On Wed, Dec 22, 2010 at 12:22 PM, Derek Gaston <derek.gaston@...> wrote:
> So... I'm currently needing things like d2xyzdxideta on nonplanar (ie slightly twisted) faces of 3d elements (ie a quad4 in 3d that is not perfectly planar). When I call fe::reinit() to get these... I get a warning message like so: "WARNING: Second derivatives are not currently correctly calculated on nonaffine elements!"
>
> My question is... is that really true in this case? It seems like for a linear quad4 d2xyzdxideta and it's like should be exactly defined.... even if the element isn't perfectly planar. But I'm sure there might be something I'm not thinking of.
>
> If there is some error involved... how much error are we talking about. I mean, will it be manageable as long as the element isn't too distorted (which none of mine will be) or is it instantaneously extremely wrong?
Hi Derek,
I don't think there should be any error in computing something like
\frac{d^2 x}{d \xi^2}, \frac{d^2 x}{d\xi d\eta}, and friends.
The errors are in second derivatives of the shape functions wrt
physical coordinates, for example (all "d's" should be treated as
partial derivs)
\frac{ d^2 \phi}{d x^2} = \frac{d \phi}{d \xi} \frac{d^2 \xi}{d x^2} +
\frac{d^2 \phi}{d \xi^2} (\frac{d \xi}{d x})^2
We currently compute \frac{d \xi}{d x} of course, but we don't compute
the \frac{d^2 \xi}{d x^2} term, though I guess you could approximate
it via finite differencing.
For a 2D bilinear element, the formulae are a bit more complicated,
it's possible this term is zero but I don't know for sure...

John
