Illustrating Important Effects of Second-Order Sensitivities on Response Uncertainties in Reactor Physics
Abstract
:1. Introduction
2. Largest 1st- and 2nd-Order Sensitivities of the Perp Benchmark’s Leakage with Respect to Perp’s Nuclear Data
- The quantity is the customary “group-flux” for group , and is the unknown state-function obtained by solving Equations (1) and (2), where is the radius of the PERP sphere while the vector denotes the outward unit normal vector at each point on the sphere’s outer boundary, denoted as .
- The total cross section for energy group and material , is computed using the following expression:
- The quantity denotes the atom density of isotope i in material m; , , where denotes the total number of isotopes, and denotes the total number of materials. The computation of uses the well-known expression:
- The quantity represents the scattering transfer cross section from energy group into energy group and is computed in terms of the -th order Legendre coefficient (of the Legendre-expanded microscopic scattering cross section from energy group into energy group for isotope ), which are tabulated parameters, in the following finite-order expansion:
- The quantity denotes the total number of spontaneous-fission isotopes. The spontaneous-fission isotopes in the PERP benchmark are “isotope 1” (239Pu) and “isotope 2” (240Pu), so , and the spontaneous fission neutron spectrum of 239Pu and 240Pu, respectively, is approximated by a Watts fission spectrum using the evaluated parameters and . The decay constant for actinide nuclide is denoted as , and denotes the fraction of decays that are spontaneous fission (the “spontaneous-fission branching fraction”).
- PARTISN [5] computes the quantity by directly using the quantities , which are provided in nuclear data files for each isotope , and energy group , as follows
- The quantity quantifies the fission spectrum in energy group .
- The numerical model of the PERP benchmark contains 7477 nonzero parameters which are subject to uncertainties, as follows: (i) 180 group-averaged microscopic total cross sections; (ii) 7101 group-averaged microscopic scattering cross sections; (iii) 60 group-averaged microscopic fission cross sections; (iv) 60 “averaged” number of neutron per fission; (v) 60 group-averaged fission spectrum constants; (vi) 10 external neutron source parameters; (vii) 6 isotopic number densities. The vector , which appears in the expression of the Boltzmann-operator , represents the “vector of uncertain model parameters”.
3. Uncertainty Quantification
3.1. Uncorrelated Total Microscopic Cross Sections
3.2. Fully Correlated Total Microscopic Cross Sections
- (i)
- The mean value of the leakage response becomes where . The quantity , where the superscript “MSC” stands for “mixed second-order correlated,” quantifies the 2nd-order contributions to the mean value of the stemming from the mixed 2nd-order sensitivities when the total cross sections are fully correlated.
- (ii)
- When the cross sections are normally distributed and fully correlated, the variance of the response is , where the contributions from the 1st-order sensitivities are contained in the term while the contributions from the 2nd-order sensitivities are contained in the term . The superscript “(FC,N)” indicates “fully correlated, normally distributed.” The contributions to involving the first-order sensitivities will be denoted as and are obtained by subtracting the uncorrelated terms from the fully correlated ones, i.e., The superscript “(1, MSC)” denotes “first-order, mixed sensitivities, correlated.” Similarly, the quantity represents the contributions to involving the 2nd-order mixed and correlated sensitivities.
4. Concluding Remarks
- The 2nd-order sensitivities become more important than the 1st-order sensitivities when the parameters’ standard deviations and correlation become larger.
- The 2nd-order sensitivities cause the mean value of the leakage response to differ from its computed value.
- The 2nd-order sensitivities cause the leakage distribution in parameter space to be skewed towards positive values relative to the response’s mean value.
- In reality, the total cross sections are partially correlated, but these correlations are currently unavailable. The results (presented in Table 3) involving the mixed 2nd-order sensitivities highlight the need for future experimental research for quantifying these currently unavailable correlations.
