Re: [Libmesh-users] Distinguising different dofs for Hermite elements

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

On Tue, 14 Dec 2010, Vetter Roman wrote:

> I'm using a 1D mesh with (cubic) Hermite shape functions to
> implement a beam. Each element comes with four dof_indices per
> variable, namely the nodal value and its derivative for each of the
> two nodes.

> In beam theory, when using Hermite shape fcts, the derivative dofs
> are treated differently than the normal ones in that their stiffness
> matrix entries are different. Thus, when building the stiffness
> matrix, I need to now which entries of the dof_indices vector
> corresponds to which type of dof.
>
> To be a bit more specific: Currently, I'm iterating over all
> active_local_elements. For each of these, I determine the
> dof_indices vector, which, for 3 variables, has a size of 12. For
> each entry in dof_indices, I would like to know three things:
> a) the variable number it belongs to,
> b) the node it belongs to,
> c) "normal" dof or derivative dof,
> so I can build the 12x12 element stiffness matrix correspondingly.
>
> I'm looking for the "intended" way to do it, i.e. such that my code
> would still be correct after doing AMR or even using different shape
> functions, both of which I'm planning to do.

If you want to be as flexibly correct as possible, the way to do it
is to write your formulation in such a way as to avoid needing to know
anything about the DoFs "types", just about their shape functions and
shape function derivatives.  Usually this bites people when they write
themselves into a corner with isoparametric-Lagrange-specific code,
but I guess with H2 problems it'd be just as easy to accidentally
write cubic-Hermite-specific code instead.

I.e. if you're solving (EI w'')'' = q, then you'd just weight that by
a test function and integrate by parts twice, and evaluating the
resulting weak formulation should give you a good approximation of w
on any C1 function space.

> I reckon that the two components per variable (n_comp(s,vn) == 2)
> precisely represent the value and the derivative, for each node,
> right? (source:
> http://www.mail-archive.com/lib...@li.../msg00844.html)

Yes.

> Is the ordering always guaranteed to be this way?

In general, NO.  As soon as you bump up to p=4, for example, you'll
end up with a new node on which the DoF doesn't correspond to either a
value or a derivative.

But if you stick to our cubic hermites you should be safe, even with
adaptive h refinement.
---
Roy