[petsc-users] Rigid body nullspace for Stokes operator

Wed Oct 30 22:20:36 CDT 2024

I think Mark mentioned this earlier, but I want to make sure that the rigid
body null vectors should be specified only when Neumann boundary conditions
are used on all boundaries of the domain, correct? Alternatively, if a
Dirichlet boundary condition is used (on any part of the domain boundary)
then there is no null space, i.e., the operator is a full rank matrix?

If the above is true, then I think I do not need to specify the rigid body
null modes because I am using Dirichlet boundary conditions for the
velocity solver.

On Wed, Oct 30, 2024 at 12:28 PM Jed Brown <jed at jedbrown.org> wrote:

> Yes to 6 null vectors in 3D, 3 null vectors in 2D. The center of mass does
> not need to be identified because you can algebraically orthogonalize
> (lines 411-420 here).
>
>
> https://urldefense.us/v3/__https://gitlab.com/petsc/petsc/-/blob/main/src/dm/impls/plex/plexfem.c?ref_type=heads*L377-425__;Iw!!G_uCfscf7eWS!ezw7zWJYwCHHUvoAmPZ4JT-HJ8l8V8aC3pq3VbTRke4EO6i5Z_AxjqmaJ1M_6BS6b99zSzfvl8FQIhZEjNHJ84-m6w$ 
>
> See also this implementation with raw coordinates. GAMG orthogonalizes
> within each aggregate (in a later phase of the algorithm) so global
> orthogonalization is not necessary.
>
>
> https://urldefense.us/v3/__https://gitlab.com/petsc/petsc/-/blob/main/src/ksp/pc/impls/gamg/agg.c?ref_type=heads*L387__;Iw!!G_uCfscf7eWS!ezw7zWJYwCHHUvoAmPZ4JT-HJ8l8V8aC3pq3VbTRke4EO6i5Z_AxjqmaJ1M_6BS6b99zSzfvl8FQIhZEjNFr-J8G0w$ 
>
> Amneet Bhalla <mail2amneet at gmail.com> writes:
>
> > 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!ezw7zWJYwCHHUvoAmPZ4JT-HJ8l8V8aC3pq3VbTRke4EO6i5Z_AxjqmaJ1M_6BS6b99zSzfvl8FQIhZEjNFHB5gzxg$ ).
> 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
>

-- 
--Amneet
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