- It has also been shown that the Second Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) conceived by Cacuci [13,14,15] is the only practical method for computing the second-order sensitivities exactly (i.e., without introducing methodological errors) for realistic and practical models of large-scale systems, which invariably involve many uncertain parameters. In particular, it has been shown that the simplest “2-point sampling method” requires for computing the 1st-order sensitivities (while introducing second-order errors!) almost as much time as the 2nd-ASAM requires for computing (without methodological errors!) all of the 2nd-order sensitivities. Additionally, it has also been shown that even the simplest “sampling approach” requires unthinkably large amount of computational time, while still producing approximate results, subject to “sampling approach errors”.
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- International Handbook of Evaluated Criticality Safety Benchmark Experiments; Nuclear Energy Agency: Paris, France, 2016.
- Brandon, E. Assembly of 239Pu Ball for Criticality Experiment; CMB-11-FAB-80-65; Los Alamos National Laboratory: Los Alamos, NM, USA, 1980. [Google Scholar]
- Miller, E.C.; Mattingly, J.K.; Clarke, S.D.; Solomon, C.J.; Dennis, B.D.; Meldrum, A.; A Pozzi, S. Computational Evaluation of Neutron Multiplicity Measurements of Polyethylene-Reflected Plutonium Metal. Nucl. Sci. Eng. 2014, 176, 167–185. [Google Scholar] [CrossRef]
- Cacuci, D.G. Application of the Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology to Compute 1st- and 2nd-Order Sensitivities of Flux Functionals in a Multiplying System with Source. Nucl. Sci. Eng. 2019, 193, 555–600. [Google Scholar] [CrossRef] [Green Version]
- Alcouffe, R.E.; Baker, R.S.; Dahl, J.A.; Turner, S.A.; Ward, R. PARTISN: A Time-Dependent, Parallel Neutral Particle Transport Code System; LA-UR-05-3925; Los Alamos National Laboratory: Los Alamos, NM, USA, 2008. [Google Scholar]
- Conlin, J.L.; Parsons, D.K.; Gardiner, S.J.; Gray, M.G.; Lee, M.B.; White, M.C. MENDF71X: Multigroup Neutron Cross-Section Data Tables Based upon ENDF/B-VII.1X; LA-UR-15-29571; Los Alamos National Laboratory: Los Alamos, NM, USA, 2013. [Google Scholar]
- Cacuci, D.G.; Fang, R.; Favorite, J.A. Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: I. Effects of Imprecisely Known Microscopic Total and Capture Cross Sections. Energies 2019, 12, 4219. [Google Scholar] [CrossRef] [Green Version]
- Fang, R.; Cacuci, D.G. Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: II. Effects of Imprecisely Known Microscopic Scattering Cross Sections. Energies 2019, 12, 4114. [Google Scholar] [CrossRef] [Green Version]
- Cacuci, D.G.; Fang, R.; Favorite, J.A.; Badea, M.C.; Di Rocco, F. Rocco Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: III. Effects of Imprecisely Known Microscopic Fission Cross Sections and Average Number of Neutrons per Fission. Energies 2019, 12, 4100. [Google Scholar] [CrossRef] [Green Version]
