k-Effective of the World - Nuclear Criticality Safety Division

X-Computational
Physics Div.
XCP-3, LANL
LA-UR-11-03593
Statistical Coverage Concerns in a
Revised k-Effective of the World
Problem
Brian Kiedrowski, Forrest Brown
Los Alamos National Laboratory
X-Computational Physics Division
1

Abstract
X-Computational
Physics Div.
XCP-3, LANL
A revised version of k-Effective of the World is defined specifically to stress
current Monte Carlo power iteration techniques. Results of numerous
independent Monte Carlo calculations show that even for batch sizes that are
typically considered reasonable (5-10K), incorrect results for k are obtained with
non-trivial probability. The causes for what makes a problem prone to having
these difficulties are identified and advice for criticality safety practitioners is
provided.
2

Overview
X-Computational
Physics Div.
XCP-3, LANL
• Review of k-Effective of the World
• The Revised Problem
• Numerical Results
• Analysis & Guidance
3

X-Computational
Physics Div.
XCP-3, LANL
Review of k-Effective of the World
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Problem Specification
X-Computational
Physics Div.
XCP-3, LANL
• Proposed by E. Whitesides in 1971
• Specification:
– 9 x 9 x 9 array of Pu-239 spheres spaced 50 cm apart in
vacuum
– The array is surrounded by a thick water reflector and is
subcritical
– Replace central sphere with one that is exactly critical by
itself
– Final result should be supercritical
5

Results
X-Computational
Physics Div.
XCP-3, LANL
• MCNP5-1.60 + ENDF/B-VII.0 data
• For uniform array of identical spheres
with surrounding water, sphere radii
adjusted to r = 3.9 cm, so that
Keff = .9328 ± .0002
• Single bare sphere, r=4.928 cm,
Keff = 1.0001 ± .0002
• Whitesides' model problem:
Replace center sphere of array
by larger (critical) sphere
Should be supercritical - is it
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Results: Then and Now
X-Computational
Physics Div.
XCP-3, LANL
• Run problem multiple times with different random
number sequence
• KENO defaults in the 1970s:
– 300 particles per cycle, skip 3, run 100 active cycles
– Results yield k as being subcritical
• Modern best practices:
– Use 5000+ particles per cycle, analyze convergence
diagnostics, start particles in each region
– Results yield correct k as supercritical
7

Results
X-Computational
Physics Div.
XCP-3, LANL
Distribution of K eff for 200 replicas, various M = neutrons/cycle
Frequency of Keff, for replicas
M=200
M=300
M=10K
M=500
Keff
8

Convergence
X-Computational
Physics Div.
XCP-3, LANL
K eff vs cycle, various M
M = neutrons/cycle
H src vs cycle, various M
M = neutrons/cycle
K eff converges in
75-100 cycles
M = 10,000
M = 10,000
H src converges in
100-150 cycles
M = 5,000
M = 5,000
Must discard 150
or more initial
cycles
M = 1,000
M = 1,000
Convergence
depends on the
dominance ratio &
source guess, NOT
on neutrons/cycle
M = 500
M = 500
100 200
100 200
Initial source guess = uniform sampling of points at sphere centers
9

Results Explained
X-Computational
Physics Div.
XCP-3, LANL
• Problem has 729 spheres, 300 per cycle not enough to
sample problem space!
– Problem of coverage
• Small batch size leads to issues in normalization
– Problem of non-conservative bias
• About 150 cycles required to find a steady source
– Problem of convergence
• Modern best practices make this problem easy to solve
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X-Computational
Physics Div.
XCP-3, LANL
The Revised Problem
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A More Difficult Problem
X-Computational
Physics Div.
XCP-3, LANL
• Simulate a flooded subcritical assembly storage vault
• Flood the area between the spheres, reduce radii and
distances between spheres to retain criticality
– Decreases coupling between spheres
• Place a thick layer of natural cadmium around the
central (most reactive) sphere
– Creates an asymmetry in coupling between the most
reactive region and the other region
• These coupling characteristics create difficulties
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Problem Specifications
X-Computational
Physics Div.
XCP-3, LANL
• Geometric Specifications:
– Radius of non-central Pu-239 spheres: 3.75 cm
– Radius of central Pu-239 sphere: 4.33 cm
– Coating thickness: 0.5 cm
– Center-to-center spacing: 20 cm
• Material specifications:
– Pure Pu-239, density: 20.0 g/cc
– Natural cadmium, density: 8.65 g/cc
– Pure water, density: 1.0 g/cc
13

