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147 lines
6.6 KiB
ReStructuredText
.. _methods_introduction:
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============
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Introduction
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============
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The physical process by which a population of particles evolves over time is
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governed by a number of `probability distributions`_. For instance, given a
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particle traveling through some material, there is a probability distribution
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for the distance it will travel until its next collision (an exponential
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distribution). Then, when it collides with a nucleus, there is an associated
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probability of undergoing each possible reaction with that nucleus. While the
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behavior of any single particle is unpredictable, the average behavior of a
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large population of particles originating from the same source is well defined.
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If the probability distributions that govern the transport of a particle are
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known, the process of single particles randomly streaming and colliding with
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nuclei can be simulated directly with computers using a technique known as
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`Monte Carlo`_ simulation. If enough particles are simulated this way, the
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average behavior can be determined to within arbitrarily small statistical
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error, a fact guaranteed by the `central limit theorem`_. To be more precise,
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the central limit theorem tells us that the variance of the sample mean of some
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physical parameter being estimated with Monte Carlo will be inversely
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proportional to the number of realizations, i.e. the number of particles we
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simulate:
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.. math::
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\sigma^2 \propto \frac{1}{N}.
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where :math:`\sigma^2` is the variance of the sample mean and :math:`N` is the
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number of realizations.
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------------------------
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Overview of Program Flow
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------------------------
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OpenMC performs a Monte Carlo simulation one particle at a time -- at no point
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is more than one particle being tracked on a single program instance. Before any
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particles are tracked, the problem must be initialized. This involves the
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following steps:
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- Read input files and building data structures for the geometry, materials,
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tallies, and other associated variables.
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- Initialize the pseudorandom number generator.
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- Read the continuous-energy or multi-group cross section data specified in
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the problem.
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- If using a special energy grid treatment such as a union energy grid or
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lethargy bins, that must be initialized as well in a continuous-energy
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problem.
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- In a multi-group problem, individual nuclide cross section information is
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combined to produce material-specific cross section data.
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- In a fixed source problem, source sites are sampled from the specified
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source. In an eigenvalue problem, source sites are sampled from some
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initial source distribution or from a source file. The source sites
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consist of coordinates, a direction, and an energy.
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Once initialization is complete, the actual transport simulation can
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proceed. The life of a single particle will proceed as follows:
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1. The particle's properties are initialized from a source site previously
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sampled.
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2. Based on the particle's coordinates, the current cell in which the particle
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resides is determined.
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3. The energy-dependent cross sections for the material that the particle is
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currently in are determined. Note that this includes the total
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cross section, which is not pre-calculated.
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4. The distance to the nearest boundary of the particle's cell is determined
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based on the bounding surfaces to the cell.
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5. The distance to the next collision is sampled. If the total material
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cross section is :math:`\Sigma_t`, this can be shown to be
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.. math::
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d = -\frac{\ln \xi}{\Sigma_t}
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where :math:`\xi` is a `pseudorandom number`_ sampled from a uniform
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distribution on :math:`[0,1)`.
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6. If the distance to the nearest boundary is less than the distance to the next
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collision, the particle is moved forward to this boundary. Then, the process
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is repeated from step 2. If the distance to collision is closer than the
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distance to the nearest boundary, then the particle will undergo a collision.
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7. The material at the collision site may consist of multiple nuclides. First,
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the nuclide with which the collision will happen is sampled based on the
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total cross sections. If the total cross section of material :math:`i` is
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:math:`\Sigma_{t,i}`, then the probability that any nuclide is sampled is
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.. math::
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P(i) = \frac{\Sigma_{t,i}}{\Sigma_t}.
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Note that the above selection of collided nuclide only applies to
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continuous-energy simulations as multi-group simulations use nuclide
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data which has already been combined in to material-specific data.
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8. Once the specific nuclide is sampled, a reaction for
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that nuclide is randomly sampled based on the microscopic cross sections. If the microscopic
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cross section for some reaction :math:`x` is :math:`\sigma_x` and the total
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microscopic cross section for the nuclide is :math:`\sigma_t`, then the
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probability that reaction :math:`x` will occur is
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.. math::
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P(x) = \frac{\sigma_x}{\sigma_t}.
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Since multi-group simulations use material-specific data, the above is
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performed with those material multi-group cross sections (i.e.,
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macroscopic cross sections for the material) instead of microscopic
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cross sections for the nuclide).
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9. If the sampled reaction is elastic or inelastic scattering, the outgoing
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energy and angle is sampled from the appropriate distribution. In
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continuous-energy simulation, reactions of type :math:`(n,xn)` are treated
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as scattering and any additional particles which may be created are added
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to a secondary particle bank to be tracked later. In a multi-group
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simulation, this secondary bank is not used but the particle weight is
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increased accordingly. The original particle then continues from step 3.
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If the reaction is absorption or fission, the particle dies and if
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necessary, fission sites are created and stored in the fission bank.
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After all particles have been simulated, there are a few final tasks that must
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be performed before the run is finished. This include the following:
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- With the accumulated sum and sum of squares for each tally, the sample mean
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and its variance is calculated.
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- All tallies and other results are written to disk.
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- If requested, a source file is written to disk.
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- Dynamically-allocated memory should be freed.
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.. _probability distributions: https://en.wikipedia.org/wiki/Probability_distribution
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.. _Monte Carlo: https://en.wikipedia.org/wiki/Monte_Carlo_method
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.. _central limit theorem: https://en.wikipedia.org/wiki/Central_limit_theorem
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.. _pseudorandom number: https://en.wikipedia.org/wiki/Pseudorandom_number_generator
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