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Old   April 24, 2012, 10:24
Default node based gradient
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lalupp
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Hi All,
In unstructured mesh strategy it is generally agreed that node based gradient gives better results than the cell centered approach(green gauss). I assume that gradients are due to spatial change in th solution or attributed to high mesh skewness or mesh grading. But is there any gradient smoothing method available which considers all these factors. In my case it produces abnormal gradient for highly skewed cells

Any suggestions are most welcome
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Old   April 24, 2012, 12:09
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Chris DeGroot
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I think you need to clarify your question more. Yes, it is true that abnormal gradients may be computed for poor quality cells. All you can do is try to maintain a good quality mesh and ensure you are using an accurate gradient reconstruction method. What method are you using? Is your code cell centred or nodal?
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Old   April 25, 2012, 09:42
Default node based gradient
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lalupp
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Thanks Chris for your kind reply

my code is cell based but for gradient estimation I used node based approach. I am using green gauss method . My case is 2d. For flux reconstruction at the face(edge) I first estimated the number of nodes surrounding a particular node in that edge and found distance weighted average. For a simple case of assumed solution distribution like phi = 6x+3 (rectangular domain) i.e d(phi)/dx =6 . But it gives abnormal gradients in highly skewed cells (value ranges from 4 to 7) . Is there any averaging method available which considers the skewness or geometric properties like area or something else other than simple distance weighing in node based formulation .

Second thing is that it is always not possible to produce high quality meshes. Assuming the mesh is bad ,question is what can be done to improve the solution. (like the reconstruction at faces ( or edges in 2d) for gradient estimation )

regards
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Old   April 25, 2012, 10:51
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A very interesting problem indeed. Unfortunately I don't know of any alternative averaging methods. I have never seen quite this type of gradient reconstruction attempted actually. I am more familiar with pure cell-centred Gauss-based methods.

You are probably right that the problem comes from the simple distance weighting. I had a thought that you could try using the current gradient estimates to extrapolate to the node and average the extrapolations. This would then become iterative as the gradients would change with each iteration. Just one idea.

Alternatively, have you looked into any higher-order cell-based Gauss-based reconstructions? The biggest problem with the typical Gauss-based methods is that they are not formally second-order accurate on non-orthogonal grids. The one that I use in my 3D unstructured code is:

Betchen, L. J., Straatman, A. G., "An accurate gradient and Hessian reconstruction method for cell-centered finite-volume discretizations on unstructured grids," Int. J. Numerical Methods in Fluids, 62(9), 945-962, 2009.

There are also some least-squares based reconstructions that appear to work well (referenced in the paper above).

In terms of the mesh quality, you are right that sometimes it is inevitably bad. But all codes will have a limit to how bad the mesh can be. I suppose my point is to first see if you can improve your mesh. You didn't really say how bad we are talking, so assuming it isn't horrible then you should work on your algorithm.
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