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LA-UR-14-24367Approved for public release; distribution is unlimited.
Title: Comparison of Iterative Time-Eigenvalue Methods with DiscreteOrdinates and Monte Carlo
Author(s): Kiedrowski, Brian C.
Intended for: American Nuclear Society Annual Meeting 2014, 2014-06-15/2014-06-19(Anaheim, California, United States)
Issued: 2014-06-13
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Comparison of Iterative Time-EigenvalueMethods with Discrete Ordinates and Monte
Carlo
Brian C. Kiedrowski
Los Alamos National Laboratory
June 17, 2014
Slide 1
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Abstract
A standard approach in Monte Carlo and deterministic methods forcomputing the α (time absorption) eigenvalue involves solving a keigenvalue problem, and then finding the what insertion of a fictitious α/vabsorber or source would make k = 1. This talk looks at using alternativemultiplicative eigenvalues, the collision c and leakage l eigenvalues, as thebasis to compute α. Results suggest that in some cases, the use of the ceigenvalue may be more efficient than using the k eigenvalue.
Slide 2
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Introduction
Methodology
Implementation (SN and Monte Carlo)
Results
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Motivation
The asymptotic, logarithmic time rate of change of the neutronpopulation (assuming no feedback) may be obtained with the αeigenvalue.
Useful in subcritical measurements and analysis of promptsupercritical excursions in simple systems.
A classic approach uses a k eigenvalue calculation and finds whatinsertion of a fictitious α/v absorber or source would make k = 1.
This work explores using a collision c and leakage l eigenvaluesinstead.
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α-Eigenvalue Equation
Assume separability of time in the neutron flux:
Ψ(r, Ω,E , t) = n(t)ψ(r, Ω,E ).
Substitute and find the prompt α-eigenvalue transport equation:
v (S + M − L− T )ψ = αψ.
The time-dependent solution is a sum of eigenfunctions andexponentials:
Ψ(r, Ω,E , t) =∞∑j=0
ψj(r, Ω,E )eαj t .
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A Classic Solution Technique
Stiffness of the problem makes the α-eigenvalue equation difficult tosolve with standard iteration methods.
Classic approach involves constructing a hybrid k-α equation:
(αv
+ L + T − S)ψ =
1
kMψ.
Guess α, solve for k using standard iteration methods (e.g., poweriteration).
Find value of α that makes k = 1.
Iterate until consistent value of k and α are found.
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Generalized Multiplicative Eigenvalue Formulation
The k eigenvalue is one of many possible multiplicative eigenvalues.
Define the generalized multiplicative eigenvalue x :
(αv
+ Hx
)ψ =
1
xGxψ.
Here Hx and Gx are operators for the left- and right-hand sides of thetransport equation, depending upon the choice of x .
E.g., for x = k, Hx = L + T − X and Gx = M.
Criticality condition is x = 1, regardless of choice of x .
Question: Are some choices of x more computationally efficient thanothers?
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Hybrid Solution Technique
Inner iteration i , solve for eigenvalue x :
xi+1 =Ni+1
Nixi .
Once x is converged, perform outer iteration j , solve for α:
αj+1 = αj +xj − 1
τx ,j,
where τx is the appropriate neutron lifetime given by
τx =
⟨1vψ
⟩〈Gxψ〉
.
Iterate until α converged.
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Collision and Leakage Eigenvalue Forms
Two forms involve a collision c and leakage l eigenvalue:
(αv
+ L + T)ψ =
1
c(S + M)ψ,
(αv
+ L)ψ =
1
l(S + M − T )ψ.
Eigenvalue c uniformly adjusts multiplication of all collisions toachieve balance.
Eigenvalue l uniformly adjusts material atomic densities to achievebalance.
k = c = l = 1 are equivalent.
Slide 9
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Discrete Ordinates (SN) Implementation
Uses standard 1-D diamond difference transport sweep.
k eigenvalue solution has standard power iteration on transport andfission source.
c and l only iterate on transport source.
In all cases, modify Σt by adding current α/v .
k scales the fission source Mψ.c scales the collision source (S + M)ψ.l scales the collision source and modified total cross section Σt + α/v .
Compute lifetime from fluxes, and iterate until α converges.
Slide 10
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Monte Carlo (MC) Implementation
Simulate batches of particles in power iteration for fixed number ofinner iterations satisfying convergence and statistical uncertainties.
For k eigenvalue, treat fission as absorption (bank neutrons for nextiteration).
For c and l eigenvalues, treat all collisions as absorption (bankingneutrons).
In all cases, modify Σt by adding current α/v .k scales the number of fission neutrons banked.c scales the number of collision neutrons banked.l scales the total cross section and number of collision neutrons banked.
Lifetime estimate with collision tallies.
Iterate until α converges, and then keep running iterations untiluncertainty on α is sufficiently small.
