uncertainty propagation in clusters

[magnoxemo]

This method basically takes the advantages of the fact that OpenMC uses batch statistics.

Let us presume that
$$\bar{ \Phi_c}$$ = average flux in a cluster.
$$\Phi_c^b$$ = $b^{th}$ batch statistics of the $i^{th}$ voxel in the cluster
$v_i$ = volume of the $i_{th}$ voxel in the cluster.
$V$ = total volume of the cluster = $\sum_{i}^{N} v_i$
$N$ = number of voxels in the cluster.
$\gamma_i$ = volumetric weight of the voxel in a cluster.

Now

$b^{th}$ batch tally in the cluster should be the sum of the weighted flux of the voxels which we could write as

$$ \Phi_c^b = \sum_{i=1}^{N} \gamma_i \Phi_i^b $$

where, $\gamma_i = \frac{v_i}{ V}$. So, average flux in a cluster after $B$ independent active batch run will be,

$$ \bar{\Phi_c} = \frac{1} {B}\sum_{b =1 }^{B} \Phi_{c}^{b} = \frac{1} {B} \sum_{b=1}^{B}\sum_{i=1 }^{N} \gamma_i \Phi_{i}^{b} $$

and the variance of the cluster will be,

$$ \sigma_c^2 = \mathbb E [ ( \Phi_c^b - \bar{ \Phi_c})^2 ]$$ $$ = \mathbb E [ (\sum_{i=1 }^{N} \gamma_i \Phi_{i}^{b} - \frac{1} {B} \sum_{b=1}^{B}\sum_{i=1 }^{N} \gamma_i \Phi_{i}^{b} )^2]$$

We know about, $\gamma_i$ from the mesh and clustering and $B$ from Cardinal or OpenMC. But in order to fully use the variance equation we would have to know about the, $\Phi_i^b$ but OpenMC only reports $\bar \Phi_i = \frac{1}{B} \sum_b^B \Phi_i^b$ to cardinal. If we can figure out a way to know $\Phi_i^b$ it will be easy to implement.

We could simplify the variance equation further and can make it a function of $\bar \Phi_i$ but that would introduce co-variance in the function.

@gonuke @pshriwise

[gonuke]

I think this looks right to me... This may be some motivation for on-the-fly amalgamation, too.

[magnoxemo]

This is mostly based on the conversation I had with @pshriwise this noon and ideas about doing mesh tally amalgamation on the fly .

add another constructor here

LibMesh(libMesh::MeshBase& input_mesh, double length_multiplier = 1.0, std::string extra_integer_name) so that from cardinal I can pass the name of the eeid or it's index to openmc.

Now let's say I have a mesh that looks like this

+---------------+
|    a    |  c  |
|------   |----
|    b    |  d  |
+--------------+

But our clustering algorithm wanted to treat a, b, d together as one single element and changed their extra element integer id to 0 ( as a is the first element which' element id = 0 ).

+-----------------+
|     a   |    c |
|         |----  |
|    b        d  |
+----------------+

From openmc side, every time a collision happens in a, b or d we can just check

if ( cluster_id != -1 ){
   # that means part of a cluster 
    bin = get_bin_from_element( mesh.elem_ptr( cluster_id ) )
}

That mean all the scores in a, b and d gets accumulated in a (even though b and d will remain zero but we can fill the solution later ). Then open mc transfer the solution to cardinal. where we can do,

if ( cluster_id != -1 ){
   # that means part of a cluster 
   #get the first element in that cluster 
   first_element_in_that_cluster = get_score_from_element( mesh.elem_ptr( cluster_id ) )

   #get score from the first element 
   tally_score = get_score (first_element_in_that_cluster)

   #store the result in the current element 
   fill_aux_variable( _current_element)
}

But another concern we are reconstructing the raw tally to any meaning full score we need to normalize it and divide it by the volume, in this case $v_a+ v_b + v_d$ so need to adjust that here as well.