[firedrake] Solve a Variational problem in a part of the domain
Anna Kalogirou
a.kalogirou at leeds.ac.uk
Fri Aug 14 11:55:52 BST 2015
Hi,
I was planning to define the heavyside function as a DG0 function.
Any ideas about solving equation (1b), which contains a time-update for
both a function and a scalar?
Thanks,
Anna.
On 13/08/15 18:35, Colin Cotter wrote:
> Hi Anna,
> It *was* interesting!
>
> How do you intend to compute the integrals against the Heaviside
> function? The Heaviside function is non-polynomial so can't be
> computed exactly by quadrature.
>
> You are right that our standard setup can't cope with this situation
> currently. What we really need are function spaces that are defined to
> be constant in an entire column. We should try to sketch out with
> David and Lawrence what might be required in the infrastructure to
> support this. I can see it being useful for many other free surface
> type problems as well.
>
> all the best
> --Colin
>
> On 13 August 2015 at 15:36, Anna Kalogirou <a.kalogirou at leeds.ac.uk
> <mailto:a.kalogirou at leeds.ac.uk>> wrote:
>
> 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 at leeds.ac.uk <mailto:a.kalogirou at 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
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>> > =wE25
>> > -----END PGP SIGNATURE-----
>> >
>> > _______________________________________________
>> > firedrake mailing list
>> > firedrake at imperial.ac.uk <mailto:firedrake at 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/>
>>
>>
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>>
>>
>>
>>
>> --
>> http://www.imperial.ac.uk/people/colin.cotter
>>
>> www.cambridge.org/9781107663916
>> <http://www.cambridge.org/9781107663916>
>>
>>
>>
>>
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>
> --
>
> 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/>
>
>
>
>
> --
> http://www.imperial.ac.uk/people/colin.cotter
>
> www.cambridge.org/9781107663916 <http://www.cambridge.org/9781107663916>
>
>
>
>
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