- Fang, R.; Cacuci, D.G. Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: IV. Effects of Imprecisely Known Source Parameters. Energies 2020, 13, 1431. [Google Scholar] [CrossRef] [Green Version]
- Fang, R.; Cacuci, D.G. Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: V. Computation of Mixed 2nd-Order Sensitivities Involving Isotopic Number Densities. Energies 2020, 13, 2580. [Google Scholar] [CrossRef]
- Cacuci, D.G.; Fang, R.; Favorite, J.A. Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark. VI: Overall Impact of 1st- and 2nd-Order Sensitivities on Response Uncertainties. Energies 2020, 13, 1674. [Google Scholar] [CrossRef] [Green Version]
- Cacuci, D.G. Second-order adjoint sensitivity analysis methodology (2nd-ASAM) for computing exactly and efficiently first- and second-order sensitivities in large-scale linear systems: I. Computational methodology. J. Comput. Phys. 2015, 284, 687–699. [Google Scholar] [CrossRef] [Green Version]
- Cacuci, D.G. Second-Order Adjoint Sensitivity Analysis Methodology (2nd-CASAM) for Large-Scale Nonlinear Systems: I. Theory. Nucl. Sci. Eng. 2016, 184, 16–30. [Google Scholar] [CrossRef]
- Cacuci, D.G. The Second-Order Adjoint Sensitivity Analysis Methodology; CRC Press: Boca Raton, FL, USA; Taylor & Francis Group: Oxfordshire, UK, 2018. [Google Scholar]
g | 1st-Order | G | 1st-Order | g | 1st-Order | g | 1st-Order | G | 1st-Order |
---|---|---|---|---|---|---|---|---|---|
1–6 | <−0.01 | 11 | −0.19 | 16 | −1.16 | 21 | −9.69 | 26 | −0.65 |
7 | −0.07 | 12 | −0.44 | 17 | −1.17 | 22 | −0.92 | 27 | −0.58 |
8 | −0.09 | 13 | −0.52 | 18 | −1.14 | 23 | −0.89 | 28 | −0.55 |
9 | −0.14 | 14 | −0.57 | 19 | −1.09 | 24 | −0.75 | 29 | −0.55 |
10 | −0.17 | 15 | −0.58 | 20 | −1.03 | 25 | −0.71 | 30 | −9.366 |
g′ = 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
g = 12 | 0.653 | 0.315 | 0.340 | 0.356 | 0.740 | 0.763 | 0.751 | 0.725 | 0.688 | 0.648 | 0.597 | 0.554 | 0.502 | 0.477 | 0.440 | 0.393 | 0.369 | 0.372 | 6.432 |
13 | 0.315 | 0.974 | 0.471 | 0.471 | 0.976 | 1.005 | 0.988 | 0.953 | 0.904 | 0.851 | 0.784 | 0.728 | 0.659 | 0.627 | 0.577 | 0.516 | 0.484 | 0.487 | 8.424 |
14 | 0.340 | 0.471 | 1.261 | 0.579 | 1.158 | 1.192 | 1.172 | 1.130 | 1.072 | 1.009 | 0.930 | 0.863 | 0.782 | 0.742 | 0.684 | 0.611 | 0.574 | 0.576 | 9.968 |
15 | 0.356 | 0.471 | 0.579 | 1.391 | 1.255 | 1.277 | 1.255 | 1.210 | 1.148 | 1.081 | 0.996 | 0.924 | 0.837 | 0.795 | 0.733 | 0.655 | 0.615 | 0.617 | 10.67 |
16 | 0.740 | 0.976 | 1.158 | 1.255 | 4.461 | 2.700 | 2.647 | 2.553 | 2.421 | 2.280 | 2.100 | 1.949 | 1.767 | 1.677 | 1.546 | 1.381 | 1.296 | 1.300 | 22.48 |
17 | 0.763 | 1.005 | 1.192 | 1.277 | 2.700 | 4.853 | 2.789 | 2.684 | 2.546 | 2.398 | 2.209 | 2.050 | 1.858 | 1.764 | 1.625 | 1.452 | 1.363 | 1.367 | 23.62 |
