SNURPL1 / 35

Monte Carlo method

May 4, 2010

Prof. Joo Han-gyu

Department of Nuclear Engineering

Reactor Numerical Analysis and Design1st Semester of 2010

Lecture Note 7

SNURPL2 / 35

Statistical Nature of Neutron Reaction

Probability of reaction for a neutron traveling unit distance

Macroscopic Cross section

tdn n dx

( ) t x

tp x e

( , ) : probability for a neutron born som ewhere in the core

to have a collision around , with , (phase space)

p p r E dVdE

r E dV dE

Probability to react within unit distance after traveling x

Mean free path

00 0

( ) t tx x

txp x dx x e dx xe

0

1t x

t

e dx

In core, collision probability

0( ) t x

n x n e

: survival probability after traveling t xe x

1p ≪

? t

SNURPL3 / 35

Statistical Nature of Neutron Reaction

Expected value?

Number of collision for N neutrons born

( , ) 1N kk

N kk N C p p

for collisions out of neutrons, possible number of occurrences k N

of : coeff of the -th order term ( 1 )N

kNa p q i q p

= probability for neutrons (out of N neutrons) to have collisionsk

Expected value of k

0 0

( , ) ?

N N

k N k

N k

k k

k k N k C p q

Expected variance ?

N p

SNURPL4 / 35

Moment Generating Function for PDF

PDF : Probability

Density

Function

0

( ) ( )

N

tx

X

k

M t e p x

The moment-generating function of a random variable is X

probability density function If has a continuous ( ) , then the M GF is given byX p x

( ) ( )tx

XM t e p x dx

( ) ( ) (0) ( ) tx

X XM t xe p x dx E X M xp x dx

( ) ( )( ) ( ) (0) ( )

n n tx n n n

X XM t x e p x dx E X M x p x dx

If has a discrete probability density function ( ) , then the MGF is given byX p x

0 0

( ) ( ) (0) ( )

N N

tx

X X

i i

M t xe p x E X M xp x

2 2 2

0 0

( ) ( ) (0) ( )

N N

tx

X X

i i

M t x e p x E X M x p x

( ) ( )tx

XM t E e

SNURPL5 / 35

Moment Generating Function for PDF

0

( )

N

tk k N k

K N k

k

M t e C p q

E k Np

1

( )N

t t

KM t N pe q pe

Expected value of using MGFk

1(0) ( )

N

KM N p q p Np

0

(0) ( )

N

K

i

E k M k p k

0

Nk

t N k

N k

k

C pe q

N

tpe q

SNURPL6 / 35

Moment Generating Function for PDF

2

2

0

,

N

N

k

k k k N

Expected variance of using MGFk

2 2

0 0 0

, 2 , ,

N N N

k k k

k k N k k k N k k N

2 2

0

2 ,

N

k

k kk k k N

2 22k k k k

0

0

,

, 1

N

k

N

k

k k N k

k N

2 2k k

22

(0) (0)K K

M M

Var k Npq

2 2 2

1N N p Np N p

2

(0) 1K

M N N p Np

2 1

2 2( ) 1

N Nt t t t

KM t N N pe q p e N pe q pe

1Np p

1

( )N

t t

KM t N pe q pe

( )N

t

KM t pe q

( ) R elative standard deviation:

Var k Npq q

k Np Np

~ 1 for 1q p

11

Np N

SNURPL7 / 35

Estimation of p

Estimation of p

Since is unknown, let's find by experimentp p

Repeated Simulation w neutrons/ci eth ycln

Let where is the number of cyclesN n C C

1

1n

i ij

j

p pn

1 if collision in phase space

cycle , neutron0 otherwise

ijp i j

1

1C

i

i

p pC

22

1

1

1

C

s i

i

p pC

2

2 i

s

pp

C

Expected Variance: of p

2

2 1

1

C

i

i

p p

pC C

1 for sam ple variance because of reduced degree of freedom

due to the use in the sam ple m ean. T w o data one deviation.

