docutils.nodesdocument)}( rawsourcechildren](htarget)}(h.. _MAVRIC:h]
attributes}(ids]classes]names]dupnames]backrefs]refidmavricutagnameh lineKparenthhhsource0/Users/john/Documents/SCALE-test/docs/MAVRIC.rstubhsection)}(hhh](htitle)}(hNMAVRIC: Monaco with Automated Variance Reduction using Importance Calculationsh]hTextNMAVRIC: Monaco with Automated Variance Reduction using Importance Calculations}(hh+hh)hhh NhNubah}(h]h]h]h]h]uhh'hh$hhh h!hKubh paragraph)}(h*D. E. Peplow and C. Celik*h]hemphasis)}(hh=h]h.D. E. Peplow and C. Celik}(hhhhAubah}(h]h]h]h]h]uhh?hh;ubah}(h]h]h]h]h]uhh9h h!hKhh$hhubh#)}(hhh](h()}(hIntroductionh]h.Introduction}(hhYhhWhhh NhNubah}(h]h]h]h]h]uhh'hhThhh h!hK ubh:)}(hX.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.h](h.XDMonte 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 }(hXDMonte 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 hhehhh NhNubsphinx.addnodespending_xref)}(hwagner_acceleration_1997h]hinline)}(hhsh]h.[wagner_acceleration_1997]}(hhhhwubah}(h]h]h]h]h]uhhuhhqubah}(h]id1ah]bibtexah]h]h] refdomaincitationreftyperef reftargethsrefwarnsupport_smartquotesuhhoh h!hKhhehhubh.. 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.}(h. 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.hhehhh NhNubeh}(h]h]h]h]h]uhh9h h!hKhhThhubh:)}(hX_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.h](h.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 }(hThe 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 hhhhh NhNubhp)}(hwagner_automated_1998h]hv)}(hhh]h.[wagner_automated_1998]}(hhhhubah}(h]h]h]h]h]uhhuhhubah}(h]id2ah]hah]h]h] refdomainhreftypeh reftargethrefwarnsupport_smartquotesuhhoh h!hK
hhhhubh. }(h hhhhh NhNubhp)}(hwagner_automated_2002h]hv)}(hhh]h.[wagner_automated_2002]}(hhhhubah}(h]h]h]h]h]uhhuhhubah}(h]id3ah]hah]h]h] refdomainhreftypeh reftargethˌrefwarnsupport_smartquotesuhhoh h!hK
hhhhubh. }(hhhhubhp)}(hhaghighat_monte_2003h]hv)}(hhh]h.[haghighat_monte_2003]}(hhhhubah}(h]h]h]h]h]uhhuhhubah}(h]id4ah]hah]h]h] refdomainhreftypeh reftargethrefwarnsupport_smartquotesuhhoh h!hK
hhhhubh. }(hhhhubhp)}(hwagner_forward-weighted_2007h]hv)}(hj
h]h.[wagner_forward-weighted_2007]}(hhhjubah}(h]h]h]h]h]uhhuhjubah}(h]id5ah]hah]h]h] refdomainhreftypeh reftargetj
refwarnsupport_smartquotesuhhoh h!hK
hhhhubh.X 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.}(hX 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.hhhhh NhNubeh}(h]h]h]h]h]uhh9h h!hK
hhThhubh:)}(hX"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.h]h.X"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.}(hj5hj3hhh NhNubah}(h]h]h]h]h]uhh9h h!hKhhThhubh:)}(hXMonaco 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.h]h.XMonaco 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.}(hjChjAhhh NhNubah}(h]h]h]h]h]uhh9h h!hKhhThhubh:)}(hXWhile 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.h]h.XWhile 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.}(hjQhjOhhh NhNubah}(h]h]h]h]h]uhh9h h!hKhhThhubeh}(h]introductionah]h]introductionah]h]uhh"hh$hhh h!hK ubh#)}(hhh](h()}(hCADIS Methodologyh]h.CADIS Methodology}(hjjhjhhhh NhNubah}(h]h]h]h]h]uhh'hjehhh h!hKubh:)}(hXyMAVRIC 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.h]h.XyMAVRIC 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.}(hjxhjvhhh NhNubah}(h]h]h]h]h]uhh9h h!hKhjehhubh#)}(hhh](h()}(hOverview of CADISh]h.Overview of CADIS}(hjhjhhh NhNubah}(h]h]h]h]h]uhh'hjhhh h!hKubh:)}(hX\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}`.h](h.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 }(hCADIS 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 hjhhh NhNubhmath)}(h+:math:`q\left(\overrightarrow{r},E \right)`h]h.#q\left(\overrightarrow{r},E \right)}(hhhjubah}(h]h]h]h]h]uhjhjubh." and a detector
response function }(h" and a detector
response function hjhhh NhNubj)}(h4:math:`\sigma_{d}\left(\overrightarrow{r},E \right)`h]h.,\sigma_{d}\left(\overrightarrow{r},E \right)}(hhhjubah}(h]h]h]h]h]uhjhjubh.,.
To determine the total detector response, }(h,.
To determine the total detector response, hjhhh NhNubh@)}(h*R*h]h.R}(hhhjubah}(h]h]h]h]h]uhh?hjubh., the forward scalar flux
}(h, the forward scalar flux
hjhhh NhNubj)}(h.:math:`\phi\left(\overrightarrow{r},E \right)`h]h.&\phi\left(\overrightarrow{r},E \right)}(hhhjubah}(h]h]h]h]h]uhjhjubh. must be known. The
response is found by integrating the product of the detector response
function and the flux over the detector volume }(h must be known. The
response is found by integrating the product of the detector response
function and the flux over the detector volume hjhhh NhNubj)}(h
:math:`V_{d}`h]h.V_{d}}(hhhjubah}(h]h]h]h]h]uhjhjubh..}(h.hjhhh NhNubeh}(h]h]h]h]h]uhh9h h!hKhjhhubh
)}(hhh]h}(h]h]h]h]h]hequation-mavric-1uhh hjhhh h!hNubh
math_block)}(hR = \int_{V_{d}}^{}{\int_{E}^{}{\sigma_{d}\left( \overrightarrow{r},E \right)}}\phi\left(\overrightarrow{r},E \right)\textit{dE dV.}h]h.R = \int_{V_{d}}^{}{\int_{E}^{}{\sigma_{d}\left( \overrightarrow{r},E \right)}}\phi\left(\overrightarrow{r},E \right)\textit{dE dV.}}(hhhjubah}(h]jah]h]h]h]docnameMAVRICnumberKlabelmavric-1nowrap xml:spacepreserveuhjh h!hK(hjhhexpect_referenced_by_name}expect_referenced_by_id}jjsubh:)}(hXAlternatively, 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}`.h](h.+Alternatively, if the adjoint scalar flux,
}(h+Alternatively, if the adjoint scalar flux,
hj+hhh NhNubj)}(h2:math:`\phi^{+}\left(\overrightarrow{r},E \right)`h]h.*\phi^{+}\left(\overrightarrow{r},E \right)}(hhhj4ubah}(h]h]h]h]h]uhjhj+ubh.F, is known from the
corresponding adjoint problem with adjoint source
}(hF, is known from the
corresponding adjoint problem with adjoint source
hj+hhh NhNubj)}(h^:math:`q^{+}\left(\overrightarrow{r},E \right) = \sigma_{d}\left(\overrightarrow{r},E \right)`h]h.Vq^{+}\left(\overrightarrow{r},E \right) = \sigma_{d}\left(\overrightarrow{r},E \right)}(hhhjGubah}(h]h]h]h]h]uhjhj+ubh.,
then the total detector response could be found by integrating the
product of the forward source and the adjoint flux over the source
volume, }(h,
then the total detector response could be found by integrating the
product of the forward source and the adjoint flux over the source
volume, hj+hhh NhNubj)}(h
