OECD Nuclear Energy Agency "Benchmarking the Accuracy of - - PDF document

NEA/NSC/DOC(2009)8 1

OECD Nuclear Energy Agency "Benchmarking the Accuracy of Solution of 3-Dimensional Transport Codes and Methods

• ver a Range in Parameter Space"

Final Meeting held in conjunction with M&C-2009

The Saratoga Hilton Hotel Saratoga Springs, New York Room: Broadway 1

6 May 2009, 6:30 to 8:30 pm Chair: Prof. Yousry Azmy Summary 1. Welcome, Introduction of participants Yousry Azmy welcomed participants to this last meeting, aiming at presenting the most recent results, some of which had been presented also at the M&C-2009 conference sessions, and to decide the next steps to complete the benchmark work and publish it. The meeting was attended by 14 participants, who introduced themselves (see Annex). 2. Status of benchmark

• a. New reference solution

Kursat B. Bekar presented the “Reference Solution Set for the NEA Suite of Benchmarks for 3 D Transport Methods and Codes over a Range in Parameter Space”. First the preliminary MCNP reference solution set was presented obtained with 2 billion particle histories. Of the 729 different configurations (each having 23 different quantities) computed by MCNP5 (with multi-group option, 1 group calculation) and using no biasing, 72 quantities were not computed (0-tally scoring for some benchmark quantities for some benchmark cases), more than 500 quantities have a statistical error larger than 5 % (poor statistics for many cases), 159 cases have unreliable results. An improved MCNP reference solution set using ADVANTG/MCNP5 with FW-CADIS (John C. Wagner, Radiation Transport & Criticality Group, ORNL) was produced. ADVANTG uses TORT driven cell-averaged scalar flux distributions to generate Monte Carlo weight windows parameters by implementing FW-CADIS methodology. Then, MCNP5 computes the benchmark quantities for all benchmark cases using the generated weight windows parameters. With this procedure the new reference solution set was

• produced. The use of ADVANTG/MCNP5 code sequence improved the reference solution

NEA/NSC/DOC(2009)8 2

• set. In fact this removed the 0-tally scoring problem, and reduced the statistical errors for

most quantities that had unreliable values without variance reduction. An additional effort to obtain the reference solution set had been provided by Alan P. Copestake, Rolls-Royce plc, using MCBEND results for a few sample cases. For these cases MCBEND and ADVANTG/MCNP results are consistent with each other for most quantities for these sample cases. Some quantities (e.g. 2.e,…,2.h) with net leakage across internal faces show larger discrepancies between the two codes. In fact, the leakage term was not calculated in the same way as with MCNP5. These cases will be recalculated and resubmitted. In conclusion the latest reference solution set computed by ADVANTG/MCNP is more reliable than the preliminary reference solution set. This set can be used by the participants to evaluate the solution of their 3D deterministic code to this benchmark.

• b. Results from participants

Presentations from some participants followed

• Yi Ce presented the results obtained with TITAN,
• Nicolas Martin presented the results obtained with DRAGON,
• Armin Seubert presented a TORT solution using very strict convergence criteria (10–7),
• Dave Barrett presented orally his results.

3. Publication of Benchmark report The results will be published in a special issue of Progress in Nuclear Energy (PNE) within a year or so. Enrico Sartori will distribute a form to participants for them to provide the relevant information on the code used, including, name, references, method used, assumptions made in the calculations and convergence criteria used. A synthesis of this will be added as an Appendix to the summary report. A report to be published by OECD/NEA summarizing the benchmark and the results

• btained, including conclusions, recommendations and lessons learned, will be prepared. The

benchmark specification, the reference solutions and the results provided by the participants will be „packaged‟ at the NEA Data Bank for distribution to participants and to others who wish to use the benchmark for testing their codes or to learn how to solve difficult cases. The PNE issue will contain

• 1. The description of the benchmark and the synthesis of the results.
• 2. The reference solutions.
• 3. The individual solutions compared with reference solutions. This part will

consist of individual articles written by participants describing in detail the methodology used and assumptions made.

NEA/NSC/DOC(2009)8 3

• a. Schedule
• Y. Azmy sends out an e-mail asking participants to vote on how they wish the data

to be presented and condensed, norms to be used for absolute and relative errors, RMS? Issues of monotonicity etc. (week of 11 May 2009)

• Participants indicate their preferences (by 21 May 2009). In case of lack of

consensus, the chairman will make the choice.