(conservative estim ation of the real standard dev of the population)

C

SNURPL8 / 35

Central Limit Theorem

the average of a large num ber of independent and identically-distributed random variables

w ill be approxim ately norm ally distributed (i.e., follow ing a G aussian distribution)

if the random var

iables have a finite variance

1 2 3

2

Let x , x , x , ... be a set of independent and identically distributed random va riables

having finite values of mean and variance > 0.

n

2

C etral lim it theorem states that

as the sam ple size increases,

the distribution of the sam ple average approaches the norm al distribution

w ith a m ean and variance irrespec tive

n

n

of the shape of the original distribution.

SNURPL9 / 35

PDF, CDF and Golden Rule

1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Pro

ba

bili

ty

Case

PDF : Probability Density Function

CDF : Cumulative Distribution Function

( ) : probabilityp x dx

x

f x dx

PDF CD F

Total Area=1

Golden Rule

W ith a CDF given, an event can be picked by a random number .

1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Pro

ba

bili

tyCase

SNURPL10 / 35

Probability of Collision at x with two regions

0 0

A A B

a sl a l

A BP x e dl e e dl

A

A B

B

,x a0

s

1 2 3 4 5

0.2

0.4

0.6

0.8

1

Aae

1 Aae

xa

1 ,

1 1 ,

A

A A B

x

a a s

e x a

e e e x a

PD F

C D F

: A x

AA p x e

Probability of collision within x

Length of Travel to first collision for a random number

0

( )x

B B

l dla s

B Be e e

( )s x a : A Ba s

BB p x e e

SNURPL11 / 35

MC Simulation of Neutron Migration in 2D with MG Xsec

Problem Geometry 2D core with assembly-wise Xsec

- No heterogeneous configuration within Assembly

- Constant xsec over each FA

Subdivision of each assembly into cells

- Flux averaged over each cell

Neutrons are moving in 3D space, but axially invariant neutron

behavior

Radial BC can be either reflective or vacuum

Cross Sections Multigroup macroscoic cross sections given with scattering

matrices

Scattering is assumed to be isotropic

1 2 1 20 ( , , ) ( , , ) for any and .x y z x y z z z

z

1w hen using diffusion solver data

3t tr

D

SNURPL12 / 35

Things to Consider in Simulating Stochastic Events

Occurring a Neutron Flight

Characterization of a neutron Position (x,y,z)

Moving direction

Energy (g)

Weight (w)

Position of collision in space How long to move from the current position?

Type of interaction absorption, scattering, fission

Scattering mechanism Deflected angle after scattering

Energy after scattering

Fission neutrons emitted Number

Location : Source point in the next simulation

( , ) or ( , , )x y z

SNURPL13 / 35

Shooting a neutron and first move

Get the position of fission source neutron from the storagei th

Determine the energy group of emission using g

Determine the angle of emission assuming isotropic emisssion

- using the angle pickup scheme on the next slide

determine the initial cell number

Determine the first travel distance

SNURPL14 / 35

Selection of Angle

2 2 21

x y z

sin cos

sin sin

cos

x

y

z

Select : 0 ~ 2

Select : 0 ~

2 0 1

( )p d Csin d

2 2

1 2 0 1

sin 1 (cos ) 1

z

z

0.5 1 1.5 2 2.5 3

0.5

1

1.5

2

0

1sin

2d

1(1 )

2z

1.0

0.5

0.75

0.25

x

y

z

x

y

z

0 0

1( ) 2

2p d C sin d C C

0

1: ( ) ( ) 1

2CDF P p d cos

1 1

1

12

z

SNURPL15 / 35

Distance to Move

C : Cummulative Distribution Functions 00

1t t t

s ss s s

tC s e ds e e

( ) (random variable)C s

1 t se

2 2sin 1 cos 1 ( )

xy zs s s s

Straight path of distance

ln(1 ) ln,

t t

s or

x

y

z

s

xys

For given , probality to have collision in after traveling :t

dx s ( ) t s

tp s dx e dx

cos ; sinx xy y xy

s s

SNURPL16 / 35

Segmented Simulation of Flight Across Cells

1 2 3 4 5

0.2

0.4

0.6

0.8

1

Aae

1 Aae

xa

A

B

( ) 1 1A A Ba a sP s e e e

as

Once-through simulation

Segmeted simulation

Requires complicated CDF for more than two cells

Stop first at surface if the first rand om flight distance is obtained longer

than the distance to surface (DTS)

what would be the probability to stop at surface?

