# Why is the FVM traditionally used in CFD, and FEM in computational structures?

Most CFD codes use FVM. Most computational structures codes use FEM. Why is the FEM not frequently used in CFD, and why is FVM not frequently used in FEM?

• There is a lot behind this question, do you have a particular question about the methods and comparisons? Otherwise a broad answer would say that FVM tends to be better for fluid flow (conservation properties and large deformations) while FEM tends to be more flexible for small deformations and matching structures. That being said both can be used for the other given appropriate approaches. – Kyle Mandli Mar 6 '19 at 17:51

The finite element method is actually quite widely used in fluid flow problems, for example for the Stokes and Navier-Stokes equations.

The delineation between the methods is more along the following line:

• The finite element method is very well suited for second order (in space) differential equations. That has something to do with the fact that trial and test spaces can be chosen the same in that case, and consequently the FEM is a tool that is well suited to problems such as elasticity and more generally structures, but also for the Stokes equation or the Navier-Stokes equations in fluids.

• Many other fluid flow problems are described by first order differential equations (think, for example, the Euler equations of gas dynamics) or equations in which the first order terms are dominant (e.g., for the compressible Navier-Stokes equations in the transonic regime). For these cases, solutions are often discontinuous -- for example in the form of shocks -- and the Galerkin method (on which the FEM is based) produces oscillatory approximations that are not useful in practice. One can fix this through stabilization techniques developed over the past twenty or so years, but at least historically the preferred approach is to base methods on the conservation properties that underlie the equations -- i.e., to use finite volume methods.

• Yeah I've became aware that FEM is also used in CFD. I would say that I see FEM used more in CFD than I see FVM used in elasticity. However, I have seen quite a few research papers using FVM in elasticity dating back to 1990s (mostly by university researchers in Europe). What issues are there attributed to FVM for structures? My guess would be that FVM doesn't naturally allow for specifying boundary conditions at nodal points like in FEM, so this could pose some challenges. But FVM would seem to be easier to implement than traditional FE. – structuralengineer Mar 14 '19 at 19:14
• If you rewrite second order equations (like the elasticity equation) as a first order system, it is not unreasonable to apply the FVM to it. – Wolfgang Bangerth Mar 14 '19 at 20:36
• Do you mean the transient form or the steady form, i.e., $\nabla \cdot \sigma = 0$? – structuralengineer Mar 14 '19 at 21:03
• Yes. $-\nabla\cdot\sigma = f$ and $C^{-1} \sigma - \nabla u = 0$. You can substitute the second into the first equation to get the usual second order differential equation of elasticity, but you don't have to and keep it as a system of first order equations. – Wolfgang Bangerth Mar 15 '19 at 1:38
• Got it! Yeah I've seen that formulation used with FVM. But I was curious about how FVM would work if you wanted to specify a boundary condition at a node? In typical FVM you typically apply boundary conditions at faces and/or at volumetric ghost cells. – structuralengineer Mar 15 '19 at 1:44

The most fundamental reason is that (static) structural problems often resolve themselves into finding minimum energy configurations and this is easily translated into minimizing the energy of a FE representation. This almost never happens for fluid flow, even if is steady, because of the convective terms. There are by now many FE codes for fluid flow (such as the discontinuous Galerkin method) that have their enthusiasts, but for most engineering purposes they are regarded as too expensive.

• Pasting a comment above: I have seen quite a few research papers using FVM in elasticity dating back to 1990s (mostly by university researchers in Europe). What issues are there attributed to FVM for structures? My guess would be that FVM doesn't naturally allow for specifying boundary conditions at nodal points like in FEM, so this could pose some challenges. But FVM would seem to be easier to implement than traditional FE. – structuralengineer Mar 14 '19 at 19:15

The most obvious advantage of FVM is that the conservation law(the flux is canceled between the interface of two neighboring cells) is satisfied automatically during the discretization process, no other special mathematical treatment needed. This property is very important for the fluid flow problem to satisfy the mass continuity equation.

• Zienkiewicz and Taylor wrote a remark to that effect in their introduction of the book "The Finite Element Method". – Dohn Joe Mar 8 '19 at 12:38