[petsc-dev] 2D finite elements in 3D ambient space

[Geoffrey Irving]

On Mon, Dec 2, 2013 at 5:44 PM, Matthew Knepley <knepley at gmail.com> wrote:
> On Mon, Dec 2, 2013 at 6:49 PM, Geoffrey Irving <irving at naml.us> wrote:
>>
>> What would be the best way to represent the rest shape of a shell (2D
>> manifold embedded in 3D) in the case that the mesh has no natural 2D
>> atlas?  The simplest example is a sphere.  I would like to give DMPlex
>> the ambient 3D coordinates so as to avoid singularities in the
>> gradient fields.
>
> As long as the elements do not take up a significant portion of the sphere,
> the regular thing is fine here.

I am aware that a sphere with a hole is isomorphic to a disk.
Obviously I was referring to an entire sphere.  As another example,
which is actually something we intend to run various simulations on,
is several periods worth of

    http://en.wikipedia.org/wiki/File:Schwarz_P_Surface.png

Please don't force me to write it as an image of a disk.

>> If I was only dealing with first order elements, presumably the
>> correct approach would be to set the topological dimension of the
>> DMPlex to 2, give it a 3D coordinate section, and fix the few places
>> required to carry through gradient information correctly.  I haven't
>> done a thorough search of missing places yet, but at least
>> DMPlexComputeLineGeometry_Internal doesn't handle 1D elements in 3D,
>> which is required at the boundary of 2D shells in 3D.
>
> Yes, this can be fixed.

Cool.

>> Unfortunately, something more is required for higher order accuracy,
>> since naively the coordinate section itself would have to be higher
>> order, and this would require lots of changes (the equivalent of
>> DMPlexComputeCellGeometry would be called once per quadrature point
>> instead of once per element).
>
> I have never been convinced that isoperimetric stuff produces enough benefit
> for its complication. Polynomials are not good approximators for the Jacobian
> of these transforms. NURBS are so much better. If I could not refine my way
> out of the problem, I would seriously consider them.

Why are NURBS better, other that their availability as the output of
standard CAD?  If polynomials are lousy for the rest shape, how could
they be good for the deformed state?

Geoffrey