• Participants submit their paper for PNE by September 1st 2009.
• Participants submit their final results by September 1st 2009 to Kursat Bekar.
• A copy of the collected files will be submitted to the NEA Data Bank by Kursat

Bekar for „packaging‟ and distribution. Distribution will be done also by RSICC. 4. Proposal for Further Benchmarks

It was proposed to continue this activity by proposing new benchmarks. One proposal was made by David Barrett, entitled “Benchmark to Assess the Accuracy of the Various Methods Used by Transport Codes to Model Material Interfaces”. Transport codes use many different spatial meshing or grid generation techniques. When faced with a configuration with curved interfaces between distinct materials codes may model these interfaces using different approximations. The idea behind this benchmark proposal is to provide a single test problem that quantifies and qualifies the effects of the different approximations. A Monte Carlo based „reference solution‟ would be used as a reference

• solution. This proposal will be submitted at the forthcoming OECD/NEA Data Bank meeting. A draft

version of the specification will be distributed to potential participants for comments. A final version and a schedule for completing the benchmark will be provided. Results could be presented at the M&C-2011 conference.

NEA/NSC/DOC(2009)8 4 Annex List of participants

CANADA MARLEAU, Guy Tel: +1 514 340 4711 ext 4204 Institut de genie nucleaire Fax: +1 514 340 4192 Ecole Polytechnique de Montreal Eml: guy.marleau@polymtl.ca Case Postale 6079

• succ. Centre-Ville

MONTREAL, QUEBEC H3C 3A7 MARTIN, Nicolas Tel: +1 514 340 4192 Ecole Polytechnique de Montreal Fax: Institut de Genie Nucleaire Eml: NICOLAS-3.MARTIN@POLYMTL.CA PO Box 6079, Station Centre-Ville 2900 boul. Edouard-Montpetit Montreal H3T 1J4 GERMANY SEUBERT, Armin Tel: +49 89 32004 469 Gesellschaft fuer Anlagen- Fax: +49 89 32004 10599 und Reaktorsicherheit (GRS) mbH Eml: armin.seubert@grs.de Forschungsinstitute D-85748 GARCHING b. Muenchen UNITED KINGDOM BARRETT, David Tel: +44 118 9826398 AWE Aldermaston Fax: +44 118 9824820 Building E3 Eml: dave.barrett@awe.co.uk READING RG7 4PR COPESTAKE, Alan Tel: +44 1 332 667124 Rolls Royce Marine Power Fax +44 1 332 622 939 P.O. Box 2000 Eml: alan.copestake@rolls-royce.com Derby DE21 7XX SMEDLEY-STEVENSON, Richard P. Tel: +44 1 18 9824173 AWE Aldermaston Fax: +44 1 18 9824820 Building E3.1, Room 206 Eml:richard.smedley-stevenson@awe.co.uk READING, RG7 4PR UNITED STATES OF AMERICA AZMY, Yousry Tel: +1 919 515 3385 Head, Department of Nuclear Engineering Fax: +1 919 515 5115 North Carolina State University Eml: yyazmy@ncsu.edu Campus Box 7909 1110 Burlington Laboratories Raleigh, NC, 27695-7909 BEKAR, Kursat B. Tel: +1 865 241 2437 OAK RIDGE NATIONAL LABORATORY Fax: +1 865 576 3513 PO Box 2008 MS6170 Eml: bekarkb@ornl.gov OAK RIDGE TN 37831-6170 DAHL, Jon A. Tel: +1 505 665 3972 Los Alamos National Laboratory Fax: +1 505 665 5538 LOS ALAMOS, NM 87544 Eml: dahl@lanl.gov HAGHIGHAT, Alireza Tel: +1 352 392 1401 x306 Nuclear and Radiological Engineering Fax: +1 352 392 3380 202 Nuclear Sciences Building Eml: haghighat@ufl.edu University of Florida Gainesville, FL 32611

NEA/NSC/DOC(2009)8 5

KIRK, Bernadette L. Tel: +1 865 574 6176 Director Fax: +1 8652414046 RSICC/ORNL Eml: kirkbl@ornl.gov PO Box 2008

• Bldg. 5700, MS 6171

Oak Ridge, TN 37831-6171 ROSA, Massimiliano Tel: +1 505 667 0869 Computational Physics (CCS-2) Fax: +1 505 665 4972 Los Alamos National Laboratory Eml: maxrosa@lanl.gov P.O. Box 1663, MS K784 Los Alamos, NM 87545 YI, Ce Tel: Nuclear and Radiological Engineering Fax: 202 Nuclear Sciences Building Eml: yice@ufl.edu University of Florida Gainesville, FL 32608 International Organisations SARTORI, Enrico Tel: +33 1 45 24 10 72 / 78 OECD/NEA Data Bank Fax: +33 1 45 24 11 10 / 28 Le Seine-Saint Germain Eml: sartori@nea.fr 12 boulevard des Iles F-92130 Issy-les-Moulineaux France

Kur ursa sat B B. . Bek Bekar ar

Radia adiation T tion Transpor ansport & Criticality Gr t & Criticality Group

• up

Oak Ridg Oak Ridge Na e National La tional Labor boratory tory

Yousry

• usry Y

Y. . Azmy Azmy

De Depar partment of tment of Nuc Nuclear Engineering lear Engineering Nor North Car th Carolina Sta

