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Page 1: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

Form 836 (7/06)

LA-UR-Approved for public release;distribution is unlimited.

Los Alamos National Laboratory, an affirmative action/equal opportunity employer, is operated by the Los Alamos National Security, LLCfor the National Nuclear Security Administration of the U.S. Department of Energy under contract DE-AC52-06NA25396. By acceptanceof this article, the publisher recognizes that the U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce thepublished form of this contribution, or to allow others to do so, for U.S. Government purposes. Los Alamos National Laboratory requeststhat the publisher identify this article as work performed under the auspices of the U.S. Department of Energy. Los Alamos NationalLaboratory strongly supports academic freedom and a researcher’s right to publish; as an institution, however, the Laboratory does notendorse the viewpoint of a publication or guarantee its technical correctness.

Title:

Author(s):

Intended for:

06-7094

Monte Carlo Eigenvalue Calculations

Forrest Brown

Monte Carlo lectures

Page 2: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

1

Monte CarloEigenvalue Calculations

Forrest Brown

Monte Carlo Codes (X-3-MCC)Los Alamos National Laboratory

[email protected]

LA-UR-06–7094

Page 3: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Abstract

Monte Carlo Eigenvalue Calculations

F Brown, X-3-MCC

This talk will cover 4 aspects of Monte Carlo eigenvalue calculations:

1. Formulation of the k- and alpha-eigenvalue equations from the time-dependent linear Boltzmann transport equation

2. The power iteration method for solving the equations & its convergencebehavior

3. The use of Shannon entropy of the fission source distribution for assessingconvergence

4. A novel application of Wielandt's method to accelerate the convergence.

LA-UR-06–7094

Page 4: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Monte Carlo Eigenvalue Calculations

• K- and -Eigenvalue Equations

• Power Iteration & Convergence

• Shannon Entropy for Convergence Analysis

• Wielandt Acceleration

Page 5: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

4

Reactor Analysis with Monte Carlo

Geometry Model (1/4) K vs cycle Hsrc vs cycle

Assembly Powers Fast Flux Thermal Flux

Page 6: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

5

K- and -Eigenvalue Equations

Page 7: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Time-dependent Transport

• Time-dependent linear Boltzmann transport equation forneutrons, with prompt fission source & external source

• This equation can be solved directly by Monte Carlo, assuming:– Each neutron history is an IID trial (independent, identically distributed)

– All neutrons must see same probability densities in all of phase space– Usual method: geometry & materials fixed over solution interval t

1v

(r,E, ,t)t

= Q(r,E, ,t) + (r,E , ,t) S(r,E E, ,t)d dE

+(r,E,t)4 F(r,E ,t) (r,E , ,t)d dE

+ T(r,E,t) (r,E, ,t)

1v

(r,E, ,t)t

= Q + [S +M] [L + T]

Without materialmotion corrections

Page 8: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Time-dependent Transport

• Monte Carlo solution (over t, with fixed geometry & materials)– Simulate time-dependent transport for a neutron history– If fission occurs, bank any secondary neutrons.– When original particle is finished, simulate secondaries till done.

– Tallies for time bins, energy bins, cells, …

• At time t, the overall neutron level is

• Alpha can be defined by: N(t) = N(0) e t

This is the "dynamic alpha", NOT an eigenvalue !

1v

(r,E, ,t)t

= Q + [S +M] [L + T]

ln N2 ln N1t2 t1

N(t) =(r,E, ˆ ,t)v

r,E, ˆ

drdEd ˆ

Page 9: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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• Random Walk for particle

• Particle History

Particle Histories

Track through geometry,- select collision site randomly- tallies

Collision physics analysis,- Select new E, randomly- tallies

SecondaryParticles

Source- select r,E,

RandomWalk

RandomWalk

RandomWalk

RandomWalk

RandomWalk

RandomWalk

Page 10: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Fixed-source Monte Carlo Calculation

