appendixc.rst.txt 74.1 KB
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Alternate normalization of the importance map and biased source --------------------------------------------------------------- The importance map and biased source implemented in MAVRIC are only functions of space and energy. The importance for a specific location and energy represents the average over all directions. For applications involving a collimated beam source, a space/energy importance map may not be representative of the true importance of the particles as they stream away from the source. As an example, consider a 14.1 MeV active interrogation beam source 1 meter from a small spherical boat containing illicit nuclear material. The objective is to compute the fission rate in the nuclear material. To create the biasing parameters, an adjoint source is located within the nuclear material and the resulting importance map is shown in :numref:fig4c1. Note that in both the air and water, the importances change with distance from the ship, but for the beam source, the importance (to causing a fission in the nuclear material) anywhere along the beam should be the same, since there is little chance a 14.1 MeV neutron will interact with the air before striking the ship. .. _fig4c1: .. figure:: figs/MAVRIC/fig41c.png :align: center Importance map computed using standard CADIS. The CADIS algorithm has done exactly what it was supposed to: it made a space/energy importance map and normalized it such that the target weight where the 14.1 MeV source particles are born is 1. The problem with this is that the source particles will stream towards the ship and strike the hull where the target weight is 0.092. Since source particles have little chance of interacting in the air, the weight windows are not used to split the particle as they travel towards the ship. When source particles cross into the ship, they are split by a factor of 11 to match the target weight. For this example, splitting each particle by a factor of 11 once they strike the ship is not so bad, but for longer distances, this will result in much larger splits. For a polyenergetic source, this could lead to undersampling of the source and could result in higher variances. To remedy this problem when using beam sources, the normalization of the importance map and biased source should not be done at the source location but instead at the point where the source particles first interact with the ship. The keyword “shiftNormPos Δx Δy Δz” will shift the source normalization position by the amounts Δx, Δy, and Δz when the biased source and importance map are developed. For the Monaco Monte Carlo calculation, the source is returned to its normal position. The source input for the above problem would then be .. highlight:: scale :: read sources src 1 title="14.1 DT neutrons - collimated" strength=1e30 sphere 0 origin x=-195 y=0 z=0 (true source position) eDistributionID=1 (a mono-energetic 14.1 MeV distribution) direction 1.0 0.0 0.0 dDistributionID=2 (a 2° beam ) shiftNormPos 107.7 0.0 0.0 (just inside the hull) end src end sources where the shift moves the source position from x = -195 to x = -87.3, just inside the hull. The resulting target weights are shown in :numref:fig4c2 The source particles are born with weight 1 in a location with a target weight 10.9. The particle weight is not checked until the particle crosses into the hull, where the target weight is 1.0. .. _fig4c2: .. figure:: figs/MAVRIC/fig4c2.png :align: center Targets weights using the “shiftNormPos” keyword. Other options to manipulate the importance map for special situations include the “mapMultiplier=\ *f*\ ” keyword (in the importanceMap block or the biasing block), which will multiply every target weight by the factor *f*, and the keyword “noCheckAtBirth” in the parameters block will prevent the weight windows from being applied to source particles when they are started. When used in the MAVRIC sequence, the “shiftNormPos” capability automatically adds “noCheckAtBirth” to the Monaco input that is created. Importance maps with directional information -------------------------------------------- In MAVRIC, the CADIS method is implemented in space and energy, but in general, it could also include particle direction as well. This formulation would be the following: True source: .. math:: q\left( \overrightarrow{r},E,\widehat{\Omega} \right) Desired response: .. math:: \sigma\left( \overrightarrow{r},E,\widehat{\Omega}\right) Adjoint flux using :math:q^{+}\left( \overrightarrow{r},E,\widehat{\Omega} \right) = \ \sigma \left( \overrightarrow{r},E,\widehat{\Omega} \right): .. math:: \psi^{+}\left( \overrightarrow{r},E,\widehat{\Omega} \right) Estimate of detector response .. math:: :label: eq4c1 R = \iiint_{}^{}{q\left( \overrightarrow{r},E,\widehat{\Omega} \right)\ \psi^{+}\left( \overrightarrow{r},E,\widehat{\Omega}\right)}d\text{Ω } dE \ dV Biased source: .. math:: :label: eq4c2 \widehat{q}\left( \overrightarrow{r},E,\widehat{\Omega} \right) = \frac{1}{R}q\left( \overrightarrow{r},E,\widehat{\Omega} \right)\ \psi^{+}\left( \overrightarrow{r},E,\widehat{\Omega} \right) Target weight windows: .. math:: :label: eq4c3 \overline{w}\left( \overrightarrow{r},E,\widehat{\Omega} \right) = \frac{R}{\psi^{+}\left( \overrightarrow{r},E,\widehat{\Omega} \right)} For a system using a deterministic method to compute the adjoint fluxes, this completely general, space/energy/angle, approach presents many difficulties in implementation, namely, a. dealing with the amount of memory required for a :math:\left( \overrightarrow{r},E,\widehat{\Omega} \right) importance map in memory, b. interpolating the importance for particle directions in between quadrature angles, and c. expressing the biased source in a form suitable for a general MC code since the above biased source is, in general, not separable. Approaches incorporating directional information ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Completely general space/energy/angle CADIS is most likely too difficult to implement and may not be necessary for most applications. In most real problems that involve directionally dependent source distributions, the directional dependence is azimuthally symmetric about some reference direction, :math:\widehat{d}. The angular distribution, :math:q_{i}\left( \widehat{\Omega} \right), can be expressed as the product of the uniform azimuthal distribution and a polar distribution about reference direction :math:{\widehat{d}}_{i} giving :math:\frac{1}{2\pi}q_{i}\left( \widehat{\Omega} \bullet {\widehat{d}}_{i} \right). The geometric size of these sources tends to be small, allowing each source distribution to be expressed as the product of two separable distributions: :math:q_{i}\left( \overrightarrow{r},E,\widehat{\Omega} \right) \cong q_{i}\left( \overrightarrow{r},E \right)\ q_{i}\left( \widehat{\Omega} \right). What is needed is a CADIS method that (1) can account for the importance of a particle traveling in a certain direction; (2) can be cast as a simple modification of the space/energy CADIS method using :math:\overline{w}\left( \overrightarrow{r},E \right) and :math:\widehat{q}\left( \overrightarrow{r},E \right); and (3) is simpler than the full space/angle/energy approach. This can be done starting with the approximation that the angular component of the adjoint flux :math:\psi^{+}\left( \overrightarrow{r},E,\widehat{\Omega} \right) is separable and symmetric about the average adjoint current direction :math:\widehat{n}\left( \overrightarrow{r},E \right), such that .. math:: \psi^{+}\left( \overrightarrow{r},E,\widehat{\Omega} \right) \cong \phi^{+}\left( \overrightarrow{r},E \right)\ \frac{1}{2\pi}f\left( \widehat{\Omega} \bullet \widehat{n} \right)\text{\ .} This is similar to the AVATAR approach :cite:van_riper_avatar_1997 but with explicitly including the azimuthal distribution so that the standard definition :math:\int_{}^{}{\phi^{+}\left( \overrightarrow{r},E \right)\ \frac{1}{2\pi}f\left( \widehat{\Omega} \bullet \widehat{n} \right)\ d\widehat{\Omega}} = \phi^{+}\left( \overrightarrow{r},E \right) applies. The probability distribution function :math:f\left( \mu \right) describing the shape of the azimuthally symmetric current at :math:\left( \overrightarrow{r},E \right) has the form of .. math:: f\left( \mu \right) = \frac{\lambda e^{\text{λμ}}}{2\ \mathrm{\sinh}\left( \lambda \right)}\ , with the single parameter :math:\lambda\left( \overrightarrow{r},E \right) determined from :math:\overline{\mu}\left( \overrightarrow{r},E \right), the average cosine of scatter. From this, we can propose that weight window targets be developed that are inversely proportional to the approximation of the adjoint angular flux: .. math:: :label: eq4c4 \overline{w}\left( \overrightarrow{r},E,\widehat{\Omega} \right) = \frac{2\pi\ k}{\phi^{+}\left( \overrightarrow{r},E \right) \ \ f\left( \widehat{\Omega} \bullet \widehat{n} \right)}\ , where :math:k is the constant of proportionality that will be adjusted to make the importance map consistent with the biased source(s). Two methods will be examined here, one without and one with biasing of the source directional dependence. For both of the methods, the S\ :sub:N code Denovo was modified to report not only the adjoint scalar fluxes, :math:\phi^{+}\left( \overrightarrow{r},E \right), but also the adjoint net currents in :math:x, :math:y, and :math:z directions: :math:J_{x}\left( \overrightarrow{r},E \right), :math:\ J_{y}\left( \overrightarrow{r},E \right), and :math:J_{z}\left( \overrightarrow{r},E \right). These currents are used to find :math:\widehat{n}\left( \overrightarrow{r},E \right) and :math:\lambda\left( \overrightarrow{r},E \right). The following methods have been developed so that the standard CADIS routines can be used to compute space/energy quantities of the response per unit source :math:R, the weight window target values :math:\overline{w}\left( \overrightarrow{r},E \right), and biased source :math:\widehat{q}\left( \overrightarrow{r},E \right) with just the adjoint scalar fluxes. These quantities are then modified by the directional information. Directionally dependent weight windows without directional source biasing ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ It is proposed that the biased source :math:\widehat{q}\left( \overrightarrow{r},E,\widehat{\Omega} \right) should