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.. _MAVRIC:

MAVRIC: Monaco with Automated Variance Reduction using Importance Calculations
==============================================================================

*D. E. Peplow and C. Celik*

Introduction
------------

Monte Carlo particle transport calculations for deep penetration problems can require very long run times in order to achieve an acceptable level of statistical uncertainty in the final answers. Discrete-ordinates codes can be faster but have limitations relative to the discretization of space, energy, and direction. Monte Carlo calculations can be modified (biased) to produce results with the same variance in less time if an approximate answer or some other additional information is already known about the problem. If an importance can be assigned to different particles based on how much they will contribute to the final answer, more time can be spent on important particles with less time devoted to unimportant particles. One of the best ways to bias a Monte Carlo code for a particular tally is to form an importance map from the adjoint flux based on that tally. Unfortunately, determining the exact adjoint flux could be just as difficult as computing the original problem itself. However, an approximate adjoint can still be very useful in biasing the Monte Carlo solution :cite:`wagner_acceleration_1997`. Discrete ordinates can be used to quickly compute that approximate adjoint. Together, Monte Carlo and discrete ordinates can be used to find solutions to thick shielding problems in reasonable times.

The MAVRIC (Monaco with Automated Variance Reduction using Importance Calculations) sequence is based on the CADIS (Consistent Adjoint Driven Importance Sampling) and FW-CADIS (Forward-Weighted CADIS) methodologies :cite:`wagner_automated_1998` :cite:`wagner_automated_2002` :cite:`haghighat_monte_2003` :cite:`wagner_forward-weighted_2007` MAVRIC automatically performs a three-dimensional, discrete-ordinates calculation using Denovo to compute the adjoint flux as a function of position and energy. This adjoint flux information is then used to construct an importance map (i.e., target weights for weight windows) and a biased source distribution that work together—particles are born with a weight matching the target weight of the cell into which they are born. The fixed-source Monte Carlo radiation transport Monaco then uses the importance map for biasing during particle transport and the biased source distribution as its source. During transport, the particle weight is compared with the importance map after each particle interaction and whenever a particle crosses into a new importance cell in the map.

For problems that do not require variance reduction to complete in a reasonable time, execution of MAVRIC without the importance map calculation provides an easy way to run Monaco. For problems that do require variance reduction to complete in a reasonable time, MAVRIC removes the burden of setting weight windows from the user and performs it automatically with a minimal amount of additional input. Note that the MAVRIC sequence can be used with the final Monaco calculation as either a multigroup (MG) or a continuous-energy (CE) calculation.

Monaco has a wide variety of tally options: it can calculate fluxes (by group) at a point in space, over any geometrical region, or for a user-defined, three-dimensional, rectangular grid. These tallies can also integrate the fluxes with either standard response functions from the cross section library or user-defined response functions. All of these tallies are available in the MAVRIC sequence.

While originally designed for CADIS, the MAVRIC sequence is also capable of creating importance maps using both forward and adjoint deterministic estimates. The FW-CADIS method can be used for optimizing several tallies at once, a mesh tally over a large region, or a mesh tally over the entire problem. Several other methods for producing importance maps are also available in MAVRIC and are explored in Appendix C.

CADIS Methodology
-----------------

MAVRIC is an implementation of CADIS (Consistent Adjoint Driven Importance Sampling) using the Denovo SN and Monaco Monte Carlo functional modules. Source biasing and a mesh-based importance map, overlaying the physical geometry, are the basic methods of variance reduction. In order to make the best use of an importance map, the map must be made consistent with the source biasing. If the source biasing is inconsistent with the weight windows that will be used during the transport process, source particles will undergo Russian roulette or splitting immediately, wasting computational time and negating the intent of the biasing.

