Variance reduction

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Variance reduction in Serpent is based on standard weight-window techniques.[1] The weight window mesh and additional parameters are defined using the wwin card. Serpent supports two mesh types:

  • Weight-window mesh generated using the built-in response matrix method-based solver (see the wgen card).
  • MCNP WWINP format weight window-mesh.[2]

This tutorial demonstrates the basic functionality of the built-in solver. The methodology is described in the related publications.[3][4]


Test case and analog simulation

The test case is comprised of an isotropic, point-wise, low-energy neutron source, enclosed inside two cylindrical shells made of steel and concrete. The geometry is in 2D for the sake of simplicity, but the same procedures apply to 3D problems as well. The geometry plot and complete input listing are provided below (click Expand to show).

VR Ref geom1.png

Completely made-up neutron shielding problem

 % --- Geometry (nested cylinders)

surf 1 cyl 0.0 0.0 100.0
surf 2 cyl 0.0 0.0 120.0
surf 3 cyl 0.0 0.0 200.0
surf 4 cyl 0.0 0.0 300.0
surf 5 sqc 0.0 0.0 400.0

cell 1 0 air      -1
cell 2 0 steel     1 -2
cell 3 0 air       2 -3
cell 4 0 concrete  3 -4
cell 5 0 air       4 -5
cell 6 0 outside   5

% --- Materials

mat steel    -8.00000E+00  rgb 100 100 100

 6000.03c    -4.00000E-04
14000.03c    -5.00000E-03
15031.03c    -2.30000E-04
16000.03c    -1.50000E-04
24000.03c    -1.90000E-01
25055.03c    -1.00000E-02
26000.03c    -7.01730E-01
28000.03c    -9.25000E-02

mat air      -1.20500E-03  rgb 255 255 220

 6000.03c    -1.24000E-04
 7014.03c    -7.55268E-01
 8016.03c    -2.31781E-01
18040.03c    -1.28270E-02

mat concrete -2.30000E+00  rgb 180 180 180

 1001.03c    -1.00000E-02
 8016.03c    -5.32000E-01
11023.03c    -2.90000E-02
13027.03c    -3.40000E-02
14000.03c    -3.37000E-01
20000.03c    -4.40000E-02
26000.03c    -1.40000E-02

% --- Source 

src 1 sp 0 0 0 se 1E-6     % Isotropic 1 eV point source
set srcrate 1              % Normalize to unit source rate

% --- Run parameters

set nps 200000 50
set gcu -1                 % Group constant generation off

% --- Geometry plot:

plot 3 500 500

% --- Mesh plot (flux)

det F1                     % Flux detector
mesh 8 -4 F1 3 500 500     % Plot detector scores

% --- Detectors

det d1 dc 3                % Flux in airspace between walls

surf s2 cyl 0.0 160.0 5.0  
det d2 dtl s2              % Flux at y = 160 (between the walls)

surf s3 cyl 0.0 350.0 5.0  
det d3 dtl s3              % Flux at y = 350 (outside the concrete wall)

Notes on the input:

  • Group constant generation is on by default in neutron transport simulations. This is switched off by set gcu -1.
  • Neutron transport simulations are normalized by default to unit loss rate. For consistency the normalization is changed to unit source rate by set srcrate 1.
  • The mesh-card is linked to a flux detector (type 8) to visualize the effect of variance reduction. The color scheme is set to logarithmic (negative entry). It should be noted that the normalization of the color scheme depends on the minimum and maximum values, so the plots produced with and without variance reduction are not similarly normalized.

In addition to the mesh plot, flux is also calculated using a cell flux detector in the airspace between the walls (d1) and two track-length detectors at a position between the walls (d2) and outside the outer concrete wall (d3). The mesh plot and detector results are presented below (click image for full-size).

VR Ref mesh1.png

Detector Score Relative statistical error
d1 (volume flux between shells) 1.59600E-01 0.16735
d2 (at y = 160) 1.04553E-04 0.73813
d3 (at y = 350) 0.00000E+00 0.00000

The results show that most neutrons are stopped by the inner shell, and none of the particles are able to penetrate the concrete.