Pick a random number for another random flight using B

AaP e

( | ) 1 A Aa sP B A e e

A=Event to move to surface in cell A

B=Event to move by s in cell B Much simpler

SNURPL17 / 35

T he four com bination of sign of x,y directional vector ( , ) determ ines

a possible surface to be reached after c ollision

x y

Determination of Distance to Surface

SNURPL18 / 35

Collision in Current Cell

Fission

Scattering

Absorption

- can occur w ith a prob. ,fg

tg

neutrons per fission

- expected number of fission neutrons per collisionfg

tg

- integer number of fission neutrons fg

tg

int

( 1 ( )) ( 1) ( 1 ( ))a a a a

E int a i P a i i P a i

- can occur w ith a prob.

- group transfer to g can occur w ith , ne ed angle selection

ag

tg

gg

sg

- if not scattering, absorption.

Terminate simulation of the current neutron

R ecord fission for every collision

for m ore spreaded fission sites than

fission only after absorption

Outscattering!

1 ( ) ( 1)( )a a a a

i a i i a i a

SNURPL19 / 35

Treatment of Boundary

Determine if the surface is external boundary

If vacuum surface, terminate migration

If reflective surface, change the moving direction

accordingly

Then sample another move

. . reflection on surface

, ,x x y y z z

e g y z

SNURPL20 / 35

Flow Chart of Simulation of a Neutron Migration

Lxy < DtsMove to surface

Pick up N

(Number of fission)

Determine free flight

distance (L)

, Projection to Plane (Lxy)

Determine distance to

surface (Dts)

n = n+1

(Record position)

Scatter ?

Externel

surface ?

Determine position ,

direction, group

Terminate

(Absorption)

Determine

destination group

Determine direction

ReflectiveTerminate

(vacuum)

Change directionNo

(Absorption)

YesYes

No

Yes

No

Yes No

(In the cell)

fg

tg

int

Record collision of

current cell , s sx x y y

x x y y

for i=1:C

for j=1:n(i)

m igrate

end

( 1)

( )

end

n ik k

n i

shoot

m igra te

collision

SNURPL21 / 35

Flux Scoring (Tally)

Reaction rate of a cell : Rc c c

g tg g cV

C Estimatorollision c

gc

g c

c tg

R

V

Accum ulate the num ber of collisions. S ince source is given per unit tim e,

the scored num ber of collisions can be regarded as collision rate.

T hus dividing it by the volum e and total X sec yields

flux.

Length EstiTrack mator

Expected number of reactions for a track of length formed within a cell:l

Total number of expected reactions within a cell:

Reaction Rate:c c c c

g tg g c tg i

i

R V l

N ote on tim e dim ension

- Although steady-state condition is s im ulated in M C calculation, w e should ad im t that source is

in fact given per unit tim e

where is the track length generated per unit time (in 3D)c

il

track den1

sity!c c

g i

c i

lV

Neutrons passing through a cell should contribute to flux, but missing

in the collision estimator if the neutron just passes a cell, not making any collision

Rl c

g tgl

c c c c

g tg i tg i

i i

R l l

SNURPL22 / 35

Fission Source Treatment

Expected number of fission neutrons at the subsequent generation

in a multiplying medium of 1 for particles simulated eff

k n

effk n

( 1)

1Adjustment of by =

ck

Avoid amplification by in advance

Fission neutron sampling

effk

( )adjust by:

fgc

i

tg

n int

( )New estimate of

ck

( ) ( )Sum up all the fission neutrons generated:

c c

i

i

n n ( )

( ) ( 1)

( 1)

c

c c

fis c

nk k

n

( 1)

1return to normal by correcting the artificial adjustment of by

ck

( ) ( ):

c c

trk i fg

i

Track n l

( )

( )

( 1)

c

c col

col c

nk

n

( ):

fgc

col

i ti

Collision n

( )

( )

( 1)

c

c trk

trk c

nk

n

SNURPL23 / 35

Inactive vs. Active cycles

need inactive cycles to change the spatial source distribution

from the initial definition to a correct distribution for the problem

Active cycle

InActive cycleCycle

avoid excessively large variance encountered in inactive cycles by avoiding tallies

in this phase

because of the unknown source distribution, initial guess of source distribution should be given

causing a lot of variation at the initial phase of cycles.