Overview of CADIS
~~~~~~~~~~~~~~~~~

CADIS has been well described in the literature, so only a
brief overview is given here. Consider a class source-detector problem
described by a unit source with emission probability distribution
function :math:`q\left(\overrightarrow{r},E \right)` and a detector
response function :math:`\sigma_{d}\left(\overrightarrow{r},E \right)`.
To determine the total detector response, *R*, the forward scalar flux
:math:`\phi\left(\overrightarrow{r},E \right)` must be known. The
response is found by integrating the product of the detector response
function and the flux over the detector volume :math:`V_{d}`.


.. math::
  :label: mavric-1

  R = \int_{V_{d}}^{}{\int_{E}^{}{\sigma_{d}\left( \overrightarrow{r},E \right)}}\phi\left(\overrightarrow{r},E \right)\textit{dE dV.}


Alternatively, if the adjoint scalar flux,
:math:`\phi^{+}\left(\overrightarrow{r},E \right)`, is known from the
corresponding adjoint problem with adjoint source
:math:`q^{+}\left(\overrightarrow{r},E \right) = \sigma_{d}\left(\overrightarrow{r},E \right)`,
then the total detector response could be found by integrating the
product of the forward source and the adjoint flux over the source
volume, :math:`V_{s}`.


.. math::
  :label: mavric-2

  R = \int_{V_{s}}^{}{\int_{E}^{}{q\left(\overrightarrow{r},E \right)}}\phi^{+}\left( \overrightarrow{r},E \right)\textit{dE dV.}

Unfortunately, the exact adjoint flux may be just as difficult to
determine as the forward flux, but an approximation of the adjoint flux
can still be used to form an importance map and a biased source
distribution for use in the forward Monte Carlo calculation.

Wagner\ :sup:`1` showed that if an estimate of the adjoint scalar flux
for the corresponding adjoint problem could be found, then an estimate
of the response *R* could be made using Eq. . The adjoint source for the
adjoint problem is typically separable and corresponds to the detector
response and spatial area of tally to be optimized:
:math:`q^{+}\left(\overrightarrow{r},E \right) = \sigma_{d}\left(E \right)g\left( \overrightarrow{r} \right)`,
where :math:`\sigma_{d}\left( E \right)` is a flux-to-dose conversion
factor and :math:`g\left( \overrightarrow{r} \right)` is 1 in the tally
volume and 0 otherwise. Then, from the adjoint flux
:math:`\phi^{+}\left( \overrightarrow{r},E \right)` and response
estimate *R*, a biased source distribution,
:math:`\widehat{q}\left( \overrightarrow{r},E \right)`, for source
sampling of the form


.. math::
  :label: mavric-3

  \widehat{q}\left(\overrightarrow{r},E \right) = \frac{1}{R}q\left(\overrightarrow{r},E\right)\phi^{+}\left( \overrightarrow{r},E \right)


and weight window target values,
:math:`\overline{w}\left( \overrightarrow{r},E \right)`, for particle
transport of the form


.. math::
  :label: mavric-4

  \overline{w}\left( \overrightarrow{r},E \right) = \frac{R}{\phi^{+}\left( \overrightarrow{r},E \right)}


could be constructed, which minimize the variance in the forward Monte
Carlo calculation of *R*.

When a particle is sampled from the biased source distribution
:math:`\widehat{q}\left( \overrightarrow{r},E \right)`, to preserve a
fair game, its initial weight is set to


.. math::
  :label: mavric-5

  w_{0}\left(\overrightarrow{r},E \right) = \frac{q\left(\overrightarrow{r},E \right)}{\widehat{q}\left( \overrightarrow{r},E \right)} = \frac{R}{\phi^{+}\left( \overrightarrow{r},E \right)}\,


which exactly matches the target weight for that particle’s position and
energy. This is the “consistent” part of CADIS—source particles are born
with a weight matching the weight window of the region/energy they are
born into. The source biasing and the weight windows work together.

CADIS has been applied to many problems—including reactor ex-core
detectors, well-logging instruments, cask shielding studies, and
independent spent fuel storage facility models—and has demonstrated very
significant speed-ups in calculation time compared to analog
simulations.

Multiple sources with CADIS
~~~~~~~~~~~~~~~~~~~~~~~~~~~

For a typical Monte Carlo calculation with multiple sources (each with a
probability distribution function
:math:`q_{i}\left( \overrightarrow{r},E \right)` and a strength
:math:`S_{i}`, giving a total source strength of
:math:`S = \sum_{}^{}S_{i}`), the source is sampled in two steps. First,
the specific source *i* is sampled with probability
:math:`p\left( i \right) = \ S_{i}/S`, and then the particle is sampled
from the specific source distribution
:math:`q_{i}\left( \overrightarrow{r},E \right)`.

The source sampling can be biased at both levels: which source to sample
from and how to sample each source. For example, the specific source can
be sampled using some arbitrary distribution,
:math:`\widehat{p}\left( i \right)`, and then the individual sources can
be sampled using distributions
:math:`{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)`. Particles
would then have a birth weight of


.. math::
  :label: mavric-6

  w_{0} \equiv \ \left(\frac{p\left( i \right)}{\widehat{p}\left( i \right)} \right)\left(\frac{q_{i}\left( \overrightarrow{r},E \right)}{{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)} \right)\text{.}


For CADIS, a biased multiple source needs to be developed so that the
birth weights of sampled particles still match the target weights of the
importance map. For a problem with multiple sources (each with a
distribution :math:`q_{i}\left( \overrightarrow{r},E \right)` and a
strength :math:`S_{i}`), the goal of the Monte Carlo calculation is to
compute some response :math:`R` for a response function
:math:`\sigma_{d}\left( \overrightarrow{r},E \right)` at a given
detector.


.. math::
  :label: mavric-7

  R = \ \int_{V}^{}{\int_{E}^{}{\sigma_{d}\left( \overrightarrow{r},E \right)\text{ϕ}\left( \overrightarrow{r},E \right)\textit{dE dV.}}}


Note that the flux :math:`\phi\left( \overrightarrow{r},E \right)` has
contributions from each source. The response, :math:`R_{i}`, from each
specific source (:math:`S_{i}` with
:math:`q_{i}\left( \overrightarrow{r},E \right)`) can be expressed using
just the flux from that source,
:math:`\phi_{i}\left( \overrightarrow{r},E \right)`, as


.. math::
  :label: mavric-8

  R_{i} = \ \int_{V}^{}{\int_{E}^{}{\sigma_{d}\left(\overrightarrow{r},E \right)\ \phi_{i}\left(\overrightarrow{r},E \right)\textit{dE dV .}}}


The total response is then found as :math:`R = \sum_{i}^{}R_{i}`.

For the adjoint problem, using the adjoint source of
:math:`q^{+}\left( \overrightarrow{r},E \right) = \sigma_{d}\left( \overrightarrow{r},E \right)`,
the response :math:`R` can also be calculated as


.. math::
  :label: mavric-9

  R = \ \int_{V}^{}{\int_{E}^{}{\left\lbrack \sum_{i}^{}{S_{i}q_{i}\left( \overrightarrow{r},E \right)} \right\rbrack\ \phi^{+}\left( \overrightarrow{r},E \right)\textit{dE dV}}},


with response contribution from each specific source being


.. math::
  :label: mavric-10

  R_{i} = \ \int_{V}^{}{\int_{E}^{}{\ {S_{i}q_{i}\left( \overrightarrow{r},E \right)\text{ϕ}}^{+}\left( \overrightarrow{r}, E \right)\textit{dE dV.}}}


The target weights
:math:`\overline{w}\left( \overrightarrow{r},E \right)` of the
importance map are found using


.. math::
  :label: mavric-11

  \overline{w}\left( \overrightarrow{r},E \right) = \frac{R/S}{\text{ϕ}^{+}\left( \overrightarrow{r},E \right)\ }.