:math:`V_{s}`h]h.V_{s}}(hhhjZubah}(h]h]h]h]h]uhjhj+ubh..}(hjhj+hhh NhNubeh}(h]h]h]h]h]uhh9h h!hK.hjhhubh
)}(hhh]h}(h]h]h]h]h]hequation-mavric-2uhh hjhhh h!hNubj)}(hR = \int_{V_{s}}^{}{\int_{E}^{}{q\left(\overrightarrow{r},E \right)}}\phi^{+}\left( \overrightarrow{r},E \right)\textit{dE dV.}h]h.R = \int_{V_{s}}^{}{\int_{E}^{}{q\left(\overrightarrow{r},E \right)}}\phi^{+}\left( \overrightarrow{r},E \right)\textit{dE dV.}}(hhhj|ubah}(h]j{ah]h]h]h]docnamej numberKlabelmavric-2nowrapj%j&uhjh h!hK7hjhhj'}j)}j{jrsubh:)}(hXUnfortunately, 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.h]h.XUnfortunately, 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.}(hjhjhhh NhNubah}(h]h]h]h]h]uhh9h h!hKhhh NhNubj)}(hI:math:`\widehat{p}\left( i \right) = \ R_{i}/\sum_{}^{}{R_{i} = R_{i}/R}`h]h.A\widehat{p}\left( i \right) = \ R_{i}/\sum_{}^{}{R_{i} = R_{i}/R}}(hhhjGubah}(h]h]h]h]h]uhjhj>ubh..}(hjhj>hhh NhNubeh}(h]h]h]h]h]uhh9h h!hKhj.hhubh:)}(hWhen 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 beh](h.HWhen using the biased distribution used to select an individual source,
}(hHWhen using the biased distribution used to select an individual source,
hj_hhh NhNubj)}(h#:math:`\widehat{p}\left( i \right)`h]h.\widehat{p}\left( i \right)}(hhhjhubah}(h]h]h]h]h]uhjhj_ubh.&, and the biased source distribution,
}(h&, and the biased source distribution,
hj_hhh NhNubj)}(h<:math:`{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)`h]h.4{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)}(hhhj{ubah}(h]h]h]h]h]uhjhj_ubh.2, the birth
weight of the sampled particle will be}(h2, the birth
weight of the sampled particle will behj_hhh NhNubeh}(h]h]h]h]h]uhh9h h!hKhj.hhubh
)}(hhh]h}(h]h]h]h]h]hequation-mavric-13uhh hj.hhh h!hNubj)}(hX* \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}h]h.X* \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}}(hhhjubah}(h]jah]h]h]h]docnamej numberK
label mavric-13nowrapj%j&uhjh h!hKhj.hhj'}j)}jjsubh:)}(hYwhich matches the target weight,
:math:`\overline{w}\left( \overrightarrow{r},E \right)`.h](h.!which matches the target weight,
}(h!which matches the target weight,
hjhhh NhNubj)}(h7:math:`\overline{w}\left( \overrightarrow{r},E \right)`h]h./\overline{w}\left( \overrightarrow{r},E \right)}(hhhjubah}(h]h]h]h]h]uhjhjubh..}(hjhjhhh NhNubeh}(h]h]h]h]h]uhh9h h!hKhj.hhubeh}(h]multiple-sources-with-cadisah]h]multiple sources with cadisah]h]uhh"hjehhh h!hK{ubh#)}(hhh](h()}(hMultiple tallies with CADISh]h.Multiple tallies with CADIS}(hjhjhhh NhNubah}(h]h]h]h]h]uhh'hjhhh h!hKubh:)}(hXPThe 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.h]h.XPThe 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.}(hjhjhhh NhNubah}(h]h]h]h]h]uhh9h h!hKhjhhubh:)}(hXhThe 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.h]h.XhThe 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.}(hjhjhhh NhNubah}(h]h]h]h]h]uhh9h h!hKhjhhubh:)}(hX 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.h]h.X 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.}(hjhj hhh NhNubah}(h]h]h]h]h]uhh9h h!hKhjhhubh:)}(hXMAVRIC 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.h]h.XMAVRIC 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.}(hjhjhhh NhNubah}(h]h]h]h]h]uhh9h h!hKhjhhubeh}(h]multiple-tallies-with-cadisah]h]multiple tallies with cadisah]h]uhh"hjehhh h!hKubh#)}(hhh](h()}(hForward-weighted CADISh]h.Forward-weighted CADIS}(hj2hj0hhh NhNubah}(h]h]h]h]h]uhh'hj-hhh h!hMubh:)}(hXIn 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.h]h.XIn 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.}(hj@hj>hhh NhNubah}(h]h]h]h]h]uhh9h h!hMhj-hhubh:)}(hXsThis 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.h]h.XsThis 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.}(hjNhjLhhh NhNubah}(h]h]h]h]h]uhh9h h!hM
hj-hhubh:)}(hXAn 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.h](h.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}(hAn 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\ hjZhhh NhNubh subscript)}(h:sub:`N`h]h.N}(hhhjeubah}(h]h]h]h]h]uhjchjZubh.Xy 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}(hX{ 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\ hjZhhh NhNubjd)}(h:sub:`N`h]h.N}(hhhjxubah}(h]h]h]h]h]uhjchjZubh.
calculation.}(h
calculation.hjZhhh NhNubeh}(h]h]h]h]h]uhh9h h!hMhj-hhubh:)}(hXFor 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 beh](h.FFor example, if the goal is to calculate a detector response function
}(hFFor example, if the goal is to calculate a detector response function
hjhhh NhNubj)}(h":math:`\sigma_{d}\left( E \right)`h]h.\sigma_{d}\left( E \right)}(hhhjubah}(h]h]h]h]h]uhjhjubh.Z (such as dose rate using
flux-to-dose-rate conversion factors) over a volume (defined by
}(hZ (such as dose rate using
flux-to-dose-rate conversion factors) over a volume (defined by
hjhhh NhNubj)}(h*:math:`g\left( \overrightarrow{r} \right)`h]h."g\left( \overrightarrow{r} \right)}(hhhjubah}(h]h]h]h]h]uhjhjubh.<) corresponding to mesh tally,
then instead of simply using
}(h<) corresponding to mesh tally,
then instead of simply using
hjhhh NhNubj)}(hd:math:`q^{+}\left( \overrightarrow{r},E \right) = \sigma_{d}\left( E \right)\ g(\overrightarrow{r})`h]h.\q^{+}\left( \overrightarrow{r},E \right) = \sigma_{d}\left( E \right)\ g(\overrightarrow{r})}(hhhjubah}(h]h]h]h]h]uhjhjubh.,
the adjoint source would be}(h,
the adjoint source would behjhhh NhNubeh}(h]h]h]h]h]uhh9h h!hMhj-hhubh
)}(hhh]h}(h]h]h]h]h]hequation-mavric-14uhh hj-hhh h!hNubj)}(h 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}}\ ,h]h. 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}}\ ,}(hhhjubah}(h]jah]h]h]h]docnamej numberKlabel mavric-14nowrapj%j&uhjh h!hM(hj-hhj'}j)}jjsubh:)}(hXcwhere :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.h](h.where }(hwhere hjhhh NhNubj)}(h/:math:`\phi\left( \overrightarrow{r},E \right)`h]h.'\phi\left( \overrightarrow{r},E \right)}(hhhj ubah}(h]h]h]h]h]uhjhjubh.Q is an estimate of
the forward flux and the energy integral is over the voxel at }(hQ is an estimate of
the forward flux and the energy integral is over the voxel at hjhhh NhNubj)}(h:math:`\overrightarrow{r}`h]h.\overrightarrow{r}}(hhhj ubah}(h]h]h]h]h]uhjhjubh.F.
The adjoint source is nonzero only where the mesh tally is defined
(}(hF.
