[Gmsh] The volume of the second order tetrahedron

Mon Mar 14 08:41:00 CET 2011

Thank you very much!
"Mesh.SecondOrderLinear=1;" is quite what I need!

Mikhail Artemiev

----- Original Message ----- 
From: "Christophe Geuzaine" <cgeuzaine at ulg.ac.be>
To: "Mikhail Artemiev" <artemiev.mikhail at ngs.ru>
Cc: <gmsh at geuz.org>
Sent: Monday, March 14, 2011 12:28 PM
Subject: Re: The volume of the second order tetrahedron

> On 14/03/11 06:26, Mikhail Artemiev wrote:
>> 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?
>>
>
> Hi Mikhail,
>
> Yes: to insert the high order nodes by linear interpolation on the 
> straight edges, set "Mesh.SecondOrderLinear=1;".
>
> If you want the high order nodes to be equispaced in parameter space, but 
> still on the curved edges, you;ll have to slightly modify the source code 
> (cf. computeEquidistantParameters in Mesh/HighOrder.cpp)
>
> Christophe
>
>> 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?
>>>>
>>>>
>>>>
>>>> _______________________________________________
>>>> gmsh mailing list
>>>> gmsh at geuz.org
>>>> http://www.geuz.org/mailman/listinfo/gmsh
>>>
>>>
>>> --
>>> Prof. Christophe Geuzaine
>>> University of Liege, Electrical Engineering and Computer Science
>>> http://www.montefiore.ulg.ac.be/~geuzaine
>>
>>
>
>
> -- 
> Prof. Christophe Geuzaine
> University of Liege, Electrical Engineering and Computer Science
> http://www.montefiore.ulg.ac.be/~geuzaine
>
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