Each biased source
:math:`{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)` pdf is
found using

.. math::
  :label: mavric-12

  {\widehat{q}}_{i}\left(\overrightarrow{r},E \right) = \frac{S_{i}}{R_{i}}{q_{i}\left( \overrightarrow{r},E \right)\text{ϕ}}^{+}\left( \overrightarrow{r}, E \right)\ ,


and the biased distribution used to select an individual source is
:math:`\widehat{p}\left( i \right) = \ R_{i}/\sum_{}^{}{R_{i} = R_{i}/R}`.

When using the biased distribution used to select an individual source,
:math:`\widehat{p}\left( i \right)`, and the biased source distribution,
:math:`{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)`, the birth
weight of the sampled particle will be


.. math::
  :label: mavric-13

   \begin{matrix}
      w_{0} & \equiv & \left( \frac{p\left( i \right)}{\widehat{p}\left( i \right)} \right)\left( \frac{q_{i}\left( \overrightarrow{r}, E \right)}{{\widehat{q}}_{i}\left(\overrightarrow{r},E \right)} \right) \\ & = & \ \left( \frac{\frac{S_{i}}{S}}{\frac{R_{i}}{R}} \right) \left( \frac{q_{i}\left( \overrightarrow{r},E \right)}{\frac{S_{i}}{R_{i}}{q_{i}\left( \overrightarrow{r},E \right)\text{ϕ}}^{+}\left( \overrightarrow{r},E \right)} \right) \\
      & = & \frac{R/S}{\text{ϕ}^{+}\left( \overrightarrow{r},E \right)\ }, \\
  \end{matrix}


which matches the target weight,
:math:`\overline{w}\left( \overrightarrow{r},E \right)`.

Multiple tallies with CADIS
~~~~~~~~~~~~~~~~~~~~~~~~~~~

The CADIS methodology works quite well for classic source/detector problems. The statistical uncertainty of the tally that serves as the adjoint source is greatly reduced since the Monte Carlo transport is optimized to spend more simulation time on those particles that contribute to the tally, at the expense of tracking particles in other parts of phase space. However, more recently, Monte Carlo has been applied to problems where multiple tallies need to all be found with low statistical uncertainties. The extension of this idea is the mesh tally—where each voxel is a tally where the user desires low statistical uncertainties. For these problems, the user must accept a total simulation time that is controlled by the tally with the slowest convergence and simulation results where the tallies have a wide range of relative uncertainties.

The obvious way around this problem is to create a separate problem for each tally and use CADIS to optimize each. Each simulation can then be run until the tally reaches the level of acceptable uncertainty. For more than a few tallies, this approach becomes complicated and time-consuming for the user. For large mesh tallies, this approach is not reasonable.

Another approach to treat several tallies, if they are in close proximity to each other, or a mesh tally covering a small portion of the physical problem is to use the CADIS methodology with the adjoint source near the middle of the tallies to be optimized. Since particles in the forward Monte Carlo simulation are optimized to reach the location of the adjoint source, all the tallies surrounding that adjoint source should converge quickly. The drawback to this approach is the difficult question of “how close.” If the tallies are too far apart, certain energies or regions that are needed for one tally may be of low importance for getting particles to the central adjoint source. This may under-predict the flux or dose at the tally sites far from the adjoint source.

MAVRIC has the capability to have multiple adjoint sources with this problem in mind. For several tallies that are far from each other, multiple adjoint sources could be used. In the forward Monte Carlo, particles would be drawn to one of those adjoint sources. The difficulty with this approach is that typically the tally that is closest to the true physical source converges faster than the other tallies—showing the closest adjoint source seems to attract more particles than the others. Assigning more strength to the adjoint source further from the true physical source helps, but finding the correct strengths so that all of the tallies converge to the same relative uncertainty in one simulation is an iterative process for the user.

Forward-weighted CADIS
~~~~~~~~~~~~~~~~~~~~~~

In order to converge several tallies to the same relative uncertainty in
one simulation, the adjoint source corresponding to each of those
tallies needs to be weighted inversely by the expected tally value. In
order to calculate the dose rate at two points—say one near a reactor
and one far from a reactor—in one simulation, then the total adjoint
source used to develop the weight windows and biased source needs to
have two parts. The adjoint source far from the reactor needs to have
more strength than the adjoint source near the reactor by a factor equal
to the ratio of the expected near dose rate to the expected far dose
rate.

This concept can be extended to mesh tallies as well. Instead of using a
uniform adjoint source strength over the entire mesh tally volume, each
voxel of the adjoint source should be weighted inversely by the expected
forward tally value for that voxel. Areas of low flux or low dose rate
would have more adjoint source strength than areas of high flux or high
dose rate.

An estimate of the expected tally results can be found by using a quick
discrete-ordinates calculation. This leads to an extension of the CADIS
method: forward-weighted CADIS (FW-CADIS).**Error! Bookmark not
defined.** First, a forward S\ :sub:`N` calculation is performed to
estimate the expected tally results. A total adjoint source is
constructed where the adjoint source corresponding to each tally is
weighted inversely by those forward tally estimates. Then the standard
CADIS approach is used—an importance map (target weight windows) and a
biased source are made using the adjoint flux computed from the adjoint
S\ :sub:`N` calculation.

