Hi Jean-Paul,
thanks for pointing that out.
However, I believe there has been a misunderstanding:
I am working here with a FE discretized energy on a seperate triangulation,
say, 'triangulation_energy', with the coordinates being the first and third
invariant of 'C'; The corresponding nodal energy values are stored in the
vector 'solution_energy' and are computed once based on a analytical energy
function.
So for each quadrature point in my geometry (first triangulation) I am
computing 'C=F^T*F', thence the first and third invariant of 'C' and store
them in a Point<2> object denoted as 'invariant_point'.
Then, I employ
'GridTools::find_active_cell_around_point(triangulation_energy,
invariant_point)' to finally create a second FEValues object with the only
purpose to compute the gradient of the energy ('solution_energy') at
'invariant_point'.
'get_function_gradients(solution_energy, grad_energy_at_ref_point ) gives
me the gradient of 'solution_energy' with respect to the first and third
invariant of 'C' because the real coordinates on the associated
triangulation are the first and third invariant of 'C'.
Now, I have all the quantities to compute the stress tensor as
S_ad = 2*(grad_energy_at_ref_point[0][0]*invariant1_wrt_C_ad
+
grad_energy_at_ref_point[0][1]*invariant3_wrt_C_ad);
My goal is to compute the 4th order tangent \frac{\partial S_ad}{\partial
C_ad}.
All that said, is it not correct to start like this?:
SymmetricTensor<2,dim> C = symmetrize(F^t *F);
ad_helper.register_independent_variable(C, C_dofs);
const SymmetricTensor<2, dim, ADNumberType> C_ad =
ad_helper.get_sensitive_variables(C_dofs);
Best,
Simon
Am Mo., 18. Juli 2022 um 22:46 Uhr schrieb Jean-Paul Pelteret <
[email protected]>:
> Hi Simon,
>
> Unfortunately I don’t have the time at this moment to give you a full
> explanation as to why the logic of your code is wrong, but in essence you
> have the sequence of operations inverted. You need to compute your energy
> based on the “sensitive” DoF values that would come from the reinit’d
> FEValues operation. You cannot compute something like C_ad ahead of time,
> like it appears that you have.
>
> Take a look at these lines in the (unfinished) tutorial that will
> illustrate the steps required to linearise an energy functional. This
> should hopefully help steer you in the right direction.
>
> https://github.com/dealii/dealii/pull/10394/files#diff-fc8b83d1370bfd7eb558ea76175bfc0e8d6305023d54b17ec9cccb0fba9b1e02R1758-R1831
>
> Best,
> Jean-Paul
>
> On 18. Jul 2022, at 19:46, Simon <[email protected]> wrote:
>
> Dear all,
>
> I want to retrieve a tangent at quadrature point level using AD - attached
> is a snippet of my code:
>
> The computation of the dependent variable 'S_ad' in line 36 involves the
> call
> get_function_gradients (solution_energy, grad_energy_at_ref_point),
> see line 35.
> 'solution_energy' is a scalar function which depends on the independent
> variable 'C_ad' as can be seen in the derivations from line 8 on.
> My issue is that I see no way to pass the NumberTypes type
> (=Differentiation::AD::NumberTypes::sacado_dfad_dfad) to the corrosponding
> FEValues object, and, as a consequence, the AD framework is not aware that
> 'solution_energy' is a function of 'C_ad'.
> In particular, 'grad_energy_at_ref_point' will be treated as a constant
> with respect to 'C_ad' in line 36f.
>
> How can I incorporate this dependency in the AD framework?
>
> Thank you,
> Simon
>
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