The adjoint source is nonzero only where the mesh tally is defined
(hjhhh NhNubj)}(h*:math:`g\left( \overrightarrow{r} \right)`h]h."g\left( \overrightarrow{r} \right)}(hhhj' ubah}(h]h]h]h]h]uhjhjubh.S), and its strength is
inversely proportional to the forward estimate of dose rate.}(hS), and its strength is
inversely proportional to the forward estimate of dose rate.hjhhh NhNubeh}(h]h]h]h]h]uhh9h h!hM-hj-hhubh:)}(hX
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)`, byh](h.XThe 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,
}(hXThe 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,
hj@ hhh NhNubj)}(h,:math:`m\left( \overrightarrow{r},E \right)`h]h.$m\left( \overrightarrow{r},E \right)}(hhhjI ubah}(h]h]h]h]h]uhjhj@ ubh.8, can be related to the true
physical particle density, }(h8, can be related to the true
physical particle density, hj@ hhh NhNubj)}(h-:math:`n\left( \overrightarrow{r},E \right),`h]h.%n\left( \overrightarrow{r},E \right),}(hhhj\ ubah}(h]h]h]h]h]uhjhj@ ubh.9
by the average Monte Carlo weight of scoring particles,
}(h9
by the average Monte Carlo weight of scoring particles,
hj@ hhh NhNubj)}(h7:math:`\overline{w}\left( \overrightarrow{r},E \right)`h]h./\overline{w}\left( \overrightarrow{r},E \right)}(hhhjo ubah}(h]h]h]h]h]uhjhj@ ubh., by}(h, byhj@ hhh NhNubeh}(h]h]h]h]h]uhh9h h!hM3hj-hhubh
)}(hhh]h}(h]h]h]h]h]hequation-mavric-15uhh hj-hhh h!hNubj)}(hn\left( \overrightarrow{r},E \right) = \ \overline{w}\left( \overrightarrow{r},E \right)\text{m}\left( \overrightarrow{r},E \right).h]h.n\left( \overrightarrow{r},E \right) = \ \overline{w}\left( \overrightarrow{r},E \right)\text{m}\left( \overrightarrow{r},E \right).}(hhhj ubah}(h]j ah]h]h]h]docnamej numberKlabel mavric-15nowrapj%j&uhjh h!hM=hj-hhj'}j)}j j subh:)}(hXIn 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.h](h.XzIn 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, }(hXzIn 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, hj hhh NhNubj)}(h,:math:`m\left( \overrightarrow{r},E \right)`h]h.$m\left( \overrightarrow{r},E \right)}(hhhj ubah}(h]h]h]h]h]uhjhj ubh.X",
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.}(hX",
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.hj hhh NhNubeh}(h]h]h]h]h]uhh9h h!hMChj-hhubh:)}(hXTwo 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`.h](h.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, }(hTwo 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, hj hhh NhNubj)}(h/:math:`\phi\left( \overrightarrow{r},E \right)`h]h.'\phi\left( \overrightarrow{r},E \right)}(hhhj ubah}(h]h]h]h]h]uhjhj ubh.. 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,}(h. 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,\ hj hhh NhNubj)}(he:math:`\int_{}^{}{\sigma_{d}\left( E \right)\text{ϕ}\left( \overrightarrow{r},E \right)}\text{\ dE}`h]h.]\int_{}^{}{\sigma_{d}\left( E \right)\text{ϕ}\left( \overrightarrow{r},E \right)}\text{\ dE}}(hhhj ubah}(h]h]h]h]h]uhjhj ubh.a.
For a tally where the total flux is desired, then the response in the
adjoint source is simply }(ha.
For a tally where the total flux is desired, then the response in the
adjoint source is simply hj hhh NhNubj)}(h&:math:`\sigma_{d}\left( E \right) = 1`h]h.\sigma_{d}\left( E \right) = 1}(hhhj ubah}(h]h]h]h]h]uhjhj ubh..}(hjhj hhh NhNubeh}(h]h]h]h]h]uhh9h h!hMOhj-hhubh:)}(hXTo 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.h](h.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
}(hTo 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
hj
hhh NhNubj)}(h*:math:`g\left( \overrightarrow{r} \right)`h]h."g\left( \overrightarrow{r} \right)}(hhhj
ubah}(h]h]h]h]h]uhjhj
ubh. and its optional response
}(h and its optional response
hj
hhh NhNubj)}(h":math:`\sigma_{d}\left( E \right)`h]h.\sigma_{d}\left( E \right)}(hhhj,
ubah}(h]h]h]h]h]uhjhj
ubh.', the standard adjoint source would be
}(h', the standard adjoint source would be
hj
hhh NhNubj)}(hv:math:`q^{+}\left( \overrightarrow{r},E \right) = \sigma_{d}\left( E \right)\text{g}\left( \overrightarrow{r} \right)`h]h.nq^{+}\left( \overrightarrow{r},E \right) = \sigma_{d}\left( E \right)\text{g}\left( \overrightarrow{r} \right)}(hhhj?
ubah}(h]h]h]h]h]uhjhj
ubh.'.
The forward-weighted adjoint source,
}(h'.
The forward-weighted adjoint source,
hj
hhh NhNubj)}(h0:math:`q^{+}\left( \overrightarrow{r},E \right)`h]h.(q^{+}\left( \overrightarrow{r},E \right)}(hhhjR
ubah}(h]h]h]h]h]uhjhj
ubh.A, depending on what
quantity is to be optimized, is listed below.}(hA, depending on what
quantity is to be optimized, is listed below.hj
hhh NhNubeh}(h]h]h]h]h]uhh9h h!hMYhj-hhubhtable)}(hhh]htgroup)}(hhh](hcolspec)}(hhh]h}(h]h]h]h]h]colwidthKuhju
hjr
ubjv
)}(hhh]h}(h]h]h]h]h]colwidthMiuhju
hjr
ubhthead)}(hhh]hrow)}(hhh](hentry)}(hhh]h:)}(h**For the calculation of**h]hstrong)}(hj
h]h.For the calculation of}(hhhj
ubah}(h]h]h]h]h]uhj
hj
ubah}(h]h]h]h]h]uhh9h h!hMehj
ubah}(h]h]h]h]h]uhj
hj
ubj
)}(hhh]h:)}(h**Adjoint source**h]j
)}(hj
h]h.Adjoint source}(hhhj
ubah}(h]h]h]h]h]uhj
hj
ubah}(h]h]h]h]h]uhh9h h!hMehj
ubah}(h]h]h]h]h]uhj
hj
ubeh}(h]h]h]h]h]uhj
hj
ubah}(h]h]h]h]h]uhj
hjr
ubhtbody)}(hhh](j
)}(hhh](j
)}(hhh]h:)}(hYEnergy and spatially dependent flux. :math:`\phi\left(\overrightarrow{r},E \right)`h](h.+Energy and spatially dependent flux. }(h+Energy and spatially dependent flux. hj
ubj)}(h.:math:`\phi\left(\overrightarrow{r},E \right)`h]h.&\phi\left(\overrightarrow{r},E \right)}(hhhj
