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Exact Monte Carlo calculation method for K-eigenvalue change using perturbation source method

Yamamoto, Toshihiro Sakamoto, Hiroki 京都大学 DOI:10.1080/00223131.2021.1883144

2021

概要

The ‘perturbation source method’ (PSM) is a Monte Carlo perturbation method that calculates an exact k-eigenvalue change caused by cross-section changes. Although the PSM, which can consider the effect of fission source perturbation, was proposed long ago, it has garnered minimal interest as a Monte Carlo perturbation method. The applicability of the PSM has not been thoroughly elucidated hitherto. This study revisits the PSM and reviews the associated Monte Carlo algorithm. Some improvements have been made to improve the efficiency. The PSM is applied to some numerical tests that involve the replacement of a fuel material with light water, a density change in a water hole, an interface shift between a fuel and reflector, and an external boundary extension. The performance of the PSM for these tests is compared with that of another exact Monte Carlo perturbation method, which is the correlated sampling method. The PSM can yield an accurate k-eigenvalue change even for large cross-section changes such as the replacement of a material with another material. The PSM used in this study is the exact method except for the approximation related to the spatial discretization for fission source perturbation. Furthermore, it exhibits superiority in terms of accuracy and computational efficiency, particularly for large perturbations added in a small region.

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参考文献

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Table captions

Table 1. Three group constants of materials (1) and (2) in Figure 2 (Cases 1 and 2).

Table 2. Three group constants of material (3) in Figure 2 (Case1).

Table 3. k-eigenvalue changes for verification (Case 1).

Table 4. Relative figure-of-merit with respect to direct method. Number in parentheses

denotes factor C in Eq. (32).

Table 5. Three group constants of material (3) in Figure 2 (Case2).

Table 6. k-eigenvalue changes for verification (Case 2).

Table 7. Three group constants of UO2 rod array (the materials (1) and (2) in Figure 2

(Case3)).

Table 8. Three group constants of light water (the material (3) in Figure 2 (Case3)).

Table 9. k-eigenvalue changes for replacement of UO2 with light water (Case 3).

Table 10. k-eigenvalue changes for light water density change (Case 4).

Table 11. Three group constants of graphite.

Table 12. k-eigenvalue changes for interface displacement by 1 cm (Case 5).

iew

Table 13. Relative figure-of-merit with respect to direct method. Number in parentheses

denotes factor C in Eq. (32).

Table 14. k-eigenvalue changes for interface displacement by 0.02 cm (Case 6).

Table 15. k-eigenvalue changes for boundary extension by 3 cm (Case 7).

Table 16. k-eigenvalue changes for boundary extension by 0.02 cm (Case 8).

Table 17. Dependence of k-eigenvalue change on spatial bin sizes in Case 2.

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Figure captions

Figure 1(a). Flow chart of unperturbed calculation mode.

Figure 1(b). Flow chart of perturbation calculation mode.

Figure 2. Geometry for perturbation calculations in Cases 1 to 4.

Figure 3. ∆k with and without source perturbation effect as a function of generation.

Figure 4. Interface shift in Case 5 (x = 1 cm) and Case 6 (x = 0.02 cm).

Figure 5. External boundary extension in Case 7 (x = 3 cm) and Case 8 (x = 0.02 cm).

iew

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Unperturbed calculation mode

Start particle

Fission source bank

Fission spectrum

perturbation

FSPB

Determine collision point

Fission source

perturbation

yes

Total cross section

perturbation

Perturbed region?

Scattering cross

section perturbation

FSPB

Kill or escape?

Perturbation

calculation mode

FSPB: Fission Source Perturbation Bank

iew

yes

Implicit capture, Russian roulette

Figure 1(a). Flow chart of unperturbed calculation mode.

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Perturbation calculation mode

Start particle

Determine collision point

FSPB

Implicit capture, Russian roulette

Kill or escape?

yes

Unperturbed calculation mode

iew

Figure 1(b). Flow chart of perturbation calculation mode.

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(1)

(2)

(3)

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Figure 2. Geometry for perturbation calculations in Cases 1 to 4.

