Hi Jean-Paul and All,
I would like to discuss the formulas in step-18 further.
- In the code, the angles are computed as follows.
For 2D,
const double curl = (grad_u[1][0] - grad_u[0][1]);
const double angle = std::atan(curl);
For 3D,
const double tan_angle = std::sqrt(curl * curl);
const double angle = std::atan(tan_angle);
I don’t really understand why we need to do the *atan* operation to get the
angle. It seems to me that half of the curl of displacement itself is the
rotation angle.
Moreover, for 2D, do we need to do the same as 3D tan_angle =
std::sqrt(curl * curl)?
I mean we first calculate the magnitude of the curl then calculate the
angle. I’m concerned about the sign of the curl in 2D angle = std::atan(curl)
(we need the magnitude of the curl).
I wonder what references I can get from these formulas. I hope I can be
fully convinced of the stuff.
- To be specific, I want to make sure where I should put the factor of ½
to make the patch. I.e., which is correct in the following?
1. const double tan_angle = std::sqrt(0.5 * curl * 0.5 * curl);
or
1. const double tan_angle = 0.5 * std::sqrt(curl * curl);
- In my opinion, the code should be as follows.
For 2D,
const double curl = (grad_u[1][0] - grad_u[0][1]);
const double angle = 1.0 / 2.0 * std::sqrt(curl * curl); // Half of the
magnitude of curl is the rotation angle.
For 3D,
const double angle = 1.0 / 2.0 * std::sqrt(curl * curl);
const double angle = std::atan(tan_angle);
Any comments will be of great help.
Best,
Michael
On Mon, Jun 28, 2021 at 2:21 AM Andrew McBride <[email protected]>
wrote:
> My view here is that the problem is quasi-static and hence time serves to
> order events. Hence a displacement increment can be viewed as equivalent to
> the velocity.
>
> For example, let’s fix the number of time-steps to 10. If the total time
> is 1s or 10s we should get the same results (assuming the timing of the
> application of loading is scaled based on the assumption that time simply
> order events).
>
> Best,
> Andrew
>
> On 27 Jun 2021, at 02:19, Michael Li <[email protected]> wrote:
>
> 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 <[email protected]>
> *Sent: *Saturday, June 26, 2021 2:57 PM
> *To: *[email protected]
> *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*(Δ*u**n*) 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 Δ*u**n* 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
> <https://antoniobilotta-structuralengineer.github.io/MyWebSite/SMbook_en/infinitesimal_sec_strain_chap_en.html>”
> 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 <[email protected]> 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 *([image: \vec{v} = \frac{d \vec{u}}{dt}]),
> since [image: \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 <[email protected]>
> 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 <[email protected]>
> 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 <[email protected]>
> 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 <[email protected]>
> 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.
>
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: [email protected]
> <[email protected]>
> www: http://www.math.colostate.edu/~bangerth/
>
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