On the other hand, Fluka dose not support parallelization. Therefore, sep-arate jobs were executed 1000 times in series with different random seed number. Fluka calculation was also executed on the same system.
Table 4.1: Code information
PHITS Fluka
Language Fortran 90 Fortran 77
Release format Source and binary Source and binary
Parallel execution Yes No
4.3.3 Variance reduction
One of the problems concerned with MC simulation is computation time required to generate enough statistical accuracy in the simulation. Despite high performance of recent computer hardwares and widespread usage of parallel computing system, computation time required for MC simulation is still too long to design and estimate the particle transport applications with complex and large geometries. In many cases of mixed radiation fields, only very a small fraction against all histories provides scored results in the re-gions of interest.
But there are several methods to reduce calculation time to practical range in the MC codes. This method is known as the variance reduction technique, which is better than increasing computation time. The variance reduction technique gives that users can improve precision of calculations and enable the quantities of interest with the reasonable statistical uncertainty in prac-tical computation time. Without the use of variance reduction techniques in complex and large geometry, MC codes have to run the calculation continu-ously for a long time and still not get statistically reliable results.
The aim of the variance reduction techniques is to improve the efficiency
of MC simulation and provide more precise results. In other word, it is to increase the precision and decrease the relative error. The variance reduc-tion parameters depend on the type of calculareduc-tion and therefore an adequate consideration is required to determine the appropriate parameters (59).
Particle Splitting and Russian roulette are most widely used classical method among the variance reduction technique and summarized as follow-ing:
1. Particle splitting
When a particle moves from a region of importance Ii to a more im-portant region Ij, (1 < Ij / Ii), the particle is split into n = Ij / Ii and weight of the identical particles w = Ii / Ij. For example, Fig 4.11 shows a schematic diagram of the particle splitting, when a particle moves from a lower importance region (Ii = 1) to a region with higher importance (Ij = 3). It means that the number of particles is increased to provide better sampling and the weight of the particle is divided.
Splitting increases computation time and decreases variance.
2. Russian roulette
Russian roulette performs completely opposite calculation with the split-ting technique. In case of moving to a less important region Ij, (1 > Ij / Ii), Russian roulette come to action and the particle is killed with prob-ability 1 - (Ij /Ii) and weightw×Ij = Ii. It means that the particles are killed to prevent wasting time. Fig 4.12 shows a schematic diagram of geometry Russian roulette, when a particle moves from a higher impor-tance region (Ii = 1) to a region with lower importance (Ij = 1/3). The objective of these techniques is to spend more time to sampling more
W = 1
W=1/3
Ii= 1 Ij= 3
W=1/3 W=1/3
Lower importance region Higher importance region
Figure 4.11: Schematic diagram of particle splitting
W = 3 W = 1
Ii= 1 Ij= 1/3
Killed
Higher importance region Lower importance region
1/3 probability
2/3 probability
Figure 4.12: Schematic diagram of Russian roulette
In general, when calculating the deep penetration inside shield, a num-ber of particles disappears in a normal simulation, but splitting technique increases the number of particles to move from a region to another. For ac-curate results, it is recommended to keep the ratio of the number of tracks moving in the desired direction to one of particles started from the source constant. Particle splitting and Russian Roulette work well only for calcu-lation without extreme angular dependence (60). In extreme cases, particles do not enter an important cell where the particles split and then the particle splitting and Russian roulette technique become useless.
Weight importance, so-called cell importance or importance biasing,
con-sists in assigning an importance value to each geometry region. The number of particles moving from a region to another will increase via particle split-ting technique or decrease via Russian roulette technique according to the ratio of importances, and the weight of particles will change inversely so that the total weight is kept to remain unchanged.
The concept of weight importance in MC calculation is described by the following equation 4.1:
Φ(E,r)= PN
i=1LiW i
n0·V , (4.1)
whereΦ(E,r) is the particle fluence determined from the scored track lengths, Li is the track length of i-th particle, Wi is the weight of i-th particle, n0 is the total history number, and V is the volume of scoring region.
In this study with both MC codes, to obtain enough statics, particle split-ting and Russian roulette of variance reduction technique were applied in the multiplication of particles crossing the boundary of the region with same conditions, referred as cell importance method in PHITS and importance bi-asing method in Fluka, which can artificially increase the probability of rare event occurrences. Since it is hard to obtain good statics due to the small detection area utilized for neutron counting.
To determine the appropriate importance parameters, several calculations were performed by applying the importance parameter from 1 ∼4 per every 40 cm. The regions applied the multiplication of particles in PHITS code are described in Fig 4.13. Rn and In are region number and importance param-eter to each cell, respectively.
Fig 4.14 shows the effect of importance parameters in PHITS code. The
R12= I12 R11= I11 R10= I10 R9= I9 R8= I8 R7= I7 R6= I6 R5= I5 R4= I4 R3= I3 R2= I2 R1= I1
Figure 4.13: Regions applied the multiplication of particles in PHITS code
code, only one value with a relative error of 100 % was obtained. It is indi-cates data can not be scored without applying the importance parameter.
Result for effect of importance parameters were listed in Table 4.2. A relative error increases when importance parameter was set to 4. Because particles are divided too much, thus, computation time was wasted without improving statistics. It is recommended to set 2 ∼ 3 for ratio of importance parameter between neighboring regions/cells (61).
In this result, the best importance value was 3. However, importance parameter was applied 2 for the calculations with PHITS and Fluka codes under same conditions since the Fluka can be set 105 of importance biasing parameter at maximum.
Table 4.2: Effect of importance parameters in PHITS code
Rn = In I = 1 I = 2 I = 3 I = 4
Number of history 2149856 1855640 1201980 138000 Time per 1 batch [s] 978 1087 1807 11505
Neutron yield 4.63E-9 1.89E-9 1.99E-9 1.89E-9 [1/cm2/source]
Relative error [%] 100 38 6 14
Neutronenergyspectra[n/sr/MeV/proton]
Neutron energy [MeV]
Concrete shield (in case of 160 cm) 10-8
10-7 10-6 10-5 10-4 10-3
10 100
Neutron flux [#/sr/MeV]
n = 1 n = 2 n = 3 n = 4
10 100
Neutron energy [MeV]
Source
PHITS
Figure 4.14: Effect of importance parameters