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0.12

0.10

Δk without source perturbation

Δk [-]

0.08

Δk source perturbation effect

0.06

0.04

0.00

0.02

Generations

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Figure 3. ∆k with and without source perturbation effect as a function of generation.

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10 cm

35 cm

Graphite

35 cm

UO2 rod

array

Interface before perturbation

Figure 4. Interface shift in Case 5 (x = 1 cm) and Case 6 (x =0.02 cm).

iew

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10 cm

35 cm

Graphite

35 cm

UO2 rod

array

Boundary before perturbation

Figure 5. External boundary extension in Case 7 (x = 3 cm) and Case 8 (x = 0.02 cm).

iew

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Table 1. Three group constants of materials (1) and (2) in Figure 2 (Cases 1 and 2).

1st group

2nd group

3rd group

𝛴𝑐(𝑐𝑚 )

𝛴𝑓(𝑐𝑚 ―1)

1.0

0.01

1.5

0.02

0.04

3.0

0.03

0.06

2.4

𝛴𝑠𝑔→𝑔(𝑐𝑚 ―1)

0.8

0.2

0.0

0.882

1.296

2.91

𝛴𝑠𝑔→𝑔 + 1(𝑐𝑚 ―1)

0.098

0.144

0.0

𝛴𝑡 (𝑐𝑚

―1

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Table 2. Three group constants of material (3) in Figure 2 (Case1).

1st group

2nd group

3rd group

1.1

0.005

0.02

2.4

1.6

0.01

0.08

2.4

3.2

0.015

0.12

2.4

0.6

0.4

0.0

𝛴𝑠𝑔→𝑔(𝑐𝑚 ―1)

0.86

1.208

3.065

𝛴𝑠𝑔→𝑔 + 1(𝑐𝑚 ―1)

0.215

0.302

0.0

𝛴𝑡 (𝑐𝑚

―1

𝛴𝑐(𝑐𝑚 )

𝛴𝑓(𝑐𝑚 ―1)

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Table 3. k-eigenvalue changes for verification (Case 1).

Δ𝑘𝑁𝑃

𝑒𝑓𝑓

Δ𝑘𝑃𝑆

𝑒𝑓𝑓

Total

PSM (C=1)

4.4574×10−2 ± 5.9×10−5

1.1139×10−1 ± 1.2×10−4

1.5596×10−1 ± 1.4×10−4

4.4517×10−2 ± 2.4×10−5

9.9430×10−2 ± 9.4×10−4

1.4395×10−1 ± 9.4×10−4

1.5585×10−1 ± 4×10−5

Direct method

1.5586×10−1

Deterministic method

iew

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Table 4. Relative figure-of-merit with respect to direct method. Number in parentheses denotes

factor C in Eq. (32).

Direct method

PSM

Case 1

Case 2

Case 3

Case 4

0.143 (C=1)

62.8 (C=1)

142 (C=1)

1092 (C=1)

0.072 (C=2)

66.4 (C=2)

184 (C=5)

4377 (C=20)

142 (C=10)

3701 (C=30)

0.004

92.0

0.238

117

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Table 5. Three group constants of material (3) in Figure 2 (Case2).

1st group

2nd group

3rd group

1.05

0.009

0.011

2.4

1.52

0.018

0.045

2.4

3.05

0.029

0.065

2.4

0.78

0.22

0.0

𝛴𝑠𝑔→𝑔(𝑐𝑚 ―1)

0.9167

1.29673

2.956

𝛴𝑠𝑔→𝑔 + 1(𝑐𝑚 ―1)

0.1133

0.16027

0.0

𝛴𝑡 (𝑐𝑚

―1

𝛴𝑐(𝑐𝑚 )

𝛴𝑓(𝑐𝑚 ―1)

iew

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Table 6. k-eigenvalue changes for verification (Case 2).

Δ𝑘𝑁𝑃

𝑒𝑓𝑓

Δ𝑘𝑃𝑆

𝑒𝑓𝑓

Total

PSM (C=1)

5.2507×10−3 ± 8.8×10−6

5.1664×10−3 ± 4.1×10−6

1.0417×10−2 ± 9.6×10−6

5.2342×10−3 ± 3.2×10−6

5.1028×10−3 ± 6.0×10−6

1.0337×10−2 ± 6.8×10−6

1.0366×10−2 ± 6.1×10−5

Direct method

1.0389×10−2

Deterministic method

iew

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Table 7. Three group constants of UO2 rod array (the materials (1) and (2) in Figure 2 (Case3)).

