Re: [firedrake] Solve a Variational problem in a part of the domain
Hi Anna, Sounds interesting. Please could you provide a bit more detail? all the best --Colin On 13 August 2015 at 12:09, Anna Kalogirou <a.kalogirou@leeds.ac.uk> wrote:
Dear all,
I have a system of equations to solve, which involves three space dependent functions phi, eta, lambda and two constants/scalars Z, W. These need to be solved simultaneously because all the equations involve at least 2 unknowns.
How do I solve that, considering that one of the scalar equations includes a spacial integral of one of the (still unknown) functions? Is it best to define the scalars as Constants?
This problem goes away when I write down the system in a standard FEM formulation, introducing the mass matrix etc. In this case, it is clear that I could solve for that function and then consecutively solve each of the remaining 4 equations. It is not that obvious how I can do that using Firedrake, that is why I thought I would have to solve simultaneously, but then I have the problem described above.
Regards,
Anna.
On 06/08/15 11:04, Lawrence Mitchell wrote:
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Hi Anna,
On 06/08/15 10:37, Anna Kalogirou wrote:
Dear all,
I have a rather simple question but I would like to get some feedback from someone in the Firedrake team.
I am now working on the problem which includes solving a variational problem in a part of the domain only (a water wave problem which includes a floating body).
I can think of a couple of possible solutions on how to solve this: 1. Define two domains and solve the problem separately in each domain. However, I will have to deal with nonzero boundary conditions on the common boundary.
2. I prefer solving the problem in the whole domain, since most of the equations/functions are valid everywhere. Then I can define a Heavyside step function which will be 0 in one part and 1 in the part of the domain I am interested in (under the floating body). I will essentially write down a variational problem valid everywhere, but will actually be zero in a part of the domain.
Is the 2nd step a good approach? The question essentially is how to split a mass matrix M_kl which is defined everywhere, and solve and integral form in a part of the domain only. I think step two is a fine approach. However, note the following issues at present. The way you would have to do this currently is as follows:
Define a DG0 field to hold your indicator function
indicator = Function(DG0)
# Set it to 1 in the appropriate part of the domain indicator.interpolate(...)
# Now use this extra field everywhere when defining your variational # problem.
However, all your integrals are still over the whole domain, you just pick up lots of zeros.
Steadily climbing our todo list (and at an increasing pace, I feel), is the ability to define proper sub domains in a mesh, and then be able to perform integrals over them. Until that time, I think approach 2 is somewhat easier to do than approach 1.
Cheers,
LAwrence -----BEGIN PGP SIGNATURE----- Version: GnuPG v1
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Dr Anna Kalogirou Research Fellow School of Mathematics University of Leeds
http://www1.maths.leeds.ac.uk/~matak/
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-- http://www.imperial.ac.uk/people/colin.cotter www.cambridge.org/9781107663916
Hi Colin, I attached a two-page document, where the system of equations I was talking about in the previous email is (1a)-(1e). I am puzzled with equation (1b) really. I can also eliminate (1b) & (1d) and solve for the remaining (1a), (1c) & (1f), but in this case I still don't know how to deal with (1f) since there is an integral into the integral, which contains the unknown function lambda^{n+1/2}. Thanks, Anna. On 13/08/15 13:47, Colin Cotter wrote:
Hi Anna, Sounds interesting. Please could you provide a bit more detail?
all the best --Colin
On 13 August 2015 at 12:09, Anna Kalogirou <a.kalogirou@leeds.ac.uk <mailto:a.kalogirou@leeds.ac.uk>> wrote:
Dear all,
I have a system of equations to solve, which involves three space dependent functions phi, eta, lambda and two constants/scalars Z, W. These need to be solved simultaneously because all the equations involve at least 2 unknowns.
How do I solve that, considering that one of the scalar equations includes a spacial integral of one of the (still unknown) functions? Is it best to define the scalars as Constants?
This problem goes away when I write down the system in a standard FEM formulation, introducing the mass matrix etc. In this case, it is clear that I could solve for that function and then consecutively solve each of the remaining 4 equations. It is not that obvious how I can do that using Firedrake, that is why I thought I would have to solve simultaneously, but then I have the problem described above.
