automatic differentiation:

[Simon]

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

[Jean-Paul Pelteret]

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

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<Code_AD.PNG>

[Simon Wiesheier]

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

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[Jean-Paul Pelteret]

Hi Simon,

Sorry that I didn't get back to you on this before now. I think that I get the gist of what you're trying to do, but I must admit that I'm a bit skeptical as to whether or not its going to work (the AD framework was not really conceptualised to be used in this manner, so it will be interesting to hear whether you end up finding success in this endeavor). Basically, what you need to do is somehow link the degrees of freedom in the found cell with those that you created C_ad with.  I guess suggests that you need to express how these values might change as your invariant point changes. If you somehow manage to form an association between these two sets of DoFs then you can use one of the various FEValuesViews get_function_gradients_from_local_dof_values() functions to compute gradients that have the required sensitivities encoded in them. This is actually what we already do in ScratchData::get_gradients(), which makes this class AD-capable.

I hope that this helps, if you haven't succeeded with this already.

Best,

Jean-Paul