Hi,

I've heard a few times that unstructured-grid solvers usually loose accuracy for tetra meshes, while higher order can be kept with hexa ones.

I've tested this in an unstructured grid solver I'm developing on my PhD thesis and I've found similar behaviour for both centered and MUSCL-limited Roe upwind schemes.

Would anyone have/suggest any "ultimate" reference on that matters?

I would also like to consult a few references on the development of flux schemes that are mesh independent. Any indication?

Thank you very much in advance for the help.

Best regards, Biga

> Would anyone have/suggest any "ultimate" reference on that matters?

A Taylor series expansion found in most maths book.

> I would also like to consult a few references on the development of flux schemes that are mesh independent.

I am not sure what you asking for here. If a scheme needs a mesh then it cannot be mesh independent.

A reconstruction procedure based on a wide computational cell is effective even on highly distorted and poor connectivity grids, which is the norm rather than an exception in unstructured grid computations. Some of the issues in 2-d are dealt in the reference :

P. Jawahar, H. Kamath, ``A High-Resolution Procedure for Euler and Navier-Stokes Computations on Unstructured Grids'', Journal of Computational Physics, Vol. 164, No. 1, pp. 165-203, 2000

Ok, thanks for the reference!

Look what I've done in the code I'm working at. I have both 2nd-order centered (Jameson) and upwind (Roe with MUSCL reconstruction) schemes.

I drive a simulation solution to a known analytical one and I compare the error between numerical and analytical results. I repeat that for successively refined grids. The error should decrease as Dx^2 for a 2nd order scheme.

For hexa meshes I get that 2nd order. For tetra meshes, however, I get lower than 1st order for the centered scheme and I get exactly first order with the upwind scheme. The difference is ONLY in the grid.

Finally, when you mention "wide computational cell", what do you exactly mean? Is it that concept of an expanded control volume where you compute the gradient, likely in the work of Barth and Jespersen. They use an expanded control volume to get their gradient instead of getting it directly from the cell itself.

Best regards, Biga

Yes, I was referring to an expanded control volume for computing the gradient apart from employing a multi-dimensional limiter based on gradients; the details can be found in the reference mentioned earlier. The L_1 and L_2 errors do exhibit second order accuracy but L_inf error(a measure of local error) is around 1.5. The limiter is quite crucial in obtaining second-order accuracy without compromising on convergence(which is a problem with Barth and Jesperson limiter).

Does their order of accuracy apply to any cell type (hexa, prism, tetra)?

I have avoided the convergence stall of Barth and Jespersen by replacing their limiter by the continuous van Albada one, still within their gradient ratio computation framework.

I'm, however, sticking with the local gradient computation instead of the expanded one. I may try that...

>For hexa meshes I get that 2nd order. For tetra meshes, >however, I get lower than 1st order for the centered scheme >and I get exactly first order with the upwind scheme. The >difference is ONLY in the grid.

Let's take a closer look at this. I would say it is impossible that the difference is ONLY in the grid. You wouldn't possibly use the exact same algorithm on structured hexa grids and tetra grids. For example: How do you obtain the second and fourth order differences for the artificial dissipation of the JST scheme? This will surely be done in different ways on structured or unstructured grids. It seems like your unstructured solver simply isn't of second order. You shouldn't merely blame that on the grid. If the scheme is implemented correctly this shouldn't happen. The grid itself doesn't change the order of accuracy. It's a combination of grid and scheme that makes the difference.There is no reason why you shouldn't be able to get a second order accurate solution on tetra grids, and I think you know that it has been done. Just don't get hung up on the grid when there is probably something wrong with your scheme.

Ok, Mani.

I never mentioned I'm comparing a structured solver with an unstructured one.

I have this unstructured code which is capable of treating hexa, prism, tetra, pyramid cells, or a combination of those. The mesh, no matter it's composed of only hexas, or tetras, or whatever, is ALWAYS UNstructured.

Therefore, the computation of gradients (for example, the Laplacian and bi-harmonic terms of the JST artificial dissipation) has EXACTLY the same algorithm no matter the cell type.

When I solve that testbed case with hexa, I get 2nd order. When I solve with tetra, I get 1st.

I guess that this behaviour is already known, and it precluded the quest for the current multidimensional upwind schemes.

I am not sure I should mention this again but a Taylor expansion really will show you what is going on and why terms which cancel with quads do not cancel with tets. It will also indicate how to go about getting quad like behaviour with long thin tets in boundary layers, how and where to calculate grid metrics to get various quantities to cancel, etc...

Humm, now I understand what you mean. Ok! But I have no idea how to apply the expansion for different cell types... you have any hint for that?

Thanx!

Of course, on a smooth hexa grid it won't matter if the solver is unstructured or structured. My point is, to get second-order accuracy on tetra grids, the algorithm will not be quite as straight forward. For example, while on a hexa grid it is good enough to consider uniform fluxes over a cell face, you may have to use a higher order (at least linear) distribution in the case of a tetra grid. Similarly you have to take special care to make those second- and fourth- order differences accurate. For a smooth hexa grid, a "bad" unstructured algorithm may still reduce to acceptable accuracy, but may not work for the higher demand of irregular grids. Maybe the following reference will give an overview over various methods that have been applied for second-order accuracy on unstructured grids. It's not even up to date, but should be good enough for a start. I am sure that this is still a hot research area, but that doesn't mean there haven't been any second-order methods for tetra grids (otherwise, unstructured grids would be pretty much useless, because 1st order accuracy is simply not acceptable). If your study should find that all the second-order methods of other researchers are indeed only of first order, that should make for a very interesting publication!

D. J. Mavriplis, "Unstructured Grid Techniques", Annual Review of Fluid Mechanics, Vol. 29, 1997

That's nice.

And does Mavriplis' reference bring all that into an implementation focus as in your reply. If not, would you have any other hints/references for that?

Say, you recommend using at least a linear property distribution for a tetra. Would you use that for the computation of the 2nd/4th differences of the centered artificial dissipation scheme? That's not what Jameson does, for example. He uses constant property, right? If I should use linear even for this scheme, that's pretty interesting! Would I need to use reconstruction, like in a MUSCL context (what I already do for the CUSP and ROE upwind scheme, and still get lower order...).

Besides, what do you mean by "uniform fluxes over a cell face"? Is it the same as constant property along the cell? And linear for linear reconstruction likely in the MUSCL method?

And what would be a "bad unstructured algorithm"? Is it the scheme or the implementation of that scheme?

Thanx!

For implementation purposes I think Blazek's book is quite nice. It will answer most of your questions:

J. Blazek, "Computational Fluid Dynamics: Principles and Applications", Elsevier, 2001