From 825d2a086471d8cab5a794465d06cbee05a78894 Mon Sep 17 00:00:00 2001 From: Paul Romano Date: Tue, 10 Feb 2015 23:51:10 -0500 Subject: [PATCH] Documentation for version 0.6.2. --- .nojekyll | 1 - _images/loss.png | Bin 0 -> 29146 bytes ...4208a5d23e58ca0506590366fd439d8bd4c9c8.png | Bin 647 -> 646 bytes ...58b57a0a7ff49829bd762992dd29db49f83d6b.png | Bin 852 -> 846 bytes ...b03dce6c6a8661c94399bda49265d52cae9c72.png | Bin 993 -> 994 bytes ...023f437f71130c12bd82bfa5604ec9ca2b0e67.png | Bin 0 -> 1233 bytes ...3e5f78508abf67433d9e0ff4b5d6f735638975.png | Bin 866 -> 866 bytes ...99a337f236d040bffecd93432b7a612a07871c.png | Bin 537 -> 537 bytes ...cae9ff7e0cfd41be2c0dcbc5d43f24ead7018b.png | Bin 592 -> 591 bytes ...f2199fb277a74a8f5f62b84516a2977ffca261.png | Bin 933 -> 932 bytes ...32656583b9d022cbfc26ee144bc86dca260bbf.png | Bin 581 -> 579 bytes ...0c60f6772c41b35ce46c7dcf54e6e19f115804.png | Bin 253 -> 253 bytes ...2dfa76529a0336c4df38e0620d170e6bfae916.png | Bin 2512 -> 2490 bytes 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a/_sources/developers.txt +++ b/_sources/developers.txt @@ -10,6 +10,8 @@ Active development of the OpenMC Monte Carlo code is currently led by: * `Bryan Herman `_ * `Nick Horelik `_ * `Adam Nelson `_ +* `Jon Walsh `_ +* `Sterling Harper `_ * `Benoit Forget `_ * `Kord Smith `_ * `Andrew Siegel `_ diff --git a/_sources/devguide/statepoint.txt b/_sources/devguide/statepoint.txt index 5f05c92c83..0fc02171c6 100644 --- a/_sources/devguide/statepoint.txt +++ b/_sources/devguide/statepoint.txt @@ -5,7 +5,7 @@ State Point Binary File Specifications ====================================== ----------- -Revision 11 +Revision 13 ----------- **integer(4) FILETYPE_STATEPOINT** @@ -49,10 +49,6 @@ Revision 11 Number of particles used per generation. -**integer(4) n_batches** - - Total number of batches (active + inactive). - **integer(4) current_batch** The number of batches already simulated. @@ -295,7 +291,304 @@ if (run_mode == MODE_EIGENVALUE and source_present) Energy of the i-th source particle. ----------- -Revision 10 +Revision 12 +----------- + +Same as revision 11, except **tallies(i) % scatt_order(j)** is now **tallies(i) +% moment_order(j)**. + +----------- +Revision 11 +----------- + +**integer(4) FILETYPE_STATEPOINT** + + Flags whether this file is a statepoint file or a particle restart file. + +**integer(4) REVISION_STATEPOINT** + + Revision of the binary state point file. Any time a change is made in the + format of the state-point file, this integer is incremented. + +**integer(4) VERSION_MAJOR** + + Major version number for OpenMC + +**integer(4) VERSION_MINOR** + + Minor version number for OpenMC + +**integer(4) VERSION_RELEASE** + + Release version number for OpenMC + +**character(19) time_stamp** + + Date and time the state point was written. + +**character(255) path** + + Absolute path to directory containing input files. + +**integer(8) seed** + + Pseudo-random number generator seed. + +**integer(4) run_mode** + + run mode used. The modes are described in constants.F90. + +**integer(8) n_particles** + + Number of particles used per generation. + +**integer(4) n_batches** + + Total number of batches (active + inactive). + +**integer(4) current_batch** + + The number of batches already simulated. + +if (run_mode == MODE_EIGENVALUE) + + **integer(4) n_inactive** + + Number of inactive batches + + **integer(4) gen_per_batch** + + Number of generations per batch for criticality calculations + + *do i = 1, current_batch \* gen_per_batch* + + **real(8) k_generation(i)** + + k-effective for the i-th total generation + + *do i = 1, current_batch \* gen_per_batch* + + **real(8) entropy(i)** + + Shannon entropy for the i-th total generation + + **real(8) k_col_abs** + + Sum of product of collision/absorption estimates of k-effective + + **real(8) k_col_tra** + + Sum of product of collision/track-length estimates of k-effective + + **real(8) k_abs_tra** + + Sum of product of absorption/track-length estimates of k-effective + + **real(8) k_combined(2)** + + Mean and standard deviation of a combined estimate of k-effective + + **integer(4) cmfd_on** + + Flag that cmfd is on + + if (cmfd_on) + + **integer(4) cmfd % indices** + + Indices for cmfd mesh (i,j,k,g) + + **real(8) cmfd % k_cmfd(1:current_batch)** + + CMFD eigenvalues + + **real(8) cmfd % src(1:G,1:I,1:J,1:K)** + + CMFD fission source + + **real(8) cmfd % entropy(1:current_batch)** + + CMFD estimate of Shannon entropy + + **real(8) cmfd % balance(1:current_batch)** + + RMS of the residual neutron balance equation on CMFD mesh + + **real(8) cmfd % dom(1:current_batch)** + + CMFD estimate of dominance ratio + + **real(8) cmfd % scr_cmp(1:current_batch)** + + RMS comparison of difference between OpenMC and CMFD fission source + +**integer(4) n_meshes** + + Number of meshes in tallies.xml file + +*do i = 1, n_meshes* + + **integer(4) meshes(i) % id** + + Unique ID of mesh. + + **integer(4) meshes(i) % type** + + Type of mesh. + + **integer(4) meshes(i) % n_dimension** + + Number of dimensions for mesh (2 or 3). + + **integer(4) meshes(i) % dimension(:)** + + Number of mesh cells in each dimension. + + **real(8) meshes(i) % lower_left(:)** + + Coordinates of lower-left corner of mesh. + + **real(8) meshes(i) % upper_right(:)** + + Coordinates of upper-right corner of mesh. + + **real(8) meshes(i) % width(:)** + + Width of each mesh cell in each dimension. + +**integer(4) n_tallies** + +*do i = 1, n_tallies* + + **integer(4) tallies(i) % id** + + Unique ID of tally. + + **integer(4) tallies(i) % n_realizations** + + Number of realizations for the i-th tally. + + **integer(4) size(tallies(i) % scores, 1)** + + Total number of score bins for the i-th tally + + **integer(4) size(tallies(i) % scores, 2)** + + Total number of filter bins for the i-th tally + + **integer(4) tallies(i) % n_filters** + + *do j = 1, tallies(i) % n_filters* + + **integer(4) tallies(i) % filter(j) % type** + + Type of tally filter. + + **integer(4) tallies(i) % filter(j) % n_bins** + + Number of bins for filter. + + **integer(4)/real(8) tallies(i) % filter(j) % bins(:)** + + Value for each filter bin of this type. + + **integer(4) tallies(i) % n_nuclide_bins** + + Number of nuclide bins. If none are specified, this is just one. + + *do j = 1, tallies(i) % n_nuclide_bins* + + **integer(4) tallies(i) % nuclide_bins(j)** + + Values of specified nuclide bins + + **integer(4) tallies(i) % n_score_bins** + + Number of scoring bins. + + *do j = 1, tallies(i) % n_score_bins* + + **integer(4) tallies(i) % score_bins(j)** + + Values of specified scoring bins (e.g. SCORE_FLUX). + + *do j = 1, tallies(i) % n_score_bins* + + **integer(4) tallies(i) % scatt_order(j)** + + Scattering Order specified scoring bins. + + **integer(4) tallies(i) % n_score_bins** + + Number of scoring bins without accounting for those added by + the scatter-pn command. + +**integer(4) source_present** + + Flag indicated if source bank is present in the file + +**integer(4) n_realizations** + + Number of realizations for global tallies. + +**integer(4) N_GLOBAL_TALLIES** + + Number of global tally scores + +*do i = 1, N_GLOBAL_TALLIES* + + **real(8) global_tallies(i) % sum** + + Accumulated sum for the i-th global tally + + **real(8) global_tallies(i) % sum_sq** + + Accumulated sum of squares for the i-th global tally + +**integer(4) tallies_on** + + Flag indicated if tallies are present in the file. + +if (tallies_on > 0) + + *do i = 1, n_tallies* + + *do k = 1, size(tallies(i) % scores, 2)* + + *do j = 1, size(tallies(i) % scores, 1)* + + **real(8) tallies(i) % scores(j,k) % sum** + + Accumulated sum for the j-th score and k-th filter of the + i-th tally + + **real(8) tallies(i) % scores(j,k) % sum_sq** + + Accumulated sum of squares for the j-th score and k-th + filter of the i-th tally + +if (run_mode == MODE_EIGENVALUE and source_present) + + *do i = 1, n_particles* + + **real(8) source_bank(i) % wgt** + + Weight of the i-th source particle + + **real(8) source_bank(i) % xyz(1:3)** + + Coordinates of the i-th source particle. + + **real(8) source_bank(i) % uvw(1:3)** + + Direction of the i-th source particle + + **real(8) source_bank(i) % E** + + Energy of the i-th source particle. + +----------- +Revision 10 ----------- **integer(4) FILETYPE_STATEPOINT** @@ -508,13 +801,13 @@ if (run_mode == MODE_EIGENVALUE) **integer(4) tallies(i) % score_bins(j)** Values of specified scoring bins (e.g. SCORE_FLUX). - + *do j = 1, tallies(i) % n_score_bins* **integer(4) tallies(i) % scatt_order(j)** Scattering Order specified scoring bins. - + **integer(4) tallies(i) % n_score_bins** Number of scoring bins without accounting for those added by @@ -551,7 +844,7 @@ if (tallies_on > 0) *do j = 1, size(tallies(i) % scores, 1)* **real(8) tallies(i) % scores(j,k) % sum** - + Accumulated sum for the j-th score and k-th filter of the i-th tally @@ -760,13 +1053,13 @@ if (run_mode == MODE_EIGENVALUE) **integer(4) tallies(i) % score_bins(j)** Values of specified scoring bins (e.g. SCORE_FLUX). - + *do j = 1, tallies(i) % n_score_bins* **integer(4) tallies(i) % scatt_order(j)** Scattering Order specified scoring bins. - + **integer(4) tallies(i) % n_score_bins** Number of scoring bins without accounting for those added by @@ -803,7 +1096,7 @@ if (tallies_on > 0) *do j = 1, size(tallies(i) % scores, 1)* **real(8) tallies(i) % scores(j,k) % sum** - + Accumulated sum for the j-th score and k-th filter of the i-th tally @@ -1008,13 +1301,13 @@ if (run_mode == MODE_EIGENVALUE) **integer(4) tallies(i) % score_bins(j)** Values of specified scoring bins (e.g. SCORE_FLUX). - + *do j = 1, tallies(i) % n_score_bins* **integer(4) tallies(i) % scatt_order(j)** Scattering Order specified scoring bins. - + **integer(4) tallies(i) % n_score_bins** Number of scoring bins without accounting for those added by @@ -1051,7 +1344,7 @@ if (tallies_on > 0) *do j = 1, size(tallies(i) % scores, 1)* **real(8) tallies(i) % scores(j,k) % sum** - + Accumulated sum for the j-th score and k-th filter of the i-th tally @@ -1240,13 +1533,13 @@ if (run_mode == MODE_EIGENVALUE) **integer(4) tallies(i) % score_bins(j)** Values of specified scoring bins (e.g. SCORE_FLUX). - + *do j = 1, tallies(i) % n_score_bins* **integer(4) tallies(i) % scatt_order(j)** Scattering Order specified scoring bins. - + **integer(4) tallies(i) % n_score_bins** Number of scoring bins without accounting for those added by @@ -1283,7 +1576,7 @@ if (tallies_on > 0) *do j = 1, size(tallies(i) % scores, 1)* **real(8) tallies(i) % scores(j,k) % sum** - + Accumulated sum for the j-th score and k-th filter of the i-th tally @@ -1504,7 +1797,7 @@ if (tallies_on > 0) *do j = 1, size(tallies(i) % scores, 1)* **real(8) tallies(i) % scores(j,k) % sum** - + Accumulated sum for the j-th score and k-th filter of the i-th tally @@ -1713,7 +2006,7 @@ if (tallies_on > 0) *do j = 1, size(tallies(i) % scores, 1)* **real(8) tallies(i) % scores(j,k) % sum** - + Accumulated sum for the j-th score and k-th filter of the i-th tally @@ -1918,7 +2211,7 @@ if (tallies_on > 0) *do j = 1, size(tallies(i) % scores, 1)* **real(8) tallies(i) % scores(j,k) % sum** - + Accumulated sum for the j-th score and k-th filter of the i-th tally @@ -2119,7 +2412,7 @@ if (tallies_on > 0) *do j = 1, size(tallies(i) % scores, 1)* **real(8) tallies(i) % scores(j,k) % sum** - + Accumulated sum for the j-th score and k-th filter of the i-th tally @@ -2240,7 +2533,7 @@ Revision 2 *do j = 1, size(tallies(i) % scores, 1)* **real(8) tallies(i) % scores(j,k) % sum** - + Accumulated sum for the j-th score and k-th filter of the i-th tally @@ -2339,7 +2632,7 @@ Revision 1 *do j = 1, size(tallies(i) % scores, 1)* **real(8) tallies(i) % scores(j,k) % sum** - + Accumulated sum for the j-th score and k-th filter of the i-th tally diff --git a/_sources/license.txt b/_sources/license.txt index e7f4b3a69d..73e3296172 100644 --- a/_sources/license.txt +++ b/_sources/license.txt @@ -4,7 +4,7 @@ License Agreement ================= -Copyright © 2011-2014 Massachusetts Institute of Technology +Copyright © 2011-2015 Massachusetts Institute of Technology Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in diff --git a/_sources/methods/cmfd.txt b/_sources/methods/cmfd.txt new file mode 100644 index 0000000000..3681df3b75 --- /dev/null +++ b/_sources/methods/cmfd.txt @@ -0,0 +1,561 @@ +.. _methods_cmfd: + +================================================================ +Nonlinear Diffusion Acceleration - Coarse Mesh Finite Difference +================================================================ + +This page section discusses how nonlinear diffusion acceleration (NDA) using +coarse mesh finite difference (CMFD) is implemented into OpenMC. Before we get +into the theory, general notation for this section is discussed. + +-------- +Notation +-------- + +Before deriving NDA relationships, notation is explained. If a parameter has a +:math:`\overline{\cdot}`, it is surface area-averaged and if it has a +:math:`\overline{\overline\cdot}`, it is volume-averaged. When describing a +specific cell in the geometry, indices :math:`(i,j,k)` are used which correspond +to directions :math:`(x,y,z)`. In most cases, the same operation is performed in +all three directions. To compactly write this, an arbitrary direction set +:math:`(u,v,w)` that corresponds to cell indices :math:`(l,m,n)` is used. Note +that :math:`u` and :math:`l` do not have to correspond to :math:`x` and +:math:`i`. However, if :math:`u` and :math:`l` correspond to :math:`y` and +:math:`j`, :math:`v` and :math:`w` correspond to :math:`x` and :math:`z` +directions. An example of this is shown in the following expression: + +.. math:: + :label: not1 + + \sum\limits_{u\in(x,y,z)}\left\langle\overline{J}^{u,g}_{l+1/2,m,n} + \Delta_m^v\Delta_n^w\right\rangle + +Here, :math:`u` takes on each direction one at a time. The parameter :math:`J` +is surface area-averaged over the transverse indices :math:`m` and :math:`n` +located at :math:`l+1/2`. Usually, spatial indices are listed as subscripts and +the direction as a superscript. Energy group indices represented by :math:`g` +and :math:`h` are also listed as superscripts here. The group :math:`g` is the +group of interest and, if present, :math:`h` is all groups. Finally, any +parameter surrounded by :math:`\left\langle\cdot\right\rangle` represents a +tally quantity that can be edited from a Monte Carlo (MC) solution. + +------ +Theory +------ + +NDA is a diffusion model that has equivalent physics to a transport model. There +are many different methods that can be classified as NDA. The CMFD method is a +type of NDA that represents second order multigroup diffusion equations on a +coarse spatial mesh. Whether a transport model or diffusion model is used to +represent the distribution of neutrons, these models must satisfy the *neutron +balance equation*. This balance is represented by the following formula for a +specific energy group :math:`g` in cell :math:`(l,m,n)`: + +.. math:: + :label: eq_neut_bal + + \sum\limits_{u\in(x,y,z)}\left(\left\langle\overline{J}^{u,g}_{l+1/2,m,n} + \Delta_m^v\Delta_n^w\right\rangle - + \left\langle\overline{J}^{u,g}_{l-1/2,m,n} + \Delta_m^v\Delta_n^w\right\rangle\right) + + + \left\langle\overline{\overline\Sigma}_{t_{l,m,n}}^g + \overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle + = \\ + \sum\limits_{h=1}^G\left\langle + \overline{\overline{\nu_s\Sigma}}_{s_{l,m,n}}^{h\rightarrow + g}\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w + \right\rangle + + + \frac{1}{k_{eff}}\sum\limits_{h=1}^G + \left\langle\overline{\overline{\nu_f\Sigma}}_{f_{l,m,n}}^{h\rightarrow + g}\overline{\overline\phi}_{l,m,n}^h + \Delta_l^u\Delta_m^v\Delta_n^w\right\rangle. + +In eq. :eq:`eq_neut_bal` the parameters are defined as: + +* :math:`\left\langle\overline{J}^{u,g}_{l\pm + 1/2,m,n}\Delta_m^v\Delta_n^w\right\rangle` --- surface area-integrated net + current over surface :math:`(l\pm 1/2,m,n)` with surface normal in direction + :math:`u` in energy group :math:`g`. By dividing this quantity by the transverse + area, :math:`\Delta_m^v\Delta_n^w`, the surface area-averaged net current can + be computed. +* :math:`\left\langle\overline{\overline\Sigma}_{t_{l,m,n}}^g + \overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle` + --- volume-integrated total reaction rate over energy group :math:`g`. +* :math:`\left\langle\overline{\overline{\nu_s\Sigma}}_{s_{l,m,n}}^{h\rightarrow + g} + \overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle` + --- volume-integrated scattering production rate of neutrons that begin with + energy in group :math:`h` and exit reaction in group :math:`g`. This reaction + rate also includes the energy transfer of reactions (except fission) that + produce multiple neutrons such as (n, 2n); hence, the need for :math:`\nu_s` + to represent neutron multiplicity. +* :math:`k_{eff}` --- core multiplication factor. +* :math:`\left\langle\overline{\overline{\nu_f\Sigma}}_{f_{l,m,n}}^{h\rightarrow + g}\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle` + --- volume-integrated fission production rate of neutrons from fissions in + group :math:`h` that exit in group :math:`g`. + +Each quantity in :math:`\left\langle\cdot\right\rangle` represents a scalar value that +is obtained from an MC tally. A good verification step when using an MC code is +to make sure that tallies satisfy this balance equation within statistics. No +NDA acceleration can be performed if the balance equation is not satisfied. + +There are three major steps to consider when performing NDA: (1) calculation of +macroscopic cross sections and nonlinear parameters, (2) solving an eigenvalue +problem with a system of linear equations, and (3) modifying MC source +distribution to align with the NDA solution on a chosen mesh. This process is +illustrated as a flow chart below. After a batch of neutrons +is simulated, NDA