Kobayashi Dog-Leg Void Benchmark#

Problem Description#

A mono-energetic, three-dimensional shielding benchmark featuring a dog-leg vacuum channel embedded in a purely scattering/absorbing shield. The problem evaluates the ability of a Monte Carlo code to transport neutrons through deep-penetration streaming paths.

It is based on the NEA steady-state fixed-source benchmark problem suite by Kobayashi et al. [Kobayashi2001].

Geometry and Materials#

The computational domain spans \(x \in [0,60]\), \(y \in [0,100]\), \(z \in [0,60]\) cm. Boundary conditions are reflective on the three symmetry planes (\(x=0\), \(y=0\), \(z=0\)) and vacuum elsewhere.

Three regions are defined:

  1. Source region — \(x \in [0,10]\), \(y \in [0,10]\), \(z \in [0,10]\) cm.

  2. Dog-leg void channel — an L-shaped duct connecting the source corner to the far side of the domain.

  3. Shield — the remaining volume.

Cross-section data (mono-energetic, cm-1)#

Region

\(\Sigma_c\)

\(\Sigma_s\)

Shield

0.05

0.05

Void channel

\(5\times10^{-5}\)

\(5\times10^{-5}\)

Physical Assumptions#

  • Mono-energetic (one-speed) neutron transport.

  • Isotropic scattering.

  • Steady-state (time-independent) fixed-source problem.

  • Implicit capture variance-reduction technique.

Numerical Setup#

Spatial mesh (tally)

\(60 \times 100 \times 60\) uniform cells (1 cm spacing)

Tally score

Scalar flux

Source particles

\(10^{3}\) (demonstration; increase for production runs)

Batches

2

Quantities of Interest#

  • Three-dimensional scalar flux distribution \(\phi(x,y,z)\).

  • Flux attenuation along the streaming channel and through the shield.

Reference Solution#

Reference solutions are tabulated in [Kobayashi2001] for several axial slices. Post-processing in MC/DC generates \((x,y)\) flux maps at selected \(z\)-planes for direct comparison.

References#

[Kobayashi2001] (1,2)

K. Kobayashi, N. Sugimura, and Y. Nagaya, “3D Radiation Transport Benchmark Problems and Results for Simple Geometries with Void Region,” Progress in Nuclear Energy, 39:2, 119–144 (2001). [link]

Step-by-Step Walkthrough#

This section walks through the input file block by block.

1. Import and Materials (lines 1–13)

 1import numpy as np
 2import mcdc
 3
 4simulation = mcdc.Simulation("Kobayashi dog-leg benchmark")
 5
 6# ======================================================================================
 7# Set model
 8# ======================================================================================
 9# Based on Kobayashi dog-leg benchmark problem
10# (PNE 2001, https://doi.org/10.1016/S0149-1970(01)00007-5)
11
12# Set materials
13m = mcdc.Material.multigroup(capture=np.array([0.05]), scatter=np.array([[0.05]]))

Two mono-energetic multigroup materials are created: m for the shield (\(\Sigma_c = \Sigma_s = 0.05\)) and m_void for the dog-leg channel (\(10^{-4}\) total).

2. Surfaces (lines 15–30)

15
16# Set surfaces
17sx1 = mcdc.Surface.PlaneX(x=0.0, boundary_condition="reflective")
18sx2 = mcdc.Surface.PlaneX(x=10.0)
19sx3 = mcdc.Surface.PlaneX(x=30.0)
20sx4 = mcdc.Surface.PlaneX(x=40.0)
21sx5 = mcdc.Surface.PlaneX(x=60.0, boundary_condition="vacuum")
22sy1 = mcdc.Surface.PlaneY(y=0.0, boundary_condition="reflective")
23sy2 = mcdc.Surface.PlaneY(y=10.0)
24sy3 = mcdc.Surface.PlaneY(y=50.0)
25sy4 = mcdc.Surface.PlaneY(y=60.0)
26sy5 = mcdc.Surface.PlaneY(y=100.0, boundary_condition="vacuum")
27sz1 = mcdc.Surface.PlaneZ(z=0.0, boundary_condition="reflective")
28sz2 = mcdc.Surface.PlaneZ(z=10.0)
29sz3 = mcdc.Surface.PlaneZ(z=30.0)
30sz4 = mcdc.Surface.PlaneZ(z=40.0)

Fifteen planar surfaces define the 3-D bounding box and the internal partitions. Reflective conditions on sx1, sy1, sz1 exploit the quarter-symmetry; vacuum on the outer faces allows leakage.

3. Cells — CSG Region Definitions (lines 32–44)

32
33# Set cells
34# Source
35source_cell = mcdc.Cell(region=+sx1 & -sx2 & +sy1 & -sy2 & +sz1 & -sz2, fill=m)
36# Voids
37channel_1 = +sx1 & -sx2 & +sy2 & -sy3 & +sz1 & -sz2
38channel_2 = +sx1 & -sx3 & +sy3 & -sy4 & +sz1 & -sz2
39channel_3 = +sx3 & -sx4 & +sy3 & -sy4 & +sz1 & -sz3
40channel_4 = +sx3 & -sx4 & +sy3 & -sy5 & +sz3 & -sz4
41void_channel = channel_1 | channel_2 | channel_3 | channel_4
42void_cell = mcdc.Cell(region=void_channel, fill=m_void)
43# Shield
44box = +sx1 & -sx5 & +sy1 & -sy5 & +sz1 & -sz5

Three cells cover the domain:

  • The source cell (a small corner cube) filled with shield material.