Source- select r,E,

RandomWalk Random

Walk

RandomWalkRandom

Walk

RandomWalk

RandomWalk

Source- select r,E,

RandomWalk Random

Walk

RandomWalk

RandomWalk

RandomWalk

Source- select r,E,

RandomWalk Random

Walk

RandomWalk

RandomWalk

RandomWalk

RandomWalk

History 1

History 2

History 3

Page 11: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Alpha Eigenvalue Equations

• For problems which are separable in space & time, it may be advantageousto solve a static eigenvalue problem, rather than a fully time-dependentproblem

• Assume:1. Fixed geometry & materials2. No external source: Q(r,E, ,t) = 03. Separability: (r,E, ,t) = (r,E, ) e t,

• Substituting into the time-dependent transport equation yields

• This is a static equation, an eigenvalue problem for and withouttime-dependence

• is often called the time-eigenvalue or time-absorption• -eigenvalue problems can be solved by Monte Carlo methods

+ T(r,E) + v(r,E, ) = (r,E , ) S(r,E E, )d dE

+(E)4 F(r,E ) (r,E , )d dE

Page 12: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Keff Eigenvalue Equations

• Another approach to creating a static eigenvalue problem from the time-dependent transport equation is to introduce Keff, a scaling factor on themultiplication ( )

• Assume:1. Fixed geometry & materials2. No external source: Q(r,E, ,t) = 03. / t = 0: /keff

• Setting / t = 0 and introducing the Keff eigenvalue gives

• This is a static equation, an eigenvalue problem for Keff and k withouttime-dependence

• Keff is called the effective multiplication factor• Keff and k should never be used to model time-dependent problems.• Keff-eigenvalue problems can be solved by Monte Carlo methods

+ T(r,E) k(r,E, ) = k(r,E , ) S(r,E E, )d dE

+1Keff

(E)4 F(r,E ) k(r,E , )d dE

Page 13: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Comments on Keff and Equations

• CriticalitySupercritical: > 0 or Keff > 1

Critical: = 0 or Keff = 1

Subcritical: < 0 or Keff < 1

• Keff vs. eigenvalue equations

– k(r,E, ) (r,E, ), except for a critical system

– eigenvalue & eigenfunction used for time-dependent problems– Keff eigenvalue & eigenfunction used for reactor design & analysis– Although = (Keff-1)/ , where = lifetime,

there is no direct relationship between k(r,E, ) and (r,E, )

• Keff eigenvalue problems can be solved directly using Monte Carlo

• eigenvalue problems are solved by Monte Carlo indirectly using aseries of Keff calculations

Page 14: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Comments on Keff and Equations

K equation [ L + T ] k = [S + 1/k M ] k

equation [ L + T + /v ] = [S + M ]

• The factor 1/k changes the relative level of the fission source

• The factor /v changes the absorption & neutron spectrum– For > 0, more absorption at low E harder spectrum– Double-density Godiva, average neutron energy causing fission:

k calculation: 1.30 MeV calculation: 1.68 MeV

• For separable problems, (r,E, ,t) = (r,E, ) e t

• No similar equation for k, since not used for time-dependence

Page 15: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Power Iteration&

Convergence

Page 16: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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K-eigenvalue equation

whereL = leakage operator S = scatter-in operatorT = collision operator M = fission multiplication operator

• Rearrange

This eigenvalue equation will be solved by power iteration

(L + T) = S +1KeffM

(L + T S) =1KeffM

=1Keff

(L + T S) 1M

=1Keff

F

(n+1)=

1Keff(n) F (n)

Page 17: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Power Iteration

Diffusion Theory orDiscrete-ordinates Transport

1. Initial guess for Keff and Keff

(0), (0)

2. Solve for (n+1)

Inner iterations over space or space/angle to solve for (n+1)

3. Compute new Keff

4. Repeat 1–3 until both Keff(n+1) and

(n+1) have converged

Monte Carlo

1. Initial guess for Keff and Keff

(0), (0)

2. Solve for (n+1)

Follow particle histories to solve for (n+1)

During histories, save fission sitesto use for source in next iteration

3. Compute new Keff

During histories for iteration (n+1), estimate Keff

(n+1)