ubah}(h]h]h]h]h]uhjhj
ubeh}(h]h]h]h]h]uhh9h h!hMghj
ubah}(h]h]h]h]h]uhj
hj
ubj
)}(hhh]j)}(hR\frac{g\left( \overrightarrow{r}\right)}{\phi\left(\overrightarrow{r},E \right)}
h]h.R\frac{g\left( \overrightarrow{r}\right)}{\phi\left(\overrightarrow{r},E \right)}
}(hhhjubah}(h]h]h]h]h]docnamej numberNlabelNnowrapj%j&uhjh h!hMghjubah}(h]h]h]h]h]uhj
hj
ubeh}(h]h]h]h]h]uhj
hj
ubj
)}(hhh](j
)}(hhh]h:)}(hnSpatially dependent total flux. :math:`\int_{}^{}{\phi\left( \overrightarrow{r},E \right)}\textit{dE}`h](h.(Spatially dependent total flux. }(h(Spatially dependent total flux. hj:ubj)}(hF:math:`\int_{}^{}{\phi\left( \overrightarrow{r},E \right)}\textit{dE}`h]h.>\int_{}^{}{\phi\left( \overrightarrow{r},E \right)}\textit{dE}}(hhhjCubah}(h]h]h]h]h]uhjhj:ubeh}(h]h]h]h]h]uhh9h h!hMihj7ubah}(h]h]h]h]h]uhj
hj4ubj
)}(hhh]j)}(hj\frac{g\left( \overrightarrow{r}\right)}{\int_{}^{}{\phi\left( \overrightarrow{r},E \right)}\textit{dE}}
h]h.j\frac{g\left( \overrightarrow{r}\right)}{\int_{}^{}{\phi\left( \overrightarrow{r},E \right)}\textit{dE}}
}(hhhj`ubah}(h]h]h]h]h]docnamej numberNlabelNnowrapj%j&uhjh h!hMihj]ubah}(h]h]h]h]h]uhj
hj4ubeh}(h]h]h]h]h]uhj
hj
ubj
)}(hhh](j
)}(hhh]h:)}(hSpatially dependent total response. :math:`\int_{}^{}{\sigma_{d}\left( E \right)\text{ϕ}\left(\overrightarrow{r},E\right)}\textit{dE}`h](h.,Spatially dependent total response. }(h,Spatially dependent total response. hjubj)}(hc:math:`\int_{}^{}{\sigma_{d}\left( E \right)\text{ϕ}\left(\overrightarrow{r},E\right)}\textit{dE}`h]h.[\int_{}^{}{\sigma_{d}\left( E \right)\text{ϕ}\left(\overrightarrow{r},E\right)}\textit{dE}}(hhhjubah}(h]h]h]h]h]uhjhjubeh}(h]h]h]h]h]uhh9h h!hMkhjubah}(h]h]h]h]h]uhj
hj~ubj
)}(hhh]j)}(h\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}}
h]h.\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}}
}(hhhjubah}(h]h]h]h]h]docnamej numberNlabelNnowrapj%j&uhjh h!hMkhjubah}(h]h]h]h]h]uhj
hj~ubeh}(h]h]h]h]h]uhj
hj
ubeh}(h]h]h]h]h]uhj
hjr
ubeh}(h]h]h]h]h]colsKuhjp
hjm
ubah}(h]h]h]h]h]aligndefaultuhjk
hj-hhh NhNubh:)}(hXThe 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.h]h.XThe 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.}(hjhjhhh NhNubah}(h]h]h]h]h]uhh9h h!hMnhj-hhubeh}(h]forward-weighted-cadisah]h]forward-weighted cadisah]h]uhh"hjehhh h!hMubeh}(h]cadis-methodologyah]h]cadis methodologyah]h]uhh"hh$hhh h!hKubh#)}(hhh](h()}(hMAVRIC Implementation of CADISh]h.MAVRIC Implementation of CADIS}(hjhjhhh NhNubah}(h]h]h]h]h]uhh'hjhhh h!hMxubh:)}(hXGWith 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.h]h.XGWith 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.}(hjhjhhh NhNubah}(h]h]h]h]h]uhh9h h!hMzhjhhubh#)}(hhh](h()}(hDenovoh]h.Denovo}(hjhjhhh NhNubah}(h]h]h]h]h]uhh'hjhhh h!hM}ubh:)}(hXDenovo 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.h]h.XDenovo 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.}(hj-hj+hhh NhNubah}(h]h]h]h]h]uhh9h h!hMhjhhubh:)}(hXDenovo 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 ash]h.XDenovo 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}(hj;hj9hhh NhNubah}(h]h]h]h]h]uhh9h h!hMhjhhubh
)}(hhh]h}(h]h]h]h]h]hequation-mavric-16uhh hjhhh h!hNubj)}(h+\mathbf{L}\psi = \mathbf{\text{MS}}\phi + qh]h.+\mathbf{L}\psi = \mathbf{\text{MS}}\phi + q}(hhhjQubah}(h]jPah]h]h]h]docnamej numberKlabel mavric-16nowrapj%j&uhjh h!hMhjhhj'}j)}jPjGsubh:)}(hXwhere **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`,h](h.where }(hwhere hjfhhh NhNubj
)}(h**L**h]h.L}(hhhjoubah}(h]h]h]h]h]uhj
hjfubh.) is the differential transport operator, }(h) is the differential transport operator, hjfhhh NhNubj
)}(h**M**h]h.M}(hhhjubah}(h]h]h]h]h]uhj
hjfubh.% is the
moment-to-discrete operator, }(h% is the
moment-to-discrete operator, hjfhhh NhNubj
)}(h**S**h]h.S}(hhhjubah}(h]h]h]h]h]uhj
hjfubh.4 is the matrix of scattering
cross-section moments, }(h4 is the matrix of scattering
cross-section moments, hjfhhh NhNubh@)}(h*q*h]h.q}(hhhjubah}(h]h]h]h]h]uhh?hjfubh.( is the external and in-scatter source,
}(h( is the external and in-scatter source,
hjfhhh NhNubj)}(h:math:`\phi`h]h.\phi}(hhhjubah}(h]h]h]h]h]uhjhjfubh., is the vector of angular flux moments, and }(h, is the vector of angular flux moments, and hjfhhh NhNubj)}(h:math:`\psi`h]h.\psi}(hhhjubah}(h]h]h]h]h]uhjhjfubh.K is
the vector of angular fluxes at discrete angles. Applying the operator
}(hK is
the vector of angular fluxes at discrete angles. Applying the operator
hjfhhh NhNubj
)}(h**D**h]h.D}(hhhjubah}(h]h]h]h]h]uhj
hjfubh., where }(h, where hjfhhh NhNubj)}(h:math:`\phi = \mathbf{D}\psi`h]h.\phi = \mathbf{D}\psi}(hhhjubah}(h]h]h]h]h]uhjhjfubh.a, and rearranging terms casts
the in-group equations in the form of a traditional linear system,
}(ha, and rearranging terms casts
the in-group equations in the form of a traditional linear system,
hjfhhh NhNubj)}(h:math:`\mathbf{A}x = b`h]h.\mathbf{A}x = b}(hhhj
ubah}(h]h]h]h]h]uhjhjfubh.,}(h,hjfhhh NhNubeh}(h]h]h]h]h]uhh9h h!hMhjhhubhblock_quote)}(hhh](h
)}(hhh]h}(h]h]h]h]h]hequation-mavric-17uhh hj"
ubj)}(hh\left( \mathbf{I} - \mathbf{D}\mathbf{L}^{- 1}\mathbf{\text{MS}} \right) = \mathbf{D}\mathbf{L}^{- 1}q .h]h.h\left( \mathbf{I} - \mathbf{D}\mathbf{L}^{- 1}\mathbf{\text{MS}} \right) = \mathbf{D}\mathbf{L}^{- 1}q .}(hhhj/
ubah}(h]j.
ah]h]h]h]docnamej numberKlabel mavric-17nowrapj%j&uhjh h!hMhj"
j'}j)}j.
