Introducing the n^th-Order Features Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (n^th-FASAM-N): II. Illustrative Example

Abstract

This work highlights the unparalleled efficiency of the “n th -Order Function/ Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n th -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 n th -FASAM-N methodology is compared to the extant “n th -Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n th -CASAM-N) showing that: (i) the 1 st -FASAM-N and the 1 st -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” (1 st -LASS); (ii) the 2 nd -FASAM-N methodology is considerably more efficient than the 2 nd -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 2 nd -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 2 nd -CASAM-N methodology requires 7 large-scale computations for obtaining these 28 second-order sensitivities; (iii) the 3 rd -FASAM-N methodology is even more efficient than the 3 rd -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 3 rd -FASAM-N methodology, while the application of the 3 rd -CASAM-N methodology requires at least 22 large-scale computations for computing the same 84 distinct third-order sensitivities. Together, the n th -FASAM-N and the n th -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.