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October 8, 1999, 17:19 
HEX VS TET

#1 
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Can anyone refer me to any recent published work where all hexahedral and all tetrahedral mesh was used to study their relative accuracy? I am NOT particularly interested in boundary layer but the flow in general under typical industrial type application. MARS scheme with denser cell density is said to provide similar accuracy for tet and hex cells (STARCD newsletter). Any comment on that?


October 9, 1999, 06:46 
Re: HEX VS TET

#2 
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... for comparable resolutions !!!! So you might end up with a lot more cells. Just read this article carefully.


October 9, 1999, 10:33 
Re: HEX VS TET: Is accuracy related to the shape of a cell? or?

#3 
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(1). Are you talking about "mesh independent solution"? or "mesh dependnet solution"? (2). Most published papers show the approach to the mesh independent solution and address the accuracy in terms of the mesh dependent solutions. (3). There are two ways to address the accuracy issue: one is to use the order of truncation error analysis, and the other is to compare the mesh independent solutions with the test results (or analytical solutions, if available). (4). My feeling is, the accuracy issue must be addressed at the absolute level, that is the mesh independent solution level. (5). So, the question is really related to the approximations made inside the cell. "Will the approximationA for the cellB used in the schemeC converge to the true solution, as the size of cellB is reduced to produce the mesh independent solution?" (6). If the approximationA for the cellB is "exact", and the approximationE for the cellF is also "exact", then both solution will be exact solutions as the cellB and the cellF are reduced to produce the mesh independent solutions. (and these two exact solutions should be the same). (7). So, you see, how you divide the computational domain into different shape cells is not the direct issue. It is the subsequent approximation made inside the cell which will affect the mesh independent solution. Obviously, a firstorder approximation will have a harder time to get accurate solution in the highlyskewed cell. But an exact method in the highlyskewed mesh will not have the same problem.


October 11, 1999, 05:11 
Re: HEX VS TET: Is accuracy related to the shape of a cell? or?

#4 
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Hi,
Not that easy, I'm afraid. For starters, I may postulate a "shape function" (i.e. the function i use to calculate the integrals) independent of the shape of the control volume (I said control volume!: this is FV specific; in FEM this is not the case). I can also go through the same discretisation gymnastics irrespective of the shape of the CV. There are two consequences: 1) The number of neighbours for a CV depends on its shape, implying the number of offdiagonal coefficients (work in the solver) AND wellposedness of the equation (I need at least 4 neighbours in 3D to calculate the gradient and a tet gives me just enough; if the neighbours are coplanar, or the cell is next to the boundary (no neighbour), I might get into trouble. 2) The second issue is the cancellation of the convection discretisation error. Remember that funny case of using Upwind Differencing (UD) on a hex mesh aligned with the flow AND getting the exact solution even though the discretisation is firstorder? What happened here? Well, let us consider the numerical diffusion of UD as a function of facetoflow angle and let's sum it up over the whole CV. If you plot the discretisation error as a function of angle, what you get is two "hills" with a zero minimum, for meshto flow alignment (see the end of the message). The equivalent graph for a triangle (2D) shows that the minimum does not go down to zero (i.e. the error cannot be completely cancelled). If you now stir in a measure of "number of cells" (for the same domain size), or "number of faces" (for the same work), or "the length of the edge" (for the same effective resolution), you get some interesting results. I did this a while ago (and never had a cheek to publish it!) but the conclusion was that ON AVERAGE (when nothing is apriori known about the flow) tets are ~9% better than hexes (in 3D) BUT the minimum error for hexes is zero (and for tets it's finite!). The conclusion is: "use your brain when buliding the mesh and you'll beat the best tet mesh every time!". This discussion could be extended to diffusion (here, hexes are better because there is more neighbour information), but I can't be bothered ;) For reference, the gnuplot graphs of the errorvsangle in 2D are: TRIANGLE: plot [0:1.577] abs(sin(x)*cos(x)) + abs(sin(2*3.14/3+x)*cos(2*3.14/3+x)) + abs(sin(4*3.14/3+x)*cos(4*3.14/3+x)) title 'triangle' SQUARE: plot [0:1.577] abs(sin(x)*cos(x)) + abs(sin(3.14/2+x)*cos(3.14/2+x)) + abs(sin(3.14+x)*cos(3.14+x)) + abs(sin(3*3.14/2+x)*cos(3*3.14/2+x)) title 'square' HEXAGON: plot [0:1.577] abs(sin(x)*cos(x)) + abs(sin(2*3.14/6+x)*cos(2*3.14/6+x))+abs(sin(4*3.14/6+x)*cos(4*3.14/6+x))+abs(sin(6*3.14/6+x)*cos(6*3.14/6+x))+abs(sin(8*3.14/6+x)*cos(8*3.14/6+x))+abs(sin(10*3.14/6+x)*cos(10*3.14/6+x)) title 'hexagon' If someone feels like picking this work up and finishing it off, please contact me and I'll give away all my notes. Hrvoje 

October 11, 1999, 10:42 
Re: HEX VS TET: Is accuracy related to the shape of a cell? or?

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The conclusion is: "use your brain when buliding the mesh and you'll beat the best tet mesh every time!". I agree with you in this matter. I have mentioned earliar in this foram that human brain is the biggest supercomputer. We use our brain power to solve creative problems. Among, structured based, Hexa is simpler than Tetra. Hexa is bricklike and Tetra is diamond like. Let us think of building a brick wall. If the carpenter uses Hexa brick then he can finish the work faster then if he uses Tetra or other more complex type of brick. If the cerpenter wants to use unstructured brick then it will take him much much more time and the brick wall will have lots of flaws. The same case holds good for mesh generations. The more complex geometries we applied the more chances are there for accumulation of errors. Even nowadays, NASA (the hometown of cfd gurus) is trying to develop simpler versatile structurebased turbulence models for their flow of interests.


October 25, 1999, 04:24 
Re: HEX VS TET: Is accuracy related to the shape of a cell? or?

#6 
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hi,
this sounds like you can help me. i'm looking for some theory 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 

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