j%
subeh}(h]h]h]h]h]uhj
hjhhh NhNubh:)}(hXThe 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.h](h.The operation }(hThe operation hjJ
hhh NhNubj)}(h:math:`\mathbf{L}^{- 1}\nu`h]h.\mathbf{L}^{- 1}\nu}(hhhjS
ubah}(h]h]h]h]h]uhjhjJ
ubh., where }(h, where hjJ
hhh NhNubj)}(h:math:`\nu`h]h.\nu}(hhhjf
ubah}(h]h]h]h]h]uhjhjJ
ubh.X3 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 }(hX3 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 hjJ
hhh NhNubhp)}(h
baker_sn_1998h]hv)}(hj{
h]h.[baker_sn_1998]}(hhhj}
ubah}(h]h]h]h]h]uhhuhjy
ubah}(h]id6ah]hah]h]h] refdomainhreftypeh reftargetj{
refwarnsupport_smartquotesuhhoh h!hMhjJ
hhubh.<. The Trilinos package is used for the GMRES
implementation }(h<. The Trilinos package is used for the GMRES
implementation hjJ
hhh NhNubhp)}(hwillenbring_trilinos_2003h]hv)}(hj
h]h.[willenbring_trilinos_2003]}(hhhj
ubah}(h]h]h]h]h]uhhuhj
ubah}(h]id7ah]hah]h]h] refdomainhreftypeh reftargetj
refwarnsupport_smartquotesuhhoh h!hMhjJ
hhubh.O Denovo stores the mesh-based scalar fluxes in a
double precision binary file (}(hO Denovo stores the mesh-based scalar fluxes in a
double precision binary file (hjJ
hhh NhNubhproblematic)}(h*h]h.*}(hhhj
ubah}(h]id9ah]h]h]h]refidid8uhj
hjJ
ubh.M.dff) called a Denovo flux file. Past
versions of SCALE/Denovo used the TORT }(hM.dff) called a Denovo flux file. Past
versions of SCALE/Denovo used the TORT hjJ
hhh NhNubhp)}(hrhoades_tort_1997h]hv)}(hj
h]h.[rhoades_tort_1997]}(hhhj
ubah}(h]h]h]h]h]uhhuhj
ubah}(h]id10ah]hah]h]h] refdomainhreftypeh reftargetj
refwarnsupport_smartquotesuhhoh h!hMhjJ
hhubh.% *.varscl file format
(DOORS package }(h& \*.varscl file format
(DOORS package hjJ
hhh NhNubhp)}(hrhoades_doors_1998h]hv)}(hj
h]h.[rhoades_doors_1998]}(hhhj
ubah}(h]h]h]h]h]uhhuhj
ubah}(h]id11ah]hah]h]h] refdomainhreftypeh reftargetj
refwarnsupport_smartquotesuhhoh h!hMhjJ
hhubh.), 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.}(h), 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.hjJ
hhh NhNubeh}(h]h]h]h]h]uhh9h h!hMhjhhubeh}(h]denovoah]h]denovoah]h]uhh"hjhhh h!hM}ubh#)}(hhh](h()}(hMonacoh]h.Monaco}(hj,hj*hhh NhNubah}(h]h]h]h]h]uhh'hj'hhh h!hMubh:)}(hXThe 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.h]h.XThe 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.}(hj:hj8hhh NhNubah}(h]h]h]h]h]uhh9h h!hMhj'hhubh:)}(hwMuch of the input to MAVRIC is the same as Monaco. More details can be
found in the Monaco chapter of the SCALE manual.h]h.wMuch of the input to MAVRIC is the same as Monaco. More details can be
found in the Monaco chapter of the SCALE manual.}(hjHhjFhhh NhNubah}(h]h]h]h]h]uhh9h h!hMhj'hhubeh}(h]monacoah]h]monacoah]h]uhh"hjhhh h!hMubh#)}(hhh](h()}(hRunning MAVRICh]h.Running MAVRIC}(hjahj_hhh NhNubah}(h]h]h]h]h]uhh'hj\hhh h!hMubh:)}(hX_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.h](h.XwThe 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
}(hXwThe 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
hjmhhh NhNubhp)}(h:numref:`Mavric-sequence`h]hliteral)}(hjxh]h.Mavric-sequence}(hhhj|ubah}(h]h](xrefstd
std-numrefeh]h]h]uhjzhjvubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarn reftargetmavric-sequenceuhhoh h!hMhjmubh.. 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.}(h. 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.hjmhhh NhNubeh}(h]h]h]h]h]uhh9h h!hMhj\hhubh:)}(hXThe 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.h](h.JThe MAVRIC sequence can be stopped after key points by using the
“parm= }(hJThe MAVRIC sequence can be stopped after key points by using the
“parm= hjhhh NhNubh@)}(h*parameter*h]h. parameter}(hhhjubah}(h]h]h]h]h]uhh?hjubh. ” operator on the “=mavric” command line, which is
the first line of the input file. The various parameters are listed in
Table }(h ” operator on the “=mavric” command line, which is
the first line of the input file. The various parameters are listed in
Table hjhhh NhNubhp)}(h:numref:`mavric-param`h]j{)}(hjh]h.mavric-param}(hhhjubah}(h]h](jstd
std-numrefeh]h]h]uhjzhjubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnjmavric-paramuhhoh h!hMhjubh.. These parameters allow the user to perform checks and make
changes to the importance map calculation before the actual Monte Carlo
calculation in Monaco.}(h. These parameters allow the user to perform checks and make
changes to the importance map calculation before the actual Monte Carlo
calculation in Monaco.hjhhh NhNubeh}(h]h]h]h]h]uhh9h h!hMhj\hhubh:)}(hXMAVRIC 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.h]h.XMAVRIC 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.}(hjhjhhh NhNubah}(h]h]h]h]h]uhh9h h!hMhj\hhubh:)}(hWhen 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.h]h.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.}(hjhjhhh NhNubah}(h]h]h]h]h]uhh9h h!hMhj\hhubh:)}(hXMAVRIC 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.h](h.yMAVRIC creates many files that use the base problem name from the output
file. For an output file called “c:path1path2\}(h~MAVRIC creates many files that use the base problem name from the output
file. For an output file called “c:\path1\path2\\\ hjhhh NhNubh@)}(h*outputName*h]h.
outputName}(hhhjubah}(h]h]h]h]h]uhh?hjubh.!.out” or
“/home/path1/path2/ }(h!.out” or
“/home/path1/path2/ hjhhh NhNubh@)}(h*outputName*h]h.