For example, if the goal is to calculate a detector response function
:math:`\sigma_{d}\left( E \right)` (such as dose rate using
flux-to-dose-rate conversion factors) over a volume (defined by
:math:`g\left( \overrightarrow{r} \right)`) corresponding to mesh tally,
then instead of simply using
:math:`q^{+}\left( \overrightarrow{r},E \right) = \sigma_{d}\left( E \right)\ g(\overrightarrow{r})`,
the adjoint source would be


.. math::
  :label: mavric-14

   q^{+}\left( \overrightarrow{r},E \right) = \frac{\sigma_{d}\left( E \right)\text{g}\left( \overrightarrow{r} \right)}{\int_{}^{}{\sigma_{d}\left( E \right)\text{ϕ}\left( \overrightarrow{r},E \right)}\textit{dE}}\ ,

where :math:`\phi\left( \overrightarrow{r},E \right)` is an estimate of
the forward flux and the energy integral is over the voxel at :math:`\overrightarrow{r}`.
The adjoint source is nonzero only where the mesh tally is defined
(:math:`g\left( \overrightarrow{r} \right)`), and its strength is
inversely proportional to the forward estimate of dose rate.

The relative uncertainty of a tally is controlled by two components:
first, the number of tracks contributing to the tally and, second, the
shape of the distribution of scores contributing to that tally. In the
Monte Carlo game, the number of simulated particles,
:math:`m\left( \overrightarrow{r},E \right)`, can be related to the true
physical particle density, :math:`n\left( \overrightarrow{r},E \right),`
by the average Monte Carlo weight of scoring particles,
:math:`\overline{w}\left( \overrightarrow{r},E \right)`, by


.. math::
  :label: mavric-15

  n\left( \overrightarrow{r},E \right) = \ \overline{w}\left( \overrightarrow{r},E \right)\text{m}\left( \overrightarrow{r},E \right).


In a typical Monte Carlo calculation, tallies are made by adding some
score, multiplied by the current particle weight, to an accumulator. To
calculate a similar quantity related to the Monte Carlo particle density
would be very close to calculating any other quantity but without
including the particle weight. The goal of FW-CADIS is to make the Monte
Carlo particle density, :math:`m\left( \overrightarrow{r},E \right)`,
uniform over the tally areas, so an importance map needs to be developed
that represents the importance to achieving uniform Monte Carlo particle
density. By attempting to keep the Monte Carlo particle density more
uniform, more uniform relative errors for the tallies should be
realized.

Two options for forward weighting are possible. For tallies over some
area where the entire group-wise flux is needed with low relative
uncertainties, the adjoint source should be weighted inversely by the
forward flux, :math:`\phi\left( \overrightarrow{r},E \right)`. The other
option, for a tally where only an energy-integrated quantity is desired,
is to weight the adjoint inversely by that energy-integrated
quantity,\ :math:`\int_{}^{}{\sigma_{d}\left( E \right)\text{ϕ}\left( \overrightarrow{r},E \right)}\text{\ dE}`.
For a tally where the total flux is desired, then the response in the
adjoint source is simply :math:`\sigma_{d}\left( E \right) = 1`.

To optimize the forward Monte Carlo simulation for the calculation of
some quantity at multiple tally locations or across a mesh tally, the
adjoint source needs to be weighted by the estimate of that quantity.
For a tally defined by its spatial location
:math:`g\left( \overrightarrow{r} \right)` and its optional response
:math:`\sigma_{d}\left( E \right)`, the standard adjoint source would be
:math:`q^{+}\left( \overrightarrow{r},E \right) = \sigma_{d}\left( E \right)\text{g}\left( \overrightarrow{r} \right)`.
The forward-weighted adjoint source,
:math:`q^{+}\left( \overrightarrow{r},E \right)`, depending on what
quantity is to be optimized, is listed below.

+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| **For the calculation of**                                                                                                                                                                                                                       | **Adjoint source**                                                                                                                                                                                                                                                                                                                                                      |
+==================================================================================================================================================================================================================================================+=========================================================================================================================================================================================================================================================================================================================================================================+
| Energy and spatially dependent flux.       :math:`\phi\left(\overrightarrow{r},E \right)`                                                                                                                                                        | .. math:: \frac{g\left( \overrightarrow{r}\right)}{\phi\left(\overrightarrow{r},E \right)}                                                                                                                                                                                                                                                                              |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| Spatially dependent total flux.         :math:`\int_{}^{}{\phi\left( \overrightarrow{r},E \right)}\textit{dE}`                                                                                                                                   | .. math:: \frac{g\left( \overrightarrow{r}\right)}{\int_{}^{}{\phi\left( \overrightarrow{r},E \right)}\textit{dE}}                                                                                                                                                                                                                                                      |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| Spatially dependent total response.         :math:`\int_{}^{}{\sigma_{d}\left( E \right)\text{ϕ}\left(\overrightarrow{r},E\right)}\textit{dE}`                                                                                                   | .. math:: \frac{\sigma_{d}\left( E \right)\text{g}\left( \overrightarrow{r} \right)}{\int_{}^{}{\sigma_{d}\left( E \right)\text{ϕ}\left( \overrightarrow{r},E \right)}\textit{dE}}                                                                                                                                                                                      |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+

The bottom line of FW-CADIS is that in order to calculate a quantity at
multiple tally locations (or across a mesh tally) with more uniform
relative uncertainties, an adjoint source needs to be developed for an
objective function that keeps some non-physical quantity—related to the
Monte Carlo particle density and similar in form to the desired
quantity—constant. FW-CADIS uses the solution of a forward
discrete-ordinates calculation to properly weight the adjoint source.
After that, the standard CADIS approach is used.