• lina State Univ

te Univer ersity sity M&C 2009, Sar M&C 2009, Saratog toga Springs a Springs, NY , NY Ma May 6, 2009 y 6, 2009

2 Managed by UT-Battelle for the Department of Energy

Pr Preliminary T eliminary TOR ORT Solutions T Solutions

• 4 le

4 level model r el model ref efinement, inement,

• Uniform mesh in each dimension, 40,80,120 and 160
• Initially started with fully-symmetric quadrature sets
• Square Legendre-Chebyshev Quadratures (S10, S14,

S18, S20)

• Angular quadrature rising order concurrent with the

mesh refinement

• θ-weighted method

3 Managed by UT-Battelle for the Department of Energy

Pr Preliminary T eliminary TOR ORT Solutions T Solutions

• 4 le

4 level model r el model ref efinement, inement,

• At the earlier blind stage, three coarser model were

compared to the finest model to test asymptoticity of the solutions

• After obtaining “Reference solution set”, all four

models compared to the reference solutions

 For most quantities for most cases  accurate TORT solutions  For some cases  iteration convergence problem  For some cases  TORT failed  Possible reason  ray effects, not using cubic mesh, using SP

TORT

 Problems in the reference solution set (0-tallies, significant

relative errors)

4 Managed by UT-Battelle for the Department of Energy

Impr Improved T ed TOR ORT Solutions T Solutions

• Primary r

Primary ray ef y effects ects  to to mitig mitigate b te by def y defining a ining a computa computational sequence with GR tional sequence with GRTUNCLD (f TUNCLD (fir irst st collision sour collision source g ce gener enerator) tor)

• Re-meshing

e-meshing  to obtain unit cubic meshes to to obtain unit cubic meshes to resolv esolve T e TOR ORT f T failur ailures es

• 64-bit arithmetic oper

64-bit arithmetic operations tions  DP T DP TOR ORT and DP T and DP GR GRTUNCL3D TUNCL3D

5 Managed by UT-Battelle for the Department of Energy

Impr Improved T ed TOR ORT Solutions T Solutions

• A T

A TOR ORT solution set w T solution set was obtained f as obtained for the benc

• r the benchmar

hmark k cases with cases with γ=0.9 (small sour 0.9 (small source r ce region, small f gion, small flux sub- lux sub- volumes

• lumes
• TOR

ORT Solutions to the benc T Solutions to the benchmar hmark cases k cases γ=0.5 has not 0.5 has not

• btained y
• btained yet

et

• For most cases
• r most cases, primary r

, primary ray ef y effects w ects was r as reduced educed  T TOR ORT results w esults wer ere impr e improved ed

• Some benc

Some benchmar hmark quantities f k quantities for some benc

• r some benchmar

hmark cases k cases still has pr still has prob

• blem (lar

lem (large discr e discrepancies when comparing to pancies when comparing to the r the ref efer erence solution) ence solution)

6 Managed by UT-Battelle for the Department of Energy

Conc Conclusions lusions

• For most cases
• r most cases, T

, TOR ORT computes the benc T computes the benchmar hmark k quantities accur quantities accurately tely

• For most cases
• r most cases, T

, TOR ORT solutions ar T solutions are in the asymptotic e in the asymptotic regime gime

7 Managed by UT-Battelle for the Department of Energy

Thank Y hank You!

• u!

Questions Questions

8 Managed by UT-Battelle for the Department of Energy

Error Calculations

• Root Mean Square(RMS) of Errors:
• Error in RMS calculation (propagated by the error in MCNP

reference results)

• Absolute and Relative Errors, (i=1,..,729)

RMS = (AE i)2

i=1 729

∑

/729

σ RMS =1/RMS ×[ ((σ mcnp

i i=1 729

∑

)2 × AE i)]

1/ 2

AE i = Rmcnp

i

− RTORT

i

RE i = AE i /Rmcnp

i

Kur ursa sat B t B. Bek . Bekar ar

Radia adiation T tion Transpor ansport & Criticality Gr t & Criticality Group

• up

Oak Ridg Oak Ridge Na e National La tional Labor boratory tory

Yousry Y

• usry Y. Azmy

. Azmy

De Depar partment of tment of Nuc Nuclear Engineering lear Engineering Nor North Car th Carolina Sta

• lina State Univ

te Univer ersity sity M&C 2009, Sar M&C 2009, Saratog toga Springs a Springs, NY , NY Ma May 4, 2009 y 4, 2009

2 Managed by UT-Battelle for the Department of Energy

Outline

• Ov

Over ervie view of w of the Benc the Benchmar hmark k

• Description of Benchmark Problems
• Suite Specification
• Benchmark Quantities
• Reference Solutions
• TOR

