FEM or FVM for CFD
Hi all,
Could anyone give strong arguments in helping me to choose between a commercial FEM or a FVMCFDpackage? (I know a lot of people think one souldn't buy commercial codes, but I don't want to discuss that point here). 1) Are there physical problems that can be solved using FVM but not using FEM? For example, I heard that FEM can not be used for turbulent calculations as in FEM it is impossible to keep epsilon positive. On the other hand ANSYS provides several keturbulence models, so I am confused....... 2) And what about future developments? I think FEM showed a lot of progress recently. How will FVM and FEM continue? 3) And calculations times? Is there any indication how FEM and FVM perform in identical cases. Thanks a lot, Astrid 
Re: FEM or FVM for CFD
(1). Once it is in the commercial domain, it is not possible to know what is in the black box package regardless of the ads put out by the vendors. (2). This is because they are trying to make a living out of it. We can only say that, the buyers beware. (3). The best way to go is to define your problem first, then discuss it with the software vendor's support engineers. In this way, you can get the first hand information related to your problem. (4). The value of commercial CFD will be determined by the market itself, so it is not something we should be worry about. (5). I would suggest that you visit the vendor's webpages first and then read some of their published papers. And we don't even know whether your problem has a solution or not. (if no one has modelled your problem before, it is likely that it will take a lot of effort to find the solution. )

Re: FEM or FVM for CFD
hi astrid,
I personally have of experience in FVM. As such it is natural i suggest FVM. The reason is that FVM has very sound physical and mathematical basis bcos it is based on conservation of mass, momentum,energy. you can expect a acceptaable solution if all of these are satisfied by a flow field. FEM may not always give conservative formulation.Besides FVM has established itself so well (at least in CFD and Heat transfer), u always have people around to help you in case u r stuck.There are lot of good FVM packages around. abhijit 
Re: FEM or FVM for CFD
People tend to like something they are familiar with, especially when they have spent a lot of time on it. The theory of FEM is try to minimize the residual projection (integral) in the subspace formed by a set of weighting functions. The weighting functions are arbitrary as long as they are orthogonal. In Galerkin FEM, the weighting functions are chosen as the shape functions. Mathematically, the only difference between FEM and FVM is that the latter uses reduced weighting functions (i.e. the Delta function). And particularly, the mass is lumped onto the nodes in FVM. That is why many people regard FVM as a reduced form of FEM. It is very easy to transform a FEM program into a FVM one. But not vice versa. If you compare them on the same mesh with same solution strategy, you may find that the results by FVM might be more diffusive due to the mass lumping, but might hence be more stable. "A more accurate algorithm unfortunately might bring a more unstable result"  quote.

Re: FEM or FVM for CFD
Plus the statment that FEM is not conservative is incorrect. As long as it is properly formulated

Re: FEM or FVM for CFD
I always say that there is no definite answer to this type of question. The only answer is we should try these codes for one same problem and compare the outcome with experimental results to find out which is better.

Re: FEM or FVM for CFD
> The reason is that FVM has very sound physical and mathematical basis bcos it is based on conservation of mass, momentum,energy. you can expect a acceptaable solution if all of these are satisfied by a flow field.
Let me interject a disagreement here ... Conservations of mass, momentum and energy are only necessary conditions but way too far from being sufficient conditions for a numerical solution to the problem to exist. Neither does satisfying these conditions qualify as "sound" mathematical basis. The best you can expect of this "soundness" is in the linear limit. Why? Because, in order for most (all) numerical methods to even begin giving us any answer (right or wrong) we have to start by locally linearising the nonlinearities of the Navier Stokes equations (e.g., the convective, the radiative, etc. nonlinearity) and then solving these _new linear_ set of equations by "sound" methods. Then, when we have convergence problems or simply get garbage we wonder where we went wrong :)))  Above meant only as food for thought  PS  the mathematics for FEM are much better developed than for FVM (and I use neither, so I'm not biased) Adrin Gharakhani 
Re: FEM or FVM for CFD
Hi, All
Good to see that i started a sort of a debate. I agree with Adrin that FEM/FVM ultimately make some approximations (locally linearize) and solve the problem. Errors introdued by numerical methods is a different issue. But I beg to disagree with you that Math for FEM is better developed, Math for FVM is very well developed. In case you get your hands on the book "Numerical Methods for Conservation Laws" by Randall LeVeque u will find that Math for FVM is well developed. Thanks. No offence meant. Abhijit Tilak 
Re: FEM or FVM for CFD
These are my observations on FVM and FEM. I am not a code developer though I have written a few simple codes.
Riemann solvers (like the Godunov scheme and its variants) do not need to linearize, they are nonlinear scheme. Such solvers mostly use FVM methods (though FDM are used sometimes). It is much easier to capture (as well as numerically possible) the characteristic nature of hyperbolic equations in FVM using flux split schemes, upwinding etc. Multidimensional upwinding turns out to be a crucial component of solvers for problems with moving interfaces (e.g. volume of fluid method). I have not seen upwinding built into a FEM, so I take it that it is rare and difficult. This I think is the reason that I have not come across a FEM that captures a shock wave. These are differences in the methodology. There may be differences in the quality of numerical solutions produced by each method. It is perhaps a good idea to compare a FVM with a control volume FEM for a reasonably complex problem. 
Re: FEM or FVM for CFD
> No offence meant.
None is taken. We are discussing a CFDrelated issue and expressing our own ideas. Adrin Gharakhani 
Re: FEM or FVM for CFD
If FVM is just a reduced form of FEM and yields less accurate results, why many people (Randall J. LeVeque, E.F.Toro, K.W.Morton, etc.) try to solve PDEs(especially hyperbolic type) via FVM ? As kalyan stated above, FVM shows good results in shockcapturing. Do you think FEM is more desirable in shockcapturing? You may know the answer. Highresolution scheme is essential in aerospace, mechanical, civi engineering. FVM or FDM is combined with it. How will you solve multidimensional shocktube problem, Dambreak problem without FVM? If you get better results using FEM, you can present your paper in Journal of Computational Physics ! What I want to say is that we cannot simply compare two methods  FVM, FEM.

