From 07bbd211cd6abfaa2e0b69c3877f411a039855bc Mon Sep 17 00:00:00 2001 From: Paul Romano Date: Sat, 28 Jul 2012 15:27:58 -0400 Subject: [PATCH] Started criticality section in documentation. --- docs/source/methods/criticality.rst | 93 ++++++++++++++++++++++++++++ docs/source/methods/index.rst | 1 + docs/source/methods/introduction.rst | 4 -- 3 files changed, 94 insertions(+), 4 deletions(-) create mode 100644 docs/source/methods/criticality.rst diff --git a/docs/source/methods/criticality.rst b/docs/source/methods/criticality.rst new file mode 100644 index 000000000..4604cec78 --- /dev/null +++ b/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). diff --git a/docs/source/methods/index.rst b/docs/source/methods/index.rst index 358a8aa99..56de78a16 100644 --- a/docs/source/methods/index.rst +++ b/docs/source/methods/index.rst @@ -9,6 +9,7 @@ Theory and Methodology :maxdepth: 3 introduction + criticality statistics geometry physics diff --git a/docs/source/methods/introduction.rst b/docs/source/methods/introduction.rst index 6b6aeded6..8f623f7e7 100644 --- a/docs/source/methods/introduction.rst +++ b/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