ORT Models and Pr T Models and Preliminary Solutions eliminary Solutions

• Impr

Improved T ed TOR ORT Solutions T Solutions

• Conc

Conclusions lusions

3 Managed by UT-Battelle for the Department of Energy

Overview of the Benchmark

• The participants are required to illustrate their

software’s performance in view of the following three criteria:

• Dependence of conclusions for given method/code on

specifics of benchmark configuration → suite suite of benchmarks

• Dependence of method/code accuracy on model

refinement level → verify that reported solution is in asymptotic regime

• Dependence of code/algorithm performance on optional

settings → report all deviations from default/standard

• ptions

4 Managed by UT-Battelle for the Department of Energy

Description of Benchmark Problems

• Outer/inner parallelepiped index 1/2:
• Square base, γ-scaled
• Vacuum BCs
• Scattering ratios: c1 & c2
• Unit source

‏)0,0,0( 1 L 1 γ L γ γ y x z ‏)0,0,0( x z

(0,0,0) (0,0,0)

5 Managed by UT-Battelle for the Department of Energy

Suite Specification and Sample Geometries

• Suite constructed by independently varying 6 parameters

(comprised of a total of 36 = 729 cases)

1.0 0.8 0.5 c2 5.0 1.0 0.1 σ2 1.0 0.8 0.5 c1 5.0 1.0 0.1 σ1 0.9 0.5 0.1 γ 5.0 1.0 0.1 L Range Parameter

L=0.1, γ=0.5

=0.5

L=1.0, γ=0.9

=0.9

L=5.0, γ=0.1

=0.1

Inner region (green) Outer region (gray/blue) Source region (yellow)

6 Managed by UT-Battelle for the Department of Energy

Benchmark Quantities

• Set of 23 Benchmark quantities per configuration:
• Region-averaged scalar flux: Over regions 1 & 2
• Net leakage out of 4 internal & 4 external faces
• Scalar flux averaged over 13 sub volumes

3.a 3.b 3.g 3.h 3.i

Sub volume (red)

7 Managed by UT-Battelle for the Department of Energy

Reference Solutions

• Preliminary reference solutions computed by MCNP5
• No biasing, NPS = 109 & NPS = 2 x 109
• 0-tally scoring for some benchmark quantities for some

benchmark cases

• Poor statistics for many cases
• With NPS = 1011 (no-biasing) → still 0-tallies for some of the

benchmark cases

• Improved reference solutions computed by ADVANTG/MCNP
• FW-CADIS methodology
• TORT driven cell-averaged fluxes → generates MC mesh-

based WW parameters

• No 0-tallies, (NPS = 107 & NPS = 108 ), poor statistics for few

cases

• Will be presented in detail soon.

8 Managed by UT-Battelle for the Department of Energy

TORT Models and Preliminary Solutions

• Four Computational Models;
• Uniform mesh: I x I x I, with I = 40, 80, 120, 160

 refinement order 1,2,3 and 4

 64 thousands to ~ 4.1 million cells)

• Square Legendre-Chebychev (SLC) angular quadrature:

 Number of non-zero-weight angles 200, 392, 648, 800  Angular quadrature rising order concurrent with the mesh

refinement

• θ - weighted method (θ = 0.9), 10–4 convergence criterion,

100 inners

• Started as a blind study, then compared to the

provided MCNP reference results

• Some cases (oblique cells) converged only to 2× 10–3
• Solutions of most cases in the suite of benchmark in the

asymptotic regime

9 Managed by UT-Battelle for the Department of Energy

Sample Quantities

Benc

Benchmar hmark quantity 1.a k quantity 1.a (Region-averaged scalar flux for region 1)

Benc

Benchmar hmark quantity 2.e k quantity 2.e (Net-leakage, internal face, bottom)

AE i = Rmcnp

i

− RTORT

i

RE i = AE i /Rmcnp

i

• Absolute and Relative Errors, (i=1,..,729)

10 Managed by UT-Battelle for the Department of Energy

Sample Quantities

Quantity 3.a

Quantity 3.a (Volume averaged scalar flux)

Quantity 3.i

Quantity 3.i (Volume averaged scalar flux)

11 Managed by UT-Battelle for the Department of Energy

Problems Reported for the Preliminary Tort Solutions

• When the aspect ratio is different from 1,

some cases failed for some of the benchmark quantities

• Iterative convergence problems
• Produced errors that do not decrease

monotonically with model refinement

12 Managed by UT-Battelle for the Department of Energy

Methodology to Mitigate ray effects

• Using the standard approach: split the solution to the

transport equation into an uncollided and fully collided flux

• GRTUNCL3D
• Generates uncollided fluxes (semi analytic method) and first

collision source

• Does not compute the uncollided fluxes at the cell boundaries

(8 benchmark quantities, 2.a,…,2.h cannot be computed)