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MAVRIC Implementation of CADIS
------------------------------

With MAVRIC, as with other shielding codes, the user defines the problem as a set of physical models—the material compositions, the geometry, the source, and the detectors (locations and response functions)—as well as some mathematical parameters on how to solve the problem (number of histories, etc.). For the variance reduction portion of MAVRIC, the only additional inputs required are (1) the mesh planes to use in the discrete-ordinates calculation(s) and (2) the adjoint source description—basically the location and the response of each tally to optimize in the forward Monte Carlo calculation. MAVRIC takes this information and constructs a Denovo adjoint problem. (The adjoint source is weighted by a Denovo forward flux or response estimate for FW-CADIS applications.)  MAVRIC then uses the CADIS methodology: it combines the adjoint flux from the Denovo calculation with the source description and creates the importance map (weight window targets) and the mesh-based biased source. Monaco is then run using the CADIS biased source distribution and the weight window targets.

Denovo
~~~~~~

Denovo is a parallel three-dimensional SN code that is used to generate adjoint (and, for FW-CADIS, forward) scalar fluxes for the CADIS methods in MAVRIC. For use in MAVRIC/CADIS, it is highly desirable that the SN code be fast, positive, and robust. The phase-space shape of the forward and adjoint fluxes, as opposed to a highly accurate solution, is the most important quality for Monte Carlo weight-window generation. Accordingly, Denovo provides a step-characteristics spatial differencing option that produces positive scalar fluxes as long as the source (volume plus in-scatter) is positive. Denovo uses an orthogonal, nonuniform mesh that is ideal for CADIS applications because of the speed and robustness of calculations on this mesh type.

Denovo uses the highly robust GMRES (Generalized Minimum Residual) Krylov method to solve the SN equations in each group. GMRES has been shown to be more robust and efficient than traditional source (fixed-point) iteration. The in-group discrete SN equations are defined as


.. math::
  :label: mavric-16

  \mathbf{L}\psi = \mathbf{\text{MS}}\phi + q

where **L** is the differential transport operator, **M** is the
moment-to-discrete operator, **S** is the matrix of scattering
cross-section moments, *q* is the external and in-scatter source,
:math:`\phi` is the vector of angular flux moments, and :math:`\psi` is
the vector of angular fluxes at discrete angles. Applying the operator
**D**, where :math:`\phi = \mathbf{D}\psi`, and rearranging terms casts
the in-group equations in the form of a traditional linear system,
:math:`\mathbf{A}x = b`,

 .. math::
  :label: mavric-17

  \left( \mathbf{I} - \mathbf{D}\mathbf{L}^{- 1}\mathbf{\text{MS}} \right) = \mathbf{D}\mathbf{L}^{- 1}q .

The operation :math:`\mathbf{L}^{- 1}\nu`, where :math:`\nu` is an
iteration vector, is performed using a traditional wave-front solve
(transport sweep). The parallel implementation of the Denovo wave-front
solver uses the well-known Koch-Baker-Alcouffe (KBA) algorithm, which is
a two-dimensional block‑spatial decomposition of a three-dimensional
orthogonal mesh :cite:`baker_sn_1998`. The Trilinos package is used for the GMRES
implementation :cite:`willenbring_trilinos_2003` Denovo stores the mesh-based scalar fluxes in a
double precision binary file (*.dff) called a Denovo flux file. Past
versions of SCALE/Denovo used the TORT :cite:`rhoades_tort_1997` \*.varscl file format
(DOORS package :cite:`rhoades_doors_1998`), but this was limited to single precision. Since
the rest of the MAVRIC sequence has not yet been parallelized, Denovo is
currently used only in serial mode within MAVRIC.

Monaco
~~~~~~

The forward Monte Carlo transport is performed using Monaco, a
fixed-source, shielding code that uses the SCALE General Geometry
Package (SGGP, the same as used by the criticality code KENO-VI) and the
standard SCALE material information processor. Monaco can use either MG
or CE cross section libraries. Monaco was originally based on the MORSE
Monte Carlo code but has been extensively modified to modernize the
coding, incorporate more flexibility in terms of sources/tallies, and
read a user-friendly block/keyword style input.

Much of the input to MAVRIC is the same as Monaco. More details can be
found in the Monaco chapter of the SCALE manual.