can take place. Each of the steps described above is described +in detail in the following sections. + +.. tikz:: Flow chart of NDA process. Note "XS" is used for cross section and + "DC" is used for diffusion coefficient. + :libs: shapes, snakes, shadows, arrows, calc, decorations.markings, patterns, fit, matrix, spy + :include: cmfd_tikz/cmfd_flow.tikz + +Calculation of Macroscopic Cross Sections +----------------------------------------- + +A diffusion model needs macroscopic cross sections and diffusion coefficients to +solve for multigroup fluxes. Cross sections are derived by conserving reaction +rates predicted by MC tallies. From Eq. :eq:`eq_neut_bal`, total, scattering +production and fission production macroscopic cross sections are needed. They are +defined from MC tallies as follows: + +.. math:: + :label: xs1 + + \overline{\overline\Sigma}_{t_{l,m,n}}^g \equiv + \frac{\left\langle\overline{\overline\Sigma}_{t_{l,m,n}}^g + \overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle} + {\left\langle\overline{\overline\phi}_{l,m,n}^g + \Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}, + +.. math:: + :label: xs2 + + \overline{\overline{\nu_s\Sigma}}_{s_{l,m,n}}^{h\rightarrow g} \equiv + \frac{\left\langle\overline{\overline{\nu_s\Sigma}}_{s_{l,m,n}}^{h\rightarrow + g}\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle} + {\left\langle\overline{\overline\phi}_{l,m,n}^h + \Delta_l^u\Delta_m^v\Delta_n^w\right\rangle} + +and + +.. math:: + :label: xs3 + + \overline{\overline{\nu_f\Sigma}}_{f_{l,m,n}}^{h\rightarrow g} \equiv + \frac{\left\langle\overline{\overline{\nu_f\Sigma}}_{f_{l,m,n}}^{h\rightarrow + g}\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle} + {\left\langle\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}. + +In order to fully conserve neutron balance, leakage rates also need to be +preserved. In standard diffusion theory, leakage rates are represented by +diffusion coefficients. Unfortunately, it is not easy in MC to calculate a +single diffusion coefficient for a cell that describes leakage out of each +surface. Luckily, it does not matter what definition of diffusion coefficient is +used because nonlinear equivalence parameters will correct for this +inconsistency. However, depending on the diffusion coefficient definition +chosen, different convergence properties of NDA equations are observed. +Here, we introduce a diffusion coefficient that is derived for a coarse energy +transport reaction rate. This definition can easily be constructed from +MC tallies provided that angular moments of scattering reaction rates can +be obtained. The diffusion coefficient is defined as follows: + +.. math:: + :label: eq_transD + + \overline{\overline D}_{l,m,n}^g = + \frac{\left\langle\overline{\overline\phi}_{l,m,n}^g + \Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}{3 + \left\langle\overline{\overline\Sigma}_{tr_{l,m,n}}^g + \overline{\overline\phi}_{l,m,n}^g + \Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}, + +where + +.. math:: + :label: xs4 + + \left\langle\overline{\overline\Sigma}_{tr_{l,m,n}}^g + \overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle + = + \left\langle\overline{\overline\Sigma}_{t_{l,m,n}}^g + \overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle + \\ - + \left\langle\overline{\overline{\nu_s\Sigma}}_{s1_{l,m,n}}^g + \overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle. + +Note that the transport reaction rate is calculated from the total reaction rate +reduced by the :math:`P_1` scattering production reaction rate. Equation :eq:`eq_transD` +does not represent the best definition of diffusion coefficients from MC; +however, it is very simple and usually fits into MC tally frameworks +easily. Different methods to calculate more accurate diffusion coefficients can +found in [Herman]_. + +CMFD Equations +-------------- + +The first part of this section is devoted to discussing second-order finite +volume discretization of multigroup diffusion equations. This will be followed +up by the formulation of CMFD equations that are used in this NDA +scheme. When performing second-order finite volume discretization of the +diffusion equation, we need information that relates current to flux. In this +numerical scheme, each cell is coupled only to its direct neighbors. Therefore, +only two types of coupling exist: (1) cell-to-cell coupling and (2) +cell-to-boundary coupling. The derivation of this procedure is referred to as +finite difference diffusion equations and can be found in literature such +as [Hebert]_. These current/flux relationships are as follows: + +* cell-to-cell coupling + +.. math:: + :label: eq_cell_cell + + \overline{J}^{u,g}_{l\pm1/2,m,n} = -\frac{2\overline{\overline + D}_{l\pm1,m,n}^g\overline{\overline + D}_{l,m,n}^g}{\overline{\overline D}_{l\pm1,m,n}^g\Delta_l^u + + \overline{\overline + D}_{l,m,n}^g\Delta_{l\pm1}^u} + \left(\pm\overline{\overline{\phi}}_{l\pm1,m,n}^g\mp + \overline{\overline{\phi}}_{l,m,n}^g\right), + +* cell-to-boundary coupling + +.. math:: + :label: eq_cell_bound + + \overline{J}^{u,g}_{l\pm1/2,m,n} = \pm\frac{2\overline{\overline + D}_{l,m,n}^g\left(1 - + \beta_{l\pm1/2,m,n}^{u,g}\right)}{4\overline{\overline + D}_{l,m,n}^g\left(1 + \beta_{l\pm1/2,m,n}^{u,g}\right) + \left(1 - + \beta_{l\pm1/2,m,n}^{u,g}\right)\Delta_l^u}\overline{\overline{\phi}}_{l,m,n}^{g}. + +In Eqs. :eq:`eq_cell_cell` and :eq:`eq_cell_bound`, the :math:`\pm` refers to +left (:math:`-x`) or right (:math:`+x`) surface in the :math:`x` direction, +back (:math:`-y`) or front (:math:`+y`) surface in the :math:`y` direction and +bottom (:math:`-z`) or top (:math:`+z`) surface in the :math:`z` direction. For +cell-to-boundary coupling, a general albedo, :math:`\beta_{l\pm1/2,m,n}^{u,g}`, +is used. The albedo is defined as the ratio of incoming (:math:`-` superscript) +to outgoing (:math:`+` superscript) partial current on any surface represented +as + +.. math:: + :label: eq_albedo + + \beta_{l\pm1/2,m,n}^{u,g} = + \frac{\overline{J}^{u,g-}_{l\pm1/2,m,n}}{\overline{J}^{u,g+}_{l\pm1/2,m,n}}. + +Common boundary conditions are: vacuum (:math:`\beta=0`), reflective +(:math:`\beta=1`) and zero flux (:math:`\beta=-1`). Both eq. :eq:`eq_cell_cell` +and eq. :eq:`eq_cell_bound` can be written in this generic form, + +.. math:: + :label: eq_dtilde + + \overline{J}^{u,g}_{l\pm1/2,m,n} = \widetilde{D}_{l,m,n}^{u,g} \left(\dots\right). + +The parameter :math:`\widetilde{D}_{l,m,n}^{u,g}` represents the linear +coupling term between current and flux. These current relationships can be +sustituted into eq. :eq:`eq_neut_bal` to produce a linear system of multigroup +diffusion equations for each spatial cell and energy group. However, a solution +to these equations is not consistent with a higher order transport solution +unless equivalence factors are present. This is because both the diffusion +approximation, governed by Fick's Law, and spatial trunction error will produce +differences. Therefore, a nonlinear parameter, +:math:`\widehat{D}_{l,m,n}^{u,g}`, is added to eqs. :eq:`eq_cell_cell` and +:eq:`eq_cell_bound`. These equations are, respectively, + +.. math:: + :label: eq_dhat_cell + + \overline{J}^{u,g}_{l\pm1/2,m,n} = -\widetilde{D}_{l,m,n}^{u,g} + \left(\pm\overline{\overline{\phi}}_{l\pm1,m,n}^g\mp + \overline{\overline{\phi}}_{l,m,n}^g\right) + \widehat{D}_{l,m,n}^{u,g} + \left(\overline{\overline{\phi}}_{l\pm1,m,n}^g + + \overline{\overline{\phi}}_{l,m,n}^g\right) + +and + +.. math:: + :label: eq_dhat_bound + + \overline{J}^{u,g}_{l\pm1/2,m,n} = \pm\widetilde{D}_{l,m,n}^{u,g} + \overline{\overline{\phi}}_{l,m,n}^{g} + \widehat{D}_{l,m,n}^{u,g} + \overline{\overline{\phi}}_{l,m,n}^{g}. + +The only unknown in each of these equations is the equivalence parameter. The +current, linear coupling term and flux can either be obtained or derived from +MC tallies. Thus, it is called nonlinear because it is dependent on the flux +which is updated on the next iteration. + +Equations :eq:`eq_dhat_cell` and :eq:`eq_dhat_bound` can be substituted into +eq. :eq:`eq_neut_bal` to create a linear system of equations that is consistent +with transport physics. One example of this equation is written for an +interior cell, + +.. math:: + :label: eq_cmfd_sys + + \sum_{u\in + x,y,x}\frac{1}{\Delta_l^u}\left[\left(-\tilde{D}_{l-1/2,m,n}^{u,g} - + \hat{D}_{l-1/2,m,n}^{u,g}\right)\overline{\overline{\phi}}_{l-1,m,n}^g\right. + + \left(\tilde{D}_{l-1/2,m,n}^{u,g} + + \tilde{D}_{l+1/2,m,n}^{u,g} - \hat{D}_{l-1/2,m,n}^{u,g} + + \hat{D}_{l+1/2,m,n}^{u,g}\right)\overline{\overline{\phi}}_{l,m,n}^g + \\ + + \left. \left(-\tilde{D}_{l+1/2,m,n}^{u,g} + + \hat{D}_{l+1/2,m,n}^{u,g}\right)\overline{\overline{\phi}}_{l+1,m,n}^g + \right] + + \overline{\overline\Sigma}_{t_{l,m,n}}^g\overline{\overline{\phi}}_{l,m,n}^g + - \sum\limits_{h=1}^G\overline{\overline{\nu_s\Sigma}}^{h\rightarrow + g}_{s_{l,m,n}}\overline{\overline{\phi}}_{l,m,n}^h = + \frac{1}{k}\sum\limits_{h=1}^G\overline{\overline{\nu_f\Sigma}}^{h\rightarrow + g}_{f_{l,m,n}}\overline{\overline{\phi}}_{l,m,n}^h. + +It should be noted that before substitution, eq. :eq:`eq_neut_bal` was divided +by the volume of the cell, :math:`\Delta_l^u\Delta_m^v\Delta_n^w`. Equation +:eq:`eq_cmfd_sys` can be represented in operator form as + +.. math:: + :label: eq_CMFDopers + + \mathbb{M}\mathbf{\Phi} = \frac{1}{k}\mathbb{F}\mathbf{\Phi}, + +where :math:`\mathbb{M}` is the neutron loss matrix operator, +:math:`\mathbb{F}` is the neutron production matrix operator, +:math:`\mathbf{\Phi}` is the multigroup flux vector and :math:`k` is the +eigenvalue. This generalized eigenvalue problem is solved to obtain fundamental +mode multigroup fluxes and eigenvalue. In order to produce consistent results +with transport theory from these equations, the neutron balance equation must +have been satisfied by MC tallies. The desire is that CMFD equations will +produce a more accurate source than MC after each fission source generation. + +CMFD Feedback +------------- + +Now that a more accurate representation of the expected source distribution is +estimated from CMFD, it needs to be communicated back to MC. The first step +in this process is to generate a probability mass function that provides +information about how probable it is for a neutron to be born in a given cell +and energy group. This is represented as + +.. math:: + :label: eq_cmfd_psrc + + p_{l,m,n}^g = + \frac{\sum_{h=1}^{G}\overline{\overline{\nu_f\Sigma}}^{h\rightarrow + g}_{f_{l,m,n}}\overline{\overline{\phi}}_{l,m,n}^h\Delta_l^u\Delta_m^v + \Delta_n^w}{\sum_n\sum_m\sum_l\sum_{h=1}^{G}\overline{ + \overline{\nu_f\Sigma}}^{h\rightarrow + g}_{f_{l,m,n}}\overline{\overline{\phi}}_{l,m,n}^h\Delta_l^u\Delta_m^v + \Delta_n^w}. + +This equation can be multiplied by the number of source neutrons to obtain an +estimate of the expected number of neutrons to be born in a given cell and +energy group. This distribution can be compared to the MC source distribution +to generate weight adjusted factors defined as + +.. math:: + :label: eq_waf + + f_{l,m,n}^g = \frac{Np_{l,m,n}^g}{\sum\limits_s w_s};\quad s\in + \left(g,l,m,n\right). + +The MC source distribution is represented on the same coarse mesh as +CMFD by summing all neutrons' weights, :math:`w_s`, in a given cell and +energy group. MC source weights can then be modified by this weight +adjustment factor so that it matches the CMFD solution on the coarse +mesh, + +.. math:: + :label: src_mod + + w^\prime_s = w_s\times f_{l,m,n}^g;\quad s\in \left(g,l,m,n\right). + +It should be noted that heterogeneous information about local coordinates and +energy remain constant throughout this modification process. + +------------------------ +Implementation in OpenMC +------------------------ + +The section describes how CMFD was implemented in OpenMC. Before the simulation +begins, a user sets up a CMFD input file that contains the following basic +information: + +* CMFD mesh (space and energy), +* boundary conditions at edge of mesh (albedos), +* acceleration region (subset of mesh, optional), +* fission source generation (FSG)/batch that CMFD should begin, and +* whether CMFD feedback should be applied. + +It should be noted that for more difficult simulations (e.g., light water +reactors), there are other options available to users such as tally resetting +parameters, effective down-scatter usage, tally estimator, etc. For more +information please see :ref:`usersguide_cmfd`. + +Of the options described above, the optional acceleration subset region is an +uncommon feature. Because OpenMC only has a structured Cartesian mesh, mesh +cells may overlay regions that don't contain fissionable material and may be so +far from the core that the neutron flux is very low. If these regions were +included in the CMFD solution, bad estimates of diffusion parameters may result +and affect CMFD feedback. To deal with this, a user can carve out an active +acceleration region from their structured Cartesian mesh. This is illustrated +in diagram below. When placing a CMFD mesh over a geometry, the boundary +conditions must be known at the global edges of the mesh. If the geometry is +complex like the one below, one may have to cover the whole geometry including +the reactor pressure vessel because we know that there is a zero incoming +current boundary condition at the outer edge of the pressure vessel. This is +not viable in practice because neutrons in simulations may not reach mesh cells +that are near the pressure vessel. To circumvent this, one can shrink the mesh +to cover just the core region as shown in the diagram. However, one must still +estimate the boundary conditions at the global boundaries, but at these +locations, they are not readily known. In OpenMC, one can carve out the active +core region from the entire structured Cartesian mesh. This is shown in the +diagram below by the darkened region over the core. The albedo boundary +conditions at the active core/reflector boundary can be tallied indirectly +during the MC simulation with incoming and outgoing partial currents. This +allows the user to not have to worry about neutrons producing adequate tallies +in mesh cells far away from the core. + +.. tikz:: Diagram of CMFD acceleration mesh + :libs: shapes, snakes, shadows, arrows, calc, decorations.markings, patterns, fit, matrix, spy + :include: cmfd_tikz/meshfig.tikz + +During an MC simulation, CMFD tallies are accumulated. The basic tallies needed +are listed in Table :ref:`tab_tally`. Each tally is performed on a spatial and +energy mesh basis. The surface area-integrated net current is tallied on every +surface of the mesh. OpenMC tally objects are created by the CMFD code +internally, and cross sections are calculated at each CMFD feedback iteration. +The first CMFD iteration, controlled by the user, occurs just after tallies are +communicated to the master processor. Once tallies are collapsed, cross +sections, diffusion coefficients and equivalence parameters are calculated. This +is performed only on the acceleration region if that option has been activated +by the user. Once all diffusion parameters are calculated, CMFD matrices are +formed where energy groups are the inner most iteration index. In OpenMC, +compressed row storage sparse matrices are used due to the sparsity of CMFD +operators. An example of this sparsity is shown for the 3-D BEAVRS model in +figures :ref:`fig_loss` and :ref:`fig_prod` [BEAVRS]_. These matrices represent +an assembly radial mesh, 24 cell mesh in the axial direction and two energy +groups. The loss matrix is 99.92% sparse and the production matrix is 99.99% +sparse. Although the loss matrix looks like it is tridiagonal, it is really a +seven banded matrix with a block diagonal matrix for scattering. The production +matrix is a :math:`2\times 2` block diagonal; however, zeros are present because +no fission neutrons appear with energies in the thermal group. + +.. _tab_tally: + +.. table:: OpenMC CMFD tally list + + +--------------------------------------------------------------------------------------------+----------------+---------------------------+ + +--------------------------------------------------------------------------------------------+----------------+---------------------------+ + | tally | score | filter | + +============================================================================================+================+===========================+ + | \ :math:`\left\langle\overline{\overline\phi}_{l,m,n}^g | flux | mesh, energy | + | \Delta_l^u\Delta_m^v\Delta_n^w\right\rangle` | | | + +--------------------------------------------------------------------------------------------+----------------+---------------------------+ + | \ :math:`\left\langle\overline{\overline\Sigma}_{t_{l,m,n}}^g | total | mesh, energy | + | \overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle` | | | + +--------------------------------------------------------------------------------------------+----------------+---------------------------+ + | \ :math:`\left\langle\overline{\overline{\nu_s\Sigma}}_{s1_{l,m,n}}^g | nu-scatter-1 | mesh, energy | + | \overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle` | | | + +--------------------------------------------------------------------------------------------+----------------+---------------------------+ + | \ :math:`\left\langle\overline{\overline{\nu_s\Sigma}}_{s_{l,m,n}}^{h\rightarrow g} | nu-scatter | mesh, energy, energyout | + | \overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle` | | | + +--------------------------------------------------------------------------------------------+----------------+---------------------------+ + | \ :math:`\left\langle\overline{\overline{\nu_f\Sigma}}_{f_{l,m,n}}^{h\rightarrow g} | nu-fission | mesh, energy, energyout | + | \overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle` | | | + +--------------------------------------------------------------------------------------------+----------------+---------------------------+ + | \ :math:`\left\langle\overline{J}^{u,g}_{l\pm 1/2,m,n}\Delta_m^v\Delta_n^w\right\rangle` | current | mesh, energy | + +--------------------------------------------------------------------------------------------+----------------+---------------------------+ + +.. _fig_loss: + +.. figure:: ../_images/loss.png + :scale: 50 + + Sparsity of Neutron Loss Operator + +.. _fig_prod: + +.. figure:: ../_images/prod.png + :scale: 50 + + Sparsity of Neutron Production Operator + +To solve the eigenvalue problem with these matrices, different source iteration +and linear solvers can be used. The most common source iteration solver used is +standard power iteration as described in [Gill]_. To accelerate these source +iterations, a Wielandt shift scheme can be used as discussed in [Park]_. PETSc +solvers were first implemented to perform the linear solution in parallel that +occurs once per source iteration. When using PETSc, different types of parallel +linear solvers and preconditioners can be used. By default, OpenMC uses an +incomplete LU preconditioner and a GMRES Krylov solver. After some initial +studies of parallelization with PETSc, it was observed that because CMFD +matrices are very sparse, solution times do not scale well. An additional +Gauss-Seidel linear solver with Chebyshev acceleration was added that is +similar to the one used for CMFD in CASMO [Rhodes]_ and [Smith]_. This solver +was implemented with a custom section for two energy groups. Because energy +group is the inner most index, a block diagonal is formed when using more than +one group. For two groups, it is easy to invert this diagonal analytically +inside the Gauss-Seidel iterative solver. For more than two groups, this +analytic inversion can still be performed, but with more computational effort. +A standard Gauss-Seidel solver is used for more than two groups. + +Besides a power iteration, a Jacobian-free Newton-Krylov method was also +implemented to obtain eigenvalue and multigroup fluxes as described in [Gill]_ +and [Knoll]_. This method is not the primary one used, but has gotten recent +attention due to its coupling advantages to other physics such as thermal +hydraulics. Once multigroup fluxes are obtained, a normalized fission source is +calculated in the code using eq. :eq:`eq_cmfd_psrc` directly. + +The next step in the process is to compute weight adjustment factors. These are +calculated by taking the ratio of the expected number of neutrons from the CMFD +source distribution to the current number of neutrons in each mesh. It is +straightforward to compute the CMFD number of neutrons because it is the +product between the total starting initial weight of neutrons and the CMFD +normalized fission source distribution. To compute the number of neutrons from +the current MC source, OpenMC sums the statistical +weights of neutrons from the source bank on a given spatial and energy mesh. +Once weight adjustment factors were calculated, each neutron's statistical +weight in the source bank was modified according to its location and energy. +Examples of CMFD simulations using OpenMC can be found in [Herman_Thesis]_. + +---------- +References +---------- + +.. [BEAVRS] Nick Horelik, Bryan Herman. *Benchmark for Evaluation And Verification of Reactor + Simulations*. Massachusetts Institute of Technology, http://crpg.mit.edu/pub/beavrs + , 2013. + +.. [Gill] Daniel F. Gill. *Newton-Krylov methods for the solution of the k-eigenvalue problem in + multigroup neutronics calculations*. Ph.D. thesis, Pennsylvania State University, 2010. + +.. [Hebert] Alain Hebert. *Applied reactor physics*. Presses Internationales Polytechnique, + Montreal, 2009. + +.. [Herman] Bryan R. Herman, Benoit Forget, Kord Smith, and Brian N. Aviles. Improved + diffusion coefficients generated from Monte Carlo codes. In *Proceedings of M&C + 2013*, Sun Valley, ID, USA, May 5 - 9, 2013. + +.. [Herman_Thesis] Bryan R. Herman. *Monte Carlo and Thermal Hydraulic Coupling using + Low-Order Nonlinear Diffusion Acceleration*. Sc.D. thesis, + Massachusetts Institute of Technology, 2014. + +.. [Knoll] D.A. Knoll, H. Park, and C. Newman. *Acceleration of k-eigenvalue/criticality + calculations using the Jacobian-free Newton-Krylov method*. Nuclear Science and + Engineering, 167:133–140, 2011. + +.. [Park] H. Park, D.A. Knoll, and C.K. Newman. *Nonlinear acceleration of transport + criticality problems*. Nuclear Science and Engineering, 172:52–65, 2012. + +.. [Rhodes] Joel Rhodes and Malte Edenius. *CASMO-4 --- A Fuel Assembly Burnup Program. + User’s Manual*. Studsvik of America, ssp-09/443-u rev 0, proprietary edition, 2001. + +.. [Smith] Kord S Smith and Joel D Rhodes III. *Full-core, 2-D, LWR core calculations with + CASMO-4E*. In Proceedings of PHYSOR 2002, Seoul, Korea, October 7 - 10, 2002. diff --git a/_sources/methods/cross_sections.txt b/_sources/methods/cross_sections.txt index a193dde96d..db2e5156ee 100644 --- a/_sources/methods/cross_sections.txt +++ b/_sources/methods/cross_sections.txt @@ -47,45 +47,28 @@ there would be for burnup calculations. Thus, there is a strong motive to implement a method of reducing the number of energy grid searches in order to speed up the calculation. -Unionized Energy Grid ---------------------- +Logarithmic Mapping +------------------- -The most naïve method to reduce the number of energy grid searches is to -construct a new energy grid that consists of the union of the energy points of -each nuclide and use this energy grid for all nuclides. This method is -computationally very efficient as it only requires one energy grid search at -each collision as well as one interpolation between cross section values since -the interpolation factor can be used for all nuclides. However, it requires -redundant storage of cross section values at points which were added to each -nuclide grid. This additional burden on memory storage can become quite -prohibitive. To lessen that burden, the unionized energy grid can be thinned -with cross sections reconstructed on the thinned energy grid. This method is -currently used by default in the Serpent Monte Carlo code. +To speed up energy grid searches, OpenMC uses logarithmic mapping technique +[Brown]_ to limit the range of energies that must be searched for each +nuclide. The entire energy range is divided up into equal-lethargy segments, and +the bounding energies of each segment are mapped to bounding indices on each of +the nuclide energy grids. By default, OpenMC uses 8000 equal-lethargy segments +as recommended by Brown. -Unionized Energy Grid with Nuclide Pointers -------------------------------------------- +Other Methods +------------- -While having a unionized grid that is used for all nuclides allows for very fast -lookup of cross sections, the burden on memory is in many circumstances -unacceptable. The OpenMC Monte Carlo code utilizes a method that allows for a -single energy grid search to be performed at every collision while avoiding the -redundant storage of cross section values. Instead of using the unionized grid -for every nuclide, the original energy grid of each nuclide is kept and a list -of pointers (of the same length as the unionized energy grid) is constructed for -each nuclide that gives the corresponding grid index on the nuclide grid for a -given grid index on the unionized grid. One must still interpolate on cross -section values for each nuclide since the interpolation factors will generally -be different. The figure below illustrates this method. All values within the -dashed box would need to be stored on a per-nuclide basis, and the union grid -would need to be stored once. This method is also referred to as *double -indexing* and is available as an option in Serpent (see paper by Leppanen_). +A good survey of other energy grid techniques, including unionized energy grids, +can be found in a paper by Leppanen_. -.. figure:: ../_images/uniongrid.* - :width: 600px - :align: center - :figclass: align-center +---------- +References +---------- - Mapping of union energy grid to nuclide energy grid through pointers. +.. [Brown] Forrest B. Brown, "New Hash-based Energy Lookup Algorithm for Monte + Carlo codes," LA-UR-14-24530, Los Alamos National Laboratory (2014). .. _MCNP: http://mcnp.lanl.gov .. _Serpent: http://montecarlo.vtt.fi diff --git a/_sources/methods/geometry.txt b/_sources/methods/geometry.txt index 1515ffa891..1ba29b79ef 100644 --- a/_sources/methods/geometry.txt +++ b/_sources/methods/geometry.txt @@ -421,7 +421,7 @@ satisfy the following equations x^2 + y^2 + z^2 - 10^2 < 0 \\ x - (-3) > 0 \\ - x - 2 < 0 + y - 2 < 0 In order to determine if a point is inside the cell, we would substitute its coordinates into equation :eq:`cell-contains-example`. If the inequalities are diff --git a/_sources/methods/index.txt b/_sources/methods/index.txt index 0ff4fb1988..1df4f324a3 100644 --- a/_sources/methods/index.txt +++ b/_sources/methods/index.txt @@ -16,3 +16,4 @@ Theory and Methodology tallies eigenvalue parallelization + cmfd diff --git a/_sources/methods/physics.txt b/_sources/methods/physics.txt index 54dc913455..db6cfd89e9 100644 --- a/_sources/methods/physics.txt +++ b/_sources/methods/physics.txt @@ -682,17 +682,20 @@ nuclear temperature, which is a function of the incoming energy of the neutron. The ACE format contains a list of nuclear temperatures versus incoming energies. The nuclear temperature is interpolated between neighboring incoming energies using a specified interpolation law. Once the temperature :math:`T` is -determined, we then calculate a candidate outgoing energy based on rule C45 in -the `Monte Carlo Sampler`_: +determined, we then calculate a candidate outgoing energy based on the algorithm +given in LA-UR-14-27694_: .. math:: :label: evaporation-E - E' = -T \log (\xi_1 \xi_2) + E' = -T \log ((1 - g\xi_1)(1 - g\xi_2)) -where :math:`\xi_1, \xi_2` are random numbers sampled on the unit -interval. The outgoing energy is only accepted according to a specified -restriction energy as in equation :eq:`maxwell-restriction`. +where :math:`g = 1 - e^{-w}`, :math:`w = (E - U)/T`, :math:`U` is the +restriction energy, and :math:`\xi_1, \xi_2` are random numbers sampled on the +unit interval. The outgoing energy is only accepted according to the restriction +energy as in equation :eq:`maxwell-restriction`. This algorithm has a much +higher rejection efficiency than the standard technique, i.e. rule C45 in the +`Monte Carlo Sampler`_. ACE Law 11 - Energy-Dependent Watt Spectrum +++++++++++++++++++++++++++++++++++++++++++ @@ -1591,6 +1594,8 @@ References .. _Monte Carlo Sampler: https://laws.lanl.gov/vhosts/mcnp.lanl.gov/pdf_files/la-9721_3rdmcsampler.pdf +.. _LA-UR-14-27694: http://permalink.lanl.gov/object/tr?what=info:lanl-repo/lareport/LA-UR-14-27694 + .. _MC21: http://www.osti.gov/bridge/servlets/purl/903083-HT5p1o/903083.pdf .. _Sutton and Brown: http://www.osti.gov/bridge/product.biblio.jsp?osti_id=307911 diff --git a/_sources/publications.txt b/_sources/publications.txt index 9d3491b8ab..99a00d13f2 100644 --- a/_sources/publications.txt +++ b/_sources/publications.txt @@ -4,6 +4,10 @@ Publications ============ +- Jonathan A. Walsh, Benoit Forget, and Kord S. Smith, "Accelerated sampling + of the free gas resonance elastic scattering kernel," *Ann. Nucl. Energy*, + **69**, 116--124 (2014). ``_ + - Benoit Forget, Sheng Xu, and Kord Smith, "Direct Doppler broadening in Monte Carlo simulations using the multipole representation," *Ann. Nucl. Energy*, **64**, 78--85 (2014). ``_ diff --git a/_sources/quickinstall.txt b/_sources/quickinstall.txt index ccbfabebe7..3fb9036d39 100644 --- a/_sources/quickinstall.txt +++ b/_sources/quickinstall.txt @@ -35,15 +35,15 @@ OpenMC from source as is described in :ref:`usersguide_install`. Installing from Source on Linux or Mac OS X ------------------------------------------- -All OpenMC source code is hosted on GitHub_. If you have git_ and the gfortran_ -compiler installed, you can download and install OpenMC be entering the -following commands in a terminal: +All OpenMC source code is hosted on GitHub_. If you have git_, the gfortran_ +compiler, and CMake_ installed, you can download and install OpenMC be entering +the following commands in a terminal: .. code-block:: sh git clone git://github.com/mit-crpg/openmc.git cd openmc/src - git checkout master + git checkout -b master origin/master make sudo make install @@ -58,3 +58,4 @@ can be replaced with a local install, e.g. .. _GitHub: https://github.com/mit-crpg/openmc .. _git: http://git-scm.com .. _gfortran: http://gcc.gnu.org/wiki/GFortran +.. _CMake: http://www.cmake.org diff --git a/_sources/releasenotes/index.txt b/_sources/releasenotes/index.txt index b40126b243..556cf518a5 100644 --- a/_sources/releasenotes/index.txt +++ b/_sources/releasenotes/index.txt @@ -10,6 +10,8 @@ bugs fixed, and known issues for each successive release. .. toctree:: :maxdepth: 1 + notes_0.6.2 + notes_0.6.1 notes_0.6.0 notes_0.5.4 notes_0.5.3 diff --git a/_sources/releasenotes/notes_0.6.1.txt b/_sources/releasenotes/notes_0.6.1.txt new file mode 100644 index 0000000000..71852bd174 --- /dev/null +++ b/_sources/releasenotes/notes_0.6.1.txt @@ -0,0 +1,65 @@ +.. _notes_0.6.1: + +============================== +Release Notes for OpenMC 0.6.1 +============================== + +------------------- +System Requirements +------------------- + +There are no special requirements for running the OpenMC code. As of this +release, OpenMC has been tested on a variety of Linux distributions, Mac OS X, +and Microsoft Windows 7. Memory requirements will vary depending on the size of +the problem at hand (mostly on the number of nuclides in the problem). + +------------ +New Features +------------ + +- Coarse mesh finite difference (CMFD) acceleration no longer requires PETSc +- Statepoint file numbering is now zero-padded +- Python scripts now compatible with Python 2 or 3 +- Ability to run particle restarts in fixed source calculations +- Capability to filter box source by fissionable materials +- Nuclide/element names are now case insensitive in input files +- Improved treatment of resonance scattering for heavy nuclides + +--------- +Bug Fixes +--------- + +- 03e890_: Check for energy-dependent multiplicities in ACE files +- 4439de_: Fix distance-to-surface calculation for general plane surface +- 5808ed_: Account for differences in URR band probabilities at different energies +- 2e60c0_: Allow zero atom/weight percents in materials +- 3e0870_: Don't use PWD environment variable when setting path to input files +- dc4776_: Handle probability table resampling correctly +- 01178b_: Fix metastables nuclides in NNDC cross_sections.xml file +- 62ec43_: Don't read tallies.xml when OpenMC is run in plotting mode +- 2a95ef_: Prevent segmentation fault on "current" score without mesh filter +- 93e482_: Check for negative values in probability tables + +.. _03e890: https://github.com/mit-crpg/openmc/commit/03e890 +.. _4439de: https://github.com/mit-crpg/openmc/commit/4439de +.. _5808ed: https://github.com/mit-crpg/openmc/commit/5808ed +.. _2e60c0: https://github.com/mit-crpg/openmc/commit/2e60c0 +.. _3e0870: https://github.com/mit-crpg/openmc/commit/3e0870 +.. _dc4776: https://github.com/mit-crpg/openmc/commit/dc4776 +.. _01178b: https://github.com/mit-crpg/openmc/commit/01178b +.. _62ec43: https://github.com/mit-crpg/openmc/commit/62ec43 +.. _2a95ef: https://github.com/mit-crpg/openmc/commit/2a95ef +.. _93e482: https://github.com/mit-crpg/openmc/commit/93e482 + +------------ +Contributors +------------ + +This release contains new contributions from the following people: + +- `Sterling Harper `_ +- `Bryan Herman `_ +- `Adam Nelson `_ +- `Paul Romano `_ +- `Jon Walsh `_ +- `Will Boyd `_ diff --git a/_sources/releasenotes/notes_0.6.2.txt b/_sources/releasenotes/notes_0.6.2.txt new file mode 100644 index 0000000000..eefc493fcf --- /dev/null +++ b/_sources/releasenotes/notes_0.6.2.txt @@ -0,0 +1,58 @@ +.. _notes_0.6.2: + +============================== +Release Notes for OpenMC 0.6.2 +============================== + +------------------- +System Requirements +------------------- + +There are no special requirements for running the OpenMC code. As of this +release, OpenMC has been tested on a variety of Linux distributions, Mac OS X, +and Microsoft Windows 7. Memory requirements will vary depending on the size of +the problem at hand (mostly on the number of nuclides in the problem). + +------------ +New Features +------------ + +- Meshline plotting capability +- Support for plotting cells/materials on middle universe levels +- Ability to model cells with no surfaces +- Compatibility with PETSc 3.5 +- Compatability with OpenMPI 1.7/1.8 +- Improved overall performance via logarithmic-mapped energy grid search +- Improved multi-threaded performance with atomic operations +- Support for fixed source problems with fissionable materials + +--------- +Bug Fixes +--------- + +- 26fb93_: Fix problem with -t, --track command-line flag +- 2f07c0_: Improved evaporation spectrum algorithm +- e6abb9_: Fix segfault when tallying in a void material +- 291b45_: Handle metastable nuclides in NNDC data and multiplicities in MT=5 data + +.. _26fb93: https://github.com/mit-crpg/openmc/commit/26fb93 +.. _2f07c0: https://github.com/mit-crpg/openmc/commit/2f07c0 +.. _e6abb9: https://github.com/mit-crpg/openmc/commit/e6abb9 +.. _291b45: https://github.com/mit-crpg/openmc/commit/291b45 + +------------ +Contributors +------------ + +This release contains new contributions from the following people: + +- `Will Boyd `_ +- `Matt Ellis `_ +- `Sterling Harper `_ +- `Bryan Herman `_ +- `Nicholas Horelik `_ +- `Anton Leontiev `_ +- `Adam Nelson `_ +- `Paul Romano `_ +- `Jon Walsh `_ +- `John Xia `_ diff --git a/_sources/usersguide/input.txt b/_sources/usersguide/input.txt index c9aa8cc866..c834bbc92e 100644 --- a/_sources/usersguide/input.txt +++ b/_sources/usersguide/input.txt @@ -134,13 +134,15 @@ should be performed. It has the following attributes/sub-elements: ------------------------- The ```` element determines the treatment of the energy grid during -a simulation. Setting this element to "nuclide" will cause OpenMC to use a -nuclide's energy grid when determining what points to interpolate between for -determining cross sections (i.e. non-unionized energy grid). To use a unionized -energy grid, set this element to "union". Note that the unionized energy grid -treatment is slightly different than that employed in Serpent. +a simulation. The valid options are "nuclide" and "logarithm". Setting this +element to "nuclide" will cause OpenMC to use a nuclide's energy grid when +determining what points to interpolate between for determining cross sections +(i.e. non-unionized energy grid). Setting this element to "logarithm" causes +OpenMC to use a logarithmic mapping technique described in LA-UR-14-24530_. - *Default*: union + *Default*: logarithm + +.. _LA-UR-14-24530: https://laws.lanl.gov/vhosts/mcnp.lanl.gov/pdf_files/la-ur-14-24530.pdf ```` Element --------------------- @@ -182,6 +184,16 @@ performed. It has the following attributes/sub-elements: *Default*: None +```` Element +--------------------------- + +The ```` element indicates the number of bins to use for the +logarithmic-mapped energy grid. Using more bins will result in energy grid +searches over a smaller range at the expense of more memory. The default is +based on the recommended value in LA-UR-14-24530_. + + *Default*: 8000 + .. _natural_elements: ```` Element @@ -260,6 +272,65 @@ or sub-elements and can be set to either "false" or "true". *Default*: true +```` Element +---------------------------------- + +The ``resonance_scattering`` element can contain one or more of the following +attributes or sub-elements: + + :scatterer: + An element with attributes/sub-elements called ``nuclide``, ``method``, + ``xs_label``, ``xs_label_0K``, ``E_min``, and ``E_max``. The ``nuclide`` + attribute is the name, as given by the ``name`` attribute within the + ``nuclide`` sub-element of the ``material`` element in ``materials.xml``, + of the nuclide to which a resonance scattering treatment is to be applied. + The ``method`` attribute gives the type of resonance scattering treatment + that is to be applied to the ``nuclide``. Acceptable inputs - none of + which are case-sensitive - for the ``method`` attribute are ``ARES``, + ``CXS``, ``WCM``, and ``DBRC``. Descriptions of each of these methods + are documented here_. The ``xs_label`` attribute gives the label for the + cross section data of the ``nuclide`` at a given temperature. The + ``xs_label_0K`` gives the label for the 0 K cross section data for the + ``nuclide``. The ``E_min`` attribute gives the minimum energy above + which the ``method`` is applied. The ``E_max`` attribute gives the + maximum energy below which the ``method`` is applied. One example would + be as follows: + + .. _here: http://dx.doi.org/10.1016/j.anucene.2014.01.017 + + .. code-block:: xml + + + + U-238 + ARES + 92238.72c + 92238.00c + 5.0e-6 + 40.0e-6 + + + Pu-239 + dbrc + 94239.72c + 94239.00c + 0.01e-6 + 210.0e-6 + + + + .. note:: If the ``resonance_scattering`` element is not given, the free gas, + constant cross section (``cxs``) scattering model, which has + historically been used by Monte Carlo codes to sample target + velocities, is used to treat the target motion of all nuclides. If + ``resonance_scattering`` is present, the ``cxs`` method is applied + below ``E_min`` and the target-at-rest (asymptotic) kernel is used + above ``E_max``. An arbitrary number of ``scatterer`` elements may + be specified, each corresponding to a single nuclide at a single + temperature. + + *Defaults*: None (scatterer), ARES (method), 0.01 eV (E_min), 1.0 keV (E_max) + ```` Element ---------------------- @@ -311,6 +382,12 @@ attributes/sub-elements: parallelepiped and the last three of which specify the upper-right corner. Source sites are sampled uniformly through that parallelepiped. + To filter a "box" spatial distribution by fissionable material, specify + "fission" tag instead of "box". The ``parameters`` should be given as six + real numbers, the first three of which specify the lower-left corner of a + parallelepiped and the last three of which specify the upper-right + corner. Source sites are sampled uniformly through that parallelepiped. + For a "point" spatial distribution, ``parameters`` should be given as three real numbers which specify the (x,y,z) location of an isotropic point source @@ -368,6 +445,13 @@ attributes/sub-elements: *Default*: 0.988 2.249 + :write_initial: + An element specifying whether to write out the initial source bank used at + the beginning of the first batch. The output file is named + "initial_source.binary(h5)" + + *Default*: false + ```` Element ------------------------- @@ -685,7 +769,10 @@ Each ```` element can have the following attributes or sub-elements: is on the negative side of surface 3 and the positive side of surface 5, the bounding surfaces would be given as "-3 5". - *Default*: None + .. note:: The surface attribute/element can be omitted to make a cell fill + its entire universe. + + *Default*: No surfaces :rotation: If the cell is filled with a universe, this element specifies the angles in @@ -947,6 +1034,16 @@ The ```` element accepts the following sub-elements: *Default*: total + :estimator: + The estimator element is used to force the use of either ``analog`` or + ``tracklength`` tally estimation. ''analog'' is generally less efficient + though it can be used with every score type. ''tracklength'' is generally + the most efficient, though its usage is restricted to tallies that do not + score particle information which requires a collision to have occured, such + as a scattering tally which utilizes outgoing energy filters. + + *Default*: ``tracklength`` but will revert to analog if necessary. + :scores: A space-separated list of the desired responses to be accumulated. Accepted options are "flux", "total", "scatter", "absorption", "fission", @@ -1020,10 +1117,10 @@ The ```` element accepts the following sub-elements: all of the harmonic moments of order 0 to N. N must be between 0 and 10. :total-YN: - Spherical harmonic expansion of the incoming particle's direction of - motion :math:`\left(\Omega\right)` of the total flux. This score will - tally all of the harmonic moments of order 0 to N. N must be between 0 - and 10. + The total reaction rate expanded via spherical harmonics about the + direction of motion of the neutron, :math:`\Omega`. + This score will tally all of the harmonic moments of order 0 to N. N must + be between 0 and 10. :current: Partial currents on the boundaries of each cell in a mesh. @@ -1098,7 +1195,7 @@ implemented in openMC: ```` Element ------------------ -Each plot must contain a combination of the following attributes or +Each plot is specified by a combination of the following attributes or sub-elements: :id: @@ -1119,6 +1216,18 @@ sub-elements: *Default*: ``cell`` + :level: + Universe depth to plot at (optional). This parameter controls how many + universe levels deep to pull cell and material ids from when setting plot + colors. If a given location does not have as many levels as specified, + colors will be taken from the lowest level at that location. For example, if + ``level`` is set to zero colors will be taken from top-level (universe zero) + cells only. However, if ``level`` is set to 1 colors will be taken from + cells in universes that fill top-level fill-cells, and from top-level cells + that contain materials. + + *Default*: Whatever the deepest universe is in the model + :origin: Specifies the (x,y,z) coordinate of the center of the plot. Should be three floats separated by spaces. @@ -1157,7 +1266,7 @@ sub-elements: attribute or sub-element: :pixels: - Specifies the number of pixes or voxels to be used along each of the basis + Specifies the number of pixels or voxels to be used along each of the basis directions for "slice" and "voxel" plots, respectively. Should be two or three integers separated by spaces. @@ -1189,7 +1298,7 @@ attributes or sub-elements. These are not used in "voxel" plots: Specifies the RGB color of the regions where no OpenMC cell can be found. Should be three integers separated by spaces. - *Default*: 0 0 0 (white) + *Default*: 0 0 0 (black) :col_spec: Any number of this optional tag may be included in each ```` element, @@ -1229,6 +1338,37 @@ attributes or sub-elements. These are not used in "voxel" plots: *Default*: None + :meshlines: + The ``meshlines`` sub-element allows for plotting the boundaries of + a tally mesh on top of a plot. Only one ``meshlines`` element is allowed per + ``plot`` element, and it must contain as attributes or sub-elements a mesh + type and a linewidth. Optionally, a color may be specified for the overlay: + + :meshtype: + The type of the mesh to be plotted. Valid options are "tally", "entropy", + "ufs", and "cmfd". If plotting "tally" meshes, the id of the mesh to plot + must be specified with the ``id`` sub-element. + + :id: + A single integer id number for the mesh specified on ``tallies.xml`` that + should be plotted. This element is only required for ``meshtype="tally"``. + + :linewidth: + A single integer number of pixels of linewidth to specify for the mesh + boundaries. Specifying this as 0 indicates that lines will be 1 pixel + thick, specifying 1 indicates 3 pixels thick, specifying 2 indicates + 5 pixels thick, etc. + + :color: + Specifies the custom color for the meshlines boundaries. Should be 3 + integers separated by whitespace. This element is optional. + + *Default*: 0 0 0 (black) + + *Default*: None + +.. _usersguide_cmfd: + ------------------------------ CMFD Specification -- cmfd.xml ------------------------------ @@ -1238,15 +1378,6 @@ Currently, it allows users to accelerate fission source convergence during inactive neutron batches. To run CMFD, the ```` element in ``settings.xml`` should be set to "true". -```` Element --------------------------- - -The ```` element controls the batch where CMFD tallies should be -reset. CMFD tallies should be reset before active batches so they are accumulated -without bias. - - *Default*: 0 - ```` Element ------------------- @@ -1268,7 +1399,25 @@ The ```` element sets one additional CMFD output column. Options are: * "source" - prints the RMS [%] between the OpenMC fission source and CMFD fission source. - *Default*: None + *Default*: balance + +```` Element +------------------------ + +The ```` element controls whether :math:`\widehat{D}` nonlinear +CMFD parameters should be reset to zero before solving CMFD eigenproblem. +It can be turned on with "true" and off with "false". + + *Default*: false + +```` Element +------------------------- + +The ```` element controls whether an effective downscatter cross +section should be used when using 2-group CMFD. It can be turned on with "true" +and off with "false". + + *Default*: false ```` Element ---------------------- @@ -1279,24 +1428,16 @@ It can be turned on with "true" and off with "false". *Default*: false -```` Element ----------------------- +```` Element +------------------------------------ -The ```` element controls if cmfd tallies should be accumulated -during inactive batches. For some applications, CMFD tallies may not be -needed until the start of active batches. This option can be turned on -with "true" and off with "false" +The ```` element specifies two parameters. The first is +the absolute inner tolerance for Gauss-Seidel iterations when performing CMFD +and the second is the relative inner tolerance for Gauss-Seidel iterations +for CMFD calculations. It is only used in the standalone CMFD power iteration +solver and not when PETSc is active. - *Default*: true - -```` Element ----------------------------- - -The ```` element controls when CMFD tallies are reset during -inactive batches. The integer set here is the interval at which this reset -occurs. The amout of resets is controlled with the ```` element. - - *Defualt*: 9999 + *Default*: 1.e-10 1.e-5 ```` Element ------------------------- @@ -1305,9 +1446,16 @@ The ```` element is used to view the convergence of linear GMRES iterations in PETSc. This option can be turned on with "true" and turned off with "false". - *Default*: false +```` Element +-------------------- + +The ```` element specifies the tolerance on the eigenvalue when performing +CMFD power iteration. + + *Default*: 1.e-8 + ```` Element ------------------ @@ -1376,14 +1524,6 @@ not impact the calculation. *Default*: 1.0 -```` Element -------------------------- - -The ```` element controls the number of CMFD tally resets that -occur during inactive CMFD batches. - - *Default*: 9999 - ```` Element --------------------------- @@ -1396,16 +1536,8 @@ This option can be turned on with "true" and turned off with "false". ------------------------- The ```` element can be turned on with "true" to have an adjoint -calculation be performed on the last batch when CMFD is active. - - *Default*: false - -```` Element --------------------------- - -The ```` element is used to view the convergence of the nonlinear SNES -function in PETSc. This option can be turned on with "true" and turned off with "false". - +calculation be performed on the last batch when CMFD is active. OpenMC should be +compiled with PETSc when using this option. *Default*: false @@ -1418,6 +1550,41 @@ By setting "power", power iteration is used and by setting "jfnk", JFNK is used. *Default*: power +```` Element +-------------------- + +The ```` element specifies an optional Wielandt shift parameter for +accelerating power iterations. It can only be used when PETSc is not active. +It is by default very large so the impact of the shift is effectively zero. + + *Default*: 1e6 + +```` Element +---------------------- + +The ```` element specifies an optional spectral radius that can be set to +accelerate the convergence of Gauss-Seidel iterations during CMFD power iteration +solve. Note this is only used in the standalone CMFD solver and does not affect +the calculation when PETSc is active. + + *Default*: power + +```` Element +------------------ + +The ```` element specifies the tolerance on the fission source when performing +CMFD power iteration. + + *Default*: 1.e-8 + +```` Element +------------------------- + +The ```` element contains a list of batch numbers in which CMFD tallies +should be reset. + + *Default*: None + ```` Element ---------------------------- @@ -1427,3 +1594,15 @@ into MATLAB using PETSc-MATLAB utilities. This option can be turned on with "true" and off with "false". *Default*: false + +------------------------------------ +ERSN-OpenMC Graphical User Interface +------------------------------------ + +A third-party Java-based user-friendly graphical user interface for creating XML +input files called ERSN-OpenMC_ is developed and maintained by members of the +Radiation and Nuclear Systems Group at the Faculty of Sciences Tetouan, Morocco. +The GUI also allows one to automatically download prerequisites for installing and +running OpenMC. + +.. _ERSN-OpenMC: https://github.com/EL-Bakkali-Jaafar/ERSN-OpenMC diff --git a/_sources/usersguide/install.txt b/_sources/usersguide/install.txt index 358937e303..7cb7ac7a45 100644 --- a/_sources/usersguide/install.txt +++ b/_sources/usersguide/install.txt @@ -66,14 +66,11 @@ Prerequisites To compile with support for parallel runs on a distributed-memory architecture, you will need to have a valid implementation of MPI installed on your machine. The code has been tested and is known to work - with the latest versions of both OpenMPI_ and MPICH_. Note that if using - OpenMPI, make sure that --with-mpi-f90-size is not set to medium or large - since this may prevent MPI calls from completing successfully in - OpenMC. OpenMPI and/or MPICH can be installed on Debian derivatives - with:: + with the latest versions of both OpenMPI_ and MPICH_. OpenMPI and/or MPICH + can be installed on Debian derivatives with:: - sudo apt-get install mpich2 libmpich2-dev - sudo apt-get install openmpi1.6-bin libopenmpi1.6-dev + sudo apt-get install mpich libmpich-dev + sudo apt-get install openmpi-bin libopenmpi1.6 libopenmpi-dev * HDF5_ Library for portable binary output format @@ -239,20 +236,22 @@ the root directory of the source code: .. code-block:: sh - cd src + mkdir src/build + cd src/build + cmake .. make - sudo make install + make install This will build an executable named ``openmc`` and install it (by default in /usr/local/bin). If you do not have administrative privileges, you can install -OpenMC locally by replacing the last command with: +OpenMC locally by specifying an install prefix when running cmake: .. code-block:: sh - make install -e prefix=$HOME/.local + cmake -DCMAKE_INSTALL_PREFIX=$HOME/.local .. -The ``prefix`` variable can be changed to any path for which you have -write-access. +The ``CMAKE_INSTALL_PREFIX`` variable can be changed to any path for which you +have write-access. Compiling on Windows -------------------- @@ -326,7 +325,7 @@ Testing Build ------------- If you have ENDF/B-VII.1 cross sections from NNDC_ you can test your build. -Make sure the **CROSS_SECTIONS** environmental variable is set to the +Make sure the **CROSS_SECTIONS** environmental variable is set to the *cross_sections.xml* file in the *data/nndc* directory. There are two ways to run tests. The first is to use the Makefile present in the source directory and run the following: diff --git a/developers.html b/developers.html index 45546f3779..1e6adcadea 100644 --- a/developers.html +++ b/developers.html @@ -15,7 +15,7 @@ + + + + + + + + + + +