4. Repeat 1–3 until both Keff(n+1) and

(n+1) have converged5. Continue iterating, to compute tallies

(L + T S) (n+1)=

1Keff(n) M (n) (L + T S) (n+1)

=1Keff(n) M (n)

Keff(n+1)

= Keff(n)

M (n+1)dr

M (n)dr

Keff(n+1)

= Keff(n) 1iM

(n+1)

1iM (n)

Page 18: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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InitialGuess

Batch 1Keff

(1)Batch 2

Keff(2)

Batch 3Keff

(3)Batch 4

Keff(4)

Batch 1Source

Batch 3Source

Batch 4Source

Batch 5Source

Batch 2Source

Power Iteration

• Power iteration for Monte Carlo k-effective calculation

Source particle generation

Monte Carlo random walkNeutron

Page 19: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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-Eigenvalue Calculations

• Eigenvalue equation with both Keff & – is a fixed number, not a variable

– Find the k-eigenvalue as function of , K( )

• Note: If < 0– Real absorption plus time absorption could be negative

– Move /v to right side to prevent negative absorption,– - /v term on right side is treated as a delta-function source

– Select a fixed value for – Solve the K-eigenvalue equations, with fixed time-absorption /v

– Select a different and solve for a new Keff– Repeat, searching for value of which results in Keff = 1

+ T(r,E) + v(r,E, ) = (r,E , ) S(r,E E, )d dE

+1Keff

(E)4 F(r,E ) (r,E , )d dE

Page 20: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Monte Carlo

Monte Carlo

K- and -Eigenvalue Calculations

• K-eigenvalue solution

Loop for Power Iteration for K• Loop over neutrons in cycle• • neutron history• • • •

• • •

• -eigenvalue solution

Loop for search iterations• Loop for Power Iteration for K• • Loop over neutrons in cycle• • • neutron history• • • • •• • • •

• • •

Find K( ), then solve for that gives K( )=1

Page 21: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Power Iteration

• Guess an initial source distribution• Iterate until converged (How do you know ???)• Then

– For Sn code: done, print the results– For Monte Carlo: start tallies,

keep running until uncertainties small enough

• Convergence? Stationarity? Bias? Statistics?

Monte CarloDeterministic (Sn)

Discard Tallies

Keff(n)

Iteration, n

Page 22: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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• Expand in terms of eigenfunctions uj(r,E, )

• Expand the initial guess in terms of the eigenmodes

• Substitute the expansion for (0) into power iteration equation

= ajujj=0

= a0u0 + a1u1 + a2u2 + a3u3 + .....

ujukdV = jk aj = ujdV

uj =1kjF uj k0 > k1 > k2 > ... k0 keffective

Power Iteration – Convergence

(0)= aj

(0)ujj=0

(n+1)=1K(n)

F (n)=1k(n)

1k(n 1)

...1k(0)

Fn (0)

=k0K(m)m=0

n

a0(0) u0 +

aj(0)

a0(0)

kjk0

n+1

ujj=1

Page 23: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Power Iteration – Convergence

• Because k0 > k1 > k2 > …, all of the red terms vanish as n– (n+1) constant u0

– K(n+1) k0

• After the initial transient, error in (n) is dominated by first mode– ( k1 / k0 ) is called the dominance ratio, DR or – Errors in (n) die off as ~ (DR)n

• For problems with a high dominance ratio (e.g., DR ~ .99),the error in Keff may be small, since the factor (k1/k0 – 1) is small.– Keff may appear converged,

even if the source distribution is not converged

(n+1) [cons tant] u0 +a1(0)

a0(0)

k1k0

n+1

u1 + ...

K(n+1) k0 1 +a1(0)

a0(0)

k1k0

nk1k0

1 G1 + ...

Page 24: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Power Iteration – Convergence

Typical K-effective convergence patterns

• Higher mode error terms die out as ( kJ / k0 )n, for n iterations

• When initial guess is concentrated in centerof reactor, initial Keff is too high(underestimates leakage)

• When initial guess is uniformly distributed,initial Keff is too low (overestimates leakage)

• The Sandwich Method uses 2 Keff calculations -one starting too high & one starting too low.Both calculations should converge to the same result.