be proportional to both the true source distribution and the space/energy component of the adjoint flux: .. math:: \widehat{q}\left( \overrightarrow{r},E,\widehat{\Omega} \right) = \frac{1}{R}\left\lbrack q\left( \overrightarrow{r},E \right)\ \frac{1}{2\pi}q\left( \widehat{\Omega} \bullet \widehat{d} \right) \right\rbrack\ \phi^{+}\left( \overrightarrow{r},E \right)\ , where the constant of proportionality, :math:R, is determined by forcing :math:\widehat{q}\left( \overrightarrow{r},E,\widehat{\Omega} \right) to be a pdf. Since the angular component of the adjoint flux is not included, the directional distribution of the biased source will be exactly the same as the true source. Note that this approach would be exact for cases where no directional biasing could be applied – beam sources. For multiple sources (each with a probability distribution function :math:q_{i}\left( \overrightarrow{r},E \right) and a strength :math:S_{i}, giving a total source strength of :math:S = \sum_{}^{}S_{i}), the user is required to provide one point in phase space :math:\left( {\overrightarrow{r}}_{i},E_{i},{\widehat{\Omega}}_{i} \right) for each source :math:i that is representative of that entire source where the biased source will match the target weight windows. For each source, a vector :math:{\widehat{n}}_{i} = \widehat{n}\left( {\overrightarrow{r}}_{i},E_{i} \right) is computed using that point. For the general case of multiple sources, the biased source sampling distribution, the biased source distributions, and the weight windows are computed using :math:R_{i} = \iint_{}^{}{q_{i}\left( \overrightarrow{r},E\right)\ \phi^{+}\left( \overrightarrow{r},E \right)} dE \ dr \ \ \ \ \ \ \ \ \ \text{(estimated response from source} \ i) :math:\widehat{p}\left( i \right) = \frac{{S_{i}R}_{i}\ f\left( {\widehat{\Omega}}_{i} \bullet {\widehat{n}}_{i} \right)}{\sum_{}^{}{{S_{i}R}_{i}\ f\left( {\widehat{\Omega}}_{i} \bullet {\widehat{n}}_{i} \right)}} \ \ \ \ \ \ \ \text{(biased sampling of source} \ i) :math:{\widehat{q}}_{i}\left(\overrightarrow{r},E,\widehat{\Omega} \right) \ = \ \frac{1}{R_{i}}q_{i}\left( \overrightarrow{r},E \right)\ \phi^{+}\left( \overrightarrow{r},E \right)\ \frac{1}{2\pi}q_{i}\left( \widehat{\Omega} \bullet {\widehat{d}}_{i} \right) \ = \ {\widehat{q}}_{i}\left( \overrightarrow{r},E\right)\ \frac{1}{2\pi}q_{i}\left( \widehat{\Omega} \bullet {\widehat{d}}_{i} \right) :math:\overline{w}\left( \overrightarrow{r},E,\widehat{\Omega} \right) \ \ \ = \ \ \ \frac{\sum_{}^{}{{S_{i}R}_{i}\ f\left( {\widehat{\Omega}}_{i} \bullet {\widehat{n}}_{i} \right)}}{S\phi^{+}\left( \overrightarrow{r},E \right)}\frac{1}{\ f\left( \widehat{\Omega} \bullet \widehat{n} \right)} \ \ \ = \ \ \ \frac{\sum_{}^{}{{S_{i}R}_{i}\ f\left( {\widehat{\Omega}}_{i} \bullet {\widehat{n}}_{i} \right)}}{\sum_{}^{}{S_{i}R}_{i}}\overline{w}\left( \overrightarrow{r},E \right)\frac{1}{f\left( \widehat{\Omega} \bullet \widehat{n} \right)} Directionally dependent weight windows with directional source biasing ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Here it is proposed that the biased source be proportional to both the true source distribution and the approximation of the adjoint angular flux. With a small geometric source, it is also assumed that there is one vector, :math:{\widehat{n}}_{0} = \widehat{n}\left( {\overrightarrow{r}}_{0},E_{0} \right), evaluated at a specific location and energy, which represents the adjoint current direction over that source. The biased source then looks like .. math:: \widehat{q}\left( \overrightarrow{r},E,\widehat{\Omega} \right) & = \frac{1}{\text{Rc}} q\left( \overrightarrow{r},E,\widehat{\Omega} \right) \ \psi^{+}\left( \overrightarrow{r},E,\widehat{\Omega} \right) & = \frac{1}{\text{Rc}}\left\lbrack q\left( \overrightarrow{r},E \right)\ \frac{1}{2\pi}q\left( \widehat{\Omega} \bullet {\widehat{d}}_{i} \right) \right\rbrack\ \left\lbrack \phi^{+}\left( \overrightarrow{r},E \right)\ \frac{1}{2\pi}\ f\left( \widehat{\Omega} \bullet {\widehat{n}}_{0} \right) \right\rbrack\ , where the constant :math:\text{Rc} is used to make :math:\widehat{q} a pdf. Note that if either the original source directional distribution :math:q\left( \widehat{\Omega} \right) or the adjoint angular flux distribution at the source is isotropic, then :math:c = \frac{1}{4\pi}. For the general case of multiple sources, the biased source sampling distribution, the biased source distributions and the weight windows are .. math:: R_{i} = \iint_{}^{}{q_{i}\left( \overrightarrow{r},E\right)\ \phi^{+}\left( \overrightarrow{r},E \right)}\text{dE}\ \text{dr} .. math:: c_{i} = \int_{}^{}{\frac{1}{2\pi}q_{i}\left( \widehat{\Omega} \bullet {\widehat{d}}_{i} \right)\ \frac{1}{2\pi}f\left( \widehat{\Omega} \bullet {\widehat{n}}_{i} \right)}d\widehat{\Omega} .. math:: \widehat{p}\left( i \right) = \frac{{S_{i}R}_{i}c_{i}}{\sum_{}^{}{{S_{i}R}_{i}c_{i}}} .. math:: {\widehat{q}}_{i}\left( \overrightarrow{r},E,\widehat{\Omega} \right) = \left\lbrack \frac{1}{R_{i}}\ q_{i}\left( \overrightarrow{r},E \right)\ \phi^{+}\left( \overrightarrow{r},E \right) \right\rbrack\ \left\lbrack \frac{1}{c_{i}}\ q_{i}\left( \widehat{\Omega} \right)\ f\left( \widehat{\Omega} \right) \right\rbrack = {\widehat{q}}_{i}\left( \overrightarrow{r},E \right)\ \frac{1}{c_{i}}\ \frac{1}{2\pi}q_{i}\left( \widehat{\Omega} \bullet {\widehat{d}}_{i} \right)\ \frac{1}{2\pi}f\left( \widehat{\Omega} \bullet {\widehat{n}}_{i} \right) .. math:: \overline{w}\left( \overrightarrow{r},E,\widehat{\Omega} \right) = \frac{\sum_{}^{}{{S_{i}R}_{i}c_{i}}}{S\phi^{+}\left( \overrightarrow{r},E \right)} \ \ \frac{2\pi}{\ f\left( \widehat{\Omega} \bullet \widehat{n} \right)} = \frac{\sum_{}^{}{{S_{i}R}_{i}c_{i}}}{\sum_{}^{}{S_{i}R}_{i}} \ \ \overline{w}\left( \overrightarrow{r},E \right)\ \frac{2\pi}{f\left( \widehat{\Omega} \bullet \widehat{n} \right)} \ . More details on the development of these methods and their application for several problems have been presented :cite:peplow_hybrid_2010,peplow_consistent_2012. Using space/energy/angle CADIS in MAVRIC ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The two angular CADIS methods that use the AVATAR-type approximation of adjoint flux are specified in MAVRIC with the “angularBiasing=” keyword in the importanceMap block. Values for this keyword are 1 or 2. Space/Energy/Angle CADIS without directional biasing (for beam sources) – This method uses one specific location, :math:{\overrightarrow{r}}_{0}, energy, :math:E_{0}, and direction, :math:{\widehat{\Omega}}_{0}, which is the reference direction of the source :math:\widehat{d}, where the weight of the biased source matches the weight window. Space/Energy/Angle CADIS with directional biasing (for general sources) – This method uses one specific energy, :math:E_{0}, to determine the adjoint current vector :math:{\widehat{n}}_{0} and the :math:\lambda_{0} parameter for the biased angular distribution for each source. With each method, the user must specify at what energy the importance map and the biased sources should be made consistent. The particle type must also be specified. This is done with the keywords “angBiasParType=” (1 for neutron or 2 for photon) and “angBiasEnergy=” (with a value in eV), also in the importanceMap block. Note that all sources should have a direction :math:\widehat{d} set, using “direction *u v w*\ ” within each source, even if the angular distribution for a given source is isotropic. The direction is used for source biasing and for aligning the weight windows and biased sources. Also note that for either angular biasing method, the Denovo S\ :sub:N calculation must use a Legendre order greater than 0. With angular biasing, a mesh angular information (\*.mai) file is produced which can be visualized with the MeshFileViewer. This file contains the space/energy-dependent :math:\lambda\left( \overrightarrow{r},E \right) values and components of the average adjoint current direction :math:\widehat{n}\left( \overrightarrow{r},E \right) = \left\langle n_{x},n_{y},n_{z} \right\rangle. An existing mesh angular information (\*.mai) file can be used in a separate MAVRIC problem by using the “meshAngInfoFile=” keyword in the biasing block. Example problem ~~~~~~~~~~~~~~~ Consider the Ueki shielding problem used as sample problems in the Monaco and MAVRIC manuals. The goal is to calculate the neutron dose on one side of a shield from a partially collimated :sup:252\ Cf source on the other side of the shield. Both of the angular approaches discussed above can be compared to analog and standard space/energy CADIS calculations. For the analog calculations, no importanceMap block is used. For the other cases, the importance map blocks are shown below. .. list-table:: :align: center * - :: CADIS - :: Angular CADIS 1 - without a biased source angular dist. - :: Angular CADIS 2 - with a biased source angular dist. * - :: read importanceMap adjointSource 1 locationID=1 responseID=5 end adjointSource gridGeometryID=7 macromaterial mmTolerance=0.01 end macromaterial end importanceMap - :: read importanceMap adjointSource 1 locationID=1 responseID=5 end adjointSource gridGeometryID=7 macromaterial mmTolerance=0.01 end macromaterial angularbiasing=1 angBiasParType=1 angBiasEnergy=2.0e6 end importanceMap - :: read importanceMap adjointSource 1 locationID=1 responseID=5 end adjointSource gridGeometryID=7 macromaterial mmTolerance=0.01 end macromaterial angularbiasing=2 angBiasParType=1 angBiasEnergy=2.0e6 end importanceMap Note that the energy at which to tie the importance map to the biased source, 2 MeV, is about the average energy of the source energy distribution. The figure-of-merit (FOM) of the calculation could change as this parameter is varied. Results, shown in :numref:tab4c1, demonstrate that the two directional approaches improved the FOM for this problem by more than a factor of 2. A larger boost is seen in method 2 where biasing is applied to the source directional distribution. In this case, the biased source distribution was an exponential distribution with a power of 2.45, significantly sampling more source neutrons in the direction of the shield than into the paraffin collimator block. Note that the space/energy/angle CADIS methods require more memory to hold the importance information. Improvements to the CADIS calculation can also be made with a judicious choice of standard source direction