[petsc-users] Rigid body nullspace for Stokes operator

[Amneet Bhalla]

I think the nullspace for the velocity operator is of the form

vnull = U + ω × r
in which U is a rigid body velocity and ω is the rigid body rotational
velocity, and r is the radius vector from the center of mass. I believe I
need to construct 6 nullspace vectors in 3D and 3 nullspace vectors in 2D.
Sounds correct? Also does the center of mass coordinates matter when
defining r?

On Wed, Oct 30, 2024 at 7:53 AM Amneet Bhalla <mail2amneet at gmail.com> wrote:

> @Mark: Is there some document/paper that I can follow to check the algebra
> of these zero eigenvectors/null space modes?
>
> @Jed : We use a projection method preconditioner to solve the coupled
> velocity pressure system as described here (
> https://urldefense.us/v3/__https://www.sciencedirect.com/science/article/pii/S0021999123004205__;!!G_uCfscf7eWS!b_U_d3oTZckny15inoAStJf_DkxTjYjj_yN7UJJPIs82UiIP-TYpBuEhMzrLsOWU6-Zv4dvrTgU9VJWe_WVwhigemQ$ ). It
> is an approximation of the Schur complement. As a part of projection
> preconditioner, we need to solve just the momentum equation separately
> without considering the pressure part.
>
> On Tue, Oct 29, 2024 at 8:03 PM Jed Brown <jed at jedbrown.org> wrote:
>
>> And to be clear, we recommend using fieldsplit Schur to separate the
>> pressure and velocity part (there are many examples). Applying AMG directly
>> to the saddle point problem will not be a good solver because the
>> heuristics assume positivity and do not preserve inf-sup stability (nor do
>> standard smoothers).
>>
>> Mark Adams <mfadams at lbl.gov> writes:
>>
>> > This is linear elasticity and there are 6 "null" vectors (if you removed
>> > Dirichlet boundary conditions these are eigenvectors with zero
>> eigenvalue):
>> > 3 translations, x, y, z, and three rotatiions xx, yy ,zz.
>> > x = (1,0,0,1,0,0,1,0 ...)
>> > and xx is something like (0, z_1, -y_1, 0, z_2, -y_2, ...) where z_1 is
>> the
>> > z coordinate of the first vertex, etc.
>> >
>> > Mark
>> >
>> > On Tue, Oct 29, 2024 at 3:47 PM Amneet Bhalla <mail2amneet at gmail.com>
>> wrote:
>> >
>> >> Hi Mark,
>> >>
>> >> Thanks! I am not sure how to construct null space and zero energy modes
>> >> manually for this operator. Is there some theory or documentation I can
>> >> follow to figure out what the null space and zero energy modes look
>> like
>> >> for this operator? Once I know what these are in symbolic form, I
>> think I
>> >> should be able to construct them manually.
>> >>
>> >> Best,
>> >> --Amneet
>> >>
>> >> On Tue, Oct 29, 2024 at 7:35 AM Mark Adams <mfadams at lbl.gov> wrote:
>> >>
>> >>> Oh my mistake. You are using staggered grids. So you don't have a
>> block
>> >>> size that hypre would use for the "nodal" methods.
>> >>> I'm not sure what you are doing exactly, but try hypre and you could
>> >>> create the null space, zero energy modes, manually, attach to the
>> matrix
>> >>> and try GAMG.
>> >>> You can run with '-info :pc' and grep on GAMG to see if GAMG is
>> picking
>> >>> the null space up (send this output if you can't figure it out).
>> >>>
>> >>> Thanks,
>> >>> Mark
>> >>>
>> >>> On Tue, Oct 29, 2024 at 9:28 AM Mark Adams <mfadams at lbl.gov> wrote:
>> >>>
>> >>>> This coordinate interface is just a shortcut for vertex based
>> >>>> discretizations with 3 dof per vertex, etc. (maybe works in 2D).
>> >>>> You will need to construct the null space vectors manually and
>> attach it
>> >>>> to the matrix. Used by GAMG.
>> >>>>
>> >>>> Note, for hypre you want to use the "nodal" options and it does not
>> use
>> >>>> these null space vectors. That is probably the way you want to go.
>> >>>> eg: -pc_hypre_boomeramg_nodal_coarsen
>> >>>>
>> >>>> I would run with hypre boomerang and -help and grep on nodal to see
>> all
>> >>>> the "nodal" options and use them.
>> >>>>
>> >>>> Thanks,
>> >>>> Mark
>> >>>>
>> >>>>
>> >>>> On Mon, Oct 28, 2024 at 8:06 PM Amneet Bhalla <mail2amneet at gmail.com
>> >
>> >>>> wrote:
>> >>>>
>> >>>>> Hi Folks,
>> >>>>>
>> >>>>> I am trying to solve the momentum equation in a projection
>> >>>>> preconditioner using GAMG or Hypre solver. The equation looks like
>> for
>> >>>>> velocity variable *v* looks like:
>> >>>>>
>> >>>>>
>> >>>>>  [image: Screenshot 2024-10-28 at 4.15.17 PM.png]
>> >>>>>
>> >>>>> Here, μ is spatially varying dynamic viscosity and λ is spatially
>> >>>>> varying bulk viscosity. I understand that I need to specify rigid
>> body
>> >>>>> nullspace modes to the multigrid solver in order to accelerate its
>> >>>>> convergence. Looking into this routine
>> MatNullSpaceCreateRigidBody() (
>> >>>>>
>> https://urldefense.us/v3/__https://petsc.org/release/manualpages/Mat/MatNullSpaceCreateRigidBody/__;!!G_uCfscf7eWS!bVR6duCoDqPhZrWS-sm1c5qxsFPjZMhdT86AqLpPzWgVy5qoRhd4_Jue2LJOIS6LRrtV2cHGrqger1Yvb-Y5f-0$
>> >>>>> <
>> https://urldefense.us/v3/__https://petsc.org/release/manualpages/Mat/MatNullSpaceCreateRigidBody/__;!!G_uCfscf7eWS!eKqgIJjCdMzIU76f7X65AmGxrU_-lC7W02BMWafJ77DNf_IuQk6O1X3qU1x9Ez8NJ20vZEL-mF6T1yNmDnwv0eWa2w$
>> >),
>> >>>>> I see that I need to provide the coordinates of each node. I am
>> using
>> >>>>> staggered grid discretization. Do I need to provide coordinates of
>> >>>>> staggered grid locations?
>> >>>>>
>> >>>>> Thanks,
>> >>>>> --
>> >>>>> --Amneet
>> >>>>>
>> >>>>>
>> >>>>>
>> >>>>>
>> >>
>> >> --
>> >> --Amneet
>> >>
>> >>
>> >>
>> >>
>>
>
>
> --
> --Amneet
>
>
>
>

-- 
--Amneet
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