outputName}(hhhj$ubah}(h]h]h]h]h]uhh?hjubh.M.inp”, spaces in the output name will
cause trouble and should not be used.}(hM.inp”, spaces in the output name will
cause trouble and should not be used.hjhhh NhNubeh}(h]h]h]h]h]uhh9h h!hMhj\hhubjl
)}(hhh](h()}(hSteps in the MAVRIC sequenceh]h.Steps in the MAVRIC sequence}(hjBhj@ubah}(h]h]h]h]h]uhh'h h!hMhj=ubjq
)}(hhh](jv
)}(hhh]h}(h]h]h]h]h]colwidthKduhju
hjNubjv
)}(hhh]h}(h]h]h]h]h]jZKduhju
hjNubj
)}(hhh](j
)}(hhh](j
)}(hhh]h:)}(h**Cross section calculation**h]j
)}(hjoh]h.Cross section calculation}(hhhjqubah}(h]h]h]h]h]uhj
hjmubah}(h]h]h]h]h]uhh9h h!hMhjjubah}(h]h]h]h]h]uhj
hjgubj
)}(hhh]h:)}(hBXSProc is used to calculate the forward cross sections for Monacoh]h.BXSProc is used to calculate the forward cross sections for Monaco}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjgubeh}(h]h]h]h]h]uhj
hjdubj
)}(hhh](j
)}(hhh]h:)}(h**Forward Denovo (optional)**h]j
)}(hjh]h.Forward Denovo (optional)}(hhhjubah}(h]h]h]h]h]uhj
hjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjubj
)}(hhh]h}(h]h]h]h]h]uhj
hjubeh}(h]h]h]h]h]uhj
hjdubj
)}(hhh](j
)}(hhh]h:)}(hCross section calculationh]h.Cross section calculation}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjubj
)}(hhh]h:)}(hBXSProc is used to calculate the forward cross sections for Denovoh]h.BXSProc is used to calculate the forward cross sections for Denovo}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjubeh}(h]h]h]h]h]uhj
hjdubj
)}(hhh](j
)}(hhh]h:)}(hForward flux calculationh]h.Forward flux calculation}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjubj
)}(hhh]h:)}(h2Denovo calculates the estimate of the forward fluxh]h.2Denovo calculates the estimate of the forward flux}(hj/hj-ubah}(h]h]h]h]h]uhh9h h!hMhj*ubah}(h]h]h]h]h]uhj
hjubeh}(h]h]h]h]h]uhj
hjdubj
)}(hhh](j
)}(hhh]h:)}(h**Adjoint Denovo (optional)**h]j
)}(hjOh]h.Adjoint Denovo (optional)}(hhhjQubah}(h]h]h]h]h]uhj
hjMubah}(h]h]h]h]h]uhh9h h!hMhjJubah}(h]h]h]h]h]uhj
hjGubj
)}(hhh]h}(h]h]h]h]h]uhj
hjGubeh}(h]h]h]h]h]uhj
hjdubj
)}(hhh](j
)}(hhh]h:)}(hCross section calculationh]h.Cross section calculation}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhj|ubah}(h]h]h]h]h]uhj
hjyubj
)}(hhh]h:)}(hBXSProc is used to calculate the adjoint cross sections for Denovoh]h.BXSProc is used to calculate the adjoint cross sections for Denovo}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjyubeh}(h]h]h]h]h]uhj
hjdubj
)}(hhh](j
)}(hhh]h:)}(hAdjoint flux calculationh]h.Adjoint flux calculation}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjubj
)}(hhh]h:)}(h2Denovo calculates the estimate of the adjoint fluxh]h.2Denovo calculates the estimate of the adjoint flux}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjubeh}(h]h]h]h]h]uhj
hjdubj
)}(hhh](j
)}(hhh]h:)}(h**CADIS (optional)**h]j
)}(hjh]h.CADIS (optional)}(hhhjubah}(h]h]h]h]h]uhj
hjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjubj
)}(hhh]h:)}(hsThe scalar flux file from Denovo is then used to create the biased source distribution and transport weight windowsh]h.sThe scalar flux file from Denovo is then used to create the biased source distribution and transport weight windows}(hjhj
ubah}(h]h]h]h]h]uhh9h h!hMhj
ubah}(h]h]h]h]h]uhj
hjubeh}(h]h]h]h]h]uhj
hjdubj
)}(hhh](j
)}(hhh]h:)}(h**Monte Carlo calculation**h]j
)}(hj/h]h.Monte Carlo calculation}(hhhj1ubah}(h]h]h]h]h]uhj
hj-ubah}(h]h]h]h]h]uhh9h h!hMhj*ubah}(h]h]h]h]h]uhj
hj'ubj
)}(hhh]h:)}(hhMonaco uses the biased source distribution and transport weight windows to calculate the various talliesh]h.hMonaco uses the biased source distribution and transport weight windows to calculate the various tallies}(hjOhjMubah}(h]h]h]h]h]uhh9h h!hMhjJubah}(h]h]h]h]h]uhj
hj'ubeh}(h]h]h]h]h]uhj
hjdubeh}(h]h]h]h]h]uhj
hjNubeh}(h]h]h]h]h]colsKuhjp
hj=ubeh}(h]mavric-sequenceah]colwidths-givenah]mavric-sequenceah]h]jcenteruhjk
hj\hhh NhNubjl
)}(hhh](h()}(h7Parameters for the MAVRIC command line (“parm=…”)h]h.7Parameters for the MAVRIC command line (“parm=…”)}(hjhjubah}(h]h]h]h]h]uhh'h h!hMhj~ubjq
)}(hhh](jv
)}(hhh]h}(h]h]h]h]h]jZK2uhju
hjubjv
)}(hhh]h}(h]h]h]h]h]jZK2uhju
hjubj
)}(hhh]j
)}(hhh](j
)}(hhh]h:)}(h Parameterh]h. Parameter}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjubj
)}(hhh]h:)}(hMAVRIC will stop afterh]h.MAVRIC will stop after}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjubeh}(h]h]h]h]h]uhj
hjubah}(h]h]h]h]h]uhj
hjubj
)}(hhh](j
)}(hhh](j
)}(hhh]h:)}(hcheckh]h.check}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjubj
)}(hhh]h:)}(hinput checkingh]h.input checking}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjubeh}(h]h]h]h]h]uhj
hjubj
)}(hhh](j
)}(hhh]h:)}(hforinph]h.forinp}(hj&hj$ubah}(h]h]h]h]h]uhh9h h!hMhj!ubah}(h]h]h]h]h]uhj
hjubj
)}(hhh]h:)}(hHForward Denovo input construction (makes ``xkba_b.inp`` in the tmp area)h](h.)Forward Denovo input construction (makes }(h)Forward Denovo input construction (makes hj;ubj{)}(h``xkba_b.inp``h]h.
xkba_b.inp}(hhhjDubah}(h]h]h]h]h]uhjzhj;ubh. in the tmp area)}(h in the tmp area)hj;ubeh}(h]h]h]h]h]uhh9h h!hMhj8ubah}(h]h]h]h]h]uhj
hjubeh}(h]h]h]h]h]uhj
hjubj
)}(hhh](j
)}(hhh]h:)}(hforwardh]h.forward}(hjqhjoubah}(h]h]h]h]h]uhh9h h!hMhjlubah}(h]h]h]h]h]uhj
hjiubj
)}(hhh]h:)}(hThe forward Denovo calculationh]h.The forward Denovo calculation}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjiubeh}(h]h]h]h]h]uhj
hjubj
)}(hhh](j
)}(hhh]h:)}(hadjinph]h.adjinp}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjubj
)}(hhh]h:)}(hHAdjoint Denovo input construction (makes ``xkba_b.inp`` in the tmp area)h](h.)Adjoint Denovo input construction (makes }(h)Adjoint Denovo input construction (makes hjubj{)}(h``xkba_b.inp``h]h.