+
+ +

+ «  8. Parallelization +   ::   + Contents +   ::   + User’s Guide  » +

+ +
+
+ + +
+

9. Nonlinear Diffusion Acceleration - Coarse Mesh Finite Difference

+

This page section discusses how nonlinear diffusion acceleration (NDA) using +coarse mesh finite difference (CMFD) is implemented into OpenMC. Before we get +into the theory, general notation for this section is discussed.

+
+

9.1. Notation

+

Before deriving NDA relationships, notation is explained. If a parameter has a +\overline{\cdot}, it is surface area-averaged and if it has a +\overline{\overline\cdot}, it is volume-averaged. When describing a +specific cell in the geometry, indices (i,j,k) are used which correspond +to directions (x,y,z). In most cases, the same operation is performed in +all three directions. To compactly write this, an arbitrary direction set +(u,v,w) that corresponds to cell indices (l,m,n) is used. Note +that u and l do not have to correspond to x and +i. However, if u and l correspond to y and +j, v and w correspond to x and z +directions. An example of this is shown in the following expression:

+
+

(1)\sum\limits_{u\in(x,y,z)}\left\langle\overline{J}^{u,g}_{l+1/2,m,n}
+\Delta_m^v\Delta_n^w\right\rangle

+

Here, u takes on each direction one at a time. The parameter J +is surface area-averaged over the transverse indices m and n +located at l+1/2. Usually, spatial indices are listed as subscripts and +the direction as a superscript. Energy group indices represented by g +and h are also listed as superscripts here. The group g is the +group of interest and, if present, h is all groups. Finally, any +parameter surrounded by \left\langle\cdot\right\rangle represents a +tally quantity that can be edited from a Monte Carlo (MC) solution.

+
+
+

9.2. Theory

+

NDA is a diffusion model that has equivalent physics to a transport model. There +are many different methods that can be classified as NDA. The CMFD method is a +type of NDA that represents second order multigroup diffusion equations on a +coarse spatial mesh. Whether a transport model or diffusion model is used to +represent the distribution of neutrons, these models must satisfy the neutron +balance equation. This balance is represented by the following formula for a +specific energy group g in cell (l,m,n):

+
+

(2)\sum\limits_{u\in(x,y,z)}\left(\left\langle\overline{J}^{u,g}_{l+1/2,m,n}
+\Delta_m^v\Delta_n^w\right\rangle -
+\left\langle\overline{J}^{u,g}_{l-1/2,m,n}
+\Delta_m^v\Delta_n^w\right\rangle\right)
++
+\left\langle\overline{\overline\Sigma}_{t_{l,m,n}}^g
+\overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle
+= \\
+\sum\limits_{h=1}^G\left\langle
+\overline{\overline{\nu_s\Sigma}}_{s_{l,m,n}}^{h\rightarrow
+g}\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w
+\right\rangle
++
+\frac{1}{k_{eff}}\sum\limits_{h=1}^G
+\left\langle\overline{\overline{\nu_f\Sigma}}_{f_{l,m,n}}^{h\rightarrow
+g}\overline{\overline\phi}_{l,m,n}^h
+\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle.

+

In eq. (2) the parameters are defined as:

+
    +
  • \left\langle\overline{J}^{u,g}_{l\pm
+1/2,m,n}\Delta_m^v\Delta_n^w\right\rangle — surface area-integrated net +current over surface (l\pm 1/2,m,n) with surface normal in direction +u in energy group g. By dividing this quantity by the transverse +area, \Delta_m^v\Delta_n^w, the surface area-averaged net current can +be computed.
  • +
  • \left\langle\overline{\overline\Sigma}_{t_{l,m,n}}^g
+\overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle +— volume-integrated total reaction rate over energy group g.
  • +
  • \left\langle\overline{\overline{\nu_s\Sigma}}_{s_{l,m,n}}^{h\rightarrow
+g}
+\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle +— volume-integrated scattering production rate of neutrons that begin with +energy in group h and exit reaction in group g. This reaction +rate also includes the energy transfer of reactions (except fission) that +produce multiple neutrons such as (n, 2n); hence, the need for \nu_s +to represent neutron multiplicity.
  • +
  • k_{eff} — core multiplication factor.
  • +
  • \left\langle\overline{\overline{\nu_f\Sigma}}_{f_{l,m,n}}^{h\rightarrow
+g}\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle +— volume-integrated fission production rate of neutrons from fissions in +group h that exit in group g.
  • +
+

Each quantity in \left\langle\cdot\right\rangle represents a scalar value that +is obtained from an MC tally. A good verification step when using an MC code is +to make sure that tallies satisfy this balance equation within statistics. No +NDA acceleration can be performed if the balance equation is not satisfied.

+

There are three major steps to consider when performing NDA: (1) calculation of +macroscopic cross sections and nonlinear parameters, (2) solving an eigenvalue +problem with a system of linear equations, and (3) modifying MC source +distribution to align with the NDA solution on a chosen mesh. This process is +illustrated as a flow chart below. After a batch of neutrons +is simulated, NDA can take place. Each of the steps described above is described +in detail in the following sections.

+
+

\begin{tikzpicture}
+	\matrix[every node/.style={draw, thick, minimum width=3cm, minimum height=1cm, align=center}, column sep=2cm, row sep=1cm] (m) {
+\node[draw, fill=red!40] (start) {Batch $i$ \\ tally NDA}; & \\
+ \node[draw, diamond, aspect=2, fill=green!40] (cmfd) {Run NDA?}; & \node[draw, fill=red!40] (end) {Batch $i + 1$ \\ tally NDA};  \\
+\node[draw, fill=blue!40] (xs) {Calculate XS \& DC}; & \node[draw, fill=blue!40] (modify) {Modify MC Source}; \\
+\node[draw, fill=blue!40] (nonlinear) {Calculate Equivalence}; &  \node[draw, fill=blue!40] (eqs) {Solve NDA eqs.};\\
+};
+
+\begin{scope}[every path/.style={->,very thick,draw}]
+        \draw (start.south) -- (cmfd.north);
+        \draw (cmfd.east)  -- node[above] {no} (end.west);
+        \draw (cmfd.south)  -- node[right] {yes} (xs.north);
+		\draw (xs.south)  --  (nonlinear.north);
+		\draw (nonlinear.east)  --  (eqs.west);
+		\draw (eqs.north)  --  (modify.south);
+		\draw (modify.north)  --  (end.south);
+    \end{scope}
+
+\end{tikzpicture}

+

Flow chart of NDA process. Note "XS" is used for cross section and + "DC" is used for diffusion coefficient.

+

9.2.1. Calculation of Macroscopic Cross Sections

+

A diffusion model needs macroscopic cross sections and diffusion coefficients to +solve for multigroup fluxes. Cross sections are derived by conserving reaction +rates predicted by MC tallies. From Eq. (2), total, scattering +production and fission production macroscopic cross sections are needed. They are +defined from MC tallies as follows:

+
+

(3)\overline{\overline\Sigma}_{t_{l,m,n}}^g \equiv
+\frac{\left\langle\overline{\overline\Sigma}_{t_{l,m,n}}^g
+\overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}
+{\left\langle\overline{\overline\phi}_{l,m,n}^g
+\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle},

+
+

(4)\overline{\overline{\nu_s\Sigma}}_{s_{l,m,n}}^{h\rightarrow g} \equiv
+\frac{\left\langle\overline{\overline{\nu_s\Sigma}}_{s_{l,m,n}}^{h\rightarrow
+g}\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}
+{\left\langle\overline{\overline\phi}_{l,m,n}^h
+\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}

+

and

+
+

(5)\overline{\overline{\nu_f\Sigma}}_{f_{l,m,n}}^{h\rightarrow g} \equiv
+\frac{\left\langle\overline{\overline{\nu_f\Sigma}}_{f_{l,m,n}}^{h\rightarrow
+g}\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}
+{\left\langle\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}.

+

In order to fully conserve neutron balance, leakage rates also need to be +preserved. In standard diffusion theory, leakage rates are represented by +diffusion coefficients. Unfortunately, it is not easy in MC to calculate a +single diffusion coefficient for a cell that describes leakage out of each +surface. Luckily, it does not matter what definition of diffusion coefficient is +used because nonlinear equivalence parameters will correct for this +inconsistency. However, depending on the diffusion coefficient definition +chosen, different convergence properties of NDA equations are observed. +Here, we introduce a diffusion coefficient that is derived for a coarse energy +transport reaction rate. This definition can easily be constructed from +MC tallies provided that angular moments of scattering reaction rates can +be obtained. The diffusion coefficient is defined as follows:

+
+

(6)\overline{\overline D}_{l,m,n}^g =
+ \frac{\left\langle\overline{\overline\phi}_{l,m,n}^g
+ \Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}{3
+ \left\langle\overline{\overline\Sigma}_{tr_{l,m,n}}^g
+ \overline{\overline\phi}_{l,m,n}^g
+ \Delta_l^u\Delta_m^v\Delta_n^w\right\rangle},

+

where

+
+

(7)\left\langle\overline{\overline\Sigma}_{tr_{l,m,n}}^g
+\overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle
+=
+\left\langle\overline{\overline\Sigma}_{t_{l,m,n}}^g
+\overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle
+\\ -
+\left\langle\overline{\overline{\nu_s\Sigma}}_{s1_{l,m,n}}^g
+\overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle.

+

Note that the transport reaction rate is calculated from the total reaction rate +reduced by the P_1 scattering production reaction rate. Equation (6) +does not represent the best definition of diffusion coefficients from MC; +however, it is very simple and usually fits into MC tally frameworks +easily. Different methods to calculate more accurate diffusion coefficients can +found in [Herman].

