Numerical Implementation in Moose

Moose is a Galerkin finite element framework which provides several plug-and-play systems to assist in the implementation of partial differential equations. Gnat takes advantage of the kernel, auxiliary kernel, Dirac kernel, boundary condition, material and action systems. The different systems used by Gnat are summarized below:

Table 1: Description of Moose systems used by Gnat.

System Name	Description
Kernel	Evaluates the non-linear residual contribution of a term in the system of coupled partial differential equations for a single non-linear variable.
Auxiliary Kernel (AuxKernel)	Evaluates the value of a variable auxiliary to the problem.
Dirac Kernel	Evaluates the non-linear residual contribution of a point source term in the system of coupled partial differential equations for a single non-linear variable.
Boundary Condition (BC)	Evaluates the non-linear residual contribution of the implicit boundary conditions in the system of coupled partial differential equations for a single non-linear variable.
Material	Provides domain-dependant material properties to other Moose systems.
Action	A system which allows Moose users to specialize and automate components of how Moose sets up, solves, and post-processes the problem.

Neutral Particle Transport Implementation

Each residual contribution in a partial differential equation can be assigned to a specific Moose system. The SAAF weak form of the Boltzmann transport equation with the S_N disretization applied in angle and P_N treatment applied to the scattering operator is:

(1)var element = document.getElementById("moose-equation-d2f0c9b2-91e1-42f9-92a3-d1edffe68161");katex.render("\\underbrace{\\Big( \\psi_{j} + \\tau_{g}\\vec{\\nabla}\\phi_{j}\\cdot\\hat{\\Omega}_{n},\\, \\frac{\\partial}{\\partial t}\\frac{\\Psi^{h}_{g,n}}{v_{g}} \\Big)_{V}}_{\\text{Kernel}} + \\underbrace{\\Big\\langle \\psi_{j},\\, \\hat{n}\\cdot\\hat{\\Omega}_{n}\\Psi^{h}_{g,n} \\Big\\rangle_{\\Gamma}}_{\\text{Boundary Condition}} + \\underbrace{\\Big( \\vec{\\nabla}\\psi_{j}\\cdot\\hat{\\Omega}_{n},\\, \\tau_{g}\\hat{\\Omega}_{n}\\cdot\\vec{\\nabla}\\Psi_{g,n}^{h} + (\\tau_{g}\\Sigma_{r,\\, g} - 1)\\Psi_{g,n}^{h} \\Big)_{V}}_{\\text{Kernel}}\\\\ + \\underbrace{\\Big( \\psi_{j},\\, \\Sigma_{r,\\,g}\\Psi^{h}_{g,n} \\Big)_{V}}_{\\text{Kernel}} - \\underbrace{\\Big( \\psi_{j} + \\tau_{g}\\vec{\\nabla}\\phi_{j}\\cdot\\hat{\\Omega}_{n},\\, \\sum_{g' = 1}^{G}\\Sigma_{s,\\, g'\\rightarrow g}\\sum_{l = 0}^{L}\\frac{2l + 1}{4\\pi} f_{g'\\rightarrow g,\\, l}\\sum_{m = -l}^{l}Y_{l,m}(\\hat{\\Omega}_{n})\\Phi_{g',l,m} \\Big)_{V}}_{\\text{Kernel}}\\\\ -\\underbrace{\\Big( \\psi_{j} + \\tau_{g}\\vec{\\nabla}\\psi_{j}\\cdot\\hat{\\Omega}_{n}, \\frac{\\chi_{g}}{4\\pi}\\sum_{g' = 1}^{G}\\nu\\Sigma_{f,\\, g'}\\Phi_{g',0,0} \\Big)_{V}}_{\\text{Kernel}} - \\underbrace{\\Big( \\psi_{j} + \\tau_{g}\\vec{\\nabla}\\phi_{j}\\cdot\\hat{\\Omega}_{n},\\, S_{g,n} \\Big)_{V}}_{\\text{Kernel or Dirac Kernel}} = 0", element, {displayMode:true,throwOnError:false});