biasing, without an increase in memory requirements. A simple distribution (where :math:\mu = 0.924 represents the edge of the cone cutout and :math:\mu = 0.974 is the center half of the shield dimension) added that to the source :: distribution 2 abscissa -1.0 0.924 .974 1.0 end truepdf 0.962 0.025 .013 end biasedpdf 0.500 0.250 0.250 end end distribution src 1 neutrons strength=4.05E+07 cuboid 0.01 0.01 0 0 0 0 eDistributionID=1 direction 1.0 0.0 0.0 dDistributionID=2 end src results in an FOM improvement of nearly 3 over the standard CADIS without the overhead of the angular CADIS methods. .. list-table:: Results of the Ueki Shielding Problem (35 cm graphite shield) :align: center :name: tab4c1 * - Calculation Method - Adj. S\ :sub:N (min) - MC (min) - dose rate (rem/hr) - relative uncert - MC FOM (/min) * - Analog - 0.0 - 152.7 - 3.998E-03 - 0.0101 - 64.7 * - CADIS - 0.2 - 9.9 - 3.998E-03 - 0.0081 - 1550 * - Directional CADIS 1, no src bias - 0.2 - 10.3 - 4.035E-03 - 0.0054 - 3390 * - Directional CADIS 2, with src bias - 0.2 - 10.0 - 4.012E-03 - 0.0049 - 4190 * - CADIS, standard src dir. bias - 0.2 - 10.0 - 3.998E-03 - 0.0047 - 4550 A series of problems was used to compare the space/energy/angle CADIS to the standard space/energy CADIS. Most of the problems saw an improvement of a factor of about 2 or 3. Some problems did not improve at all, and some photon problems actually performed worse. In that case (a photon litho-density gauge), it could be that the angular approximation for importance, an exponential function in :math:\mu, cannot adequately describe the true importance. Since the space/energy/angle CADIS methods are more difficult to explain (more theory, adjustable parameters set by the user), require more memory than standard CADIS, and may not offer any improvement over standard space/energy CADIS, they have not been made part of the main MAVRIC manual and have been left as an advanced/developing feature. These methods were not removed from the MAVRIC code since they may be helpful to future problems. University of Michigan methods for global variance reduction ------------------------------------------------------------ The use of hybrid deterministic/Monte Carlo methods, particularly for global variance reduction, has been an active area of research by the transport team at the University of Michigan for a long time. One of the first approaches studied was a way to develop Monte Carlo weight window target values that were proportional to deterministically estimated values of the forward flux :cite:cooper_automated_2001. For global problems using isotropic weight windows, this reference argues that in order to get uniform relative uncertainties in the Monte Carlo calculation, the weight windows should be set such that the number density of Monte Carlo particles, :math:m\left( \overrightarrow{r} \right), is constant. The physical particle density, :math:n\left( \overrightarrow{r} \right), is related by the average weight, :math:\overline{w}\left( \overrightarrow{r} \right), to the Monte Carlo particle density by .. math:: n\left( \overrightarrow{r} \right) = \overline{w}\left( \overrightarrow{r} \right)\ m\left( \overrightarrow{r} \right) \ . To make :math:m\left( \overrightarrow{r} \right) constant over the geometry, the weight window targets :math:\overline{w}\left( \overrightarrow{r} \right) need to be proportional to the physical particle density. Cooper and Larsen calculate the weight window targets from an estimate of the forward scalar flux :math:\phi\left( \overrightarrow{r} \right) to be .. math:: \overline{w}\left( \overrightarrow{r} \right) = \frac{\phi\left( \overrightarrow{r} \right)}{\mathrm{\max}\left( \phi\left( \overrightarrow{r} \right) \right)} \ . Two approaches have since been developed :cite:becker_application_2009, based on what global information the user desires from the simulation: global flux weight windows, for obtaining every energy group at every location, and global response weight windows, for obtaining an energy-integrated response at every spatial location. Both of these methods are designed for calculating the “global solution” – everywhere in the geometry of the problem – with nearly uniform statistics. Note that none of the University of Michigan methods discussed here included the development of biased sources. These methods have all been extended to include a consistent biased source by ORNL during their implementation in the MAVRIC sequence of SCALE. The methods have also been extended by ORNL for multiple sources. Weight windows using only forward estimates of flux ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Global flux weight windows ^^^^^^^^^^^^^^^^^^^^^^^^^^ This method keeps the Monte Carlo particle distribution uniform in space and energy. Note that this is a space/energy version of the original space-only Cooper’s Method. The target weight windows, :math:\overline{w}\left( \overrightarrow{r},E \right), should be proportional to the estimate of the forward scalar flux, :math:\phi\left( \overrightarrow{r},E \right). .. math:: \overline{w}\left( \overrightarrow{r},E \right) = c\ \phi\left( \overrightarrow{r},E \right) \ . A biased source