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Old   January 31, 2002, 02:37
Default tetraeder or hexaeder grids
uwe schrottke
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Hello together,

has anyone of you references to literature, papers or personal investigations that give me more information about the accuracy of tetraeder grids versus hexaeder grids and information where it is allowable to use one or the other?


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Old   January 31, 2002, 09:17
Default Re: tetraeder or hexaeder grids
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There is no general answer to this question but a specific answer: the numerical properties of the particular solver you are using.

For example, for tetrahedrals with large angles or hexahedra which strongly depart from orthogonality most numerical schemes will generate negative off diagonal coefficients for the diffusion terms. In the presence of strong gradients in, say, dissipation of kinetic energy in shear layers this will lead to negative diffusion coefficients and the solution procedure rapidly diverging ("blowing up"). If nothing is done to "fix" the problem then one must avoid large angles and/or departure from othogonality in various parts of the flow field. If something is done to improve robustness, it is can (and often is) inconsistent in a mathematical sense (i.e. the numerical scheme is not an approximation of the diffusion terms anymore) and this leaves us where? Not necessarily with rubbish as an answer but one needs to take this into account when interpretting the solution. It is important know this sort of thing and with commercial codes the information is often not available.

The discretization for a hexahedral element may be a simple unstructured one which does not handle strong anisotropy in sensible manner. This is fairly common and largely defeats the point of using anisotropic elements like hexahedrals and prism in place of tetrahedrals. If you are using such a scheme it is useful to know.

Of course, there are a host of other factors such as the smoothing, what the adaptation scheme can handle, etc...

The above does not mean that the gridding guidelines you are seeking do not exist but that they should come from the supplier of your solver. Guidelines for other solvers may not read across accurately and general solver-independent guidelines want interpretting with some caution.

By far the best guide you can have is for your solver to provide a conservative estimate of the error in the solution. Unfortunately, most commercial suppliers seem reluctant to do this. I wonder why?

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Old   February 1, 2002, 06:03
Default Re: tetraeder or hexaeder grids
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Dear andy, Do you have any idea or reference on fixing the problmes of approximating diffusion term on highly skewed element ?
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Old   February 1, 2002, 09:00
Default Re: tetraeder or hexaeder grids
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You may have to look hard to find references. People rarely write papers about their kludges. Internal reports and Ph.D. theses might be a place to look. It tends to be the kind of information that resides in stable experienced research groups.

The type of kludge depends on the type of element, the type of numerical scheme and what one wants to achieve. The use of highly skewed elements generally indicates a low weighting given to accuracy. In which case writing the diffusion terms as the sum of orthogonal terms plus cross terms and simply reducing the cross terms when you are in trouble is a simple pragmatic approach. It is inconsistent though. Another approach might be to use two consistent schemes a "+" scheme and a "x" scheme. In cases where a single nodal value is causing trouble in an element (most cases) switch to the alternative scheme. Consistency maintained but can cause conservation problems. Also can be beaten by several awkwardly placed large values in the stencil.

If accuracy is required then replacing the skewed element is almost always going to be the wisest move.

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