[Gmsh] The volume of the second order tetrahedron

[Mikhail Artemiev]

Hello.
Dear Christophe,
is there any parameter in gmsh to handle the distribution of the second 
order additional nodes?
I mean - can I change any parameter to stand these additional nodes at the 
middles of the edges regadless curvilinear geometry?

Thanks
Mikhail Artemiev

----- Original Message ----- 
From: "Christophe Geuzaine" <cgeuzaine at ulg.ac.be>
To: "Geordie McBain" <gdmcbain at freeshell.org>
Cc: "Mikhail Artemiev" <artemiev.mikhail at ngs.ru>; <gmsh at geuz.org>
Sent: Wednesday, March 09, 2011 2:37 PM
Subject: Re: [Gmsh] The volume of the second order tetrahedron


> On 08/03/11 04:46, Geordie McBain wrote:
>> 2011/3/7 Mikhail Artemiev<artemiev.mikhail at ngs.ru>:
>>>>> Hello, Geordie.
>>>>> Thank you for reply.
>>>>> I used "Tools - Visibility - Numeric - Mesh - Hide all elements - Show
>>>>> Element (for instance, 1365)" to draw that figure.
>>>>
>>>> Oh right, O.K.  It's a funny looking shape, isn't it.  Perhaps that's
>>>> just how Gmsh depicts nonlinear elements?
>>>
>>> It's a very important question!
>>
>> Indeed.  I don't know much about the internals of the Gmsh, but let's
>> see if we can't make some progress.
>>
>>> Please, look at 2 figures:
>>> this is a shpere that was approximated by first order tets
>>> http://saveimg.ru/show-image.php?id=47cdfa0e6f7a01f5dee8880c7081e151
>>> this is a sphere that was approximated by second order tets
>>> http://saveimg.ru/show-image.php?id=fa66051392be4e36e2262055272170e1
>>> These 2 meshes was created by gmsh from one geo file (and with the same
>>> characteristic lengths).
>>
>> It looks as though all the nodes on the outer six-node triangular
>> faces of the ten-node tetrahedra lie on the geometric sphere.  Is that
>> right?  If so, that's good, and your quadratic mesh is a better
>> representation of the geometry than the linear one.
>>
>>> I will wonder if it is a feature of visualization.
>>
>> Why?  I don't know how the visualizer works internally but if (say)
>> all it can do is depict triangles, then what it's showing is a
>> reasonable representation of a quadratic tetrahedron, isn't it?
>>
>
> Hello - The visualization of high-order mesh elements can be enhanced with 
> the following "Mesh.NumSubEdges" option (default=2),. For example:
>
> gmsh demos/sphere.geo -clscale 4 -order 4 -string "Mesh.NumSubEdges=10;"
>
>
>
>
>>> Standard 10-node second order tetrahedron differs from 4-node first 
>>> order
>>> tetrahedron by adding 6 node in the middles of the edges of tetrahedron.
>>
>> No, the additional six nodes don't have to be on the midpoints of the
>> edges of the tetrahedron.  They can be, and probably will be if you're
>> only meshing a polyhedron, but in general no.  There are some
>> restrictions, they can't be just anywhere (or the Jacobian of the
>> transformation from the canonical element will change sign within the
>> element), but they can move around a bit, and indeed this is most
>> desirable when meshing a curved geometry.
>>
>>> I think that gmsh not only adds new 6 nodes but changes the coordinates 
>>> of
>>> these nodes too.
>>
>> Perhaps to make them lie on the geometric sphere?
>>
>>> Therefore we have nonstandard 10-node quadratic tetrahedron and the 
>>> formulae
>>> of the shape functions defined on the standard one don't work.
>>
>> I think they will. I'm still hopeful this is a standard quadratic
>> tetrahedron.  In terms of figure 4.3.1 on p. 228 of Ciarlet's book,
>> referred to earlier, I trust that these are are `isoparametric'
>> tetrahedra `of type (2)'; you'll also find drawings of `three
>> isoparametric tetrahedra of type (2)' in figure 4.4.2 on p. 251.
>>
>>> Am I wrong?
>>
>> I'm not sure, but I'm hopeful Gmsh is correctly approximating `a
>> curved boundary with isoparametric finite element' as described by
>> Ciarlet pp. 248 ff.
>>
>> I presume the same thing happens in two-dimensions, e.g. using 6-node
>> triangles to mesh a sector.  I've tried this, as attached.  It looks
>> good.  In sector.png, I've gotten Gmsh to number the nodes, as they
>> appear in sector.msh, which was generated by "gmsh -2 -order 2
>> sector.geo". The edges of each triangle along the geometric perimeter
>> aren't drawn as curves (parabolas), but I think that's just economy of
>> depiction, and we're free to treat the elements as isoparametric
>> triangles of type (2), no?
>>
>>
>>
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>
>
> -- 
> Prof. Christophe Geuzaine
> University of Liege, Electrical Engineering and Computer Science
> http://www.montefiore.ulg.ac.be/~geuzaine