+ 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
+
, it is surface area-averaged and if it has a
+
, it is volume-averaged. When describing a
+specific cell in the geometry, indices 
are used which correspond
+to directions 
. In most cases, the same operation is performed in
+all three directions. To compactly write this, an arbitrary direction set
+
that corresponds to cell indices 
is used. Note
+that 
and 
do not have to correspond to 
and
+
. However, if and correspond to 
and
+
, 
and 
correspond to and 
+directions. An example of this is shown in the following expression:
(1)
Here, takes on each direction one at a time. The parameter 
+is surface area-averaged over the transverse indices 
and 
+located at 
. Usually, spatial indices are listed as subscripts and
+the direction as a superscript. Energy group indices represented by 
+and 
are also listed as superscripts here. The group is the
+group of interest and, if present, is all groups. Finally, any
+parameter surrounded by 
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 in cell :
(2)
In eq. (2) the parameters are defined as:
+-
+
— surface area-integrated net
+current over surface 
with surface normal in direction
+ in energy group . By dividing this quantity by the transverse
+area, 
, the surface area-averaged net current can
+be computed.
+
+— volume-integrated total reaction rate over energy group .
+
+— volume-integrated scattering production rate of neutrons that begin with
+energy in group and exit reaction in group . This reaction
+rate also includes the energy transfer of reactions (except fission) that
+produce multiple neutrons such as (n, 2n); hence, the need for 
+to represent neutron multiplicity.
+
— core multiplication factor.
+
+— volume-integrated fission production rate of neutrons from fissions in
+group that exit in group .
+
Each quantity in 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}](../_images/tikz-2415a321261cca6307913eacc8e1d6b53ce536bf.png)
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)
(4)
and
+(5)
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)
where
+(7)
Note that the transport reaction rate is calculated from the total reaction rate
+reduced by the 
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)
-
+
- cell-to-boundary coupling +
(9)
In Eqs. (8) and (9), the 
refers to
+left (
) or right (
) surface in the direction,
+back (
) or front (
) surface in the direction and
+bottom (
) or top (
) surface in the direction. For
+cell-to-boundary coupling, a general albedo, 
,
+is used. The albedo is defined as the ratio of incoming (
superscript)
+to outgoing (
superscript) partial current on any surface represented
+as
(10)
Common boundary conditions are: vacuum (
), reflective
+(
) and zero flux (
). Both eq. (8)
+and eq. (9) can be written in this generic form,
(11)
The parameter 
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,
+
, is added to eqs. (8) and
+(9). These equations are, respectively,
(12)
and
+(13)
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.](../_images/math/49d4475e4ce6612d284b21992ac713097dce851f.png)
It should be noted that before substitution, eq. (2) was divided
+by the volume of the cell, 
. Equation
+(14) can be represented in operator form as
(15)
where 
is the neutron loss matrix operator,
+
is the neutron production matrix operator,
+
is the multigroup flux vector and 
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)
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)
The MC source distribution is represented on the same coarse mesh as
+CMFD by summing all neutrons’ weights, 
, 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)
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}](../_images/tikz-e6b509a58476b271ffa1e096f52c41de26d3371d.png)
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 
block diagonal; however, zeros are present because
+no fission neutrons appear with energies in the thermal group.
| + | + | + |
|---|---|---|
| tally | +score | +filter | +
 |
+flux | +mesh, energy | +
|
+total | +mesh, energy | +
 |
+nu-scatter-1 | +mesh, energy | +
 |
+nu-scatter | +mesh, energy, energyout | +
 |
+nu-fission | +mesh, energy, energyout | +
 |
+current | +mesh, energy | +
+Sparsity of Neutron Loss Operator
+
+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. |