Pde on a 2 dimensional surface

On Tue, Nov 6, 2012 at 3:01 PM, Nicolas Thorstensen
<email address hidden> wrote:
> New question #213453 on FEniCS Project:
> https://answers.launchpad.net/fenics/+question/213453
>
> Hi
>
> I wish to solve a Pde on a 2 dimensional surface embedded in R^3.
>
>
> i) Is there a way to load a closed triangle mesh (that is without border) from a file (e.g.*.off) into FENICS?

No. you need to convert your mesh to the DOLFIN XML format. The
dolfin-convert can perform the conversion for a number of common
formats.

> ii) Are are the differential operators for surfaces in Euclidean 3 space implemented ?
>

No yet.

Garth

>
> Regards,
> Nicolas
>
>
>
>
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For a surface embedded in R^3, the gradient is the 3D gradient minus the normal projection (nabla_grad(u) - dot(nabla_grad(u), n)*n) *ds where n is the surface normal and ds is the surface element. Actually, depending on how the assembly works, it might actually not be necessary to modify the gradient at all! (that is, it is possible that the correct operator to use is in fact the 3D gradient). I have written code, though for different software, that simply allows the usual mapping of a master element to a mesh element to vary in R^3 instead of R^2. The way I write down the inverse of the mapping is to extend the original mapping to a 3D space by using the normal vector, and conveniently, then leaving off the third component is equivalent to doing said projection. That said, it probably is still useful to know, conceptually.

So the question is, then, is the functionality for assembling systems with triangles (not tetrahedra) in R^3 already there? I'm in the process of trying this now, since dolfin-convert doesn't seem to handle off format as far as I can tell, and when I convert it to mesh format, it insists on zeroing out the third component since the dimension is 2 -- it's a problem of not distinguishing manifold with embedding dimension). Since the converters are all in python, I can dig in and tweak things directly. Will update soon.