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addressed @paulromano comments
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Bryan Herman
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2 changed files with 38 additions and 160 deletions
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@ -37,7 +37,7 @@ the direction as a superscript. Energy group indices represented by :math:`g`
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and :math:`h` are also listed as superscripts here. The group :math:`g` is the
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group of interest and, if present, :math:`h` is all groups. Finally, any
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parameter surrounded by :math:`\left\langle\cdot\right\rangle` represents a
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tally quantity that can be edited from an MC solution.
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tally quantity that can be edited from a Monte Carlo (MC) solution.
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------
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Theory
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@ -83,7 +83,7 @@ In eq. :eq:`eq_neut_bal` the parameters are defined as:
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* :math:`\left\langle\overline{\overline\Sigma}_{t_{l,m,n}}^g
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\overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle`
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--- volume-integrated total reaction rate over energy group :math:`g`.
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* :math:`\left\langle\overline{\overline{\nu_s\Sigma}}_{s_{l,m,n}}^{h\rightarrow
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* :math:`\left\langle\overline{\overline{\nu_s\Sigma}}_{s_{l,m,n}}^{h\rightarrow
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g}
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\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle`
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--- volume-integrated scattering production rate of neutrons that begin with
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@ -98,7 +98,7 @@ In eq. :eq:`eq_neut_bal` the parameters are defined as:
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group :math:`h` that exit in group :math:`g`.
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Each quantity in :math:`\left\langle\cdot\right\rangle` represents a scalar value that
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is obtained from an MC tally. A good verification step when using an MC is
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is obtained from an MC tally. A good verification step when using an MC code is
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to make sure that tallies satisfy this balance equation within statistics. No
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NDA acceleration can be performed if the balance equation is not satisfied.
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@ -110,7 +110,8 @@ illustrated as a flow chart below. After a batch of neutrons
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is simulated, NDA can take place. Each of the steps described above is described
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in detail in the following sections.
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.. tikz:: Flow chart of NDA process
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.. tikz:: Flow chart of NDA process. Note "XS" is used for cross section and
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"DC" is used for diffusion coefficient.
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:libs: shapes, snakes, shadows, arrows, calc, decorations.markings, patterns, fit, matrix, spy
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:include: cmfd_tikz/cmfd_flow.tikz
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@ -234,9 +235,9 @@ as [Hebert]_. These current/flux relationships are as follows:
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\beta_{l\pm1/2,m,n}^{u,g}\right)\Delta_l^u}\overline{\overline{\phi}}_{l,m,n}^{g}.
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In Eqs. :eq:`eq_cell_cell` and :eq:`eq_cell_bound`, the :math:`\pm` refers to
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left (-:math:`x`) or right (+:math:`x`) surface in the :math:`x` direction,
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back (-:math:`y`) or front (+:math:`y`) surface in the :math:`y` direction and
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bottom (-:math:`z`) or top (+:math:`z`) surface in the :math:`z` direction. For
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left (:math:`-x`) or right (:math:`+x`) surface in the :math:`x` direction,
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back (:math:`-y`) or front (:math:`+y`) surface in the :math:`y` direction and
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bottom (:math:`-z`) or top (:math:`+z`) surface in the :math:`z` direction. For
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cell-to-boundary coupling, a general albedo, :math:`\beta_{l\pm1/2,m,n}^{u,g}`,
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is used. The albedo is defined as the ratio of incoming (:math:`-` superscript)
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to outgoing (:math:`+` superscript) partial current on any surface represented
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@ -395,7 +396,7 @@ information:
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It should be noted that for more difficult simulations (e.g., light water
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reactors), there are other options available to users such as tally resetting
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parameters, effective down-scatter usage, tally estimator, etc. For more
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information please see [Herman_Thesis]_.
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information please see :ref:`usersguide_cmfd`.
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Of the options described above, the optional acceleration subset region is an
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uncommon feature. Because OpenMC only has a structured Cartesian mesh, mesh
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@ -425,26 +426,26 @@ in mesh cells far away from the core.