• Latest GRTUNCL3D was modified to compute the uncollided

flux at the cell boundaries

13 Managed by UT-Battelle for the Department of Energy

GTort Computation Sequence

14 Managed by UT-Battelle for the Department of Energy

GTort Solution for a Sample Case

Coar Coarsest model (40x40x40, sest model (40x40x40, SL SLC-10) f C-10) for L=5.0,

• r L=5.0, γ=0.9

=0.9

• GTORT sequence

mitigates the primary ray effects in the solutions

• It's semi-analytic

methodology is poor in the source region

15 Managed by UT-Battelle for the Department of Energy

MGTort Computation Sequence

• GRTUNCL3D produces unacceptable results due to its inability to

accurately perform ray-tracing within a source cell

16 Managed by UT-Battelle for the Department of Energy

Improved TORT Solutions (40x40x40, SLC-10)‏

17 Managed by UT-Battelle for the Department of Energy

TORT Solutions with MGTORT Computational Sequence

Coar Coarsest model (40x40x40, SL sest model (40x40x40, SLC-10) C-10)

18 Managed by UT-Battelle for the Department of Energy

TORT Solutions with MGTORT Computational Sequence

Benchmark Quantity 3.e for Four Models

19 Managed by UT-Battelle for the Department of Energy

Resolving Convergence Problem

• Modify the mesh so as to use cubic cells (unit aspect ratio)
• Non-linear θ-weighted method works well if the cell aspect ratio is

close to 1

• Using 400x400x40, 40x40x40, and 40x40x200 mesh structures (cubic

cells) for the benchmark cases L=0.1, 1.0, and, 5.0 solved the convergence problem for straight TORT calculations (except five cases for L=5.0)

• Did not help for the calculation performed by MGTORT sequence.

Why?

20 Managed by UT-Battelle for the Department of Energy

TORT Convergence Problem in MGTORT Sequence

• Convergence problem becomes evident for the TORT

calculations in MGTORT sequence

• Dominant part of the flux is uncollided flux
• The collided flux comprises too small numbers
• Using cubic cells is not always a realistic option
• The finest computational model needs an extremely large

amount of memory and disk space (for both TORT and GRTUNCL3D)

• Solution: Using 64-bit arithmetic operations in MGTORT

sequence

• Longer computation time and almost 2 times larger disk/

memory space requirement

• All cases converged

21 Managed by UT-Battelle for the Department of Energy

Change in Convergence Rate for Two TORT Executables

L = 5.0,γ = 0.5,σ1,2 = 5.0,c1 = 0.5,c2 =1.0

22 Managed by UT-Battelle for the Department of Energy

Execution Times for Two TORT Executables

Calcula Calculations with Model-2 f tions with Model-2 for all benc

• r all benchmar

hmark cases k cases

23 Managed by UT-Battelle for the Department of Energy

Conclusion

• Solutions for most cases in the suite of benchmarks as

computed by TORT are reasonably accurate

• Generating the first collision source and supplying this source

to drive the TORT calculations of the fully collided flux yields more accurate results

• Introducing double-precision versions of TORT and

GRTUNCL3D completely resolved the convergence problem

• Improved TORT solutions and ADVANTG/MCNP reference

results are in good agreement except some cases;

• MGTORT sequence only mitigates the primary ray effects
• Secondary ray effects evident for some cases.

24 Managed by UT-Battelle for the Department of Energy

Thank Y hank You!

• u!

Questions Questions

25 Managed by UT-Battelle for the Department of Energy

Error Calculations

• Root Mean Square(RMS) of Errors:
• Error in RMS calculation (propagated by the error in MCNP

reference results)

• Absolute and Relative Errors, (i=1,..,729)

RMS = (AE i)2

i=1 729

∑

/729

σ RMS =1/RMS ×[ ((σ mcnp

i i=1 729

∑

)2 × AE i)]

1/ 2

AE i = Rmcnp

i

− RTORT

i

RE i = AE i /Rmcnp

i

1

Preliminary TORT Results on the 3-D Transport Accuracy Benchmark over a Range in Parameter Space

• A. Seubert

Gesellschaft für Anlagen- und Reaktorsicherheit (GRS) mbH Forschungsinstitute D-85748 Garching M&C 2009 – 6 May 2009

TORT calculation details



Considered cases  model refinement:



Equal mesh sizes in each spatial dimension



Chebychev-Legendre quadrature

–

Number of ordinates with non-zero weights: 4608 for S48,12800 for S80



80x80x80-S48 calculation: 512.000 spatial meshes

–

Taken as preliminary reference for model refinement studies until S80 calculation has finished



Convergence criteria (TORT-64):