1st group

2nd group

3rd group

0.29829

3.2674×10−3

3.0586×10−3

2.4

0.83334

9.7371×10−3

2.1579×10−3

2.4

1.6389

0.029252

0.056928

2.4

0.87820

0.12180

0.0

𝛴𝑠𝑔→𝑔(𝑐𝑚 ―1)

0.22106

0.77764

1.5527

𝛴𝑠𝑔→𝑔 + 1(𝑐𝑚 ―1)

0.073843

0.043803

0.0

𝛴𝑡 (𝑐𝑚

―1

𝛴𝑐(𝑐𝑚 )

𝛴𝑓(𝑐𝑚 ―1)

iew

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Table 8. Three group constants of light water (the material (3) in Figure 2 (Case3)).

𝛴𝑡 (𝑐𝑚 ―1)

―1

𝛴𝑐(𝑐𝑚 )

𝛴𝑠𝑔→𝑔(𝑐𝑚 ―1)

𝛴𝑠𝑔→𝑔 + 1(𝑐𝑚

―1

1st group

2nd group

3rd group

0.33207

3.0500×10−4

1.1265

3.6990×10−4

2.7812

0.018250

0.22713

1.0281

2.7630

0.10464

0.097961

0.0

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Table 9. k-eigenvalue changes for replacement of UO2 with light water (Case 3).

Δ𝑘𝑁𝑃

𝑒𝑓𝑓

Δ𝑘𝑃𝑆

𝑒𝑓𝑓

Total

PSM (C=5)

9.1827×10−5 ± 2.52×10−6

2.2405×10−4 ± 3.80×10−6

3.1587×10−4 ± 4.56×10−6

1.4112×10−4 ± 4.64×10−5

3.1462×10−4 ± 1.66×10−4

4.5574×10−4 ± 1.72×10−4

3.3500×10−4 ± 1.96×10−5

Direct method

3.1018×10−4

Deterministic method

iew

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Table 10. k-eigenvalue changes for light water density change (Case 4).

Δ𝑘𝑁𝑃

𝑒𝑓𝑓

Δ𝑘𝑃𝑆

𝑒𝑓𝑓

Total

PSM (C=20)

−7.7306×10−7 ± 5.53×10−7

−1.0834×10−4 ± 2.20×10−7

−1.0911×10−4 ± 5.95×10−7

−1.2908×10−6 ± 1.50×10−6

−1.0825×10−4 ± 3.81×10−6

−1.0955×10−4 ± 4.10×10−6

−1.3600×10−4 ± 2.19×10−5

Direct method

−1.0915×10−4

Deterministic method

iew

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Table 11. Three group constants of graphite.

𝛴𝑡 (𝑐𝑚 ―1)

―1

𝛴𝑐(𝑐𝑚 )

𝛴𝑠𝑔→𝑔(𝑐𝑚 ―1)

𝛴𝑠𝑔→𝑔 + 1(𝑐𝑚 ―1)

1st group

2nd group

3rd group

0.23264

4.1681×10−5

0.22122

0.011379

0.41793

7.6738×10−6

0.41327

4.6514×10−3

0.42354

2.1311×10−4

2.5313

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Table 12. k-eigenvalue changes for interface displacement by 1 cm (Case 5).

Δ𝑘𝑁𝑃

𝑒𝑓𝑓

Δ𝑘𝑃𝑆

𝑒𝑓𝑓

Total

PSM (C=3)

1.1793×10−2 ± 1.2×10−5

−7.7237×10−3 ± 9.6×10−6

4.0692×10−3 ± 1.5×10−5

4.0220×10−3 ± 2.16×10−5

Direct method

4.0182×10−3

Deterministic method

iew

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Table 13. Relative figure-of-merit with respect to direct method. Number in parentheses denotes

factor C in Eq. (32).

