[aspect-devel] Internal heating in aspect (Ludovic Jeanniot)
Scott King
sdk at vt.edu
Thu Aug 30 07:25:25 PDT 2018
Submitted for your consideration…
ASPECT results from low-Ra, steady-state, thermal convection with default and zero beta factors on various grids
Several things stand out:
1) results are always closer to other codes with beta=cR=0 than the default beta and cR
2) mean quantities are in excellent agreement with beta=cR=0 (better than 1%) even on a very coarse mesh for A1
3) even with beta=cR=0 the top/bottom Nusselt numbers are significantly off (10% for C1 but that is better than 40% with default), the flux calculation seems to be a separate issue (as already noted)
4) the results from a uniform radial mesh spacing are always better than from the default uniform aspect ratio mesh
More results are coming. I’ve asked Grant to difference the fields from the default and zero beta cases to see if the pattern correlates with the art. diff. pattern Max showed
> On Aug 29, 2018, at 3:49 PM, Max Rudolph <maxrudolph at ucdavis.edu> wrote:
>
> Rene,
> Thank you for summarizing the current state of affairs. I would suggest that when you make the changes in (1)-(2), you also make ASPECT print at least a warning when the entropy viscosity exceeds, say, 1% of the thermal conductivity. It might be even better for the code to exit in with an error unless a parameter is explicitly set to ignore such a condition.
>
> It also seems that one way around (1) is to use DG elements for composition. Is this what most users are doing anyways?
>
> Thank you for addressing these issues so quickly. I am happy to help with testing as needed.
>
> Best,
> Max
>
> On Wed, Aug 29, 2018 at 12:16 PM Rene Gassmoeller <rene.gassmoeller at mailbox.org <mailto:rene.gassmoeller at mailbox.org>> wrote:
> 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 <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 <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/>
>>
>>
>>
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> --
> Rene Gassmoeller
> https://gassmoeller.github.io/ <https://gassmoeller.github.io/>_______________________________________________
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