[aspect-devel] Internal heating in aspect (Ludovic Jeanniot)
Rene Gassmoeller
rene.gassmoeller at mailbox.org
Wed Aug 29 12:16:22 PDT 2018
Hi all,
I think by now we have a pretty good understanding of the problem,
however there is no clear path forward yet. I have done some further
testing with the shell_simple_3d cookbook:
- As Max pointed out the artificial viscosity at resolution <= 2
global refinement in spherical models is bigger than the natural
diffusion, although it reduces drastically with resolution (compare to a
natural conductivity of ~4 W/m*K).
Global refinement / Maximum artificial viscosity timestep 0 / timestep 2:
1 (4 radial elements): 94.1 W/m*K / 59.9 W/m*K
2 (8 radial elements): 48.9 W/m*K / 6.12 W/m*K
3 (16 radial elements): 24.8 W/m*K / 0.92 W/m*K
There are some pitfalls here, in particular the artificial viscosity in
the output of timestep 0 is in general much bigger than in later
timesteps (and it only scales linearly with cell size), because we do
not have a velocity solution yet. Timestep 1 is somewhat unreliable as
well, because the BDF2 scheme is not fully initialized. Nevertheless,
for later timesteps the artificial viscosity scales at least with cell
size squared (h^2) as expected (actually slightly better, because it is
not only h^2, but also based on the residual of the equation, which
reduces as well, ideally with h^3). This means higher resolutions should
significantly reduce the total amount of artificial diffusion, although
it might still be significant (>1% of total diffusion) up to resolution
4 or so, which seems unacceptably expensive. Moreover, as Scott and
Cedric mentioned there will always be an imbalance between the heat
fluxes at the top and bottom boundary with this method, unless we use
radially uniformly spaced elements, or disable the stabilization,
although the magnitude of the imbalance should reduce with increased
resolution. Also the thermal conductivity within the domain will be
unequally distributed (artificial conductivity in the upper mantle will
be higher than in the lower mantle).
While the above point suggests that the artificial viscosity scheme is
correctly implemented, there still exist three points for possible
improvements:
- As Wolfgang mentioned the parameters that were originally chosen for
the artificial viscosity scheme were different from what they are today,
because we noticed that the original smoothing was too weak to stabilize
oscillations in compositional fields. It is completely possible that the
original values were sufficient for the temperature (which already has
natural diffusion), and in hindsight I should have thought of just using
different parameters for composition and temperature (though that was 6
years ago, and I was just a 2nd year PhD student at the time I worked on
modifying the parameters for stabilizing the composition equation). I
can easily make a change that allows for different parameters for
temperature and composition, which would allow everyone to test their
favorite values, without risking oscillations in compositional fields.
- Even if we can reduce the parameters it is of course possible that as
Max pointed out the anisotropic nature of the diffusion in SUPG is more
appropriate / "less wrong" than the isotropic entropy viscosity for our
problem. As Juliane and Timo pointed out we could reimplement the
content of https://github.com/geodynamics/aspect/pull/412 and see if
SUPG performs better for low resolutions. Does anyone know if the
"artificial viscosity" of SUPG also scales with h^2? Because if it is
linear, we might implement something that is better for low resolutions,
but worse for high resolutions.
- As Max mentioned: We should not need a stabilization for pure
conduction problems (where velocity is 0), and should modify the
algorithm accordingly.
So the next steps could be:
1. Allow for different stabilization parameters for temperature and
composition, and check which values are still stable.
2. Do not stabilize advection/diffusion solutions where the velocity is
zero (because it is only a diffusion equation).
3. Reimplement the SUPG based on
https://github.com/geodynamics/aspect/pull/412 and see how it performs
(at low and high resolutions).
Does that summarize our discussion appropriately? I can easily make the
code adjustments 1 and 2 (they are easy and useful in any case), and
could also look into creating an initial version of 3 (although it would
take a bit of time), but I currently do not have the time for much
testing of the methods, so I would be greatful if someone else could do
the testing and benchmarking of the methods.
Best,
Rene
On 08/29/2018 09:19 AM, Max Rudolph wrote:
> On Tue, Aug 28, 2018 at 8:01 PM Wolfgang Bangerth
> <bangerth at colostate.edu <mailto:bangerth at colostate.edu>> wrote:
>
> On 08/28/2018 05:33 PM, Max Rudolph wrote:
> > From this, it is very obvious why the solution to the
> convection problem at
> > low resolution is very diffusive and also why the interior
> temperature is much
> > closer to the surface temperature than to the CMB temperature
> because the
> > artificial viscosity is on the order of 20 times larger than the
> thermal
> > conductivity near the surface.
>
> Would it be easy to verify whether the artificial viscosity
> ("artificial
> conductivity") decreases at the expected rate with mesh refinement?
>
>
> What is the most helpful way for me to show this? Visualization of a
> couple of slices from the 3D conduction model? I tried to get the
> depth average of artificial viscosity but the postprocessor is not
> implemented.
>
> > For the conduction problem, the default values of the artificial
> viscosity are
> > also much larger than the thermal conductivity.
>
> I think that's the point worth investigating. Since in this case
> the velocity
> is zero, one would expect the artificial viscosity to also be at
> least quite
> small. Why is it not?
>
>
> *Maybe the spherical and 2D annulus geometry models are returning an
> unhelpful length scale, like planetary radius instead of layer depth?*
>
> aspect/source/simulator/entropy_viscosity.cc (starting line 191):
> // If the velocity is 0 we have to assume a sensible velocity to calculate
> // an artificial diffusion. We choose similar to nondimensional
> // formulations: v ~ thermal_diffusivity / length_scale, which cancels
> // the density and specific heat from the entropy formulation. It
> seems
> // surprising at first that only the conductivity remains, but
> remember
> // that this actually *is* an additional artificial diffusion.
> if (std::abs(global_u_infty) < 1e-50)
> return parameters.stabilization_beta *
> max_conductivity / geometry_model->length_scale() *
> cell_diameter;
>
>
> Best
> W.
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: bangerth at colostate.edu
> <mailto:bangerth at colostate.edu>
> www:
> http://www.math.colostate.edu/~bangerth/
> <http://www.math.colostate.edu/%7Ebangerth/>
>
>
>
> _______________________________________________
> Aspect-devel mailing list
> Aspect-devel at geodynamics.org
> http://lists.geodynamics.org/cgi-bin/mailman/listinfo/aspect-devel
--
Rene Gassmoeller
https://gassmoeller.github.io/
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.geodynamics.org/pipermail/aspect-devel/attachments/20180829/e3136035/attachment.html>
More information about the Aspect-devel
mailing list