K

Iteration, n

K

Iteration, n

Page 25: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Power Iteration – Convergence

• Keff is an integral quantity – converges faster than sourceshape

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 100 200 300 400 500 600 700 800 900 1000

cycle

va

lue

div

ide

d b

y t

rue

me

an

keff

source inright slab

Keff calculation for 2 nearly symmetric slabs, with Dominance Ratio = .9925

Page 26: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Shannon Entropy of theFission Source DistributionFor Assessing Convergence

Page 27: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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The Challenge

– In the old days, when people used Monte Carlojust to compute K-effective, plots of kcycle vscycle were adequate to judge convergence

– Today, for computing power distributions &localized reaction rates, new tools are needed tojudge local convergence of source distribution

• K-effective converges before the sourcedistribution converges

• How do you tell if a 3D distribution hasconverged ?

Page 28: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Source Distribution Convergence

Geometry Model (1/4) K vs cycle Hsrc vs cycle

Assembly Powers Fast Flux Thermal Flux

Page 29: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Keff Calculations

• Initial cycles of a Monte Carlo K-effective calculation should bediscarded, to avoid contaminating results with errors from initial guess– How many cycles should be discarded?– How do you know if you discarded enough cycles?

• Analysis of the power iteration method shows that Keff is not a reliableindicator of convergence — Keff can converge faster than the sourceshape

• Based on concepts from information theory (not physics),Shannon entropy of the source distribution is useful forcharacterizing the convergence of the source distribution

Discard Tallies

Keff(n)

Iteration, n

Page 30: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Keff Calculations – Stationarity Diagnostics

• Divide the fissionable regions of the problem into NS spatial bins– Spatial bins should be consistent with problem symmetry– Typical choices: — 1 bin for each assembly

— regular grid superimposed on core– Use dozens or hundreds of bins, not thousands

• During the random walks for a cycle, tally the fission source points ineach bin– Provides a discretized approximation to the source distribution– { pJ, J=1,NS }

• Shannon entropy of the source distribution

H(S) = pJ ln2(pJ ), where pJ =(# source particles in bin J)

(total # source particles in all bins)J=1

NS

Page 31: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Keff Calculations – Stationarity Diagnostics

• Shannon entropy of the source distribution

– 0 H(S) ln2( NS )

– For a uniform source distribution, H(S) = ln2( NS )since p1 = p2 = … = pNs = 1/NS

– For a point source (in a single bin), H(S) = 0

• H(S(n)) provides a single number to characterize the source distribution for iteration n (no physics!)

As the source distribution converges in 3D space,a line plot of H(S(n)) vs. n (the iteration number) converges

H(S) = pJ ln2(pJ ), where pJ =(# source particles in bin J)

(total # source particles in all bins)J=1

NS

Page 32: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Keff Calculations – Stationarity Diagnostics

• Example – Reactor core (Problem inp24)

K(n) vs cycle

H( fission source )

Keff

50

80

Page 33: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Keff Calculations – Stationarity Diagnostics

• Example – Loosely-coupled array of spheres (Problem test4s)

K(n) vs cycle

H( fission source )

Keff

75

85

Page 34: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Keff Calculations – Stationarity Diagnostics

• Example – Fuel Storage Vault (Problem OECD_bench1)

K(n) vs cycle

H( fission source )

20 ?

2000

Page 35: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Keff Calculations – Stationarity Diagnostics

• Example – PWR 1/4-Core (Napolitano)

K(n) vs cycle

H( fission source )

25

50

Page 36: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Keff Calculations – Stationarity Diagnostics

• Example – 2D PWR (Ueki)

K(n) vs cycle

H( fission source )

25

50

Page 37: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Source Entropy & MCNP5

• Grid for computing Hsrc

– User can specify a rectangular grid in inputhsrc nx xmin xmax ny ymin ymax nz zmin zmax

example: hsrc 5 0. 100. 5 0. 100. 1 -2. 50.