Re: FEM or FVM for CFD
(1). I thought the war among FEM , FDM, and FVM was long over back in 70's. (2). If one just look at the stepbystep process of each method, it is not difficult to see that they are not related to one another. (that does not mean that there is no one trying to unify these methods into one unified theory.)(3). I think, for Laplace equation, it is perfectly all right to use any one of the methods. Under the structured, rectangular grid, it is likely that the final algebraic equations for the Laplace equation will end up the same regardless of the approach used. (4). For nonlinear NavierStokes equations, one still have to show that the FEM (or a FEM) can obtain the true solution. From my point of view, this is less a problem for the FVM, and it is normally considered a nonproblem for the FDM. (5). In other words, the FDM allows you to obtain a solution to the Navier_Stokes equations, and formally force you to reduce the mesh size to reach the mesh independent solution. So, there is no escape there but to reduce the mesh systematically. (6). For the FVM, this requirement is slightly relaxed, that means, you have the luxury to use coarse mesh and still at the same time to satisfy the cell conservation laws. So, if one is satisfied at the mass or momentum conservation level, then he does not have to go all the way to use very fine meshes. This is just an illusion, because when the wall skin friction and the heat transfer become the main issue, fine meshes still must be used to obtain the accurate result of the field variable, such as velocity, and temperature variables. (7). For FVM, one just follow the procedure and obtain the solution. The solution should be acceptable regardless of the number of elements used. The reason why we have to refine the mesh comes from our experience of the real flow field solution, not from the FEM procedure. In other words, FEM does not force you to refine the mesh systematically. It is much harder to evaluate the accuracy of the FEM solution, especially when applied to the NavierStokes equations. (8). By the way, even back in 70's, I think, there were studies done by researcher using FEM to handle discontinuous solutions, through various schemes. Since I am not using FEM currently, it is hard for me to make any comments on the current state of the art of FEM. There is no question about the fact that FEM requires much more math operations than the FDM and FVM. (9). The original advantage of using the FEM, is now replaced by the systematic mesh refinement approach of the FVM, which can handle also the complex geometry through structured and unstructured meshes. The original advantage of FEM over the FDM's rigid single block structured mesh is also not as attractive, becuase FDM can also handle complex geometry through blocked structured meshes. (10). So, I think, we need to look at these issues , especially the issue of the particular equation to be solved, instead of the overall impression as to which method is better.

Errors: should read (7). For FEM,
"(7).For FVM,... " should read "(7). For FEM,..." . Sorry for the typing errors.

Re: FEM or FVM for CFD
I'm not very kind with numerical scheme, but as far as I was concerned (develloping turbulence models) it's easier to use FEM for the form of the diffusive term it give, but it is hard to deal with discontinuities with this method.
If you want to compute a compressible/transonic flow, then the FVM give more stability near a discontinuity, because of the upwinding capability of the scheme. That the reason why some people have develloped a mixing FE/FVM using cell vertex on unstructured mesh (see works from INRIA). I apologize for my poor english and hope this could help you. Sylvain 
Re: FEM or FVM for CFD
So nobody has mentioned spectral methods which is superior to all these. It appears to be a more capable for turbulence modelling etc. somebody object to this ?