Problem Specification
X-Computational
Physics Div.
XCP-3, LANL
• Source specification
– Uniformly sample points in each sphere
– 500 inactive cycles, 1000 total
– Vary size of batch
• Criticality results for variants:
– Uniform array: keff = 0.95
– Larger, coated sphere in infinite water bath: keff = 1.0
– Composite system should be supercritical
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Problem Specification
X-Computational
Physics Div.
XCP-3, LANL
15

X-Computational
Physics Div.
XCP-3, LANL
Numerical Results
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Reliability Tests
X-Computational
Physics Div.
XCP-3, LANL
• Main Observation: MCNP (and any other package using
power iteration) produces unreliable results
– Sometimes correct (k > 1.0), sometimes wrong (k = 0.95)
– Reliability increases with size of batch
– A “good” starting source is important
• Reliability is defined as the probability of getting the
correct answer
– Determine with 100 independent MCNP trials with different
random number sequences
• Try to explain why…
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Probability Density of Results
X-Computational
Physics Div.
XCP-3, LANL
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
5K
20K
10K
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Reliability Curve
X-Computational
Physics Div.
XCP-3, LANL
1.0
0.9
Frequency of Incorrect k
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0 5 10 15 20 25
Batch Size (1000 neutrons/cycle)
19

X-Computational
Physics Div.
XCP-3, LANL
Analysis & Guidance
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Discussion of Results
X-Computational
Physics Div.
XCP-3, LANL
• Best practices
– Batch size 5K or more may be too small
– May require 100K+ for “difficult” problems
• Probability of getting incorrect k decreases
exponentially with batch size
• No observable impact of bias
– Mean value for “correct” peak has consistent average k
– Problem is coverage not bias
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Discussion of Results
X-Computational
Physics Div.
XCP-3, LANL
• Diagnostics often fail to detect coverage
– Convergence in k plot is false
– Shannon entropy offers little to no indication
– Possible to sample central sphere, but not initiate a sustaining
fission chain
• Prescribed source guess inadequate
– Source guess points in central sphere guarantees
convergence to correct k
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Recipe for a “Difficult” Problem
X-Computational
Physics Div.
XCP-3, LANL
• Loosely-coupled regions
• Varying importance
– Most important region MUST be adequately sampled
• Asymmetric coupling
– Arises from material or geometric effects
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Asymmetric Coupling
X-Computational
Physics Div.
XCP-3, LANL
24

Practical Guidance
X-Computational
Physics Div.
XCP-3, LANL
• Understand the problem
– Exercise caution for problems with discrete zones
– Identify most important region(s)
• Good parameters defeats many problems
– Large batch sizes (100K+) preferable
– Source guess focus on most important region(s)
• Analyze results closely
– Check population tables to ensure adequate sampling
– Does value of k match your intuition
– For suspicious outcomes, run again with different random
number seeds
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References
X-Computational
Physics Div.
XCP-3, LANL
• X-5 MONTE CARLO TEAM, “MCNP – A
General N-Particle Transport Code, Version
5 – Volume I: Overview and Theory,” LA-
UR-03-1987, Los Alamos National
Laboratory (2003).
• G.E. WHITESIDES, "Difficulty in Computing
the k-effective of the World," Trans. Am.
Nucl. Soc., 14, No. 2, 680 (1971).
• F.B. BROWN, "Review of Best Practices for
Monte Carlo Criticality Calculations", ANS
NCSD-2009, Richland, WA, Sept 13-17
(2009).
• F.B. Brown, “’K-Effective Of The World’
And Other Concerns For Monte Carlo
Eigenvalue Calculations”, SNA+MC-2010,
Tokyo, Oct 17-20 [also, LA-UR-10-05548]
• R.N. Blomquist, et al., "Source
Convergence in Criticality Safety Analysis,
Phase I: Results of Four Test Problems,"
OECD Nuclear Energy Agency, OECD NEA
No. 5431 (2006).
• R.N. Blomquist, et al., "NEA Expert Group
on Source Convergence Phase II:
Guidance for Criticality Calculations", 8 th
International International Conference on
Criticality Safety, St. Petersburg, Russia,
May 28 – June 1, 2007 (May 2007).
• F.B. Brown, “Revisiting the ‘K-effective of
the World’ Problem”, Trans. Am. Nuc. Soc,
102, June 2010, [also, LA-UR-10-00189]
(2010).
• F.B. Brown, B.C. Kiedrowski, J.S. Bull,
"MCNP5-1.60 Release Notes", LA-UR-10-
06235 (2010).
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X-Computational
Physics Div.
XCP-3, LANL
Questions
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