# Applying external pressure on a large deformed media

**URL:** <https://fenicsproject.discourse.group/t/applying-external-pressure-on-a-large-deformed-media/2577>\
**Category:** variational formulation\
**Created:** [March 6, 2020, 6:44pm UTC](https://fenicsproject.discourse.group/t/applying-external-pressure-on-a-large-deformed-media/2577 "2020-03-06T18:44:10Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![amir](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/amir/32/683_2.png) [@amir](https://fenicsproject.discourse.group/u/amir)\
**Post date:** [March 6, 2020, 6:44pm UTC](https://fenicsproject.discourse.group/t/applying-external-pressure-on-a-large-deformed-media/2577/1 "2020-03-06T18:44:10Z")

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I need to solve a hyperelastic problem in the form seen in the [FEniCS tutorial.](https://fenicsproject.org/docs/dolfin/2019.1.0/python/demos/hyperelasticity/demo_hyperelasticity.py.html)

\Pi = \int\_{\Omega} \psi(u) \, {\rm d} x - \int\_{\Omega} B \cdot u \, {\rm d} x - \int\_{\partial\Omega} T \cdot u \, {\rm d} s.

The problem I have with this variational formulation is that it considers that the traction force is applied to the reference area. Since the Neumann boundary conditions in my code are in the form of external pressures, I will need the traction force to be applied to the deformed area, not the undeformed one. I just wanted to ask how do I need to modify the equation above to be compatible with my simulations.

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**Author:** ![kamensky](https://avatars.discourse-cdn.com/v4/letter/k/e95f7d/32.png) [@kamensky](https://fenicsproject.discourse.group/u/kamensky)\
**Post date:** [March 6, 2020, 7:24pm UTC](https://fenicsproject.discourse.group/t/applying-external-pressure-on-a-large-deformed-media/2577/2 "2020-03-06T19:24:49Z")

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You would need to use Nanson’s formula (as derived clearly in a [Wikipedia article](https://en.wikipedia.org/wiki/Finite_strain_theory#Transformation_of_a_surface_and_volume_element)) to obtain \mathbf{n}da, where \mathbf{n} is the deformed normal (i.e., the direction in which a pressure traction would act) and da is the measure for integration over the deformed surface. In the notation of the linked Wikipedia article, “dA” would correspond to `ds` in a total Lagrangian implementation (like the hyperelasticity demo), and \mathbf{N} would be `FacetNormal(mesh)`.

Also, you would have to add the traction term directly to the residual to get a problem like

D\_\mathbf{v}\Pi - \int\_{\partial\Omega}\mathbf{t}\cdot\mathbf{v}\,da = 0\text{ ,}

instead of adding \int\_{\partial\Omega}\mathbf{T}\cdot\mathbf{u}\,dA to \Pi. Note that

D\_{\mathbf{v}}\int\_{\partial\Omega}\mathbf{t}\cdot\mathbf{u}\,da \neq \int\_{\partial\Omega}\mathbf{t}\cdot\mathbf{v}\,da\text{ ,}

because \mathbf{t} and da also depend on \mathbf{u}.

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**Author:** ![amir](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/amir/32/683_2.png) [@amir](https://fenicsproject.discourse.group/u/amir)\
**Post date:** [March 9, 2020, 10:27pm UTC](https://fenicsproject.discourse.group/t/applying-external-pressure-on-a-large-deformed-media/2577/3 "2020-03-09T22:27:50Z")

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Thanks a lot, I did exactly as you said and yeah it works perfectly.

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**Author:** ![swatisharmapanda](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/swatisharmapanda/32/109_2.png) [@swatisharmapanda](https://fenicsproject.discourse.group/u/swatisharmapanda)\
**Post date:** [April 1, 2021, 5:45am UTC](https://fenicsproject.discourse.group/t/applying-external-pressure-on-a-large-deformed-media/2577/4 "2021-04-01T05:45:08Z")

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Hi,

I am also encountering the same issue. Could you please send me the corrected FEnics code?

Thanks for raising the issue.

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**Author:** ![PreGMjr](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/pregmjr/32/4762_2.png) [@PreGMjr](https://fenicsproject.discourse.group/u/PreGMjr)\
**Post date:** [May 30, 2023, 10:00pm UTC](https://fenicsproject.discourse.group/t/applying-external-pressure-on-a-large-deformed-media/2577/5 "2023-05-30T22:00:14Z")

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I have a question please, I wasn’t able to derive this part :

> [@kamensky](#):
>
> D\_v Π − ∫\_{∂Ω} \pmb t⋅\pmb v da=0

How was the second term derived on the current configuration, since the derivation was done on Π defined on the original configuration, unless we have the property ∫\_{∂Ω{(t)}} \pmb t⋅\pmb v da = ∫\_{∂Ω\_0} \pmb T⋅\pmb v dA ? (in which case we don’t need a transformation ?)

What I could find is \pmb T = \pmb P . \pmb N = J ∣ \pmb F ^{-T} \pmb N ∣ \pmb t on ∂Ω\_0 (also same formula is found in [Cardiovascular Mathematics, 2009](https://link.springer.com/book/10.1007/978-88-470-1152-6) ) using also Nanson’s formula, giving eventually

D\_v Π − ∫\_{∂Ω\_0} J ∣ \pmb F ^{-T} \pmb N ∣ \pmb t dA = 0 .  
Do you find the same formula ? (But that assumes that D\_v Π − ∫\_{∂Ω\_0} \pmb T⋅\pmb v dA=0 and not D\_v Π − ∫\_{∂Ω} \pmb t⋅\pmb v da=0

I appreciate the help. Thank you in advance.  
@kamensky