Length scale for CFL in unstructured grids
 

Hi.

How do you calculate the length scale for CFL in unstructured grids? Since the grid is not always aligned with the flow, I'm using the square root of the control volume area - in 2D. Is that correct?

TIA

Re: Length scale for CFL in unstructured grids
 

Hi Marcio,

I'm not sure for CVFEM formulation but I think there isn't a more or less correct way for computing it.

I've already implemented a 2D FEM incompressible flow program using:

he=sqrt(2*Ae), where he is the element "length" and Ae is the area of the element. This element size is based on the side of the square with 2 times the area of the triangle.

but I´ve also tested other approaches, for example:

he=sqrt(4*Ae/pi), element size based on the diameter of the circle with the same area.

and

he = max(e_1, e_2, e_3), where e_i are the length of the edges of the element (3 for triangles, of course)

Besides the CFL number, I use the element size to compute the stabilization terms of the formulation (SUPG and PSPG) and all these approaches produced similar results.

Nowadays, I´ve been working in a 3D solver based on the same formulation (stabilized FEM) and using the element length based on the diameter of the sphere with the same volume of the element but, as I've cited before for the 2D case, there are also other ways of computing it (cubic square of 6 times the volume for tetrahedra for example).

I know that I haven't helped you but it is always a good subject to discuss because I think that the research community has not found the best way of computing it yet.

Thus, for the time step, sometimes I do a roughly approximation taking the minor element size from the mesh and computing the time step for the desired CFL.

Regards

Renato.

Re: Length scale for CFL in unstructured grids
 

Dear Marcio,

There is no unique way of computing a length scale in unstructured meshes, and sqrt(A) is one way of doing it. Although, other alternatives are possible, the present one which you are using does give a reasonable measure, except in certain cases of heavy skewness etc..., so I presume that you can proceed with the same .

Regards,

Ganesh

Re: Length scale for CFL in unstructured grids
 

Apply the finite volume scheme to a linear convection/diffusion equation and do a stability analysis to get the CFL condition. Then generalize this to your case by replacing the wave-speed of the linear equation with the eigenvalues of the system case. The resulting formula will turn out to be costly and people usually make approximations. Any representative measure of the element size like side of smallest length, smallest altitude, etc. will work but may not give the least restrictive condition. See Blazek for some discussions on this topic.

Re: Length scale for CFL in unstructured grids
 

Thank you all for your answers. I've tested this scheme with CVFEM this weekend and it worked. The difference from the usual finite element scheme is that here I use the control volume area instead of the element area - so the length scale is roughly 1/3 of the traditional FEM for a 9-node finite element. Also, for skewed elements I'm going to test the smallest edge as the length scale, but I've had no problem with it when solving a laminar channel (Re = 500) flow with streamwise-skewed elements with a aspect ratio = 2 and 3.

Regards

Márcio Ricardo