+
+
+

9.2.2. CMFD Equations

+

The first part of this section is devoted to discussing second-order finite +volume discretization of multigroup diffusion equations. This will be followed +up by the formulation of CMFD equations that are used in this NDA +scheme. When performing second-order finite volume discretization of the +diffusion equation, we need information that relates current to flux. In this +numerical scheme, each cell is coupled only to its direct neighbors. Therefore, +only two types of coupling exist: (1) cell-to-cell coupling and (2) +cell-to-boundary coupling. The derivation of this procedure is referred to as +finite difference diffusion equations and can be found in literature such +as [Hebert]. These current/flux relationships are as follows:

+
    +
  • cell-to-cell coupling
  • +
+
+

(8)\overline{J}^{u,g}_{l\pm1/2,m,n} = -\frac{2\overline{\overline
+D}_{l\pm1,m,n}^g\overline{\overline
+D}_{l,m,n}^g}{\overline{\overline D}_{l\pm1,m,n}^g\Delta_l^u +
+\overline{\overline
+D}_{l,m,n}^g\Delta_{l\pm1}^u}
+\left(\pm\overline{\overline{\phi}}_{l\pm1,m,n}^g\mp
+\overline{\overline{\phi}}_{l,m,n}^g\right),

+
    +
  • cell-to-boundary coupling
  • +
+
+

(9)\overline{J}^{u,g}_{l\pm1/2,m,n} = \pm\frac{2\overline{\overline
+D}_{l,m,n}^g\left(1 -
+\beta_{l\pm1/2,m,n}^{u,g}\right)}{4\overline{\overline
+D}_{l,m,n}^g\left(1 + \beta_{l\pm1/2,m,n}^{u,g}\right) + \left(1 -
+\beta_{l\pm1/2,m,n}^{u,g}\right)\Delta_l^u}\overline{\overline{\phi}}_{l,m,n}^{g}.

+

In Eqs. (8) and (9), the \pm refers to +left (-x) or right (+x) surface in the x direction, +back (-y) or front (+y) surface in the y direction and +bottom (-z) or top (+z) surface in the z direction. For +cell-to-boundary coupling, a general albedo, \beta_{l\pm1/2,m,n}^{u,g}, +is used. The albedo is defined as the ratio of incoming (- superscript) +to outgoing (+ superscript) partial current on any surface represented +as

+
+

(10)\beta_{l\pm1/2,m,n}^{u,g} =
+\frac{\overline{J}^{u,g-}_{l\pm1/2,m,n}}{\overline{J}^{u,g+}_{l\pm1/2,m,n}}.

+

Common boundary conditions are: vacuum (\beta=0), reflective +(\beta=1) and zero flux (\beta=-1). Both eq. (8) +and eq. (9) can be written in this generic form,

+
+

(11)\overline{J}^{u,g}_{l\pm1/2,m,n} = \widetilde{D}_{l,m,n}^{u,g} \left(\dots\right).

+

The parameter \widetilde{D}_{l,m,n}^{u,g} represents the linear +coupling term between current and flux. These current relationships can be +sustituted into eq. (2) to produce a linear system of multigroup +diffusion equations for each spatial cell and energy group. However, a solution +to these equations is not consistent with a higher order transport solution +unless equivalence factors are present. This is because both the diffusion +approximation, governed by Fick’s Law, and spatial trunction error will produce +differences. Therefore, a nonlinear parameter, +\widehat{D}_{l,m,n}^{u,g}, is added to eqs. (8) and +(9). These equations are, respectively,

+
+

(12)\overline{J}^{u,g}_{l\pm1/2,m,n} = -\widetilde{D}_{l,m,n}^{u,g}
+\left(\pm\overline{\overline{\phi}}_{l\pm1,m,n}^g\mp
+\overline{\overline{\phi}}_{l,m,n}^g\right) + \widehat{D}_{l,m,n}^{u,g}
+\left(\overline{\overline{\phi}}_{l\pm1,m,n}^g +
+\overline{\overline{\phi}}_{l,m,n}^g\right)

+

and

+
+

(13)\overline{J}^{u,g}_{l\pm1/2,m,n} = \pm\widetilde{D}_{l,m,n}^{u,g}
+\overline{\overline{\phi}}_{l,m,n}^{g} + \widehat{D}_{l,m,n}^{u,g}
+\overline{\overline{\phi}}_{l,m,n}^{g}.

+

The only unknown in each of these equations is the equivalence parameter. The +current, linear coupling term and flux can either be obtained or derived from +MC tallies. Thus, it is called nonlinear because it is dependent on the flux +which is updated on the next iteration.

+

Equations (12) and (13) can be substituted into +eq. (2) to create a linear system of equations that is consistent +with transport physics. One example of this equation is written for an +interior cell,

+
+

(14)\sum_{u\in
+x,y,x}\frac{1}{\Delta_l^u}\left[\left(-\tilde{D}_{l-1/2,m,n}^{u,g} -
+\hat{D}_{l-1/2,m,n}^{u,g}\right)\overline{\overline{\phi}}_{l-1,m,n}^g\right.
++ \left(\tilde{D}_{l-1/2,m,n}^{u,g} +
+\tilde{D}_{l+1/2,m,n}^{u,g} - \hat{D}_{l-1/2,m,n}^{u,g} +
+\hat{D}_{l+1/2,m,n}^{u,g}\right)\overline{\overline{\phi}}_{l,m,n}^g
+\\ +
+\left. \left(-\tilde{D}_{l+1/2,m,n}^{u,g} +
+\hat{D}_{l+1/2,m,n}^{u,g}\right)\overline{\overline{\phi}}_{l+1,m,n}^g
+\right] +
+\overline{\overline\Sigma}_{t_{l,m,n}}^g\overline{\overline{\phi}}_{l,m,n}^g
+- \sum\limits_{h=1}^G\overline{\overline{\nu_s\Sigma}}^{h\rightarrow
+g}_{s_{l,m,n}}\overline{\overline{\phi}}_{l,m,n}^h =
+\frac{1}{k}\sum\limits_{h=1}^G\overline{\overline{\nu_f\Sigma}}^{h\rightarrow
+g}_{f_{l,m,n}}\overline{\overline{\phi}}_{l,m,n}^h.

+

It should be noted that before substitution, eq. (2) was divided +by the volume of the cell, \Delta_l^u\Delta_m^v\Delta_n^w. Equation +(14) can be represented in operator form as

+
+

(15)\mathbb{M}\mathbf{\Phi} = \frac{1}{k}\mathbb{F}\mathbf{\Phi},

+

where \mathbb{M} is the neutron loss matrix operator, +\mathbb{F} is the neutron production matrix operator, +\mathbf{\Phi} is the multigroup flux vector and k is the +eigenvalue. This generalized eigenvalue problem is solved to obtain fundamental +mode multigroup fluxes and eigenvalue. In order to produce consistent results +with transport theory from these equations, the neutron balance equation must +have been satisfied by MC tallies. The desire is that CMFD equations will +produce a more accurate source than MC after each fission source generation.

+
+
+

9.2.3. CMFD Feedback

+

Now that a more accurate representation of the expected source distribution is +estimated from CMFD, it needs to be communicated back to MC. The first step +in this process is to generate a probability mass function that provides +information about how probable it is for a neutron to be born in a given cell +and energy group. This is represented as

+
+

(16)p_{l,m,n}^g =
+\frac{\sum_{h=1}^{G}\overline{\overline{\nu_f\Sigma}}^{h\rightarrow
+g}_{f_{l,m,n}}\overline{\overline{\phi}}_{l,m,n}^h\Delta_l^u\Delta_m^v
+\Delta_n^w}{\sum_n\sum_m\sum_l\sum_{h=1}^{G}\overline{
+\overline{\nu_f\Sigma}}^{h\rightarrow
+g}_{f_{l,m,n}}\overline{\overline{\phi}}_{l,m,n}^h\Delta_l^u\Delta_m^v
+\Delta_n^w}.

+

This equation can be multiplied by the number of source neutrons to obtain an +estimate of the expected number of neutrons to be born in a given cell and +energy group. This distribution can be compared to the MC source distribution +to generate weight adjusted factors defined as

+
+

(17)f_{l,m,n}^g = \frac{Np_{l,m,n}^g}{\sum\limits_s w_s};\quad s\in
+\left(g,l,m,n\right).

+

The MC source distribution is represented on the same coarse mesh as +CMFD by summing all neutrons’ weights, w_s, in a given cell and +energy group. MC source weights can then be modified by this weight +adjustment factor so that it matches the CMFD solution on the coarse +mesh,

+
+

(18)w^\prime_s = w_s\times f_{l,m,n}^g;\quad s\in \left(g,l,m,n\right).

+

It should be noted that heterogeneous information about local coordinates and +energy remain constant throughout this modification process.

+
+
+
+

9.3. Implementation in OpenMC

+

The section describes how CMFD was implemented in OpenMC. Before the simulation +begins, a user sets up a CMFD input file that contains the following basic +information:

+
    +
  • CMFD mesh (space and energy),
  • +
  • boundary conditions at edge of mesh (albedos),
  • +
  • acceleration region (subset of mesh, optional),
  • +
  • fission source generation (FSG)/batch that CMFD should begin, and
  • +
  • whether CMFD feedback should be applied.
  • +
+

It should be noted that for more difficult simulations (e.g., light water +reactors), there are other options available to users such as tally resetting +parameters, effective down-scatter usage, tally estimator, etc. For more +information please see CMFD Specification – cmfd.xml.

+

Of the options described above, the optional acceleration subset region is an +uncommon feature. Because OpenMC only has a structured Cartesian mesh, mesh +cells may overlay regions that don’t contain fissionable material and may be so +far from the core that the neutron flux is very low. If these regions were +included in the CMFD solution, bad estimates of diffusion parameters may result +and affect CMFD feedback. To deal with this, a user can carve out an active +acceleration region from their structured Cartesian mesh. This is illustrated +in diagram below. When placing a CMFD mesh over a geometry, the boundary +conditions must be known at the global edges of the mesh. If the geometry is +complex like the one below, one may have to cover the whole geometry including +the reactor pressure vessel because we know that there is a zero incoming +current boundary condition at the outer edge of the pressure vessel. This is +not viable in practice because neutrons in simulations may not reach mesh cells +that are near the pressure vessel. To circumvent this, one can shrink the mesh +to cover just the core region as shown in the diagram. However, one must still +estimate the boundary conditions at the global boundaries, but at these +locations, they are not readily known. In OpenMC, one can carve out the active +core region from the entire structured Cartesian mesh. This is shown in the +diagram below by the darkened region over the core. The albedo boundary +conditions at the active core/reflector boundary can be tallied indirectly +during the MC simulation with incoming and outgoing partial currents. This +allows the user to not have to worry about neutrons producing adequate tallies +in mesh cells far away from the core.