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:libs: shapes, snakes, shadows, arrows, calc, decorations.markings, patterns, fit, matrix, spy
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:include: cmfd_tikz/meshfig.tikz
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During an MC simulation, CMFD tallies are accumulated. The basic tallies
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needed are listed in Table :ref:`tab_tally`. Each tally is performed on a
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spatial and energy mesh basis. The surface area-integrated net current is
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tallied on every surface of the mesh. OpenMC tally objects are created by
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the CMFD code internally, and cross sections are calculated at each
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CMFD feedback iteration. The first CMFD iteration, controlled by the
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user, occurs just after tallies are communicated to the master processor. Once
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tallies are collapsed, cross sections, diffusion coefficients and equivalence
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parameters are calculated. This is performed only on the acceleration region if
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that option has been activated by the user. Once all diffusion parameters
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are calculated, CMFD matrices are formed where energy groups are the inner
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most iteration index. In OpenMC, compressed row storage sparse matrices are
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used due to the sparsity of CMFD operators. An example of this sparsity is
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shown for the 3-D BEAVRS model in figures :ref:`fig_loss` and :ref:`fig_prod`. These matrices
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represent an assembly radial mesh, 24 cell mesh in the axial direction and two
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energy groups. The loss matrix is 99.92% sparse and the production matrix is
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99.99% sparse. Although the loss matrix looks like it is tridiagonal, it is
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really a seven banded matrix with a block diagonal matrix for scattering. The
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production matrix is a :math:`2\times 2` block diagonal; however, zeros are present
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because no fission neutrons appear with energies in the thermal group.
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During an MC simulation, CMFD tallies are accumulated. The basic tallies needed
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are listed in Table :ref:`tab_tally`. Each tally is performed on a spatial and
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energy mesh basis. The surface area-integrated net current is tallied on every
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surface of the mesh. OpenMC tally objects are created by the CMFD code
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internally, and cross sections are calculated at each CMFD feedback iteration.
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The first CMFD iteration, controlled by the user, occurs just after tallies are
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communicated to the master processor. Once tallies are collapsed, cross
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sections, diffusion coefficients and equivalence parameters are calculated. This
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is performed only on the acceleration region if that option has been activated
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by the user. Once all diffusion parameters are calculated, CMFD matrices are
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formed where energy groups are the inner most iteration index. In OpenMC,
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compressed row storage sparse matrices are used due to the sparsity of CMFD
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operators. An example of this sparsity is shown for the 3-D BEAVRS model in
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figures :ref:`fig_loss` and :ref:`fig_prod` [BEAVRS]_. These matrices represent
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an assembly radial mesh, 24 cell mesh in the axial direction and two energy
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groups. The loss matrix is 99.92% sparse and the production matrix is 99.99%
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sparse. Although the loss matrix looks like it is tridiagonal, it is really a
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seven banded matrix with a block diagonal matrix for scattering. The production
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matrix is a :math:`2\times 2` block diagonal; however, zeros are present because
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no fission neutrons appear with energies in the thermal group.
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.. _tab_tally:
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@ -518,145 +519,20 @@ source distribution to the current number of neutrons in each mesh. It is
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straightforward to compute the CMFD number of neutrons because it is the
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product between the total starting initial weight of neutrons and the CMFD
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normalized fission source distribution. To compute the number of neutrons from
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the current MC source, a subroutine was implemented to sum the statistical
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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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-------------------
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Toy Problem Example
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-------------------
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Before applying CMFD to a large reactor, a simple 1-D slab toy problem was
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analyzed to understand how CMFD works. Table :ref:`tab_1Dtoyinput` presents
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data used to construct this problem. For MFD, the mesh was 2 cm over the
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geometry with one energy group. A comparison of fission source convergence
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using Shannon entropy is shown in figure :ref:`fig_1Dentropy`. The figure
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illustrates that it takes about 150 FSGs (equivalent to batches) to converge
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the source with standard MC without CMFD. For the case with CMFD, it is
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activated at batch 11 and directly affects the fission source for batch 12.
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Convergence of the fission source is almost immediately reached.
