FV discretisation on tet meshes
hi,
i'm looking for some theoretical and analytical stuff on finite volume formulation/discretisation on tet meshes. for example how to formulate ud or central schemes on tets, how to estimate the truncation errors and so on. so can you tell me some papers dealing with these things. peter 
Re: FV discretisation on tet meshes
hi peter or (susanne?),
you are looking for quite exotic stuff. can you please specify you discretisation method. we are working on nonconforming elements for the velocity and constant or P1iso_P1bubble discretisation for the pressure, respectively. this discretisation is very well adapted to the needs of LES. for this discretisation we have internal reports or pdffiles available. please contact me if you are interested. a good reference on finite volumes is always r.a.nicolaides. you can find information in the web salut ulrich 
Re: FV discretisation on tet meshes
this isn't very high tech, i'd say you can find this in any relatively advanced CFD book. try Hirsch's book which most people seem to like. or you can try Tannehill Anderson and Pletcher's CFD book.

Re: FV discretisation on tet meshes
Hello Peter,
simple upwind and central schemes are easily formulated on tetrahedral grids with the finitevolume method. The central scheme on the mediandual control volume on a tetrahedral grid is equivalent (shown by many people) to the Galerkin finiteelement discretisation with linear basis functions. To get simple schemes implemented on tetrahedral grids is actually quite simple. If you want more accurate ones, it becomes quite tricky. In general, as long as the tetrahedra are not too highly stretched, the discretisation will work very well. It's when the tetrahedra are highly stretched, to capture boundary layers, for example, that numerical difficulties appear. There is quite a large body of literature on discretisation on tetrahedral grids. Have you tried searching for relevant keywords on the NASA Technical Report Server (http://techreports.larc.nasa.gov/cgibin/NTRS)? You might find the following reports/papers of interest: M.B. Giles. `Stability analysis of a Galerkin/RungeKutta NavierStokes discretisation on unstructured tetrahedral grids'. Journal of Computational Physics, 132:201214, 1997 (Also available in ps format from Prof. Giles' homepage: http://users.comlab.ox.ac.uk/mike.giles/index.html) T.J. Barth. Aspects of Unstructured Grids and FiniteVolume Solvers for the Euler and NavierStokes Equations, High Resolution Upwind and TVD Methods for the Compressible Flow Equations, Von Karman Inst. for Fluid Dynamics, 1994. T.J. Barth. Aspects of unstructured grids and finitevolume solvers for the Euler and NavierStokes equations. AGARD, Special Course on Unstructured Grid Methods for Advection Dominated Flows, R787, May 1992 Martin Berzins. Mesh Quality: A Function of Geometry, Error Estimates or Both? 7th International Meshing Roundtable, October 1998, Available from: http://www.andrew.cmu.edu/user/sowen/imr7.html. (When you're at the site, check also the other years of the conference for relevant articles by Berzins and other people.) If you have any more questions, just email me. Greetings Andreas 
Re: FV discretisation on tet meshes
hello ulrich,
at first thanks for your help. i'm just looking for how to make upwind differencing on tets (to compare this with the hex formulation  so the question is how to deal with the missing faces) peter 
Re: FV discretisation on tet meshes
hello
i'm sure that it isn't very high tech  it's implemented in every cfd code i know, but nevertheless i can't find it. hirsch's books is alway a good place to start, if you don't have too much questions. but in hirsch's books there are only hexahedral meshes (and schemes are always denoted for the cartesian ones)  but maybe my versions of his books are too old  i'll check this. the other book you mentioned i did't know, so mybe you can tell me the title. thanks a lot peter 
Re: FV discretisation on tet meshes
hello andreas,
thanks a lot for your answer. i'll check all these papers as well as i will take a look on the nasa database and after this i may come back to you. concerning the first remark you made  do you have such a paper where the equivalence of fem and fv is shown in that case. thanks again greetings peter 
Re: FV discretisation on tet meshes
Hello Peter,
for the equivalence of the diffusion terms, you can have a look at the references by Barth from 1992 and 1994. For the equivalence of the convection terms, you can show that yourself very quickly. You should see that the linear Galerkin approach gives the same as the centred finitevolume method on the median dual. If the discretisation is noncentred (i.e., an upwind scheme), I cannot remember having seen a proof of equivalence, although there might have been some work by French research groups, especially at INRIA. I guess that there is an equivalence as well in certain cases. Again, if you have any more questions, just email me. Greetings Andreas 
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