  • The void channel — four rectangular segments combined with the | (union) operator to form the L-shaped duct.

  • The shield — the full box minus the void channel, using the ~ (complement) operator.

4. Source (lines 50–57)

50# ======================================================================================
51source = mcdc.Source(
52    x=[0.0, 10.0],
53    y=[0.0, 10.0],
54    z=[0.0, 10.0],
55    isotropic=True,
56    energy=0,
57)

An isotropic, uniformly distributed source fills the \(10 \times 10 \times 10\) cm corner cube.

5. Tallies, Settings, Techniques, and Run (lines 63–74)

63
64# Tallies
65mesh = mcdc.MeshUniform(x=(0.0, 1.0, 60), y=(0.0, 1.0, 100), z=(0.0, 1.0, 60))
66tally = mcdc.Tally(mesh=mesh, scores=["flux"])
67simulation.set_tallies([tally])
68
69# Settings
70simulation.settings.N_particle = 1000
71simulation.settings.N_batch = 2
72
73# Techniques
74simulation.technique.implicit_capture()

  • A uniform \(60 \times 100 \times 60\) mesh tally records scalar flux.

  • 1 000 source particles in 2 batches (increase for production).

  • Implicit capture prevents particles from being absorbed prematurely.

  • simulation.run() launches the simulation.

What to try:

  • Increase N_particle to \(10^5\) or more for smoother flux maps.

  • Change void-channel cross sections to see how attenuation changes.

  • Add a time grid to the tally for a transient variant (see the TD example).

Full Input#

Click here to view the input file: examples/kobayashi/input.py.

The complete input used for this example is embedded below:

 1import numpy as np
 2import mcdc
 3
 4simulation = mcdc.Simulation("Kobayashi dog-leg benchmark")
 5
 6# ======================================================================================
 7# Set model
 8# ======================================================================================
 9# Based on Kobayashi dog-leg benchmark problem
10# (PNE 2001, https://doi.org/10.1016/S0149-1970(01)00007-5)
11
12# Set materials
13m = mcdc.Material.multigroup(capture=np.array([0.05]), scatter=np.array([[0.05]]))
14m_void = mcdc.Material.multigroup(capture=np.array([5e-5]), scatter=np.array([[5e-5]]))
15
16# Set surfaces
17sx1 = mcdc.Surface.PlaneX(x=0.0, boundary_condition="reflective")
18sx2 = mcdc.Surface.PlaneX(x=10.0)
19sx3 = mcdc.Surface.PlaneX(x=30.0)
20sx4 = mcdc.Surface.PlaneX(x=40.0)
21sx5 = mcdc.Surface.PlaneX(x=60.0, boundary_condition="vacuum")
22sy1 = mcdc.Surface.PlaneY(y=0.0, boundary_condition="reflective")
23sy2 = mcdc.Surface.PlaneY(y=10.0)
24sy3 = mcdc.Surface.PlaneY(y=50.0)
25sy4 = mcdc.Surface.PlaneY(y=60.0)
26sy5 = mcdc.Surface.PlaneY(y=100.0, boundary_condition="vacuum")
27sz1 = mcdc.Surface.PlaneZ(z=0.0, boundary_condition="reflective")
28sz2 = mcdc.Surface.PlaneZ(z=10.0)
29sz3 = mcdc.Surface.PlaneZ(z=30.0)
30sz4 = mcdc.Surface.PlaneZ(z=40.0)
31sz5 = mcdc.Surface.PlaneZ(z=60.0, boundary_condition="vacuum")
32
33# Set cells
34# Source
35source_cell = mcdc.Cell(region=+sx1 & -sx2 & +sy1 & -sy2 & +sz1 & -sz2, fill=m)
36# Voids
37channel_1 = +sx1 & -sx2 & +sy2 & -sy3 & +sz1 & -sz2
38channel_2 = +sx1 & -sx3 & +sy3 & -sy4 & +sz1 & -sz2
39channel_3 = +sx3 & -sx4 & +sy3 & -sy4 & +sz1 & -sz3
40channel_4 = +sx3 & -sx4 & +sy3 & -sy5 & +sz3 & -sz4
41void_channel = channel_1 | channel_2 | channel_3 | channel_4
42void_cell = mcdc.Cell(region=void_channel, fill=m_void)
43# Shield
44box = +sx1 & -sx5 & +sy1 & -sy5 & +sz1 & -sz5
45shield_cell = mcdc.Cell(region=box & ~void_channel, fill=m)
46simulation.set_model([source_cell, void_cell, shield_cell])
47
48# ======================================================================================
49# Set source
50# ======================================================================================
51source = mcdc.Source(
52    x=[0.0, 10.0],
53    y=[0.0, 10.0],
54    z=[0.0, 10.0],
55    isotropic=True,
56    energy=0,
57)
58simulation.set_sources([source])
59
60# ======================================================================================
61# Set tallies, settings, techniques, and run MC/DC
62# ======================================================================================
63
64# Tallies
65mesh = mcdc.MeshUniform(x=(0.0, 1.0, 60), y=(0.0, 1.0, 100), z=(0.0, 1.0, 60))
66tally = mcdc.Tally(mesh=mesh, scores=["flux"])
67simulation.set_tallies([tally])
68
69# Settings
70simulation.settings.N_particle = 1000
71simulation.settings.N_batch = 2
72
73# Techniques
74simulation.technique.implicit_capture()
75
76# Run
77simulation.run()

How to Run#

From inside examples/kobayashi run:

python input.py

Expected Output#

A mesh tally HDF5 file with 3-D flux data and example plotting using the companion process-output.py in the same directory.