Regards,
Anna.
On 06/08/15 11:04, Lawrence Mitchell wrote: > -----BEGIN PGP SIGNED MESSAGE----- > Hash: SHA1 > > Hi Anna, > > On 06/08/15 10:37, Anna Kalogirou wrote: >> Dear all, >> >> I have a rather simple question but I would like to get some >> feedback from someone in the Firedrake team. >> >> I am now working on the problem which includes solving a >> variational problem in a part of the domain only (a water wave >> problem which includes a floating body). >> >> I can think of a couple of possible solutions on how to solve this: >> 1. Define two domains and solve the problem separately in each >> domain. However, I will have to deal with nonzero boundary >> conditions on the common boundary. >> >> 2. I prefer solving the problem in the whole domain, since most of >> the equations/functions are valid everywhere. Then I can define a >> Heavyside step function which will be 0 in one part and 1 in the >> part of the domain I am interested in (under the floating body). I >> will essentially write down a variational problem valid everywhere, >> but will actually be zero in a part of the domain. >> >> Is the 2nd step a good approach? The question essentially is how >> to split a mass matrix M_kl which is defined everywhere, and solve >> and integral form in a part of the domain only. > I think step two is a fine approach. However, note the following > issues at present. The way you would have to do this currently is as > follows: > > Define a DG0 field to hold your indicator function > > indicator = Function(DG0) > > # Set it to 1 in the appropriate part of the domain > indicator.interpolate(...) > > # Now use this extra field everywhere when defining your variational > # problem. > > However, all your integrals are still over the whole domain, you just > pick up lots of zeros. > > Steadily climbing our todo list (and at an increasing pace, I feel), > is the ability to define proper sub domains in a mesh, and then be > able to perform integrals over them. Until that time, I think > approach 2 is somewhat easier to do than approach 1. > > Cheers, > > LAwrence > -----BEGIN PGP SIGNATURE----- > Version: GnuPG v1 > > iQEcBAEBAgAGBQJVwzElAAoJECOc1kQ8PEYvGswIAN+Mr6hD+U0NmsnZAGminaq+ > j/SqmI+o7GlbXRSgPqEfrpyWwFiiFVN6DnPmTboRooZr3h93ZhGDhECQ/O12XKA/ > 00yNuCiDn4YBaXJamWBM6u8ILdZYGMyVNTG1JOaZZIMBoiIkrTyYpXatGApcE4sD > 4T+DoAYSUEY7u6MV4eJmRc1lzZEd8Rw43YP+Viy0itCs2jKV+HsDpvpvu1DU/FSQ > zMeE/cRR8It6aImk1L3OON8Zo2ZB0HNirK6YuD3sMkIoqLEYANnNl8qaeU3LL227 > 3TL4ddkALXQLGaYGo5ETgtMKI6afNDIsfjwGUcVRhTTxHM3GKOMQgWV6miNtgv4= > =wE25 > -----END PGP SIGNATURE----- > > _______________________________________________ > firedrake mailing list > firedrake@imperial.ac.uk <mailto:firedrake@imperial.ac.uk> > https://mailman.ic.ac.uk/mailman/listinfo/firedrake
--
Dr Anna Kalogirou Research Fellow School of Mathematics University of Leeds
http://www1.maths.leeds.ac.uk/~matak/ <http://www1.maths.leeds.ac.uk/%7Ematak/>
_______________________________________________ firedrake mailing list firedrake@imperial.ac.uk <mailto:firedrake@imperial.ac.uk> https://mailman.ic.ac.uk/mailman/listinfo/firedrake
-- http://www.imperial.ac.uk/people/colin.cotter
www.cambridge.org/9781107663916 <http://www.cambridge.org/9781107663916>
_______________________________________________ firedrake mailing list firedrake@imperial.ac.uk https://mailman.ic.ac.uk/mailman/listinfo/firedrake
-- Dr Anna Kalogirou Research Fellow School of Mathematics University of Leeds http://www1.maths.leeds.ac.uk/~matak/
participants (2)
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Anna Kalogirou
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Colin Cotter