Variance reduction, simple approach

The simplest approach to applying variance reduction is to divide the calculation in two parts. The first run generates the importance mesh, and the second run produces the final results by applying the weight-window technique. The weigh window generator is invoked by adding the wgen card:

wwgen card

% --- Weight window generation

wwgen 1                    % wwgen identifier
1E-9 10000                 % convergence criteria
1 -1                       % single detector, no energy binning
1                          % cartesian mesh
-400.01 400.01 51          % x-dimensions
-400.01 400.01 51          % y-dimensions
-1E18 1E18 1               % z-dimensions
d2 1.0                     % detetor name and weight factor

Complete input

 % --- Geometry (nested cylinders)

surf 1 cyl 0.0 0.0 100.0
surf 2 cyl 0.0 0.0 120.0
surf 3 cyl 0.0 0.0 200.0
surf 4 cyl 0.0 0.0 300.0
surf 5 sqc 0.0 0.0 400.0

cell 1 0 air      -1
cell 2 0 steel     1 -2
cell 3 0 air       2 -3
cell 4 0 concrete  3 -4
cell 5 0 air       4 -5
cell 6 0 outside   5

% --- Materials

mat steel    -8.00000E+00  rgb 100 100 100

 6000.03c    -4.00000E-04
14000.03c    -5.00000E-03
15031.03c    -2.30000E-04
16000.03c    -1.50000E-04
24000.03c    -1.90000E-01
25055.03c    -1.00000E-02
26000.03c    -7.01730E-01
28000.03c    -9.25000E-02

mat air      -1.20500E-03  rgb 255 255 220

 6000.03c    -1.24000E-04
 7014.03c    -7.55268E-01
 8016.03c    -2.31781E-01
18040.03c    -1.28270E-02

mat concrete -2.30000E+00  rgb 180 180 180

 1001.03c    -1.00000E-02
 8016.03c    -5.32000E-01
11023.03c    -2.90000E-02
13027.03c    -3.40000E-02
14000.03c    -3.37000E-01
20000.03c    -4.40000E-02
26000.03c    -1.40000E-02

% --- Source 

src 1 sp 0 0 0 se 1E-6     % Isotropic 1 eV point source
set srcrate 1              % Normalize to unit source rate

% --- Run parameters

set nps 200000 50
set gcu -1                 % Group constant generation off

% --- Geometry plot:

plot 3 500 500

% --- Mesh plot (flux)

det F1                     % Flux detector
mesh 8 -4 F1 3 500 500     % Plot detector scores

% --- Detectors

det d1 dc 3                % Flux in airspace between walls

surf s2 cyl 0.0 160.0 5.0  
det d2 dtl s2              % Flux at y = 160 (between the walls)

surf s3 cyl 0.0 350.0 5.0  
det d3 dtl s3              % Flux at y = 350 (outside the concrete wall)

% --- Weight window generation

wwgen 1                    % wwgen identifier
1E-9 10000                 % convergence criteria
1 -1                       % single detector, no energy binning
1                          % Cartesian mesh
-400.01 400.01 51          % x-dimensions
-400.01 400.01 51          % y-dimensions
-1E18 1E18 1               % z-dimensions
d2 1.0                     % detecor name and weight factor

Notes:

  • The geometry is covered by a Cartesian 51x51x1 mesh.
  • An uneven number of cells is selected to prevent the point source (at origin) coinciding with the mesh boundaries.
  • The mesh should be slightly larger (here 0.01 cm) than the geometry to prevent problems with numerical precision.
  • In 2D calculations the axial limits should be set to large values to prevent particles escaping the mesh.
  • The importances are calculated with respect to detector d2, positioned between the two shells.

When Serpent is run, the run-time output shows some information on the response matrix solution:

output

Running response matrix solver...

Number of iterations : 10000
Convergence limit : 1.00E-09

Iterating adjoint solution:

  5% of iterations completed: frac = 1.00E-09

Completed in 88 milliseconds. Convergence reached after 566 iterations.
Relative difference to total response: -1.00E-09.

Normalizing importances...
OK.

Minimum source importance: 1.00000E+00
Maximum source importance: 1.00000E+00

Minimum cell importance: 9.10517E-03
Maximum cell importance: 3.36720E+04

If the solution does not converge, the reason is most likely poor statistics.

