TITLE:
Introducing the nth-Order Features Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-FASAM-N): II. Illustrative Example
AUTHORS:
Dan Gabriel Cacuci
KEYWORDS:
Nordheim-Fuchs Reactor Safety Model, Feature Functions of Model Parameters, High-Order Response Sensitivities to Parameters, Adjoint Sensitivity Systems
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.14 No.1,
March
27,
2024
ABSTRACT: This work highlights the unparalleled efficiency of
the “nth-Order Function/ Feature Adjoint Sensitivity Analysis Methodology for
Nonlinear Systems” (nth-FASAM-N) by considering the well-known
Nordheim-Fuchs reactor dynamics/safety model. This model describes a short-time
self-limiting power excursion in a nuclear reactor system having a negative
temperature coefficient in which a large amount of reactivity is suddenly
inserted, either intentionally or by accident. This nonlinear paradigm model is
sufficiently complex to model realistically self-limiting power excursions for
short times yet admits closed-form exact expressions for the time-dependent neutron
flux, temperature distribution and energy released during the transient power
burst. The nth-FASAM-N methodology is compared to the extant “nth-Order
Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems”
(nth-CASAM-N) showing that: (i) the 1st-FASAM-N and the 1st-CASAM-N
methodologies are equally efficient for computing the first-order
sensitivities; each methodology requires a single large-scale computation for
solving the “First-Level Adjoint Sensitivity System” (1st-LASS);
(ii) the 2nd-FASAM-N methodology is considerably more efficient than
the 2nd-CASAM-N methodology for computing the second-order
sensitivities since the number of feature-functions is much smaller than the
number of primary parameters; specifically for the Nordheim-Fuchs model, the 2nd-FASAM-N
methodology requires 2 large-scale computations to obtain all of the exact
expressions of the 28 distinct second-order response sensitivities with respect
to the model parameters while the 2nd-CASAM-N methodology requires 7
large-scale computations for obtaining these 28 second-order sensitivities;
(iii) the 3rd-FASAM-N methodology is even more efficient than the 3rd-CASAM-N
methodology: only 2 large-scale computations are needed to obtain the exact
expressions of the 84 distinct third-order response sensitivities with respect
to the Nordheim-Fuchs model’s parameters when applying the 3rd-FASAM-N
methodology, while the application of the 3rd-CASAM-N methodology
requires at least 22 large-scale computations for computing the same 84
distinct third-order sensitivities. Together, the nth-FASAM-N and
the nth-CASAM-N methodologies are the most practical methodologies
for computing response sensitivities of any order comprehensively and
accurately, overcoming the curse of dimensionality in sensitivity analysis.