+
+

% these dimensions are determined in arrow_dimms.ods
+
+    \def\scale{1.0}
+
+    \def\latWidth{0.2808363589*\scale}
+    
+    \def\RPVOR{3*\scale}
+    \def\rectW{0.75*\scale}
+    \def\RPVIR{2.8694005485*\scale}
+    \def\BarrelIR{2.4547472901*\scale}
+    \def\BarrelOR{2.5293848766*\scale}
+    \def\ShieldOR{2.6040224631*\scale}
+
+    \def\bafCIRx{0.9829272561*\scale}
+    \def\bafCIRy{2.1062726917*\scale}
+    \def\bafCORx{1.0119529842*\scale}
+    \def\bafCORy{2.1352984197*\scale}
+    \def\bafMIRx{1.8254363328*\scale}
+    \def\bafMIRy{1.5445999739*\scale}
+    \def\bafMORx{1.8544620609*\scale}
+    \def\bafMORy{1.573625702*\scale}
+    
+    \tikzset{Assembly/.style={
+        inner sep=0pt,
+        text width=\latWidth in,
+        minimum size=\latWidth in,
+        draw=black,
+        align=center
+        }
+    }
+    
+    \def\tkzRPV{(0,0) circle (\RPVIR) (0,0) circle (\RPVOR)}
+    \def\tkzBarrel{(0,0) circle (\BarrelIR) (0,0) circle (\BarrelOR)}
+    \def\tkzShields{(0,0) circle (\BarrelOR) (0,0) circle (\ShieldOR)}
+    
+    \def\tkzBaffCOR{(-\bafCORx, -\bafCORy) rectangle (\bafCORx, \bafCORy)}
+    \def\tkzBaffCIR{(-\bafCIRx, -\bafCIRy) rectangle (\bafCIRx, \bafCIRy)} 
+    \def\tkzBaffMOR{(-\bafMORx, -\bafMORy) rectangle (\bafMORx, \bafMORy)}
+    \def\tkzBaffMIR{(-\bafMIRx, -\bafMIRy) rectangle (\bafMIRx, \bafMIRy) }
+    \def\tkzBaffleC{ \tkzBaffCIR \tkzBaffCOR }
+    \def\tkzBaffleM{ \tkzBaffMIR \tkzBaffMOR }
+
+    \def\tkzBaffCClip{\tkzBaffCIR (-\RPVOR, -\RPVOR) rectangle (\RPVOR, \RPVOR)}
+    \def\tkzBaffMClip{\tkzBaffMIR (-\RPVOR, -\RPVOR) rectangle (\RPVOR, \RPVOR)}
+
+    \def\highenr{blue!50}
+    \def\midenr{yellow!50}
+    \def\lowenr{red!50}
+    \def\lightgray{black!25}
+    \def\darkgray{black!80}
+
+      \begin{tikzpicture}[x=1in,y=1in, xshift=3in]
+\scalebox{0.6}{
+        % draw RPV, barrel, and shield panels
+        
+        \path[fill=black,even odd rule] \tkzRPV;
+        \path[fill=black,even odd rule] \tkzBarrel;
+        \begin{scope}
+          \clip[rotate around={45:(0,0)}] (-\RPVOR, -\rectW) rectangle (\RPVOR, \rectW) (-\rectW, \RPVOR) rectangle (\rectW, -\RPVOR);
+          \path[fill=black,even odd rule] \tkzShields;
+        \end{scope}
+
+        
+        % draw assembly row/column headers
+        
+        \draw[red, thick] ($(-7*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {R} -- ($(-7*\latWidth,4*\latWidth)$);
+        \draw[red, thick] ($(-6*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {P} -- ($(-6*\latWidth,6*\latWidth)$);
+        \draw[red, thick] ($(-5*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {N} -- ($(-5*\latWidth,7*\latWidth)$);
+        \draw[red, thick] ($(-4*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {M} -- ($(-4*\latWidth,7*\latWidth)$);
+        \draw[red, thick] ($(-3*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {L} -- ($(-3*\latWidth,8*\latWidth)$);
+        \draw[red, thick] ($(-2*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {K} -- ($(-2*\latWidth,8*\latWidth)$);
+        \draw[red, thick] ($(-1*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {J} -- ($(-1*\latWidth,8*\latWidth)$);
+        \draw[red, thick] ($(-0*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {H} -- ($(-0*\latWidth,8*\latWidth)$);
+        \draw[red, thick] ($(1*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {G} -- ($(1*\latWidth,8*\latWidth)$);
+        \draw[red, thick] ($(2*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {F} -- ($(2*\latWidth,8*\latWidth)$);
+        \draw[red, thick] ($(3*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {E} -- ($(3*\latWidth,8*\latWidth)$);
+        \draw[red, thick] ($(4*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {D} -- ($(4*\latWidth,7*\latWidth)$);
+        \draw[red, thick] ($(5*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {C} -- ($(5*\latWidth,7*\latWidth)$);
+        \draw[red, thick] ($(6*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {B} -- ($(6*\latWidth,6*\latWidth)$);
+        \draw[red, thick] ($(7*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[above, anchor=south] {A} -- ($(7*\latWidth,4*\latWidth)$);
+        
+        \begin{scope}[rotate=90]
+          \draw[red, thick] ($(-7*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {15} -- ($(-7*\latWidth,4*\latWidth)$);
+          \draw[red, thick] ($(-6*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {14} -- ($(-6*\latWidth,6*\latWidth)$);
+          \draw[red, thick] ($(-5*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {13} -- ($(-5*\latWidth,7*\latWidth)$);
+          \draw[red, thick] ($(-4*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {12} -- ($(-4*\latWidth,7*\latWidth)$);
+          \draw[red, thick] ($(-3*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {11} -- ($(-3*\latWidth,8*\latWidth)$);
+          \draw[red, thick] ($(-2*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {10} -- ($(-2*\latWidth,8*\latWidth)$);
+          \draw[red, thick] ($(-1*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {9} -- ($(-1*\latWidth,8*\latWidth)$);
+          \draw[red, thick] ($(-0*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {8} -- ($(-0*\latWidth,8*\latWidth)$);
+          \draw[red, thick] ($(1*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {7} -- ($(1*\latWidth,8*\latWidth)$);
+          \draw[red, thick] ($(2*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {6} -- ($(2*\latWidth,8*\latWidth)$);
+          \draw[red, thick] ($(3*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {5} -- ($(3*\latWidth,8*\latWidth)$);
+          \draw[red, thick] ($(4*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {4} -- ($(4*\latWidth,7*\latWidth)$);
+          \draw[red, thick] ($(5*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {3} -- ($(5*\latWidth,7*\latWidth)$);
+          \draw[red, thick] ($(6*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {2} -- ($(6*\latWidth,6*\latWidth)$);
+          \draw[red, thick] ($(7*\latWidth,\RPVOR/\latWidth*\latWidth)$) node[left, anchor=east] {1} -- ($(7*\latWidth,4*\latWidth)$);
+        \end{scope}
+        
+        % draw fuel assembly nodes
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-7*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-6*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-5*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-4*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-3*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-2*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-1*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-0*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 1*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 2*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 3*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 4*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 5*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 6*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 7*\latWidth,8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,8*\latWidth)$) {};
+        
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-7*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-6*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-5*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-4*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-3*\latWidth,7*\latWidth)$) {}; % L1
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-2*\latWidth,7*\latWidth)$) {6}; % K1
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-1*\latWidth,7*\latWidth)$) {}; % J1
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-0*\latWidth,7*\latWidth)$) {6}; % H1
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 1*\latWidth,7*\latWidth)$) {}; % G1
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 2*\latWidth,7*\latWidth)$) {6}; % F1
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 3*\latWidth,7*\latWidth)$) {}; % E1
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 4*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 5*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 6*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 7*\latWidth,7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,7*\latWidth)$) {};
+
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-7*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-6*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-5*\latWidth,6*\latWidth)$) {}; % N2
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-5*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-4*\latWidth,6*\latWidth)$) {}; % M2
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-4*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-3*\latWidth,6*\latWidth)$) {16}; % L2
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-2*\latWidth,6*\latWidth)$) {}; % K2
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-1*\latWidth,6*\latWidth)$) {20}; % J2
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-0*\latWidth,6*\latWidth)$) {}; % H2
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 1*\latWidth,6*\latWidth)$) {20}; % G2
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 2*\latWidth,6*\latWidth)$) {}; % F2
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 3*\latWidth,6*\latWidth)$) {16}; % E2
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 4*\latWidth,6*\latWidth)$) {}; % D2
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 4*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 5*\latWidth,6*\latWidth)$) {}; % C2
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 5*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 6*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 7*\latWidth,6*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,6*\latWidth)$) {};
+
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-7*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-6*\latWidth,5*\latWidth)$) {}; % P3
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-6*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-5*\latWidth,5*\latWidth)$) {15}; % N3
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-5*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-4*\latWidth,5*\latWidth)$) {16}; % M3
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-4*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-3*\latWidth,5*\latWidth)$) {}; % L3
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-2*\latWidth,5*\latWidth)$) {16}; % K3
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-1*\latWidth,5*\latWidth)$) {}; % J3
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-0*\latWidth,5*\latWidth)$) {16}; % H3
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 1*\latWidth,5*\latWidth)$) {}; % G3
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 2*\latWidth,5*\latWidth)$) {16}; % F3
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 3*\latWidth,5*\latWidth)$) {}; % E3
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 4*\latWidth,5*\latWidth)$) {16}; % D3
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 4*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 5*\latWidth,5*\latWidth)$) {15}; % C3
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 5*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 6*\latWidth,5*\latWidth)$) {}; % B3
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 6*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 7*\latWidth,5*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,5*\latWidth)$) {};
+
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-7*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-6*\latWidth,4*\latWidth)$) {}; % P4
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-6*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-5*\latWidth,4*\latWidth)$) {16}; % N4
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-5*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-4*\latWidth,4*\latWidth)$) {}; % M4
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-4*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-3*\latWidth,4*\latWidth)$) {16}; % L4
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-2*\latWidth,4*\latWidth)$) {}; % K4
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-1*\latWidth,4*\latWidth)$) {12}; % J4
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-0*\latWidth,4*\latWidth)$) {}; % H4
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 1*\latWidth,4*\latWidth)$) {12}; % G4
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 2*\latWidth,4*\latWidth)$) {}; % F4
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 3*\latWidth,4*\latWidth)$) {16}; % E4
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 4*\latWidth,4*\latWidth)$) {}; % D4
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 4*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 5*\latWidth,4*\latWidth)$) {16}; % C4
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 5*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 6*\latWidth,4*\latWidth)$) {}; % B4
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 6*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 7*\latWidth,4*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,4*\latWidth)$) {};
+
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-7*\latWidth,3*\latWidth)$) {}; % R5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-7*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-6*\latWidth,3*\latWidth)$) {16}; % P5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-6*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-5*\latWidth,3*\latWidth)$) {}; % N5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-5*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-4*\latWidth,3*\latWidth)$) {16}; % M5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-4*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-3*\latWidth,3*\latWidth)$) {}; % L5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-2*\latWidth,3*\latWidth)$) {12}; % K5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-1*\latWidth,3*\latWidth)$) {}; % J5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-0*\latWidth,3*\latWidth)$) {12}; % H5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 1*\latWidth,3*\latWidth)$) {}; % G5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 2*\latWidth,3*\latWidth)$) {12}; % F5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 3*\latWidth,3*\latWidth)$) {}; % E5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 4*\latWidth,3*\latWidth)$) {16}; % D5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 4*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 5*\latWidth,3*\latWidth)$) {}; % C5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 5*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 6*\latWidth,3*\latWidth)$) {16}; % B5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 6*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 7*\latWidth,3*\latWidth)$) {}; % A5
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 7*\latWidth,3*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,3*\latWidth)$) {};
+
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-7*\latWidth,2*\latWidth)$) {6}; % R6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-7*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-6*\latWidth,2*\latWidth)$) {}; % P6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-6*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-5*\latWidth,2*\latWidth)$) {16}; % N6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-5*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-4*\latWidth,2*\latWidth)$) {}; % M6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-4*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-3*\latWidth,2*\latWidth)$) {12}; % L6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-2*\latWidth,2*\latWidth)$) {}; % K6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-1*\latWidth,2*\latWidth)$) {12}; % J6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-0*\latWidth,2*\latWidth)$) {}; % H6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 1*\latWidth,2*\latWidth)$) {12}; % G6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 2*\latWidth,2*\latWidth)$) {}; % F6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 3*\latWidth,2*\latWidth)$) {12}; % E6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 4*\latWidth,2*\latWidth)$) {}; % D6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 4*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 5*\latWidth,2*\latWidth)$) {16}; % C6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 5*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 6*\latWidth,2*\latWidth)$) {}; % B6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 6*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 7*\latWidth,2*\latWidth)$) {6}; % A6
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 7*\latWidth,2*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,2*\latWidth)$) {};
+
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-7*\latWidth,1*\latWidth)$) {}; % R7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-7*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-6*\latWidth,1*\latWidth)$) {20}; % P7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-6*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-5*\latWidth,1*\latWidth)$) {}; % N7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-5*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-4*\latWidth,1*\latWidth)$) {12}; % M7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-4*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-3*\latWidth,1*\latWidth)$) {}; % L7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-2*\latWidth,1*\latWidth)$) {12}; % K7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-1*\latWidth,1*\latWidth)$) {}; % J7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-0*\latWidth,1*\latWidth)$) {16}; % H7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 1*\latWidth,1*\latWidth)$) {}; % G7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 2*\latWidth,1*\latWidth)$) {12}; % F7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 3*\latWidth,1*\latWidth)$) {}; % E7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 4*\latWidth,1*\latWidth)$) {12}; % D7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 4*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 5*\latWidth,1*\latWidth)$) {}; % C7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 5*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 6*\latWidth,1*\latWidth)$) {20}; % B7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 6*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 7*\latWidth,1*\latWidth)$) {}; % A7
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 7*\latWidth,1*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,1*\latWidth)$) {};
+
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-7*\latWidth,0*\latWidth)$) {6}; % R8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-7*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-6*\latWidth,0*\latWidth)$) {}; % P8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-6*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-5*\latWidth,0*\latWidth)$) {16}; % N8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-5*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-4*\latWidth,0*\latWidth)$) {}; % M8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-4*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-3*\latWidth,0*\latWidth)$) {12}; % L8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-2*\latWidth,0*\latWidth)$) {}; % K8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-1*\latWidth,0*\latWidth)$) {16}; % J8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-0*\latWidth,0*\latWidth)$) {}; % H8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 1*\latWidth,0*\latWidth)$) {16}; % G8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 2*\latWidth,0*\latWidth)$) {}; % F8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 3*\latWidth,0*\latWidth)$) {12}; % E8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 4*\latWidth,0*\latWidth)$) {}; % D8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 4*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 5*\latWidth,0*\latWidth)$) {16}; % C8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 5*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 6*\latWidth,0*\latWidth)$) {}; % B8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 6*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 7*\latWidth,0*\latWidth)$) {6}; % A8
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 7*\latWidth,0*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,0*\latWidth)$) {};
+
+
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-7*\latWidth,-1*\latWidth)$) {}; % R9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-7*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-6*\latWidth,-1*\latWidth)$) {20}; % P9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-6*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-5*\latWidth,-1*\latWidth)$) {}; % N9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-5*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-4*\latWidth,-1*\latWidth)$) {12}; % M9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-4*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-3*\latWidth,-1*\latWidth)$) {}; % L9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-2*\latWidth,-1*\latWidth)$) {12}; % K9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-1*\latWidth,-1*\latWidth)$) {}; % J9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-0*\latWidth,-1*\latWidth)$) {16}; % H9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 1*\latWidth,-1*\latWidth)$) {}; % G9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 2*\latWidth,-1*\latWidth)$) {12}; % F9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 3*\latWidth,-1*\latWidth)$) {}; % E9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 4*\latWidth,-1*\latWidth)$) {12}; % D9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 4*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 5*\latWidth,-1*\latWidth)$) {}; % C9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 5*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 6*\latWidth,-1*\latWidth)$) {20}; % B9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 6*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 7*\latWidth,-1*\latWidth)$) {}; % A9
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 7*\latWidth,-1*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,-1*\latWidth)$) {};
+
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-7*\latWidth,-2*\latWidth)$) {6}; % R10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-7*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-6*\latWidth,-2*\latWidth)$) {}; % P10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-6*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-5*\latWidth,-2*\latWidth)$) {16}; % N10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-5*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-4*\latWidth,-2*\latWidth)$) {}; % M10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-4*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-3*\latWidth,-2*\latWidth)$) {12}; % L10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-2*\latWidth,-2*\latWidth)$) {}; % K10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-1*\latWidth,-2*\latWidth)$) {12}; % J10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-0*\latWidth,-2*\latWidth)$) {}; % H10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 1*\latWidth,-2*\latWidth)$) {12}; % G10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 2*\latWidth,-2*\latWidth)$) {}; % F10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 3*\latWidth,-2*\latWidth)$) {12}; % E10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 4*\latWidth,-2*\latWidth)$) {}; % D10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 4*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 5*\latWidth,-2*\latWidth)$) {16}; % C10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 5*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 6*\latWidth,-2*\latWidth)$) {}; % B10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 6*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 7*\latWidth,-2*\latWidth)$) {6}; % A10
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 7*\latWidth,-2*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,-2*\latWidth)$) {};
+
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-7*\latWidth,-3*\latWidth)$) {}; % R11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-7*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-6*\latWidth,-3*\latWidth)$) {16}; % P11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-6*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-5*\latWidth,-3*\latWidth)$) {}; % N11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-5*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-4*\latWidth,-3*\latWidth)$) {16}; % M11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-4*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-3*\latWidth,-3*\latWidth)$) {}; % L11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-2*\latWidth,-3*\latWidth)$) {12}; % K11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-1*\latWidth,-3*\latWidth)$) {}; % J11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-0*\latWidth,-3*\latWidth)$) {12}; % H11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 1*\latWidth,-3*\latWidth)$) {}; % G11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 2*\latWidth,-3*\latWidth)$) {12}; % F11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 3*\latWidth,-3*\latWidth)$) {}; % E11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 4*\latWidth,-3*\latWidth)$) {16}; % D11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 4*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 5*\latWidth,-3*\latWidth)$) {}; % C11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 5*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 6*\latWidth,-3*\latWidth)$) {16}; % B11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 6*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 7*\latWidth,-3*\latWidth)$) {}; % A11
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 7*\latWidth,-3*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,-3*\latWidth)$) {};
+
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-7*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-6*\latWidth,-4*\latWidth)$) {}; % P12
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-6*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-5*\latWidth,-4*\latWidth)$) {16}; % N12
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-5*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-4*\latWidth,-4*\latWidth)$) {}; % M12
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-4*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-3*\latWidth,-4*\latWidth)$) {16}; % L12
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-2*\latWidth,-4*\latWidth)$) {}; % K12
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-1*\latWidth,-4*\latWidth)$) {12}; % J12
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-0*\latWidth,-4*\latWidth)$) {}; % H12
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 1*\latWidth,-4*\latWidth)$) {12}; % G12
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 2*\latWidth,-4*\latWidth)$) {}; % F12
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 3*\latWidth,-4*\latWidth)$) {16}; % E12
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 4*\latWidth,-4*\latWidth)$) {}; % D12
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 4*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 5*\latWidth,-4*\latWidth)$) {16}; % C12
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 5*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 6*\latWidth,-4*\latWidth)$) {}; % B12
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 6*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 7*\latWidth,-4*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,-4*\latWidth)$) {};
+
+
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-7*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-6*\latWidth,-5*\latWidth)$) {}; % P13
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-6*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-5*\latWidth,-5*\latWidth)$) {15}; % N13
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-5*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-4*\latWidth,-5*\latWidth)$) {16}; % M13
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-4*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-3*\latWidth,-5*\latWidth)$) {}; % L13
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-2*\latWidth,-5*\latWidth)$) {16}; % K13
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-1*\latWidth,-5*\latWidth)$) {}; % J13
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($(-0*\latWidth,-5*\latWidth)$) {16}; % H13
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 1*\latWidth,-5*\latWidth)$) {}; % G13
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 2*\latWidth,-5*\latWidth)$) {16}; % F13
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 3*\latWidth,-5*\latWidth)$) {}; % E13
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\midenr] at ($( 4*\latWidth,-5*\latWidth)$) {16}; % D13
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 4*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 5*\latWidth,-5*\latWidth)$) {15}; % C13
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 5*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 6*\latWidth,-5*\latWidth)$) {}; % B13
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 6*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 7*\latWidth,-5*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,-5*\latWidth)$) {};
+
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-7*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-6*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-5*\latWidth,-6*\latWidth)$) {}; % N14
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-5*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-4*\latWidth,-6*\latWidth)$) {}; % M14
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-4*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-3*\latWidth,-6*\latWidth)$) {16}; % L14
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-2*\latWidth,-6*\latWidth)$) {}; % K14
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-1*\latWidth,-6*\latWidth)$) {20}; % J14
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($(-0*\latWidth,-6*\latWidth)$) {}; % H14
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 1*\latWidth,-6*\latWidth)$) {20}; % G14
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\lowenr] at ($( 2*\latWidth,-6*\latWidth)$) {}; % F14
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 3*\latWidth,-6*\latWidth)$) {16}; % E14
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 4*\latWidth,-6*\latWidth)$) {}; % D14
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 4*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 5*\latWidth,-6*\latWidth)$) {}; % C14
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 5*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 6*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 7*\latWidth,-6*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,-6*\latWidth)$) {};
+
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-7*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-6*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-5*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-4*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-3*\latWidth,-7*\latWidth)$) {}; % L15
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-3*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-2*\latWidth,-7*\latWidth)$) {6}; % K15
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-2*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-1*\latWidth,-7*\latWidth)$) {}; % J15
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-1*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($(-0*\latWidth,-7*\latWidth)$) {6}; % H15
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($(-0*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 1*\latWidth,-7*\latWidth)$) {}; % G15
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 1*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 2*\latWidth,-7*\latWidth)$) {6}; % F15
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 2*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\highenr] at ($( 3*\latWidth,-7*\latWidth)$) {}; % E15
+        \node [Assembly, fill=\darkgray, opacity=0.7] at ($( 3*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 4*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 5*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 6*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 7*\latWidth,-7*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,-7*\latWidth)$) {};
+        
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-8*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-7*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-6*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-5*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-4*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-3*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-2*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-1*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($(-0*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 1*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 2*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 3*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 4*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 5*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 6*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 7*\latWidth,-8*\latWidth)$) {};
+        \node [Assembly, fill=\lightgray, opacity=0.3] at ($( 8*\latWidth,-8*\latWidth)$) {};
+
+        % draw baffle north/south
+        
+        \begin{scope}[even odd rule]
+          \clip[rotate=90] \tkzBaffMClip;
+          \path[fill=black] \tkzBaffleC;
+        \end{scope}
+        \begin{scope}[even odd rule]
+          \clip \tkzBaffCClip;
+          \clip \tkzBaffMClip;
+          \path[fill=black, rotate=90] \tkzBaffleM;
+        \end{scope}
+        
+        % draw baffle east/west
+        
+        \begin{scope}[rotate=90]
+          \begin{scope}[even odd rule]
+            \clip[rotate=90] \tkzBaffMClip;
+            \path[fill=black] \tkzBaffleC;
+          \end{scope}
+          \begin{scope}[even odd rule]
+            \clip \tkzBaffCClip;
+            \clip \tkzBaffMClip;
+            \path[fill=black, rotate=90] \tkzBaffleM;
+          \end{scope}
+        \end{scope}}
+      \end{tikzpicture}

+

Diagram of CMFD acceleration mesh

During an MC simulation, CMFD tallies are accumulated. The basic tallies needed +are listed in Table OpenMC CMFD tally list. Each tally is performed on a spatial and +energy mesh basis. The surface area-integrated net current is tallied on every +surface of the mesh. OpenMC tally objects are created by the CMFD code +internally, and cross sections are calculated at each CMFD feedback iteration. +The first CMFD iteration, controlled by the user, occurs just after tallies are +communicated to the master processor. Once tallies are collapsed, cross +sections, diffusion coefficients and equivalence parameters are calculated. This +is performed only on the acceleration region if that option has been activated +by the user. Once all diffusion parameters are calculated, CMFD matrices are +formed where energy groups are the inner most iteration index. In OpenMC, +compressed row storage sparse matrices are used due to the sparsity of CMFD +operators. An example of this sparsity is shown for the 3-D BEAVRS model in +figures Sparsity of Neutron Loss Operator and Sparsity of Neutron Production Operator [BEAVRS]. These matrices represent +an assembly radial mesh, 24 cell mesh in the axial direction and two energy +groups. The loss matrix is 99.92% sparse and the production matrix is 99.99% +sparse. Although the loss matrix looks like it is tridiagonal, it is really a +seven banded matrix with a block diagonal matrix for scattering. The production +matrix is a 2\times 2 block diagonal; however, zeros are present because +no fission neutrons appear with energies in the thermal group.