The calculation produces a binary weight-window mesh file: [input].wwd, which can be read into another simulation using the wwin card:

wwin card

% --- Weight windows

wwin 1 wf "YOUR_FILE.wwd" 1     % Read from previous run

Complete input

% --- Geometry (nested cylinders)

surf 1 cyl 0.0 0.0 100.0
surf 2 cyl 0.0 0.0 120.0
surf 3 cyl 0.0 0.0 200.0
surf 4 cyl 0.0 0.0 300.0
surf 5 sqc 0.0 0.0 400.0

cell 1 0 air      -1
cell 2 0 steel     1 -2
cell 3 0 air       2 -3
cell 4 0 concrete  3 -4
cell 5 0 air       4 -5
cell 6 0 outside   5

% --- Materials

mat steel    -8.00000E+00  rgb 100 100 100

 6000.03c    -4.00000E-04
14000.03c    -5.00000E-03
15031.03c    -2.30000E-04
16000.03c    -1.50000E-04
24000.03c    -1.90000E-01
25055.03c    -1.00000E-02
26000.03c    -7.01730E-01
28000.03c    -9.25000E-02

mat air      -1.20500E-03  rgb 255 255 220

 6000.03c    -1.24000E-04
 7014.03c    -7.55268E-01
 8016.03c    -2.31781E-01
18040.03c    -1.28270E-02

mat concrete -2.30000E+00  rgb 180 180 180

 1001.03c    -1.00000E-02
 8016.03c    -5.32000E-01
11023.03c    -2.90000E-02
13027.03c    -3.40000E-02
14000.03c    -3.37000E-01
20000.03c    -4.40000E-02
26000.03c    -1.40000E-02

% --- Source 

src 1 sp 0 0 0 se 1E-6     % Isotropic 1 eV point source
set srcrate 1              % Normalize to unit source rate

% --- Run parameters

set nps 200000 50
set gcu -1                 % Group constant generation off
set bala 1                 % Use OMP load balancing

% --- Geometry plot:

plot 35 1E-4 1E4 -1 500 500

% --- Mesh plot (flux)

det F1                     % Flux detector
mesh 8 -4 F1 3 500 500     % Plot detector scores

% --- Detectors

det d1 dc 3                % Flux in airspace between walls

surf s2 cyl 0.0 160.0 5.0  
det d2 dtl s2              % Flux at y = 160 (between the walls)

surf s3 cyl 0.0 350.0 5.0  
det d3 dtl s3              % Flux at y = 350 (outside the concrete wall)

% --- Weight windows

wwin 1 wf "YOUR_FILE.wwd" 1     % Read from previous run

Notes on the input:

  • Variance reduction causes splitting of the particle histories, which may result in deterioration of OpenMP parallel scalability. To improve performance, it is recommended to switch OpenMP load balancing on by set bala 1.
  • The importance mesh can be plotted on top of the geometry using the plot card. The additional input parameters include importance boundaries and energy (for energy-dependent calculations).
  • The current version of Serpent (2.1.31) does not plot the mesh grid for the importance distribution, but this can be enabled by removing the "wwp = NO;" statement on line 116 of geometryplotter.c

The mesh plot and detector results using variance reduction are presented below (click image for full-size).

VR Wwin1 geom1.png VR Wwin1 mesh1.png

Detector Score Relative statistical error
d1 (volume flux between shells) 1.18215E-01 0.14545
d2 (at y = 160) 1.50203E-04 0.10000
d3 (at y = 350) 0.00000E+00 0.00000

The importances were calculated with respect to detector d2, which recieves much better statistics compared to the analog calculation. The mesh plot shows how the calculated flux is concentrated around the detectors, as particles less likely to contribute to the result are killed by Russian roulette.

Global variance reduction, fixed mesh

One of the challenges with large and heavily shielded geometries is that the particle histories may not reach the region of interest, in which case the Monte Carlo simulation fails to provide the coupling coefficients for the response matrix method-based importance solver. The solution is to apply global variance reduction (GVR), which means producing a weight-window mesh that uniformly populates the entire geometry. The calculation proceeds by iterations. Each cycle allows the particles to wonder beyond the region charted for the weight-window mesh, and the collected new data is used to extend the mesh deeper into the geometry. This option is invoked by setting the MOD parameter to 3 in the wgen card. Also the detector entry is left out.

