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Merge pull request #306 from smpark7/sn-method
SN method & hybrid SN-diffusion method
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‎doc/content/source/auxkernels/DelayedNeutronSourceAux.md‎

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# DelayedNeutronSourceAux
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!syntax description /AuxKernels/DelayedNeutronSourceAux
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## Overview
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Computes the cumulative delayed neutron source rate distribution, $\sum_i^I \lambda_i C_i$.
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## Example Input File Syntax
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!! Describe and include an example of how to use the DelayedNeutronSourceAux object.
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!syntax parameters /AuxKernels/DelayedNeutronSourceAux
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!syntax inputs /AuxKernels/DelayedNeutronSourceAux
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!syntax children /AuxKernels/DelayedNeutronSourceAux

‎doc/content/source/auxkernels/FissionRateAux.md‎

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# FissionRateAux
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!syntax description /AuxKernels/FissionRateAux
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## Overview
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Computes the cumulative fission rate distribution, $\sum_g^G \Sigma_{f,g} \phi_g$.
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## Example Input File Syntax
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!! Describe and include an example of how to use the FissionRateAux object.
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!syntax parameters /AuxKernels/FissionRateAux
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!syntax inputs /AuxKernels/FissionRateAux
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!syntax children /AuxKernels/FissionRateAux

‎doc/content/source/auxkernels/GroupDriftAux.md‎

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# GroupDriftAux
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!syntax description /AuxKernels/GroupDriftAux
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## Overview
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Computes the neutron drift coefficient for nonlinear diffusion acceleration of the $S_N$ method or
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the hybrid $S_N$-diffusion method.
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!equation
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\vec{D}_g = \frac{\sum^{N_d}_{d=1}w_d\left(\tau_g\hat{\Omega}_d\hat{\Omega}_d\cdot\nabla\Psi_{g,d}
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+ \left(\tau_g\Sigma_{t,g}-1\right)\hat{\Omega}_d\Psi_{g,d}
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- \tau_g\sum^G_{g'=1}\Sigma^{g'\rightarrow g}_{s,1}\hat{\Omega}_d\Psi_{g',d}
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- D_g\nabla\Psi_{g,d}\right)}{\sum^{N_d}_{d=1}w_d\Psi_{g,d}}
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## Example Input File Syntax
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!! Describe and include an example of how to use the GroupDriftAux object.
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!syntax parameters /AuxKernels/GroupDriftAux
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!syntax inputs /AuxKernels/GroupDriftAux
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!syntax children /AuxKernels/GroupDriftAux

‎doc/content/source/auxkernels/NeutronSourceAux.md‎

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# NeutronSourceAux
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!syntax description /AuxKernels/NeutronSourceAux
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## Overview
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Computes the cumulative fission neutron source distribution,
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$\sum^G_g \nu\Sigma_{f,g}\phi_g$.
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## Example Input File Syntax
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!! Describe and include an example of how to use the NeutronSourceAux object.
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!syntax parameters /AuxKernels/NeutronSourceAux
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!syntax inputs /AuxKernels/NeutronSourceAux
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!syntax children /AuxKernels/NeutronSourceAux

‎doc/content/source/auxkernels/SNScalarFluxAux.md‎

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# SNScalarFluxAux
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!syntax description /AuxKernels/SNScalarFluxAux
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## Overview
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Computes the scalar neutron group flux from the angular neutron group flux,
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$\phi_g=\sum_{d=1}^{N_d}w_d\Psi_{g,d}$.
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## Example Input File Syntax
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!! Describe and include an example of how to use the SNScalarFluxAux object.
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!syntax parameters /AuxKernels/SNScalarFluxAux
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!syntax inputs /AuxKernels/SNScalarFluxAux
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!syntax children /AuxKernels/SNScalarFluxAux

‎doc/content/source/auxkernels/VacuumCoefAux.md‎

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# VacuumCoefAux
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!syntax description /AuxKernels/VacuumCoefAux
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## Overview
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Computes the boundary correction coefficient for nonlinear diffusion acceleration of the $S_N$
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method or the hybrid $S_N$-diffusion method.
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!equation
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\gamma_g =
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\frac{\sum_{\hat{\Omega}_d\cdot\hat{n}_b > 0}w_d |\hat{\Omega}_d\cdot\hat{n}_b |
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\Psi_{g,d}}{\sum^{N_d}_{d=1}w_d\Psi_{g,d}}
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## Example Input File Syntax
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!! Describe and include an example of how to use the VacuumCoefAux object.
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!syntax parameters /AuxKernels/VacuumCoefAux
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!syntax inputs /AuxKernels/VacuumCoefAux
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!syntax children /AuxKernels/VacuumCoefAux