The flux moments $var element = document.getElementById("moose-equation-b312af66-20c5-4375-9d6d-9f87bde1b035");katex.render("\\Phi_{g,l,m}", element, {displayMode:false,throwOnError:false});$ are computed using an angular quadrature to treat the integral:

(2)var element = document.getElementById("moose-equation-3eb37645-6b44-401f-b409-d5900cfa88a0");katex.render("\\Phi_{g,l,m} = \\sum_{i = 1}^{N} w_{n} Y_{l,m}(\\hat{\\Omega}_{n}) \\Psi_{g,n}", element, {displayMode:true,throwOnError:false});

Neutral Particle Transport Physics Objects

The the MooseObjects which implement each term in Eq. (1) and Eq. (2) are summarized below in Table 2.

Table 2: MooseObjects corresponding to terms in Eq. (1).

Object Name	Type	Implemented Term	Used If
ParticleFluxMoment	AuxKernel	$var element = document.getElementById("moose-equation-ee8621c1-7f6b-4a84-86d1-65acac91f4a4");katex.render("\\Phi_{g,l,m} = \\sum_{i = 1}^{N} w_{n} Y_{l,m}(\\hat{\\Omega}_{n}) \\Psi_{g,n}", element, {displayMode:false,throwOnError:false});$	SAAF
SNSourceBC	BC	$var element = document.getElementById("moose-equation-1deb89d7-922f-4859-a8e8-e0c913068491");katex.render("\\Big\\langle \\psi_{j},\\, \\hat{n}\\cdot\\hat{\\Omega}_{n}\\Psi^{h}_{g,n} \\Big\\rangle_{\\Gamma}", element, {displayMode:false,throwOnError:false});$	SAAF Fixed source boundary
SNReflectiveBC	BC	$var element = document.getElementById("moose-equation-21b0f7bf-0c08-4768-b634-4e0845274c57");katex.render("\\Big\\langle \\psi_{j},\\, \\hat{n}\\cdot\\hat{\\Omega}_{n}\\Psi^{h}_{g,n} \\Big\\rangle_{\\Gamma}", element, {displayMode:false,throwOnError:false});$	SAAF Reflective boundary
SNVacuumBC	BC	$var element = document.getElementById("moose-equation-1c3d261d-0425-4bb1-8a5b-0f625d2dd091");katex.render("\\Big\\langle \\psi_{j},\\, \\hat{n}\\cdot\\hat{\\Omega}_{n}\\Psi^{h}_{g,n} \\Big\\rangle_{\\Gamma}", element, {displayMode:false,throwOnError:false});$	SAAF Non-reentrant boundary
SNRemoval	Kernel	$var element = document.getElementById("moose-equation-82f61c17-9041-4752-81ec-73753e081d02");katex.render("\\Big( \\psi_{j},\\, \\Sigma_{r,\\,g}\\Psi^{h}_{g,n} \\Big)_{V}", element, {displayMode:false,throwOnError:false});$	SAAF
SAAFTimeDerivative	Kernel	$var element = document.getElementById("moose-equation-f5ec63f5-d46f-4d66-9868-8b1f55476a93");katex.render("\\Big( \\psi_{j} + \\tau_{g}\\vec{\\nabla}\\psi_{j}\\cdot\\hat{\\Omega}_{n},\\, \\frac{\\partial}{\\partial t}\\frac{\\Psi^{h}_{g,n}}{v_{g}} \\Big)_{V}", element, {displayMode:false,throwOnError:false});$	SAAF Transient
SAAFStreaming	Kernel	$var element = document.getElementById("moose-equation-fba5ac5b-3caf-4322-a364-c0e38ede8541");katex.render("\\Big( \\vec{\\nabla}\\psi_{j}\\cdot\\hat{\\Omega}_{n},\\, \\tau_{g}\\hat{\\Omega}_{n}\\cdot\\vec{\\nabla}\\Psi_{g,n}^{h} + (\\tau_{g}\\Sigma_{r,\\, g} - 1)\\Psi_{g,n}^{h} \\Big)_{V}", element, {displayMode:false,throwOnError:false});$	SAAF