SNURPL24 / 35

Statistical Processing

Sample mean and variance after active cycle simulations C

( )

1

1

ia

ia

N C

c

col col

c N

k kC

2 ( ) 2

,

1

1 ( )

1

ia

ia

N C

c

k col col col

c N

s k kC

Standard deviation of the sample mean

,

,

k col

k col

s

C

- C onfidence level

( ) 2

1

1 ( ) ( 1)

ia

ia

N C

c

col col

c N

k k C CC

: 68.3%

2 : 95.4%

3 : 99.7%

k

k

k

SNURPL25 / 35

Control Rod

Fission

Absorption

Scattering &

Moderation

Fission chamber

WaterFuel Assembly

Fuel Assembly

Fuel Assembly

Fuel Assembly

Example of particle tracing

L336C 5G 2>

SNURPL26 / 35

Example of particle tracing

L336C 5G 2>

SNURPL27 / 35

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0 50 100 150 200 250 300 350 400 450 500

Cycle

K-eff variation for C5G2 Model Problem

20, 000batch

n

SNURPL28 / 35

Variance Reduction Methods

Need and Approach Reduce the variances of tallies by modifying neutron behavior

- Save neutrons as much as possible instead of killing them while

conserving the mean values of tallies

Analogue vs. Non-Analogue Monte Carlo

Typical MethodsWeight Window Method

Implicit Capture

SNURPL29 / 35

Particle Weight

Concept A single particle being traced in MC simulation is not a single

neutron, rather a group of neutrons

The number of neutrons may be adjusted during the simulation

by introducing weight for each particle

Weight Relative number of neutrons represented by a particle

1

1

1 0

Fission

10

0

Fission10

10

1

10

11

1eff

k 10

110

effk

SNURPL30 / 35

Implicit Capture

Background When an absorption reaction occurs, all the neutrons represented by

the particle disappears at once in analogue MC. This can magnify the

variance of MC calculation

Implementation Do not kill the particle when absorption is selected. Instead, make it

survive with less weight

Weight reduction with survival probability

Collision / Track length / Fission Neutron adjustment

. .f

fis

t

e g P w

= 1sa

t t

w w w

SNURPL31 / 35

Termination of Implicit Capture

Deterministic weight reduction after each collision

Termination Insignificant contribution to tallies if w is too low

Kill if w<w_low, but with a certain probability

Weight adjustment to conserve the weight

- Adjusted weight can potentially lower then the limit

Fixed Adjusted Weight (w_0)

- Adjust survival probability to

DTC

w

s

t

w

2

s

t

w

3

s

t

w

0

0

X

,

lowif w w and

otherw i w

w

w

se w

(0.25) (0.4)low S

w w and

S

ww

( ) 0 (1 )

S S

S

wE w

0

0 0 0

( ) 0 (1 )w w w

E w ww w w

n

s

t

w

02 0.5w if w

SNURPL32 / 35

Modified Calculation Flow for Implicit Capture

Starts

Migration

Routine

Determine

Fission neutron

n = int

( λ*(νΣf/Σt)*w + ζ )

w < 0.25 ?

End

Energy group

change

Direction

change

ζ > w/0.5 ?

Weight reduction

w=w*(Σs/Σt)

Recover the w

w=0.5

Russian Roulette

n = 0 ?Save the fission

site

Starts

Calc

DTC, DTS

Move to

position

DTC < DTS ?Move to

surface

System

Boundary ?

Reflective?

End(leakage)

Flips

Collision

Routine

Transition to

Next region

No

Updates

xsec

migration routine> collision routine>

N

Y

N

Y

SNURPL33 / 35

Effectiveness of Implicit Capture

0.930

0.932

0.934

0.936

0.938

0.940

0.942

0.944

0.946

0.948

0.950

Implicit capture

<Case2 : n=10,000>

Estim

ate

d k

eff

<Case1 : n=1,000>

Reference k=0.93960

Analogue MC

R esult of L336C 5G 2>