– If hsrc card is absent, MCNP5 will choose a grid based on the fissionsource points, expanding it if needed during the calculation

• MCNP5 prints Hsrc for each cycle

• MCNP5 can plot Hsrc vs cycle

• Convergence check at end of problem– MCNP5 computes the average Hsrc and its population variance H

2

for the last half of the cycles– Then, finds the first cycle where Hsrc is within the band <Hsrc> ± 2 H

– Then, checks to see if at least that many cycles were discarded

Page 38: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Summary

• Local errors in the source distribution decay as ( kJ/k0 )n

– Higher eigenmodes die out rapidly, convergence dominated by k1/k0

– High DR slow convergence– High DR large correlation large error in computed variances

• Errors in Keff decay as (kJ/k0 – 1) * ( kJ/k0 )n

– High DR kJ/k0 ~ 1 small error

• Keff errors die out faster than local source errors– Keff is an integral quantity – positive & negative fluctuations cancel

• Shannon entropy of the fission source distribution (Hsrc) is an effectivediagnostic for source convergence– Now part of standard MCNP5 (beginning with version 1.40, November 2005)– Basis for initial source convergence tests — more are coming

If local tallies are important (e.g., assembly power, pin power, …),examine convergence using Hsrc - not just Keff convergence

Page 39: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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WielandtAcceleration

Page 40: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Wielandt Method

• Basic transport equation for eigenvalue problems

L = loss to leakage S = gain from scatter-inT = loss to collisions M = gain from fission multiplication

• Define a fixed parameter ke such that ke > k0 (k0 = exacteigenvalue)

• Subtract from each side of the transport equation

• Solve the modified transport equation by power iteration

(L + T S) =1KeffM

1keM

(L + T S 1keM) = ( 1

Keff1ke)M

(L + T S 1keM) (n+1)

= ( 1Keff(n)

1ke)M (n)

Page 41: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Wielandt Method

• Power iteration for modified transport equation

• How to choose ke

– ke must be larger than k0 (but, don't know k0!)– ke must be held constant for all of the histories in a batch,

but can be adjusted between batches• Typically, guess a large initial value for ke, such as ke=5 or ke=2• Run a few batches, keeping ke fixed, to get an initial estimate of Keff

• Adjust ke to a value slightly larger than the estimated Keff

• Run more batches, possibly adjusting ke if the estimated Keff changes

(L + T S 1keM) (n+1)

= ( 1Keff(n)

1ke)M (n)

(n+1)= ( 1

Keff(n)

1ke) (L + T S 1

keM) 1M (n)

(n+1)=

1K(n)

F (n)

where K(n)= ( 1

Keff(n)

1ke) 1 or Keff

(n)= ( 1

K(n)+

1ke) 1

Page 42: JetForm:836 - mcnp.lanl.gov · 06-7094 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo lectures. 1 Monte Carlo Eigenvalue Calculations Forrest Brown Monte Carlo Codes

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Wielandt Method

• Convergence– Eigenfunctions for the Wielandt method are same as for basic power iteration– Eigenvalues are shifted:

– Expand the initial guess, substitute into Wielandt method, rearrange to:

– Additional factor (ke-k0)/(ke-k1) is less than 1 and positive, so that the redterms die out faster than for standard power iteration

(n+1) [cons tant] u0 +a1(0)

a0(0)

ke k0ke k1

k1k0

n+1

u1 + ...

K(n+1) k0 1 +a1(0)

a0(0)

ke k0ke k1

k1k0

nke k0ke k1

k1k0

1 G1 + ...

kJ =

1kJ

1ke

1ke > k0 > k1 > ...

