Hi Matthew,
I'm glad that you find step-44 to be a useful tutorial! Let me try to
answer your questions directly.
/My first question deals with the statement "The Euler-Lagrange
equations corresponding to the residual"/
/
/
/Directly above this sentence is the residual, whose derivation I
understand. Where I am lost is that s tact on to equations. I
have//only one residual equation. I cannot bridge that disconnect. /
This is just another way of saying that the three equations listed there
(these E-L equations) are the strong form of the governing equations.
Basically, if you take each of these equations (along the way, modifying
the definition of the stress in the equilibrium equation), test them
with the appropriate test function and sum up the three residual
contributions then you recover the (total) residual, or stationary point
of the residual, that is listed above. The point is that it not
necessarily so straight forward to go from the strong form to the weak
form for this mixed formulation, so identifying the conservation
equations a-postori is a helpful sanity check here. They seem to align
with what we're trying to do here.
/I am completely lost here. What is the significance of p and J not
having derivatives on them that makes it "easy" to solve for those
terms in isolation?
/
Well, that's a valid point. I can't quite recall what exactly we were
trying to identify with this comment. I'll think about it, as that seems
to be a point that we could improve on in the documentation. That there
is no K_{pp} contribution is significant, because it makes condensing
out the p and J fields easier. Maybe we meant to refer to the lack of
contribution to K_{pp} (as there is no second derivative involving a
variation and linearisation of p.
/I am also confused how K_pJ, K_Jp and K_JJ form a block diagonal
matrix. I could get there if I ignore the top row but then the
equations below do not make sense I think. Some detail on this part
of the process would be great. /
So this is more easy to explain. We specifically choose discontinuous
shape function to discretise these fields. As there are no
interface/flux contributions, all local element contributions for these
terms will remain local and you therefore end up with an assembled
matrix for these contributions that has a block-like structure. The
K_{JJ} matrix is evidently block-diagonal, as it is a field that couples
with itself. As for the coupling matrices K_{pJ} and K_{Jp}, they are
block-diagonal because we chose *exactly* the same discretisation for
both fields (i.e. the shape functions and polynomial order match).
Does that make sense?
Best,
Jean-Paul
On 2022/09/30 19:16, Matthew Rich wrote:
Hi,
Step-44 has a lot going on and really sets you up to tackle a variety
of real world problems. There is a significant jump in complexity
between this tutorial and steps 8,17 & 18.
I have been reading the references and I am about there but need few
clarifications for why things are the way they are.
Any assistance would be much appreciated...
My first question deals with the statement "The Euler-Lagrange
equations corresponding to the residual"
Directly above this sentence is the residual, whose derivation I
understand. Where I am lost is that s tact on to equations. I have
only one residual equation. I cannot bridge that disconnect.
My other question has to deal with the solving procedure for K (see
snippet below)
Ksolve.png
I am completely lost here. What is the significance of p and J not
having derivatives on them that makes it "easy" to solve for those
terms in isolation? I am also confused how K_pJ, K_Jp and K_JJ form a
block diagonal matrix. I could get there if I ignore the top row but
then the equations below do not make sense I think. Some detail on
this part of the process would be great.
Thanks in advance,
Matt
--
The deal.II project is located at http://www.dealii.org/
For mailing list/forum options, see
https://groups.google.com/d/forum/dealii?hl=en
---
You received this message because you are subscribed to the Google
Groups "deal.II User Group" group.
To unsubscribe from this group and stop receiving emails from it, send
an email to dealii+unsubscr...@googlegroups.com.
To view this discussion on the web visit
https://groups.google.com/d/msgid/dealii/a090a200-fa7a-44d9-ba59-6c5e6064fd5bn%40googlegroups.com
<https://groups.google.com/d/msgid/dealii/a090a200-fa7a-44d9-ba59-6c5e6064fd5bn%40googlegroups.com?utm_medium=email&utm_source=footer>.
--
The deal.II project is located at http://www.dealii.org/
For mailing list/forum options, see
https://groups.google.com/d/forum/dealii?hl=en
---
You received this message because you are subscribed to the Google Groups "deal.II User Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to dealii+unsubscr...@googlegroups.com.
To view this discussion on the web visit
https://groups.google.com/d/msgid/dealii/b5c0a491-7b87-8a56-6c08-36ff5d345377%40gmail.com.
