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

[Matthew Knepley]

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.


> 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.


> 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.

  Matt


> Is there a better clean way to support FE PDEs on spheres or other
> nontrivial surfaces in 3D?
>
> Thanks,
> Geoffrey
>



-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
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