[aspect-devel] Averaging material properties

[Bruno Turcksin]

Wolfgang,

On 04/20/2015 12:08 PM, Wolfgang Bangerth wrote:
> I think that in reactor physics, you need to homogenize because you 
> can't resolve the features. But you can assume that they are somewhat 
> periodic. In the geosciences, you have these discontinuities, but 
> they're not everywhere: say, they're just a plate subducting beneath 
> another, in an otherwise rather smooth medium. I don't know how to 
> apply homogenization theory in such cases. What do people in your 
> field do? 
Usually in reactor physics, you have quick variations of some physical 
properties that you cannot represent because your mesh is too coarse 
(typically, you have different materials in one cell). In this case, you 
can create a new averaged material property such that when you solve the 
equation on the coarse mesh, you get the right solution, i.e, the exact 
average solution. For example, in reactor physics, we are interested in 
the quantity Sigma*Phi*V, where Sigma is a material property, Phi the 
flux, and V the volume. We want this quantity to be the same on the 
coarse mesh and the fine mesh. So we define Sigma_avg such that
(I assume that I have two materials, 1 and 2):
Sigma_avg = (Sigma_1 Phi_1 V_1 + Sigma_2 Phi_2 V_2)/(Phi_1 V_1 + Phi_2 
V_2) = (Sigma_1 + Sigma_2 (V_2/V_1) X)/(1+(V_2/V_1) X)
X = Phi_2/Phi_1 is the parameter that you try to determine using an 
analytical approximation or using a computation on a fine mesh (this is 
useful only if you can reuse the result).

Best,

Bruno