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"Gauss linear" gradient makes OpenFOAM zerothorder accurate on unstructured meshes 

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July 20, 2017, 04:16 

#21  
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Anton Kidess
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For completeness, ESi also have their own bugtracker:
http://www.openfoam.com/code/bugreporting.php Quote:
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July 21, 2017, 07:00 

#22  
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Arjun
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For example StarCCM+ has their default scheme set to Least Square gradients and they also encourage people to use for the reasons (more accuracy) you mentioned. My own code favours least square method. Starccm is known to be more stable (even with least square method) compared to openFOAM with Gauss scheme. (I think the same about FVUS with LSQR being more stable than openfoam with green gauss). Last Guass method and its improvements are very important because not only the gradients but the whole finite volume method is dependent on Gauss integration. 

July 21, 2017, 08:00 

#23 
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Ansys (probably the largest user base by a margin) also favours Gauss scheme for general meshes.
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July 21, 2017, 09:15 

#24  
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Alexandros Syrakos
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Quote:
https://www.sharcnet.ca/Software/Ans...rivatives.html I don't know if the Gauss gradient was originally the default but they switched to the leastsquares at some point in time. 

July 21, 2017, 10:26 

#25 
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I looked it up in the Ansys AIM help, which I thought was based on Fluent and thus used the same settings (apparently I was wrong). The "Automatic" setting uses GreenGauss node based when the mesh contains tets or wedges.
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July 21, 2017, 10:45 

#26  
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Alexandros Syrakos
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The iterative Gauss method (calc_grad_iGSl.f) is based on a very simple idea (see eq. 22 of my arXiv paper): the iterative loop such as the "iterloop" loop in calc_grad_GSl is replaced by the outer iterations of the flow solver, which are performed anyway. For example, if the SIMPLE solver is used, the gradient is calculated once at the beginning of each SIMPLE outer iteration; with calc_grad_iGSl the gradient is computed in one pass (no iterations) using the gradient estimate of the previous SIMPLE iteration to account for skewness. At convergence the gradient at the new SIMPLE iteration should be the same as that of the previous SIMPLE iteration. Hence the array "grad", holding the new gradient estimate, has intent(inout) in calc_grad_iGSl (as opposed to intent(out) in calc_grad_GSl). This exploitation of the outer iteration involves a relatively small penalty, in that the outer iterations converge somewhat more slowly (see table 1 of the paper). To be honest, I have used this new scheme only for the purposes of this paper, i.e. only for solving a Poisson equation, not a flow problem. I normally use the leastsquares gradient. I've tested it only in 2D, but the subroutine should be exactly the same in 3D. The scheme you are referring to is probably the one depicted in figure 6, where indeed extension to 3D would be more involved. Last edited by alexandross; July 22, 2017 at 11:16. 

July 21, 2017, 23:19 

#27  
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Arjun
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It is not that clear cut. Least square gradients come in Fluent very very late and they have possibly not bothered to change the default. It is probable that they did not need to change the default. Do you know that their default convective scheme is first order upwind scheme, while Starccm always had Second order upwind scheme as default. It does not mean that they Fluent favours first order upwind scheme over second order upwind. Their thinking is that they provide simplest default and user specelizes the solver later. Fluent people have made presentations etc showing how least square is a better scheme. All this does not point that they favour green gauss. We can only say their stance is not known. (They don't make categorical statements like Green Gauss is more stable etc like Henry did). Also the Starccm comment was to show that people are divided over it and there is no clear cut one side to chose green gauss. (their user base is not small). 

July 21, 2017, 23:22 

#28  
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Arjun
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This is very interesting comment. Because Fluent says that node based green gauss of similar accuracy as least square gradients. I think the possible reason is stability problems of LSQR method when skew and aspect is bad. 

July 22, 2017, 11:25 

#29 
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Alexandros Syrakos
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January 8, 2018, 12:07 
Published paper

#30 
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Alexandros Syrakos
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Happy new year to everyone.
I would like to inform the readers of this thread that a revised version of our paper has now been published in: Physics of Fluids 29, 127103 (2017) http://dx.doi.org/10.1063/1.4997682 (the texts in arXiv and ResearchGate have also been updated). 

January 9, 2018, 09:16 

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Very interesting topic, thank you for bringing it to our attention. From now on, I'll only use least squares thanks to your work.


January 13, 2020, 04:52 

#32 
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Dear alexandross,
Would you please tell us whether you can release the source code used for the publication, or not? It would be very useful for the community I believe, and the OpenFOAM maintainers would love to test it.
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January 14, 2020, 07:34 

#33  
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Alexandros Syrakos
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Thank you for your interest. I used my own custom Fortran code for most of the experiments in the publication (except for those with OpenFOAM, in which I only used the existing gradient schemes "Gauss linear" and "leastSquares", so there's nothing new there). I haven't made my own code publicly available because it's not very efficient and I would like to make some improvements first. Due to my use of custom data structures, it would not be easy to just extract the gradient routines from it and use them elsewhere. However, I did upload (in post #26 of this thread) my GreenGauss subroutine and the corresponding iterative version that splits the iterations among the SIMPLE iterations (denoted \nabla^{d \infty} in the paper). Nevertheless, your question gives me the opportunity to mention that very recently we have written another paper (it is currently under review, but is available in arXiv, see below) where we propose a new family of gradient methods that are akin to the leastsquares methods but also bear some resemblance to the GreenGauss gradient, but are always at least firstorder accurate. The paper can be found at https://arxiv.org/abs/1912.08064 or in my ResearchGate page. The experiments there were performed again with my own Fortran code, but my coauthors from the company Engys work with OpenFOAM and hopefully soon the corresponding OpenFOAM routines will be made available (it is up to them though). I will post a message here when this happens, informing everyone about where they can get the routines from. Best regards, Alexandros 

Tags 
finite volume method, gauss gradient, gradient accuracy, leastsquares, unstructured grid 
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