[Gmsh] The volume of the second order tetrahedron

[Geordie McBain]

2011/3/5 Mikhail Artemiev <artemiev.mikhail at ngs.ru>:
> Hi all!
> I know how to calculate the volume of the first order tetrahedron.
> But in the case of second order tetrahedra creating by gmsh for my
> complicated geometry (see one of these tetrahedra
> http://saveimg.ru/show-image.php?id=0971dc74b46b676f549960aebeae3e4b) I
> encountered a problem of finding the right volume of the tetrahedron.
>
> Can anybody help me?

Hello.  Although it's not drawn like one, isn't your second order
tetrahedron really just a standard ten-node quadratic tetrahedron;
i.e. a quadratic isoparametric element?  If so, there are formulae for
the shape functions in many of the standard texts on the finite
element method, e.g. (2.2.9) of Ciarlet (2002, p. 47) or at the bottom
of p. 22 of Ern & Guermond (2004)

Ciarlet, P. G. (2002). The Finite Element Method for Elliptic Problems
(2nd ed.), Volume 40 of Classics in Applied Mathematics. SIAM: Society
for Industrial and Applied Mathematics.
http://books.google.com.au/books?id=isEEyUXW9qkC

Ern, A. and J.-L. Guermond (2004). Theory and Practice of Finite
Elements, Volume 159 of Applied Mathematical Sciences. New York:
Springer
http://books.google.fr/books?id=CCjm79FbJbcC