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

[Oliver Sander]

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?

Sorry, but I don't know enough about pdelab to answer this.

Best,
Oliver

> 
> Any help (in German or English) would be immensely appreciated!
> 
> Thanks and best wishes,
> Alessandro
> [RVC Logo - link to RVC Website]<http://www.rvc.ac.uk>    [Twitter icon - link to RVC (Official) Twitter] <http://twitter.com/RoyalVetCollege>     [Facebook icon - link to RVC (Official) Facebook] <http://www.facebook.com/theRVC>     [YouTube icon - link to RVC YouTube] <http://www.youtube.com/user/RoyalVetsLondon?feature=mhee>     [Pinterest icon - link to RVC Pinterest] <http://pinterest.com/royalvetcollege/>     [Instagram icon - link to RVC Instagram] <http://instagram.com/royalvetcollege>
> 
> This message, together with any attachments, is intended for the stated addressee(s) only and may contain privileged or confidential information. Any views or opinions presented are solely those of the author and do not necessarily represent those of the Royal Veterinary College (RVC). If you are not the intended recipient, please notify the sender and be advised that you have received this message in error and that any use, dissemination, forwarding, printing, or copying is strictly prohibited. Unless stated expressly in this email, this email does not create, form part of, or vary any contractual or unilateral obligation. Email communication cannot be guaranteed to be secure or error free as information could be intercepted, corrupted, amended, lost, destroyed, incomplete or contain viruses. Therefore, we do not accept liability for any such matters or their consequences. Communication with us by email will be taken as acceptance of the risks inherent in doing so.
> 
> _______________________________________________
> dune-pdelab mailing list
> dune-pdelab at dune-project.org
> http://lists.dune-project.org/mailman/listinfo/dune-pdelab
> 


-------------- next part --------------
A non-text attachment was scrubbed...
Name: signature.asc
Type: application/pgp-signature
Size: 473 bytes
Desc: OpenPGP digital signature
URL: <https://lists.dune-project.org/pipermail/dune-pdelab/attachments/20150504/5f11a106/attachment.sig>