Hello all, I am wondering what are the differences between FEM and FVM. They are two different methods and FEM as far as I now, is good for structural problems and FVM is good for fluid flow problems. Is that true, if yes or no why? In addition what are the advantages of one to other in terms of solution. Thanking you in advance

(1). In structure problems, the finite-element method is very popular. (2). In CFD, the finite-volume method is now very popular. (3). But the finite-element method has been used in CFD since 70's. (4). Most commercial CFD codes use finite-volume method. (5). Some commercial CFD codes use finite-element method. (6). You will have to run both codes on the same problem to find out the pros-and-cons of the methods. (7). There have been a few discussions and messages posted here before. Why not dig out thoese first?

I am happened to be the person who knows a little about both solid and fluid. I would like to share with everyone.

In solid mechanics, Lagrangian configuration is widely used, and FEM is the most popular numerical method. I did numerical simulation of metal forming process before. I think there are 2 most difficult things which still remain today:

one is the incompressibility, which results in the singularity of the linear system. The widely used technique is the using of the reduced integration or the selective integration. But I think this technique only alleviates the problem, didn't solve the problem. For large deformation, pressure mode will certainly appear. Another way is to use the nonconforming element. I read few papers about this technique. (For fluid mechanics, there are tons of papers about this.)

Another is the contact problem. Even with the simple Coulumb(?) assumption: f = u N, all the existing softwares still have big problems. The most widely used technique is Augmented Lagrangian Method. But this method still has some big problems there (No papers discussed them, only we did code implementation suffer such bitter feelings). Additionally, such technique requires the contact surface has continuity of C2 (second-order derivative smooth). Too bad. All these are the truth I am telling you. No paper will tell you such things. All the paper published on the famous journals are bragging how wonderful their methods are.

Now I am doing 3D fluid simulations with low Reynolds number. Also the incompressible problem is the grand challange. No conforming element can pass due to LBB conditions. But with nonconforming element, with few elements, error could be very large. I tried both FVM and FEM. They are almost the same. The major difference is that FVM is easier to code, and easier to understand. In mathematical opinion, these 2 methods could be considered as 2 similar least square methods. Only the criteria are different. (If you use linear conforming element, these two methods will give you the exact linear system. And none of them will produce results since incompressible problem.) Several papers addressed bubble function to be added for FEM method, but few papers or no paper addressed bubble function to be added for FVM method. I tried using same bubble function of FEM on FVM, the results are not very good. Many people like FVM method, since using FVM method, SIMPLE like or Pressure Schur complement method are easy to enforced, which split the pressure and velocity, thus breaking the whole linear system into 4 sub-linear system.

Essentially speaking, solid mechanics and fluid mechanics (with low Reynolds number) are very similar. The most difficult thing is to solve the huge linear system efficiently. Since in 3D, all the stiffness matris are unsymmetric, so currently CGStab and GMRES + multigrid are viewed as the most efficient way to solve 3D problems.

All these are my personal opinion. If you have novel ideas, please don't hesitate to tell us.

in truth their is no theoretical difference between the two and finite volume technique can be shown to be a special case of FEM. indeed you'll find that the terminology used is similar and the schemes are equivalent whether the flow is compressible or incompressible.

I am new comer in CFD field.

How 's the finite volume technique can be shown to be a special case of FEM?

Thank you.

if you remember you FEM theory: if D(w,x,t)=0 is a differential equation for w(x,t) then the weak form of the PDE is 'integral over volume of v(x,t)*D(w,x,t) dV = 0' where v(x,t) are unrestricted weight functions. the FE method approximates the differential operator D(w,x,t) by substituting an approximation wn(x,t) = o(x)q(t) where o(x) are shape functions.

the shape functions o(x) are restricted while v(x,t) are unrestricted. but we can trade reducing restrictions in o(x) for increasing restrictions on v(x,t). example the Galerkin formulation lets o(x) = v(x). in any case if the weight functions are appropriately selected the form is equivalent to the finite volume formulation. i don't know what choice of weight functions are required (i think they might be constants with x) because i haven't learned FE for fluids just structural dynamics. but the process is similar. the fact that the two methods are equivalent is useful because much of the mathematical knowledge base about finite elements (which is considerable) can then be applied to finite volumes

I am no expert, but my understanding is that FEM is best used for structural problems and FVM is best for fluid and thermal problems. FEM was originally derived from physical principles using energy methods. Later, practitioners noticed that the derivations can also be made using the calculus of variations. They then used this as a springboard to solve other problems using the calculus of variations, including thermal and fluid problems. However, when solving these problems using FEM, the global conservation equations are not necessarily conserved. I personally find this to be a major drawback. I once solved a conduction problem and found that the total heat flux into the system was significantly different than the heat flux out of the system (it was more than round-off error could account for and there were no internal heat sources). I'll admit that it could have been a bug in the solver. The sum of external forces and reactions will always equal zero using FEM on a steady structural problem (within roundoff). I remember my professor proving it in a FEM class I once took, but I forget the reasoning. Similarly, the conservation equations will be satisfied over the entire domain when using FVM since the conservation equations are satisfied for each cell when the equations are derived. I believe this is one of the reasons why FVM has become the method of choice in CFD.

Dear Collegues

In general both methods are efficient and robust, being capable to solve a wide class of partial differential equations. Several publications already reported some comparative studies among those methods. The general conclusions is always the same: both present a good acurate and robustness. An important subject is WHI do the GIGBOYS of CFD packages such as FLUENT adopted the FVM? In general the codes that use FVM are faster than those that use FEM. This is truth due to two reasons (1) in the FEM code it is necessary the numeric integration in the gauss points for all the mesh elements, what requests a considerable processing effort, (2) in the FVM code the discretized equation is implemented directly and, still, the system of equation to be solved presents fixed bands, being solved more quickly. Reminding that the "bottle neck" of the most numerical methods is usually the solver.

A great deficiency that FVM presented was the mesh generation for complex geometries, however in the last decade, with the birth of MVF in triangular meshes, there is no more problem related with mesh generation.

I take advantage of the opportunity to congratulate this discutition forum. Today my research subject is 3D FVM in tetraedral meshes for conjugated heat transmition and fluid flwo.

You'll find JUST such a comparison in my book, "INCOMPRESSIBLE FLOW AND THE FINITE ELEMENT METHOD" By P.M. Gresho and R.L.Sani, John Wiley & Sons, 1999; with a new paperback edition to be availabe in 1-2 months.