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Convergence of Leakage Eigenvalue
The l eigenvalue is not guaranteed to exist (i.e., no global densityadjustment can make the transport equation balance).
When α step is too large such that l does not exist, take smaller stepsuntil it does.
Empirically, the l eigenvalue can exhibit oscillatory and very slowconvergence (possibly a complex dominance ratio?).
Simple acceleration is to take midpoint of oscillation (improvesconvergence rate by factor of 2 or more).
α will also oscillate, and midpoint technique improves convergencethere as well.
Slide 12
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SN Results
Three multigroup slab test problems:
Bare, fast (4-group), vary slab thicknessReflected, fast (4-group), vary reflector thicknessReflected, thermal (8-group), vary fuel/moderator ratio.
Use S64 Gauss-Legendre quadature, 1000 total spatial elements.
Speedup (wall-clock time ratio) to assess performance relative to keigenvalue:
Speedup =WallTimex
WallTimek.
Slide 13
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SN, Bare-Fast Case
0
0.5
1
1.5
2
2.5
18 20 22 24 26 28 30
Spe
edup
Core Thickness (cm)
c-al-a
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SN, Reflected-Fast Case
0
0.5
1
1.5
2
2.5
3
3.5
4
1 2 3 4 5 6 7 8 9 10
Spe
edup
Reflector Thickness (cm)
c-al-a
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SN, Reflected-Thermal Case
0
0.5
1
1.5
2
2.5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Spe
edup
Fuel Atomic Fraction
c-al-a
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MC Results
Same three multigroup problems:
Bare, fast (4-group), vary slab thicknessReflected, fast (4-group), vary reflector thicknessReflected, thermal (8-group), vary fuel/moderator ratio.
Use 10,000 neutron histories per inner iteration, 50 skip, 500 activeinner iterations per outer iteration.
250 active outer iterations were used.
Figure of merit (1/R2T ) ratio relative to k eigenvalue:
Performance =FOMx
FOMk.
Slide 17
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MC, Bare-Fast Case
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
18 20 22 24 26 28 30
Rel
ativ
e P
erfo
rman
ce
Core Thickness (cm)
c-al-a
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MC, Reflected-Fast Case
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
1 2 3 4 5 6 7 8 9 10
Rel
ativ
e P
erfo
rman
ce
Reflector Thickness (cm)
c-al-a
Slide 19
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MC, Reflected-Thermal Case
0
1
2
3
4
5
6
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Rel
ativ
e P
erfo
rman
ce
Fuel Atomic Fraction
c-al-a
Slide 20
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Discussion
SN results suggest greater efficiency gains.
Need to compare against state-of-the-art SN methods for computing α.
Unfortunately, multigroup MC results show lower efficiency at worstand margin gains at best.
How does this compare in continuous energy?
Slide 21
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MCNP Continuous-Energy Results
About two years ago, implemented prototype for c-α methods inresearch version of MCNP6.
Since all histories are a single track, may use Newton-Rhapson tocompute α that makes c = 1 after α source is converged.
Significant improvements observed over MCNP’s internal k-α method,but is an“apples to oranges” comparison of methods because:
Different approaches used. Difficult to do Newton-Rhapson withmultiple track histories.k eigenvalue routines in MCNP are inherently more complicated havingmore capabilities.Correlation effects of the c eigenvalue are not well understood (mostlikely a larger effect than k).
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Noise in c Versus k (3D-PWR)
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
0 50 100 150 200 250 300 350 400 450 500
Eig
enva
lue
Col
lisio
n E
stim
ate
Cycle
kc
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Continuous-Energy MC Test Problems
Four test problems in order of complexity:
1. Supercritical, bare HEU sphere (Godiva with elevated density)
2. Supercritical, Be-reflected HEU sphere
3. Slightly subcritical can of Pu-Nitrate solution
4. Nearly critical 3-D Pressurized Water Reactor (Hoogenboom-Martinbenchmark)
Slide 24
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Continuous-Energy MC Results
k-α c-α
Case α FOM α FOM
1 0.176 9.2× 103 0.176 2.0× 104
2 1.468 5.9× 104 1.474 5.4× 105
3 Failed — −3.181× 10−4 2604 2.425× 10−5 19 1.944× 10−5 44
Slide 25
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Summary & Future Work
α eigenvalue iterations explored with multiplication k, collision c , andleakage l eigenvalues.
SN results show moderate improvement with using c as opposed to k.
MC results on equivalent multigroup problems shows generaldisadvantage to using other eigenvalues over k.
MC results for CE problems are suggestive, but are not a faircomparison. More investigation needed.
Slide 26
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Acknowledgments
Funding provided by the U.S. DOE/NNSA Advanced ScientificComputing Program and Nuclear Criticality Safety Program.
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Questions?
Slide 28
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