18 | 0.751 | 0.988 | 1.172 | 1.255 | 2.647 | 2.789 | 4.828 | 2.689 | 2.546 | 2.399 | 2.210 | 2.051 | 1.859 | 1.764 | 1.626 | 1.453 | 1.363 | 1.367 | 23.62 |
19 | 0.725 | 0.953 | 1.130 | 1.210 | 2.553 | 2.684 | 2.689 | 4.619 | 2.498 | 2.349 | 2.165 | 2.010 | 1.822 | 1.729 | 1.594 | 1.424 | 1.336 | 1.340 | 23.15 |
20 | 0.688 | 0.904 | 1.072 | 1.148 | 2.421 | 2.546 | 2.546 | 2.498 | 4.284 | 2.266 | 2.085 | 1.936 | 1.755 | 1.666 | 1.535 | 1.372 | 1.287 | 1.290 | 22.29 |
21 | 0.648 | 0.851 | 1.009 | 1.081 | 2.280 | 2.398 | 2.399 | 2.349 | 2.266 | 3.937 | 2.004 | 1.857 | 1.684 | 1.599 | 1.474 | 1.317 | 1.236 | 1.238 | 21.40 |
22 | 0.597 | 0.784 | 0.930 | 0.996 | 2.100 | 2.209 | 2.210 | 2.165 | 2.085 | 2.004 | 3.515 | 1.760 | 1.593 | 1.512 | 1.394 | 1.246 | 1.169 | 1.171 | 20.24 |
23 | 0.554 | 0.728 | 0.863 | 0.924 | 1.949 | 2.050 | 2.051 | 2.010 | 1.936 | 1.857 | 1.760 | 3.177 | 1.521 | 1.440 | 1.328 | 1.187 | 1.114 | 1.116 | 19.28 |
24 | 0.502 | 0.659 | 0.782 | 0.837 | 1.767 | 1.858 | 1.859 | 1.822 | 1.755 | 1.684 | 1.593 | 1.521 | 2.792 | 1.358 | 1.249 | 1.117 | 1.048 | 1.049 | 18.13 |
25 | 0.477 | 0.627 | 0.742 | 0.795 | 1.677 | 1.764 | 1.764 | 1.729 | 1.666 | 1.599 | 1.512 | 1.440 | 1.358 | 2.604 | 1.214 | 1.082 | 1.016 | 1.017 | 17.58 |
26 | 0.440 | 0.577 | 0.684 | 0.733 | 1.546 | 1.625 | 1.626 | 1.594 | 1.535 | 1.474 | 1.394 | 1.328 | 1.249 | 1.214 | 2.349 | 1.037 | 0.971 | 0.972 | 16.79 |
27 | 0.393 | 0.516 | 0.611 | 0.655 | 1.381 | 1.452 | 1.453 | 1.424 | 1.372 | 1.317 | 1.246 | 1.187 | 1.117 | 1.082 | 1.037 | 2.039 | 0.913 | 0.912 | 15.76 |
28 | 0.369 | 0.484 | 0.574 | 0.615 | 1.296 | 1.363 | 1.363 | 1.336 | 1.287 | 1.236 | 1.169 | 1.114 | 1.048 | 1.016 | 0.971 | 0.913 | 1.885 | 0.888 | 15.30 |
29 | 0.372 | 0.487 | 0.576 | 0.617 | 1.300 | 1.367 | 1.367 | 1.340 | 1.290 | 1.238 | 1.171 | 1.116 | 1.049 | 1.017 | 0.972 | 0.912 | 0.888 | 1.891 | 15.39 |
30 | 6.432 | 8.424 | 9.97 | 10.67 | 22.48 | 23.62 | 23.62 | 23.15 | 22.29 | 21.40 | 20.24 | 19.28 | 18.13 | 17.58 | 16.79 | 15.76 | 15.30 | 15.39 | 429.6 |
Fully Correlated Cross Sections | Uncorrelated Cross Sections | ||||
---|---|---|---|---|---|
Rel. St. Dev. | 10% | 5% | Rel. St. Dev. | 10% | 5% |
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Cacuci, D.G. Illustrating Important Effects of Second-Order Sensitivities on Response Uncertainties in Reactor Physics. J. Nucl. Eng. 2021, 2, 114-123. https://0-doi-org.brum.beds.ac.uk/10.3390/jne2020012
Cacuci DG. Illustrating Important Effects of Second-Order Sensitivities on Response Uncertainties in Reactor Physics. Journal of Nuclear Engineering. 2021; 2(2):114-123. https://0-doi-org.brum.beds.ac.uk/10.3390/jne2020012
Chicago/Turabian StyleCacuci, Dan G. 2021. "Illustrating Important Effects of Second-Order Sensitivities on Response Uncertainties in Reactor Physics" Journal of Nuclear Engineering 2, no. 2: 114-123. https://0-doi-org.brum.beds.ac.uk/10.3390/jne2020012