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August 7, 2001, 10:02 
checkerboard in FEM

#1 
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I am using a finite element code to solve a thin film problem. The result appears to be repeating bands of high and low pressure. I suspect a checkerboarding instability. The griding options are tet meshes and a 27 noded brick elements both of which result in the same problem. Any suggestions? Thanks


August 7, 2001, 14:05 
Re: checkerboard in FEM

#2 
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(1). From the computed result, calculate the average pressure (between neighboring hi and low points) distribution. (2). Find out whether this average pressure distribution is "useful" or not. (3). If yes, use the average pressure. If no, ask the author of the code to remove the oscillations.


August 7, 2001, 15:51 
Re: checkerboard in FEM

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NS equations are nonlinear. The filtered pressure (average of two adjacent) points makes no sense beyond the first iteration (not time step).
You have to have a compatible bases for the pressure and velocity fields to remove the oscillations. There is a criterion which determines the compatibilty and I can not remember it off hand since I do not work in FEM. You should be able to find it in any CFD book with FEM methods. 

August 7, 2001, 16:19 
Re: checkerboard in FEM

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You have to have a compatible bases for the pressure and velocity fields to remove the oscillations. <blockquote>
Absolutely, although some Galerkin Least Squares formulations advertise somewhat less restrictive conditions on the velocity and pressure spaces than what a pure Galerkin weighted residual formulation would require. At the least, insure your pressure is interpolated with one order less than your velocity. </blockquote> There is a criterion which determines the compatibilty and I can not remember it off hand since I do not work in FEM. You should be able to find it in any CFD book with FEM methods. <blockquote>That would be the LBB condition or the "infsup" condition.</blockquote> 

August 7, 2001, 17:03 
Re: checkerboard in FEM

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(1). The pressure averaging was actually used get get useful pressure information from a 3D code which was producing checkerboard solution. It worked in my case. (2). If you don't like it, you have a bigger problem to solve.


August 8, 2001, 08:13 
Re: checkerboard in FEM

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1) The spaces of approximation for the velocity and the pressure must be different.
2) A suggestion: Linear interpolation for all components of the velocity. Constant approximation by element for the pressure. 3) Another suggestion: Turek elements for the velocity (dof on the edges see: http://www.mathematik.unidortmund.d...featflow/ture/ ). And the pressure constant on all elements. 

August 8, 2001, 11:58 
Re: checkerboard in FEM

#7 
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The pressure is always troublesome. Smoothing indeed gives reasonable pressure, sometimes, as John mentioned. The staggered or collocated grids in the FDM or the least squares in FEM are actually averaging the pressure. However, in the projection FEM, you do can use equalorder approximations for v and p. Have a look at the recent paper by R. Codina (JCP 170, 112140, 2001).


August 8, 2001, 12:57 
Re: checkerboard in FEM

#8 
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"The staggered ... grids in the FDM ... are actually averaging the pressure."
Using a staggered grid, the pressure in the incompressible case appears only as a difference from cell center to cell center in the momentum equations. If the pressure were averaged, wouldn't it appear as a weighted sum of the values at cell centers? In the case of colocated variables, it would seem that averaging is used. I have no experience with collocation of variables. 

August 8, 2001, 13:14 
Re: checkerboard in FEM

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(1). I must say that the "pressure average" was my invention, and it was carried out only on the converged solution with wiggles in it. (2). I was forced to do so, and I was able to use the solution with wiggles in a positive way. (in other words, it's not random solution) (3). So, the suggestion was, if you have a converged solution with wiggles, you can find the average pressure distribution (between neighboring points) and check if it will give you some consistent information. (4). To eliminate the wiggles or oscillations, you will have to go all the way back to the algorithm and formulation.


August 8, 2001, 15:14 
Re: checkerboard in FEM

#10 
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Jim is right. The collocation methods are the ones that use averaging (perhaps biased interpolation is a better word). You do not need any averaging in staggered schemes.
You can use the pressure averaged over two adjacent points and then compute the pressure gradient in the momentum equation in collocated schemes. However, the pressure would only be first order accurate in space which can actually reduce the formal spatial accuracy of the whole scheme. Note that some claim pressure update can be first order accurate even though the velocity field evolves with second order accuracy in some schemes. They are talking about time accuracy and has nothing to spatial accuracy discussed above. If your converged solution with wiggles is "useful", I must say that is sheer luck. Usually it makes no sense. Anyway all these averaging ideas make sense only on structured mesh where the oscillation have a chessboard type pattern. How do you filter oscillations from a triangular or mixed triangular/rectangular mesh which is quite likely the case since the original question was regarding FEM. 

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