Running MAVRIC
~~~~~~~~~~~~~~

The objective of a SCALE sequence is to execute several codes, passing
the output from one to the input of the next, in order to perform some
analysis—things that users typically had to do in the past. MAVRIC does
this for difficult shielding problems by running approximate
discrete-ordinates calculations, constructing an importance map and
biased source for one or more tallies that the user wants to optimize in
the Monte Carlo calculation, and then using those in a forward Monaco
Monte Carlo calculation. MAVRIC also prepares the forward and adjoint
cross sections when needed. The steps of a MAVRIC sequence are listed in
:numref:`Mavric-sequence`. The user can instruct MAVRIC to run this whole sequence of
steps or just some subset of the steps—in order to verify the
intermediate steps or to reuse previously calculated quantities in a new
analyses.

The MAVRIC sequence can be stopped after key points by using the
“parm= *parameter* ” operator on the “=mavric” command line, which is
the first line of the input file. The various parameters are listed in
Table :numref:`mavric-param`. These parameters allow the user to perform checks and make
changes to the importance map calculation before the actual Monte Carlo
calculation in Monaco.

MAVRIC also allows the sequence to start at several different points. If
an importance map and biased source have already been computed, they can
be used directly. If the adjoint scalar fluxes are known, they can
quickly be used to create the importance map and biased source and then
begin the forward Monte Carlo. All of the different combinations of
starting MAVRIC with some previously calculated quantities are listed in
the following section detailing the input options.

When using MG cross-section libraries that do not have flux-to-dose-rate
conversion factors, use “parm=nodose” to prevent the cross section
processing codes from trying to move these values into the working
library.

MAVRIC creates many files that use the base problem name from the output
file. For an output file called “c:\path1\path2\\\ *outputName*.out” or
“/home/path1/path2/ *outputName*.inp”, spaces in the output name will
cause trouble and should not be used.

.. list-table:: Steps in the MAVRIC sequence
   :name: Mavric-sequence
   :widths: 100 100
   :header-rows: 0
   :align: center

   * - **Cross section calculation**
     - XSProc is used to calculate the forward cross sections for Monaco
   * - **Forward Denovo (optional)**
     -
   * -  Cross section calculation
     - XSProc is used to calculate the forward cross sections for Denovo
   * -  Forward flux calculation
     - Denovo calculates the estimate of the forward flux
   * - **Adjoint Denovo (optional)**
     -
   * -  Cross section calculation
     - XSProc is used to calculate the adjoint cross sections for Denovo
   * -  Adjoint flux calculation
     - Denovo calculates the estimate of the adjoint flux
   * - **CADIS (optional)**
     - The scalar flux file from Denovo is then used to create the biased source distribution and transport weight windows
   * - **Monte Carlo calculation**
     - Monaco uses the biased source distribution and transport weight windows to calculate the various tallies

.. list-table:: Parameters for the MAVRIC command line (“parm=…”)
   :name: mavric-param
   :widths: 50 50
   :header-rows: 1
   :align: center

   * - Parameter
     - MAVRIC will stop after
   * - check
     - input checking
   * - forinp
     - Forward Denovo input construction (makes ``xkba_b.inp`` in the tmp area)
   * - forward
     - The forward Denovo calculation
   * - adjinp
     - Adjoint Denovo input construction (makes ``xkba_b.inp`` in the tmp area)
   * - adjoint
     - The adjoint Denovo calculation
   * - impmap
     - Calculation of importance map and biased source

MAVRIC input
------------

The input file for MAVRIC consists of three lines of text (“=mavric”
command line with optional parameters, the problem title, and SCALE
cross section library name) and then several blocks, with each block
starting with “read xxxx” and ending with “end xxxx”. There are three
required blocks and nine optional blocks. Material and geometry blocks
must be listed first and in the specified order. Other blocks may be
listed in any order.

Blocks (must be in this order):

-  Composition – (required) SCALE standard composition, list of materials used in the problem

-  Celldata – SCALE resonance self-shielding

-  Geometry – (required) SCALE general geometry description

-  Array – optional addition to the above geometry description

-  Volume – optional calculation or listing of region volumes

-  Plot – create 2D slices of the SGGP geometry

Other Blocks (any order, following the blocks listed above):

-  Definitions – defines locations, response functions, and grid geometries used by other blocks

-  Sources – (required) description of the particle source spatial, energy, and directional distributions

-  Tallies – description of what to calculate: point detector tallies, region tallies, or mesh tallies

-  Parameters – how to perform the simulation (random number seed, how many histories, etc.)

-  Biasing – data for reducing the variance of the simulation

-  ImportanceMap – instructions for creating an importance map based on a discrete-ordinates calculation

The material blocks (Composition and Celldata) and the physical model
blocks (Geometry, Array, Volume, and Plot) follow the standard SCALE
format. See the other SCALE references as noted in the following
sections for details. The Biasing block and ImportanceMap block cannot
both be used.

For the other six blocks, scalar variables are set by “keyword=value”,
fixed-length arrays are set with “keyword value\ :sub:`1` ...
value\ :sub:`N`\ ”, variable-length arrays are set with “keyword
value\ :sub:`1` ... value\ :sub:`N` end”, and some text and filenames
are read in as quoted strings. Single keywords to set options are also
used in some instances. The indention, comment lines, and
upper/lowercase shown in this document are not required— they are used
in the examples only for clarity. Except for strings in quotes (like
filenames), SCALE is case insensitive.

After all input blocks are listed, a single line with “end data” should be listed.
A final “end” should also be listed, to signify the end of all MAVRIC input.
Nine of the blocks are the same input blocks used by the functional module Monaco,
with a few extra keywords only for use with MAVRIC. These extra keywords are highlighted here,
without relisting all of the standard Monaco keywords for those blocks.
See :numref:`input-format` for an overview of MAVRIC input file structure.

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Composition block
~~~~~~~~~~~~~~~~~

Material information input follows the standard SCALE format for
material input. Basic materials known to the SCALE library may be used
as well as completely user-defined materials (using isotopes with known
cross sections). Input instructions are located in the XSProc chapter in
the SCALE manual. The Standard Composition Library chapter lists the
different cross section libraries and the names of standard materials.
An example is as follows:

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.. code:: rest
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   read composition

       uo2 1 0.2 293.0 92234 0.0055 92235 3.5 92238 96.4945 end

       orconcrete 2 1.0 293.0 end

       ss304 3 1.0 293.0 end

   end composition

Details on the cell data block are also included in the XSProc chapter.
When using different libraries for the importance map production (listed
at the top of the input) and the final Monte Carlo calculation (listed
in the parameters block, if different), make sure that the materials are
present in both libraries.