–

Fission rate pointwise: 5.010-7

–

Flux pointwise: 1.010-7

2 nx = ny = nz S8 S16 S32 S48 S80 40     80     (ref.) () 120    160   

3

Geometrical configurations  L = 0.1 cm

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

inner parallelepiped source volume 3.a 3.b 3.c 3.d 3.e 3.f 3.g 3.h 3.i 3.j 3.k 3.l 3.m

• uter parallelepiped

inner parallelepiped source volume 3.a 3.b 3.c 3.d 3.e 3.f 3.g 3.h 3.i 3.j 3.k 3.l 3.m

• uter parallelepiped

inner parallelepiped source volume 3.a 3.b 3.c 3.d 3.e 3.f 3.g 3.h 3.i 3.j 3.k 3.l 3.m

4

Geometrical configurations  L = 1.0 cm

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

inner parallelepiped source volume 3.a 3.b 3.c 3.d 3.e 3.f 3.g 3.h 3.i 3.j 3.k 3.l 3.m

• uter parallelepiped

inner parallelepiped source volume 3.a 3.b 3.c 3.d 3.e 3.f 3.g 3.h 3.i 3.j 3.k 3.l 3.m

• uter parallelepiped

inner parallelepiped source volume 3.a 3.b 3.c 3.d 3.e 3.f 3.g 3.h 3.i 3.j 3.k 3.l 3.m

5

Geometrical configurations  L = 5.0 cm

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

inner parallelepiped source volume 3.a 3.b 3.c 3.d 3.e 3.f 3.g 3.h 3.i 3.j 3.k 3.l 3.m

• uter parallelepiped

inner parallelepiped source volume 3.a 3.b 3.c 3.d 3.e 3.f 3.g 3.h 3.i 3.j 3.k 3.l 3.m

• uter parallelepiped

inner parallelepiped source volume 3.a 3.b 3.c 3.d 3.e 3.f 3.g 3.h 3.i 3.j 3.k 3.l 3.m

3.a 3.b 3.j 3.k 3.f 3.e 3.l 3.m 3.c 3.d 3.g 3.h 3.i

General findings

6

Items  = 0.1  = 0.5  = 0.9 1

reasonable reasonable reasonable 2 reasonable reasonable reasonable 3 reasonable c-f, h, i, k c-f, h-i

Items  = 0.1  = 0.5  = 0.9 1

very good very good good 2 very good very good good 3 very good very good e-h

Items  = 0.1  = 0.5  = 0.9 1

good good good 2 good good reasonable 3 good good d-i L = 0.1cm L = 1.0cm L = 5.0cm

7

TORT 80x80x80 S48  Item 1.a

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

L = 0.1 cm L = 1.0 cm L = 5.0 cm

 = 0.9  = 0.5  = 0.1  = 0.9  = 0.5  = 0.1  = 0.9  = 0.5  = 0.1

8

TORT 80x80x80 S48  Item 2.a

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

L = 0.1 cm L = 1.0 cm L = 5.0 cm

 = 0.9  = 0.5  = 0.1  = 0.9  = 0.5  = 0.1  = 0.9  = 0.5  = 0.1

Dependence on model refinement: Quadrature order

9

Items 1.a-b Items 2.a-h Items 3.a-f Items 3.g-m

Dependence on model refinement: Spatial meshing



Dependence on angular refinement more pronounced than on spatial refinement 10

Items 1.a-b Items 2.a-h Items 3.a-f Items 3.g-m

11

TORT 80x80x80 S48  Item 3.c

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

L = 0.1 cm L = 1.0 cm L = 5.0 cm

 = 0.9  = 0.5  = 0.1  = 0.9  = 0.5  = 0.1  = 0.9  = 0.5  = 0.1

12

TORT 80x80x80 S48  Item 3.e

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

L = 0.1 cm L = 1.0 cm L = 5.0 cm

 = 0.9  = 0.5  = 0.1  = 0.9  = 0.5  = 0.1  = 0.9  = 0.5  = 0.1

13

TORT 80x80x80 S48  Item 3.i

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

L = 0.1 cm L = 1.0 cm L = 5.0 cm

 = 0.9  = 0.5  = 0.1  = 0.9  = 0.5  = 0.1  = 0.9  = 0.5  = 0.1

14

MCNP  Item 3.i  Errors

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

• uter parallelepiped

• uter parallelepiped

• uter parallelepiped

 = 0.1  = 0.5  = 0.9

L = 0.1 cm L = 1.0 cm L = 5.0 cm

 = 0.9  = 0.5  = 0.1  = 0.9  = 0.5  = 0.1  = 0.9  = 0.5  = 0.1

Dependence on inner parallelepiped material properties

 Consider item 3.i for the case with L = 5.0 cm,  = 0.9, 1 = 0.1 cm-1

15

c1 = 0.5 c1 = 0.8 c1 = 1.0

Dependence on inner parallelepiped material properties

 Consider item 3.i for the case with L = 5.0 cm,  = 0.9, 1 = 1.0 cm-1

16

c1 = 0.5 c1 = 0.8 c1 = 1.0

DRAGON solutions to the 3D transport benchmark over a range in parameter space

Nicolas Martin, Alain H´ ebert, Guy Marleau Institut de G´ enie Nucl´ eaire ´ Ecole Polytechnique de Montr´ eal

