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ngj December 19, 2012 09:58

Reference on the Lack of Uniqueness of the Cell Volume
Good afternoon to all of you,

I have for a while been looking for a reference, which discuss the volume of a general hexahedral cell, i.e. non-planar faces.

I have found some references which hint that the volume of such a hexahedral cell is not uniquely defined, and based on some practical tests with different methods to compute the volume, I have reached the same conclusion.

Nonetheless, I would be grateful if anyone of you know about an article, which directly addresses the non-uniqueness in evaluating the volume of a general hexahedral cell.

Thank you and Merry Christmas,


deepsterblue December 19, 2012 14:03

Why just hexahedral cells? Any general (non-simplical) polyhedral cell with non-planar faces would apply.
It depends on how you decompose the cell during the calculation. I've seen that a face-and-cell decomposition gives the best results.

kmooney December 19, 2012 14:14

If every face is decomposed consistently, I believe the errors in volume will at least cancel out globally.

ngj December 19, 2012 18:09

Good evening,

Thanks for your replies.

Right now I am working on an article, where I have found myself in the need to compute the volume of a general hexahedral in order to prove a conservation property, however, since the volume appears to be non-uniquely defined, I find myself unable to give a full proof. Therefore, I would like to be able to refer to a firm/non-arguable statement/proof/article that the volume is indeed non-unique.

Kind regards,


kmooney December 19, 2012 20:05

Hi Niels,

This paper:
mentions that without an expression for the surface that comprises a non-planar face, there is no information available to define it (pg. 5) and you need to make something up and approximate before decomposition. The fact that you need to base your decomposition on some approximation, and volume is a function of that decomposition, tell me that it is not unique.

This paper (sec. 4.2):

Mentions that you need to be consistent in decomposition. Since you have different options when decomposing a polyhedra, I'm thinking this implies that you can get different volume calculations.

If you search around for "tetrahedral subdivisions" or words like that you should find some more useful info. Sorry I can't give you a concrete answer on this.



ngj December 20, 2012 05:24

Hi Kyle,

Thank you, I think those references should do the trick - at least they lend sufficient support to my arguments :)

Merry Christmas,


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