Direct method

PSM

Case 5

Case 6

Case 7

Case 8

5.17 (C=1)

341 (C=1)

54.1 (C=1)

2.33×105 (C=1)

13.4 (C=3)

2114 (C=40)

301 (C=20)

7.20×106 (C=3500)

12.1 (C=5)

1479 (C=60)

241 (C=40)

6.00×106 (C=5000)

135

1.92×104

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Table 14. k-eigenvalue changes for interface displacement by 0.02 cm (Case 6).

Δ𝑘𝑁𝑃

𝑒𝑓𝑓

Δ𝑘𝑃𝑆

𝑒𝑓𝑓

Total

PSM (C=40)

2.3001×10−4 ± 2.4×10−7

−1.4561×10−4 ± 8.3×10−7

8.4398×10−5 ± 8.69×10−7

4.40×10−5 ± 1.55×10−5

Direct method

8.3940×10−5

Deterministic method

iew

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Table 15. k-eigenvalue changes for boundary extension by 3 cm (Case 7).

Δ𝑘𝑁𝑃

𝑒𝑓𝑓

Δ𝑘𝑃𝑆

𝑒𝑓𝑓

Total

PSM (C=20)

4.0950×10−3 ± 3.7×10−6

−1.2387×10−3 ± 3.5×10−6

2.8563×10−3 ± 5.1×10−6

4.0875×10−3 ± 4.6×10−6

−1.2572×10−3 ± 6.8×10−6

2.8302×10−3 ± 8.2×10−6

2.8810×10−3 ± 1.92×10−5

Direct method

2.8564×10−3

Deterministic method

iew

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Table 16. k-eigenvalue changes for boundary extension by 0.02 cm (Case 8).

Δ𝑘𝑁𝑃

𝑒𝑓𝑓

Δ𝑘𝑃𝑆

𝑒𝑓𝑓

Total

PSM (C=3500)

3.3091×10−5 ± 2.3×10−8

−9.8679×10−6 ± 2.16×10−8

2.3223×10−5 ± 3.2×10−8

3.3132×10−5 ± 3.34×10−7

−1.0123×10−5 ± 5.08×10−7

2.3009×10−5 ± 6.08×10−7

Direct method

2.3790×10−5

Deterministic method

iew

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Table 17. Dependence of k-eigenvalue change on spatial bin sizes in Case 2.

Spatial

bins

Δ𝑘𝑁𝑃

𝑒𝑓𝑓

Δ𝑘𝑃𝑆

𝑒𝑓𝑓

Total

*This

(300+100+300)×

(300+100+300)*

(15+5+15)×(15+5+5)

(7+3+7)×(7+3+7)

5.2403×10−3 ± 8.6×10−6

5.1561×10−3 ± 6.8×10−6

1.0396×10−2 ± 1.1×10−5

5.2396×10−3 ±8.5×10−6

5.1982×10−3 ±6.8×10−6

1.0438×10−2 ±1.1×10−5

5.1796×10−3 ±8.2×10−6

5.1985×10−3 ±2.4×10−6

1.0378×10−2 ±8.5×10−6

is the same as in Table 6.

Table 17 (continued).

Δ𝑘𝑃𝑆

𝑒𝑓𝑓

4.9725×10−3 ±8.3×10−6

5.1395×10−3 ±2.5×10−6

1.0112×10−2 ±8.7×10−6

(3+2+3)×(3+2+3)

4.5344×10−3 ±8.2×10−6

4.7738×10−3 ±2.4×10−6

9.3082×10−3 ±8.5×10−6

iew

Total

(4+2+4)×(4+2+4)

Δ𝑘𝑁𝑃

𝑒𝑓𝑓

Spatial

bins

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Exact Monte Carlo calculation method for K-eigenvalue change using perturbation source method

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Exact Monte Carlo calculation method for K-eigenvalue change using perturbation source method

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関連論文

Monte Carlo perturbation calculation for geometry change in fixed source problems with the perturbation source method

A multireference coupled-electron pair approximation combined with complete-active space perturbation theory in local pair-natural orbital framework

Geometry of configuration space in Markov chain Monte Carlo methods and the worldvolume approach to the tempered Lefschetz thimble method

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