distribution, :math:\widehat{q}\left( \overrightarrow{r},E \right), that is consistent with the target weight windows can be found from the true source distribution, :math:q\left( \overrightarrow{r},E \right), and the forward flux to be .. math:: \widehat{q}\left( \overrightarrow{r},E \right) = \ \frac{1}{c}\ \frac{q\left( \overrightarrow{r},E \right)}{\phi\left( \overrightarrow{r},E \right)}\ , where the constant *c* can be determined so that the biased source distribution is a probability distribution function, .. math:: c = \int_{V}^{}{\int_{E}^{}{\ \frac{q\left( \overrightarrow{r},E \right)}{\phi\left( \overrightarrow{r},E \right)}\ \text{dE}\ d\overrightarrow{r}}} \ . Thus, using the estimate of forward flux, we can first compute *c* and then form a consistent set of weight window target values and a biased source distribution. Global response weight windows ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For an energy-integrated response (such as dose) desired at all spatial locations, this method keeps the Monte Carlo particle distribution proportional to the integral of the product of the response function, :math:\sigma_{d}\left( \overrightarrow{r},E \right), and the estimate of the forward flux, :math:\phi\left( \overrightarrow{r},E \right). The energy-integrated response, :math:D\left( \overrightarrow{r} \right), is found from the estimate of the forward flux as .. math:: D\left( \overrightarrow{r} \right) = \int_{E}^{}{\sigma_{d}\left( \overrightarrow{r},E \right)\ \phi\left( \overrightarrow{r},E \right) DE } . The target weight windows, :math:\overline{w}\left( \overrightarrow{r},E \right), should be .. math:: \overline{w}\left( \overrightarrow{r},E \right) = c\ \frac{D\left( \overrightarrow{r} \right)}{\sigma_{d}\left( \overrightarrow{r},E \right)} \ . A biased source distribution, :math:\widehat{q}\left( \overrightarrow{r},E \right), that is consistent with the target weight windows can be found from the true source distribution, :math:q\left( \overrightarrow{r},E \right), and the forward dose estimate to be .. math:: \widehat{q}\left( \overrightarrow{r},E \right) = \ \frac{1}{c}\ \frac{q\left( \overrightarrow{r},E \right)\ \sigma_{d}\left( \overrightarrow{r},E \right)}{D\left( \overrightarrow{r} \right)} using a proportionality constant of .. math:: c = \int_{V}^{}{\frac{1}{D\left( \overrightarrow{r} \right)}\int_{E}^{}{q\left( \overrightarrow{r},E \right)\ \sigma_{d}\left( \overrightarrow{r},E \right)\ \text{dE}\ d\overrightarrow{r}}}\text{\ .} Implementation in MAVRIC ^^^^^^^^^^^^^^^^^^^^^^^^ The global flux weight windows (GFWW) approach and the global response weight windows (GRWW) are both triggered by specifying an importance map block without any adjoint sources. For the GRWW approach, the response of interest is listed in the importance map block. If none is listed, GFWW is used. For problems with multiple sources (each with probability distribution function :math:q_{i}\left( \overrightarrow{r},E \right) and strength :math:S_{i}, giving a total source strength of :math:S = \sum_{}^{}S_{i}), the GWW methods require a biased mesh source for each. This was implemented in MAVRIC in a manner similar to the multiple source CADIS routines. Each biased source, :math:{\widehat{q}}_{i}\left( \overrightarrow{r},E \right), is developed as .. math:: \widehat{q}_{i}\left( \vec{r},E \right) = \begin{cases} \frac{1}{c_{i}} \frac{q_{i}\left( \vec{r},E \right)}{\phi\left( \vec{r},E \right)} \ \ \ \ \ & \text{global flux weight windows} \\ \frac{1}{c_{i}} \frac{q_{i}\left( \vec{r},E \right) \sigma_{d} \left( \vec{r},E \right)}{D \left( \vec{r}\right)} \ \ \ & \text{global response weight windows} \end{cases} where :math:c_{i} is a normalization constant. The weight windows are then set to .. math:: \overline{w}\left( \overrightarrow{r},E \right) = \left\{ \begin{matrix} \frac{\sum_{}^{}c_{i}}{\sum_{}^{}S_{i}}\ \phi\left( \overrightarrow{r},E \right) & \text{global flux weight windows} \\ \frac{\sum_{}^{}c_{i}}{\sum_{}^{}S_{i}}\ \frac{D\left( \overrightarrow{r} \right)}{\sigma_{d}\left( \overrightarrow{r},E \right)} & \text{global response weight windows} \\ \end{matrix} \right.\ \ \ \ . In the final Monte Carlo, the specific source *i* is sampled with probability :math:p\left( i \right) = \ S_{i}/S, and then the particle is sampled from the biased mesh source :math:{\widehat{q}}_{i}\left( \overrightarrow{r},E \right). Unlike the CADIS method for multiple sources, there is no way to develop a biased probability distribution for which source to sample without knowing the contribution to the global estimate from each source separately. For multiple source problems where the expected contribution from each source is very different from the true strengths of those sources, it may be more efficient to run a series of problems with one source each for different amounts of time. The resulting mesh tallies can then be added together using the mesh tally adder (part of the MAVRIC utilities). Methods using forward and adjoint estimates ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Becker :cite:becker_hybrid_2009 proposed three methods for developing weight windows based on both forward and adjoint deterministic