RE: [deal.II] Questions on Step-18

Michael Li Sat, 26 Jun 2021 18:26:24 -0700

Hi Jean-Paul,

Thank you for the nice explanation and information! It cleared out my concern and helped me understand the related concepts of vorticity and rotation. Vorticity (1/2 curl of velocity) stands for the rotation rate (angular velocity) but the infinitesimal rotation tensor gives the rotation angle (curl of displacement). Do correct me if I’m wrong.

Now I’m excited to learn how to make my first patch.

- Michael

From: Jean-Paul Pelteret
Sent: Saturday, June 26, 2021 2:57 PM
To: dealii@googlegroups.com
Subject: Re: [deal.II] Questions on Step-18

Hi Michael,

What’s written in the implementation suggests that computing the curl of the displacement (increments) is the right thing to do in this instance:

Nevertheless, if the material was prestressed in a certain direction, then this direction will be rotated along with the material. To this end, we have to define a rotation matrix R(Δun) that describes, in each point the rotation due to the displacement increments. It is not hard to see that the actual dependence of R on Δun can only be through the curl of the displacement, rather than the displacement itself or its full gradient (as mentioned above, the constant components of the increment describe translations, its divergence the dilational modes, and the curl the rotational modes).

I think that this is because the displacement field for solids is analogous to the velocity field for fluids, for which that equation in the Wiki link was expressly written (Compare incompressible linear elasticity to Stokes flow — the equations have the same form). Furthermore, having dug through my continuum mechanics notes, I now see that this specific rotation matrix is called the “infinitesimal rotation tensor” and is defined as the antisymmetric part of \grad u. Since step-18 is dealing with incremental updates, the incremental form of the rotation tensor is computed.

Sorry for the confusion. So, to my understanding it’s just the factor of 1/2 that needs to be added.

Best,

Jean-Paul

On 26. Jun 2021, at 16:14, Michael Lee <lianxi...@gmail.com> wrote:

I would be happy to do the comparison and make the fix. But before doing that, I want to make sure I understand the formula correctly.

When we calculate the rotation matrix, it seems that the du (incremental displacement) is used.
          // Then initialize the <code>FEValues</code> object on the present
          // cell, and extract the gradients of the displacement at the
          // quadrature points for later computation of the strains
          fe_values.reinit(cell);
          fe_values.get_function_gradients(incremental_displacement,
                                           displacement_increment_grads);

Should we use the velocity ( $\vec{v} = \frac{d \vec{u}}{dt}$ ), since $\vec{\omega} = \frac{1}{2}\nabla \times \vec{v} = \frac{1}{2}\nabla \times \frac{d\vec{u}}{dt}$ ?

Can you check this as well?

Best,
Michael

On Sat, Jun 26, 2021 at 1:48 AM Jean-Paul Pelteret <jppelte...@gmail.com> wrote:
Following Andrew’s explanation, I suppose that this is relation for which we’re lacking the factor of 1/2, right?

https://en.wikipedia.org/wiki/Angular_velocity#Angular_velocity_as_a_vector_field

If so, then maybe we should link to this equation in the tutorial documentation too.

If this is wrong in the deal.II code, would you be interested in writing a patch to correct this? It should be an easy fix, and a good first patch to contribute to the library!

I concur — it would be great if you’d be willing to write a patch that fixes this! It would be interesting to see the effect on the results of the tutorial.

Best,
Jean-Paul

On 26. Jun 2021, at 05:58, Wolfgang Bangerth <bange...@colostate.edu> wrote:

On 6/24/21 6:32 PM, Michael Li wrote:

Andrew, thanks for confirming that. The missing 1/2 does not affect the demonstration of functionalities of deal.II but it may change the results.

If this is wrong in the deal.II code, would you be interested in writing a patch to correct this? It should be an easy fix, and a good first patch to contribute to the library!

If you're curious about ho to write a patch, take a look at
https://github.com/dealii/dealii/wiki/Contributing

Best
W.

On Sat, Jun 26, 2021 at 1:48 AM Jean-Paul Pelteret <jppelte...@gmail.com> wrote:
Following Andrew’s explanation, I suppose that this is relation for which we’re lacking the factor of 1/2, right?

https://en.wikipedia.org/wiki/Angular_velocity#Angular_velocity_as_a_vector_field

If so, then maybe we should link to this equation in the tutorial documentation too.

If this is wrong in the deal.II code, would you be interested in writing a patch to correct this? It should be an easy fix, and a good first patch to contribute to the library!

I concur — it would be great if you’d be willing to write a patch that fixes this! It would be interesting to see the effect on the results of the tutorial.

Best,
Jean-Paul

On 26. Jun 2021, at 05:58, Wolfgang Bangerth <bange...@colostate.edu> wrote:

On 6/24/21 6:32 PM, Michael Li wrote:

Andrew, thanks for confirming that. The missing 1/2 does not affect the demonstration of functionalities of deal.II but it may change the results.

If this is wrong in the deal.II code, would you be interested in writing a patch to correct this? It should be an easy fix, and a good first patch to contribute to the library!

If you're curious about ho to write a patch, take a look at
https://github.com/dealii/dealii/wiki/Contributing

Best
W.

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

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RE: [deal.II] Questions on Step-18

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