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

[Matthew Knepley]

On Mon, Dec 2, 2013 at 8:01 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:

> Matthew Knepley <knepley at gmail.com> writes:
> >> 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.
>
> These issues are orthogonal.  If the mapping is not affine, you need
> separate Jacobians at each quadrature point.  Non-affine elements are
> required for high-order accuracy with curved boundaries, and in many
> cases when using quad and hex elements.
>

I agree that non-affine stuff requires the linearization of the mapping at
each
quadrature. The interface Geoff suggested is the one I have been using for
that. I am fine with it.


> The value of NURBS is that (a) some coordinate transformations can be
> represented exactly and (b) for certain problems, the rest solution can
> be represented exactly in the ansatz space.  Quadrature error does not
> magically vanish.
>

My point is that trying to resolve particular geometry with polynomials is
very slowly convergent. NURBS are much better. It depends on how
complicated your geometry is.

   Matt

-- 
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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