[dune-pdelab] Basic question about shape function derivatives in PDELab

Oliver Sander sander at igpm.rwth-aachen.de
Mon May 4 10:56:06 CEST 2015

Hi Alessandro,

> However, now I am trying to find the derivatives (i.e. the strains at
> the element midpoints) of my solution vector (the nodal displacements).
> Mathematically, if I understand correctly, you would need to get the
> gradient of the reference element shape functions, multiply these from
> the left with the inverse transposed Jacobian, and then accordingly
> multiply the result with the corresponding nodal values (u_i, v_i, w_i)
> i=0,1,2,3 (for a linear tetrahedron).

Yes, that sounds correct.

>
> In textbooks, this element-wise Cauchy strain that I am looking for is
> often denoted as
>
> \vec{\epsilon_e(x,y,z)} = B \vec{u_e(x,y,z)}
>

Not sure I understand your notation here.  Is B the inverse transposed Jacobian
of the map from the reference element to your element?  Or is it the
linearized strain operator?

> For P1 elements, I think B is constant per element (so in the worst case
> I could stitch it together myself), but I might want to use P2 or other
> elements in the future..

It is for this reason that the inverse transposed Jacobian depends on
the position.  You wrote that you want the strain at the element midpoint.
So simply call inverseJacobianTransposed with the element midpoint as the
argument, und you will get the correct transformation also on p2 grids.

>
> Is there an easy and generic way to get the strain epsilon or the
> strain-displacement matrix B in PDELab? Or how would one do this?

Best,
Oliver

>
> Any help (in German or English) would be immensely appreciated!
>
> Thanks and best wishes,
> Alessandro
>
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