.. list-table:: Overall input format
   :widths: 30 30
   :header-rows: 1
   :align: center
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   :name: input-format
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   * - input file
     - Comment
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   * - .. code:: rest
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         =mavric
         Some title for this problem
         v7-27n19g
         read composition
            ...
         end composition
         read celldata
            ...
         end celldata
         read geometry
            ...
         end geometry
         read array
            ...
         end array
         read volume
            ...
         end volume
         read plot
            ...
         end plot
         read definitions
            ...
         end definitions
         read sources
            ...
         end sources
         read tallies
            ...
         end tallies
         read parameters
            ...
         end parameters
         read biasing
            ...
         end biasing
         read importanceMap
            ...
         end importanceMap
         end data
         end
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     - .. code:: rest
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          name of sequence
          title
          cross section library name
          SCALE material compositions
              [required block]

          SCALE resonance self-shielding
              [optional block]

          SCALE SGGP geometry
              [required block]

          SCALE SGGP arrays
              [optional block]

          SCALE SGGP volume calc
              [optional block]

          SGGP Plots
              [optional block]

          Definitions
              [possibly required]

          Sources definition
              [required block]

          Tally specifications
              [optional block]

          Monte Carlo parameters
              [optional block]

          Biasing information
              [optional block]

          Importance map
              [optional block]

          end of all blocks
          end of MAVRIC input

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SGGP geometry blocks
~~~~~~~~~~~~~~~~~~~~

MAVRIC uses the functional module Monaco for the forward Monte Carlo calculation. Monaco tracks particles through the physical geometry described by the SGGP input blocks as well as through the mesh importance map and any mesh tallies, which are defined in the global coordinates and overlay the physical geometry. Because Monaco must track through all of these geometries at the same time, users should not use the reflective boundary capability in the SGGP geometry.

For more details on each SGGP Geometry block, see the following sections of the KENO-VI chapter of the SCALE Manual.

    Geometry – *Geometry Data*

    Array – *Array Data*

    Volume – *Volume Data*

    Plot – *Plot Data*

Other blocks shared with Monaco
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The definitions, sources, tallies, and biasing blocks are all the same
as Monaco. They are all fully described in the Monaco chapter of the
SCALE Manual.

   Definitions – *Definitions Block*

   Sources – *Sources Block*

   Tallies – *Tallies Block*

   Biasing – *Biasing Block*

In the parameters block, there are several extra keywords compared to
Monaco (see the *Parameter Block* section of the Monaco chapter) which
are used when the cross section library used in the importance
calculations is different from the library used in the final forward
Monaco Monte Carlo calculation. The library listed at the beginning of
the MAVRIC input file will be used for the importance calculations
(forward and adjoint Denovo calculation, formation of the importance
map, and biased sources). To use a different MG library in the final
Monaco simulation, use the keyword “library=” with the cross section
library name in quotes. A cross section library for Monaco will be made
using csas-mg. If there are any extra parameters to use (“parm=” in the
“=csas-mg” line of the csas-mg input), they can be passed along using
the keyword “parmString=” with the extra information in quotes. For
example, the following input file would use a coarse-group library for
the importance calculations and a fine-group library for the final
Monaco, each with CENTRM processing.

.. code:: rest

    =mavric parm=centrm
    v7-27n19g


    read parameters

        library=”v7-200n47g” parmString=”centrm”



    end parameters



    end data

    end


To use a CE cross section in the final Monaco step, use the keyword “ceLibrary=” with the cross section library name in quotes. When using the “library=” or “ceLibrary=” keywords, they should precede the “neutron”, “photon”, “noNeutron”, and “noPhoton” keywords. :numref:`extra-keywords` summarizes all of the keywords in the MAVRIC parameter block.

When using two different cross section libraries, be sure that the responses and distributions are defined in ways that do not depend on the cross section library. For example, any response that is just a list of n values (corresponding to a cross section library of n groups) needs to have the group energies specifically listed so that it can be evaluated properly on the other group structure.

.. csv-table:: Extra keywords for the parameters block
  :file: csv-tables/table4.1.04.csv
  :header-rows: 1
  :name: extra-keywords

Importance map block
~~~~~~~~~~~~~~~~~~~~

The importance map block is the “heart and soul” of MAVRIC. This block lists the parameters for creating an importance map and biased source from one (adjoint) or two (forward, followed by adjoint) Denovo discrete-ordinates calculations. Without an importance map block, MAVRIC can be used to run Monaco and use its conventional types of variance reduction. If both the importance map and biasing blocks are specified, only the importance map block will be used. There are a variety of ways to use the importance map block, as explained in the subsections below. Keywords for this block are summarized at the end of this section, in
:numref:``

Constructing a mesh for the S\ :sub:`N` calculation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

All of the uses of the importance map block that run the
discrete-ordinates code require the use of a grid geometry that overlays
the physical geometry. Grid geometries are defined in the definitions
block of the MAVRIC input. The extent and level of detail needed in a
grid geometry are discussed in the following paragraphs.

When using S\ :sub:`N` methods alone for solving radiation transport in
shielding problems, a good rule of thumb is to use mesh cell sizes on
the order of a meanfree path of the particle. For complex shielding
problems, this could lead to an extremely large number of mesh cells,
especially when considering the size of the meanfree path of the lowest
energy neutrons and photons in common shielding materials.

In MAVRIC, the goal is to use the S\ :sub:`N` calculation for a quick
approximate solution. Accuracy is not paramount—just getting an idea of
the overall shape of the true importance map will help accelerate the
convergence of the forward Monte Carlo calculation. The more accurate
the importance map, the better the forward Monte Carlo acceleration will
be. At some point there is a time trade-off when the computational time
for calculating the importance map followed by the Monte Carlo
calculation exceeds that of a standard analog Monte Carlo calculation.
Large numbers of mesh cells, coming from using very small mesh sizes,
for S\ :sub:`N` calculations also use a great deal of computer memory.

Because the deterministic solution(s) for CADIS and FW-CADIS can have
moderate fidelity and still provide variance reduction parameters that
substantially accelerate the Monte Carlo solution, mesh cell sizes in
MAVRIC applications can be larger than what most S\ :sub:`N` practioners
would typically use. The use of relatively coarse mesh reduces memory
requirements and the run time of the deterministic solution(s). Some
general guidelines to keep in mind when creating a mesh for the
importance map/biased source are:

-  The true source regions should be included in the mesh with mesh
   planes at their boundaries.