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Wielandt Method

• The dominance ratio for this modified power iteration is

– Since ke > k0 and k0 > k1, DR' < DR– DR of Wielandt method is always smaller than standard power iteration

• Wielandt acceleration improves the convergence rate of the poweriteration method for solving the k-eigenvalue equation

Weilandt method converges at a faster rate than power iteration

DR =k1k0

=[ 1k1

1ke] 1

[ 1k01ke] 1

=ke k0ke k1

k1k0

=ke k0ke k1

DR

Standard power iteration

K(n)

Iteration, n

Power iteration with Wielandt acceleration

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Wielandt Method

• Monte Carlo procedure for Wielandt acceleration

• For standard Monte Carlo (power iteration) in generation n+1– When a collision occurs, the expected number of fission neutrons produced is

– Store nF copies of particle in the fission-bank for the next generation (n+2)

• For Monte Carlo Wielandt method in generation n+1– When a collision occurs, compute 2 expected numbers of fission neutrons

– Note that E[ n'F + n'e ] = E[ nF ]– Follow n'e copies of the particle in the current generation (n+1)– Store n'F copies of particle in the fission-bank for the next generation (n+2)

(L + T S 1keM) (n+1)

= ( 1Keff(n)

1ke)M (n)

nF = wgt F

T

1K(n)

+

nF = wgt F

T

1K(n)

1ke

+ ne = wgt F

T

1ke

+

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InitialGuess

Batch 1Keff

(1)Batch 2

Keff(2)

Batch 3Keff

(3)Batch 4

Keff(4)

Batch 1Source

Batch 3Source

Batch 4Source

Batch 5Source

Batch 2Source

Wielandt Method

• Power iteration for Monte Carlo k-effective calculation

Source particle generation

Monte Carlo random walkNeutron

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InitialGuess

Batch 1Keff

(1)Batch 2

Keff(2)

Batch 3Keff

(3)Batch 4

Keff(4)

Batch 1Source

Batch 3Source

Batch 4Source

Batch 5Source

Batch 2Source

Wielandt Method

• Wielandt method for Monte Carlo k-effective calculation

Source particle generation

Monte Carlo random walk

Neutron

Additional Monte Carlo random walks within generation due to Wielandt method

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46

MCNP5 Testing of Wielandt's Method

• Wielandt shift parameterKe

(n+1) = K(n)collision +

Convergence of Hsrc vs Iterations for convergence

= -- black

= 1 -- red

= .1 -- blue

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Wielandt Method

Summary

• Wielandt Method has a lower DR than power iteration– Faster convergence rate than power iteration fewer iterations

– Some of the particle random walks are moved from the nextgeneration into the current generation more work per iteration

– Same total number of random walks no reduction in CPU time

• Advantages– Reduced chance of false convergence for very slowly converging

problems

– Reduced inter-generation correlation effects on variance

– Fission source distribution spreads more widely in a generation (dueto the additional particle random walks), which should result in moreinteractions for loosely-coupled problems

• Wielandt method will be included in next version of MCNP5

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48

References

F. B. Brown, "Fundamentals of Monte Carlo Particle Transport," LA-UR-05-4983, available at http://mcnp.lanl.gov (2005).

Monte Carlo k-effective Calculations

J. Lieberoth, "A Monte Carlo Technique to Solve the Static Eigenvalue Problemof the Boltzmann Transport Equation," Nukleonik 11,213 (1968).

M. R. Mendelson, "Monte Carlo Criticality Calculations for Thermal Reactors,"Nucl. Sci Eng. 32, 319–331 (1968).

H. Rief and H. Kschwendt, "Reactor Analysis by Monte Carlo," Nucl. Sci. Eng.,30, 395 (1967).

W. Goad and R. Johnston, "A Monte Carlo Method for Criticality Problems,"Nucl. Sci. Eng. 5, 371–375 (1959).

Superhistory Method

R.J. Brissenden and A.R. Garlick, "Biases in the Estimation of Keff and Its Errorby Monte Carlo Methods," Ann. Nucl. Energy, Vol 13, No. 2, 63–83 (1986).

Wielandt Method

T Yamamoto & Y Miyoshi, "Reliable Method for Fission Source Convergence ofMonte Carlo Criticality Calculation with Wielandt's Method", J. Nuc. Sci. Tech.,41, No. 2, 99–107 (Feb 2004).

S Nakamura, Computational Methods in Engineering and Science, R. E. KriegerPub. Company, Malabar, FL (1986).