Re: [Libmesh-users] second order problems on quads

[Roy S. <roy...@ic...>]

On Fri, 4 Sep 2009, Hanjoerg Seybold wrote:

> Hello,
> we would like to solve a biharmonic problem on a tube.
> For this we would like to use a Hex mesh (not rectilinear)
>
> For the Hermite elements we get the warning.
> -WARNING: Second derivatives are not currently correctly calculated on
> non-affine elements!
>
> Does this mean that Hermite elements can only be used on rectilinear/octree
> meshes or just that the approximation of the boundary is not correct
> in the mapping.

There are two problems that will affect you here:

One is that, as you suspected, Hermite elements can only be used on
rectilinear/octree meshes.  Parallelepipeds should also work but
haven't been tested.  This is a conceptual problem with the Hermite
elements themselves, not just a software limitation; the way these
elements maintain continuity is to enforce equality of a few (but not
all) mixed second derivative terms at nodes, and that breaks unless
those terms match up from element to element.

Two is that, as the library is warning you, we don't calculate second
derivatives correctly on curved elements.  This is just a software
limitation; I was only running problems on polygonal and rectilinear
meshes when I added the second derivative support, so I lazily avoided
adding the extra terms that would come from non-affine geometric
mappings.

The way to handle fourth-order problems on general 3D domains would be
to add these terms and to also add a class of general C1 3D elements.
If you're motivated enough to do this, I will be thrilled.

But if you're as lazy as I am, and if your domains are all simple
tubes, the easier way to do things would be to recast your problem in
polar coordinates, then solve it on a rectilinear domain with periodic
boundary conditions.
---
Roy