A question to the researchers out there...

I was discussing things with a Senior Lecturer in my post-grad classes. He is expert in the FEA field. The topic debated was that FEA is a more modern approach than FV, which has its roots strongly in the Finite Difference (old) technique...

This discussion follows on from an earlier posting of mine regarding the use of the matrix-building techniques of FEA, compared to the lumped 'source & non-linearity' approach.

Any ideas?

Regards, diaw...

Yes. Tell him to buy a book. It would be fairer to say that, recognising the failures of the standard Galerkin FEM formulation due to conservation problems, researchers in the FEM have chosen the path of Discontinuous Galerkin, which is simply a higher order (re)formulation of the FVM.

As for the other details, treatment of non-linearities etc, I really recommend some literature because a formu is not really suited for this level of technical detail.

Oh, by the way, I've got both the FVM and FEM experience

Hi 'Hrvoje Jasak',

Your name rings a bell...

What books could you recommend which could do justice to the positive & negative aspects of both approaches? Feel free to email me directly, if you would like.

Thanks...

Regards, diaw...

If one uses divergence-free elements, there is no conservation of mass problem. The formulation of incompressible flow problems is then very simple. With some discontinuous Galerkin methods such as O. Karakashian, the discontinuities on the element level can be solved by inspection yielding divergence-free elements. These divergence-free elements can be written as the curl of stream function or vector potential elements. Using Hermite-type divergence-free elements where the degrees-of-freedom are the stream function and the velocity field components, one uses do-nothing boundary conditions at the outflow (and even the inflow) so arguments about pressure b.c. are mute. For outflow such as the backwards-facing step, the mesh can be cut off in the middle of the recirculation region without any change in the visual flow pattern. The approximations are invariant under affine transformations. Invertible transformations can be generalized to arbitrary convex shapes without destroying the solenoidal properties of the elements. Highly curved elements with high aspect ratios seem stable, but my experience in this area is limited to Poiseulle flow.

Yes, I know that one, but this is really a re-formulation of the equation from pressure-velocity to vorticity formulation hidden in a shape function.

As an example of my point, consider a mildly compressible flow (call it variable density on a low Mach number): something like a compression stroke in an internal combustion engine. Mass conservation and conservative fluxes are still essential, the fluid density varies (otherwise it cannot be compressed) but the vorticity formulation (to my knowledge) cannot do this.

Hi Jonas,

Thanks for your kind contribution to the discussion. Could you point me towards specific literature & recent Research Work in that field?

Thanks very much.

Regards, diaw...

How about more modern variants, such as SUPG, least-square Galerkin and the Characteristics-Based FEM? They seem to work much better than the "standard Galerkin" (Bubnov-Galerkin, isn't it?).

What I am refering to is NOT related to any vorticity formulation any more than the Navier-Stokes equation is. If the velocity is divergence-free, it can be written as the curl of a stream function psi or of a vector potential. Substitute curl psi for u in the Navier-Stokes equation and you have a third order differential equation for psi. At this point, the Navier-Stokes equation can be identified as part of a mixed method, where one is using separate approximations for psi and its curl, u. But that is not the point. If on the left of the equation we have d(u)/dt or d(curl psi)/dt which is divergence-free, then by the Helmholtz decomposition, which states the orthogonality of solenoidal and irrotational functions, the right side must also be divergence-free. On the right, one collects the solenoidal and the rotational parts. Solenoidal parts include non-conservative body forces and part of the convection term. The irrotational part includes the pressure gradient, conservative parts of any body force (such as gravity) and part of the convection term. The irrotational part must itself be zero, and this gives rise to an equation for pressure gradient that is similar to the pressure Poisson equation except that it doesn't have a Laplacian operating on the pressure. The remaining solenoidal part is a pressureless governing equation for the velocity. This is known as the Leray decomposition. The velocity equation looks like a Burgers equation except that one must must use only the solenoidal part of the right side. The projection is done by a Greens function, but this leads to an integro-differential equation. Pressure projection methods are an attempt to do the projection a different way. If one uses divergence-free test functions, one finds a weak formulation which doesn't require projection operators because of the orthogonality. But weak formulations are the basis of the Galerkin finite element formulation. For this reason, the Galerkin finite element method is naturally suited to solving incompressible flow. The problems with the pressure and incompressibility have been removed and one needs only deal with the nonlinearity associated with the convection term.

The nature of the problem changes in the incompressibile limit, so it is difficult to make arguments based on the compressible regime.

Just as in the invisid limit where the boundary conditions change, in the incompressible limit the boundary conditions change. The new boundary conditions include the stream function (replacing the pressure), reflecting the fact that we are dealing with a third order equation for the stream function. One CAN use pressure boundary conditions to influence the flow, but then the pressure is acting as a control parameter.

Thanks Jonas,

That's getting me back to my books somewhat... excellent discussion... thank you...

diaw...