-  For point or very small sources, place them in the center of a mesh
   cell, not on the mesh planes.

-  Any region of the geometry where particles could eventually
   contribute to the tallies (the “important” areas) should be included
   in the mesh.

-  Point adjoint sources (corresponding to point detector locations) in
   standard CADIS calculations do not have to be included inside the
   mesh. For FW-CADIS, they must be in the mesh and should be located at
   a mesh cell center, not on any of the mesh planes.

-  Volumetric adjoint sources should be included in the mesh with mesh
   planes at their boundaries.

-  Mesh planes should be placed at significant material boundaries.

-  Neighboring cell sizes should not be drastically different.

-  Smaller cell sizes should be used where the adjoint flux is changing
   rapidly, for example, toward the surfaces of adjoint sources and
   shields (rather than their interiors).

Another aspect to keep in mind is that the source in the forward Monaco
Monte Carlo calculation will be a biased, mesh-based source. Source
particles will be selected by first sampling which mesh cell to use and
then sampling a position uniformly within that mesh cell that meets the
user criteria of “unit=”, “region=”, or “mixture=” if specified. The
mesh should have enough resolution that the mesh source will be an
accurate representation of the true source.

The geometry for the Denovo calculation is specified using the keyword
“gridGeometryID=” and the identification number of a grid geometry that
was defined in the definitions block. The material assigned to each
voxel of the mesh is determined by testing the center point in the SGGP
geometry (unless the macro-material option is used – see below).

Macromaterials for S\ :sub:`N` geometries
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Part of the advantage of the CADIS method is that the adjoint
discrete-ordinates calculation only needs to be approximate in order to
form a reasonable importance map and biased source. This usually means
that the mesh used is much coarser than the mesh that would be used if
the problem were to be solved only with a discrete-ordinates code. This
coarse mesh may miss significant details (especially curves) in the
geometry and produce a less-than-optimal importance map.

In order to get more accurate solutions from a coarse-mesh
discrete-ordinates calculation, Denovo can represent the material in
each voxel of the mesh as a volume-weighted mixture of the real
materials, called macromaterials, in the problem. When constructing the
Denovo input, the Denovo EigenValue Calculation (DEVC, see section SECTIONREFERENCE)
sequence can estimate the volume fraction occupied by each real
material in each voxel by a sampling method. The user can specify
parameters for how to sample the geometry. Note that finer sampling
makes more accurate estimates of the material fraction but requires more
setup time to create the Denovo input. Users should understand how the
macromaterials are sampled and consider that when constructing a mesh
grid. This is especially important for geometries that contain arrays.
Careful consideration should be given when overlaying a mesh on a
geometry that contains arrays of arrays.

Because the list of macromaterials could become large, the user can also
specify a tolerance for how close two different macromaterials can be to
be considered the same, thereby reducing the total number of
macromaterials. The macromaterial tolerance, “``mmTolerance=``”, is used for
creating a different macromaterial from the ones already created by
looking at the infinity norm between two macromaterials.
The number of macromaterials does not appreciably impact Denovo run time
or memory requirements.

Two different sampling methods are available—point testing :cite:`ibrahim_improving_2009` with
the keyword ``mmPointTest`` and ray tracing :cite:`johnson_fast_2013` with the keyword
``mmRayTest``.

Ray Tracing
'''''''''''

This method estimates the volume of different materials in the Denovo mesh grid elements by tracing rays through the SGGP geometry and computing the average track lengths through the each material. Rays are traced in all three dimensions to better estimate the volume fractions of materials within each voxel. The mmSubCell parameter controls how many rays to trace in each voxel in each dimension. For example, if mmSubCell= n, then when tracing rays in the z dimension, each column of voxels uses a set of n×n rays starting uniformly spaced in the x  and y  dimensions. With rays being cast from all three orthogonal directions, a total of 3n2 rays are used to sample each voxel. One can think of subcells as an equally spaced sub-mesh with a single ray positioned at each center. The number of subcells in each direction, and hence the number of rays, can be explicitly given with mmSubCells ny nz nx nz nx ny end keyword for rays parallel to the x axis, y axis, and z axis.
:numref:`ray-positions` shows different subcell configurations (in two dimensions) for a given voxel.

.. _ray-positions:

.. figure:: figs/fig4.1.01_rayTrace6.png
  :width: 500
  :align: center

  Ray positions within a voxel with different mmSubCells parameters.

Ray tracing is a more robust method compared to the simple point testing
method used in previous versions of SCALE/MAVRIC; however, it requires
more memory than point testing. Ray tracing gives more accurate
estimates of volume fractions because track lengths across a voxel give
more information than a series of test points. Ray tracing is also much
faster than point testing because the particle tracking routines are
optimized for quickly determining lists of materials and distance along
a given ray.

Ray tracing operates on the grid geometry supplied by the user and
shoots rays in all three directions starting from the lower bounds of
the mesh grid. An example of an arbitrary assembly geometry is shown in
:numref:`geom-model`. A ray consists of a number of steps that each correspond
to crossing a material boundary along the path of the ray. Ratios of
each step’s length to the voxel length in the ray’s direction determine
the material volume fraction of that step in that voxel, and summation
of the same material volume fractions gives the material volume fraction
of that material in that voxel. Ray tracing through a single voxel that
contains a fuel pin is illustrated in Figure 4.1.3.

.. _geom-model:

.. figure:: figs/fig4.1.02_kenoDenovo.png
  :width: 600
  :align: center

  Geometry model (left) and the Denovo representation (right) of an assembly using macromaterials determined by ray tracing.











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.. bibliography:: /bibs/mavric.bib