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March 3, 2000, 06:26 
FVM: vertexcentered, cellcentered, or cellvertex?

#1 
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Hello,
For finitevolume formulation (for Euler!) on unstructured triangular meshes, there are "perhaps" mainly three different approaches. (1)Vertexcentered (2)Cellcentered (3)Cellvertex Which approach is the most popular one in realworld applications? And is there any specific reason for that? (or is it just a matter of personal preference?) I've just finished writing my own cellcentered FV code which I found the easiest to write. And I'm curious about other approaches. Thank you very much, Nishikawa 

March 3, 2000, 07:23 
Re: FVM: vertexcentered, cellcentered, or cellvertex?

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Between cell based and vertex based schemes, the main difference is in the storage requirements. Typically, the number of vertices in a tetrahedral mesh is 1/6th the number of cells, so storage decreases by that much. However, the counter argument is that you have 6 times more resolution with cell centered schemes. so you can argue both ways.
I have some experience with vertex based schemes. The problem there are the viscous fluxes. Using edge data structures and vertex storage obtaining mesh transparency for viscous fluxes is ellusive. THis shows up really badly for turbulent flow calculations using twoequation models. I really have not seen any two equation model calculations using cell based schemes. if you or anybody else out thre is aware of such methods do let me know! thanks srinivasan 

March 3, 2000, 11:48 
Re: FVM: vertexcentered, cellcentered, or cellvertex?

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Could you explain a bit more about the second part?
Vertexbased method is not good for NavierStokes? Twoequation models are hard to implement for cellbased method? Thank you, Nishikawa 

March 3, 2000, 13:42 
Re: FVM: vertexcentered, cellcentered, or cellvertex?

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Finally I found someone wich have interest in work with that. Recently I made a big review about this subject. Today, there are reserchers working with both cellcentered and vertexcentered schemes, but the number of publications that use the cellvertex schemes directly are smaller. I finish to implement the convectivedifusive unsteady equation with both of those methodologies. Whit this work I could verify that the cellcentered aproach is more acurated and also with the finite element method. I sugest you read the review of Frink, N.T. (Upwind Scheme for Solving the Euler Equation on Unstructured Tetrahedral Meshes, AIAA J., vol. 30, p. 7077, 1992). In this paper the author make a good discussion about the most popular schemes. I also found some othes articles about Euler and NavierStokes simulation with both of those methodologies. Thank's the opportunity.
Marcio Aredes Martins Doctoral Student Mechanical Engineering Departament UNIVERSIDADE FEDERAL DE MINAS GERAIS 

March 3, 2000, 15:22 
Re: FVM: vertexcentered, cellcentered, or cellvertex?

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Thanks for your information. I'll take a look at the Frink's paper.
So, you say that the cellcentered scheme is the most accurate one among the three and FEM. Is it mainly because of the larger number of data points as Mr. Arunajatesan said in the second message? Or are there any other explanations? (such as boundary conditions) Thank you! Nishikawa 

March 4, 2000, 07:21 
Re: FVM: vertexcentered, cellcentered, or cellvertex?

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Hi!
Will cell centered schemes, you get better accuracy because of the increased number of data points. Also there are other accuracy issues as well. With cell based schemes, the data point is always at the centroid of the control volume. With vertex based schemes, depending on the triangulation, the point can be very far from the centroid of the control volume enclosed by the faces on which the fluxes are computed. (see Tim Barth's papers to better understand how this happens). Another issue is the alignment of the normal to the control volume face to the edge. These two may not always be aligned very well. THis can happen in cell based schemes also. But, with the increased number of data points this is less crucial in cell based schemes. with the viscous fluxes, for accuracy, these fluxes have to be computed on the cell centered ans distributed back to the nodes, directly computing the fluxes on the control volumen face is not accurate enough. with the highly non linear behaviour seen in two equation turbulence models, this becomes much more pronounced. hope this clarifies, srinivasan 

March 4, 2000, 16:03 
Re: FVM: vertexcentered, cellcentered, or cellvertex?

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Dear Collegues
I agree with Mr. (or Dr. ?) Srinivasan Arunajatesan about the cellcentered scheme. If we considered only the first order schemes, the error analisys betwen cellcentered, vertexcentered and FEM show that the first one is more acurated. I started an analisis about the matrix builded by those methods and the most obviously conclusion is that the matrix builded by the cellcentered method is the bigger, becouse it is based in the number of mesh volumes. This fact could be an bad characteristic, becouse the "bottle neck" of the code ís the solver. But, in the frist order approach, the matrix has only four nonzero position for each line while the FEM and vertexcenterd matrixes are smaller, but they have much more nonzero position. Another very interesting comment is that the cellcentered scheme is the only of then that is the "real" mass and energy balance at the cell boundary, wich is completly coerent with the traditional finite volume method. Marcio Aredes Martins Doctoral Student  Mechanical Engineering Department UNIVERSIDADE FEDERAL DE MINAS GERAIS  Brazil 

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