xkba_b.inp}(hhhjubah}(h]h]h]h]h]uhjzhjubh. in the tmp area)}(h in the tmp area)hjubeh}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhj
hjubeh}(h]h]h]h]h]uhj
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)}(hhh](j
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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.h]h.XThe 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.}(hjhjhhh NhNubah}(h]h]h]h]h]uhh9h h!hMhjhhubh:)}(hBlocks (must be in this order):h]h.Blocks (must be in this order):}(hjhjhhh NhNubah}(h]h]h]h]h]uhh9h h!hMhjhhubhbullet_list)}(hhh](h list_item)}(h]Composition – (required) SCALE standard composition, list of materials used in the problem
h]h:)}(h\Composition – (required) SCALE standard composition, list of materials used in the problemh]h.\Composition – (required) SCALE standard composition, list of materials used in the problem}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhjhjhhh h!hNubj)}(h,Celldata – SCALE resonance self-shielding
h]h:)}(h+Celldata – SCALE resonance self-shieldingh]h.+Celldata – SCALE resonance self-shielding}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhjhjhhh h!hNubj)}(h;Geometry – (required) SCALE general geometry description
h]h:)}(h:Geometry – (required) SCALE general geometry descriptionh]h.:Geometry – (required) SCALE general geometry description}(hjhjubah}(h]h]h]h]h]uhh9h h!hMhjubah}(h]h]h]h]h]uhjhjhhh h!hNubj)}(h>Array – optional addition to the above geometry description
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h]h:)}(h summarizes all of the keywords in the MAVRIC parameter block.}(h> summarizes all of the keywords in the MAVRIC parameter block.hjhhh NhNubeh}(h]h]h]h]h]uhh9h h!hM
hjIhhubh:)}(hXWhen 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.h]h.XWhen 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.}(hjOhjMhhh NhNubah}(h]h]h]h]h]uhh9h h!hMhjIhhubjl
)}(hhh](h()}(h'Extra keywords for the parameters blockh]h.'Extra keywords for the parameters block}(hj`hj^ubah}(h]h]h]h]h]uhh'h h!hMhj[ubjq
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hj[ubeh}(h]extra-keywordsah]h]extra-keywordsah]h]jjuhjk
hjIhhh jVhNubeh}(h]other-blocks-shared-with-monacoah]h]other blocks shared with monacoah]h]uhh"hjhhh h!hMubh#)}(hhh](h()}(hImportance map blockh]h.Importance map block}(hj hj hhh NhNubah}(h]h]h]h]h]uhh'hj hhh h!hMubh:)}(hXThe 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:``h]h.XThe 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:``}(hj hj hhh NhNubah}(h]h]h]h]h]uhh9h h!hMhj hhubh#)}(hhh](h()}(h3Constructing a mesh for the S\ :sub:`N` calculationh](h.Constructing a mesh for the S}(hConstructing a mesh for the S\ hj hhh NhNubjd)}(h:sub:`N`h]h.N}(hhhj ubah}(h]h]h]h]h]uhjchj ubh. calculation}(h calculationhj hhh NhNubeh}(h]h]h]h]h]uhh'hj hhh h!hMubh:)}(hXFAll 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.h]h.XFAll 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.}(hj hj hhh NhNubah}(h]h]h]h]h]uhh9h h!hMhj hhubh:)}(hXWhen 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.h](h.When using S}(hWhen using S\ hj hhh NhNubjd)}(h:sub:`N`h]h.N}(hhhj!ubah}(h]h]h]h]h]uhjchj ubh.X 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.}(hX 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.hj hhh NhNubeh}(h]h]h]h]h]uhh9h h!hM"hj hhubh:)}(hXIn 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.h](h.#In MAVRIC, the goal is to use the S}(h%In MAVRIC, the goal is to use the S\ hj!hhh NhNubjd)}(h:sub:`N`h]h.N}(hhhj$!ubah}(h]h]h]h]h]uhjchj!ubh.XQ 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}(hXS 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\ hj!hhh NhNubjd)}(h:sub:`N`h]h.N}(hhhj7!ubah}(h]h]h]h]h]uhjchj!ubh.7 calculations also use a great deal of computer memory.}(h7 calculations also use a great deal of computer memory.hj!hhh NhNubeh}(h]h]h]h]h]uhh9h h!hM)hj hhubh:)}(hXBecause 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:h](h.XBecause 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}(hXBecause 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\ hjP!hhh NhNubjd)}(h:sub:`N`h]h.N}(hhhjY!ubah}(h]h]h]h]h]uhjchjP!ubh. 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:}(h 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:hjP!hhh NhNubeh}(h]h]h]h]h]uhh9h h!hM4hj hhubj)}(hhh](j)}(h]The true source regions should be included in the mesh with mesh
planes at their boundaries.
h]h:)}(h\The true source regions should be included in the mesh with mesh
planes at their boundaries.h]h.\The true source regions should be included in the mesh with mesh
planes at their boundaries.}(hj{!hjy!ubah}(h]h]h]h]h]uhh9h h!hM=hju!ubah}(h]h]h]h]h]uhjhjr!hhh h!hNubj)}(hbFor point or very small sources, place them in the center of a mesh
cell, not on the mesh planes.
h]h:)}(haFor point or very small sources, place them in the center of a mesh
cell, not on the mesh planes.h]h.aFor point or very small sources, place them in the center of a mesh
cell, not on the mesh planes.}(hj!hj!ubah}(h]h]h]h]h]uhh9h h!hM@hj!ubah}(h]h]h]h]h]uhjhjr!hhh h!hNubj)}(hAny region of the geometry where particles could eventually
contribute to the tallies (the “important” areas) should be included
in the mesh.
h]h:)}(hAny region of the geometry where particles could eventually
contribute to the tallies (the “important” areas) should be included
in the mesh.h]h.Any region of the geometry where particles could eventually
contribute to the tallies (the “important” areas) should be included
in the mesh.}(hj!hj!ubah}(h]h]h]h]h]uhh9h h!hMChj!ubah}(h]h]h]h]h]uhjhjr!hhh h!hNubj)}(hXPoint 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.
h]h:)}(hPoint 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.h]h.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.}(hj!hj!ubah}(h]h]h]h]h]uhh9h h!hMGhj!ubah}(h]h]h]h]h]uhjhjr!hhh h!hNubj)}(h`Volumetric adjoint sources should be included in the mesh with mesh
planes at their boundaries.
h]h:)}(h_Volumetric adjoint sources should be included in the mesh with mesh
planes at their boundaries.h]h._Volumetric adjoint sources should be included in the mesh with mesh
planes at their boundaries.}(hj!hj!ubah}(h]h]h]h]h]uhh9h h!hMLhj!ubah}(h]h]h]h]h]uhjhjr!hhh h!hNubj)}(hAMesh planes should be placed at significant material boundaries.
h]h:)}(h@Mesh planes should be placed at significant material boundaries.h]h.@Mesh planes should be placed at significant material boundaries.}(hj!hj!ubah}(h]h]h]h]h]uhh9h h!hMOhj!ubah}(h]h]h]h]h]uhjhjr!hhh h!hNubj)}(h%ubah}(h]h]h]h]h]jJ$uhj;$hj;%ubh:)}(hhh](h.Ahmad}(hAhmadhjK%ubjU$h.M. Ibrahim, Douglas}(hM. Ibrahim, DouglashjK%ubjU$h.E. Peplow, Thomas}(hE. Peplow, ThomashjK%ubjU$h.M. Evans, John}(hM. Evans, JohnhjK%ubjU$h.C. Wagner, and Paul}(hC. Wagner, and PaulhjK%ubjU$h.
PH Wilson.}(h
PH Wilson.hjK%ubh. }(h hjK%ubh.Improving the }(hImproving the hjK%ubh.Mesh}(hMeshhjK%ubh. }(h hjK%ubh.
Generation}(h
GenerationhjK%ubh. }(h hjK%ubh.Capabilities}(hCapabilitieshjK%ubh. in the }(h in the hjK%ubh.SCALE}(hSCALEhjK%ubh. }(h hjK%ubh.Hybrid}(hHybridhjK%ubh. }(h hjK%ubh. Shielding}(h ShieldinghjK%ubh. }(h hjK%ubh.Analysis}(hAnalysishjK%ubh. }(hjp%hjK%ubh.Sequence}(hSequencehjK%ubh..}(hjhjK%ubjy$h@)}(hhh]h.Trans. Am. Nucl. Soc}(hTrans. Am. Nucl. Sochj%ubah}(h]h]h]h]h]uhh?hjK%ubh., 100:302, 2009.}(h, 100:302, 2009.hjK%ubeh}(h]h]h]h]h]uhh9hj;%ubeh}(h]ibrahim-improving-2009ah]hah]ibrahim_improving_2009ah]h]j"aj$j uhhhj4$j$Kubj7$)}(hhh](j<$)}(hhh]h.johnson_fast_2013}(hhhj%ubah}(h]h]h]h]h]jJ$uhj;$hj%ubh:)}(hhh](h.Seth}(hSethhj%ubjU$h.R. Johnson.}(hR. Johnson.hj%ubjy$h.8Fast mix table construction for material discretization.}(h8Fast mix table construction for material discretization.hj%ubjy$h.In }(hIn hj%ubh@)}(hhh](h.Proceedings of the 2013 }(hProceedings of the 2013 hj&ubh.
International}(h
Internationalhj&ubh. }(hhhj&ubh.
Conference}(h
Conferencehj&ubh. on }(h on hj&ubh.Mathematics}(hMathematicshj&ubh. and }(h and hj&ubh.
