-John
Luca Antiga writes:
> Hi guys,
> as promised we took a few images of the 3D lid-driven cavity
> showing the difference it makes
> to impose L2 vs lumped penalty boundary conditions.
> You can take a look at them here
>
> http://villacamozzi.marionegri.it/~luca/HexLid3dL2.png
> http://villacamozzi.marionegri.it/~luca/HexLid3dLumped.png
> http://villacamozzi.marionegri.it/~luca/TetLid3dL2.png
> http://villacamozzi.marionegri.it/~luca/TetLid3dlumped.png
>
> where the names of the images explain which case is which.
> As you can see, imposing lumped boundary conditions does make a
> difference.
> In particular, apart from the spurious non-zero velocities on the
> side walls, the top
> velocity is 1 for lumped bcs, while it overshoots by 20% in case of
> L2 in case of Tet10
> elements. We get a bit less pronounced overshoots (15%) with hexas
> and L2.
>
> We tried lowering the absolute threshold of the linear system
> solvers, with no appreciable
> changes.
>
> Luca
>
> --
> Luca Antiga, PhD
> Head, Medical Imaging Unit,
> Bioengineering Department,
> Mario Negri Institute
> email: antiga@marionegri.it
> web: http://villacamozzi.marionegri.it/~luca
> mail: Villa Camozzi, 24020, Ranica (BG), Italy
> phone: +39 035 4535-381
>
>
> On Apr 4, 2007, at 5:09 PM, John Peterson wrote:
>
> > Luca Antiga writes:
> >> Hi,
> >>> One thing that seemed to help a bit (at least in 2D) was to use the
> >>> "lumped" L2 projection (e.g. ex13). If memory serves, that solution
> >>> "looked" slightly better (at least in 2D).
> >> We confirm that this helps also in 3D. We'll show you images as soon
> >> as Lorenzo is around.
> >> The situation changes sensibly, you see much fewer non-zero vectors
> >> on the no-slip faces.
> >>
> >> Ultimately we'll work with blood vessels, so we won't have
> >> discontinuous boundary conditions.
> >> However, I can tell you that we tried on a vessel with parabolic
> >> inlet conditions (that go to zero
> >> at the wall, so no discontinuity). With L2 boundary conditions, the
> >> first ring of refined nodes on the
> >> no-slip side wall connected to the inlet face had non-zero velocity
> >> vectors pointing opposite
> >> to the inlet flow direction.
> >
> > Hmm... and the lumped L2-projection looks slightly better than that?
> > I will try implementing the "pinwheel" type forcing function for the
> > Stokes problem with homogeneous zero Dirichlet BCs in 2D using
> > adaptivity,
> > and see what that gives us.
> >
> >>>> Yes, but then Gibbs' type ringing usually is large. What's
> >>>> worse: as
> >>>> you refine the mesh the boundary condition approximation will
> >>>> converge
> >>>> in the L2 norm, but it can't converge in L_infinity..
> >>
> >> I don't know the details, so I'm not sure it makes sense, but it
> >> looks to me like the presence and size of
> >> over-undershoots depends on the size of the coarse-level mesh. Could
> >> this be related to the way
> >> hanging node constraints are handled (L2)?
> >> Just to get more insight: will a uniformly AMR refined mesh behave
> >> differently than a non-refined
> >> mesh with the same sized elements as the fine AMR ones? How much of
> >> an influence will the coarse-level
> >> mesh have?
> >
> > It shouldn't depend on what mesh you start with, unless it was coarse
> > enough to have Galerkin-instability type oscillations. In that case,
> > local cell-Reynolds number violations can cause over/undershoots (for
> > Navier-Stokes obviously, not Stokes flow).
> >
> > In answer to your second question, a "uniformly-refined AMR mesh"
> > should behave identically to a non-refined mesh with the same size
> > elements. I would be interested in seeing a case where it does not,
> > if you have one.
> >
> > -J
> >
> >
> >
> >> Thanks a lot for your time
> >>
> >> Luca
> >>
> >> --
> >> Luca Antiga, PhD
> >> Head, Medical Imaging Unit,
> >> Bioengineering Department,
> >> Mario Negri Institute
> >> email: antiga@marionegri.it
> >> web: http://villacamozzi.marionegri.it/~luca
> >> mail: Villa Camozzi, 24020, Ranica (BG), Italy
> >> phone: +39 035 4535-381
> >>
> >>
> >> On Apr 4, 2007, at 4:22 PM, John Peterson wrote:
> >>
> >>> Roy Stogner writes:
> >>>> On Wed, 4 Apr 2007, Luca Antiga wrote:
> >>>>
> >>>>> I could live with non zero values on the zero velocity faces, but
> >>>>> aren't
> >>>>> those a bit large (10% of the top face velocity)?
> >>>>
> >>>> Yes, but then Gibbs' type ringing usually is large. What's
> >>>> worse: as
> >>>> you refine the mesh the boundary condition approximation will
> >>>> converge
> >>>> in the L2 norm, but it can't converge in L_infinity..
> >>>>
> >>>>> I'm just worried that the situation might go out of control in
> >>>>> complicated geometries, so that's why I'm a bit picky on this
> >>>>> problem.
> >>>>
> >>>> It's not the geometry that's controlling the problem, it's the
> >>>> discontinuity in the boundary conditions.
> >>>
> >>>> I think John's had somewhat smoother results by using an H1 instead
> >>>> of L2 boundary penalty, but when you try to force a continuous
> >>>> approximation function to take on discontinuous values, there's
> >>>> really no good way for it to react.
> >>>
> >>> Actually, I couldn't get the H1 projection to work (in 2D). I tried
> >>> penalizing the tangential derivatives, to e.g. enforce du/dx=dv/dx=0
> >>> along the lid.
> >>>
> >>> One thing that seemed to help a bit (at least in 2D) was to use the
> >>> "lumped" L2 projection (e.g. ex13). If memory serves, that solution
> >>> "looked" slightly better (at least in 2D).
> >>>
> >>>> Many people give up and just regularize the problem boundary
> >>>> conditions.
> >>>
> >>> Indeed, the hyperbolic tangent-type regularized boundary conditions
> >>> seem to be
> >>> fairly standard in the literature for this type of problem. If we
> >>> just
> >>> want to be sure Stokes+AMR+Tets is working, let's try a test problem
> >>> with continuous data?
> >>>
> >>> -J