot!

[sci]

I read these in a book, might be helpful for you.

The Finite Volume method can accommodate any type of grid, so it is suitable for complex geometries. The grid defines only the control volume boundaries
and need not be related to a coordinate system. The method is conservative
by construction, so long as surface integrals (which represent convective and
diffusive fluxes) are the same for the control volume's sharing the boundary.

An important advantage of finite element methods is the ability to deal
with arbitrary geometries; there is an extensive literature devoted to the
construction of grids for finite element methods. The grids are easily refined; each element is simply subdivided. Finite element methods are relatively easy to analyze mathematically and can be shown to have optimality properties for certain types of equations. The principal drawback, which is shared by any method that uses unstructured grids, is that the matrices of the linearized equations are not as well structured as those for regular grids making it more difficult to find efficient solution methods.

[dbarca]

If I may add...

From my experience they provide the same quality of results.

The software COMSOL is based on FEM and solves both fluid and structure (and other stuff). By the way, here you have another advantage of FEM: if you want to solve multi-physics FEM is well suited for it.

[Rami]

Originally Posted by Kiffer

hello, may I ask what the advantadges and disadvantadges of FEM and FVM in CFD? And why the mainstream commercial CFD codes are based on FVM? Thanks a lot!

Roughly speaking, FVM is a special case of FVM where the shape functions are piecewise constant. However, usually additional assumptions are used in the FVM (e.g., treatment of flux at faces), while the FEM is usually more consistent to the used shape and test functions.

FVM is easier to formulate, and there is a large and long volume of research using it. Another reason for its larger spread is conformism of the CFD community.

There is another point. the FVM is conservative by constuction, while the FEM is not necessarily so. There is a debate how important this property is for better accuracy. To the best of my knowledge, FEM is usually more accurate for a given grid.

[cfdnewbie]

plus you can't do real good shock capturing with a globally continuous method! I guess that's the main reason why FV is the standard.

[FMDenaro]

As already stated, FV can be seen as a particula type of FEM for a specific choice of the shape function.
I would add that often the term FV is used for the discretization of the integral form of the equation as well as for the differential form in divergence form. In my opinion this abitude is not correct, FV should be uniquely addressed for the discretization of the integral form, that has a physical background expressed by the transport (Reynolds) theorem. The integral form is the only physically admissable for any case and FV are the congruent translation in the discrete sense.

[Rami]

Originally Posted by cfdnewbie

plus you can't do real good shock capturing with a globally continuous method! I guess that's the main reason why FV is the standard.

I would argue on this. The usual shock capturing tricks used for FDM and FVM are also used for FEM. As an example, the artificial viscosity concept is of common use - see, e.g., my paper and example of a shock wave in a 1D solid:

R. Ben-Zvi, M. Perl, A. Libai, A Curved Axisymmetric Shell Element for Nonlinear Dynamic Elastoplastic Problems, Part II - Implementation and Results, Computers and Structures, Vol. 42, No. 4, pp. 641-648, 1992.

A newer concept in FEM is the discontinuous Galerkin metod. A nice review, showing very nice examples for shocks and other discontinuities is

B. Cockburn and C.-W. Shu, Runge-Kutta Galerkin Methods for Convection-Dominated Problems, J of Scientific Computing, Vol 16 No. 3, pp. 173-261, 2001.

[cfdnewbie]

Originally Posted by Rami

I would argue on this. The usual shock capturing tricks used for FDM and FVM are also used for FEM. As an example, the artificial viscosity concept is of common use - see, e.g., my paper and example of a shock wave in a 1D solid:

R. Ben-Zvi, M. Perl, A. Libai, A Curved Axisymmetric Shell Element for Nonlinear Dynamic Elastoplastic Problems, Part II - Implementation and Results, Computers and Structures, Vol. 42, No. 4, pp. 641-648, 1992.

A newer concept in FEM is the discontinuous Galerkin metod. A nice review, showing very nice examples for shocks and other discontinuities is

B. Cockburn and C.-W. Shu, Runge-Kutta Galerkin Methods for Convection-Dominated Problems, J of Scientific Computing, Vol 16 No. 3, pp. 173-261, 2001.

Agreed, artifical viscosity is one way of doing shock capturing, but it is not what I would consider a good one The Shu TVB is - if I remember correctly - only a third order limiter, so that's too low for DG in my opinion...

Maybe I put it wrongly, I should have said: High order shock capturing is the problem!

Oh and in my opinion: What makes DG better for compressible flows is the fact that it is discontinuous - a property borrowed from FV, not FEM.

Thanks for posting you references up there. Is the paper available freely`?

[Rami]

I don't think the papers are free. I can e-mail you a copy of mine (if you are not afraid of shell theory...).

[leflix]

Originally Posted by Rami

Roughly speaking, FVM is a special case of FVM

FVM is a special case of FEM !!!

[cfdnewbie]

Originally Posted by Rami

I don't think the papers are free. I can e-mail you a copy of mine (if you are not afraid of shell theory...).

thanks, I will check tomorrow if we have access to the journal (should be), else I will contact you!