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April 24, 2012, 10:24 
node based gradient

#1 
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lalupp
Join Date: Jul 2010
Location: India
Posts: 38
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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 

April 24, 2012, 12:09 

#2 
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Chris DeGroot
Join Date: Nov 2011
Location: Canada
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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?


April 25, 2012, 09:42 
node based gradient

#3 
Member
lalupp
Join Date: Jul 2010
Location: India
Posts: 38
Rep Power: 8 
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 

April 25, 2012, 10:51 

#4 
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Chris DeGroot
Join Date: Nov 2011
Location: Canada
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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 cellcentred Gaussbased 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 higherorder cellbased Gaussbased reconstructions? The biggest problem with the typical Gaussbased methods is that they are not formally secondorder accurate on nonorthogonal 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 cellcentered finitevolume discretizations on unstructured grids," Int. J. Numerical Methods in Fluids, 62(9), 945962, 2009. There are also some leastsquares 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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