+ + +++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
OpenMC CMFD tally list
   
tallyscorefilter
\left\langle\overline{\overline\phi}_{l,m,n}^g
+\Delta_l^u\Delta_m^v\Delta_n^w\right\ranglefluxmesh, energy
\left\langle\overline{\overline\Sigma}_{t_{l,m,n}}^g
+\overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangletotalmesh, energy
\left\langle\overline{\overline{\nu_s\Sigma}}_{s1_{l,m,n}}^g
+\overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\ranglenu-scatter-1mesh, energy
\left\langle\overline{\overline{\nu_s\Sigma}}_{s_{l,m,n}}^{h\rightarrow g}
+\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\ranglenu-scattermesh, energy, energyout
\left\langle\overline{\overline{\nu_f\Sigma}}_{f_{l,m,n}}^{h\rightarrow g}
+\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\ranglenu-fissionmesh, energy, energyout
\left\langle\overline{J}^{u,g}_{l\pm 1/2,m,n}\Delta_m^v\Delta_n^w\right\ranglecurrentmesh, energy
+
+../_images/loss.png +

Sparsity of Neutron Loss Operator

+
+
+../_images/prod.png +

Sparsity of Neutron Production Operator

+
+

To solve the eigenvalue problem with these matrices, different source iteration +and linear solvers can be used. The most common source iteration solver used is +standard power iteration as described in [Gill]. To accelerate these source +iterations, a Wielandt shift scheme can be used as discussed in [Park]. PETSc +solvers were first implemented to perform the linear solution in parallel that +occurs once per source iteration. When using PETSc, different types of parallel +linear solvers and preconditioners can be used. By default, OpenMC uses an +incomplete LU preconditioner and a GMRES Krylov solver. After some initial +studies of parallelization with PETSc, it was observed that because CMFD +matrices are very sparse, solution times do not scale well. An additional +Gauss-Seidel linear solver with Chebyshev acceleration was added that is +similar to the one used for CMFD in CASMO [Rhodes] and [Smith]. This solver +was implemented with a custom section for two energy groups. Because energy +group is the inner most index, a block diagonal is formed when using more than +one group. For two groups, it is easy to invert this diagonal analytically +inside the Gauss-Seidel iterative solver. For more than two groups, this +analytic inversion can still be performed, but with more computational effort. +A standard Gauss-Seidel solver is used for more than two groups.

+

Besides a power iteration, a Jacobian-free Newton-Krylov method was also +implemented to obtain eigenvalue and multigroup fluxes as described in [Gill] +and [Knoll]. This method is not the primary one used, but has gotten recent +attention due to its coupling advantages to other physics such as thermal +hydraulics. Once multigroup fluxes are obtained, a normalized fission source is +calculated in the code using eq. (16) directly.

+

The next step in the process is to compute weight adjustment factors. These are +calculated by taking the ratio of the expected number of neutrons from the CMFD +source distribution to the current number of neutrons in each mesh. It is +straightforward to compute the CMFD number of neutrons because it is the +product between the total starting initial weight of neutrons and the CMFD +normalized fission source distribution. To compute the number of neutrons from +the current MC source, OpenMC sums the statistical +weights of neutrons from the source bank on a given spatial and energy mesh. +Once weight adjustment factors were calculated, each neutron’s statistical +weight in the source bank was modified according to its location and energy. +Examples of CMFD simulations using OpenMC can be found in [Herman_Thesis].

+
+
+

9.4. References

+ + + + + +
[BEAVRS]Nick Horelik, Bryan Herman. Benchmark for Evaluation And Verification of Reactor +Simulations. Massachusetts Institute of Technology, http://crpg.mit.edu/pub/beavrs +, 2013.
+ + + + + +
[Gill](1, 2) Daniel F. Gill. Newton-Krylov methods for the solution of the k-eigenvalue problem in +multigroup neutronics calculations. Ph.D. thesis, Pennsylvania State University, 2010.
+ + + + + +
[Hebert]Alain Hebert. Applied reactor physics. Presses Internationales Polytechnique, +Montreal, 2009.
+ + + + + +
[Herman]Bryan R. Herman, Benoit Forget, Kord Smith, and Brian N. Aviles. Improved +diffusion coefficients generated from Monte Carlo codes. In Proceedings of M&C +2013, Sun Valley, ID, USA, May 5 - 9, 2013.
+ + + + + +
[Herman_Thesis]Bryan R. Herman. Monte Carlo and Thermal Hydraulic Coupling using +Low-Order Nonlinear Diffusion Acceleration. Sc.D. thesis, +Massachusetts Institute of Technology, 2014.
+ + + + + +
[Knoll]D.A. Knoll, H. Park, and C. Newman. Acceleration of k-eigenvalue/criticality +calculations using the Jacobian-free Newton-Krylov method. Nuclear Science and +Engineering, 167:133–140, 2011.
+ + + + + +
[Park]H. Park, D.A. Knoll, and C.K. Newman. Nonlinear acceleration of transport +criticality problems. Nuclear Science and Engineering, 172:52–65, 2012.
+ + + + + +
[Rhodes]Joel Rhodes and Malte Edenius. CASMO-4 — A Fuel Assembly Burnup Program. +User’s Manual. Studsvik of America, ssp-09/443-u rev 0, proprietary edition, 2001.
+ + + + + +
[Smith]Kord S Smith and Joel D Rhodes III. Full-core, 2-D, LWR core calculations with +CASMO-4E. In Proceedings of PHYSOR 2002, Seoul, Korea, October 7 - 10, 2002.
+
+
+ + +
+
+ +

+ «  8. Parallelization +   ::   + Contents +   ::   + User’s Guide  » +

+ +
+ + + + + + + \ No newline at end of file diff --git a/methods/cross_sections.html b/methods/cross_sections.html index a6f8adfa94..31b16594e2 100644 --- a/methods/cross_sections.html +++ b/methods/cross_sections.html @@ -15,7 +15,7 @@ - + @@ -43,7 +43,7 @@   ::   Contents   ::   - User’s Guide  » + 9. Nonlinear Diffusion Acceleration - Coarse Mesh Finite Difference  »

@@ -517,7 +517,7 @@ Radiation Penetration Calculations on a Parallel Computer,”   ::   Contents   ::   - User’s Guide  » + 9. Nonlinear Diffusion Acceleration - Coarse Mesh Finite Difference  »

diff --git a/methods/physics.html b/methods/physics.html index af261d6919..d1d9a9e847 100644 --- a/methods/physics.html +++ b/methods/physics.html @@ -15,7 +15,7 @@ - + @@ -42,7 +42,7 @@   ::   Contents   ::   - Release Notes for OpenMC 0.6.0  » + Release Notes for OpenMC 0.6.2  »

@@ -55,6 +55,8 @@ bugs fixed, and known issues for each successive release.

diff --git a/releasenotes/notes_0.4.0.html b/releasenotes/notes_0.4.0.html index 9a3fbf3d9d..41c0da33f2 100644 --- a/releasenotes/notes_0.4.0.html +++ b/releasenotes/notes_0.4.0.html @@ -15,7 +15,7 @@ + + + + + + + + + + +
+ + + +
+
+ +

+ «  Release Notes for OpenMC 0.6.2 +   ::   + Contents +   ::   + Release Notes for OpenMC 0.6.0  » +

+ +
+
+ + +
+

Release Notes for OpenMC 0.6.1

+
+

System Requirements

+

There are no special requirements for running the OpenMC code. As of this +release, OpenMC has been tested on a variety of Linux distributions, Mac OS X, +and Microsoft Windows 7. Memory requirements will vary depending on the size of +the problem at hand (mostly on the number of nuclides in the problem).

+
+
+

New Features

+
    +
  • Coarse mesh finite difference (CMFD) acceleration no longer requires PETSc
  • +
  • Statepoint file numbering is now zero-padded
  • +
  • Python scripts now compatible with Python 2 or 3
  • +
  • Ability to run particle restarts in fixed source calculations
  • +
  • Capability to filter box source by fissionable materials
  • +
  • Nuclide/element names are now case insensitive in input files
  • +
  • Improved treatment of resonance scattering for heavy nuclides
  • +
+
+
+

Bug Fixes

+
    +
  • 03e890: Check for energy-dependent multiplicities in ACE files
  • +
  • 4439de: Fix distance-to-surface calculation for general plane surface
  • +
  • 5808ed: Account for differences in URR band probabilities at different energies
  • +
  • 2e60c0: Allow zero atom/weight percents in materials
  • +
  • 3e0870: Don’t use PWD environment variable when setting path to input files
  • +
  • dc4776: Handle probability table resampling correctly
  • +
  • 01178b: Fix metastables nuclides in NNDC cross_sections.xml file
  • +
  • 62ec43: Don’t read tallies.xml when OpenMC is run in plotting mode
  • +
  • 2a95ef: Prevent segmentation fault on “current” score without mesh filter
  • +
  • 93e482: Check for negative values in probability tables
  • +
+
+
+

Contributors

+

This release contains new contributions from the following people:

+ +
+
+ + +
+
+ +

+ «  Release Notes for OpenMC 0.6.2 +   ::   + Contents +   ::   + Release Notes for OpenMC 0.6.0  » +

+ +
+ + + + + + + \ No newline at end of file diff --git a/releasenotes/notes_0.6.2.html b/releasenotes/notes_0.6.2.html new file mode 100644 index 0000000000..0ba14e0746 --- /dev/null +++ b/releasenotes/notes_0.6.2.html @@ -0,0 +1,136 @@ + + + + + + + + Release Notes for OpenMC 0.6.2 — OpenMC Documentation + + + + + + + + + + + + + + + + +
+ + + +
+
+ +

+ «  Release Notes +   ::   + Contents +   ::   + Release Notes for OpenMC 0.6.1  » +

+ +
+
+ + +
+

Release Notes for OpenMC 0.6.2

+
+

System Requirements

+

There are no special requirements for running the OpenMC code. As of this +release, OpenMC has been tested on a variety of Linux distributions, Mac OS X, +and Microsoft Windows 7. Memory requirements will vary depending on the size of +the problem at hand (mostly on the number of nuclides in the problem).

+
+
+

New Features

+
    +
  • Meshline plotting capability
  • +
  • Support for plotting cells/materials on middle universe levels
  • +
  • Ability to model cells with no surfaces
  • +
  • Compatibility with PETSc 3.5
  • +
  • Compatability with OpenMPI 1.7/1.8
  • +
  • Improved overall performance via logarithmic-mapped energy grid search
  • +
  • Improved multi-threaded performance with atomic operations
  • +
  • Support for fixed source problems with fissionable materials
  • +
+
+
+

Bug Fixes

+
    +
  • 26fb93: Fix problem with -t, –track command-line flag
  • +
  • 2f07c0: Improved evaporation spectrum algorithm
  • +
  • e6abb9: Fix segfault when tallying in a void material
  • +
  • 291b45: Handle metastable nuclides in NNDC data and multiplicities in MT=5 data
  • +
+
+
+

Contributors

+

This release contains new contributions from the following people:

+ +
+
+ + +
+
+ +

+ «  Release Notes +   ::   + Contents +   ::   + Release Notes for OpenMC 0.6.1  » +

+ +
+ + + + + + + \ No newline at end of file diff --git a/search.html b/search.html index 6cd6a649ee..0a71356be5 100644 --- a/search.html +++ b/search.html @@ -15,7 +15,7 @@ - +
@@ -38,7 +38,7 @@

- «  8. Parallelization + «  9. Nonlinear Diffusion Acceleration - Coarse Mesh Finite Difference   ::   Contents   ::   @@ -96,22 +96,24 @@ essential aspects of using OpenMC to perform neutronic simulations.

  • 3.2.5. <energy_grid> Element
  • 3.2.6. <entropy> Element
  • 3.2.7. <fixed_source> Element
  • -
  • 3.2.8. <natural_elements> Element
  • -
  • 3.2.9. <no_reduce> Element
  • -
  • 3.2.10. <output> Element
  • -
  • 3.2.11. <output_path> Element
  • -
  • 3.2.12. <ptables> Element
  • -
  • 3.2.13. <run_cmfd> Element
  • -
  • 3.2.14. <seed> Element
  • -
  • 3.2.15. <source> Element
  • -
  • 3.2.16. <state_point> Element
  • -
  • 3.2.17. <source_point> Element
  • -
  • 3.2.18. <survival_biasing> Element
  • -
  • 3.2.19. <threads> Element
  • -
  • 3.2.20. <trace> Element
  • -
  • 3.2.21. <track> Element
  • -
  • 3.2.22. <uniform_fs> Element
  • -
  • 3.2.23. <verbosity> Element
  • +
  • 3.2.8. <log_grid_bins> Element
  • +
  • 3.2.9. <natural_elements> Element
  • +
  • 3.2.10. <no_reduce> Element
  • +
  • 3.2.11. <output> Element
  • +
  • 3.2.12. <output_path> Element
  • +
  • 3.2.13. <ptables> Element
  • +
  • 3.2.14. <resonance_scattering> Element
  • +
  • 3.2.15. <run_cmfd> Element
  • +
  • 3.2.16. <seed> Element
  • +
  • 3.2.17. <source> Element
  • +
  • 3.2.18. <state_point> Element
  • +
  • 3.2.19. <source_point> Element
  • +
  • 3.2.20. <survival_biasing> Element
  • +
  • 3.2.21. <threads> Element
  • +
  • 3.2.22. <trace> Element
  • +
  • 3.2.23. <track> Element
  • +
  • 3.2.24. <uniform_fs> Element
  • +
  • 3.2.25. <verbosity> Element
  • 3.3. Geometry Specification – geometry.xml
      @@ -136,23 +138,27 @@ essential aspects of using OpenMC to perform neutronic simulations.

  • 3.7. CMFD Specification – cmfd.xml
  • +
  • 3.8. ERSN-OpenMC Graphical User Interface
  • 4. Data Processing and Visualization
      @@ -198,7 +204,7 @@ essential aspects of using OpenMC to perform neutronic simulations.

      - «  8. Parallelization + «  9. Nonlinear Diffusion Acceleration - Coarse Mesh Finite Difference   ::   Contents   ::   diff --git a/usersguide/input.html b/usersguide/input.html index ea6b995b3b..e25c0fba2a 100644 --- a/usersguide/input.html +++ b/usersguide/input.html @@ -15,7 +15,7 @@