Abstract
This work extends to fourth-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 7477 imprecisely known (uncertain) model parameters which have nonzero values. These parameters are as follows: 180 microscopic total cross sections; 7101 microscopic scattering sections; 60 microscopic fission cross sections; 60 parameters that characterize the average number of neutrons per fission; 60 parameters that characterize the fission spectrum; 10 parameters that characterize the fission source; and 6 parameters that characterize the isotope number densities. Previous works have used the adjoint sensitivity analysis methodology to compute exactly and efficiently all of the 7477 first-order and 27,956,503 second-order sensitivities of the PERP benchmark’s leakage response to all of the benchmark’s uncertain parameters. These works showed that largest response sensitivities involve the total microscopic cross sections, which motivated the recent computation of all of the (180)3 third-order sensitivities of the PERP leakage response with respect to these total microscopic cross sections. It turned out that some of these 3rd-order cross sections were far larger than the corresponding 2nd-order ones, thereby having the largest impact on the uncertainties induced in the PERP benchmark’s response. This finding has motivated the development of the original 4th-order formulas presented in this work, which are valid not only for the PERP benchmark but can also be used for computing the 4th-order sensitivities of 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 the largest fourth-order sensitivities of the PERP benchmark’s response to the total microscopic cross section and use them for a pioneering fourth-order uncertainty analysis of the PERP benchmark’s response.
Highlights
Until recently, only the first-order sensitivities of a computational model’s responses to the respective model’s imprecisely known parameters have been considered when assessing the uncertainties induced in the respective responses by the parameter uncertainties
There are three methods for computing deterministically the response sensitivities to model and response parameters, as follows: 1) The so-called “brute-force re-computations” method, which uses finitedifference formulas to approximate the derivative that expresses the response sensitivity to the parameter under consideration; this method will be presented in Subsection 3.1; 2) The Forward Sensitivity Analysis Methodology (FSAM), which will be applied in Subsection 3.2; 3) The 1st-Order Comprehensive Adjoint Sensitivity Analysis Methodology (CASAM) conceived by Cacuci [21], which will be applied in Subsection 3.3
The total computational resources and times (CPU) time needed for computing all of the 2nd-order sensitivities using the 2nd-CASAM was ca. 929 hours, comprising 735 hours used for the 14843 PARTISN computations (S256 angular quadrature; ISN = 256) and 194 hours used for performing the integrations needed to compute the respective unmixed and mixed second-order sensitivities
Summary
Only the first-order sensitivities (i.e., functional derivatives) of a computational model’s responses (i.e., quantities of interest) to the respective model’s imprecisely known (i.e., uncertain) parameters have been considered when assessing the uncertainties induced in the respective responses by the parameter uncertainties. To enable the computation of such 3rd-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 applied this general theory to the PERP benchmark in order to derive the exact analytical expressions of the 3rd-order sensitivities of the PERP benchmark’s leakage response with respect to this benchmark’s microscopic total cross sections [13] [14] [15]. These results underscore the fact that the 3rd-CASAM is exact, introducing no intrinsic methodological errors in the computation of sensitivities, and is by far more efficient than any other method.
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