Re: FEM or FVM for CFD
Dear Mr. Kang
(1) Thx for ur reply. (2) The statement about the ˇ°reduced formˇ± is really not mine, though mathematically I think it is true. Pls read ˇ°Incompressible Flow and the Finite Element Methodˇ±, Gresho, PM and Sani, RL (1998), John Wiley & Sons Ltd. I believe they have published papers in the journal u mentioned. So I think they are in the position to discuss with u. (3) CFD is not nothing but capturing shocks. Extending ˇ°the white/black cat theoryˇ±, I bet u would say, no matter what kind of method, it must be the best as long as it can capture the shocks? (4) The numerical methods we are talking about (FDM/FEM/FVM) is based on the finite and continuous theories. They may all have problems in trying to simulate the discontinuity. Pls distinguish the techniques and the mathematics. (4) Searching the ejournals, everyone can make a long list of big names in his own field. (5) Indeed, we cannot simply compare this two methods, but we are, anyway. (6) I wanted to send this massage to u personally, but could not find ur email address. Regards, Tony 
Re: FEM or FVM for CFD
Thanks for your reply.
I know and have the book you said  "Incompressible Flow and the Finite Element Method" by P.M,Gresho, R.L,Sani, M.S,Engelman. In addition, I had guessed that you cited the word "reduce form" from the book when I first read your messages because I also have read the article ( pp.1821, Chapter 1.7  Why Finite Elements? Why not Finite Volumes? ) before. You are really misunderstanding my intention. I have never said 'shock capturing ability' means the superiority of the numerical scheme. You said I mentioned as "CFD is nothing but capturing shocks." I didn't say it. I know CFD is more than shock capturing. But the fact that shock capturing is an important part in CFD is incontrovertible. The reason that I cited "shock capturing" is because you insisted as if FEM does everything what FVM does. (You said "It is very easy to transform a FEM program into a FVM one. But not vice versa") > I do not agree. (1) You said "The numerical methods we are talking about (FDM/FEM/FVM) is based on the finite and continuous theories." But unfortunately FVM is not based on continuous theories. It perfroms integration over a control volume and therefore allows discontinuous solution. It is well explained in basic text books related to hyperbolic conservation laws. You said "They may all have problems in trying to simulate the discontinuity." Yes it's right. They all have difficulties. But the degree of difficulty is diffrent. It is well know FVM is the most proper method. In combination with high resolution scheme( ENO, MUSCL or PPM reconstruction ) FVM yields nearly accurate solution near discontinuity comparing with analytic one. (2) You said "many people regard FVM as a reduced form of FEM". Who are the 'many people'?? Not many people but some people. The article you cited is just an opinion of the authors. Don't generalize the private opinions into public opinion. (3) You said "Authors of the book are in the position to discuss with me". Sure, anyone can discuss with me. They hardly mentioned the advantages of FVM in their book. They just pointed out bad things of FVM. It is biased. (4) And CFD is more than mathematics. If it takes 10yrs to execute a code, but yields perfect solution, it is useless. 
Re: FEM or FVM for CFD
Capturing turbulence accurately does indeed present a significant challenge in CFD. In light of the extended discussion we are having here, it seems pertinent to point out that schemes which can capture shocks (or any discontinuities) are usually too dissipative for capturing turbulence.
If turbulence spectrum is what you want, spectral methods indeed do better (though there ought be dealiasing). However, spectral methods often use regular meshes (axisymmetric meshes with some standard basis functions or cartesian meshes that use tensor products of onedimensional basis functions) which tend to limit their potential. This is a practical limitation. Also, finite difference approximations based on compact (6th or higher order) schemes have spectra that is almost as good as spectral approximations. Although nonlocal effects are important in turbulence, there is some strong local character in turbulence. Spectral methods use global transforms that tend to overlook the local nature. Spectral element methods seem to be the right comprimise here. One step further and you could be using wavelet based methods. It appears that they can capture strong local phenomena as well as nonlocal effects in addition to their multiresolution capability. They have also been used effectively for adaptive grid methods (see Vasilyev et al.). The claims of wavelet methods are not fully substantiated yet and I am not fully informed about the practical limitations. But, they seem promising and their use in turbulence is likely to grow. 
Re: FEM or FVM for CFD
I am agree that there is no point in justifying which method is good or bad. I am using FEM, there are some advantages and so the difficulties also. For example implementation of natural B.C. is very easy in FEM, unlike FVM we need not to deal with "ghosts"!!. Regarding upwinding, it is also very easy if we are going for TaylorGalerkin or for that matter Characteristic Galerkin method, pl. look at the recent publications of Zienkiewicz's. But I don't think, if any commercial code has all the recent features. But FEM codes are slow. In my view we must look for something, which contains best of every thing. And this is where future research must concentrate on. I am sorry, I have not included Spectral methods, elsewhere in the same discussion forum, while expressing my view on "future cfd research". Its indeed a powerful numerical tool. But it has go a long way.
GS 
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