Re: [Libmesh-users] Subdivision surface based FEM

[Benjamin K. <ben...@na...>]

>> Declare a 'geometry system' which uses some C1 fe basis (clough-tocher
>> does come to mind...).
> 
> Clough-Tocher may not be ideal.  Since they're not h-hierarchic, you
> can't refine without (very slightly) changing the result.  That's not
> a problem for my applications but it may be for this one.
> 
> I'd suggest implementing Argyris or Powell-Sabin-Heindl elements
> instead.
> 
> But (and this is embarrassing, since I know some of these elements
> started out precisely for use in geometric approximation) I'm not
> certain what your global degrees of freedom would look like this way.
> Right now we assume that "x" and "y" are well defined globally by the
> Lagrange mapping, and we have C1 global dofs that are gradients or
> fluxes in xy space.  How does that work if "x" and "y" are only
> defined by the C1 mapping?  xi and eta aren't well defined globally,
> and I don't see how to define something similar without making
> limiting assumptions that wouldn't handle arbitrary manifold
> topologies.

Well, what we can provide now is

phi(X) = phi(X(xi)) and
(dphi/dX)(X) = [(dphi/dxi)(dxi/dX)](X(xi))

Where phi is whatever your finite element says it is and X(xi) is the C0 map
provided by the Lagrange basis.  (dxi/dX) is obtained directly by inverting
the (dX/dxi) transformation map.  My understanding of the issue is that this
is no good because X(xi) needs to be C1 so that the curvature is square
integrable...

So why not compute instead

phi(Xc) = phi(Xc(X)) = phi(Xc(X(xi))) and
(dphi/dXc)(Xc) = [(dphi/dxi)(dxi/dX)(dX/dXc)](Xc(X(xi)))

Where Xc(X) is the C1 geometry representation provided by the "geometry
system" described previously.  The additional terms needed to compute (what
I think is) the right map are (dX/dXc) at the quadrature points, which can
be constructed analytically from (dXc/dX), which the user computes from the
"geometry system."

So in some sense there ate two jacobian transformations which are
required...

Since Roy's worked with a lot more C1 systems than me (read: 0), I
ultimately defer to him as to whether this is possible or just a bunch of
nonsense...

-Ben