Third-Order Adjoint Sensitivity Analysis of an OECD/NEA Reactor Physics Benchmark: I. Mathematical Framework

Abstract

This work extends to third-order previously published work on developing the adjoint sensitivity and uncertainty analysis of the numerical model of a polyethylene-reflected plutonium (acronym: PERP) OECD/NEA reactor physics benchmark. The PERP benchmark comprises 21,976 imprecisely known (uncertain) model parameters. Previous works have used the adjoint sensitivity analysis methodology to compute exactly and efficiently all of the 21,976 first-order and (21,976) 2 second-order sensitivities of the PERP benchmark's leakage response to all of the benchmark's uncertain parameters, showing that the largest and most consequential 1 stand 2 nd-order response sensitivities are with respect to the total microscopic cross sections. These results have motivated extending the previous adjoint-based derivations to third-order, leading to the derivation, in this work, of the exact mathematical expressions of the (180) 3 third-order sensitivities of the PERP leakage response with respect to these total microscopic cross sections. The formulas derived in this work are valid not only for the PERP benchmark but can also be used for computing the 3 rd-order sensitivities of the leakage response of any nuclear system involving fissionable material and internal or external neutron sources. Subsequent works will use the adjoint-based mathematical expressions obtained in this work to compute exactly and efficiently the numerical values of these (180) 3 third-order sensitivities (which turned out to be very large and consequential) and use them for a third-order uncertainty analysis of the PERP benchmark's leakage response.

Fourth-Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: I. Computed Sensitivities

“…The formulas presented in this work for computing the 4th-order sensitivities of the leakage response to the benchmark's cross sections, along with the formulas presented in [3][4][5][6][7][8][9][10][11][12][13] for computing the 1st-, 2nd-and 3rd-order sensitivities of the PERP benchmark to cross sections can be implemented in both deterministic and Monte-Carlo codes for solving the neutron transport equations. These formulas could be used for sensitivity analysis of reaction-rate responses in subcritical reactors and for subsequent quantification of uncertainties induced in such responses by the uncertainties in the underlying cross sections.…”

“…These formulas could be used for sensitivity analysis of reaction-rate responses in subcritical reactors and for subsequent quantification of uncertainties induced in such responses by the uncertainties in the underlying cross sections. The 4th-order sensitivity analysis methodology used to produce the results presented in this work, along with the previous sensitivity analysis results reported in [3][4][5][6][7][8][9][10][11][12][13], cannot be obtained by any other methods. The computational times that would be required even by simple-minded finite-difference schemes have been shown to be prohibitive for problems involving as many parameters (21,976) as involved in the PERP benchmark.…”

“…Since the results obtained in [4][5][6][7][8][9] indicated that the effects of the 2nd-order sensitivities of the PERP leakage response to the total microscopic group cross sections are much larger than the effects of the corresponding 1st-order sensitivities, Cacuci and Fang [10][11][12] have extended the concepts underlying the 2nd-ASAM in order to compute the (180) 3 3rd-order sensitivities of the PERP benchmark's total leakage response with respect to the microscopic total cross sections. It turned out that some of these 3rd-order sensitivities were far larger than the corresponding 2nd-order ones, thereby having the largest impact on the uncertainties induced in the PERP benchmark's leakage response.…”

Fourth-Order Adjoint Sensitivity Analysis of an OECD/NEA Reactor Physics Benchmark: I. Mathematical Expressions and CPU-Time Comparisons for Computing 1<sup>st</sup>-, 2<sup>nd</sup>- and 3<sup>rd</sup>-Order Sensitivities

“…To enable the computation of such 3 rd -order sensitivities, Cacuci [12] has recently conceived the "third order adjoint sensitivity analysis methodology for reaction rate responses in a multiplying nuclear system with source" and then applied this general theory to the PERP benchmark in order to derive the exact analytical expressions of the 3 rd -order sensitivities of the PERP benchmark's leakage response with respect to this benchmark's microscopic total cross sections [13] [14] [15]. These works [13] [14] [15] have shown that the largest 3 rd -order sensitivities involve the microscopic total cross section for the lowest (30 th ) energy group of isotope 1 H (i.e., 30 ,6 t σ ). In particular, the largest overall 3 rd -order sensitivity is the mixed 3 rd -order sensitivity ( ) ( ) S σ σ σ…”

“…, 429.6 The results obtained in [5]- [10] and [13] [14] [15] indicated that the total microscopic cross section of isotopes 1 H and 239 Pu are the most important parameters affecting the PERP benchmark's leakage response, since they are responsible for the largest 1 st -, 2 nd -and 3 rd -order sensitivities.…”