In this example global variance reduction is used with weight window iterations. The mesh type is changed to cylindrical. The input is listed below.


wwgen card

% --- Weight window generation

wwgen 1                    % wwgen identifier
1E-9 10000                 % convergence criteria
3 -1                       % global variance reduction, no energy binning
2                          % cylindrical mesh
0.0 565.70 50              % radial dimensions
0.0 360.00 1               % azimuthal dimensions
-1E18 1E18 1               % axial dimensions

% --- GVR iterations

wwin 1
wi 1 3                     % 3 iterations using the same mesh
1 1.0
1 1.0
1 1.0

Complete input

% --- Geometry (nested cylinders)

surf 1 cyl 0.0 0.0 100.0
surf 2 cyl 0.0 0.0 120.0
surf 3 cyl 0.0 0.0 200.0
surf 4 cyl 0.0 0.0 300.0
surf 5 sqc 0.0 0.0 400.0

cell 1 0 air      -1
cell 2 0 steel     1 -2
cell 3 0 air       2 -3
cell 4 0 concrete  3 -4
cell 5 0 air       4 -5
cell 6 0 outside   5

% --- Materials

mat steel    -8.00000E+00  rgb 100 100 100

 6000.03c    -4.00000E-04
14000.03c    -5.00000E-03
15031.03c    -2.30000E-04
16000.03c    -1.50000E-04
24000.03c    -1.90000E-01
25055.03c    -1.00000E-02
26000.03c    -7.01730E-01
28000.03c    -9.25000E-02

mat air      -1.20500E-03  rgb 255 255 220

 6000.03c    -1.24000E-04
 7014.03c    -7.55268E-01
 8016.03c    -2.31781E-01
18040.03c    -1.28270E-02

mat concrete -2.30000E+00  rgb 180 180 180

 1001.03c    -1.00000E-02
 8016.03c    -5.32000E-01
11023.03c    -2.90000E-02
13027.03c    -3.40000E-02
14000.03c    -3.37000E-01
20000.03c    -4.40000E-02
26000.03c    -1.40000E-02

% --- Source 

src 1 sp 0 0 0 se 1E-6     % Isotropic 1 eV point source
set srcrate 1              % Normalize to unit source rate

% --- Run parameters

set nps 200000 50
set gcu -1                 % Group constant generation off
set bala 1                 % Use OMP load balancing

% --- Geometry plot:

plot 35 1E-9 1E9 -1 500 500

% --- Mesh plot (flux)

det F1                     % Flux detector
mesh 8 -4 F1 3 500 500     % Plot detector scores

% --- Detectors

det d1 dc 3                % Flux in airspace between walls

surf s2 cyl 0.0 160.0 5.0  
det d2 dtl s2              % Flux at y = 160 (between the walls)

surf s3 cyl 0.0 350.0 5.0  
det d3 dtl s3              % Flux at y = 350 (outside the concrete wall)

% --- Weight window generation

wwgen 1                    % wwgen identifier
1E-9 10000                 % convergence criteria
3 -1                       % global variance reduction, no energy binning
2                          % cylindrical mesh
0.0 565.70 50              % radial dimensions
0.0 360.00 1               % azimuthal dimensions
-1E18 1E18 1               % axial dimensions

% --- GVR iterations

wwin 1
wi 1 3                     % 3 iterations using the same mesh
1 1.0
1 1.0
1 1.0

References

  1. ^ Lux, I. and Koblinger, L. "Monte Carlo Particle Transport Methods: Neutron and Photon Calculations." CRC-Press, 1991. (available on-line).
  2. ^ D. Pelowitz (ed.), “MCNP6 User’s Manual: Appendix C Mesh-Based WWINP, WWOUT, and WWONE File Format,” LA-CP-13-00634, Los Alamos National Laboratory (2013).
  3. ^ J. Leppänen "Response Matrix Method–Based Importance Solver and Variance Reduction Scheme in the Serpent 2 Monte Carlo Code." Nucl. Technol. (in press)
  4. ^ J. Leppänen and M. Jokipii. "Global Variance Reduction Scheme with Self-Adaptive Weight-Window Mesh in the Serpent 2 Monte Carlo Code." In Proc. M&C2019, Portland, OR, Aug. 25-29, 2019.