[firedrake] Solve a Variational problem in a part of the domain

Thu Aug 13 18:35:52 BST 2015

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> 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>
> 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 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/
>>
>>
>> _______________________________________________
>> firedrake mailing list
>> firedrake at imperial.ac.uk
>> https://mailman.ic.ac.uk/mailman/listinfo/firedrake
>>
>
>
>
> --
> http://www.imperial.ac.uk/people/colin.cotter
>
> www.cambridge.org/9781107663916
>
>
>
>
> _______________________________________________
> firedrake mailing listfiredrake at imperial.ac.ukhttps://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://www.imperial.ac.uk/people/colin.cotter

www.cambridge.org/9781107663916
-------------- next part --------------
HTML attachment scrubbed and removed