Computational}(h
Computationalhj&ubh. }(hhhj&ubh.Methods}(hMethodshj&ubh. }(hhhj&ubh.Applied}(hAppliedhj&ubh. to }(h to hj&ubh.Nuclear}(hNuclearhj&ubh. }(hhhj&ubh.Science}(hSciencehj&ubh. and }(h and hj&ubh.Engineering}(hEngineeringhj&ubh.-}(hjJhj&ubh.M}(hMhj&ubh. and }(hj0&hj&ubh.C}(hChj&ubh. 2013}(h 2013hj&ubeh}(h]h]h]h]h]uhh?hj%ubh.. 2013.}(h. 2013.hj%ubeh}(h]h]h]h]h]uhh9hj%ubeh}(h]johnson-fast-2013ah]hah]johnson_fast_2013ah]h]j#aj$j uhhhj4$j$Kubj7$)}(hhh](j<$)}(hhh]h.rhoades_doors_1998}(hhhj&ubah}(h]h]h]h]h]jJ$uhj;$hj&ubh:)}(hhh](h.W.}(hW.hj&ubjU$h.A. Rhoades and R.}(hA. Rhoades and R.hj&ubjU$h.
L. Childs.}(h
L. Childs.hj&ubjy$h.DOORS}(hDOORShj&ubh.\ 3.2, one-, two-, three-dimensional discrete ordinates neutron/photon transport code system.}(h\ 3.2, one-, two-, three-dimensional discrete ordinates neutron/photon transport code system.hj&ubjy$h@)}(hhh]h.$RSICC, Oak Ridge National Laboratory}(h$RSICC, Oak Ridge National Laboratoryhj&ubah}(h]h]h]h]h]uhh?hj&ubh., 300:650, 1998.}(h, 300:650, 1998.hj&ubeh}(h]h]h]h]h]uhh9hj&ubeh}(h]rhoades-doors-1998ah]hah]rhoades_doors_1998ah]h]j
aj$j uhhhj4$j$Kubj7$)}(hhh](j<$)}(hhh]h.rhoades_tort_1997}(hhhj&ubah}(h]h]h]h]h]jJ$uhj;$hj&ubh:)}(hhh](h.Wayne}(hWaynehj&ubjU$h.A. Rhoades and D.}(hA. Rhoades and D.hj&ubjU$h.B. Simpson.}(hB. Simpson.hj&ubjy$h.The }(hThe hj&ubh.TORT}(hTORThj&ubh.E three-dimensional discrete ordinates neutron/photon transport code (}(hE three-dimensional discrete ordinates neutron/photon transport code (hj&ubh.TORT}(hj
'hj&ubh. version 3).}(h version 3).hj&ubjy$h.DTechnical Report, Oak Ridge National Lab., TN (United States), 1997.}(hDTechnical Report, Oak Ridge National Lab., TN (United States), 1997.hj&ubeh}(h]h]h]h]h]uhh9hj&ubeh}(h]rhoades-tort-1997ah]hah]rhoades_tort_1997ah]h]j
aj$j uhhhj4$j$Kubj7$)}(hhh](j<$)}(hhh]h.wagner_acceleration_1997}(hhhj2'ubah}(h]h]h]h]h]jJ$uhj;$hj/'ubh:)}(hhh](h.John}(hJohnhj?'ubjU$h.
C. Wagner.}(h
C. Wagner.hj?'ubjy$h@)}(hhh](h.Acceleration of }(hAcceleration of hjL'ubh.Monte}(hMontehjL'ubh. }(hhhjL'ubh.Carlo}(hCarlohjL'ubh.^ shielding calculations with an automated variance reduction technique and parallel processing}(h^ shielding calculations with an automated variance reduction technique and parallel processinghjL'ubeh}(h]h]h]h]h]uhh?hj?'ubh..}(hjhj?'ubjy$h.0PhD thesis, Pennsylvania State University, 1997.}(h0PhD thesis, Pennsylvania State University, 1997.hj?'ubeh}(h]h]h]h]h]uhh9hj/'ubeh}(h]wagner-acceleration-1997ah]hah]wagner_acceleration_1997ah]h]haj$j uhhhj4$j$Kubj7$)}(hhh](j<$)}(hhh]h.wagner_automated_2002}(hhhj'ubah}(h]h]h]h]h]jJ$uhj;$hj'ubh:)}(hhh](h.John}(hJohnhj'ubjU$h.
C. Wagner.}(h
C. Wagner.hj'ubh. }(h hj'ubh.)KsRparse_messages]j:*atransform_messages](j9*)}(hhh]h:)}(hhh]h.,Hyperlink target "mavric" is not referenced.}(hhhj*ubah}(h]h]h]h]h]uhh9hj~*ubah}(h]h]h]h]h]levelKtypeINFOsourceh!lineKuhj8*ubj9*)}(hhh]h:)}(hhh]h.7Hyperlink target "equation-mavric-1" is not referenced.}(hhhj*ubah}(h]h]h]h]h]uhh9hj*ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.7Hyperlink target "equation-mavric-2" is not referenced.}(hhhj*ubah}(h]h]h]h]h]uhh9hj*ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.7Hyperlink target "equation-mavric-3" is not referenced.}(hhhj*ubah}(h]h]h]h]h]uhh9hj*ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.7Hyperlink target "equation-mavric-4" is not referenced.}(hhhj*ubah}(h]h]h]h]h]uhh9hj*ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.7Hyperlink target "equation-mavric-5" is not referenced.}(hhhj+ubah}(h]h]h]h]h]uhh9hj*ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.7Hyperlink target "equation-mavric-6" is not referenced.}(hhhj+ubah}(h]h]h]h]h]uhh9hj+ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.7Hyperlink target "equation-mavric-7" is not referenced.}(hhhj2+ubah}(h]h]h]h]h]uhh9hj/+ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.7Hyperlink target "equation-mavric-8" is not referenced.}(hhhjK+ubah}(h]h]h]h]h]uhh9hjH+ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.7Hyperlink target "equation-mavric-9" is not referenced.}(hhhjd+ubah}(h]h]h]h]h]uhh9hja+ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.8Hyperlink target "equation-mavric-10" is not referenced.}(hhhj}+ubah}(h]h]h]h]h]uhh9hjz+ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.8Hyperlink target "equation-mavric-11" is not referenced.}(hhhj+ubah}(h]h]h]h]h]uhh9hj+ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.8Hyperlink target "equation-mavric-12" is not referenced.}(hhhj+ubah}(h]h]h]h]h]uhh9hj+ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.8Hyperlink target "equation-mavric-13" is not referenced.}(hhhj+ubah}(h]h]h]h]h]uhh9hj+ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.8Hyperlink target "equation-mavric-14" is not referenced.}(hhhj+ubah}(h]h]h]h]h]uhh9hj+ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.8Hyperlink target "equation-mavric-15" is not referenced.}(hhhj+ubah}(h]h]h]h]h]uhh9hj+ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.8Hyperlink target "equation-mavric-16" is not referenced.}(hhhj,ubah}(h]h]h]h]h]uhh9hj,ubah}(h]h]h]h]h]levelKtypej*sourceh!uhj8*ubj9*)}(hhh]h:)}(hhh]h.8Hyperlink target "equation-mavric-17" is not referenced.}(hhhj,,ubah}(h]h]h]h]h]uhh9hj),ubah}(h]h]h]h]h]levelKtypej*sourceh!lineM}uhj8*ubj9*)}(hhh]h:)}(hhh]h.3Hyperlink target "ray-positions" is not referenced.}(hhhjF,ubah}(h]h]h]h]h]uhh9hjC,ubah}(h]h]h]h]h]levelKtypej*sourceh!lineMuhj8*ubj9*)}(hhh]h:)}(hhh]h.0Hyperlink target "geom-model" is not referenced.}(hhhj`,ubah}(h]h]h]h]h]uhh9hj],ubah}(h]h]h]h]h]levelKtypej*sourceh!lineMuhj8*ubetransformerN
decorationNhhub.