3D Transport Benchmark meeting, M&C 2009

DRAGON solutions to the 3D transport benchmark over a range in parameter space – 1/10

Status of DRAGON results 1

Comparative study of MoC and SN solutions to be published in ANE. Results are displayed using the methodology proposed by Bekar and Azmy: The average relative error by region for all the 729 cases. The RMS error per quantity (with propapaged MCNP uncertainty. Results are encouraging, relative errors seem to be similar to those of TORT results (as published in ANE paper by Bekar and Azmy).

3D Transport Benchmark meeting, M&C 2009

DRAGON solutions to the 3D transport benchmark over a range in parameter space – 2/10

Computational models 1

Spatial discretization: Basic mesh is

• 0, 1−γ

4 , 1−γ 2 , 1 2, 1+γ 2 , 3+γ 4 , 1

• and for the z axis,
• 0, L(1−γ)

4

, L(1−γ)

2

, L

2 , L(1+γ) 2

, L(3+γ)

4

, L

• .

Relies on the optical thickness of the medias, i.e., refined by 1 N × Σi , N = {2, 3, 4}. Angular quadrature: Fully symmetric, Legendre Chebyshev and Quadruple range quadratures were tested. For the MoC calculations : 500 tracks cm−2.

3D Transport Benchmark meeting, M&C 2009

DRAGON solutions to the 3D transport benchmark over a range in parameter space – 3/10

SN computational model

1

1. 1 2 × Σi discretization (maximum of 22400 regions), S16 fully symmetric quadrature, 2. 1 3 × Σi discretization (maximum of 75600 regions), S18 fully symmetric quadrature, 3. 1 4 × Σi discretization (maximum of 179200 regions), S20 fully symmetric quadrature. 2 additional runs performed to test different angular quadratures: N = 2 and S44 Pn − Tn quadrature, N = 2 and S54 QRn (Quadruple Range) quadrature. Parabolic Diamond Differencing scheme by default: 32 moments of the flux per computational cell.

3D Transport Benchmark meeting, M&C 2009

DRAGON solutions to the 3D transport benchmark over a range in parameter space – 4/10

MoC computational model 1

1 A discretization of the geometry by a factor of 1 2 × Σi with track density of 5 × 102 integration lines in cm−2, and an angular quadrature of type Pn-Tn with n=16. 2 A discretization of the geometry by a factor of 1 3 × Σi with a track density of 5 × 102 integration lines in cm−2, and an angular quadrature of type Pn-Tn with n=24. 3 A discretization of the geometry by a factor of 1 4 × Σi with a track density of 1 × 103 integration lines in cm−2, and an angular quadrature of type Pn-Tn with n=32.

3D Transport Benchmark meeting, M&C 2009

DRAGON solutions to the 3D transport benchmark over a range in parameter space – 5/10

SN relative error

1

Benchmark quantity 1.a Benchmark quantity 1.b Benchmark quantity 3.a Benchmark quantity 3.m

3D Transport Benchmark meeting, M&C 2009

DRAGON solutions to the 3D transport benchmark over a range in parameter space – 6/10

MoC relative error 1

Benchmark quantity 1.a Benchmark quantity 1.b Benchmark quantity 3.a Benchmark quantity 3.m

3D Transport Benchmark meeting, M&C 2009

DRAGON solutions to the 3D transport benchmark over a range in parameter space – 7/10

SN RMS error

1

Benchmark quantity 1.a Benchmark quantity 1.b Benchmark quantity 3.a Benchmark quantity 3.m

3D Transport Benchmark meeting, M&C 2009

DRAGON solutions to the 3D transport benchmark over a range in parameter space – 8/10

MoC RMS error 1

Benchmark quantity 1.a Benchmark quantity 1.b Benchmark quantity 3.a Benchmark quantity 3.m

3D Transport Benchmark meeting, M&C 2009

DRAGON solutions to the 3D transport benchmark over a range in parameter space – 9/10

Conclusions 1

MoC and SN results exhibit similar behavior Scalar fluxes for sub volumes close to the source are generally well computed. Relative errors grow when the sub volume dimension is reduced and when the distance from the source is important in term of mean free path. Difficult for both methods to obtain spatially uniform convergence Model refinement fails for some sub regions. A possible solution can be the use of ξ - biased angular quadratures. Use of 64-bit arithmetic precision to encompass memory limitations.

