Re: [GiNaC-list] Differentiation of a function with respect to a tensor

11 Feb 2011


      But do remember, if E is symmetric, that you have to somehow tell GiNaC 
that e12=e21.  I do a lot of similar stuff in Mathematica, and I start 
with a statement like
E = {{e11,e12,e13},{e12,e22,e23},{e13,e23,e33}}
so one way you could do this in GiNaC is to create a variable E:
expr E[3][3]
and set E[i][j] = to eij or eji as appropriate.  And then for something 
like E:E you get
expr EddE = 0;
for (i=0;i<3;i++) for (j=0;j<3;j++)
   EddE += E[i][j]*E[i][j];
(In Mathematica you could more easily do
EddE = Tr[E.E]
but Mathematica costs a fortune unless you are part of a university with 
a site license.)

Bernardo Rocha wrote:
...
Dear Stephen,
thanks a lot for your support. I was considering if doing it this way it
would work, now I got my confirmation. Thanks a lot again.
Best regards,
Bernardo
2011/2/11 Stephen Montgomery-Smith <stephen@missouri.edu
<mailto:stephen@missouri.edu>>
Bernardo Rocha wrote:
Hi everyone,
I've recently discovered GiNaC and I'm really excited about its
        capabilities. There is one thing that I would like to know if it
        is able
        to do that I haven't found in the tutorial.pdf or in any other place
        that I've searched.
I would like to know if, given a function \Psi=\Psi(E), like the
        strain
        energy function for the St. Venant-Kirchhoff material
\Psi(E) = 0.5 * \lambda * (tr E)^2 + \mu E:E
is it possible to differentiate it with respect to E, that is i
        would
        like to compute \frac{\partial \Psi}{\partial E}. If this is
        possible,
        could someone please send some examples or maybe point to which
        classes
        should I use to do that?
That's all for now. Many thanks in advance.
Best regards,
        Bernardo M. R.
Couldn't you do it this way?  Write \Psi(E) as an expression
    involving the variables e11,e12,e13,...,e33 which are the entries of
    E.  Then compute the partial derivatives \frac{\partial
    \Psi}{\partial eij} for 1<=i,j<=3.  (Presumably you suppose that E
    is symmetric so only six partial derivatives need to be computed,
    but even if it is not necessarily symmetric you still only need 9
    partial derivatives.)  Just store this as something like:
expr dPsi_dE[3][3]
or
vector<expr> dPsi_dE
or something similar.