solutions. These three methods correspond to the portion of the phase space over which uniform relative uncertainties are desired: a small “detector” region, a region comprising a significant portion of the entire problem, and the global problem. In this discussion, only a brief outline of each method, focusing on its implementation into MAVRIC, will be given. Source/detector problems ^^^^^^^^^^^^^^^^^^^^^^^^ For a small detector of volume :math:V_{D}\ where we want to optimize the MC calculation of the detector response .. math:: R = \int_{V_{D}}^{}{\int_{0}^{\infty}{\sigma\left( \overrightarrow{r},E \right)\ \phi\left( \overrightarrow{r},E \right)}\text{dE}\ \text{dV}} or optimize for the energy dependent flux at the detector, the following is used: forward flux estimate .. math:: :label: eq4c5 \phi\left( \overrightarrow{r},E \right) adjoint source for flux .. math:: q^{+}\left( \overrightarrow{r},E \right) = \frac{1}{\phi\left( \overrightarrow{r},E \right)} or for response .. math:: q^{+}\left( \overrightarrow{r},E \right) = \sigma\left( \overrightarrow{r},E \right) adjoint flux .. math:: \phi^{+}\left( \overrightarrow{r},E \right) contributon flux .. math:: \phi^{c}\left( \overrightarrow{r},E \right) = \phi\left( \overrightarrow{r},E \right)\ \phi^{+}\left( \overrightarrow{r},E \right) normalization constant .. math:: C_{\mathrm{\text{norm}}} = \frac{V_{D}}{\int_{V_{D}}^{}{\int_{0}^{\infty}{\phi^{c}\left( \overrightarrow{r},E \right)} dE} \ dV} space-only contributon flux .. math:: {\widetilde{\phi}}^{c}\left( \overrightarrow{r} \right) = \ C_{\mathrm{\text{norm}}}\int_{0}^{\infty}{\phi^{c}\left( \overrightarrow{r},E \right)} dE spatial parameter .. math:: \alpha\left( \overrightarrow{r} \right) = \left\lbrack 1 + exp\left( \frac{{\widetilde{\phi}}_{\mathrm{\max} \in V_{D}}^{c}}{{\widetilde{\phi}}^{c}\left( \overrightarrow{r} \right)} - \frac{{\widetilde{\phi}}^{c}\left( \overrightarrow{r} \right)}{{\widetilde{\phi}}_{\mathrm{\max} \in V_{D}}^{c}} \right) \right\rbrack^{- 1} spatial parameter .. math:: B\left( \overrightarrow{r} \right) = \ \alpha\left( \overrightarrow{r} \right){\widetilde{\phi}}^{c}\left( \overrightarrow{r} \right) + \ 1 - \ \alpha\left( \overrightarrow{r} \right) weight windows .. math:: :label: eq4c6 \overline{w}\left( \overrightarrow{r},E \right) = \frac{B\left( \overrightarrow{r} \right)}{\phi^{+}\left( \overrightarrow{r},E \right)} Source-region problems ^^^^^^^^^^^^^^^^^^^^^^ For a detector of volume :math:V_{D} and surface area :math:A_{D} (smaller than the entire problem) where we want to optimize the MC calculation of the detector response .. math:: R\left( \overrightarrow{r} \right) = \int_{0}^{\infty}{\sigma\left( \overrightarrow{r},E \right)\ \phi\left( \overrightarrow{r},E \right)} dE \ \ \ \ \ \ \ \ \ \overrightarrow{r} \in V_{D} or optimize for the energy dependent flux in the region, the following is used: forward flux estimate .. math:: :label: eq4c7 \phi\left( \overrightarrow{r},E \right) adjoint source for flux .. math:: q^{+}\left( \overrightarrow{r},E \right) = \frac{1}{\phi\left( \overrightarrow{r},E \right)} adjoint source for response .. math:: q^{+}\left( \overrightarrow{r},E \right) = \frac{\sigma\left( \overrightarrow{r},E \right)}{\int_{0}^{\infty}{\sigma\left( \overrightarrow{r},E \right)\ \phi\left( \overrightarrow{r},E \right)} dE} adjoint flux estimate .. math:: \phi^{+}\left( \overrightarrow{r},E \right) contributon flux .. math:: \phi^{c}\left( \overrightarrow{r},E \right) = \phi\left( \overrightarrow{r},E \right)\ \phi^{+}\left( \overrightarrow{r},E \right) normalization constant .. math:: C_{\mathrm{\text{norm}}} = \frac{A_{D}}{\int_{A_{D}}^{}{\int_{0}^{\infty}{\phi^{c}\left( \overrightarrow{r},E \right)} dE} \ dA} space-only contributon flux .. math:: {\widetilde{\phi}}^{c}\left( \overrightarrow{r} \right) = \ C_{\mathrm{\text{norm}}}\int_{0}^{\infty}{\phi^{c}\left( \overrightarrow{r},E \right)} dE spatial parameter .. math:: \alpha\left( \overrightarrow{r} \right) = \left\lbrack 1 + exp\left( \frac{{\widetilde{\phi}}_{\mathrm{\max} \in V_{D}}^{c}}{{\widetilde{\phi}}^{c}\left( \overrightarrow{r} \right)} - \frac{{\widetilde{\phi}}^{c}\left( \overrightarrow{r} \right)}{{\widetilde{\phi}}_{\mathrm{\max} \in V_{D}}^{c}} \right) \right\rbrack^{- 1} spatial parameter .. math:: :label: eq4c8 B\left( \vec{r} \right) = \begin{cases} \tilde{\phi^{c}}\left(\vec{r}\right) & \vec{r} \in V_{D} \alpha\left(\vec{r}\right)\tilde{\phi^{c}}\left(\vec{r}\right) + 1 - \alpha\left(\vec{r}\right) & \text{otherwise} \end{cases} weight windows .. math:: \overline{w}\left( \overrightarrow{r},E \right) = \frac{B\left( \overrightarrow{r} \right)}{\phi^{+}\left( \overrightarrow{r},E \right)} Note that :math:A_{D} does not include surfaces of :math:V_{D} which are on the boundary of the problem. Global response problem ^^^^^^^^^^^^^^^^^^^^^^^ For optimizing the Monte Carlo calculation of a detector response everywhere in phase space .. math:: R(\overrightarrow{r}) = \int_{0}^{\infty}{\sigma\left( \overrightarrow{r},E \right)\ \phi\left( \overrightarrow{r},E \right)}\text{dE} or optimizing for the energy-dependent flux everywhere, the following is used: forward flux estimate .. math:: :label: