Started criticality section in documentation.

docs/source/methods/criticality.rst

@@ -0,0 +1,93 @@
+.. _methods_criticality:
+
+========================
+Criticality Calculations
+========================
+
+A criticality calculation is a transport simulation wherein the source of
+neutrons includes a fissionable material. Some common criticality calculations
+include the simulation of nuclear reactors, spent fuel pools, nuclear weapons,
+and other fissile systems. The term criticality calculation is also synonymous
+with the term eigenvalue calculation. The reason for this is that the transport
+equation becomes an eigenvalue value equation if a fissionable source is present
+since then the source of neutrons will depend on the flux of neutrons
+itself. Criticality simulations using Monte Carlo methods are becoming
+increasingly common with the advent of high-performance computing.
+
+This section will explore the theory behind and implementation of criticality
+calculations in a Monte Carlo code.
+
+--------------------------------
+Method of Successive Generations
+--------------------------------
+
+-------------------------
+Source Convergence Issues
+-------------------------
+
+Diagnosing Convergence with Shannon Entropy
+-------------------------------------------
+
+As discussed earlier, it is necessary to converge both :math:`k_{eff}` and the
+source distribution before any tallies can begin. Moreover, the convergence rate
+of the source distribution is in general slower than that of
+:math:`k_{eff}`. One should thus examine not only the convergence of
+:math:`k_{eff}` but also the convergence of the source distribution in order to
+make decisions on when to start active batches.
+
+However, the representation of the source distribution makes it a bit more
+difficult to analyze its convergence. Since :math:`k_{eff}` is a scalar
+quantity, it is easy to simply look at a line plot of :math:`k_{eff}` versus the
+number of batches and this should give the user some idea about whether it has
+converged. On the other hand, the source distribution at any given batch is a
+finite set of coordinates in Euclidean space. In order to analyze the
+convergence, we would either need to use a method for assessing convergence of
+an N-dimensional quantity or transform our set of coordinates into a scalar
+metric. The latter approach has been developed considerably over the last decade
+and a method now commonly used in Monte Carlo criticality calculations is to use
+a metric called the `Shannon entropy`_, a concept borrowed from information
+theory.
+
+To compute the Shannon entropy of the source distribution, we first need to
+discretize the source distribution rather than having a set of coordinates in
+Euclidean space. This can be done by superimposing a structured mesh over the
+geometry (containing at least all fissionable materials). Then, the fraction of
+source sites that are present in each mesh element is counted:
+
+.. math::
+    :label: fraction-source
+
+    S_i = \frac{\text{Source sites in $i$-th mesh element}}{\text{Total number of
+    source sites}}
+
+The Shannon entropy is then computed as
+
+.. math::
+    :label: shannon-entropy
+
+    H = - \sum_{i=1}^N S_i \log_2 S_i
+
+where :math:`N` is the number of mesh elements. With equation
+:eq:`shannon-entropy`, we now have a scalar metric that we can use to assess the
+convergence of the source distribution by observing line plots of the Shannon
+entropy versus the number of batches.
+
+In recent years, researchers have started looking at ways of automatically
+assessing source convergence to relieve the burden on the user of having to look
+at plots of :math:`k_{eff}` and the Shannon entropy. A number of methods have
+been proposed (see e.g. [Romano]_, [Ueki]_), but each of these is not without
+problems.
+
+-----------------------
+Fission and Source Bank
+-----------------------
+
+.. _Shannon entropy: https://laws.lanl.gov/vhosts/mcnp.lanl.gov/pdf_files/la-ur-06-3737_entropy.pdf
+
+.. [Romano] Paul K. Romano, "Application of the Stochastic Oscillator to Assess
+   Source Convergence in Monte Carlo Criticality Calculations,"
+   *Proc. International Conference on Mathematics, Computational Methods, and
+   Reactor Physics*, Saratoga Springs, New York (2009).
+
+.. [Ueki] Taro Ueki, "On-the-Fly Judgments of Monte Carlo Fission Source
+   Convergence," *Trans. Am. Nucl. Soc.*, **98**, 512 (2008).

docs/source/methods/index.rst

@@ -9,6 +9,7 @@ Theory and Methodology
     :maxdepth: 3
 
     introduction
+    criticality
     statistics
     geometry
     physics

docs/source/methods/introduction.rst

@@ -125,10 +125,6 @@ be performed before the run is finished. This include the following:
 
   - All allocatable arrays are deallocated.
 
----------------------
-Criticality Algorithm
----------------------
-
 .. _probability distributions: http://en.wikipedia.org/wiki/Probability_distribution
 .. _Monte Carlo: http://en.wikipedia.org/wiki/Monte_Carlo_method
 .. _central limit theorem: http://en.wikipedia.org/wiki/Central_limit_theorem