3D Transport Benchmark meeting, M&C 2009

DRAGON solutions to the 3D transport benchmark over a range in parameter space – 10/10

C. Yi and A. Haghighat

Accuracy curacy of TI TITA TAN N Based d on a Ne New w OECD CD-NE NEA A Benchm nchmark ark over a Ra Range e in Parameter meter Space ce

ANS M&C 2009

Contents

Description of the TITAN code

1

g Introduction on Benchmark

2

Calculation results and Comparison

3

Conclusions

4

Hybrid approach

 Hybrid Discrete Ordinate and Characteristics Method

• Discrete Ordinates (Sn)

Method in regular regions

• Characteristics method in

low-scattering regions

Benefit: To solve problems that contain regions of low- scattering materials more efficiently

Coarse-mesh-oriented approach

Sn or characteristics solver can be assigned to different coarse meshes

TITAN code

1.

Written from scratch in Fortran 90 with some features in Fortran 2003 features.

2.

• bject oriented programming, dynamic memory

allocation, and layered code structure

3.

Benchmarked on a number of problems:

a)

C5G7 MOX problem

b)

Kobayashi problems

c)

SPECT and CT models

TIT ITAN AN Det eterm erminis inistic tic Code de

Characteristic Solver Hybrid Approach Localized meshing and quadrature set Coarse Mesh-oriented Paradigm Sn Solver High computational Efficiency

TITAN

Ben enchmar marki king ng the e Accuracy uracy of Solut ution ion of 3-D D Transpo port t Code des and d Met ethods ds over er a R Range ge in in Par Parameter er Spa pace

“The geometric configuration and xs data are intentionally simple and unsophisticated to avoid diverting the participants’ attention and efforts toward modeling details”

• Number of cases : 729
• Calculating targets
• Pure scattering

Easy points: Hard points:

Ben enchmar mark k Geo eometr try

Parameter Range L 0.1 1.0 5.0 γ 0.1 0.5 0.9 σ1 0.1 1.0 5.0 c1 0.5 0.8 1.0 σ2 0.1 1.0 5.0 c2 0.5 0.8 1.0 Total cases: 36=729 Case Numbering: 111111 to 333333 e.g. Case 123123 will be: L=0.1, γ=0.5 σ1=5.0 c1=0.5 σ2=1.0 c2=1.0

Calculating Targets

There are 23 benchmark quantities to be calculated:

 Set 1: Scalar fluxes in

material regions (2 values)

 Set 2: Net leakages (8 values)  Set 3: Scalar fluxes in some

small boxes (13 values)

Most of the difficult values are in Set 3. Reference solutions are provided with the benchmark (MCNP)

TITAN model

 3x3x3 coarse meshes  Fine meshing is automatically adjusted

case by case

 A Python script to drive all the cases  Added subroutines to calculate required

quantities

Batch Run Quadrature Meshing 1 Serial S50 from 1,728 for Case 111111 to 22,400 for Case 333333 2 Parallel S60 from 5,832 for Case 111111 to 75,600 for Case 333333

S50 0 and S60 Qu Quadrature ure Set et

S50 Quadrature set: ~2500 direction S60 Quadrature set: ~3600 direction

Quantity 1.a -averaged flux in outside box

Second Batch First Batch

• 6.00%
• 5.00%
• 4.00%
• 3.00%
• 2.00%
• 1.00%

0.00% 1.00% 100 200 300 400 500 600 700 800

Quantity 2.a – net leakage at left boundary

Second Batch First Batch

• 1.00%
• 0.80%
• 0.60%
• 0.40%
• 0.20%

0.00% 0.20% 0.40% 0.60% 100 200 300 400 500 600 700 800

3a

Quantity 3.a – Avg. Flux over part of the source box

Second Batch First Batch

• 2.50%
• 2.00%
• 1.50%
• 1.00%
• 0.50%

0.00% 100 200 300 400 500 600 700 800

Case 33333 quantities

0.20% 0.39% 0.78% 1.56% 3.13% 6.25% 12.50% 25.00% 50.00% 100.00% 1.a 1.b 2.a 2.b 2.c 2.d 2.e 2.f 2.g 2.h 3.a 3.b 3.c 3.d 3.e 3.f 3.g 3.h 3.i 3.j 3.k 3.l 3.m

Conclusions

• Some quantities are usually less than 1% difference.

Including: 1.a 2.a 2.c 2.d 2.g 3.a 3.g

• Most of the rest quantities are within 5% difference.

Some quantities for low scattering cases are up to 10% different, including 3.c and 3.m

• The most difficult quantities to calculate are 3.f and 3.i.
• Running time

Batch Run Quadrature Time 1 Serial S50 30 hrs 2 Parallel 20 CPU S60 5 hrs

Thank You!