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Fixes to documentation
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6 changed files with 52 additions and 2391 deletions
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@ -28,7 +28,7 @@ directions. An example of this is shown in the following expression:
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:label: not1
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\sum\limits_{u\in(x,y,z)}\left\langle\overline{J}^{u,g}_{l+1/2,m,n}
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\Delta_m^v\Delta_n^w\right\rangle
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\Delta_m^v\Delta_n^w\right\rangle
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Here, :math:`u` takes on each direction one at a time. The parameter :math:`J`
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is surface area-averaged over the transverse indices :math:`m` and :math:`n`
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@ -142,7 +142,7 @@ defined from MC tallies as follows:
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{\left\langle\overline{\overline\phi}_{l,m,n}^h
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\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}
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and
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and
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.. math::
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:label: xs3
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@ -306,7 +306,7 @@ interior cell,
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+ \left(\tilde{D}_{l-1/2,m,n}^{u,g} +
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\tilde{D}_{l+1/2,m,n}^{u,g} - \hat{D}_{l-1/2,m,n}^{u,g} +
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\hat{D}_{l+1/2,m,n}^{u,g}\right)\overline{\overline{\phi}}_{l,m,n}^g
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\\ +
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\\ +
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\left. \left(-\tilde{D}_{l+1/2,m,n}^{u,g} +
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\hat{D}_{l+1/2,m,n}^{u,g}\right)\overline{\overline{\phi}}_{l+1,m,n}^g
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\right] +
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@ -318,7 +318,7 @@ interior cell,
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It should be noted that before substitution, eq. :eq:`eq_neut_bal` was divided
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by the volume of the cell, :math:`\Delta_l^u\Delta_m^v\Delta_n^w`. Equation
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:eq:`eq_cmfd_sys` can be represented in operator form as
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:eq:`eq_cmfd_sys` can be represented in operator form as
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.. math::
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:label: eq_CMFDopers
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@ -349,7 +349,7 @@ and energy group. This is represented as
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p_{l,m,n}^g =
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\frac{\sum_{h=1}^{G}\overline{\overline{\nu_f\Sigma}}^{h\rightarrow
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g}_{f_{l,m,n}}\overline{\overline{\phi}}_{l,m,n}^h\Delta_l^u\Delta_m^v
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\Delta_n^w}{\sum_n\sum_m\sum_l\sum_{h=1}^{G}\overline{
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\Delta_n^w}{\sum_n\sum_m\sum_l\sum_{h=1}^{G}\overline{
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\overline{\nu_f\Sigma}}^{h\rightarrow
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g}_{f_{l,m,n}}\overline{\overline{\phi}}_{l,m,n}^h\Delta_l^u\Delta_m^v
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\Delta_n^w}.
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@ -478,14 +478,14 @@ no fission neutrons appear with energies in the thermal group.
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.. figure:: ../_images/loss.png
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:scale: 50
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Sparsity of Neutron Loss Operator
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Sparsity of Neutron Loss Operator
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.. _fig_prod:
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.. figure:: ../_images/prod.png
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:scale: 50
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Sparsity of Neutron Production Operator
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Sparsity of Neutron Production Operator
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To solve the eigenvalue problem with these matrices, different source iteration
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and linear solvers can be used. The most common source iteration solver used is
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@ -511,7 +511,7 @@ implemented to obtain eigenvalue and multigroup fluxes as described in [Gill]_
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and [Knoll]_. This method is not the primary one used, but has gotten recent
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attention due to its coupling advantages to other physics such as thermal
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hydraulics. Once multigroup fluxes are obtained, a normalized fission source is
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calculated in the code using eq. :eq:`eq_cmfd_psrc` directly.
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calculated in the code using eq. :eq:`eq_cmfd_psrc` directly.
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The next step in the process is to compute weight adjustment factors. These are
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calculated by taking the ratio of the expected number of neutrons from the CMFD
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@ -523,7 +523,7 @@ the current MC source, OpenMC sums the statistical
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weights of neutrons from the source bank on a given spatial and energy mesh.
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Once weight adjustment factors were calculated, each neutron's statistical
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weight in the source bank was modified according to its location and energy.
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Examples of CMFD simulations using OpenMC can be found in [Herman_Thesis]_.
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Examples of CMFD simulations using OpenMC can be found in [HermanThesis]_.
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----------
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References
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@ -536,23 +536,23 @@ References
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.. [Gill] Daniel F. Gill. *Newton-Krylov methods for the solution of the k-eigenvalue problem in
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multigroup neutronics calculations*. Ph.D. thesis, Pennsylvania State University, 2010.
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.. [Hebert] Alain Hebert. *Applied reactor physics*. Presses Internationales Polytechnique,
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.. [Hebert] Alain Hebert. *Applied reactor physics*. Presses Internationales Polytechnique,
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Montreal, 2009.
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.. [Herman] Bryan R. Herman, Benoit Forget, Kord Smith, and Brian N. Aviles. Improved
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diffusion coefficients generated from Monte Carlo codes. In *Proceedings of M&C
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2013*, Sun Valley, ID, USA, May 5 - 9, 2013.
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.. [Herman_Thesis] Bryan R. Herman. *Monte Carlo and Thermal Hydraulic Coupling using
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Low-Order Nonlinear Diffusion Acceleration*. Sc.D. thesis,
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Massachusetts Institute of Technology, 2014.
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.. [HermanThesis] Bryan R. Herman. *Monte Carlo and Thermal Hydraulic Coupling using
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Low-Order Nonlinear Diffusion Acceleration*. Sc.D. thesis,
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Massachusetts Institute of Technology, 2014.
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.. [Knoll] D.A. Knoll, H. Park, and C. Newman. *Acceleration of k-eigenvalue/criticality
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calculations using the Jacobian-free Newton-Krylov method*. Nuclear Science and
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Engineering, 167:133–140, 2011.
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.. [Park] H. Park, D.A. Knoll, and C.K. Newman. *Nonlinear acceleration of transport
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criticality problems*. Nuclear Science and Engineering, 172:52–65, 2012.
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criticality problems*. Nuclear Science and Engineering, 172:52–65, 2012.
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.. [Rhodes] Joel Rhodes and Malte Edenius. *CASMO-4 --- A Fuel Assembly Burnup Program.
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User’s Manual*. Studsvik of America, ssp-09/443-u rev 0, proprietary edition, 2001.
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