SAAFScattering	Kernel	$var element = document.getElementById("moose-equation-e4baa4de-c731-4e07-aaed-9961b6e79dd7");katex.render("-\\Big( \\psi_{j} + \\tau_{g}\\vec{\\nabla}\\psi_{j}\\cdot\\hat{\\Omega}_{n},\\,\\\\ \\sum_{g' = 1}^{G}\\Sigma_{s,\\, g'\\rightarrow g}\\sum_{l = 0}^{L}\\frac{2l + 1}{4\\pi} f_{g'\\rightarrow g,\\, l}\\sum_{m = -l}^{l}Y_{l,m}(\\hat{\\Omega}_{n})\\Phi_{g',l,m} \\Big)_{V}", element, {displayMode:false,throwOnError:false});$	SAAF User enables off-diagonal jacobian contributions
SAAFMomentScattering	Kernel	$var element = document.getElementById("moose-equation-b341b589-2ecc-4ea7-9339-e09eab902ded");katex.render("-\\Big( \\psi_{j} + \\tau_{g}\\vec{\\nabla}\\psi_{j}\\cdot\\hat{\\Omega}_{n},\\\\ \\sum_{g' = 1}^{G}\\Sigma_{s,\\, g'\\rightarrow g}\\sum_{l = 0}^{L}\\frac{2l + 1}{4\\pi} f_{g'\\rightarrow g,\\, l}\\sum_{m = -l}^{l}Y_{l,m}(\\hat{\\Omega}_{n})\\Phi_{g',l,m} \\Big)_{V}", element, {displayMode:false,throwOnError:false});$	SAAF User disables off-diagonal jacobian contributions
SAAFMomentFission	Kernel	$var element = document.getElementById("moose-equation-11f38e15-1c9c-40a0-90dc-ca811516b251");katex.render("-\\Big( \\psi_{j} + \\tau_{g}\\vec{\\nabla}\\psi_{j}\\cdot\\hat{\\Omega}_{n}, \\frac{\\chi_{g}}{4\\pi}\\sum_{g' = 1}^{G}\\nu\\Sigma_{f,\\, g'}\\Phi_{g',0,0} \\Big)_{V}", element, {displayMode:false,throwOnError:false});$	SAAF Particle is neutron User enables fission User disables off-diagonal jacobian contributions
SAAFVolumeSource	Kernel	$var element = document.getElementById("moose-equation-1a524869-6483-49ca-a660-97ad1ec11f71");katex.render("- \\Big( \\psi_{j} + \\tau_{g}\\vec{\\nabla}\\psi_{j}\\cdot\\hat{\\Omega}_{n},\\, S_{g,n} \\Big)_{V}", element, {displayMode:false,throwOnError:false});$	SAAF Block contains a constant volumetric source
SAAFFieldSource	Kernel	$var element = document.getElementById("moose-equation-e731ba8f-2230-491f-b1d0-326ce158a0fe");katex.render("- \\Big( \\psi_{j} + \\tau_{g}\\vec{\\nabla}\\psi_{j}\\cdot\\hat{\\Omega}_{n},\\, S_{g,n}(\\vec{r}, t) \\Big)_{V}", element, {displayMode:false,throwOnError:false});$	SAAF Block contains a particle source that is a function of another variable
SAAFPointSource	Dirac Kernel	$var element = document.getElementById("moose-equation-6a8021a5-10c1-4c96-8275-69c659afc62b");katex.render("- \\Big( \\psi_{j} + \\tau_{g}\\vec{\\nabla}\\psi_{j}\\cdot\\hat{\\Omega}_{n},\\, \\delta(\\vec{r} - \\vec{r}_{s})S_{g,n} \\Big)_{V}", element, {displayMode:false,throwOnError:false});$	SAAF Problem contains a constant point source

More information of the implementation of Moose Actions and Materials for Gnat can be found in the transport system syntax page and the material source pages.

Neutral Particle Transport Solver Objects

Trace Species Transport

Neutral Particle Transport Implementation
Trace Species Transport