ABOUT THE DEFINITION OF RESIDUAL IN STEP-31

[Xiang Sun]

Hi,

In step-31, why did you define the residual by multiplying T^(a-1)? What does "a" mean? Thank you very much.

Best,

Xiang

[Wolfgang Bangerth]

On 10/22/19 10:41 AM, Xiang Sun wrote:
>
> In step-31, why did you define the residual by multiplying T^(a-1)? What does
> "a" mean? Thank you very much.

I assume you mean here?
https://dealii.org/developer/doxygen/deal.II/step_31.html#Stabilizationweakformandspacediscretizationforthetemperatureequation

To understand the form stated there, you have to go back to the paper by
Guermond and Popov. In essence, this goes back to some deep theory about
hyperbolic equations that says that a weak solution is a physical solution
only if it satisfies an entropy inequality. There are many ways to define an
entropy pair (a pair of functions), and one of these is to choose E=T^a. The
differential inequality E then has to satisfy is obtained by multiplying the
equation for T by T^{a-1}.

The theory is not particularly simple, and requires substantial background of
hyperbolic equations. Going to the papers by Guermond and coauthors will give
you the necessary insight.

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Xiang Sun]

Hi, Wolfgang,

Thank you for your reply. I am learning dealii to build up a geothermal model, which includes a Laplace equation and a heat transfer considering both convection and diffusion. The Laplace equation describes a pressure distibution in a site with several wellbores. The heat transfer equation describes the site temperature change with the change in the wellbore temperature. I think the model can be established by combining Step-4 and Step-31. Do you think these two example code is good for our model? Thank you very much.

Best,

Xiang

[Wolfgang Bangerth]

On 10/22/19 10:57 PM, Xiang Sun wrote:
>
> Thank you for your reply. I am learning dealii to build up a geothermal model,
> which includes a Laplace equation and a heat transfer considering both
> convection and diffusion. The Laplace equation describes a pressure
> distibution in a site with several wellbores. The heat transfer equation
> describes the site temperature change with the change in the wellbore
> temperature. I think the model can be established by combining Step-4 and
> Step-31. Do you think these two example code is good for our model? Thank you
> very much.

This sounds about right, but without knowing what the exact equations are,
it's hard to tell.

That said, there are certainly several tutorial programs that already
implement pieces of what you want to do. You have probably noticed already
that step-32 is a more capable version of step-31.

[Xiang Sun]

Hi, Wolfgang,

Thank you for your reply. I will try it. The equations are:

where T is temperature, p is pressure, u is flow rate. K is hydraulic conductivity which is constant, κ and λ are the coefficients of heat transfer. T1 means the temperature at boundary 1. Tw and T0 are constant. 

The model is a column with a hole. Boundary 1 is the surface of the hole. Boundary 2 is the outer surface of the column. 

Best,

Xiang

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[Wolfgang Bangerth]

> Thank you for your reply. I will try it. The equations are: > [...]

Both step 31 and 32 will be good starting points. The only difference is
that you have Darcy flow instead of Stokes flow, but that should be easy
to incorporate into these programs.

[Xiang Sun]

Thank you very much.

Best,

Xiang

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