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

[Felder, Alessandro]

Hi everyone,

I have a pretty basic question, unfortunately: I'm having a lot of fun
solving a linear elasticity problem with PDELab on a 3d tetrahedral mesh
with conforming P1 finite elements. Roughly, I am doing something like:

        ...

        typedef Dune::PDELab::PkLocalFiniteElementMap<GV,Coord,Real,1> FEM;
        FEM fem(gv);
        ...
typedef
Dune::PDELab::VectorGridFunctionSpace<GV,FEM,dim,VBE,VBE,CON,Dune::PDELab::EntityBlockedOrderingTag>GFS;

        GFS gfs(gv,fem);

        ... //set up gridoperator and constraints

        typedef Dune::PDELab::StationaryLinearProblemSolver<GO,LS,U> SLP;
        SLP slp(go,ls,uold,1e-10);
        slp.apply();

        ... //write u to vtk file


So far, so good.

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).

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)}

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..

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?

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.