Second-Order Adjoint Sensitivity and Uncertainty Analysis of a Heat Transport Benchmark Problem—I: Analytical Results

This work presents an illustrative application of the second-order adjoint sensitivity analysis methodology (2^nd-ASAM) to a heat transport benchmark problem that simulates the radial heat conduction in a heated fuel rod together with the axial heat transport along the coolant channel surrounding the rod. This paradigm benchmark is sufficiently simple to admit an exact solution, thereby making transparent the mathematical derivations underlying the 2^nd-ASAM, and also for performing solution verification of computational fluid dynamics simulation tools such as the FLUENT Adjoint Solver, a code that has been used for computing thermal-hydraulic processes within the G4M Reactor. The G4M Reactor is an innovative small modular fast reactor cooled by lead-bismuth eutectic.

The benchmark problem has six thermal-hydraulic parameters representative of the G4M Reactor, as well as of a test section that is under design for analyzing thermal-hydraulic phenomena expected within the reactor’s core. Thus, the benchmark exhibits 6 first-order and 21 second-order sensitivities for the temperature distribution at any location within the heated rod (and/or on its surface) and just as many sensitivities for the temperature distribution at any location within the coolant. Locations of particular importance are those where the rod temperature attains its maximum along the rod’s axis or on the rod’s surface, as well as the coolant’s exit, where the coolant temperature attains its maximum.

The general theory underlying the 2^nd-ASAM indicates that for a physical system comprising N_α parameters, the computation of all of the first- and second-order response sensitivities requires N_α large-scale computations involving correspondingly constructed adjoint systems, which we called second-level adjoint sensitivity systems (2^nd-LASS). In practice, however, the actual number of large-scale adjoint computations can be significantly smaller; in particular, all 54 first- and second-order rod and coolant temperature sensitivities for the benchmark presented in this work were obtained using only seven independent adjoint computations. Furthermore, the construction and solution of the 2^nd-LASS require very little additional effort beyond the construction of the first-level adjoint sensitivity system (1^st-LASS) needed for computing the first-order sensitivities. Very significantly, only the sources on the right side of the differential operator needed to be modified; the left side of the respective differential equations (and hence the solver in large-scale practical applications) remained unchanged from the 1^st-LASS and from the original equations when these are linear in the unknown state function.

This work also shows that the second-order sensitivities have the following impacts on the computed moments of the response distribution: (a) they cause the expected value of the response to differ from the computed nominal value of the response, (b) they contribute to the response variances and covariances, and (c) they contribute decisively to causing asymmetries in the response distribution. Indeed, neglecting the second-order sensitivities would nullify the third-order response correlations stemming from normally distributed parameters and hence would nullify the skewness of the response; consequently, events occurring in a response’s long and/or short tails could be missed.