Question about Step 44 shape function gradient computations

[Claire]

Dear all,

When going through the Step 44 tutorial, there is one thing that I was not able to understand well.

In the shape function gradient computations, the inverse of the deformation gradient is used, and the provided explanation specifies:
"Note that the shape function gradients are defined with regard to the current configuration"

If I got it well, a total Lagrangian formulation is used, meaning that the reference configuration is the initial (i.e undeformed) one and that the mesh is not moved (unlike in step 18)

So, why would the shape function gradients be computed with respect to the current (i.e. deformed) configuration?

Thank you very much by advance for any help that I may get.

Best,

Claire

[Jean-Paul Pelteret]

Dear Claire,

You are right that this is a total Lagrangian formulation but that doesn't mean that one is restricted to defining the problem in terms of fully referential quantities. 

One can arrive at the same conclusion from a number of starting points, but ultimately its because we'd chosen to integrate spatial quantities on the reference configuration. Along with that, following from the weak form we need shape functions defined in the spatial configuration in order to perform the integration correctly. These can be computed using the chain rule: d/dx_{j} [N^{I}] = d/dX_{K} [N^{i}] . dX_{K}/dX_{j} =  d/dx_{j} [N^{I}] . F^{-1}_{jK}.

Does that help at all? Its a good exercise to derive the variational problem with a fully referential or two-point description, and then with some relatively simple (although tedious) manipulations you would end up with the formulation adopted in the tutorial.

Best,

Jean-Paul

[Claire]

Dear Jean-Paul,

Thank you very much for your answer.
I still have to figure out some of the calculations you suggested me to do, but I got the concepts and the reason why the gradients are defined on the deformed configuration rather than on the initial one.

Best,

Claire

[Jean-Paul Pelteret]

Dear Claire,

You're welcome. If you go through the formulation and still don't have full clarity, then let me know. I do have this derivation documented somewhere...

Regards,

Jean-Paul

[Wolfgang Bangerth]

On 11/08/2016 01:36 AM, Jean-Paul Pelteret wrote:
>
> You're welcome. If you go through the formulation and still don't have full
> clarity, then let me know. I do have this derivation documented somewhere...

Claire -- If, on the other hand, you do figure it out, write it up into a
one-page latex document that we'll be happy to help you convert into something
we could add to the documentation of that tutorial program! If this issue
wasn't clear to you, it's likely also not going to be clear to others.

Cheers
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Claire]

Dear Wolfgang and Jean-Paul,

I will try to do my best to produce the document and give a contribution (even small) to the library.
I am not sure when exactly I will be able to have time to make it though...
The problem that I personally try to solve is simpler and does not require a three field formulation... I actually ask the question in the first place to satisfy my curiosity!
However I'll be glad to think about it, it will certainly be a good way to practice and improve my derivation skills.
But in that case I will accept your offer Jean-Paul, I will for sure use a documented derivation.

Thank you very much by advance,

Best,

Claire

[Andrew McBride]

Dear Claire

You might also be interested in the one-field version of step-44 in the code gallery https://dealii.org/developer/doxygen/deal.II/code_gallery_Quasi_static_Finite_strain_Compressible_Elasticity.html 

A

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[Anup Basak]

Hi all,

The attached pdf might be useful. There is some notational difference from step44, but it is defined here.

Thanks,

Anup

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