eq4c9 \phi\left( \overrightarrow{r},E \right) adjoint source for flux .. math:: q^{+}\left( \overrightarrow{r},E \right) = \frac{1}{\phi\left( \overrightarrow{r},E \right)} adjoint source for response .. math:: q^{+}\left( \overrightarrow{r},E \right) = \frac{\sigma\left( \overrightarrow{r},E \right)}{\int_{0}^{\infty}{\sigma\left( \overrightarrow{r},E \right)\ \phi\left( \overrightarrow{r},E \right)} dE } adjoint flux estimate .. math:: \phi^{+}\left( \overrightarrow{r},E \right) contributon flux .. math:: \phi^{c}\left( \overrightarrow{r},E \right) = \phi\left( \overrightarrow{r},E \right)\ \phi^{+}\left( \overrightarrow{r},E \right) space-only contributon flux .. math:: \phi^{c}\left( \overrightarrow{r} \right) = \int_{0}^{\infty}{\phi^{c}\left( \overrightarrow{r},E \right)} dE spatial parameter .. math:: B\left( \overrightarrow{r} \right) = \phi^{c}\left( \overrightarrow{r} \right) weight windows .. math:: \overline{w}\left( \overrightarrow{r},E \right) = \frac{B\left( \overrightarrow{r} \right)}{\phi^{+}\left( \overrightarrow{r},E \right)} Implementation in MAVRIC ~~~~~~~~~~~~~~~~~~~~~~~~ Like CADIS and FW-CADIS, the Denovo S\ :sub:N code is used to calculate the forward flux estimate, :math:\phi\left( \overrightarrow{r},E \right), and the estimate of the adjoint flux, :math:\phi^{+}\left( \overrightarrow{r},E \right), for all of the Michigan weight window methods. None of the above discussions of the University of Michigan methods include information on how the weight window target values were adjusted to match the source sampling. When implemented into MAVRIC, each of the above problem types will compute a biased source, :math:\widehat{q}\left( \overrightarrow{r},E \right), along with the target weight, :math:\overline{w}\left( \overrightarrow{r},E \right), that are produced. For a problem with a single source of strength :math:S and distribution :math:q\left( \overrightarrow{r},E \right), the biased source distribution\ :math:\ \widehat{q}\left( \overrightarrow{r},E \right) is found by using .. math:: \widehat{q}\left( \overrightarrow{r},E \right) = \frac{q\left( \overrightarrow{r},E \right)}{\overline{w}\left( \overrightarrow{r},E \right)}\ . The weight windows are multiplied by a factor of :math:R/S, where :math:R is defined as .. math:: R = \iint_{}^{}{\widehat{q}\left( \overrightarrow{r},E \right)}\phi^{+}\left( \overrightarrow{r},E \right)\ dE \ dV . Sampled source particles will then be born with a weight that matches the weight window of the phase space where they are born. For multiple sources, each with strength :math:S_{i} and distribution :math:q_{i}\left( \overrightarrow{r},E \right), each biased source distribution\ :math:\ {\widehat{q}}_{i}\left( \overrightarrow{r},E \right) is found by using .. math:: {\widehat{q}}_{i}\left( \overrightarrow{r},E \right) = \frac{q_{i}\left( \overrightarrow{r},E \right)}{\overline{w}\left( \overrightarrow{r},E \right)} and the response from each source being .. math:: R_{i} = \iint_{}^{}{{\widehat{q}}_{i}\left( \overrightarrow{r},E \right)}dE \ dV \ . The individual sources are sampled with a biased probability of :math:\widehat{p}\left( i \right) = \ R_{i}/\sum_{}^{}R_{i}. The weight windows are then multiplied by a factor of .. math:: \frac{\sum_{i}^{}R_{i}}{\sum_{i}^{}S_{i}} to match the source birth weights. To use one of the Becker methods in MAVRIC, the keyword “beckerMethod=” is used with values of 1, 2, or 3 for the source/detector, source/region, or global method. Adjoint sources are described just like standard MAVRIC CADIS and FW-CADIS problems. To switch between optimizing flux in every group or optimizing a response, the keywords “fluxWeighting” and “respWeighting” are used. Just like FW‑CADIS, the response listed in each adjoint source is the response that is optimized. Note that even when starting a calculation with known forward and adjoint flux files, the adjoint source(s) still need to be listed since they are used in the final normalization of the weight windows. Example problems ^^^^^^^^^^^^^^^^ The first Becker method (source/detector) is demonstrated using the Ueki shielding problem (used as sample problems in the Monaco and MAVRIC manuals and above in the space/energy/angle CADIS example). The goal is to calculate the neutron dose on one side of a shield from a partially collimated :sup:252\ Cf source on the other side of the shield. For the analog calculations, no importanceMap block is used. For the other cases, the importance map blocks are shown below. .. list-table:: :align: center :width: 80 * - :: CADIS - :: Becker 1 – flux optimization - :: Becker 2 – response optimization * - :: read importanceMap adjointSource 1 locationID=1 responseID=5 end adjointSource gridGeometryID=7 macromaterial mmTolerance=0.01 end macromaterial end importanceMap - :: read importanceMap adjointSource 1 locationID=1 responseID=5 end adjointSource gridGeometryID=7 macromaterial mmTolerance=0.01 end macromaterial beckerMethod=1 fluxWeighting end importanceMap - :: read importanceMap adjointSource 1 locationID=1 responseID=5 end adjointSource gridGeometryID=7 macromaterial mmTolerance=0.01 end macromaterial beckerMethod=1 respWeighting end importanceMap