‎doc/content/source/bcs/SNDiffusionBC.md‎

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# SNDiffusionBC
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!syntax description /BCs/SNDiffusionBC
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## Overview
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This object applies the boundary source condition for the $S_N$ neutron transport method using the
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$P_1$ approximation from scalar flux distributions from the neutron diffusion method.
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!equation
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\langle\bm{\Psi}^*,\bm{\Psi}\rangle^+ - \langle\bm{\Psi}^*,\bm{\Psi}^\text{inc}\rangle^- =
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\begin{cases}
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\sum^G_{g=1}\sum^{N_d}_{d=1}w_d\left(\Psi^*_{g,d},
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\hat{\Omega}_d\cdot\hat{n}_b\Psi_{g,d}\right)_{\partial\mathcal{D}},
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& \hat{\Omega}\cdot\hat{n}_b>0,\vec{r}\in\partial\mathcal{D} \\
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\sum^G_{g=1}\sum^{N_d}_{d=1}w_d\left(\Psi^*_{g,d},
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\hat{\Omega}_d\cdot\hat{n}_b\Psi^\text{inc}_{g,d}\right)_{\partial\mathcal{D}},
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& \hat{\Omega}\cdot\hat{n}_b<0,\vec{r}\in\partial\mathcal{D}
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\end{cases}
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where
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!equation
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\hat{n}_b = \text{ boundary unit vector,} \\
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\Psi^\text{inc}_{g,d} = \frac{1}{\sum_{d=1}^{N_d}w_d}
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\left(\phi^\text{diff}_g-3\hat{\Omega}_d\cdot D_g\nabla\phi^\text{diff}_g\right).
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## Example Input File Syntax
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!! Describe and include an example of how to use the SNDiffusionBC object.
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!syntax parameters /BCs/SNDiffusionBC
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!syntax inputs /BCs/SNDiffusionBC
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!syntax children /BCs/SNDiffusionBC

‎doc/content/source/bcs/SNReflectingBC.md‎

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# SNReflectingBC
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!syntax description /BCs/SNReflectingBC
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## Overview
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This object applies the reflecting boundary condition for the $S_N$ neutron transport method.
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This BC can only be applied to plane boundaries perpendicular to the x-, y-, or z-axis.
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!equation
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\langle\bm{\Psi}^*,\bm{\Psi}\rangle^+ - \langle\bm{\Psi}^*,\mathbb{B}\bm{\Psi}\rangle^- =
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\begin{cases}
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\sum^G_{g=1}\sum^{N_d}_{d=1}w_d\left(\Psi^*_{g,d},
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\hat{\Omega}_d\cdot\hat{n}_b\Psi_{g,d}\right),
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& \hat{\Omega}\cdot\hat{n}_b>0,\vec{r}\in\partial\mathcal{D}_s \\
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\sum^G_{g=1}\sum^{N_d}_{d=1}w_d\left(\Psi^*_{g,d},
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\hat{\Omega}_d\cdot\hat{n}_b\Psi_{g,d_r}\right),
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& \hat{\Omega}\cdot\hat{n}_b<0,\vec{r}\in\partial\mathcal{D}_s
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\end{cases},
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where
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!equation
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\hat{n}_b = \text{ boundary unit vector}, \\
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\hat{\Omega}_{d_r} = \hat{\Omega}_d - 2(\hat{\Omega}_d\cdot\hat{n}_b)\hat{n}_b = \text{ direction
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of incident neutron travel.}
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## Example Input File Syntax
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!! Describe and include an example of how to use the SNReflectingBC object.
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!syntax parameters /BCs/SNReflectingBC
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!syntax inputs /BCs/SNReflectingBC
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!syntax children /BCs/SNReflectingBC

‎doc/content/source/bcs/SNVacuumBC.md‎

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# SNVacuumBC
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!syntax description /BCs/SNVacuumBC
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## Overview
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This object applies the vacuum boundary condition for the $S_N$ neutron transport method.
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!equation
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\langle\bm{\Psi}^*,\bm{\Psi}\rangle^+ - \langle\bm{\Psi}^*,\bm{\Psi}^\text{inc}\rangle^- =
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\begin{cases}
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\sum^G_{g=1}\sum^{N_d}_{d=1}w_d\left(\Psi^*_{g,d},
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\hat{\Omega}_d\cdot\hat{n}_b\Psi_{g,d}\right)_{\partial\mathcal{D}},
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& \hat{\Omega}\cdot\hat{n}_b>0,\vec{r}\in\partial\mathcal{D} \\
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0,
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& \hat{\Omega}\cdot\hat{n}_b<0,\vec{r}\in\partial\mathcal{D}
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\end{cases},
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where
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!equation
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\hat{n}_b = \text{ boundary unit vector.}
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## Example Input File Syntax
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!! Describe and include an example of how to use the SNVacuumBC object.
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!syntax parameters /BCs/SNVacuumBC
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!syntax inputs /BCs/SNVacuumBC
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!syntax children /BCs/SNVacuumBC

‎doc/content/source/bcs/VacuumCorrectionBC.md‎

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# VacuumCorrectionBC
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!syntax description /BCs/VacuumCorrectionBC
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## Overview
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This object applies boundary correction along vacuum boundaries to the neutron diffusion equation
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for nonlinear diffusion acceleration of the $S_N$ method or the hybrid $S_N$-diffusion method.
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!equation
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\left(\phi^*,\gamma_g\phi_g\right)_{\partial\mathcal{D}}
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where
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!equation
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\gamma_g =
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\frac{\sum_{\hat{\Omega}_d\cdot\hat{n}_b > 0}w_d |\hat{\Omega}_d\cdot\hat{n}_b |
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\Psi_{g,d}}{\sum^{N_d}_{d=1}w_d\Psi_{g,d}}
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## Example Input File Syntax
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!! Describe and include an example of how to use the VacuumCorrectionBC object.
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!syntax parameters /BCs/VacuumCorrectionBC
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!syntax inputs /BCs/VacuumCorrectionBC
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!syntax children /BCs/VacuumCorrectionBC

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