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.. _tab_1Dtoyinput:
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.. table:: Input data for 1-D slab toy problem
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+--------------------------------------------+-------------------+
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+--------------------------------------------+-------------------+
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| Slab Length | 200 cm |
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+============================================+===================+
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| Homogeneous material of :math:`\rm UO_2` | 19 g/cc density |
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+--------------------------------------------+-------------------+
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| U-235 weight percent | 0.21 |
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+--------------------------------------------+-------------------+
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| U-238 weight percent | 0.68 |
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+--------------------------------------------+-------------------+
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| O-16 weight percent | 0.11 |
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+--------------------------------------------+-------------------+
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| Number of particles per | 4,000,000 |
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+--------------------------------------------+-------------------+
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| Number of inactive | 400 |
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+--------------------------------------------+-------------------+
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.. _fig_1Dentropy:
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.. figure:: ../_images/entropy_1Dslab.png
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:scale: 10
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Source convergence comparison for 1-D slab toy problem
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To further show this convergence, source distributions were edited at various
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batches and compared. Figures :ref:`fig_toy6` to :ref:`fig_toy200` compare the
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OpenMC source distribution from the no CMFD case, the OpenMC source
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distribution from the CMFD case and the CMFD source distribution for six
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different batches. Figure :ref:`fig_toy6` compares the distributions at batch
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6. Here, CMFD has not been activated yet, so it is just plotted at zero. As
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expected, the OpenMC source distributions from the two cases are equal because
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CMFD has not yet affected it. From this plot, one can also see that the source
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distribution is very flat because the initial guess was uniform over space.
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Because the dominance ratio is close to unity, the source will slowly converge
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to a cosine-like shape. Results from batch 10 are presented in figure
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:ref:`fig_toy10`. The same information is shown in this plot, but the source is
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slightly more converged. It is plotted here to show how little the source
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changes in four batches. Batch 11 is the first time a CMFD source is
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calculated. Right away, it appears as a smooth cosine-like shape as shown in
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figure :ref:`fig_toy11`. Thus, when CMFD is fed back, its source shape will
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modify the OpenMC source bank to preserve this distribution on the CMFD mesh.
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On the next batch, shown in :ref:`fig_toy12`, the OpenMC source and the CMFD
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source from the CMFD case match, while the OpenMC source from the no CMFD case
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lags behind. Two more batches are shown in figure :ref:`fig_toy40` and figure
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:ref:`fig_toy200` to illustrate that all source distributions eventually line
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up at batch 200.
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.. _fig_toy6:
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.. figure:: ../_images/statepoint6.png
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:scale: 10
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Source at FSG 6
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.. _fig_toy10:
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.. figure:: ../_images/statepoint10.png
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:scale: 10
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Source at FSG 10
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.. _fig_toy11:
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.. figure:: ../_images/statepoint11.png
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:scale: 10
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Source at FSG 11
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.. _fig_toy12:
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.. figure:: ../_images/statepoint12.png
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:scale: 10
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Source at FSG 12
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.. _fig_toy40:
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.. figure:: ../_images/statepoint40.png
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:scale: 10
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Source at FSG 40
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.. _fig_toy200:
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.. figure:: ../_images/statepoint200.png
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:scale: 10
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Source at FSG 200
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This simple illustration shows the power of NDA. Because of the nature of the
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diffusion equation, it can propagate and dampen higher harmonics much faster
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than the MC transport solution. It should be noted here that this is a very
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simplified problem where the dominance ratio was increased by changing the size
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of the slab. Also, the source distribution is very smooth and there was only
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one homogeneous material. The problem becomes more difficult to solve when
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expanding to more spatial dimensions and complex materials.
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Examples of CMFD simulations using OpenMC can be found in [Herman_Thesis]_.
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----------
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References
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----------
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.. [BEAVRS] Nick Horelik, Bryan Herman. *Benchmark for Evaluation And Verification of Reactor
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Simulations*. Massachusetts Institute of Technology, http://crpg.mit.edu/pub/beavrs
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, 2013.
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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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@ -1301,6 +1301,8 @@ attributes or sub-elements. These are not used in "voxel" plots:
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*Default*: None
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.. _usersguide_cmfd:
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------------------------------
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CMFD Specification -- cmfd.xml
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------------------------------
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@ -1361,7 +1363,7 @@ It can be turned on with "true" and off with "false".
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*Default*: false
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``<gauss_seidel_tolerance>`` Element
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--------------------
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------------------------------------
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The ``<gauss_seidel_tolerance>`` element specifies two parameters. The first is
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the absolute inner tolerance for Gauss-Seidel iterations when performing CMFD
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